the uncertainty principle determines the non-locality of quantum … · 2013. 1. 15. · the...

The uncertainty principle determines the non-locality of quantum

mechanics

Jonathan Oppenheim1 Stephanie Wehner23

1DAMTP University of Cambridge CB3 0WA Cambridge UK2Institute for Quantum Information Caltech Pasadena CA 91125 USA

3Centre for Quantum Technologies National University of Singapore 117543 Singapore

Two central concepts of quantum mechanics are Heisenbergrsquos uncertaintyprinciple and a subtle form of non-locality that Einstein famously calledldquospooky action at a distancerdquo These two fundamental features havethus far been distinct concepts Here we show that they are inextricablyand quantitatively linked Quantum mechanics cannot be more non-localwith measurements that respect the uncertainty principle In fact thelink between uncertainty and non-locality holds for all physical theoriesMore specifically the degree of non-locality of any theory is determined bytwo factors ndash the strength of the uncertainty principle and the strengthof a property called ldquosteeringrdquo which determines which states can beprepared at one location given a measurement at another

A measurement allows us to gain information about the state of a physical system Forexample when measuring the position of a particle the measurement outcomes correspond topossible locations Whenever the state of the system is such that we can predict this positionwith certainty then there is only one measurement outcome that can occur Heisenberg [16]observed that quantum mechanics imposes strict restrictions on what we can hope to learn ndashthere are incompatible measurements such as position and momentum whose results cannot besimultaneously predicted with certainty These restrictions are known as uncertainty relationsFor example uncertainty relations tell us that if we were able to predict the momentum of aparticle with certainty then when measuring its position all measurement outcomes occur withequal probability That is we are completely uncertain about its location

Non-locality can be exhibited when performing measurements on two or more distant quan-tum systems ndash the outcomes can be correlated in way that defies any local classical descrip-tion [6] This is why we know that quantum theory will never by superceded by a local classicaltheory Nevertheless even quantum correlations are restricted to some extent ndash measurementresults cannot be correlated so strongly that they would allow signalling between two distantsystems However quantum correlations are still weaker than what the no-signalling principledemands [26 27 28] So why are quantum correlations strong enough to be non-local yet not

1

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ivs

ubm

it01

4892

5 [

quan

t-ph

] 1

9 N

ov 2

010

as strong as they could be Is there a principle that determines the degree of this non-localityInformation-theory [33 24] communication complexity [38] and local quantum mechanics [3]provided us with some rationale why limits on quantum theory may exist But evidence sug-gests that many of these attempts provide only partial answers Here we take a very differentapproach and relate the limitations of non-local correlations to two inherent properties of anyphysical theory

Uncertainty relations

At the heart of quantum mechanics lies Heisenbergrsquos uncertainty principle [16] Traditionallyuncertainty relations have been expressed in terms of commutators

∆A∆B ge 1

2|〈ψ|[AB]|ψ〉| (1)

with standard deviations ∆X =radic〈ψ|X2|ψ〉 minus 〈ψ|X|ψ〉2 for X isin AB However the more

modern approach is to use entropic measures Let p(x(t)|t)σ denote the probability that weobtain outcome x(t) when performing a measurement labelled t when the system is prepared inthe state σ In quantum theory σ is a density operator while for a general physical theory weassume that σ is simply an abstract representation of a state The well-known Shannon entropyH(t)σ of the distribution over measurement outcomes of measurement t on a system in state σis thus

H(t)σ = minussumx(t)

p(x(t)|t)σ log p(x(t)|t)σ (2)

In any uncertainty relation we wish to compare outcome distributions for multiple measure-ments In terms of entropies such relations are of the formsum

tisinTp(t) H(t)σ ge cT D (3)

where p(t) is any probability distribution over the set of measurements T and cT D is somepositive constant determined by T and the distribution D = p(t)t To see why (3) forms anuncertainty relation note that whenever cT D gt 0 we cannot predict the outcome of at least oneof the measurements t with certainty ie H(t)σ gt 0 Such relations have the great advantagethat the lower bound cT D does not depend on the state σ [12] Instead cT D depends only onthe measurements and hence quantifies their inherent incompatibility It has been shown thatfor two measurements entropic uncertainty relations do in fact imply Heisenbergrsquos uncertaintyrelation [7] providing us with a more general way of capturing uncertainty (see [42] for a survey)

One may consider many entropic measures instead of the Shannon entropy For examplethe min-entropy

Hinfin(t)σ = minus log maxx(t)

p(x(t)|t)σ (4)

used in [12] plays an important role in cryptography and provides a lower bound on H(t)σEntropic functions are however a rather coarse way of measuring the uncertainty of a set of

2

measurements as they do not distinguish the uncertainty inherent in obtaining any combinationof outcomes x(t) for different measurements t It is thus useful to consider much more fine-graineduncertainty relations consisting of a series of inequalities one for each combination of possibleoutcomes which we write as a string ~x = (x(1) x(n)) isin Btimesn with n = |T | 1 That is foreach ~x a set of measurements T and distribution D = p(t)t

P cert(σ ~x) =nsumt=1

p(t) p(x(t)|t)σ le ζ~x(T D) (5)

For a fixed set of measurements the set of inequalities

U =

nsumt=1

p(t) p(x(t)|t)σ le ζ~x | forall~x isin Btimesn

(6)

thus forms a fine-grained uncertainty relation as it dictates that one cannot obtain a measure-ment outcome with certainty for all measurements simultaneously whenever ζ~x lt 1 The valuesof ζ~x thus confine the set of allowed probability distributions and the measurements have un-certainty if ζ~x lt 1 for all ~x To characterise the ldquoamount of uncertaintyrdquo in a particular physicaltheory we are thus interested in the values of

ζ~x = maxσ

nsumt=1

p(t)p(x(t)|t)σ (7)

where the maximization is taken over all states allowed on a particular system (for simplicitywe assume it can be attained in the theory considered) We will also refer to the state ρ~x thatattains the maximum as a ldquomaximally certain staterdquo However we will also be interested in thedegree of uncertainty exhibited by measurements on a set of states Σ quantified by ζΣ

~x definedwith the maximisation in Eq 7 taken over σ isin Σ Fine-grained uncertainty relations are directlyrelated to the entropic ones and have both a physical and an information processing appeal (seeappendix) As an example consider the binary spin-12 observables Z and X If we can obtaina particular outcome x(Z) given that we measured Z with certainty ie p(x(Z)|Z) = 1 then thecomplementary observable must be completely uncertain ie p(x(X)|X) = 12 If we choosewhich measurement to make with probability 12 then this notion of uncertainty is captured bythe relations

1

2p(x(X)|X) +

1

2p(x(Z)|Z) le ζ~x =

1

2+

1

2radic

2for all ~x = (x(X) x(Z)) isin 0 12 (8)

where the maximally certain states are given by the eigenstates of (X+Z)radic

2 and (XminusZ)radic

2

Non-local correlations

We now introduce the concept of non-local correlations Instead of considering measurementson a single system we consider measurements on two (or more) space-like separated systems

1Without loss of generality we assume that each measurement has the same set of possible outcomes since wemay simply add additional outcomes which never occur

3

traditionally named Alice and Bob We label Bobrsquos measurements using t isin T and use b isin B tolabel his measurement outcomes For Alice we use s isin S to label her measurements and a isin Ato label her outcomes (recall that wlog we can assume all measurements have the same numberof outcomes) When Alice and Bob perform measurements on a shared state σAB the outcomesof their measurements can be correlated Let p(a b|s t)σAB be the joint probability that theyobtain outcomes a and b for measurements s and t We can now again ask ourselves whatcorrelations are possible in nature In other words what probability distributions are allowed

Quantum mechanics as well as classical mechanics obeys the no-signalling principle meaningthat information cannot travel faster than light This means that the probability that Bobobtains outcome b when performing measurement t cannot depend on which measurement Alicechooses to perform (and vice versa) More formally

suma p(a b|s t)σAB = p(b|t)σB for all s

where σB = trA(σAB) Curiously however this is not the only constraint imposed by quantummechanics [26] and finding other constraints is a difficult task [21 14] Here we find that theuncertainty principle imposes the other limitation

To do so let us recall the concept of so-called Bell inequalities [6] which are used to describelimits on such joint probability distributions These are most easily explained in their moremodern form in terms of a game played between Alice and Bob Let us choose questions s isin Sand t isin T according to some distribution p(s t) and send them to Alice and Bob respectivelywhere we take for simplicity p(s t) = p(s)p(t) The two players now return answers a isin A andb isin B Every game comes with a set of rules that determines whether a and b are winninganswers given questions s and t Again for simplicity we thereby assume that for every st anda there exists exactly one winning answer b for Bob (and similarly for Alice) That is for everysetting s and outcome a of Alice there is a string ~xsa = (xsa

(1) xsa(n)) isin Btimesn of length

n = |T | that determines the correct answer b = xsa(t) for question t for Bob (Fig 1) We

say that s and a determine a ldquorandom access codingrdquo [40] meaning that Bob is not trying tolearn the full string ~xsa but only the value of one entry The case of non-unique games is astraightforward but cumbersome generalisation

As an example the Clauser-Horne-Shimony-Holt (CHSH) inequality [11] one of the simplestBell inequalities whose violation implies non-locality can be expressed as a game in whichAlice and Bob receive binary questions s t isin 0 1 respectively and similarly their answersa b isin 0 1 are single bits Alice and Bob win the CHSH game if their answers satisfy aoplusb = smiddottWe can label Alicersquos outcomes using string ~xsa and Bobrsquos goal is to retrieve the t-th element ofthis string For s = 0 Bob will always need to give the same answer as Alice in order to winand hence we have ~x00 = (0 0) and ~x01 = (1 1) For s = 1 Bob needs to give the same answerfor t = 0 but the opposite answer if t = 1 That is ~x10 = (0 1) and ~x11 = (1 0)

Alice and Bob may agree on any strategy ahead of time but once the game starts theirphysical distance prevents them from communicating For any physical theory such a strategyconsists of a choice of shared state σAB as well as measurements where we may without lossof generality assume that the players perform a measurement depending on the question theyreceive and return the outcome of said measurement as their answer For any particular strategywe may hence write the probability that Alice and Bob win the game as

P game(S T σAB) =sumst

p(s t)suma

p(a b = xsa(t)|s t)σAB (9)

4

Figure 1 Any game where for every choice of settings s and t and answer a of Alice there isonly one correct answer b for Bob can be expressed using strings ~xsa such that b = xsa

(t) is thecorrect answer for Bob [40] for s and a The game is equivalent to Alice outputting ~xsa andBob outputting xsa

(t)

To characterize what distributions are allowed we are generally interested in the winningprobability maximized over all possible strategies for Alice and Bob

P gamemax = max

ST σAB

P game(S T σAB) (10)

which in the case of quantum theory we refer to as a Tsirelsons type bound for the game [35]For the CHSH inequality we have P game

max = 34 classically P gamemax = 12 + 1(2

radic2) quantum

mechanically and P gamemax = 1 for a theory allowing any nonsignalling correlations P game

max quan-tifies the strength of nonlocality for any theory with the understanding that a theory possessesgenuine nonlocality when it differs from the value that can be achieved classically The connec-tion we will demonstrate between uncertainty relations and nonlocality holds even before thisoptimization

Steering

The final concept we need in our discussion is steerability which determines what states Alicecan prepare on Bobrsquos system remotely Imagine Alice and Bob share a state σAB and considerthe reduced state σB = trA(σAB) on Bobrsquos side In quantum mechanics as well as many othertheories in which Bobrsquos state space is a convex set S the state σB isin S can be decomposed inmany different ways as a convex sum

σB =suma

p(a|s) σsa with σsa isin S (11)

corresponding to an ensemble Es = p(a|s) σsaa It was Schrodinger [30 31] who noted thatin quantum mechanics for all s there exists a measurement on Alicersquos system that allows her toprepare Es = p(a|s) σsaa on Bobrsquos site (Fig 2) That is for measurement s Bobrsquos system will

5

Figure 2 When Alice performs a measurement labelled s and obtains outcome a with proba-bility p(a|s) she effectively prepares the state σsa on Bobrsquos system This is known as ldquosteeringrdquo

be in the state σsa with probability p(a|s) Schrodinger called this steering to the ensemble Esand it does not violate the no-signalling principle because for each of Alicersquos setting the stateof Bobrsquos system is the same once we average over Alicersquos measurement outcomes

Even more he observed that for any set of ensembles Ess that respect the no-signallingconstraint ie for which there exists a state σB such that Eq 11 holds we can in fact find abipartite quantum state σAB and measurements that allow Alice to steer to such ensembles Wecan imagine theories in which our ability to steer is either more or maybe even less restricted(some amount of signalling is permitted) Our notion of steering thus allows forms of steering notconsidered in quantum mechanics [30 31 43 18] or other restricted classes of theories [4] Ourability to steer however is a property of the set of ensembles we consider and not a propertyof one single ensemble

Main result

We are now in a position to derive the relation between how non-local a theory is and howuncertain it is For any theory we will find that the uncertainty relation for Bobrsquos measurements(optimal or otherwise) acting on the states that Alice can steer to is what determines the strengthof non-locality (See Eq (14)) We then find that in quantum mechanics for all games wherethe optimal measurements are known the states which Alice can steer to are identical to themost certain states and it is thus only the uncertainty relations of Bobrsquos measurements whichdetermine the outcome (see Eq (15))

First of all note that we may express the probability (9) that Alice and Bob win the gameusing a particular strategy (Fig 1) as

P game(S T σAB) =sums

p(s)suma

p(a|s)P cert(σsa ~xsa) (12)

where σsa is the reduced state of Bobrsquos system for setting s and outcome a of Alice and p(t)in P cert(middot) is the probability distribution over Bobrsquos questions T in the game This immediately

6

suggests that there is indeed a close connection between games and our fine-grained uncertaintyrelations In particular every game can be understood as giving rise to a set of uncertaintyrelations and vice versa It is also clear from Eq 5 for Bobrsquos choice of measurements T anddistribution D over T that the strength of the uncertainty relations imposes an upper bound onthe winning probability

P game(S T σAB) lesums

p(s)suma

p(a|s) ζ~xsa(T D) le max~xsa

ζ~xsa(T D) (13)

where we have made explicit the functional dependence of ζ~x on the set of measurements Thisseems curious given that we know from [44] that any two incompatible observables lead to aviolation of the CHSH inequality and that to achieve Tsirelsonrsquos bound Bob has to measuremaximally incompatible observables [25] that yield stringent uncertainty relations Howeverfrom Eq 13 we may be tempted to conclude that for any theory it would be in Bobrsquos bestinterest to perform measurements that are very compatible and have weak uncertainty relationsin the sense that the values ζ~xsa can be very large But can Alice prepare states σsa that attainζ~xsa for any choice of Bobrsquos measurements

The other important ingredient in understanding non-local games is thus steerability Wecan think of Alicersquos part of the strategy S σAB as preparing the ensemble p(a|s) σsaa onBobrsquos system whenever she receives question s Thus when considering the optimal strategy fornon-local games we want to maximise P game(S T σAB) over all ensembles Es = p(a|s) σsaathat Alice can steer to and use the optimal measurements Topt for Bob That is

P gamemax = max

Ess

sums

p(s)suma

p(a|s) ζσsa~x (ToptD) (14)

and hence the probability that Alice and Bob win the game depends only on the strength of theuncertainty relations with respect to the sets of steerable states To achieve the upper boundgiven by Eq 13 Alice needs to be able to prepare the ensemble p(a|s) ρ~xsaa of maximallycertain states on Bobrsquos system In general it is not clear that the maximally certain states forthe measurements which are optimal for Bob in the game can be steered to

It turns out that in quantum mechanics this can be achieved in cases where we knowthe optimal strategy For all XOR games 2 that is correlation inequalities for two outcomeobservables (which include CHSH as a special case) as well as other games where the optimalmeasurements are known we find that the states which are maximally certain can be steeredto (see appendix) The uncertainty relations for Bobrsquos optimal measurements thus give a tightbound

P gamemax =

sums

p(s)suma

p(a|s)ζ~x(ToptD) (15)

where we recall that ζ~x is the bound given by the maximization over the full set of allowed stateson Bobrsquos system It is an open question whether this holds for all games in quantum mechanics

An important consequence of this is that any theory that allows Alice and Bob to win witha probability exceeding P game

max requires measurements which do not respect the fine-grained

2In an XOR game the answers a isin 0 1 and b isin 0 1 of Alice and Bob respectively are single bits and thedecision whether Alice and Bob win or not depends only on the XOR of the answers aoplus b = a+ b mod 2

7

uncertainty relations given by ζ~x for the measurements used by Bob (the same argument canbe made for Alice) Even more it can lead to a violation of the corresponding min-entropicuncertainty relations (see appendix) For example if quantum mechanics were to violate theTsirelson bound using the same measurements for Bob it would need to violate the min-entropicuncertainty relations [12] This relation holds even if Alice and Bob were to perform altogetherdifferent measurements when winning the game with a probability exceeding P game

max for thesenew measurements there exist analogous uncertainty relations on the set Σ of steerable statesand a higher winning probability thus always leads to a violation of such an uncertainty relationConversely if a theory allows any states violating even one of these fine-grained uncertaintyrelations for Bobrsquos (or Alicersquos) optimal measurements on the sets of steerable states then Aliceand Bob are able to violate the Tsirelsonrsquos type bound for the game

Although the connection between non-locality and uncertainty is more general we examinethe example of the CHSH inequality in detail to gain some intuition on how uncertainty relationsof various theories determine the extent to which the theory can violate Tsirelsonrsquos bound (seeappendix) To briefly summerize in quantum theory Bobrsquos optimal measurements are X and Zwhich have uncertainty relations given by ζ~xsa = 12 + 1(2

radic2) of Eq 8 Thus if Alice could

steer to the maximally certain states for these measurements they would be able to achieve awinning probability given by P game

max = ζ~xsa ie the degree of non-locality would be determinedby the uncertainty relation This is the case ndash if Alice and Bob share the singlet state then Alicecan steer to the maximally certain states by measuring in the basis given by the eigenstates of(X + Z)

radic2 or of (X minus Z)

radic2 For quantum mechanics our ability to steer is only limited by

the no-signalling principle but we encounter strong uncertainty relationsOn the other hand for a local hidden variable theory we can also have perfect steering but

only with an uncertainty relation given by ζ~xsa = 34 and thus we also have the degree of non-locality given by P game

max = 34 This value of non-locality is the same as deterministic classicalmechanics where we have no uncertainty relations on the full set of deterministic states but ourabilities to steer to them are severely limited In the other direction there are theories that aremaximally non-local yet remain no-signalling [26] These have no uncertainty ie ζ~xsa = 1but unlike in the classical world we still have perfect steering so they win the CHSH game withprobability 1

For any physical theory we can thus consider the strength of non-local correlations to be atradeoff between two aspects steerability and uncertainty In turn the strength of non-localitycan determine the strength of uncertainty in measurements However it does not determinethe strength of complementarity of measurements (see appendix) The concepts of uncertaintyand complementarity are usually linked but we find that one can have theories that are justas non-local and uncertain as quantum mechanics but that have less complementarity Thissuggests a rich structure relating these quantities which may be elucidated by further researchin the direction suggested here

Acknowledgments The retrieval game used was discovered in collaboration with AndrewDoherty JO is supported by the Royal Society SW is supported by NSF grants PHY-04056720and PHY-0803371 the National Research Foundation and the Ministry of Education SingaporePart of this work was done while JO was visiting Caltech (Pasadena USA) and while JO andSW were visiting KITP (Santa Barbara USA) funded by NSF grant PHY-0551164

8

References

[1] J Allcock N Brunner M Pawlowski and V Scarani Recovering part of the quantumboundary from information causality httparxivorgabs09063464 2009

[2] A Ambainis A Nayak A Ta-Shma and U Vazirani Quantum dense coding and a lowerbound for 1-way quantum finite automata In Proceedings of 31st ACM STOC pages376ndash383 1999 httparxivorgabsquant-ph9804043

[3] H Barnum S Beigi S Boixo M Elliot and S Wehner Local quantum measurement andno-signaling imply quantum correlations Phys Rev Lett 104140401 2010

[4] H Barnum CP Gaebler and A Wilce Ensemble Steering Weak Self-Duality and theStructure of Probabilistic Theories Imprint 2009

[5] J Barrett Information processing in generalized probabilistic theories Phys Rev A3032304 2007

[6] J S Bell On the Einstein-Podolsky-Rosen paradox Physics 1195ndash200 1965

[7] I Bialynicki-Birula and J Mycielski Uncertainty relations for information entropy in wavemechanics Communications in Mathematical Physics 44129ndash132 1975

[8] N Bohr Can quantum-mechanical description of physical reality be considered completePhys Rev 48(8)696ndash702 Oct 1935

[9] G Brassard H Buhrman N Linden A Methot A Tapp and F Unger Limit onnonlocality in any world in which communication complexity is not trivial Phys RevLett 96(25)250401 2006

[10] H Buhrman and S Massar Causality and Cirelrsquoson bounds httparxivorgabs

quant-ph0409066 2004

[11] JF Clauser MA Horne A Shimony and RA Holt Proposed experiment to test localhidden-variable theories Physical Review Letters 23(15)880ndash884 1969

[12] D Deutsch Uncertainty in quantum measurements Phys Rev Lett 50631ndash633 1983

[13] K Dietz Generalized bloch spheres for m-qubit states Journal of Physics A Math Gen36(6)1433ndash1447 2006

[14] A C Doherty Y Liang B Toner and S Wehner The quantum moment problem andbounds on entangled multi-prover games In Proceedings of IEEE Conference on Compu-tational Complexity pages 199ndash210 2008

[15] D Gopal and S Wehner On the use of post-measurement information in state discrimina-tion httparxivorgabs10030716 2010

[16] W Heisenberg Uber den anschaulichen Inhalt der quantentheoretischen Kinematik undMechanik Zeitschrift fur Physik 43172ndash198 1927

9

[17] S Ji J Lee J Lim K Nagata and H Lee Multisetting bell inequality for qudits PhysRev A 78052103 2008

[18] S J Jones H M Wiseman and ACDoherty Entanglement epr-correlations bell-nonlocality and steering Phys Rev A 76052116 2007

[19] P Jordan and E Wigner Uber das paulische aquivalenzverbot Zeitschrift fur Physik47631 1928

[20] Y Liang C Lim and D Deng Reexamination of a multisetting bell inequality for quditsPhys Rev A 80052116 2009

[21] M Navascues S Pironio and A Acin Bounding the set of quantum correlations PhysRev Lett 98010401 2007

[22] M Navascues S Pironio and A Acin A convergent hierarchy of semidefinite programscharacterizing the set of quantum correlations httparxivorgabs08034290 2008

[23] A Nayak Optimal lower bounds for quantum automata and random access codes InProceedings of 40th IEEE FOCS pages 369ndash376 1999 httparxivorgabsquant-ph9904093

[24] M Pawlowski T Paterek D Kaszlikowski V Scarani A Winter and M ZukowskiInformation causality as a physical principle Nature 461(7267)1101ndash1104 2009

[25] A Peres Quantum Theory Concepts and Methods Kluwer Academic Publishers 1993

[26] S Popescu and D Rohrlich Quantum nonlocality as an axiom Foundations of Physics24(3)379ndash385 1994

[27] S Popescu and D Rohrlich Nonlocality as an axiom for quantum theory In A Mannand M Revzen editors The dilemma of Einstein Podolsky and Rosen 60 years laterInternational symposium in honour of Nathan Rosen Israel Physical Society Haifa Israel1996 httparxivorgabsquant-ph9508009

[28] S Popescu and D Rohrlich Causality and nonlocality as axioms for quantum mechanicsIn Geoffrey Hunter Stanley Jeffers and Jean-Pierre Vigier editors Proceedings of theSymposium of Causality and Locality in Modern Physics and Astronomy Open Questionsand Possible Solutions page 383 Kluwer Academic Publishers DordrechtBostonLondon1997 httparxivorgabsquant-ph9709026

[29] P Rastall Locality Bellrsquos theorem and quantum mechanics Foundations of Physics 15(9)1985

[30] E Schrodinger Discussion of probability relations between separated systems Proc Cam-bridge Phil Soc 31553 1935

[31] E Schrodinger Probability relations between separated systems Proc Cambridge PhilSoc 32446 1936

10

[32] RW Spekkens In defense of the epistemic view of quantum states a toy theory http

arxiv org abs quant-ph 0401052 2004

[33] G Ver Steeg and S Wehner Relaxed uncertainty relations and information processingQIC 9(9)801 2009

[34] B Toner and F Verstraete Monogamy of Bell correlations and Tsirelsonrsquos bound http

arxivorgabsquant-ph0611001 2006

[35] B Tsirelson Quantum generalizations of Bellrsquos inequality Letters in Mathematical Physics493ndash100 1980

[36] B Tsirelson Quantum analogues of Bell inequalities The case of two spatially separateddomains Journal of Soviet Mathematics 36557ndash570 1987

[37] BS Tsirelson Some results and problems on quantum Bell-type inequalities Hadronic Jour-nal Supplement 8(4)329ndash345 1993 httpwwwtauacil~tsireldownloadhadronhtml

[38] W van Dam Impossible consequences of superstrong nonlocality quant-ph0501159 2005

[39] S Wehner Tsirelson bounds for generalized Clauser-Horne-Shimony-Holt inequalitiesPhysical Review A 73022110 2006 httparxivorgabsquant-ph0510076

[40] S Wehner M Christandl and A C Doherty A lower bound on the dimension of aquantum system given measured data Phys Rev A 78062112 2008

[41] S Wehner and A Winter Higher entropic uncertainty relations for anti-commuting ob-servables Journal of Mathematical Physics 49062105 2008

[42] S Wehner and A Winter Entropic uncertainty relations ndash A survey New Journal ofPhysics 12025009 2010 Special Issue on Quantum Information and Many-Body Theory

[43] HM Wiseman SJ Jones and AC Doherty Steering entanglement nonlocality and theepr paradox Phys Rev Lett 98140402 2007

[44] MM Wolf D Perez-Garcia and C Fernandez Measurements incompatible in quantumtheory cannot be measured jointly in any other local theory Phys Rev Lett 1032304022009

[45] W K Wootters and D M Sussman Discrete phase space and minimum-uncertainty stateshttparxivorgabs07041277 2007

Supplementary Material

In this appendix we provide the technical details used or discussed in the main part In Section Awe show that for the case of XOR games Alice can always steer to the maximally certain statesof Bobrsquos optimal measurement operators That is the relation between uncertainty relations and

11

non-local games does not depend on any additional steering constraints Hence a violation ofthe Tsirelsonrsquos bound implies a violation of the corresponding uncertainty relation Converselya violation of the uncertainty relation leads to a violation of the Tsirelson bound as long thetheory allows Alice to steer to the new maximally certain states The famous CHSH gameis a particular example of an XOR game and in Sections A3 and B we find that Tsirelsonrsquosbound [35] is violated if and only if Deutschrsquo min-entropic uncertainty relation [12] is violatedwhenever steering is possible In fact for an even wider class of games called retrieval games aviolation of Tsirelsonrsquos bound implies a violation of the min-entropic uncertainty relations forBobrsquos optimal measurement derived in [41]

In Section B we show that our fine-grained uncertainty relations are not only directly relatedto entropic uncertainty relations

nsumt=1

Hinfin(t)σ ge minus log max~x

ζ~x (16)

but they are particularly appealing from both a physical as well as an information processingperspective For measurements in a full set of so-called mutually unbiased bases the ζ~x are theextrema of the discrete Wigner function used in physics [45] From an information processingperspective we may think of the string ~x as being encoded into quantum states σ~x where wecan perform the measurement t to retrieve the t-th entry from ~x Such an encoding is knownas a random access encoding of the string ~x [2 23] The value of ζ~x can now be understood asan upper bound on the probability that we retrieve the desired entry correctly averaged overour choice of t which is just P cert(σ~x) This upper bound is attained by the maximally certainstate ρ~x Bounds on the performance of random access encodings thus translate directly intouncertainty relations and vice versa

In Section C we discuss a number of example theories in the context of CHSH in orderto demonstrate the interplay between uncertainty and non-locality This includes quantummechanics classical mechanics local hidden variable models and theories which are maximallynon-local but still non-signalling In Section D we show that although non-locality mightdetermine the extent to which measurements in a theory are uncertain it does not determinehow complementary the measurements are We distinguish these concepts and give an examplein the case of the CHSH inequality of a theory which is just as non-local and uncertain asquantum mechanics but where the measurements are less complementary Finally in SectionE we show how the form of any uncertainty relation can be used to construct a non-local game

A XOR-games

In this section we concentrate on showing that for the case of XOR-games the relation betweenuncertainty relations and non-local games does not involve steering constraints since the steeringis only limited by the no-signalling condition Quantum mechanics allows Alice to steer Bobrsquosstates to those which are maximally certain for his optimal measurement settings For quantummechanics to become more non-local it must have weaker uncertainty relations for steerablestates

12

First of all let us recall some important facts about XOR games These games are equivalentto Bell inequalities for observables with two outcomes ie bipartite correlation inequalities fordichotomic observables They form the only general class of games which are truly understoodat present Not only do we know the structure of the measurements that Alice and Bob willperform but we can also find them efficiently using semidefinite programming for any XORgame [39] which remains a daunting problem for general games [14 21 22]

A1 Structure of measurements

We first recall the structure of the optimal measurements used by Alice and Bob In an XORgame the answers a isin 0 1 and b isin 0 1 of Alice and Bob respectively are single bitsand the decision whether Alice and Bob win or not depends only on the XOR of the answersaoplusb = a+b mod 2 The winning condition can thus be written in terms of a predicate that obeysV (c = aoplusb|s t) = 1 if and only if aoplusb = c are winning answers for Alice and Bob given settings sand t (otherwise V (c|s t) = 0) Let Aas and Bb

t denote the measurement operators correspondingto measurement settings s and t giving outcomes a and b of Alice and Bob respectively Withoutloss of generality we may thereby assume that these are projective measurements satisfyingAasA

aprimes = δaaprimeA

as and similarly for Bob Since we have only two measurement outcomes we may

view this measurement as measuring the observables

As = A0s minusA1

s (17)

Bt = B0t minusB1

t (18)

with outcomes plusmn1 where we label the outcome rsquo+1rsquo as rsquo0rsquo and the outcome rsquominus1rsquo as rsquo1rsquoTsirelson [35 36] has shown that the optimal winning probability for Alice and Bob can beachieved using traceless observables of the form

As =sumj

a(j)s Γj (19)

Bt =sumj

b(j)t Γj (20)

where ~as = (a(1)s a

(N)s ) isin RN and~bt = (b

(1)t b

(N)t ) isin RN are real unit vectors of dimension

N = min(|S| |T |) and Γ1 ΓN are the anti-commuting generators of a Clifford algebra Thatis we have

Γj Γk = ΓjΓk + ΓkΓj = 0 for all j 6= k (21)

(Γj)2 = I for all j (22)

In dimension d = 2n we can find at most N = 2n+1 such operators which can be obtained usingthe well known Jordan-Wigner transform [19] Since we have A0

s +A1s = I and B0

t +B1t = I we

may now use Eqs 17 and 18 to express the measurement operators in terms of the observablesas

Aas =1

2(I + (minus1)aAs) (23)

Bbt =

1

2

(I + (minus1)bBt

) (24)

13

Tsirelson furthermore showed that the optimal state that Alice and Bob use with said observablesis the maximally entangled state

|ψ〉 =1radicd

dminus1sumk=0

|k〉|k〉 (25)

of total dimension d = (2bN2c)2To see how XOR games are related to Bell correlation inequalities note that we may use

Eqs 23 and 24 to rewrite the winning probability of the game in terms of the observables as


p(s t)sumab

V (a b|s t)〈ψ|Aas otimesBbt |ψ〉 (26)

=1

2

(sumst

p(s t)sumc

V (c|s t) (1 + (minus1)c〈ψ|As otimesBt|ψ〉)

) (27)

For example for the well known CHSH inequality given by

〈ψ|CHSH|ψ〉 = 〈ψ|A0 otimesB0 +A0 otimesB1 +A1 otimesB0 minusA1 otimesB1|ψ〉 (28)

with p(s t) = p(s)p(t) = 14 we have

P game(S T σAB) =1

2

(1 +〈ψ|CHSH|ψ〉

4

)(29)

=1

4

sumst

sumab


where V (a b|s t) = 1 if and only if aoplus b = s middot t

A2 Steering to the maximally certain states

Now that we know the structure of the measurements that achieve the maximal winning prob-ability in an XOR game we can write down the corresponding uncertainty operators for Bob(the case for Alice is analogous) Once we have done that we will see that it is possible for Aliceto steer Bobrsquos state to the maximally certain ones First of all note that in the quantum casewe have that the set of measurement operators Btt is each decomposable in terms of POVMelements Bb

tb and thus ζ~x corresponds to the largest eigenvalue of the uncertainty operator

Q~xsa =sumt

p(t)Bx(t)sa

t (31)

The Q are positive operators so sets of them together with an error operator form a positivevalued operator and can be measured In the case of XOR games we thus have

Q~xsa =sumt

p(t)sumb

V (ab|st)=1

Bbt (32)

14

and the Bell polynomial can be expressed as

β =sumstab

p(s)p(t)V (ab|st)Aas otimesBbt (33)

=sumsa

p(s)Aas otimesQ~xsa (34)

For the special case of XOR-games we can now use Eq 24 to rewrite Bobrsquos uncertainty operatorusing the short hand ~v middot ~Γ =

sumj v

(j)Γj as

Q~xsa =1

2

sumt

p(t)sumb

V (ab|st)=1

(I + (minus1)bBt

)

=1

2

[csaI + ~vsa middot ~Γ

] (35)

where

csa =sumt

p(t)|sumb

V (a b|s t)| (36)

~vsa =sumt

p(t)sumb

V (ab|st)=1

(minus1)b ~bt (37)

What makes XOR games so appealing is that we can now easily compute the maximallycertain states Except for the case of two measurements and rank one measurement operatorsthis is generally not so easy to do even though bounds may of course be found Finding a gen-eral expression for such maximally certainty states allows us easily to compute the extrema ofthe discrete Wigner function as well as to prove tight entropic uncertainty relations for all mea-surements This has been an open problem since Deutschrsquos initial bound for two measurementsusing rank one operators [12 42] In particular we recall for completeness [15] that

Claim A1 For any XOR game we have that for any uncertainty operator Q~xsa of observablesof the form Eqs 19 and 20

ζ~xsa = maxρge0tr ρ=1

tr(ρQ~xsa

)= tr(ρ~xsaQ~xsa) =

1

2(csa + ~vsa2) (38)

with

ρ~xsa =1

d

I +sumj

r(j)saΓj

(39)

and

~rsa = ~vsa~vsa2 (40)

15

Proof We first note that the set of operators IcupΓjjcupiΓjΓkjkcup forms an orthonormal 3

basis for dtimes d Hermitian operators [13] We may hence write any state ρ as

ρ =1

d

I +sumj

r(j)saΓj +

(41)

for some real coefficients r(j)sa Using the fact that S forms an orthonormal basis we then obtain

from Eq 35 that

tr(ρQ~xsa) =1

2(csa + ~rsa middot ~vsa) (42)

Since for any quantum state ρ we must have ~rsa2 le 1 [41] maximising the left hand sidecorresponds to maximising the right hand side over vectors ~rsa of unit length The claim nowfollows by noting that if ~rsa has unit length then ρ~xsa is also a valid quantum state [41]

Note that the states ρ~xsa are highly mixed outside of dimension d = 2 but still correspondto the maximally certain states We now claim that that for any setting s Alice can in factsteer to the states which are optimal for Bobrsquos uncertainty operator

Lemma A2 For any XOR game we have for all s s isin Ssuma

p(a|s) ρ~xsa =suma

p(a|s) ρsa (43)

where ρ~xsa is defined as in Eq 39

Proof First of all note that it is easy to see in any XOR game the marginals of Alice (andanalogously Bob) are uniform That is the probability that Alice obtains outcome a givensetting s obeys

p(a|s) = 〈ψ|Aas otimes I|ψ〉 (44)

=1

2(〈ψ|Iotimes I|ψ〉+ (minus1)a〈ψ|As otimes I|ψ〉)

=1

2

(1 + tr(ATs )

)=

1

2

Second it follows from the fact that we are considering an XOR game that ~vsa = minus~vs1minusa forall s and a Hence by Claim A1 we have ~rsa = minus~rs1minusa and hence

1

2

(ρ~xs0 + ρ~xs1

)=

Id (45)

from which the statement follows

3With respect to the Hilbert-Schmidt inner product

16

Unrelated to our problem it is interesting to note that the observables As = ~asmiddot~Γ employed byAlice are determined exactly by the vectors rsa above Let σas denote Bobrsquos post-measurementstate which are steered to when Alice performs the measurement labelled by s and obtainsoutcome a We then have that for any of Bobrsquos measurement operators

tr(σsaQ~xsa

)=〈ψ|Aas otimesQ~xsa |ψ〉

p(a|s)(46)

=1

2(csa + (minus1)a~as middot ~vsa) (47)

Since Alicersquos observables should obey (As)2 = I we would like that ~as2 = 1 When trying to

find the optimal measurements for Alice we are hence faced with exactly the same task as whentrying to optimize over states ρ~xsa in Claim A1 That is letting ~as = (minus1)arsa gives us theoptimal observables for Alice

A3 Retrieval games and entropic uncertainty relations

As a further example we now consider a special class of non-local games that we call a retrievalgames which are a class of XOR games This class includes the famous CHSH game Retrievalgames have the further property that not only are the fine-grained uncertainty relations violatedif quantum mechanics would be more non-local but also the min-entropic uncertainty relationderived in [41] would be violated The class of retrieval games are those which corresponddirectly to the retrieval of a bit from an n-bit string More precisely we label the settings ofAlice using all the n-bit strings starting with a rsquo0rsquo That is

S = (0 s(2) s(n)) | s(j) isin 0 1 (48)

where we will choose each setting with uniform probability p(s) = 12nminus1 For each settings Alice has two outcomes which we label using the strings s and s where s is the bitwisecomplement of the string s More specifically we let

~xs0 = s (49)

~xs1 = s (50)

Bobrsquos settings are labelled by the indices T = 1 n and chosen uniformly at randomp(t) = 1n Alice and Bob win the game if and only if Bob outputs the bit at position t of thestring that Alice outputs In terms of the predicate the game rules can be written as

V (a b|s t) =

1 if b = ~x

(t)sa

0 otherwise (51)

We call such a game a retrieval game of length n It is not hard to see that the CHSH gameabove is a retrieval game of length n = 2 [40]

We now show that for the case of retrieval games not only can Alice steer perfectly to theoptimal uncertainty states but Bobrsquos optimal measurements are in fact very incompatible inthe sense that his measurement operators necessarily anti-commute In particular this meansthat we obtain very strong uncertainty relations for Bobrsquos optimal measurements [41] as we willdiscuss in more detail below

17

Lemma A3 Let Bt = B0t minus B1

t denote Bobrsquos optimal dichotomic observables in dimensiond = 2n Then for any retrieval game

Bt Btprime = 0 for all t 6= tprime isin T (52)

Proof First of all note that using Claim A1 we can write the winning probability for arbitraryobservables Bt = ~bt middot ~Γ in terms of the maximum uncertainty states ρ~xsa as

P game(S T σAB) =1

2n

sumsa

tr(ρ~xsaQ~xsa

)(53)

=1

2n

sumsa

1

2(1 + ~vsa2) (54)

=1

2

(1 +

1

2n

sumsa

radic~vsa middot ~vsa

)(55)

le 1

2

1 +

radic1

2n

sumsa

sumttprime

(minus1)~x(t)sa(minus1)~x

(tprime)sa~bt middot~btprime

(56)

=1

2+

1

2radicn (57)

where the inequality follows from the concavity of the square-root function and Jensenrsquos inequal-ity and the final inequality from the fact that ~bt is of unit length Equality with this upperbound can be achieved by choosing ~bt middot~btprime = 0 for all t 6= tprime which is equivalent to Bt Btprime = 0for all t 6= tprime

For our discussions about uncertainty relations in the next section it will be useful to notethat the above implies that λmax(Q~xsa) = λmax(Q~xsprimeaprime ) for all settings s sprime isin S and outcomes

a aprime isin A A useful consequence of the fact that ~bt middot~btprime = 0 for t 6= tprime is also that the probabilitythat Bob retrieves a bit of the string ~xsa correctly is the same for all bits

Corollary A4 For a retrieval game we have for the optimal strategy that for all settings s isin Sand outcomes a isin A

tr

(ρ~xsaB

~x(t)sa

t

)= tr

(ρ~xsaB

~x(tprime)sa

tprime

)for all t tprime isin T (58)

The fact that min-entropic uncertainty relations are violated if quantum mechanics were todo better than the Tsirelsonrsquos bound for retrieval games will be shown in Section B

It is an interesting open question whether Alice can steer to the maximally certain states forBobrsquos optimal measurements for all games This is for example possible for the mod 3-gamesuggested in [10] to which the optimal measurements were derived in [17 20] Unfortunately thestructure of measurement operators that are optimal for Bob is ill-understood for most gameswhich makes this a more difficult task

18

B Min-entropic uncertainty relations

We now discuss the relationship between fine-grained uncertainty relations and min-entropicones The relation between retrieval games and min-entropic uncertainty relations will followfrom that Recall that the min-entropy of the distribution obtained by measuring a state ρ usingthe measurement given by the observables Bt can be written as

Hinfin(Bt)ρ = minus log maxx(t)isin01

tr(Bx(t)

t ρ) (59)

A min-entropic uncertainty relation for the measurements in an n-bit retrieval game can bebounded as

1

n

nsumt=1

Hinfin(Bt)ρ = minus 1

n

sumt

log maxx(t)isin01

tr(Bx(t)

t ρ)

(60)

ge minus log max~xisinBtimesn

sumt

tr(Bx(t)

t ρ)

(61)

= minus log maxsa

λmax(Q~xsa) (62)

where the inequality follows from Jensenrsquos inequality and the concavity of the log Ie thefine-grained relations provide a lower bound on the min-entropic uncertainty relations

Now we note that for the case of retrieval games not only do we have

λmax(Q~xsa) =1

2+

1

2radicn (63)

but by Corollary A4 we have that the inequality 61 is in fact tight where equality is achievedfor any of the maximum uncertainty states ρ~xsa from the retrieval game We thus have

Theorem B1 For any retrieval game a violation of the Tsirelsonrsquos bound implies a violationof the min-entropic uncertainty relations for Bobrsquos optimal measurements A violation of themin-entropic uncertainty relation implies a violation of the Tsirelsonrsquos bound as long as steeringis possible

Note that since the CHSH game is a retrieval game with n = 2 we have that as longas steering is possible Tsirelsonrsquos bound [35] is violated if and only if Deutschrsquo min-entropicuncertainty relation [12] is violated for Bobrsquos (or Alicersquos) measurements

C An example the CHSH inequality in general theories

Probably the most well studied Bell inequality is the CHSH inequality and previous attemptsto understand the strength of quantum non-locality have been with respect to it [24 38 9 1]Although the connections between non-locality and uncertainty are more general we can usethe CHSH inequality as an example and show how the uncertainty relations of various theoriesdetermine the extent to which the theory can violate it We will see in this example hownon-locality requires steering (which is what prevents classical mechanics from violating a Bell

19

inequality despite having maximal certainty) Furthermore we will see that local-hidden variabletheories can have increased steering ability but donrsquot violate a Bell inequality because theyexactly compensate by having more uncertainty Quantum mechanics has perfect steering andso itrsquos non-locality is limited only by the uncertainty principle We will also discuss theorieswhich have the same degree of steering as quantum theory but greater non-locality because theyhave greater certainty (so-called ldquoPR-boxesrdquo [26 27 28] being an example)

The CHSH inequality can be expressed as a game in which Alice and Bob receive binaryquestions s t isin 0 1 respectively and similarly their answers a b isin 0 1 are single bits Aliceand Bob win the CHSH game if their answers satisfy aoplus b = s middot t The CHSH game thus belongsto the class of XOR games and any other XOR game could be used as a similar example

Note that we may again rephrase the game in the language of random access coding [40]where we label Alicersquos outcomes using string ~xsa and Bobrsquos goal is to retrieve the t-th elementof this string For s = 0 Bob will always need to give the same answer as Alice in order to winindependent of t and hence we have ~x00 = (0 0) and ~x01 = (1 1) For s = 1 Bob needs togive the same answer for t = 0 but the opposite answer if t = 1 That is ~x10 = (0 1) and~x11 = (1 0)

To gain some intuition of the tradeoff between steerability and uncertainty we consider an(over)simplified example in Figure 3 We examine quantum classical and a theory allowingmaximal non-locality below(i) Quantum mechanics

As we showed in Section A we have for any XOR game that we can always steer to themaximally certain states ρ~xsa and hence Alicersquos and Bobrsquos winning probability depend onlyon the uncertainty relations For CHSH Bobrsquos optimal measurement are given by the binaryobservables B0 = Z and B1 = X The amount of uncertainty we observe for Bobrsquos optimalmeasurements is given by

ζ~xsa =1

2+

1

2radic

2for all ~xsa isin 0 12 (64)



2Alice can steer Bobrsquos states to the eigenstates of these operators by measuring in the basis ofthese operators on the state

|ψminus〉 =1radic2

(|00〉AB + |11〉AB) (65)

We hence have P gamemax = ζ~xsa = 12 + 1(2

radic2) which is Tsirelsonrsquos bound [35 36] If Alice

and Bob could obtain a higher value for the same measurements at least one of the fine-graineduncertainty relations is violated We also saw above that a larger violation for CHSH also impliesa violation of Deutschrsquo min-entropic uncertainty relation(ii) Classical mechanics amp local hidden variable theories

For classical theories let us first consider the case where we use a deterministic strategysince classically there is no fundamental restriction on how much information can be gainedie if we optimize the uncertainty relations over all classical states we have ζ~xsa = 1 for anyset of measurements However for the states which are maximally certain there is no steering

20

070 075 080 085 090 095 100070

075

080

085

090

095

100

Steering

Unc

erta

inty

Figure 3 A simplified example Imagine a world in which the only steerable states are themaximally certain states of the uncertainty relations we consider and we have an all or nothingform of steering Ie either Alice can steer to all ensembles with probability psteer or failsentirely The vertical axis denotes the certainty pcert (that is the lack of uncertainty) andthe horizontal axis psteer Lighter colours indicate a larger winning probability which in thissimplified case is just P game(S T σAB) = psteerpcert The solid line denotes the case of P game

max =34 which can be achieved classically The point on the line denotes the combination of valuesfor a classical deterministic theory there is no uncertainty (ζ~xsa = 1 for all ~xsa) and no steeringother than the trivial one to the state Alice and Bob already share as part of their strategy whichyields 34 on average The dashed line denotes the value P game

max = 12 + 1(2radic

2) achievableby a quantum strategy The point on the line denotes the point reached quantumly there isuncertainty but we can steer perfectly to the maximally certain states Finally the point at(1 1) denotes the point achievable by ldquoPR-boxesrdquo there is no uncertainty but neverthelessperfect steering

21

0

Figure 4 The hidden variable states ρ~x which encode each of the four random access stringson Bobrsquos site are depicted above The grey region corresponds to the hidden variable beingsuch that Bob can correctly retrieve both bits The blue and red regions correspond to theregion where Bob incorrectly retrieves the first or second bit respectively These regions mustbe included if Alice is to be able to steer to the states

property either because in the no-signalling constraint the density matrix has only one term init A deterministic state cannot be written as a convex sum of any other states ndash it is an extremalpoint of a convex set The best deterministic strategy is thus to prepare a particular state atBobrsquos site (eg an encoding of the bit-string 00) This results in P game(S T σAB) = 34 Aprobabilistic strategy cannot do better since it would be a convex combination of deterministicstrategies and one might as well choose the best one However it will be instructive to considerthe non-deterministic case

Although the maximally certain states cannot be steered to we can steer to states whichare mixtures of deterministic states (cf [32]) This corresponds to using a non-deterministicstrategy or if the non-determinism is fundamental to a local hidden variable theory Howevermeasurements on the states which can be steered to will not have well-defined outcomes If weoptimize the uncertainty relations with respect to the non-deterministic states we will find thatthere is a substantial uncertainty in measurement outcomes on these states We will find that theability to steer is exactly compensated by an inability to obtain certain measurement outcomesthus the probability of winning the non-local game will again be P game(S T σAB) = 34

Figure 4 depicts an optimal local hidden variable theory for CHSH where the local hiddenvariable is a point on the unit circle labelled by an angle Ω and the states are probabilitydistributions

ρ~xsa =

int Ωprime~xsa

Ω~xsa

p(Ω)σΩdΩ (66)

with σΩ denoting the state of the hidden variable and p(Ω) uniform Alice can prepare thesestates at Bobrsquos site if they initially share the maximally correlated state ψMC

AB =int 2π

0 p(Ω)σΩAσ

ΩBdΩ

and she makes a partial measurement on her state which determines that the value of her hiddenvariable lies within Ω~xsa and Ωprime~xsa

A measurement by Bob corresponds to cutting the circle in half along some angle and thendetermining whether the hidden variable lies above or below the cut Ie a coarse graineddetermination of the bounds of integration in ρ~xsa to within π Bobrsquos measurement for t = 0thereby corresponds to determining whether the hidden variable lies above or below the cut

22

along the equator while his measurement for t = 1 corresponds to determining whether thehidden variable lies above or below the cut labelled by the angle θ1 If the hidden variable isabove the cut we label the outcome as 1 and 0 otherwise

First of all note that if the hidden variable lies in the region shaded in grey as depicted inFigure 4 then Bob will be able to retrieve both bits correctly because it lies in the region suchthat the results for Bobrsquos measurements would match the string that the state encodes Forexample for the ρ00 state the hidden variable in the grey region lies below both cuts while forthe ρ11 state the hidden variable lies above both cuts If we optimize the uncertainty relationsover states which are chosen from the grey region then ζ~xsa = 1 If ρ~xsa only includes the greyregion then it is a maximally certain state

Whether steering is possible to a particular set of ρ~xsa is determined by the no-signallingcondition If the bounds of integration Ω~xsa and Ωprime~xsa in Equation (66) only included the grey

regions then we would not be able to steer to the states ρ~xsa since p(00)ρ00 + p(11)ρ11 6=p(01)ρ01 + p(10)ρ10 That is steering to the maximally certain states is forbidden by the no-signalling principle if they lie in the grey region The states only become steerable if the ρ~xsainclude the red and blue areas depicted in Figure 4 Then Alice making measurements on hershare of ψMC

AB which only determine the hidden variable to within an angle π will be able toprepare the appropriate states on Bobrsquos site Alicersquos measurement angles are labelled by theangles φ0 for the s = 0 partition and φ1 for the s = 1 partition

However if the hidden variable lies in the blue area then Bob retrieves the first bit incorrectlyand if it is in the red region he retrieves the second bit incorrectly If the ρ~xsa include a convexcombination of hidden variables which include the grey blue and red regions then the statesare now steerable but the uncertainty relations with respect to these states give ζ00 = ζ11 =1minus θB2π and ζ01 = ζ10 = 1minus (πminus θB)2π since the error probabilities are just proportional tothe size of the red and blue regions Hence we obtain P game(S T σAB) = ζ~xsa = 34(iii) No-signalling theories with maximal non-locality

It is possible to construct a probability distribution which not only obeys the no-signallingconstraint and violates the CHSH inequality [29 37] but also violates it more strongly thanquantum theory and in fact is maximally non-local [26] Objects which have this propertywe call PR-boxes and in particular they allow us to win the CHSH game with probability 1Note that this implies that there is no uncertainty in measurement outcomes ζ~xsa = 1 for all~xsa isin 0 12 where ρ~xsa is the maximally certain state for ~xsa [5 33] Conditional on Alicersquosmeasurement setting and outcome Bobrsquos answer must be correct for either of his measurementslabelled t = 0 and t = 1 and described by the following probability distributions p(b|t) formeasurements t isin 0 1

ρ00 = p(0|0) = 1 p(0|1) = 1 (67)

ρ01 = p(1|0) = 1 p(1|1) = 1 (68)

ρ10 = p(0|0) = 1 p(1|1) = 1 (69)

ρ11 = p(1|0) = 1 p(0|1) = 1 (70)

The only constraint that needs to be imposed on our ability to steer to these states is givenby the no-signalling condition and indeed Alice may steer to the ensembles 12 ρ~x0aa and

23

12 ρ~x1aa at will while still satisfying the no-signalling condition We hence have P game(S T σAB) =ζ~xsa = 1

D Uncertainty and complementarity

Uncertainty and complementarity are often conflated However even though they are closelyrelated they are nevertheless distinct concepts Here we provide a simple example that illustratestheir differences a full discussion of their relationship is outside the scope of this work Recallthat the uncertainty principle as used here is about the possible probability distribution ofmeasurement outcomes p(b|tj) when one measurement t1 is performed on one system and anothermeasurement t2 is performed on an identically prepared system Complementarity on the otherhand is the notion that you can only perform one of two incompatible measurements (see forexample [8] since one measurement disturbs the possible measurement outcomes when bothmeasurements are performed on the same system in succession We will find that althoughthe degree of non-locality determines how uncertain measurements are this is not the case forcomplementarity ndash there are theories which have less complementarity than quantum mechanicsbut the same degree of non-locality and uncertainty

Let τtb denote the state of the system after we performed the measurement labelled t and ob-tained outcome b After the measurement we are then interested in the probability distributionp(bprime|tprime)τtb of obtaining outcome bprime when performing measurement tprime on the post-measurementstate As before we can consider the term

η~x =sumtprime

p(tprime)p(bprime|tprime)τtb (71)

This quantity has a similar form as ζ~x and in the case where a measurement is equivalent to apreparation we clearly have that

η~x le ζ~x (72)

since post-measurement state when obtaining outcome b is just a particular preparation whileζ~x is a maximisation over all preparations

There is however a different way of looking at complementarity which is about the extractionof information In this sense one would say that two measurements are complementary if thesecond measurement can extract no more information about the preparation procedure than thefirst measurement and visa versa We refer to this as information complementarity Note thatquantum mechanically this does not necessarily have to do with whether two measurementscommute For example if the first measurement is a complete Von Neumann measurementsthen all subsequent measurements gain no new information than the first one whether theycommute or otherwise

D1 Quantum mechanics could be less complementarity with the same degreeof non-locality

We now consider a simple example that illustrates the differences between complementarityand uncertainty Recall from Section A3 that the CHSH game is an instance of a retrieval

24

game where Bob is challenged to retrieve either the first or second bit of a string ~xsa isin 0 12prepared by Alice For our example we then imagine that the initial state σ~xsa of Bobrsquos systemcorresponds to an encoding of a string ~xsa isin 0 12 and fix Bobrsquos measurements to be the twopossible optimal measurements he performs in the CHSH game when given questions t = 0 andt = 1 respectively When considering complementarity between the measurements labelled byt = 0 and t = 1 we are interested in the probabilities p(bprime|tprime)τtb of decoding the second bit fromthe post-measurement state τtb after Bob performed the measurement t and obtained outcomeb We say that there is no-complementarity if p(bprime|tprime)τtb = p(bprime|tprime)σ~xsa for all tprime and bprime That is

the probabilities of obtaining outcomes bprime when performing tprime are the same as if Bob had notmeasured t at all Note that if the measurements that Bob (and Alice) perform in a non-localgame had no-complementarity then their statistics could be described by a LHV model sinceone can assign a fixed probability distribution to the outcomes of each measurement As a resultif there is no complementarity there cannot be a violation of the CHSH inequality

We now ask what are the allowed values for p(bprime|tprime)τtb subject to the restrictions imposed byEq 72 and no-signalling For the case of CHSH where the notions of uncertainty and non-localityare equivalent the no-signalling principle dictates that Bob can never learn the parity of thestring ~xsa as this would allow him to determine Alicersquos measurement setting s Since Bob mightuse his two measurements to determine the parity this imposes an additional constraint on the

allowed probabilities p(bprime|tprime)τtb For clarity we will write p(right|t ~xsa) = p(b = ~x(t)sa|t)σ~xsa for

the probability that Bob correctly retrieves the t-th bit of the string ~xsa on the initial state and

p(right|tprime right) = p(~x(tprime)sa |tprime)τ

~x(t)sat

and p(right|tprime wrong) = p(~x(tprime)sa |tprime)τ

~x(1minust)sa t

for the probabilities

that he correctly retrieves the tprime-th bit given that he previously retrieved the t-th bit correctly orincorrectly respectively For any t 6= tprime the fact that Bob can not learn the parity by performingthe two measurements in succession can then be expressed as

p(right|t ~xsa)p(right|tprime right)+ p(wrong|t ~xsa)p(wrong|tprime wrong) =1

2(73)

since retrieving both bits incorrectly will also lead him to correctly guess the parity The tradeoffbetween p(right|tprime right) and p(wrong|tprime wrong) dictated by Eq 73 is captured by Figure 5

As previously observed the ldquoamountrdquo of uncertainty ζ~xsa directly determines the violation

of the CHSH inequality In the quantum setting we have for all ~xsa that ζ~xsa = 12+1(2radic

2) asymp0853 where for the individual probabilities we have for all s a t and b

p(right|t ~xsa) =1

2+

1

2radic

2 (74)

We also have that p(right|tprime right) = p(right|tprime wrong) = 12 for all tprime 6= t That is Bobrsquos opti-mal measurements are maximally complementary Once Bob performed the first measurementhe can do no better than to guess the second bit Note however that neither Eq 73 nor lettingζ~xsa = 12 + 1(2

radic2) demands that quantum mechanics be maximally complementary without

referring to the Hilbert space formalism In particular consider the average probability thatBob retrieves the tprime-th bit correctly after the measurement t has already been performed whichis given by

psecond = p(right|t ~xsa)p(right|tprime right)+ p(wrong|t ~xsa)p(right|tprime wrong) (75)

25

00 02 04 06 08 1000

02

04

06

08

10

pHrightEgravetrightL

pHw

rong

Egravetw

rong

L

Figure 5 Allowed values for (p(right|tprime right) p(wrong|tprime wrong)) for p(right|t ~xsa) = 12(dot dashed light blue line) 34 (dashed green line) 12 + 1(2

radic2) (dotted red line) and 1

(solid blue line)

We can use Eq 73 to determine p(right|tprime wrong) = 1 minus p(wrong|tprime wrong) Using Eq 74we can now maximize psecond over the only free remaining variable p(right|tprime right) such that0 le p(wrong|tprime wrong) le 1 This gives us psecond = 1 minus 1(2

radic2) asymp 065 which is attained for

p(right|tprime right) = 2 minusradic

2 asymp 059 and p(right|tprime wrong) = 1 However for the measurementsused by Bobrsquos optimal quantum strategy we only have psecond = 12 We thus see that it maybe possible to have a physical theory which is as non-local and uncertain as quantum mechanicsbut at the same time less complementary We would like to emphasize however that in quantumtheory it is known [25] that Bobrsquos observables must be maximally complementary in order toachieve Tsirelsonrsquos bound which is a consequence of the Hilbert space formalism

Another interesting example is the case of a PR-boxes and other less non-local boxes Herewe can have p(right|t ~xsa) = 1 minus ε for all t ~xsa and 12 le ε le 1 Note that for ε rarr 1maximising Eq 75 gives us psecond = 12 with p(right|tprime right) = 12 Figure 6 shows thevalue of psecond in terms of p(right|t ~xsa) If there is no uncertainty we thus have maximalcomplementarity However as we saw from the example of quantum mechanics for intermediatevalues of uncertainty the degree of complementarity is not uniquely determined

It would be interesting to examine the role of comlementarity and its relation to non-locality

26

05 06 07 08 09 10pHrightEgravetxL05

06

07

08

09

10pHsecondL

Figure 6 Maximum value of psecond subject to Eq 73 in terms of p(right|t ~xsa)

in more detail The existance of general monogamy relations [34] for example could be under-stood as a special form of complementarity when particular measurements are applied

E From uncertainty relations to non-local games

In the body of the paper we showed how every game corresponds to an uncertainty relationHere we show that every uncertainty relation gives rise to a game To construct the game froman uncertainty relation we can simply go the other way

We start with the set of inequalities

U =

nsumt=1

p(t) p(bt|t)σ le ζ~x | forall~x isin Btimesn

which form the uncertainty relation Additionally they can be thought of as the average successprobability that Bob will be able to correctly output the trsquoth entry from a string ~x 4

Here he is performing a measurement on the state σ which encodes the string ~x Themaximisation of these relations

P succΣ (~x) = ζ~x = max

σisinΣ

nsumt=1

p(t)p(bt|t)σ (76)

will play a key role where here the maximisation is taken over a set Σ determined by thetheoryrsquos steering properties

Consider the set of all strings ~x induced by the above uncertainty relation We construct thegame by choosing some partitioning of these strings into sets P1 PM such that cupPs = BtimesL

4Indeed the left hand side of the inequality corresponds to a set of operatorssumn

t=1 p(t) Mbtt which together

form a complete Postive Operator-Valued Measure (POVM) and can be used to measure this average

27

We we will challenge Alice to output one of the strings in set s and Bob to output a certainentry t of that string Alicersquos settings are given by S = 1 M that is each setting willcorrespond to a set Ps The outcomes for a setting s isin S are simply the strings contained in Pswhich using the notation from the main paper we will label ~xsa

Bobrsquos settings in the game are in one-to-one correspondence to the measurements for whichwe have an uncertainty relation That is we label his settings by his choices of measurementsT = 1 n and his outcomes by the alphabet of the string We furthermore let the distri-butions over measurements be given by p(t) as in the case of the uncertainty relation Note thatthe distribution p(s) over Alicersquos measurement settings is not yet defined and may be chosenarbitrarily The predicate is now simply defined as V (a b|s t) = 1 if and only if bt is the trsquothentry in the string ~xsa

Note that due to our construction Alice will be able to steer Bobrsquos part of the state intosome state σ~xsa encoding the string ~x for all settings s We are then interested in the set ofensembles I = Ess that the theory allows Alice to steer to In no-signalling theories thiscorresponds to probability distributions p(~x|s)s and an average state ρ such thatsum

~xisinPs

p(~x|s) σ~x = ρ

As before the states we can steer to in a particular ensemble s we denote by ΣsNow that we have defined the game this way we obtain the following lower bound on the

value ω(G) of the game sums

suma

p(s)p(~xsa|s)P succΣs

(~xsa) le P gamemax (77)

Whereas the measurements of our uncertainty relation do form a possible strategy for Alice andBob which due to the steering property can indeed be implemented they may not be optimalIndeed there may exist an altogether different strategy consisting of different measurements anda different state in possibly much larger dimension that enables them to do significantly better

However having defined the game we may now again consider a new uncertainty relation interms of the optimal states and measurements for this game For these optimal measurementsa violation of the uncertainty relations then lead to a violation of the corresponding Tsirelsonrsquosbound and vice versa

28

A XOR-games







D1 Quantum mechanics could be less complementarity with the same degree of non-locality


as strong as they could be Is there a principle that determines the degree of this non-localityInformation-theory [33 24] communication complexity [38] and local quantum mechanics [3]provided us with some rationale why limits on quantum theory may exist But evidence sug-gests that many of these attempts provide only partial answers Here we take a very differentapproach and relate the limitations of non-local correlations to two inherent properties of anyphysical theory

Uncertainty relations

At the heart of quantum mechanics lies Heisenbergrsquos uncertainty principle [16] Traditionallyuncertainty relations have been expressed in terms of commutators

∆A∆B ge 1

2|〈ψ|[AB]|ψ〉| (1)

with standard deviations ∆X =radic〈ψ|X2|ψ〉 minus 〈ψ|X|ψ〉2 for X isin AB However the more

modern approach is to use entropic measures Let p(x(t)|t)σ denote the probability that weobtain outcome x(t) when performing a measurement labelled t when the system is prepared inthe state σ In quantum theory σ is a density operator while for a general physical theory weassume that σ is simply an abstract representation of a state The well-known Shannon entropyH(t)σ of the distribution over measurement outcomes of measurement t on a system in state σis thus

H(t)σ = minussumx(t)

p(x(t)|t)σ log p(x(t)|t)σ (2)

In any uncertainty relation we wish to compare outcome distributions for multiple measure-ments In terms of entropies such relations are of the formsum

tisinTp(t) H(t)σ ge cT D (3)

where p(t) is any probability distribution over the set of measurements T and cT D is somepositive constant determined by T and the distribution D = p(t)t To see why (3) forms anuncertainty relation note that whenever cT D gt 0 we cannot predict the outcome of at least oneof the measurements t with certainty ie H(t)σ gt 0 Such relations have the great advantagethat the lower bound cT D does not depend on the state σ [12] Instead cT D depends only onthe measurements and hence quantifies their inherent incompatibility It has been shown thatfor two measurements entropic uncertainty relations do in fact imply Heisenbergrsquos uncertaintyrelation [7] providing us with a more general way of capturing uncertainty (see [42] for a survey)

One may consider many entropic measures instead of the Shannon entropy For examplethe min-entropy

Hinfin(t)σ = minus log maxx(t)

p(x(t)|t)σ (4)

used in [12] plays an important role in cryptography and provides a lower bound on H(t)σEntropic functions are however a rather coarse way of measuring the uncertainty of a set of

2





U =

nsumt=1


(6)


ζ~x = maxσ

nsumt=1

p(t)p(x(t)|t)σ (7)



1

2p(x(X)|X) +

1


1

2+

1

2radic




2




3












p(s t)suma


4



(t)


P gamemax = max

ST σAB




radic2) quantum



Steering


σB =suma



5




Main result




p(s)suma



6



p(s)suma


ζ~xsa(T D) (13)



P gamemax = max

Ess

sums

p(s)suma




P gamemax =

sums

p(s)suma






7














8

References











quant-ph0409066 2004







9
















10



















11



nsumt=1


ζ~x (16)



A XOR-games


12








As = A0s minusA1

s (17)

Bt = B0t minusB1

t (18)


As =sumj

a(j)s Γj (19)

Bt =sumj

b(j)t Γj (20)



(1)t b






s +A1s = I and B0

t +B1t = I we


Aas =1

2(I + (minus1)aAs) (23)

Bbt =

1

2

(I + (minus1)bBt

) (24)

13


|ψ〉 =1radicd

dminus1sumk=0

|k〉|k〉 (25)




p(s t)sumab


=1

2

(sumst

p(s t)sumc


) (27)




P game(S T σAB) =1

2


4

)(29)

=1

4

sumst

sumab






Q~xsa =sumt

p(t)Bx(t)sa

t (31)


Q~xsa =sumt

p(t)sumb

V (ab|st)=1

Bbt (32)

14


β =sumstab


=sumsa



sumj v

(j)Γj as

Q~xsa =1

2

sumt

p(t)sumb

V (ab|st)=1

(I + (minus1)bBt

)

=1

2


] (35)

where

csa =sumt

p(t)|sumb

V (a b|s t)| (36)

~vsa =sumt

p(t)sumb

V (ab|st)=1

(minus1)b ~bt (37)




tr(ρQ~xsa


1

2(csa + ~vsa2) (38)

with

ρ~xsa =1

d

I +sumj

r(j)saΓj

(39)

and


15



ρ =1

d

I +sumj

r(j)saΓj +

(41)


from Eq 35 that

tr(ρQ~xsa) =1





p(a|s) ρ~xsa =suma

p(a|s) ρsa (43)




=1


=1

2

(1 + tr(ATs )

)=

1

2


1

2

(ρ~xs0 + ρ~xs1

)=

Id (45)



16


tr(σsaQ~xsa


p(a|s)(46)

=1






S = (0 s(2) s(n)) | s(j) isin 0 1 (48)


~xs0 = s (49)

~xs1 = s (50)


V (a b|s t) =

1 if b = ~x

(t)sa

0 otherwise (51)



17





P game(S T σAB) =1

2n

sumsa

tr(ρ~xsaQ~xsa

)(53)

=1

2n

sumsa

1

2(1 + ~vsa2) (54)

=1

2

(1 +

1

2n

sumsa


)(55)

le 1

2

1 +

radic1

2n

sumsa

sumttprime



(56)

=1

2+

1

2radicn (57)





tr

(ρ~xsaB

~x(t)sa

t

)= tr

(ρ~xsaB

~x(tprime)sa

tprime




18




tr(Bx(t)

t ρ) (59)


1

n

nsumt=1


n

sumt

log maxx(t)isin01

tr(Bx(t)

t ρ)

(60)


sumt

tr(Bx(t)

t ρ)

(61)

= minus log maxsa

λmax(Q~xsa) (62)



λmax(Q~xsa) =1

2+

1

2radicn (63)






19






ζ~xsa =1

2+

1

2radic






(|00〉AB + |11〉AB) (65)





20

070 075 080 085 090 095 100070

075

080

085

090

095

100

Steering

Unc

erta

inty



max = 12 + 1(2radic


21

0





ρ~xsa =

int Ωprime~xsa

Ω~xsa

p(Ω)σΩdΩ (66)


AB =int 2π

0 p(Ω)σΩAσ

ΩBdΩ



22








ρ00 = p(0|0) = 1 p(0|1) = 1 (67)

ρ01 = p(1|0) = 1 p(1|1) = 1 (68)

ρ10 = p(0|0) = 1 p(1|1) = 1 (69)

ρ11 = p(1|0) = 1 p(0|1) = 1 (70)


23





η~x =sumtprime



η~x le ζ~x (72)





24







~x(t)sat


~x(1minust)sa t




2(73)





p(right|t ~xsa) =1

2+

1

2radic

2 (74)





25

00 02 04 06 08 1000

02

04

06

08

10


pHw

rong

Egravetw

rong

L



(solid blue line)







26


06

07

08

09

10pHsecondL






U =

nsumt=1





σisinΣ

nsumt=1

p(t)p(bt|t)σ (76)






27




~xisinPs

p(~x|s) σ~x = ρ



suma





28

A XOR-games













U =

nsumt=1


(6)


ζ~x = maxσ

nsumt=1

p(t)p(x(t)|t)σ (7)



1

2p(x(X)|X) +

1


1

2+

1

2radic




2




3












p(s t)suma


4



(t)


P gamemax = max

ST σAB




radic2) quantum



Steering


σB =suma



5




Main result




p(s)suma



6



p(s)suma


ζ~xsa(T D) (13)



P gamemax = max

Ess

sums

p(s)suma




P gamemax =

sums

p(s)suma






7














8

References











quant-ph0409066 2004







9
















10



















11



nsumt=1


ζ~x (16)



A XOR-games


12








As = A0s minusA1

s (17)

Bt = B0t minusB1

t (18)


As =sumj

a(j)s Γj (19)

Bt =sumj

b(j)t Γj (20)



(1)t b






s +A1s = I and B0

t +B1t = I we


Aas =1

2(I + (minus1)aAs) (23)

Bbt =

1

2

(I + (minus1)bBt

) (24)

13


|ψ〉 =1radicd

dminus1sumk=0

|k〉|k〉 (25)




p(s t)sumab


=1

2

(sumst

p(s t)sumc


) (27)




P game(S T σAB) =1

2


4

)(29)

=1

4

sumst

sumab






Q~xsa =sumt

p(t)Bx(t)sa

t (31)


Q~xsa =sumt

p(t)sumb

V (ab|st)=1

Bbt (32)

14


β =sumstab


=sumsa



sumj v

(j)Γj as

Q~xsa =1

2

sumt

p(t)sumb

V (ab|st)=1

(I + (minus1)bBt

)

=1

2


] (35)

where

csa =sumt

p(t)|sumb

V (a b|s t)| (36)

~vsa =sumt

p(t)sumb

V (ab|st)=1

(minus1)b ~bt (37)




tr(ρQ~xsa


1

2(csa + ~vsa2) (38)

with

ρ~xsa =1

d

I +sumj

r(j)saΓj

(39)

and


15



ρ =1

d

I +sumj

r(j)saΓj +

(41)


from Eq 35 that

tr(ρQ~xsa) =1





p(a|s) ρ~xsa =suma

p(a|s) ρsa (43)




=1


=1

2

(1 + tr(ATs )

)=

1

2


1

2

(ρ~xs0 + ρ~xs1

)=

Id (45)



16


tr(σsaQ~xsa


p(a|s)(46)

=1






S = (0 s(2) s(n)) | s(j) isin 0 1 (48)


~xs0 = s (49)

~xs1 = s (50)


V (a b|s t) =

1 if b = ~x

(t)sa

0 otherwise (51)



17





P game(S T σAB) =1

2n

sumsa

tr(ρ~xsaQ~xsa

)(53)

=1

2n

sumsa

1

2(1 + ~vsa2) (54)

=1

2

(1 +

1

2n

sumsa


)(55)

le 1

2

1 +

radic1

2n

sumsa

sumttprime



(56)

=1

2+

1

2radicn (57)





tr

(ρ~xsaB

~x(t)sa

t

)= tr

(ρ~xsaB

~x(tprime)sa

tprime




18




tr(Bx(t)

t ρ) (59)


1

n

nsumt=1


n

sumt

log maxx(t)isin01

tr(Bx(t)

t ρ)

(60)


sumt

tr(Bx(t)

t ρ)

(61)

= minus log maxsa

λmax(Q~xsa) (62)



λmax(Q~xsa) =1

2+

1

2radicn (63)






19






ζ~xsa =1

2+

1

2radic






(|00〉AB + |11〉AB) (65)





20

070 075 080 085 090 095 100070

075

080

085

090

095

100

Steering

Unc

erta

inty



max = 12 + 1(2radic


21

0





ρ~xsa =

int Ωprime~xsa

Ω~xsa

p(Ω)σΩdΩ (66)


AB =int 2π

0 p(Ω)σΩAσ

ΩBdΩ



22








ρ00 = p(0|0) = 1 p(0|1) = 1 (67)

ρ01 = p(1|0) = 1 p(1|1) = 1 (68)

ρ10 = p(0|0) = 1 p(1|1) = 1 (69)

ρ11 = p(1|0) = 1 p(0|1) = 1 (70)


23





η~x =sumtprime



η~x le ζ~x (72)





24







~x(t)sat


~x(1minust)sa t




2(73)





p(right|t ~xsa) =1

2+

1

2radic

2 (74)





25

00 02 04 06 08 1000

02

04

06

08

10


pHw

rong

Egravetw

rong

L



(solid blue line)







26


06

07

08

09

10pHsecondL






U =

nsumt=1





σisinΣ

nsumt=1

p(t)p(bt|t)σ (76)






27




~xisinPs

p(~x|s) σ~x = ρ



suma





28

A XOR-games




















p(s t)suma


4



(t)


P gamemax = max

ST σAB




radic2) quantum



Steering


σB =suma



5




Main result




p(s)suma



6



p(s)suma


ζ~xsa(T D) (13)



P gamemax = max

Ess

sums

p(s)suma




P gamemax =

sums

p(s)suma






7














8

References











quant-ph0409066 2004







9
















10



















11



nsumt=1


ζ~x (16)



A XOR-games


12








As = A0s minusA1

s (17)

Bt = B0t minusB1

t (18)


As =sumj

a(j)s Γj (19)

Bt =sumj

b(j)t Γj (20)



(1)t b






s +A1s = I and B0

t +B1t = I we


Aas =1

2(I + (minus1)aAs) (23)

Bbt =

1

2

(I + (minus1)bBt

) (24)

13


|ψ〉 =1radicd

dminus1sumk=0

|k〉|k〉 (25)




p(s t)sumab


=1

2

(sumst

p(s t)sumc


) (27)




P game(S T σAB) =1

2


4

)(29)

=1

4

sumst

sumab






Q~xsa =sumt

p(t)Bx(t)sa

t (31)


Q~xsa =sumt

p(t)sumb

V (ab|st)=1

Bbt (32)

14


β =sumstab


=sumsa



sumj v

(j)Γj as

Q~xsa =1

2

sumt

p(t)sumb

V (ab|st)=1

(I + (minus1)bBt

)

=1

2


] (35)

where

csa =sumt

p(t)|sumb

V (a b|s t)| (36)

~vsa =sumt

p(t)sumb

V (ab|st)=1

(minus1)b ~bt (37)




tr(ρQ~xsa


1

2(csa + ~vsa2) (38)

with

ρ~xsa =1

d

I +sumj

r(j)saΓj

(39)

and


15



ρ =1

d

I +sumj

r(j)saΓj +

(41)


from Eq 35 that

tr(ρQ~xsa) =1





p(a|s) ρ~xsa =suma

p(a|s) ρsa (43)




=1


=1

2

(1 + tr(ATs )

)=

1

2


1

2

(ρ~xs0 + ρ~xs1

)=

Id (45)



16


tr(σsaQ~xsa


p(a|s)(46)

=1






S = (0 s(2) s(n)) | s(j) isin 0 1 (48)


~xs0 = s (49)

~xs1 = s (50)


V (a b|s t) =

1 if b = ~x

(t)sa

0 otherwise (51)



17





P game(S T σAB) =1

2n

sumsa

tr(ρ~xsaQ~xsa

)(53)

=1

2n

sumsa

1

2(1 + ~vsa2) (54)

=1

2

(1 +

1

2n

sumsa


)(55)

le 1

2

1 +

radic1

2n

sumsa

sumttprime



(56)

=1

2+

1

2radicn (57)





tr

(ρ~xsaB

~x(t)sa

t

)= tr

(ρ~xsaB

~x(tprime)sa

tprime




18




tr(Bx(t)

t ρ) (59)


1

n

nsumt=1


n

sumt

log maxx(t)isin01

tr(Bx(t)

t ρ)

(60)


sumt

tr(Bx(t)

t ρ)

(61)

= minus log maxsa

λmax(Q~xsa) (62)



λmax(Q~xsa) =1

2+

1

2radicn (63)






19






ζ~xsa =1

2+

1

2radic






(|00〉AB + |11〉AB) (65)





20

070 075 080 085 090 095 100070

075

080

085

090

095

100

Steering

Unc

erta

inty



max = 12 + 1(2radic


21

0





ρ~xsa =

int Ωprime~xsa

Ω~xsa

p(Ω)σΩdΩ (66)


AB =int 2π

0 p(Ω)σΩAσ

ΩBdΩ



22








ρ00 = p(0|0) = 1 p(0|1) = 1 (67)

ρ01 = p(1|0) = 1 p(1|1) = 1 (68)

ρ10 = p(0|0) = 1 p(1|1) = 1 (69)

ρ11 = p(1|0) = 1 p(0|1) = 1 (70)


23





η~x =sumtprime



η~x le ζ~x (72)





24







~x(t)sat


~x(1minust)sa t




2(73)





p(right|t ~xsa) =1

2+

1

2radic

2 (74)





25

00 02 04 06 08 1000

02

04

06

08

10


pHw

rong

Egravetw

rong

L



(solid blue line)







26


06

07

08

09

10pHsecondL






U =

nsumt=1





σisinΣ

nsumt=1

p(t)p(bt|t)σ (76)






27




~xisinPs

p(~x|s) σ~x = ρ



suma





28

A XOR-games











(t)


P gamemax = max

ST σAB




radic2) quantum



Steering


σB =suma



5




Main result




p(s)suma



6



p(s)suma


ζ~xsa(T D) (13)



P gamemax = max

Ess

sums

p(s)suma




P gamemax =

sums

p(s)suma






7














8

References











quant-ph0409066 2004







9
















10



















11



nsumt=1


ζ~x (16)



A XOR-games


12








As = A0s minusA1

s (17)

Bt = B0t minusB1

t (18)


As =sumj

a(j)s Γj (19)

Bt =sumj

b(j)t Γj (20)



(1)t b






s +A1s = I and B0

t +B1t = I we


Aas =1

2(I + (minus1)aAs) (23)

Bbt =

1

2

(I + (minus1)bBt

) (24)

13


|ψ〉 =1radicd

dminus1sumk=0

|k〉|k〉 (25)




p(s t)sumab


=1

2

(sumst

p(s t)sumc


) (27)




P game(S T σAB) =1

2


4

)(29)

=1

4

sumst

sumab






Q~xsa =sumt

p(t)Bx(t)sa

t (31)


Q~xsa =sumt

p(t)sumb

V (ab|st)=1

Bbt (32)

14


β =sumstab


=sumsa



sumj v

(j)Γj as

Q~xsa =1

2

sumt

p(t)sumb

V (ab|st)=1

(I + (minus1)bBt

)

=1

2


] (35)

where

csa =sumt

p(t)|sumb

V (a b|s t)| (36)

~vsa =sumt

p(t)sumb

V (ab|st)=1

(minus1)b ~bt (37)




tr(ρQ~xsa


1

2(csa + ~vsa2) (38)

with

ρ~xsa =1

d

I +sumj

r(j)saΓj

(39)

and


15



ρ =1

d

I +sumj

r(j)saΓj +

(41)


from Eq 35 that

tr(ρQ~xsa) =1





p(a|s) ρ~xsa =suma

p(a|s) ρsa (43)




=1


=1

2

(1 + tr(ATs )

)=

1

2


1

2

(ρ~xs0 + ρ~xs1

)=

Id (45)



16


tr(σsaQ~xsa


p(a|s)(46)

=1






S = (0 s(2) s(n)) | s(j) isin 0 1 (48)


~xs0 = s (49)

~xs1 = s (50)


V (a b|s t) =

1 if b = ~x

(t)sa

0 otherwise (51)



17





P game(S T σAB) =1

2n

sumsa

tr(ρ~xsaQ~xsa

)(53)

=1

2n

sumsa

1

2(1 + ~vsa2) (54)

=1

2

(1 +

1

2n

sumsa


)(55)

le 1

2

1 +

radic1

2n

sumsa

sumttprime



(56)

=1

2+

1

2radicn (57)





tr

(ρ~xsaB

~x(t)sa

t

)= tr

(ρ~xsaB

~x(tprime)sa

tprime




18




tr(Bx(t)

t ρ) (59)


1

n

nsumt=1


n

sumt

log maxx(t)isin01

tr(Bx(t)

t ρ)

(60)


sumt

tr(Bx(t)

t ρ)

(61)

= minus log maxsa

λmax(Q~xsa) (62)



λmax(Q~xsa) =1

2+

1

2radicn (63)






19






ζ~xsa =1

2+

1

2radic






(|00〉AB + |11〉AB) (65)





20

070 075 080 085 090 095 100070

075

080

085

090

095

100

Steering

Unc

erta

inty



max = 12 + 1(2radic


21

0





ρ~xsa =

int Ωprime~xsa

Ω~xsa

p(Ω)σΩdΩ (66)


AB =int 2π

0 p(Ω)σΩAσ

ΩBdΩ



22








ρ00 = p(0|0) = 1 p(0|1) = 1 (67)

ρ01 = p(1|0) = 1 p(1|1) = 1 (68)

ρ10 = p(0|0) = 1 p(1|1) = 1 (69)

ρ11 = p(1|0) = 1 p(0|1) = 1 (70)


23





η~x =sumtprime



η~x le ζ~x (72)





24







~x(t)sat


~x(1minust)sa t




2(73)





p(right|t ~xsa) =1

2+

1

2radic

2 (74)





25

00 02 04 06 08 1000

02

04

06

08

10


pHw

rong

Egravetw

rong

L



(solid blue line)







26


06

07

08

09

10pHsecondL






U =

nsumt=1





σisinΣ

nsumt=1

p(t)p(bt|t)σ (76)






27




~xisinPs

p(~x|s) σ~x = ρ



suma





28

A XOR-games












Main result




p(s)suma



6



p(s)suma


ζ~xsa(T D) (13)



P gamemax = max

Ess

sums

p(s)suma




P gamemax =

sums

p(s)suma






7














8

References











quant-ph0409066 2004







9
















10



















11



nsumt=1


ζ~x (16)



A XOR-games


12








As = A0s minusA1

s (17)

Bt = B0t minusB1

t (18)


As =sumj

a(j)s Γj (19)

Bt =sumj

b(j)t Γj (20)



(1)t b






s +A1s = I and B0

t +B1t = I we


Aas =1

2(I + (minus1)aAs) (23)

Bbt =

1

2

(I + (minus1)bBt

) (24)

13


|ψ〉 =1radicd

dminus1sumk=0

|k〉|k〉 (25)




p(s t)sumab


=1

2

(sumst

p(s t)sumc


) (27)




P game(S T σAB) =1

2


4

)(29)

=1

4

sumst

sumab






Q~xsa =sumt

p(t)Bx(t)sa

t (31)


Q~xsa =sumt

p(t)sumb

V (ab|st)=1

Bbt (32)

14


β =sumstab


=sumsa



sumj v

(j)Γj as

Q~xsa =1

2

sumt

p(t)sumb

V (ab|st)=1

(I + (minus1)bBt

)

=1

2


] (35)

where

csa =sumt

p(t)|sumb

V (a b|s t)| (36)

~vsa =sumt

p(t)sumb

V (ab|st)=1

(minus1)b ~bt (37)




tr(ρQ~xsa


1

2(csa + ~vsa2) (38)

with

ρ~xsa =1

d

I +sumj

r(j)saΓj

(39)

and


15



ρ =1

d

I +sumj

r(j)saΓj +

(41)


from Eq 35 that

tr(ρQ~xsa) =1





p(a|s) ρ~xsa =suma

p(a|s) ρsa (43)




=1


=1

2

(1 + tr(ATs )

)=

1

2


1

2

(ρ~xs0 + ρ~xs1

)=

Id (45)



16


tr(σsaQ~xsa


p(a|s)(46)

=1






S = (0 s(2) s(n)) | s(j) isin 0 1 (48)


~xs0 = s (49)

~xs1 = s (50)


V (a b|s t) =

1 if b = ~x

(t)sa

0 otherwise (51)



17





P game(S T σAB) =1

2n

sumsa

tr(ρ~xsaQ~xsa

)(53)

=1

2n

sumsa

1

2(1 + ~vsa2) (54)

=1

2

(1 +

1

2n

sumsa


)(55)

le 1

2

1 +

radic1

2n

sumsa

sumttprime



(56)

=1

2+

1

2radicn (57)





tr

(ρ~xsaB

~x(t)sa

t

)= tr

(ρ~xsaB

~x(tprime)sa

tprime




18




tr(Bx(t)

t ρ) (59)


1

n

nsumt=1


n

sumt

log maxx(t)isin01

tr(Bx(t)

t ρ)

(60)


sumt

tr(Bx(t)

t ρ)

(61)

= minus log maxsa

λmax(Q~xsa) (62)



λmax(Q~xsa) =1

2+

1

2radicn (63)






19






ζ~xsa =1

2+

1

2radic






(|00〉AB + |11〉AB) (65)





20

070 075 080 085 090 095 100070

075

080

085

090

095

100

Steering

Unc

erta

inty



max = 12 + 1(2radic


21

0





ρ~xsa =

int Ωprime~xsa

Ω~xsa

p(Ω)σΩdΩ (66)


AB =int 2π

0 p(Ω)σΩAσ

ΩBdΩ



22








ρ00 = p(0|0) = 1 p(0|1) = 1 (67)

ρ01 = p(1|0) = 1 p(1|1) = 1 (68)

ρ10 = p(0|0) = 1 p(1|1) = 1 (69)

ρ11 = p(1|0) = 1 p(0|1) = 1 (70)


23





η~x =sumtprime



η~x le ζ~x (72)





24







~x(t)sat


~x(1minust)sa t




2(73)





p(right|t ~xsa) =1

2+

1

2radic

2 (74)





25

00 02 04 06 08 1000

02

04

06

08

10


pHw

rong

Egravetw

rong

L



(solid blue line)







26


06

07

08

09

10pHsecondL






U =

nsumt=1





σisinΣ

nsumt=1

p(t)p(bt|t)σ (76)






27




~xisinPs

p(~x|s) σ~x = ρ



suma





28

A XOR-games











p(s)suma


ζ~xsa(T D) (13)



P gamemax = max

Ess

sums

p(s)suma




P gamemax =

sums

p(s)suma






7














8

References











quant-ph0409066 2004







9
















10



















11



nsumt=1


ζ~x (16)



A XOR-games


12








As = A0s minusA1

s (17)

Bt = B0t minusB1

t (18)


As =sumj

a(j)s Γj (19)

Bt =sumj

b(j)t Γj (20)



(1)t b






s +A1s = I and B0

t +B1t = I we


Aas =1

2(I + (minus1)aAs) (23)

Bbt =

1

2

(I + (minus1)bBt

) (24)

13


|ψ〉 =1radicd

dminus1sumk=0

|k〉|k〉 (25)




p(s t)sumab


=1

2

(sumst

p(s t)sumc


) (27)




P game(S T σAB) =1

2


4

)(29)

=1

4

sumst

sumab






Q~xsa =sumt

p(t)Bx(t)sa

t (31)


Q~xsa =sumt

p(t)sumb

V (ab|st)=1

Bbt (32)

14


β =sumstab


=sumsa



sumj v

(j)Γj as

Q~xsa =1

2

sumt

p(t)sumb

V (ab|st)=1

(I + (minus1)bBt

)

=1

2


] (35)

where

csa =sumt

p(t)|sumb

V (a b|s t)| (36)

~vsa =sumt

p(t)sumb

V (ab|st)=1

(minus1)b ~bt (37)




tr(ρQ~xsa


1

2(csa + ~vsa2) (38)

with

ρ~xsa =1

d

I +sumj

r(j)saΓj

(39)

and


15



ρ =1

d

I +sumj

r(j)saΓj +

(41)


from Eq 35 that

tr(ρQ~xsa) =1





p(a|s) ρ~xsa =suma

p(a|s) ρsa (43)




=1


=1

2

(1 + tr(ATs )

)=

1

2


1

2

(ρ~xs0 + ρ~xs1

)=

Id (45)



16


tr(σsaQ~xsa


p(a|s)(46)

=1






S = (0 s(2) s(n)) | s(j) isin 0 1 (48)


~xs0 = s (49)

~xs1 = s (50)


V (a b|s t) =

1 if b = ~x

(t)sa

0 otherwise (51)



17





P game(S T σAB) =1

2n

sumsa

tr(ρ~xsaQ~xsa

)(53)

=1

2n

sumsa

1

2(1 + ~vsa2) (54)

=1

2

(1 +

1

2n

sumsa


)(55)

le 1

2

1 +

radic1

2n

sumsa

sumttprime



(56)

=1

2+

1

2radicn (57)





tr

(ρ~xsaB

~x(t)sa

t

)= tr

(ρ~xsaB

~x(tprime)sa

tprime




18




tr(Bx(t)

t ρ) (59)


1

n

nsumt=1


n

sumt

log maxx(t)isin01

tr(Bx(t)

t ρ)

(60)


sumt

tr(Bx(t)

t ρ)

(61)

= minus log maxsa

λmax(Q~xsa) (62)



λmax(Q~xsa) =1

2+

1

2radicn (63)






19






ζ~xsa =1

2+

1

2radic






(|00〉AB + |11〉AB) (65)





20

070 075 080 085 090 095 100070

075

080

085

090

095

100

Steering

Unc

erta

inty



max = 12 + 1(2radic


21

0





ρ~xsa =

int Ωprime~xsa

Ω~xsa

p(Ω)σΩdΩ (66)


AB =int 2π

0 p(Ω)σΩAσ

ΩBdΩ



22








ρ00 = p(0|0) = 1 p(0|1) = 1 (67)

ρ01 = p(1|0) = 1 p(1|1) = 1 (68)

ρ10 = p(0|0) = 1 p(1|1) = 1 (69)

ρ11 = p(1|0) = 1 p(0|1) = 1 (70)


23





η~x =sumtprime



η~x le ζ~x (72)





24







~x(t)sat


~x(1minust)sa t




2(73)





p(right|t ~xsa) =1

2+

1

2radic

2 (74)





25

00 02 04 06 08 1000

02

04

06

08

10


pHw

rong

Egravetw

rong

L



(solid blue line)







26


06

07

08

09

10pHsecondL






U =

nsumt=1





σisinΣ

nsumt=1

p(t)p(bt|t)σ (76)






27




~xisinPs

p(~x|s) σ~x = ρ



suma





28

A XOR-games






















8

References











quant-ph0409066 2004







9
















10



















11



nsumt=1


ζ~x (16)



A XOR-games


12








As = A0s minusA1

s (17)

Bt = B0t minusB1

t (18)


As =sumj

a(j)s Γj (19)

Bt =sumj

b(j)t Γj (20)



(1)t b






s +A1s = I and B0

t +B1t = I we


Aas =1

2(I + (minus1)aAs) (23)

Bbt =

1

2

(I + (minus1)bBt

) (24)

13


|ψ〉 =1radicd

dminus1sumk=0

|k〉|k〉 (25)




p(s t)sumab


=1

2

(sumst

p(s t)sumc


) (27)




P game(S T σAB) =1

2


4

)(29)

=1

4

sumst

sumab






Q~xsa =sumt

p(t)Bx(t)sa

t (31)


Q~xsa =sumt

p(t)sumb

V (ab|st)=1

Bbt (32)

14


β =sumstab


=sumsa



sumj v

(j)Γj as

Q~xsa =1

2

sumt

p(t)sumb

V (ab|st)=1

(I + (minus1)bBt

)

=1

2


] (35)

where

csa =sumt

p(t)|sumb

V (a b|s t)| (36)

~vsa =sumt

p(t)sumb

V (ab|st)=1

(minus1)b ~bt (37)




tr(ρQ~xsa


1

2(csa + ~vsa2) (38)

with

ρ~xsa =1

d

I +sumj

r(j)saΓj

(39)

and


15



ρ =1

d

I +sumj

r(j)saΓj +

(41)


from Eq 35 that

tr(ρQ~xsa) =1





p(a|s) ρ~xsa =suma

p(a|s) ρsa (43)




=1


=1

2

(1 + tr(ATs )

)=

1

2


1

2

(ρ~xs0 + ρ~xs1

)=

Id (45)



16


tr(σsaQ~xsa


p(a|s)(46)

=1






S = (0 s(2) s(n)) | s(j) isin 0 1 (48)


~xs0 = s (49)

~xs1 = s (50)


V (a b|s t) =

1 if b = ~x

(t)sa

0 otherwise (51)



17





P game(S T σAB) =1

2n

sumsa

tr(ρ~xsaQ~xsa

)(53)

=1

2n

sumsa

1

2(1 + ~vsa2) (54)

=1

2

(1 +

1

2n

sumsa


)(55)

le 1

2

1 +

radic1

2n

sumsa

sumttprime



(56)

=1

2+

1

2radicn (57)





tr

(ρ~xsaB

~x(t)sa

t

)= tr

(ρ~xsaB

~x(tprime)sa

tprime




18




tr(Bx(t)

t ρ) (59)


1

n

nsumt=1


n

sumt

log maxx(t)isin01

tr(Bx(t)

t ρ)

(60)


sumt

tr(Bx(t)

t ρ)

(61)

= minus log maxsa

λmax(Q~xsa) (62)



λmax(Q~xsa) =1

2+

1

2radicn (63)






19






ζ~xsa =1

2+

1

2radic






(|00〉AB + |11〉AB) (65)





20

070 075 080 085 090 095 100070

075

080

085

090

095

100

Steering

Unc

erta

inty



max = 12 + 1(2radic


21

0





ρ~xsa =

int Ωprime~xsa

Ω~xsa

p(Ω)σΩdΩ (66)


AB =int 2π

0 p(Ω)σΩAσ

ΩBdΩ



22








ρ00 = p(0|0) = 1 p(0|1) = 1 (67)

ρ01 = p(1|0) = 1 p(1|1) = 1 (68)

ρ10 = p(0|0) = 1 p(1|1) = 1 (69)

ρ11 = p(1|0) = 1 p(0|1) = 1 (70)


23





η~x =sumtprime



η~x le ζ~x (72)





24







~x(t)sat


~x(1minust)sa t




2(73)





p(right|t ~xsa) =1

2+

1

2radic

2 (74)





25

00 02 04 06 08 1000

02

04

06

08

10


pHw

rong

Egravetw

rong

L



(solid blue line)







26


06

07

08

09

10pHsecondL






U =

nsumt=1





σisinΣ

nsumt=1

p(t)p(bt|t)σ (76)






27




~xisinPs

p(~x|s) σ~x = ρ



suma





28

A XOR-games









References











quant-ph0409066 2004







9
















10



















11



nsumt=1


ζ~x (16)



A XOR-games


12








As = A0s minusA1

s (17)

Bt = B0t minusB1

t (18)


As =sumj

a(j)s Γj (19)

Bt =sumj

b(j)t Γj (20)



(1)t b






s +A1s = I and B0

t +B1t = I we


Aas =1

2(I + (minus1)aAs) (23)

Bbt =

1

2

(I + (minus1)bBt

) (24)

13


|ψ〉 =1radicd

dminus1sumk=0

|k〉|k〉 (25)




p(s t)sumab


=1

2

(sumst

p(s t)sumc


) (27)




P game(S T σAB) =1

2


4

)(29)

=1

4

sumst

sumab






Q~xsa =sumt

p(t)Bx(t)sa

t (31)


Q~xsa =sumt

p(t)sumb

V (ab|st)=1

Bbt (32)

14


β =sumstab


=sumsa



sumj v

(j)Γj as

Q~xsa =1

2

sumt

p(t)sumb

V (ab|st)=1

(I + (minus1)bBt

)

=1

2


] (35)

where

csa =sumt

p(t)|sumb

V (a b|s t)| (36)

~vsa =sumt

p(t)sumb

V (ab|st)=1

(minus1)b ~bt (37)




tr(ρQ~xsa


1

2(csa + ~vsa2) (38)

with

ρ~xsa =1

d

I +sumj

r(j)saΓj

(39)

and


15



ρ =1

d

I +sumj

r(j)saΓj +

(41)


from Eq 35 that

tr(ρQ~xsa) =1





p(a|s) ρ~xsa =suma

p(a|s) ρsa (43)




=1


=1

2

(1 + tr(ATs )

)=

1

2


1

2

(ρ~xs0 + ρ~xs1

)=

Id (45)



16


tr(σsaQ~xsa


p(a|s)(46)

=1






S = (0 s(2) s(n)) | s(j) isin 0 1 (48)


~xs0 = s (49)

~xs1 = s (50)


V (a b|s t) =

1 if b = ~x

(t)sa

0 otherwise (51)



17





P game(S T σAB) =1

2n

sumsa

tr(ρ~xsaQ~xsa

)(53)

=1

2n

sumsa

1

2(1 + ~vsa2) (54)

=1

2

(1 +

1

2n

sumsa


)(55)

le 1

2

1 +

radic1

2n

sumsa

sumttprime



(56)

=1

2+

1

2radicn (57)





tr

(ρ~xsaB

~x(t)sa

t

)= tr

(ρ~xsaB

~x(tprime)sa

tprime




18




tr(Bx(t)

t ρ) (59)


1

n

nsumt=1


n

sumt

log maxx(t)isin01

tr(Bx(t)

t ρ)

(60)


sumt

tr(Bx(t)

t ρ)

(61)

= minus log maxsa

λmax(Q~xsa) (62)



λmax(Q~xsa) =1

2+

1

2radicn (63)






19






ζ~xsa =1

2+

1

2radic






(|00〉AB + |11〉AB) (65)





20

070 075 080 085 090 095 100070

075

080

085

090

095

100

Steering

Unc

erta

inty



max = 12 + 1(2radic


21

0





ρ~xsa =

int Ωprime~xsa

Ω~xsa

p(Ω)σΩdΩ (66)


AB =int 2π

0 p(Ω)σΩAσ

ΩBdΩ



22








ρ00 = p(0|0) = 1 p(0|1) = 1 (67)

ρ01 = p(1|0) = 1 p(1|1) = 1 (68)

ρ10 = p(0|0) = 1 p(1|1) = 1 (69)

ρ11 = p(1|0) = 1 p(0|1) = 1 (70)


23





η~x =sumtprime



η~x le ζ~x (72)





24







~x(t)sat


~x(1minust)sa t




2(73)





p(right|t ~xsa) =1

2+

1

2radic

2 (74)





25

00 02 04 06 08 1000

02

04

06

08

10


pHw

rong

Egravetw

rong

L



(solid blue line)







26


06

07

08

09

10pHsecondL






U =

nsumt=1





σisinΣ

nsumt=1

p(t)p(bt|t)σ (76)






27




~xisinPs

p(~x|s) σ~x = ρ



suma





28

A XOR-games
























10



















11



nsumt=1


ζ~x (16)



A XOR-games


12








As = A0s minusA1

s (17)

Bt = B0t minusB1

t (18)


As =sumj

a(j)s Γj (19)

Bt =sumj

b(j)t Γj (20)



(1)t b






s +A1s = I and B0

t +B1t = I we


Aas =1

2(I + (minus1)aAs) (23)

Bbt =

1

2

(I + (minus1)bBt

) (24)

13


|ψ〉 =1radicd

dminus1sumk=0

|k〉|k〉 (25)




p(s t)sumab


=1

2

(sumst

p(s t)sumc


) (27)




P game(S T σAB) =1

2


4

)(29)

=1

4

sumst

sumab






Q~xsa =sumt

p(t)Bx(t)sa

t (31)


Q~xsa =sumt

p(t)sumb

V (ab|st)=1

Bbt (32)

14


β =sumstab


=sumsa



sumj v

(j)Γj as

Q~xsa =1

2

sumt

p(t)sumb

V (ab|st)=1

(I + (minus1)bBt

)

=1

2


] (35)

where

csa =sumt

p(t)|sumb

V (a b|s t)| (36)

~vsa =sumt

p(t)sumb

V (ab|st)=1

(minus1)b ~bt (37)




tr(ρQ~xsa


1

2(csa + ~vsa2) (38)

with

ρ~xsa =1

d

I +sumj

r(j)saΓj

(39)

and


15



ρ =1

d

I +sumj

r(j)saΓj +

(41)


from Eq 35 that

tr(ρQ~xsa) =1





p(a|s) ρ~xsa =suma

p(a|s) ρsa (43)




=1


=1

2

(1 + tr(ATs )

)=

1

2


1

2

(ρ~xs0 + ρ~xs1

)=

Id (45)



16


tr(σsaQ~xsa


p(a|s)(46)

=1






S = (0 s(2) s(n)) | s(j) isin 0 1 (48)


~xs0 = s (49)

~xs1 = s (50)


V (a b|s t) =

1 if b = ~x

(t)sa

0 otherwise (51)



17





P game(S T σAB) =1

2n

sumsa

tr(ρ~xsaQ~xsa

)(53)

=1

2n

sumsa

1

2(1 + ~vsa2) (54)

=1

2

(1 +

1

2n

sumsa


)(55)

le 1

2

1 +

radic1

2n

sumsa

sumttprime



(56)

=1

2+

1

2radicn (57)





tr

(ρ~xsaB

~x(t)sa

t

)= tr

(ρ~xsaB

~x(tprime)sa

tprime




18




tr(Bx(t)

t ρ) (59)


1

n

nsumt=1


n

sumt

log maxx(t)isin01

tr(Bx(t)

t ρ)

(60)


sumt

tr(Bx(t)

t ρ)

(61)

= minus log maxsa

λmax(Q~xsa) (62)



λmax(Q~xsa) =1

2+

1

2radicn (63)






19






ζ~xsa =1

2+

1

2radic






(|00〉AB + |11〉AB) (65)





20

070 075 080 085 090 095 100070

075

080

085

090

095

100

Steering

Unc

erta

inty



max = 12 + 1(2radic


21

0





ρ~xsa =

int Ωprime~xsa

Ω~xsa

p(Ω)σΩdΩ (66)


AB =int 2π

0 p(Ω)σΩAσ

ΩBdΩ



22








ρ00 = p(0|0) = 1 p(0|1) = 1 (67)

ρ01 = p(1|0) = 1 p(1|1) = 1 (68)

ρ10 = p(0|0) = 1 p(1|1) = 1 (69)

ρ11 = p(1|0) = 1 p(0|1) = 1 (70)


23





η~x =sumtprime



η~x le ζ~x (72)





24







~x(t)sat


~x(1minust)sa t




2(73)





p(right|t ~xsa) =1

2+

1

2radic

2 (74)





25

00 02 04 06 08 1000

02

04

06

08

10


pHw

rong

Egravetw

rong

L



(solid blue line)







26


06

07

08

09

10pHsecondL






U =

nsumt=1





σisinΣ

nsumt=1

p(t)p(bt|t)σ (76)






27




~xisinPs

p(~x|s) σ~x = ρ



suma





28

A XOR-games



























11



nsumt=1


ζ~x (16)



A XOR-games


12








As = A0s minusA1

s (17)

Bt = B0t minusB1

t (18)


As =sumj

a(j)s Γj (19)

Bt =sumj

b(j)t Γj (20)



(1)t b






s +A1s = I and B0

t +B1t = I we


Aas =1

2(I + (minus1)aAs) (23)

Bbt =

1

2

(I + (minus1)bBt

) (24)

13


|ψ〉 =1radicd

dminus1sumk=0

|k〉|k〉 (25)




p(s t)sumab


=1

2

(sumst

p(s t)sumc


) (27)




P game(S T σAB) =1

2


4

)(29)

=1

4

sumst

sumab






Q~xsa =sumt

p(t)Bx(t)sa

t (31)


Q~xsa =sumt

p(t)sumb

V (ab|st)=1

Bbt (32)

14


β =sumstab


=sumsa



sumj v

(j)Γj as

Q~xsa =1

2

sumt

p(t)sumb

V (ab|st)=1

(I + (minus1)bBt

)

=1

2


] (35)

where

csa =sumt

p(t)|sumb

V (a b|s t)| (36)

~vsa =sumt

p(t)sumb

V (ab|st)=1

(minus1)b ~bt (37)




tr(ρQ~xsa


1

2(csa + ~vsa2) (38)

with

ρ~xsa =1

d

I +sumj

r(j)saΓj

(39)

and


15



ρ =1

d

I +sumj

r(j)saΓj +

(41)


from Eq 35 that

tr(ρQ~xsa) =1





p(a|s) ρ~xsa =suma

p(a|s) ρsa (43)




=1


=1

2

(1 + tr(ATs )

)=

1

2


1

2

(ρ~xs0 + ρ~xs1

)=

Id (45)



16


tr(σsaQ~xsa


p(a|s)(46)

=1






S = (0 s(2) s(n)) | s(j) isin 0 1 (48)


~xs0 = s (49)

~xs1 = s (50)


V (a b|s t) =

1 if b = ~x

(t)sa

0 otherwise (51)



17





P game(S T σAB) =1

2n

sumsa

tr(ρ~xsaQ~xsa

)(53)

=1

2n

sumsa

1

2(1 + ~vsa2) (54)

=1

2

(1 +

1

2n

sumsa


)(55)

le 1

2

1 +

radic1

2n

sumsa

sumttprime



(56)

=1

2+

1

2radicn (57)





tr

(ρ~xsaB

~x(t)sa

t

)= tr

(ρ~xsaB

~x(tprime)sa

tprime




18




tr(Bx(t)

t ρ) (59)


1

n

nsumt=1


n

sumt

log maxx(t)isin01

tr(Bx(t)

t ρ)

(60)


sumt

tr(Bx(t)

t ρ)

(61)

= minus log maxsa

λmax(Q~xsa) (62)



λmax(Q~xsa) =1

2+

1

2radicn (63)






19






ζ~xsa =1

2+

1

2radic






(|00〉AB + |11〉AB) (65)





20

070 075 080 085 090 095 100070

075

080

085

090

095

100

Steering

Unc

erta

inty



max = 12 + 1(2radic


21

0





ρ~xsa =

int Ωprime~xsa

Ω~xsa

p(Ω)σΩdΩ (66)


AB =int 2π

0 p(Ω)σΩAσ

ΩBdΩ



22








ρ00 = p(0|0) = 1 p(0|1) = 1 (67)

ρ01 = p(1|0) = 1 p(1|1) = 1 (68)

ρ10 = p(0|0) = 1 p(1|1) = 1 (69)

ρ11 = p(1|0) = 1 p(0|1) = 1 (70)


23





η~x =sumtprime



η~x le ζ~x (72)





24







~x(t)sat


~x(1minust)sa t




2(73)





p(right|t ~xsa) =1

2+

1

2radic

2 (74)





25

00 02 04 06 08 1000

02

04

06

08

10


pHw

rong

Egravetw

rong

L



(solid blue line)







26


06

07

08

09

10pHsecondL






U =

nsumt=1





σisinΣ

nsumt=1

p(t)p(bt|t)σ (76)






27




~xisinPs

p(~x|s) σ~x = ρ



suma





28

A XOR-games











nsumt=1


ζ~x (16)



A XOR-games


12








As = A0s minusA1

s (17)

Bt = B0t minusB1

t (18)


As =sumj

a(j)s Γj (19)

Bt =sumj

b(j)t Γj (20)



(1)t b






s +A1s = I and B0

t +B1t = I we


Aas =1

2(I + (minus1)aAs) (23)

Bbt =

1

2

(I + (minus1)bBt

) (24)

13


|ψ〉 =1radicd

dminus1sumk=0

|k〉|k〉 (25)




p(s t)sumab


=1

2

(sumst

p(s t)sumc


) (27)




P game(S T σAB) =1

2


4

)(29)

=1

4

sumst

sumab






Q~xsa =sumt

p(t)Bx(t)sa

t (31)


Q~xsa =sumt

p(t)sumb

V (ab|st)=1

Bbt (32)

14


β =sumstab


=sumsa



sumj v

(j)Γj as

Q~xsa =1

2

sumt

p(t)sumb

V (ab|st)=1

(I + (minus1)bBt

)

=1

2


] (35)

where

csa =sumt

p(t)|sumb

V (a b|s t)| (36)

~vsa =sumt

p(t)sumb

V (ab|st)=1

(minus1)b ~bt (37)




tr(ρQ~xsa


1

2(csa + ~vsa2) (38)

with

ρ~xsa =1

d

I +sumj

r(j)saΓj

(39)

and


15



ρ =1

d

I +sumj

r(j)saΓj +

(41)


from Eq 35 that

tr(ρQ~xsa) =1





p(a|s) ρ~xsa =suma

p(a|s) ρsa (43)




=1


=1

2

(1 + tr(ATs )

)=

1

2


1

2

(ρ~xs0 + ρ~xs1

)=

Id (45)



16


tr(σsaQ~xsa


p(a|s)(46)

=1






S = (0 s(2) s(n)) | s(j) isin 0 1 (48)


~xs0 = s (49)

~xs1 = s (50)


V (a b|s t) =

1 if b = ~x

(t)sa

0 otherwise (51)



17





P game(S T σAB) =1

2n

sumsa

tr(ρ~xsaQ~xsa

)(53)

=1

2n

sumsa

1

2(1 + ~vsa2) (54)

=1

2

(1 +

1

2n

sumsa


)(55)

le 1

2

1 +

radic1

2n

sumsa

sumttprime



(56)

=1

2+

1

2radicn (57)





tr

(ρ~xsaB

~x(t)sa

t

)= tr

(ρ~xsaB

~x(tprime)sa

tprime




18




tr(Bx(t)

t ρ) (59)


1

n

nsumt=1


n

sumt

log maxx(t)isin01

tr(Bx(t)

t ρ)

(60)


sumt

tr(Bx(t)

t ρ)

(61)

= minus log maxsa

λmax(Q~xsa) (62)



λmax(Q~xsa) =1

2+

1

2radicn (63)






19






ζ~xsa =1

2+

1

2radic






(|00〉AB + |11〉AB) (65)





20

070 075 080 085 090 095 100070

075

080

085

090

095

100

Steering

Unc

erta

inty



max = 12 + 1(2radic


21

0





ρ~xsa =

int Ωprime~xsa

Ω~xsa

p(Ω)σΩdΩ (66)


AB =int 2π

0 p(Ω)σΩAσ

ΩBdΩ



22








ρ00 = p(0|0) = 1 p(0|1) = 1 (67)

ρ01 = p(1|0) = 1 p(1|1) = 1 (68)

ρ10 = p(0|0) = 1 p(1|1) = 1 (69)

ρ11 = p(1|0) = 1 p(0|1) = 1 (70)


23





η~x =sumtprime



η~x le ζ~x (72)





24







~x(t)sat


~x(1minust)sa t




2(73)





p(right|t ~xsa) =1

2+

1

2radic

2 (74)





25

00 02 04 06 08 1000

02

04

06

08

10


pHw

rong

Egravetw

rong

L



(solid blue line)







26


06

07

08

09

10pHsecondL






U =

nsumt=1





σisinΣ

nsumt=1

p(t)p(bt|t)σ (76)






27




~xisinPs

p(~x|s) σ~x = ρ



suma





28

A XOR-games
















As = A0s minusA1

s (17)

Bt = B0t minusB1

t (18)


As =sumj

a(j)s Γj (19)

Bt =sumj

b(j)t Γj (20)



(1)t b






s +A1s = I and B0

t +B1t = I we


Aas =1

2(I + (minus1)aAs) (23)

Bbt =

1

2

(I + (minus1)bBt

) (24)

13


|ψ〉 =1radicd

dminus1sumk=0

|k〉|k〉 (25)




p(s t)sumab


=1

2

(sumst

p(s t)sumc


) (27)




P game(S T σAB) =1

2


4

)(29)

=1

4

sumst

sumab






Q~xsa =sumt

p(t)Bx(t)sa

t (31)


Q~xsa =sumt

p(t)sumb

V (ab|st)=1

Bbt (32)

14


β =sumstab


=sumsa



sumj v

(j)Γj as

Q~xsa =1

2

sumt

p(t)sumb

V (ab|st)=1

(I + (minus1)bBt

)

=1

2


] (35)

where

csa =sumt

p(t)|sumb

V (a b|s t)| (36)

~vsa =sumt

p(t)sumb

V (ab|st)=1

(minus1)b ~bt (37)




tr(ρQ~xsa


1

2(csa + ~vsa2) (38)

with

ρ~xsa =1

d

I +sumj

r(j)saΓj

(39)

and


15



ρ =1

d

I +sumj

r(j)saΓj +

(41)


from Eq 35 that

tr(ρQ~xsa) =1





p(a|s) ρ~xsa =suma

p(a|s) ρsa (43)




=1


=1

2

(1 + tr(ATs )

)=

1

2


1

2

(ρ~xs0 + ρ~xs1

)=

Id (45)



16


tr(σsaQ~xsa


p(a|s)(46)

=1






S = (0 s(2) s(n)) | s(j) isin 0 1 (48)


~xs0 = s (49)

~xs1 = s (50)


V (a b|s t) =

1 if b = ~x

(t)sa

0 otherwise (51)



17





P game(S T σAB) =1

2n

sumsa

tr(ρ~xsaQ~xsa

)(53)

=1

2n

sumsa

1

2(1 + ~vsa2) (54)

=1

2

(1 +

1

2n

sumsa


)(55)

le 1

2

1 +

radic1

2n

sumsa

sumttprime



(56)

=1

2+

1

2radicn (57)





tr

(ρ~xsaB

~x(t)sa

t

)= tr

(ρ~xsaB

~x(tprime)sa

tprime




18




tr(Bx(t)

t ρ) (59)


1

n

nsumt=1


n

sumt

log maxx(t)isin01

tr(Bx(t)

t ρ)

(60)


sumt

tr(Bx(t)

t ρ)

(61)

= minus log maxsa

λmax(Q~xsa) (62)



λmax(Q~xsa) =1

2+

1

2radicn (63)






19






ζ~xsa =1

2+

1

2radic






(|00〉AB + |11〉AB) (65)





20

070 075 080 085 090 095 100070

075

080

085

090

095

100

Steering

Unc

erta

inty



max = 12 + 1(2radic


21

0





ρ~xsa =

int Ωprime~xsa

Ω~xsa

p(Ω)σΩdΩ (66)


AB =int 2π

0 p(Ω)σΩAσ

ΩBdΩ



22








ρ00 = p(0|0) = 1 p(0|1) = 1 (67)

ρ01 = p(1|0) = 1 p(1|1) = 1 (68)

ρ10 = p(0|0) = 1 p(1|1) = 1 (69)

ρ11 = p(1|0) = 1 p(0|1) = 1 (70)


23





η~x =sumtprime



η~x le ζ~x (72)





24







~x(t)sat


~x(1minust)sa t




2(73)





p(right|t ~xsa) =1

2+

1

2radic

2 (74)





25

00 02 04 06 08 1000

02

04

06

08

10


pHw

rong

Egravetw

rong

L



(solid blue line)







26


06

07

08

09

10pHsecondL






U =

nsumt=1





σisinΣ

nsumt=1

p(t)p(bt|t)σ (76)






27




~xisinPs

p(~x|s) σ~x = ρ



suma





28

A XOR-games










|ψ〉 =1radicd

dminus1sumk=0

|k〉|k〉 (25)




p(s t)sumab


=1

2

(sumst

p(s t)sumc


) (27)




P game(S T σAB) =1

2


4

)(29)

=1

4

sumst

sumab






Q~xsa =sumt

p(t)Bx(t)sa

t (31)


Q~xsa =sumt

p(t)sumb

V (ab|st)=1

Bbt (32)

14


β =sumstab


=sumsa



sumj v

(j)Γj as

Q~xsa =1

2

sumt

p(t)sumb

V (ab|st)=1

(I + (minus1)bBt

)

=1

2


] (35)

where

csa =sumt

p(t)|sumb

V (a b|s t)| (36)

~vsa =sumt

p(t)sumb

V (ab|st)=1

(minus1)b ~bt (37)




tr(ρQ~xsa


1

2(csa + ~vsa2) (38)

with

ρ~xsa =1

d

I +sumj

r(j)saΓj

(39)

and


15



ρ =1

d

I +sumj

r(j)saΓj +

(41)


from Eq 35 that

tr(ρQ~xsa) =1





p(a|s) ρ~xsa =suma

p(a|s) ρsa (43)




=1


=1

2

(1 + tr(ATs )

)=

1

2


1

2

(ρ~xs0 + ρ~xs1

)=

Id (45)



16


tr(σsaQ~xsa


p(a|s)(46)

=1






S = (0 s(2) s(n)) | s(j) isin 0 1 (48)


~xs0 = s (49)

~xs1 = s (50)


V (a b|s t) =

1 if b = ~x

(t)sa

0 otherwise (51)



17





P game(S T σAB) =1

2n

sumsa

tr(ρ~xsaQ~xsa

)(53)

=1

2n

sumsa

1

2(1 + ~vsa2) (54)

=1

2

(1 +

1

2n

sumsa


)(55)

le 1

2

1 +

radic1

2n

sumsa

sumttprime



(56)

=1

2+

1

2radicn (57)





tr

(ρ~xsaB

~x(t)sa

t

)= tr

(ρ~xsaB

~x(tprime)sa

tprime




18




tr(Bx(t)

t ρ) (59)


1

n

nsumt=1


n

sumt

log maxx(t)isin01

tr(Bx(t)

t ρ)

(60)


sumt

tr(Bx(t)

t ρ)

(61)

= minus log maxsa

λmax(Q~xsa) (62)



λmax(Q~xsa) =1

2+

1

2radicn (63)






19






ζ~xsa =1

2+

1

2radic






(|00〉AB + |11〉AB) (65)





20

070 075 080 085 090 095 100070

075

080

085

090

095

100

Steering

Unc

erta

inty



max = 12 + 1(2radic


21

0





ρ~xsa =

int Ωprime~xsa

Ω~xsa

p(Ω)σΩdΩ (66)


AB =int 2π

0 p(Ω)σΩAσ

ΩBdΩ



22








ρ00 = p(0|0) = 1 p(0|1) = 1 (67)

ρ01 = p(1|0) = 1 p(1|1) = 1 (68)

ρ10 = p(0|0) = 1 p(1|1) = 1 (69)

ρ11 = p(1|0) = 1 p(0|1) = 1 (70)


23





η~x =sumtprime



η~x le ζ~x (72)





24







~x(t)sat


~x(1minust)sa t




2(73)





p(right|t ~xsa) =1

2+

1

2radic

2 (74)





25

00 02 04 06 08 1000

02

04

06

08

10


pHw

rong

Egravetw

rong

L



(solid blue line)







26


06

07

08

09

10pHsecondL






U =

nsumt=1





σisinΣ

nsumt=1

p(t)p(bt|t)σ (76)






27




~xisinPs

p(~x|s) σ~x = ρ



suma





28

A XOR-games










β =sumstab


=sumsa



sumj v

(j)Γj as

Q~xsa =1

2

sumt

p(t)sumb

V (ab|st)=1

(I + (minus1)bBt

)

=1

2


] (35)

where

csa =sumt

p(t)|sumb

V (a b|s t)| (36)

~vsa =sumt

p(t)sumb

V (ab|st)=1

(minus1)b ~bt (37)




tr(ρQ~xsa


1

2(csa + ~vsa2) (38)

with

ρ~xsa =1

d

I +sumj

r(j)saΓj

(39)

and


15



ρ =1

d

I +sumj

r(j)saΓj +

(41)


from Eq 35 that

tr(ρQ~xsa) =1





p(a|s) ρ~xsa =suma

p(a|s) ρsa (43)




=1


=1

2

(1 + tr(ATs )

)=

1

2


1

2

(ρ~xs0 + ρ~xs1

)=

Id (45)



16


tr(σsaQ~xsa


p(a|s)(46)

=1






S = (0 s(2) s(n)) | s(j) isin 0 1 (48)


~xs0 = s (49)

~xs1 = s (50)


V (a b|s t) =

1 if b = ~x

(t)sa

0 otherwise (51)



17





P game(S T σAB) =1

2n

sumsa

tr(ρ~xsaQ~xsa

)(53)

=1

2n

sumsa

1

2(1 + ~vsa2) (54)

=1

2

(1 +

1

2n

sumsa


)(55)

le 1

2

1 +

radic1

2n

sumsa

sumttprime



(56)

=1

2+

1

2radicn (57)





tr

(ρ~xsaB

~x(t)sa

t

)= tr

(ρ~xsaB

~x(tprime)sa

tprime




18




tr(Bx(t)

t ρ) (59)


1

n

nsumt=1


n

sumt

log maxx(t)isin01

tr(Bx(t)

t ρ)

(60)


sumt

tr(Bx(t)

t ρ)

(61)

= minus log maxsa

λmax(Q~xsa) (62)



λmax(Q~xsa) =1

2+

1

2radicn (63)






19






ζ~xsa =1

2+

1

2radic






(|00〉AB + |11〉AB) (65)





20

070 075 080 085 090 095 100070

075

080

085

090

095

100

Steering

Unc

erta

inty



max = 12 + 1(2radic


21

0





ρ~xsa =

int Ωprime~xsa

Ω~xsa

p(Ω)σΩdΩ (66)


AB =int 2π

0 p(Ω)σΩAσ

ΩBdΩ



22








ρ00 = p(0|0) = 1 p(0|1) = 1 (67)

ρ01 = p(1|0) = 1 p(1|1) = 1 (68)

ρ10 = p(0|0) = 1 p(1|1) = 1 (69)

ρ11 = p(1|0) = 1 p(0|1) = 1 (70)


23





η~x =sumtprime



η~x le ζ~x (72)





24







~x(t)sat


~x(1minust)sa t




2(73)





p(right|t ~xsa) =1

2+

1

2radic

2 (74)





25

00 02 04 06 08 1000

02

04

06

08

10


pHw

rong

Egravetw

rong

L



(solid blue line)







26


06

07

08

09

10pHsecondL






U =

nsumt=1





σisinΣ

nsumt=1

p(t)p(bt|t)σ (76)






27




~xisinPs

p(~x|s) σ~x = ρ



suma





28

A XOR-games











ρ =1

d

I +sumj

r(j)saΓj +

(41)


from Eq 35 that

tr(ρQ~xsa) =1





p(a|s) ρ~xsa =suma

p(a|s) ρsa (43)




=1


=1

2

(1 + tr(ATs )

)=

1

2


1

2

(ρ~xs0 + ρ~xs1

)=

Id (45)



16


tr(σsaQ~xsa


p(a|s)(46)

=1






S = (0 s(2) s(n)) | s(j) isin 0 1 (48)


~xs0 = s (49)

~xs1 = s (50)


V (a b|s t) =

1 if b = ~x

(t)sa

0 otherwise (51)



17





P game(S T σAB) =1

2n

sumsa

tr(ρ~xsaQ~xsa

)(53)

=1

2n

sumsa

1

2(1 + ~vsa2) (54)

=1

2

(1 +

1

2n

sumsa


)(55)

le 1

2

1 +

radic1

2n

sumsa

sumttprime



(56)

=1

2+

1

2radicn (57)





tr

(ρ~xsaB

~x(t)sa

t

)= tr

(ρ~xsaB

~x(tprime)sa

tprime




18




tr(Bx(t)

t ρ) (59)


1

n

nsumt=1


n

sumt

log maxx(t)isin01

tr(Bx(t)

t ρ)

(60)


sumt

tr(Bx(t)

t ρ)

(61)

= minus log maxsa

λmax(Q~xsa) (62)



λmax(Q~xsa) =1

2+

1

2radicn (63)






19






ζ~xsa =1

2+

1

2radic






(|00〉AB + |11〉AB) (65)





20

070 075 080 085 090 095 100070

075

080

085

090

095

100

Steering

Unc

erta

inty



max = 12 + 1(2radic


21

0





ρ~xsa =

int Ωprime~xsa

Ω~xsa

p(Ω)σΩdΩ (66)


AB =int 2π

0 p(Ω)σΩAσ

ΩBdΩ



22








ρ00 = p(0|0) = 1 p(0|1) = 1 (67)

ρ01 = p(1|0) = 1 p(1|1) = 1 (68)

ρ10 = p(0|0) = 1 p(1|1) = 1 (69)

ρ11 = p(1|0) = 1 p(0|1) = 1 (70)


23





η~x =sumtprime



η~x le ζ~x (72)





24







~x(t)sat


~x(1minust)sa t




2(73)





p(right|t ~xsa) =1

2+

1

2radic

2 (74)





25

00 02 04 06 08 1000

02

04

06

08

10


pHw

rong

Egravetw

rong

L



(solid blue line)







26


06

07

08

09

10pHsecondL






U =

nsumt=1





σisinΣ

nsumt=1

p(t)p(bt|t)σ (76)






27




~xisinPs

p(~x|s) σ~x = ρ



suma





28

A XOR-games










tr(σsaQ~xsa


p(a|s)(46)

=1






S = (0 s(2) s(n)) | s(j) isin 0 1 (48)


~xs0 = s (49)

~xs1 = s (50)


V (a b|s t) =

1 if b = ~x

(t)sa

0 otherwise (51)



17





P game(S T σAB) =1

2n

sumsa

tr(ρ~xsaQ~xsa

)(53)

=1

2n

sumsa

1

2(1 + ~vsa2) (54)

=1

2

(1 +

1

2n

sumsa


)(55)

le 1

2

1 +

radic1

2n

sumsa

sumttprime



(56)

=1

2+

1

2radicn (57)





tr

(ρ~xsaB

~x(t)sa

t

)= tr

(ρ~xsaB

~x(tprime)sa

tprime




18




tr(Bx(t)

t ρ) (59)


1

n

nsumt=1


n

sumt

log maxx(t)isin01

tr(Bx(t)

t ρ)

(60)


sumt

tr(Bx(t)

t ρ)

(61)

= minus log maxsa

λmax(Q~xsa) (62)



λmax(Q~xsa) =1

2+

1

2radicn (63)






19






ζ~xsa =1

2+

1

2radic






(|00〉AB + |11〉AB) (65)





20

070 075 080 085 090 095 100070

075

080

085

090

095

100

Steering

Unc

erta

inty



max = 12 + 1(2radic


21

0





ρ~xsa =

int Ωprime~xsa

Ω~xsa

p(Ω)σΩdΩ (66)


AB =int 2π

0 p(Ω)σΩAσ

ΩBdΩ



22








ρ00 = p(0|0) = 1 p(0|1) = 1 (67)

ρ01 = p(1|0) = 1 p(1|1) = 1 (68)

ρ10 = p(0|0) = 1 p(1|1) = 1 (69)

ρ11 = p(1|0) = 1 p(0|1) = 1 (70)


23





η~x =sumtprime



η~x le ζ~x (72)





24







~x(t)sat


~x(1minust)sa t




2(73)





p(right|t ~xsa) =1

2+

1

2radic

2 (74)





25

00 02 04 06 08 1000

02

04

06

08

10


pHw

rong

Egravetw

rong

L



(solid blue line)







26


06

07

08

09

10pHsecondL






U =

nsumt=1





σisinΣ

nsumt=1

p(t)p(bt|t)σ (76)






27




~xisinPs

p(~x|s) σ~x = ρ



suma





28

A XOR-games













P game(S T σAB) =1

2n

sumsa

tr(ρ~xsaQ~xsa

)(53)

=1

2n

sumsa

1

2(1 + ~vsa2) (54)

=1

2

(1 +

1

2n

sumsa


)(55)

le 1

2

1 +

radic1

2n

sumsa

sumttprime



(56)

=1

2+

1

2radicn (57)





tr

(ρ~xsaB

~x(t)sa

t

)= tr

(ρ~xsaB

~x(tprime)sa

tprime




18




tr(Bx(t)

t ρ) (59)


1

n

nsumt=1


n

sumt

log maxx(t)isin01

tr(Bx(t)

t ρ)

(60)


sumt

tr(Bx(t)

t ρ)

(61)

= minus log maxsa

λmax(Q~xsa) (62)



λmax(Q~xsa) =1

2+

1

2radicn (63)






19






ζ~xsa =1

2+

1

2radic






(|00〉AB + |11〉AB) (65)





20

070 075 080 085 090 095 100070

075

080

085

090

095

100

Steering

Unc

erta

inty



max = 12 + 1(2radic


21

0





ρ~xsa =

int Ωprime~xsa

Ω~xsa

p(Ω)σΩdΩ (66)


AB =int 2π

0 p(Ω)σΩAσ

ΩBdΩ



22








ρ00 = p(0|0) = 1 p(0|1) = 1 (67)

ρ01 = p(1|0) = 1 p(1|1) = 1 (68)

ρ10 = p(0|0) = 1 p(1|1) = 1 (69)

ρ11 = p(1|0) = 1 p(0|1) = 1 (70)


23





η~x =sumtprime



η~x le ζ~x (72)





24







~x(t)sat


~x(1minust)sa t




2(73)





p(right|t ~xsa) =1

2+

1

2radic

2 (74)





25

00 02 04 06 08 1000

02

04

06

08

10


pHw

rong

Egravetw

rong

L



(solid blue line)







26


06

07

08

09

10pHsecondL






U =

nsumt=1





σisinΣ

nsumt=1

p(t)p(bt|t)σ (76)






27




~xisinPs

p(~x|s) σ~x = ρ



suma





28

A XOR-games












tr(Bx(t)

t ρ) (59)


1

n

nsumt=1


n

sumt

log maxx(t)isin01

tr(Bx(t)

t ρ)

(60)


sumt

tr(Bx(t)

t ρ)

(61)

= minus log maxsa

λmax(Q~xsa) (62)



λmax(Q~xsa) =1

2+

1

2radicn (63)






19






ζ~xsa =1

2+

1

2radic






(|00〉AB + |11〉AB) (65)





20

070 075 080 085 090 095 100070

075

080

085

090

095

100

Steering

Unc

erta

inty



max = 12 + 1(2radic


21

0





ρ~xsa =

int Ωprime~xsa

Ω~xsa

p(Ω)σΩdΩ (66)


AB =int 2π

0 p(Ω)σΩAσ

ΩBdΩ



22








ρ00 = p(0|0) = 1 p(0|1) = 1 (67)

ρ01 = p(1|0) = 1 p(1|1) = 1 (68)

ρ10 = p(0|0) = 1 p(1|1) = 1 (69)

ρ11 = p(1|0) = 1 p(0|1) = 1 (70)


23





η~x =sumtprime



η~x le ζ~x (72)





24







~x(t)sat


~x(1minust)sa t




2(73)





p(right|t ~xsa) =1

2+

1

2radic

2 (74)





25

00 02 04 06 08 1000

02

04

06

08

10


pHw

rong

Egravetw

rong

L



(solid blue line)







26


06

07

08

09

10pHsecondL






U =

nsumt=1





σisinΣ

nsumt=1

p(t)p(bt|t)σ (76)






27




~xisinPs

p(~x|s) σ~x = ρ



suma





28

A XOR-games














ζ~xsa =1

2+

1

2radic






(|00〉AB + |11〉AB) (65)





20

070 075 080 085 090 095 100070

075

080

085

090

095

100

Steering

Unc

erta

inty



max = 12 + 1(2radic


21

0





ρ~xsa =

int Ωprime~xsa

Ω~xsa

p(Ω)σΩdΩ (66)


AB =int 2π

0 p(Ω)σΩAσ

ΩBdΩ



22








ρ00 = p(0|0) = 1 p(0|1) = 1 (67)

ρ01 = p(1|0) = 1 p(1|1) = 1 (68)

ρ10 = p(0|0) = 1 p(1|1) = 1 (69)

ρ11 = p(1|0) = 1 p(0|1) = 1 (70)


23





η~x =sumtprime



η~x le ζ~x (72)





24







~x(t)sat


~x(1minust)sa t




2(73)





p(right|t ~xsa) =1

2+

1

2radic

2 (74)





25

00 02 04 06 08 1000

02

04

06

08

10


pHw

rong

Egravetw

rong

L



(solid blue line)







26


06

07

08

09

10pHsecondL






U =

nsumt=1





σisinΣ

nsumt=1

p(t)p(bt|t)σ (76)






27




~xisinPs

p(~x|s) σ~x = ρ



suma





28

A XOR-games









070 075 080 085 090 095 100070

075

080

085

090

095

100

Steering

Unc

erta

inty



max = 12 + 1(2radic


21

0





ρ~xsa =

int Ωprime~xsa

Ω~xsa

p(Ω)σΩdΩ (66)


AB =int 2π

0 p(Ω)σΩAσ

ΩBdΩ



22








ρ00 = p(0|0) = 1 p(0|1) = 1 (67)

ρ01 = p(1|0) = 1 p(1|1) = 1 (68)

ρ10 = p(0|0) = 1 p(1|1) = 1 (69)

ρ11 = p(1|0) = 1 p(0|1) = 1 (70)


23





η~x =sumtprime



η~x le ζ~x (72)





24







~x(t)sat


~x(1minust)sa t




2(73)





p(right|t ~xsa) =1

2+

1

2radic

2 (74)





25

00 02 04 06 08 1000

02

04

06

08

10


pHw

rong

Egravetw

rong

L



(solid blue line)







26


06

07

08

09

10pHsecondL






U =

nsumt=1





σisinΣ

nsumt=1

p(t)p(bt|t)σ (76)






27




~xisinPs

p(~x|s) σ~x = ρ



suma





28

A XOR-games









0





ρ~xsa =

int Ωprime~xsa

Ω~xsa

p(Ω)σΩdΩ (66)


AB =int 2π

0 p(Ω)σΩAσ

ΩBdΩ



22








ρ00 = p(0|0) = 1 p(0|1) = 1 (67)

ρ01 = p(1|0) = 1 p(1|1) = 1 (68)

ρ10 = p(0|0) = 1 p(1|1) = 1 (69)

ρ11 = p(1|0) = 1 p(0|1) = 1 (70)


23





η~x =sumtprime



η~x le ζ~x (72)





24







~x(t)sat


~x(1minust)sa t




2(73)





p(right|t ~xsa) =1

2+

1

2radic

2 (74)





25

00 02 04 06 08 1000

02

04

06

08

10


pHw

rong

Egravetw

rong

L



(solid blue line)







26


06

07

08

09

10pHsecondL






U =

nsumt=1





σisinΣ

nsumt=1

p(t)p(bt|t)σ (76)






27




~xisinPs

p(~x|s) σ~x = ρ



suma





28

A XOR-games
















ρ00 = p(0|0) = 1 p(0|1) = 1 (67)

ρ01 = p(1|0) = 1 p(1|1) = 1 (68)

ρ10 = p(0|0) = 1 p(1|1) = 1 (69)

ρ11 = p(1|0) = 1 p(0|1) = 1 (70)


23





η~x =sumtprime



η~x le ζ~x (72)





24







~x(t)sat


~x(1minust)sa t




2(73)





p(right|t ~xsa) =1

2+

1

2radic

2 (74)





25

00 02 04 06 08 1000

02

04

06

08

10


pHw

rong

Egravetw

rong

L



(solid blue line)







26


06

07

08

09

10pHsecondL






U =

nsumt=1





σisinΣ

nsumt=1

p(t)p(bt|t)σ (76)






27




~xisinPs

p(~x|s) σ~x = ρ



suma





28

A XOR-games













η~x =sumtprime



η~x le ζ~x (72)





24







~x(t)sat


~x(1minust)sa t




2(73)





p(right|t ~xsa) =1

2+

1

2radic

2 (74)





25

00 02 04 06 08 1000

02

04

06

08

10


pHw

rong

Egravetw

rong

L



(solid blue line)







26


06

07

08

09

10pHsecondL






U =

nsumt=1





σisinΣ

nsumt=1

p(t)p(bt|t)σ (76)






27




~xisinPs

p(~x|s) σ~x = ρ



suma





28

A XOR-games















~x(t)sat


~x(1minust)sa t




2(73)





p(right|t ~xsa) =1

2+

1

2radic

2 (74)





25

00 02 04 06 08 1000

02

04

06

08

10


pHw

rong

Egravetw

rong

L



(solid blue line)







26


06

07

08

09

10pHsecondL






U =

nsumt=1





σisinΣ

nsumt=1

p(t)p(bt|t)σ (76)






27




~xisinPs

p(~x|s) σ~x = ρ



suma





28

A XOR-games









00 02 04 06 08 1000

02

04

06

08

10


pHw

rong

Egravetw

rong

L



(solid blue line)







26


06

07

08

09

10pHsecondL






U =

nsumt=1





σisinΣ

nsumt=1

p(t)p(bt|t)σ (76)






27




~xisinPs

p(~x|s) σ~x = ρ



suma





28

A XOR-games










06

07

08

09

10pHsecondL






U =

nsumt=1





σisinΣ

nsumt=1

p(t)p(bt|t)σ (76)






27




~xisinPs

p(~x|s) σ~x = ρ



suma





28

A XOR-games












~xisinPs

p(~x|s) σ~x = ρ



suma





28

A XOR-games









the uncertainty principle determines the non-locality of quantum … · 2013. 1. 15. · the...

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