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1 From the Foundations of Quantum Theory

to Quantum Technology an Introduction

Gernot Alber

Nowadays, the new technological prospects of processing quantum informa-tion in quantum cryptography [1], quantum computation [2] and quantumcommunication [3] attract not only physicists but also researchers from otherscientic communities, mainly computer scientists, discrete mathematiciansand electrical engineers. Current developments demonstrate that character-istic quantum phenomena which appear to be surprising from the point ofview of classical physics may enable one to perform tasks of practical interestbetter than by any other known method. In quantum cryptography, the no-cloning property of quantum states [4] or the phenomenon of entanglement[5] helps in the exchange of secret keys between various parties, thus en-suring the security of one-time-pad cryptosystems [6]. Quantum parallelism[7], which relies on quantum interference and which typically also involvesentanglement [8], may be exploited for accelerating computations. Quantumalgorithms are even capable of factorizing numbers more eciently than anyknown classical method is [9], thus challenging the security of public-key cryp-tosystems such as the RSA system [6]. Classical information and quantuminformation based on entangled quantum systems can be used for quantumcommunication purposes such as teleporting quantum states [10, 11].

Owing to signicant experimental advances, methods for processing quan-tum information have developed rapidly during the last few years.1 Basicquantum communication schemes have been realized with photons [10, 11],and basic quantum logical operations have been demonstrated with trappedions [13, 14] and with nuclear spins of organic molecules [15]. Also, cavityquantum electrodynamical setups [16], atom chips [17], ultracold atoms inoptical lattices [18, 19], ions in an array of microtraps [20] and solid-statedevices [21, 22, 23] are promising physical systems for future developmentsin this research area. All these technologically oriented, current developmentsrely on fundamental quantum phenomena, such as quantum interference, themeasurement process and entanglement. These phenomena and their distinc-tive dierences from basic concepts of classical physics have always been ofcentral interest in research on the foundations of quantum theory. However,in emphasizing their technological potential, the advances in quantum infor-1 Numerous recent experimental and theoretical achievements are discussed in [12].

G. Alber, T. Beth, M. Horodecki, P. Horodecki, R. Horodecki, M. Rotteler, H. Weinfurter,R. Werner, A. Zeilinger: Quantum Information, STMP 173, 113 (2001)c Springer-Verlag Berlin Heidelberg 2001

2 Gernot Alber

mation processing reect a profound change in the general attitude towardsthese fundamental phenomena. Thus, after almost two decades of impressivescientic achievements, it is time to retrace some of those signicant early de-velopments in quantum physics which are at the heart of quantum technologyand which have shaped its present-day appearance.

1.1 Early Developments

Many of the current methods and developments in the processing of quantuminformation have grown out of a long struggle of physicists with the foun-dations of modern quantum theory. The famous considerations by Einstein,Podolsky and Rosen (EPR) [24] on reality, locality and completeness of phys-ical theories are an early example in this respect. The critical questions raisedby these authors inspired many researchers to study quantitatively the essen-tial dierence between quantum physics and the classical concepts of realityand locality. The breakthrough was the discovery by J.S. Bell [25] that thestatistical correlations of entangled quantum states are incompatible with thepredictions of any theory which is based on the concepts of reality and lo-cality of EPR. The constraints imposed on statistical correlations within theframework of a local, realistic theory (LRT) are expressed by Bells inequality[25]. As the concept of entanglement and its peculiar correlation propertieshave been of fundamental signicance for the development of quantum infor-mation processing, it is worth recalling some of its most elementary featuresin more detail.

1.1.1 Entanglement and Local, Realistic Theories

In order to clarify the characteristic dierences between quantum mechan-ical correlations originating from entangled states and classical correlationsoriginating from local, realistic theories, let us consider the following basicexperimental setup (Fig. 1.1). A quantum mechanical two-particle system,

ii

2

+1+1

A B

s-1 -1

1

1

2

Fig. 1.1. Basic experimental setup for testing Bells inequality; the choices of thedirections of polarization on the Bloch sphere for optimal violation of the CHSHinequality (1.3) correspond to = /4 for spin-1/2 systems

1 From the Foundations of Quantum Theory to Quantum Technology 3

such as a photon pair, is produced by a source s. Polarization properties ofeach of these particles are measured subsequently by two distant observers Aand B. Observers A and B perform polarization measurements by randomlyselecting one of the directions 1 or 2, and 1 or 2, respectively, in eachexperiment. Furthermore, let us assume that for each of these directions onlytwo measurement results are possible, namely +1 or 1. In the case of pho-tons these measurement results would correspond to horizontal or verticalpolarization.

What are the restrictions imposed on correlations of the measurementresults if the physical process can be described by an underlying LRT withunknown (hidden) parameters? For this purpose, let us rst of all summarizethe minimal set of conditions any LRT should fulll.

1. The state of the two-particle system is determined uniquely by a parame-ter , which may denote an arbitrary set of discrete or continuous labels.Thus the most general observable of observer A or B for the experimentalsetup depicted in Fig. 1.1 is a function of the variables (i,j , ). If theactual value of the parameter is unknown (hidden), the state of thetwo-particle system has to be described by a normalized probability dis-tribution P (), i.e.

dP () = 1, where characterizes the set of all

possible states. The state variable determines all results of all possiblemeasurements, irrespective of whether these measurements are performedor not. It represents the element of physical reality inherent in the ar-guments of EPR: If, without in any way disturbing a system, we canpredict with certainty the value of a physical quantity, then there existsan element of physical reality corresponding to this physical quantity[24].

2. The measurement results of each of the distant (space-like separated)observers are independent of the choice of polarizations of the other ob-server. This assumption reects the locality concept inherent in the argu-ments of EPR: The real factual situation of the system A is independentof what is done with the system B, which is spatially separated from theformer [24]. Thus, taking into account also this locality requirement, themost general observable of observer A for the experimental setup depictedin Fig. 1.1 can depend on the variables i and (for B, j and ) only.

These two assumptions, which reect fundamental notions of classical physicsas used in the arguments of EPR, restrict signicantly the possible cor-relations of measurements performed by both distant observers. Accord-ing to these assumptions, the following measurement results are possible:a(i, ) ai = 1 (i = 1, 2) for observer A, and b(i, ) bi = 1(i = 1, 2) for observer B. For a given value of the state variable , all thesepossible measurement results of the dichotomic (two-valued) variables ai andbi (i = 1, 2) can be combined in the single relation

|(a1 + a2)b1 + (a2 a1)b2| = 2 . (1.1)

4 Gernot Alber

It should be mentioned that this relation is counterfactual [26] in the sensethat it involves both results of actually performed measurements and pos-sible results of unperformed measurements. All these measurement resultsare determined uniquely by the state variable . If this state variable is un-known (hidden), (1.1) has to be averaged over the corresponding probabilitydistribution P (). This yields an inequality for the statistical mean values,

aibjLRT =

d P ()a(i, )b(j , ) (i, j = 1, 2), (1.2)

which is a variant of Bells inequality and which is due to Clauser, Horne,Shimony and Holt (CHSH) [27], namely

| a1b1LRT + a2b1LRT + a2b2LRT a1b2LRT | 2 . (1.3)This inequality characterizes the restrictions imposed on the correlations be-tween dichotomic variables of two distant observers within the framework ofany LRT. There are other, equivalent forms of Bells inequality, one of whichwas proposed by Wigner [28] and will be discussed in Chap. 3.

Quantum mechanical correlations can violate this inequality. For this pur-pose let us consider, for example, the spin-entangled singlet state

| = 12(|+ 1A| 1B | 1A|+ 1B) , (1.4)

where | 1A and | 1B denote the eigenstates of the Pauli spin operatorsAz and

Bz , with eigenvalues 1. Quantum mechanically, the measurement

of the dichotomic polarization variables ai and bi is represented by the spinoperators ai = iA and bi = iB. (A, for example, denotes the vector ofPauli spin operators referring to observer A, i.e. A =

i=x,y,z

Ai ei, where

ei are the unit vectors.) The corresponding quantum mechanical correlationsentering the CHSH inequality (1.3) are given by

aibjQM = |aibj | = i j . (1.5)Choosing the directions of the polarizations (1,1), (1,2), (2,2) onthe Bloch sphere so that they involve an angle of /4 (see Fig. 109), one ndsa maximal violation of inequality (1.3), namely

| a1b1QM + a2b1QM + a2b2QM a1b2QM |= 22 > 2 . (1.6)

Thus, for this entangled state, the quantum mechanical correlations betweenthe measurement results of the distant observers A and B are stronger thanany possible correlation within the framework of an LRT. Obviously, thesecorrelations are incompatible with the classical notions of reality and local-ity of any LRT. It is these peculiar quantum correlations originating fromentanglement which have been of central interest in research on the founda-tions of quantum theory and which are also of central interest for quantuminformation processing.


So far, numerous experiments testing and supporting violations of Bellsinequality [29, 30, 31] have been performed.2 However, from a strictly logicalpoint of view, the results of all these experiments could still be explainedby an LRT, owing to two loopholes, namely the locality and the detectionloopholes. The locality loophole concerns violations of the crucial locality as-sumption underlying the derivation of Bells inequality. According to this as-sumption one has to ensure that any signaling between two distant observersA and B is impossible. The recently performed experiment of G. Weihs etal. [31] succeeded in fullling this locality requirement by choosing the sep-aration between these observers to be suciently large. However, so far allexperiments have involved low detection eciencies, so that in principle theobserved correlations which violate Bells inequality can still be explained byan LRT [32, 33]. This latter detection loophole constitutes a major experi-mental challenge, and it is one of the current experimental aims to close boththe detection loophole and the locality loophole simultaneously [34, 35, 36].

The concepts of physical reality and locality which lead to inequality (1.3)can also lead to logical contradictions with quantum theory which are not ofstatistical origin. This becomes particularly apparent when one considers anentangled three-particle state of the form

|GHZ = 12(|+ 1A|+ 1B|+ 1C | 1A| 1B| 1C) , (1.7)

a so-called GreenbergerHorneZeilinger (GHZ) state [37]. Again | 1A,| 1B, and | 1C denote the eigenstates of the Pauli spin operators Az ,Bz , and Cz , with eigenvalues 1. Similarly to Fig. 109, let us assume thatthe polarization properties of this entangled quantum state are investigatedby three distant (space-like separated) observers A, B and C. Each of theseobservers chooses his or her direction of polarization randomly along eitherthe x or the y axis.

What are the consequences an LRT would predict? As the three observersare space-like separated, the locality assumption implies that a polarizationmeasurement by one of these observers cannot inuence the results of theother observers. Following the notation of Fig. 109, the possible results of thepolarization measurements of observers A, B and C along directions i, jand k are ai = 1, bj = 1 and ck = 1. Let us now consider four pos-sible coincidence measurements of these three distant observers, with results(ax, bx, cx), (ax, by, cy), (ay, bx, cy) and (ay, by, cx). As we are dealing withdichotomic variables, within an LRT the product of all these measurementresults is always given by

RLRT = (axbxcx)(axbycy)(aybxcy)(aybycx) = a2xb2xc

2xa

2yb

2yc

2y = 1 . (1.8)

What are the corresponding predictions of quantum theory? In quantumtheory the variables ai, bj and ck are replaced by the Pauli spin operators

2 For a comprehensive discussion of experiments performed before 1989, see [29].

6 Gernot Alber

ai = i A, bj = j B and ck = k C. The GHZ state of (1.7) fulllsthe relations

axbxcx|GHZ = |GHZ ,

axbycy|GHZ = aybxcy|GHZ = ayby cx|GHZ = |GHZ . (1.9)Therefore the quantum mechanical result for the product of (1.8) is given by

RQM|GHZ = (axbxcx)(axby cy)(ay bxcy)(ay by cx)|GHZ= (1)|GHZ (1.10)

and contradicts the corresponding result of an LRT. These peculiar quantummechanical predictions have recently been observed experimentally [38]. Theentanglement inherent in these states oers interesting perspectives on thepossibility of distributing quantum information between three parties [39].

1.1.2 Characteristic Quantum Eects for Practical Purposes

According to a suggestion of Feynman [40], quantum systems are not only ofinterest for their own sake but might also serve specic practical purposes.Simple quantum systems may be used, for example, for simulating other, morecomplicated quantum systems. This early suggestion of Feynman emphasizespossible practical applications and thus indicates already a change in theattitude towards characteristic quantum phenomena.

In the same spirit, but independently, Wiesner suggested in the 1960s theuse of nonorthogonal quantum states for the practical purpose of encodingsecret classical information [41].3 The security of such an encoding procedureis based on a characteristic quantum phenomenon which does not involveentanglement, namely the impossibility of copying (or cloning) nonorthogonalquantum states [4]. This impossibility becomes apparent from the followingelementary consideration. Let us imagine a quantum process which is capableof copying two nonorthogonal quantum states, say |0 and |1, with 0

| a >

| x >

| a f(x)>

U

Fig. 1.2. Basic operation of a quantum oracle Uf which evaluates a Boolean func-tion f : x Zn2 f(x) Z12 {0, 1}; |x is the input state of an n-qubit quantumsystem; |a is a one-qubit state and denotes addition modulo 2

Quantum mechanically, the situation is dierent. The 2n possible binaryn-bit strings x can be represented by quantum states |x, which form a ba-sis in a 2n-dimensional Hilbert space H2n , which is the state space of nqubits. Furthermore, we imagine that the classical oracle is replaced by acorresponding quantum oracle (Fig. 1.2). This is a unitary transformationUf which maps basis states of the form |x|a, where a {0, 1}, to outputstates of the form |x|a f(x) in a single step. Here, |a denotes the quan-tum state of an ancilla qubit and denotes addition modulo 2. If the initialstate is |x|0, for example, the quantum oracle performs an evaluation off(x), resulting in the nal state |x|f(x). However, as this transformationis unitary, it can perform this task also for any linear combination of possi-ble basis states in a single step. This is the key idea of quantum parallelism[7]. Deutschs quantum algorithm obtains the solution to the problem posedabove by the following steps (Fig. 1.3):

1. The n-qubit quantum system and the ancilla system are prepared instates |0 and (|0 |1)/2. Then a Hadamard transformation

H : |0 12(|0+ |1) ,

|1 12(|0 |1) (1.12)

10 Gernot Alber

H

H

H

H

| 0 > < 0 |

| 0 > < 0 |

m e a s u r e m e n t

U f.

.

.

.

.

.

.

.

.

1 2

> >| |

n

(| 0 > - | 1 >)

| 0 >

| 0 >2

1

Fig. 1.3. Schematic representation of Deutschs quantum algorithm

is applied to all of the rst n qubits. We denote by H(i) the applicationof H to the ith qubit. Thus, the separable quantum state

|1 12

[( ni=1

H(i))|0](|0 |1) = 12n+1

x2n

|x(|0 |1) (1.13)

is prepared.2. A single application of the quantum oracle Uf to state |1 yields the

quantum state

|2 Uf |1 = 12n+1

x2n

(1)f(x)|x(|0 |1) . (1.14)

3. Subsequently a quantum measurement is performed to determine whetherthe system is in state |1 or not. With the help of n Hadamard trans-formations (as in step 1), this quantum measurement can be reduced toa measurement of whether the rst n qubits of the quantum system arein state |0 or not.

If in step 3 the quantum system is found in state |1, f is constant, otherwisef is balanced. One of these two possibilities is observed with unit probability.The probability p of observing the quantum system in state |1 is given by

p | 1|2 |2= 12n |x2n

(1)f(x) |2 . (1.15)

Taking into account the single application of the quantum oracle in step 2and the application of the Hadamard transformations in the preparation andmeasurement processes, Deutschs quantum algorithm requires O(n) steps toobtain the nal answer, in contrast to any classical algorithm, which needsan exponential number of steps. Thus Deutschs quantum algorithm leads toan exponential speedup.

A key element of this quantum algorithm and of those discovered later isthe quantum parallelism involved in step 2, where the linear superposition


of the rst n qubits comprises the requested global information about thefunction f . For most of the possible functions f this intermediate quantumstate is expected to be entangled. An exception is the case of a constant func-tion f , for which the quantum state |2 is separable. Furthermore, it is alsocrucial for the success of this quantum algorithm that the nal measurementin step 3 yielding the required answer can be implemented by a fast quantummeasurement whose complexity is polynomial in n. This is a requirementfullled by all other known fast quantum algorithms. The quantum algo-rithm described above was the rst example demonstrating that quantumphenomena may speed up computations in such a way that an exponentialgap appears between the complexity class of the quantum problem and thecomplexity class of the corresponding classical probabilistic problem.

Continuing this development initiated by Deutsch, other, new fast quan-tum algorithms were discovered in the subsequent years. The most prominentexamples are Simons quantum algorithm [57], Shors celebrated algorithm[9] for factorizing numbers, and Grovers search algorithm [58]. (Quantum al-gorithms are discussed in detail in Chap. 4.) In addition, possible realizationsof quantum computing devices were suggested which were based on trappedions [59] and on cavity quantum electrodynamical setups [60]. These devel-opments called for new methods for stabilizing quantum algorithms againstperturbing environmental inuences, which tend to destroy quantum inter-ference and quantum entanglement [61]. This led to the development of therst error-correcting codes [62, 63, 64, 65, 66] by adaptation of classical error-correcting techniques to the quantum domain. An introduction to the theoryof quantum error correction is presented in Chap. 4.

1.2 Quantum Physics and Information Processing

What are the common features of these early developments? The commonelement of these early developments in quantum cryptography and quan-tum computation is that they all involve the practical processing of informa-tion and they are all founded on and facilitated by characteristic quantumphenomena. These phenomena, among which the most prominent is entan-glement, are in conict with the classical concepts of physical reality andlocality. Obviously, these early developments hint at a profound connectionbetween the concept of information and some fundamental concepts of quan-tum theory, which is also promising from the technological point of view.It is these technologically oriented aspects of quantum information theory[67, 68, 69] which are at the heart of quantum information processing.

Methods for processing quantum information have developed rapidly dur-ing the last few years [12]. Owing to signicant experimental advances, ba-sic interference and entanglement phenomena which are of central interestfor processing quantum information have been realized in the laboratory invarious physical systems. Basic schemes for quantum communication have

12 Gernot Alber

been demonstrated with photons [10, 11, 49, 70]. Realizations of elementaryquantum logical operations have been based on trapped ions [13, 14] and onnuclear magnetic resonance [15]. Recent experiments indicate that besidescavity quantum electrodynamical setups [16], trapped neutral atoms whichare guided along magnetic wires (atom chips) might also be useful for quan-tum information processing [17]. There have also been theoretical proposalson using ultracold atoms in optical lattices [18, 19], on ions in an array ofmicrotraps [20] and on solid-state devices [21, 22, 23] for the implementationof quantum logical gates.

By now, quantum information processing has become an interdisciplinarysubject which attracts not only physicists but also researchers from othercommunities. The common interest is the practical, technologically orientedapplication of characteristic quantum phenomena. At this stage of develop-ment, it appears necessary to examine recent achievements and to emphasizethe underlying, general, basic concepts, which have been developing gradu-ally and which are now commonly adopted by all researchers in this eld.This is one of the main intentions of the rest of the book.

In Chap. 2, Werner introduces the basic concepts of quantum informationtheory and describes the fundamental mathematical structures underlying re-cent and current developments. In particular, this chapter addresses a naturalquestion appearing in connection with Feynmans suggestion, namely whatcan be done with the help of quantum systems and what cannot be done. Arst example of an impossible quantum process, the copying of nonorthogonalquantum states, has already been mentioned. Other examples of possible andimpossible quantum processes are discussed in detail in this contribution.

First experimental realizations of basic quantum communication schemesbased on entangled photon pairs are discussed in Chap. 3 by Weinfurter andZeilinger. These rst experiments on entanglement-based quantum cryptog-raphy, dense coding and quantum teleportation demonstrate the importantrole photons play in current experiments. Furthermore, these experimentsalso emphasize once again the fundamental signicance of entanglement forquantum information processing.

The basic theoretical concepts of quantum computation and the mathe-matical structure underlying quantum algorithms are discussed in Chap. 4by Beth and Rotteler. In particular, it is demonstrated how recent results inthe theory of signal processing can be used for the development of new fastquantum algorithms. A short introduction to the theory of quantum errorcorrection is also presented.

A comprehensive account of the mathematical structure of entanglementand of the signicance of mixed entangled states for quantum informationprocessing is presented in Chap. 5 by M. Horodecki, P. Horodecki and R.Horodecki. One of the most surprising recent developments in this contexthas been the discovery of bound entanglement [71]. Though much is stillunknown, this section gives a state-of-the-art presentation of what is known


about this new form of entanglement and its implications for processing quan-tum information.

2 Quantum Information Theory

an Invitation

Reinhard F. Werner

2.1 Introduction

Quantum information and quantum computers have received a lot of publicattention recently. Quantum computers have been advertised as a kind ofwarp drive for computing, and indeed the promise of the algorithms of Shorand Grover is to perform computations which are extremely hard or evenprovably impossible on any merely classical computer. On the experimentalside, perhaps the most remarkable feat of quantum information processingwas the realization of quantum teleportation, which once again has sciencection overtones.

In some sense these miracles are an extension of the strangeness of quan-tum mechanics those unresolved questions in the foundations of quantummechanics, which most physicists know about, but few try to tackle directlyin their research. However, trying to build an explanation of quantum in-formation on the literature about the foundations of quantum mechanics ismore likely to mystify than to clarify. It would also give a wrong idea ofhow discussions in this new eld are conducted. Because, just as physicistswith widely diering convictions on foundational matters can usually agreequite easily on what the predictions of quantum mechanics are in a partic-ular experimental setup, researchers in quantum information can agree onwhether a device should work, no matter what they may think about thedeeper meaning of the wave function. For example, one of the founders of theeld is an outspoken proponent of the many-worlds interpretation of quan-tum mechanics (which I, personally, nd useless and bizarre). But, whateverthe intuitions leading him to his discoveries about quantum computing mayhave been, these discoveries make sense in every other interpretation.

In this article I shall give an account of the basic concepts of quantuminformation theory, staying as much as possible in the area of general agree-ment. So, in order to enter this new eld, plain quantum mechanics is enough,and no new, perhaps obscure, views are needed. There is, of course, a charac-teristic shift in emphasis expressed by the word information, and we shallhave to explore the consequences of this shift.

The article is divided into two parts. The rst (up to the end of Sect. 2.5)is mostly in plain English, centered around the exploration of what can orcannot be done with quantum systems as information carriers. The secondG. Alber, T. Beth, M. Horodecki, P. Horodecki, R. Horodecki, M. Rotteler, H. Weinfurter,R. Werner, A. Zeilinger: Quantum Information, STMP 173, 1457 (2001)c Springer-Verlag Berlin Heidelberg 2001

2 Quantum Information Theory an Invitation 15

part, Sect. 2.6, then gives a description of the mathematical structures andof some of the tools needed to develop the theory.

2.2 What is Quantum Information?

Let us start with a preliminary denition:

Quantum information is that kind of information which is carriedby quantum systems from the preparation device to the measuringapparatus in a quantum mechanical experiment.

So a transmitter of quantum information is nothing but a device preparingquantum particles, and a receiver is just a measuring device. Of course, thisis not saying much. But even so, it is a strange statement from the point ofview of classical information theory: in that theory one usually does not careabout the physical carrier of the information, or else one would have to dis-tinguish electrodynamical information, printed information, magneticinformation and many more. In fact, the success of (classical) informationtheory depends largely on abstracting from the physical carrier, and goinginstead for the general principles underlying any information exchange. Sowhy should quantum information be any dierent?

A moments reection makes clear why the abstraction from the physicalcarrier of information leads to a successful theory: the reason is that it is soeasy to convert information between all such carriers. The conversion frombytes on a hard disk, to currents in a chip, to signals on a cable, to radiowaves via satellite and maybe, nally, to an image on a computer screen inanother continent all happens essentially without loss, and if there are losses,they are well understood, and it is known how to correct for them. Therefore,the crucial question is: can quantum information in the above loose sensealso be converted to those standard classical kinds of information, and back,without loss? Or: are there fundamental limitations to such a translation,and is quantum information hence really a new kind of information?

This book would not have been written if the answer to the last questionwere not armative: quantum information is indeed a new kind of informa-tion. But to make this precise, let us see what would be required of a suc-cessful translation. Let us begin with the conversion of quantum informationto classical information: a device for this conversion would take a quantumsystem and produce as its output some classical information. This is nothingbut a complicated way of saying measurement. The reverse translation,from classical to quantum information, obviously involves some preparationof quantum systems. The classical input to such a device is used to controlthe settings of the preparing device, and any dependence of the preparationprocess on classical information is admissible. There are two kinds of de-vices we can combine from these two elements. Let us rst consider a devicegoing from classical to quantum to classical information. This is a rather

16 Reinhard F. Werner

M P

Fig. 2.1. Classical teleportation. Here and in the following diagrams, a wavy arrowstands for quantum systems, and a straight arrow for the flow of classical informa-tion

commonplace operation. For example, one can encode one classical bit inthe polarization degree of freedom of a photon (clearly a quantum system),by choosing one of two orthogonal polarizations for the photon, dependingon the value of the classical bit. The readout is done by a photomultipliercombined with a polarization lter in one of the corresponding directions. Inprinciple, this allows a perfect transmission. In some sense every transmis-sion of classical information is of this kind, because every physical systemultimately obeys the laws of quantum mechanics, even if we can often dis-regard this fact and treat it classically. Hence classical information can betranslated into quantum information (and back).

But what about the converse? This hypothetical (and in fact, impossi-ble) process has come to be known as classical teleportation (see Fig. 2.1). Itwould involve a measuring device M, operating on some input quantum sys-tems. The results of the measurements are subsequently fed into a preparingdevice P, which produces the nal output of the combined device. The taskis to set things up such that the outputs of the combined device are indistin-guishable from the quantum inputs. Of course, we have to say precisely whatindistinguishable should mean. Clearly, this cannot mean that the samesystem comes out at the other end. In the classical case this is not demandedeither. What can only be meant in quantum mechanics is that no statisticaltest will see the dierence. In other words, no matter what the preparationof the input systems is and no matter what observable we measure on theoutputs of the teleportation device, we shall always get the same probabilitydistribution of results as if the inputs had been directly measured. Note alsothat this criterion does not involve the states of individual systems, but onlystates in the form of the distribution parameters of ensembles of identicallyprepared systems.

The impossibility of classical teleportation will be treated extensively inthe following section, where it is related to a hierarchy of impossible machines.For a mathematical statement of this impossibility in the standard quantumformalism of quantum mechanics, see the remark after (2.7). For the moment,however, let us take it for granted, and see what all this says about the newconcept of quantum information.

First of all, we are concerned here with problems of transmission, not withcontent or meaning. This is exactly the same as in classical information the-ory. There, too, it is often not easy to avoid confusion with a dierent concept


of information used in everyday language, namely the kind available at aninformation desk. Information theory does not care whether a TV channelis used for misinformation, but can say everything about what it takesto ensure the technical quality of the nal images. Hence the quantitativemeasures of information all relate to storage and transmission capacity, tothe possibilities of compression and error correction and so on. In the samevein, quantum information theory will not tell us what the meaning of aquantum message is, and this is probably meaningless anyway, because amessage that has been read is classical almost by denition. But quantuminformation theory has precise notions of the resources needed to transmitsuch information faithfully.

Secondly, transmission of quantum information is not at all an exoticconcept in the context of modern physics. It can be paraphrased in various,perhaps more familiar ways, for example as transmission of intact quantumstates, as coherent transmission of quantum systems or as transmissionpreserving all interference possibilities of the system. Nevertheless the in-formation metaphor is useful, not only because it suggests new applications,but also because it leads one to ask new questions, and leads to quantitativenotions where previously there was only a qualitative understanding. Andpossibly this even provides a way to see in a sharper light the old conun-drums of the foundations of quantum mechanics.

2.3 Impossible Machines

The usefulness of considering impossible machines is well known from ther-modynamics: the second law of thermodynamics is often stated as the impos-sibility of a perpetual-motion machine. The theorem of the impossibility ofclassical teleportation is likewise a fundamental law of quantum mechanics,and a lot can be learned from analyzing it. Typically, the impossible ma-chines of quantum theory are perfectly possible in classical physics, so theirimpossibility does not follow supercially from their description, but rathercarries a connotation of paradox.

We shall discuss a range of impossible tasks, consisting of

teleportation copying (cloning) joint measurement Bells telephone.

As we shall see, teleportation is the most powerful of these, in the sense thatif we had a teleportation device, we could build a quantum copier, from whichwe could in turn construct joint measurements and, nally, a device knownas Bells Telephone, by which we could set up superluminal communication.Hence, if we uphold the principle of causality, which forbids the weakest


machine in this hierarchy, we are certain that teleportation is likewise im-possible. In this section we shall follow this line of reasoning to prove theimpossibility of teleportation. Of course, there are other, more direct ways ofproving this result from the structure of quantum mechanics. However, theseusually require more of the quantum formalism and give less insight into thedierences between classical and quantum information.

2.3.1 The Quantum Copier

This is the machine referred to in the well-known paper of Wootters andZurek entitled A single quantum cannot be cloned [4]. By denition, acopier would be a device taking one quantum system as input and turning outtwo systems of the same type. The condition for calling this a (faithful) copieris that we would not be able to distinguish a system coming from the outputfrom the input system by any statistical test, i.e. by means of the probabilitiesmeasured for any observable, and for any preparation of the initial state.Hence the device has to operate on arbitrary unknown states. It is clearthat a copier in the ordinary sense, e.g. a mail relay distributing email toseveral recipients, indeed satises this condition in the domain of classicalinformation. Note that we are not so unreasonable as to demand what thepaper quoted above suggests, namely that we could test this device on singleevents, or even assume some ontological identity of input and output: thecriterion for faithful copying is atly statistical, and can be veried by astraightforward collection of statistical tests.

Given a teleportation device, building a copier is quite easy (see Fig. 2.2).All we have to do is to remember that the classical information obtained inthe intermediate stage of the teleportation process can be copied perfectly.Hence we can apply the measuring device of the teleportation line to theinput system, copy the results, and simply run the reconstructing preparationprocess on each of these copies.

M

P

P

C=

Fig. 2.2. Making a copier from a classical teleportation line


2.3.2 Joint Measurement

This is the task of combining two separate measuring devices into a singledevice, or the simultaneous measurement of two quantum observables Aand B. Thus, a joint measuring device A&B is a device giving a pair (a, b)of classical outputs each time it is operated, such that a is a possible outputof A, and b is a possible output of B. (We use the symbol A to denote both anobservable and a device that measures this observable, and similar for B.) Werequire that the statistics of the a outcomes alone are the same as for deviceA, and similarly for B. Note that once again our criterion is statistical, andcan be tested without recourse to counterfactual conditionals such as theresult which would have resulted if B rather than A had been measured onthis particular quantum particle.

Many quantum observables are not jointly measurable in this sense. Themost famous examples, position and momentum, dierent components ofangular momentum, and positions of a free particle at dierent times, areprobably contained in every quantum mechanics course. Hence the impossi-bility of joint measurements is nothing but a precise statement of an aspectof complementarity.

Nevertheless, a joint measurement device for any of these could readilybe constructed given a functioning quantum copier (see Fig. 2.3): one wouldsimply run the copier C on the quantum system, and then apply the two givenmeasuring devices, A and B, to the copies. It is easy to see that the denitionof the copier then guarantees that the statistics of a and b separately comeout right. In other words, a copier can be seen as a universal joint measuringdevice.

=C

Fig. 2.3. Obtaining joint measurements from a copier

2.3.3 Bells Telephone

This is not named after a certain phone company, but after John S. Bell,who never proposed it in this form, but might have. It refers to a project of


performing superluminal communication using only correlations of the typetested by Bells inequalities. Without going into details for the moment, thebasic setup would consist of a source producing pairs of particles and sendingone member of each pair to each of the two communicating parties, conven-tionally named Alice and Bob. Each of them has a collection of dierentmeasuring devices to choose from, and the idea is for Alice to do some-thing which creates a noticeable change in the probabilities measured byBob. Clearly, this is a paradoxical task, because no particle or other physi-cal carrier of information actually goes from Alice to Bob. Therefore, if theparticles move suciently far apart from one another, this device transmitssuperluminally.

It is maybe useful to point out here a common confusion concerning suchsuperluminal eects, which sometimes even aicts otherwise reliable profes-sional writers. The mistake can usually be spotted easily by a device I callthe ping-pong ball test. It goes like this:

Take an authors explanation of Bells inequalities, and substituteping-pong balls for every quantum particle. Then if whatever theauthor is selling as paradoxical remains true, he/she hasnt under-stood a thing.

Here is an example: imagine a box containing a ping-pong ball; the box canbe separated into two parts, without anyone looking at the ball. One part isshipped to Tokyo or Alpha Centauri, without anyone looking inside. Then ifI open the other box I know instantly, i.e. at superluminal speed, whetherthe ball is at the distant location or not. Of course, this is true but hardlyparadoxical, and is totally useless for sending a message either way. To repeat:there is nothing paradoxical in statistical correlations per se between distantsystems with a common past, even if the correlation is perfect.

If Alice wants to send a message to Bob, correlations between two mea-suring devices are useless, because they cannot even be detected withoutcomparing the results, which requires exactly the communication the Tele-phone was intended for. Only if something Alice does has an eect on themeasurement results at Bobs end can we speak of communication. The onlything Alice can do in the standard setup is to choose a measuring device, andBells Telephone can be said to work if these choices have an inuence on theprobabilities measured by Bob (who has no access to Alices measurementresults). If there is no physical system traveling from Alice to Bob, however,this will be impossible.

To be fair, this can hardly be counted as an impossible machine of quan-tum mechanics, since the argument has nothing to do with quantum theory.What makes it t into the hierarchy described here is the following: if weassume that Bob has a joint measuring device for two yes/no measurements,and Bells inequalities are violated, we can design a strategy for Alice to sendsignals to Bob with better than chance results. Hence the joint measurementof suitable observables can provide a device suciently strong to achieve a


S=?

Fig. 2.4. Building Bells Telephone from a joint measurement

task forbidden by causality, and hence is impossible in general. This is thelink between the last two elements in the hierarchy of impossible machinesmentioned at the beginning of Sect. 2.3.

The proof of this step amounts to yet another derivation of Bells inequal-ities, but since it emphasizes the communication aspect it ts well into ourcontext, and we shall at least sketch it. This step will be rather more technicalthan the rest of this section, but does not require any quantum theory. Theargument can be skipped without loss as far as later sections are concerned.

So let us assume that Alice and Bob each have at their disposal twomeasuring devices, say A1, A2 and B1, B2, respectively. Each of these can givea result of either +1 or 1. We shall denote by P(a, b | Ai, Bj) the probabilityfor Alice to obtain a and Bob to obtain b in a correlation experiment in whichAlice uses measuring device Ai and Bob uses Bj . By

C(Ai, Bj) =a,b

ab P(a, b | Ai, Bj)

we shall denote the correlation coecient, which lies between 1 and +1.The combination

= C(A1, B1) + C(A1, B2) + C(A2, B1) C(A2, B2) (2.1)

carries special signicance, as we shall see below. Because the inequality 2is known as the Bell inequality (see Sect. 1.1.1), we shall call the Bell corre-lation for this choice of four observables. It is a quantity directly accessible toexperiment. Note that Bob usually cannot tell from his data which apparatus(A1 or A2) Alice chose. This is reected by the equation

a

P(a, b | A1, Bj) =a

P(a, b | A2, Bj) P(b | Bj) ,

and is borne out by all known experimental data. Now suppose Bob has ajoint measuring device for his B1 and B2, which we shall denote by B1&B2,which produces pair outcomes (b1, b2) (see Fig. 2.4). We can then determine


the probabilities pi(ai, b1, b2) = P(ai, (b1, b2) | Ai, B1&B2). The conditionthat this is really a joint measurement is expressed by the equations

b1

pi(ai, b1, b2) = P(ai, b2 | Ai, B2) and (2.2)b2

pi(ai, b1, b2) = P(ai, b1 | Ai, B1) , (2.3)

each for i = 1, 2. The basic rule for the information transmission is thefollowing:

Alice encodes the bit she wants to send by choosing either apparatusA1 or apparatus A2. Then Bob looks at his readout and interprets itas A1 whenever the two displays coincide (b1 = b2), and as A2if they are dierent.

We can then estimate the probability pok for Bob to be right, assumingthat the choices A1 and A2 are made with the same frequency. Assume rstthat Alice chooses A1. Then Bob is right with probability

a1,b1,b2

b1 + b22 |a1| p1(a1, b1, b2) ,

where the rst factor takes into account the condition b1 = b2, and the secondis introduced for later convenience. Combining this with a second term ofsimilar kind for Alices choice A2, and taking into account the probability1/2 for each of these choices, we obtain the overall probability pok for Bob tobe correct as

pok =12

a1,b1,b2

b1 + b22 |a1| p1(a1, b1, b2)

+12

a2,b1,b2

b1 b22 |a2| p2(a2, b1, b2)

14

a1,b1,b2

(b1 + b2)a1 p1(a1, b1, b2)

+14

a2,b1,b2

(b1 b2)a1 p2(a2, b1, b2)

=14

(C(A1, B1) + C(A1, B2) + C(A2, B1) C(A2, B2)

)=

4. (2.4)

Bob is right with a better probability than chance if pok > 1/2, which, by thiscomputation, can be guaranteed if > 2, i.e. if the classical Bell inequality(in ClauserHorneShimonyHolt form [72]) is violated. But this is indeed the


case in experiments conducted to determine (e.g. [73]), which give roughly 22 2.8. If we believe these experiments, the only conclusion can bethat the joint measurability of the B1 and B2 used in the experiment wouldbe sucient to make Bells Telephone work, which was our claim.

2.3.4 Entanglement, Mixed-State Analyzersand Correlation Resolvers

Violations of Bells inequalities can also be seen to prove the existence of anew class of correlations between quantum systems, known as entanglement.This concept is as fundamental to the eld of quantum information theoryas the idea of quantum information itself. So rather than organizing thisintroduction as an answer to the the question why quantum information isdierent from classical information, we could have followed the line whyentanglement is dierent from classical correlation. There are impossiblemachines in this line of approach, too, and we shall now describe briey howthey t in.

Consider a correlation experiment of the kind used in the study of Bellsinequalities (see Sect. 2.3.3). If Bob looks at his particles, and makes mea-surements on them without any communication from Alice, he will nd thattheir statistics are described by a certain mixed state. The state must bemixed, because if he now listens to Alice and sorts his particles accordingto Alices measurement results, he will get two subensembles, which are ingeneral dierent. In the usual ideal 2-qubit situation, in which one obtainsthe maximal violation of Bells inequalities, these subensembles are describedby pure states.

This is very satisfying for people who see the occurrence of mixed statesin quantum mechanics merely as a result of ignorance, as opposed to thedeeper kind of randomness encoded in pure states. This view usually comeswith an individual-state interpretation of quantum mechanics, in which eachindividual system can be assigned a pure state (a single vector in Hilbertspace), and a general preparation procedure is given not just by its densitymatrix, but by a specic probability distribution of pure states. Let us usethe term mixed-state analyzer for a hypothetical device which can see thedierence, i.e. a measuring device whose output after many measurementson a given ensemble is not just a collection of expectations of quantum ob-servables, but the distribution of pure states in the ensemble. In the case of acorrelation experiment, where Bob sees a mixed state only because he is igno-rant about Alices results, this machine would nd for him the decompositionof his mixed state into two pure states.

The problem is, of course, that Alice has several choices of measuring de-vices, and that the decomposition of Bobs mixed state depends, accordingly,on Alices choice. Hence she could signal to Bob, and we would have anotherinstance of Bells Telephone. There would be a way out if joint measurements


were available (to Alice in this case): then we could say that the two decom-positions were just the rst step in an even ner decomposition, a furtherreduction of ignorance, which would be brought to light if Alice were to ap-ply her joint measurement. Presumably the mixed-state analyzer would thenyield this ner decomposition, because the operation of this device would notdepend on how closely Alice cared to look at her particles.

But just as two quantum observables are often not jointly measurable, twodecompositions of mixed states often have no common renement (actually,in the formalism of quantum theory, these are two variants of the same the-orem). In particular, the two decompositions belonging to Alices choices inan experiment demonstrating a violation of Bells inequalities have no com-mon renement, and any mixed-state analyzer could be used for superluminalcommunication in this situation.

Another device, which is suggested by the individual-state interpretation,arises from a naive extrapolation of this view to the parts of a compositesystem: if every single system could be assigned a pure state, a compositesystem could be assigned a pair of pure states, one for each subsystem. Acorrelated state should therefore be given by a probability distribution ofsuch pairs. A device which represented an arbitrary state of a compositesystem as a mixture of uncorrelated pure product states might be called acorrelation resolver. It could be built given a classical teleportation line: whenone applies teleportation to one of the subsystems and applies conditionson the classical measurement results of the intermediate stage, one obtainsprecisely a representation of an arbitrary state in this form. But it is easy tosee that any state which can be so analyzed automatically satises all Bell-type inequalities, and hence once again the experimental violations of Bellsinequalities show that such a correlation resolver cannot exist. Hence wehave here a second line of reasoning in favor of the no-teleportation theorem:a teleportation device would allow classical correlation resolution, which isshown to be impossible by the Bell experiments.

The distinction between resolvable states and their complement is oneof the starting points of entanglement theory, where the resolvable statesare called separable, or classically correlated, and all others are calledentangled. For a more detailed treatment and an up-to date overview, thereader is referred to Chap. 5.

Without going into philosophical discussions about the foundations ofquantum mechanics, I should like to comment briey on the individual-stateinterpretation, which has suggested the two impossible machines discussed inthis subsection. First, this view is not at all uncommon, and it is quite possi-ble to read some passages from the masters of the Copenhagen interpretationas an endorsement of this view. Secondly, if we dene a hidden-variable theoryas a theory in which individual systems are described by classical parame-ters, whose distribution is responsible for the randomness seen in quantumexperiments, we have no choice but to call the individual-state interpreta-


tion a hidden-variable theory. The hidden variable in this theory is usuallydenoted by . And sure enough, as we have just pointed out, this theoryhas all the diculties with locality such a theory is known to have on gen-eral grounds. Thirdly, avoiding an individual-state interpretation, and withit some of its misleading intuitions, is easy enough. In practice this is doneanyhow, by concentrating on those aspects of the theory which have somedirect statistical meaning, and not on these involving hypothetical, and usu-ally impossible devices. This common ground is the statistical interpretationof quantum mechanics, in which states (pure or mixed) are the analoguesof classical probability distributions, and are not seen as a property of anindividual system, but of a specic way of preparing the system.

2.4 Possible Machines

2.4.1 Operations on Multiple Inputs

The no-teleportation theorem derived in the previous section says that thereis no way to measure a quantum state in such a way that the measuringresults suce to reconstruct the state. At rst sight this seems to deny thatthe notion of quantum states has an operational meaning at all. But there isno contradiction, and we shall resolve the apparent conict in this subsection,if only to sharpen the statement of the no-teleportation theorem.

Let us recall the operational denition of quantum states, according tothe statistical interpretation of quantum mechanics. A state is a descriptionof a way of preparing quantum systems, and in all its aspects it is related tocomputing expectation values. We might also say that it is the assignment ofan expectation value to every observable of the system. So to the extent thatexpectation values can be measured, it is possible to determine the state bytesting it on suciently many observables. What is crucial, however, is thateven the determination of a single expectation value is a statistical measure-ment. Hence such a determination requires a repetition of the experimentmany times, using many systems prepared according to the same procedure.In contrast, the above description of teleportation demands that it workswith a single quantum system as input, and that the measuring device doesnot accumulate results from several input systems. Expressed in the currentjargon, teleportation is required to be a one-shot operation. Note that thisdoes not contradict our statistical criteria for the success of teleportation andof other devices, which involve a statistics of independent single shots.

If we have available many identically prepared systems, many operationswhich are otherwise impossible become easy. Let us begin with classical tele-portation. Its multiinput analogue is the state estimation problem: how canwe design a measurement operating on samples of many (say, N) systemsfrom the same preparing device, such that the measurement result in eachcase is a collection of classical parameters forming a Hermitian matrix which


on average is close to the density matrix describing the initial preparation.This is symbolized in Fig. 2.5 (with the box T omitted for the moment):the box P at the end represents a repreparation of systems according to theestimated density matrix. The overall output will then be a quantum system,which can be directly compared with the inputs in statistical experiments. Itis clear that the state cannot be determined exactly from a sample with -nite N , but the determination becomes arbitrarily good in the limit N .Optimal estimation observables are known in the case when the inputs areguaranteed to be pure [74], but in the case of general mixed states there areno clear-cut theorems yet, partly owing to the fact that it is less clear whatgure of merit best describes the quality of such an estimator.

T P

Estim

ate

Fig. 2.5. Classical teleportation with multiple inputs, or state estimation

Given a good estimator we can, of course, proceed to good cloning by justrepeating the repreparation P as often as desired. The surprise here [75] isthat if only a xed number M of outputs is required, it is possible to obtainbetter clones with devices that stay entirely in the quantum world than bygoing via classical estimation. Again, the problem of optimal cloning is fullyunderstood for pure states [76], but work has only just begun to understandthe mixed-state case.

Another operation which becomes accessible in this way is the universalnot operation, assigning to each pure qubit state the unique pure state or-thogonal to it. Like time reversal, this is just a special case of an antiunitarysymmetry operation. In this case, a strategy using a classical estimation asan intermediate step can be shown to be optimal [77]. In this sense universalnot is a harder task than cloning.

More generally, we can look at schemes such as those in Fig. 2.5, where Trepresents any transformation of the density matrix data, whether or not thistransformation corresponds to a physically realizable transformation of quan-tum states. A further interesting application is to the purication of states.In this problem it is assumed that the input states were once pure, but werelater corrupted in some noisy environment (the same for all inputs). The taskis to reconstruct the original pure states. Usually, the noise corresponds to aninvertible linear transformation of the density matrices, but its inverse is nota possible operation, because it transforms some density matrices to opera-


tors with negative eigenvalues. So the reversal of noise is not possible with aone-shot device, but is easy to perform to high accuracy when many equallyprepared inputs are available. In the simplest case of a so-called depolarizingchannel, this problem is well understood [78]; it is also well understood in theversion requiring many outputs, as in the optimal-cloning problem [79].

2.4.2 Quantum Cryptography

It may seem impossible to nd applications of impossible machines. But thatis not quite true: sometimes the impossibility of a certain task is preciselywhat is called for in an application. A case in point is cryptography: hereone tries to make the deciphering of a code impossible. So if we could designa code whose breaking would require one of the machines described in theprevious section, we could guarantee its security with the certainty of naturallaw. This is precisely what quantum cryptography sets out to do. Becauseonly small quantum systems are involved it is one of the easiest applica-tions of quantum information ideas, and was indeed the rst to be realizedexperimentally. For a detailed description we refer to Chap. 3. Here we justdescribe in what sense it is the application of an impossible machine.

As always in cryptography, the basic situation is that two parties, Aliceand Bob, say, want to communicate without giving an evil eavesdropper,conventionally named Eve, a chance to listen in. What classical eavesdrop-pers do is to tap the transmission line, make a copy of what they hear forlater analysis, and otherwise let the signal pass undisturbed to the legitimatereceiver (Bob). But if the signal is quantum, the no-cloning theorem tellsus that faithful copying is impossible. So either Eves copy or Bobs copy iscorrupted. In the rst case Eve wont learn anything, and hence there wasno eavesdropping anyway. In the second case Bob will know that somethingmay have gone wrong, and will tell Alice that they must discard that partof the secret key which they were exchanging. Of course, intermediate situa-tions are possible, and one has to show very carefully that there is an exacttrade-o between the amount of information Eve can get and the amount ofperturbation she must inict on the channel.

2.4.3 Entanglement-Assisted Teleportation

This is arguably the rst major discovery in the eld of quantum informa-tion. The no-cloning and no-teleportation theorems, although they had notbeen formulated in such terms, would hardly have come as a surprise to peo-ple working on the foundations of quantum mechanics in the 1960s, say. Butentanglement assistance was really an unexpected turn. It was rst seen byBennett et al. [52], who also coined the term teleportation. It is gratifyingto see, though it is hardly a surprise on the same scale, that this prediction ofquantum mechanics has also been implemented experimentally. The experi-ments are another interesting story, which will no doubt be told much better


M P

S

Alice Bob1 qubit 2 bit 1 qubit

1 ebit

Fig. 2.6. Entanglement-assisted teleportation

in Chap. 3 by Weinfurter and Zeilinger, who represent one team in which amajor breakthrough in this regard was achieved.

The teleportation scheme is shown in Fig. 2.6. What makes it so surpris-ing is that it combines two machines whose impossibility was discussed in theprevious section: omitting the distribution of the entangled state (the lowerhalf of Fig. 2.6), we get the impossible process of classical teleportation. Onthe other hand, if we omit the classical channel, we get an attempt to transmitinformation by means of correlations alone, i.e. a version of Bells Telephone.Since the time dimension is not represented in this diagram, let us considerthe steps in the proper order. The rst step is that Alice and Bob each receiveone half of an entangled system. The source can be a third party or can beBobs lab. The last choice is maybe best for illustrative purposes, because itmakes clear that no information is owing from Alice to Bob at this stage.Alice is next given the quantum system whose state (which is unknown toher) she is to teleport. Alice then makes a measurement on a system madeby combining the input and her half of the entangled system. She sends theresults via a classical channel to Bob, who uses them to adjust the settingson his device, which then performs some unitary transformation on his halfof the entangled system. The resulting system is the output, and if every-thing is chosen in the right way, these output systems are indeed statisticallyindistinguishable from the outputs. To see just how the entangled state S,the measurement M and the repreparation P have to be chosen requires themathematical framework of quantum theory. In the standard example oneteleports a state of one qubit, using up one maximally entangled two-qubitsystem (in the current jargon, 1 ebit) and sending two classical bits fromAlice to Bob. A general characterization of the teleportation schemes forqubits and higher-dimensional systems is given below in Sect. 2.6.6.


2.4.4 Superdense Coding

It is easy to see and in fact is a commonplace occurrence that classical in-formation can be transmitted on quantum channels. For example, one bit ofclassical information can be coded in any two-level system, such as the polar-ization degree of freedom of a photon. It is not entirely trivial to prove, buthardly surprising that one cannot do better than one bit per qubit. Can webeat this bound using the idea of entanglement assistance? It turns out thatone can. In fact, one can double the amount of classical information carriedby a quantum channel (two bits per qubit). Remarkably, the setups for do-ing this are closely related to teleportation schemes, and in the simplest casesAlice and Bob just have to swap their equipment for entanglement assistedteleportation. This is explained in detail in Sect. 2.6.6.

2.4.5 Quantum Computation

Again, we shall be very brief on this subject, although it is certainly centralto the eld. After all, it is partly the promise of a fantastic new class ofcomputers which has boosted the interest in quantum information in recentyears. But since in this book computation is covered in Chap. 4, we shall onlymake a few remarks connecting this subject to the theme of possible versusimpossible machines.

So can quantum computers perform otherwise impossible tasks? Not re-ally, because in principle we can solve the dynamical equations of quantummechanics on a classical computer and simulate all the results. Hence classi-cally unsolvable problems such as the halting problem for Turing machinesand the word problem in group theory cannot be solved on quantum com-puters either. But this argument only shows the possibility of emulating allquantum computations on a classical computer, and omits the possibilitythat the eciency of this procedure may be terrible. The great promise ofquantum computation lies therefore in the reduction of running time, fromexponential to polynomial time in the case of Shors factorization algorithm[80]. This reduction is comparable to replacing the task of counting all theway up to a 137 digit number by just having to write it. No matter what theconstants are in the growth laws for the computing time (and they will prob-ably not be very favorable for the quantum contestant), the polynomial timeis going to win if we are really interested in factoring very large numbers.

A word of caution is necessary here concerning the impossible/possibledistinction. While it is true that no polynomial-time classical factoring algo-rithm is known, and this is what counts from a practical point of view, thereis no proof that no such algorithm exists. This is a typical state of aairsin complexity theory, because the nonexistence of an algorithm is a state-ment about the rather unwieldy set of all Turing machine programs. A proofby inspecting all of them is obviously out, so it would have to be based on


some principle of conservation of diculties, which rarely exists for real-life problems. One problem in which this is possible is that of identifyingwhich (unique) element of a large list has a certain property (needle in ahaystack). In this case the obvious strategy of inspecting every element inturn can be shown to be the optimal classical one, and has a running timeproportional to the length N of the list. But Grovers quantum algorithm[58] performs the task in the order of

N steps, an amazing gain even if it

is not exponential. Hence there are problems for which quantum computersare provably faster than any classical computer.

So what makes the reduction of running time work? This is not so easy toanswer, even after working through Shors algorithm and verifying the claimof exponential speedup. Massive entanglement is used in the algorithm, sothis is certainly one important element. Then there is a technique known asquantum parallelism, in which a quantum computation is run on a coherentsuperposition of all possible classical inputs, and, in a sense, all values of afunction are computed simultaneously. A catchy paraphrase, attributed toD. Deutsch, is to call this a computation in the parallel worlds of the many-worlds interpretation.

But perhaps the best way to nd out what powers quantum computationis to to turn it around and to really try the classical emulation. The practi-cal diculty which then becomes apparent immediately is that the Hilbertspace dimensions grow extremely fast. For N qubits (two-level systems), onehas to operate in a Hilbert space of 2N dimensions. The corresponding spaceof density matrices has 22N dimensions. For classical bits one has instead aconguration space of 2N discrete points, and the analogue of the density ma-trices, the probability densities, live in a merely 2N -dimensional space. Bruteforce simulations of the whole system therefore tend to grind to a halt even onfairly small systems. Feynman was the rst to turn this around: maybe onlya quantum system can be used to simulate a quantum system, and maybe,while we are at it, we can go beyond simulation and do some interestingcomputations as well. So, putting it positively, in a quantum system we haveexponentially more dimensions to work with: there is lots of room in Hilbertspace. The added complexity of quantum versus classical correlations, i.e. thephenomenon of entanglement, is also a consequence of this.

But it is not so easy to use those extra dimensions. For example, fortransmission of classical information an N -qubit system is no better thana classical N -bit system. Only the entanglement assistance of superdensecoding brings out the additional dimensions. Similarly, quantum computersdo not speed up every computation, but are good only at specic tasks wherethe extra dimensions can be brought into play.

2.4.6 Error Correction

Again, we shall only make a few remarks related to the possible/impossibletheme, and refer the reader to Chap. 4 for a deeper discussion. First of all,


error correction is absolutely crucial for the implementation of quantum com-puters. Very early in the development of the subject the suspicion was raisedthat exponential speedup was only possible if all component parts of the com-puter were realized with exponentially high (and hence practically unattain-able) precision.

In a classical computer the solution to this problem is digitization: everybit is realized by a bistable circuit, and any deviation from the two wantedstates is restored by the circuit at the expense of some energy and with someheat generation. This works separately for every bit, so in a sense every bithas its own heat bath. But this strategy will not work for quantum comput-ers: to begin with, there is now a continuum of pure states which would haveto be stabilized for every qubit, and, secondly, one heat bath per qubit wouldquickly destroy entanglement and hence make the quantum computation im-possible. There are many indications that entanglement is indeed more easilydestroyed by thermal noise and other sources of errors; this is summarily re-ferred to as decoherence. For example, a Gaussian channel (this is a specialtype of innite-dimensional channel) has innite capacity for classical infor-mation, no matter how much noise we add. But its quantum capacity dropsto zero if we add more classical noise than that specied by the Heisenberguncertainty relations [81].

A standard technique for stabilizing classical information is redundancy:just send a classical bit three times, and decide at the end by majority votewhich bit to take. It is easy to see that this reduces the probability of errorfrom order to order 2. But quantum mechanically, this procedure is for-bidden by the no-cloning theorem: we simply cannot make three copies tostart the process.

Fortunately, quantum error correction is possible in spite of all thesedoubts [82]. Like classical error correction, it also works by distributing thequantum information over several parallel channels, but it does this in a muchmore subtle way than copying. Using ve parallel channels, one can obtain asimilar reduction of errors from order to order 2 [63]. Much more has beendone, but many open questions remain, for which I refer once again to Chap.4.

2.5 A Preview of the Quantum Theory of Information

Before we go on in the next section to turn some of the heuristic descriptionsof the previous sections into rigorous mathematical statements, I shall tryto give a avor of the theory to be constructed, and of its motivations andcurrent state of development.

Theoretical physics contributes to the eld of quantum information pro-cessing in two distinct, though interrelated ways. One of these ways is theconstruction of theoretical models of the systems which are being set up ex-perimentally as candidates for quantum devices. Of course, any such system


will have very many degrees of freedom, of which only very few are singledout as the qubits on which the quantum computation is performed. Henceit is necessary to analyze to what degree and on what timescales it is justi-ed to treat the qubit degrees of freedom separately, and with what errorsthe desired quantum operation can be realized in the given system. Thesequestions are crucial for the realization of any quantum devices, and requirespecialized in-depth knowledge of the appropriate theory, e.g. quantum op-tics, solid-state theory or quantum chemistry (in the case of NMR quantumcomputing). However, these problems are not what we want to look at in thischapter. The other way in which theoretical physics contributes to the eld ofquantum information processing is in the form of another kind of theoreticalwork, which could be called the abstract quantum theory of information.Recall the arguments in Sect. 2.2, where the possibility of translating be-tween dierent carriers of (classical) information was taken as the justica-tion for looking at an abstracted version, the classical theory of information,as founded by Shannon. While it is true that quantum information cannotbe translated into this framework, and is hence a new kind of information,translation is often possible (at least in principle) between dierent carriersof quantum information. Therefore, we can make a similar abstraction in thequantum case. To this abstract theory all qubits are the same, whether theyare realized as polarizations of photons, nuclear spins, excited states of ions ina trap, modes of a cavity electromagnetic eld or whatever other realizationmay be feasible. A large amount of work is currently being devoted to thisabstract branch of quantum information theory, so I shall list some of thereasons for this eort.

Abstract quantum theoretical reasoning is how it all started. In the earlypapers of Feynman and Deutsch, and in the papers by Bennett and co-workers, it is the structure of quantum theory itself which opens up allthese new possibilities. No hint from experiment and no particular the-oretical diculty in the description of concrete systems prompted thisdevelopment. Since the technical realizations are lagging behind so much,the eld will probably remain theory driven for some time to come.

If we want to transfer ideas from the classical theory of information tothe quantum theory, we shall always get abstract statements. This worksquite well for importing good questions. Unfortunately, however, the an-swers are most of the time not transferred so easily.

The reason for this diculty with importing classical results is that someof the standard probabilistic techniques, such as conditioning, do notwork in quantum theory, or work only sporadically. This is the sameproblem that the statistical mechanics of quantum many-particle sys-tems faces in comparison with its classical sister. The cure can only bethe development of new, genuinely quantum techniques. Preferably theseshould work in the widest (and hence most abstract) possible setting.


One of the fascinating aspects of quantum information is that featuresof quantum mechanics which were formerly seen only as paradoxical orcounterintuitive are now turned into an asset: these are precisely thefeatures one is trying to utilize now. But this means that naive intu-itive reasoning tends to lead to wrong results. Until we know much moreabout quantum information, we shall need rigorous guidance from a solidconceptual and mathematical foundation of the theory.

When we take as a selling point for, say, quantum cryptography thatsecrets are protected with the security of natural law, the argument isonly as convincing as the proof that reduces this claim to rst principles.Clearly this requires abstract reasoning, because it must be independentof the physical implementation of the device the eavesdropper uses. Theargument must also be completely rigorous in the mathematical sense.

Because it does not care about the physical realization of its qubits,the abstract quantum theory of information is applicable to a wide rangeof seemingly very dierent systems. Consider, for example, some abstractquantum gate like the controlled not (C-NOT). From the abstract the-ory, we can hope to obtain relevant quality criteria, such as the minimaldelity with which this device has to be implemented for some algorithmto work. So systems of quite dierent types can be checked according tothe same set of criteria, and a direct competition becomes possible (andinteresting) between dierent branches of experimental physics.

So what will be the basic concepts and features of the emerging quantumtheory of information? The information-theoretical perspective typically gen-erates questions like

How can a given task of quantum information processing be performedoptimally with the given resources?

We have already seen a few typical tasks of quantum information process-ing in the previous section and, of course, there are more. Typical resourcesrequired for cryptography, quantum teleportation and dense coding are en-tangled states, quantum channels and classical channels. In error correctionand computing tasks, the resources are the size of the quantum memoryand the number of quantum operations. Hence all these notions take on aquantitative meaning.

For example, in entanglement-assisted teleportation the entangled pairsare used up (one maximally entangled qubit pair is needed for every qubitteleported). If we try to run this process with less than maximally entangledstates, we may still ask how many pairs from a given preparation device areneeded per qubit to teleport a message of many qubits, say, with an error lessthan . This quantity is clearly a measure of entanglement. But other tasksmay lead to dierent quantitative measures of entanglement. Very often it ispossible to nd inequalities between dierent measures of entanglement, andestablishing these inequalities is again a task of quantum information theory.


The direct denition of an entanglement measure based on teleportation,of the quantum information capacity of a channel and of many similar quan-tities requires an optimization with respect to all codings and decodings ofasymptotically long quantum messages, which is extremely hard to evaluate.In the classical case, however, there is a simple formula for the capacity of anoisy channel, called Shannons coding theorem, which allows us to computethe capacity directly from the transition probabilities of a channel. Findingquantum analogues of the coding theorem (and similar formulas for entan-glement resources) is still one of the great challenges in quantum informationtheory.

2.6 Elements of Quantum Information Theory

It is probably too early to write a denitive account of quantum informationtheory there are simply too many open questions. But the basic conceptsare clear enough, and it will be the task of the remainder of this chapter toexplain them, and use the precise denitions to state some of the interestingopen questions in the eld. In the limited space available this cannot be donein textbook style, with many examples and full proofs of all the things usedon the way (or even full references of them). So I shall try to emphasize themain lines and to set up the basic denitions using as few primitive conceptsas possible. For example, the capacities of a channel for either classical orquantum information will be dened on exactly the same pattern. This willmake it easier to establish the relations between these concepts.

The following sections begin with material which every physicist knowsfrom quantum mechanics courses, although maybe not in this form. We needto go over this material, though, in order to establish the notation.

2.6.1 Systems and States

The systems occurring in the theory can be either quantum or classical, orcan be hybrids composed of a classical and a quantum part. Therefore, weneed a mathematical framework covering all these cases. A good choice is tocharacterize each type of system by its algebra of observables. In this chapterall algebras of observables will be taken to be nite-dimensional for simplicity.Extensions to innite dimensionality are mostly straightforward, though, andin fact a strength of the algebraic approach to quantum theory is that it dealsnot just with innite-dimensional algebras, but also with systems of innitelymany degrees of freedom, as in quantum eld theory [83, 84] and statisticalmechanics [85].

The rst main type of system consists of purely classical systems, whosealgebra of observables is commutative, and can hence be considered as a spaceof complex-valued functions on a setX . Our assumption of niteness requires


that X is a nite set, and the algebra of observables A will be C(X), the spaceof all functions f : X C. A single classical bit corresponds to the choiceX = {0, 1}. On the other hand, a purely quantum system is determined by thechoice A = B(H), the algebra of all bounded linear operators on the Hilbertspace H. The niteness assumption requires that H has a nite dimensiond, so A is just the space Md of complex d d matrices. A qubit is given byA =M2.

The basic statistical interpretation of the algebra of observables is thesame in the quantum and classical cases, and hinges on the cone of positiveelements in the algebra. Here Y is called positive (in symbols, Y 0) if itcan be written in the form Y = XX . Then Y Md is positive exactlyif it is given by a positive semidenite matrix, and f C(X) is positive if(x) 0 for all x. In any algebra of observables A, we shall denote by 1I Athe identity element.

A state on A is a positive normalized linear functional on A. That is, : A C is linear, with (XX) 0 and (1I) = 1. Each state describesa way of preparing systems, in all the details that are relevant to subsequentstatistical measurements on the systems. The measurements are described byassigning to each outcome from a device an eect F A, i.e. an element with0 F 1I. The prediction of the theory for the probability of that outcome,measured on systems prepared according to the state , is then (F ).

For explicit computations we shall often need to expand states and ele-ments of A in a basis. The standard basis in C(X) consists of the functionsex, x X , such that ex(y) = 1 for x = y and zero otherwise. Similarly,if H is an orthonormal basis of the Hilbert space of a quantum sys-tem, we denote by e = |ee | B(H) the corresponding matrix units.Then a state p on the classical algebra C(X) is characterized by the numberspx p(ex), which form a probability distribution on X , i.e. p(x) 0 and

x p(x) = 1. Similarly, a quantum state on B(H) is given by the numbers (e), which form the so-called density matrix. If we interpret themas the expansion coecients of an operator =

e , the density

operator of , we can also write (A) = tr(A).A state is called pure if it is extremal in the convex set of all states, i.e.

if it cannot be written as a convex combination + (1 ) of otherstates. These are the states which contain as little randomness as possible.In the classical case, the only pure states are those concentrated on a singlepoint z X , i.e. pz = 1, or p(f) = f(z). The pure states in the quantumcase are determined by wave vectors H such that (A) = ,A,and = ||. Thus, in the simplest case of a classical bit, there are justtwo extreme points, whereas in the case of a qubit the extreme points forma sphere in three dimensions and are given by the expectations of the three


Fig. 2.7. State spaces as convex sets: left, one classical bit; right, one quantum bit(qubit)

Pauli matrices:

=12

(1 + x3 x1 ix2x1 + ix2 1 x3

)=

12(1I + x) ,

xk = (k) . (2.5)

Then positivity requires |x|2 1, with equality when is pure. This is shownin Fig. 2.7.

Thus, in addition to the north pole |1 and the south pole |0, whichroughly correspond to the extremal states of the classical bit, we have theircoherent superpositions corresponding to the wave vectors |1+|0, where, C, and ||2 + ||2 = 1. This additional freedom becomes even moredramatic in higher-dimensional systems, and is crucial for the possibility ofentanglement.

Entanglement is a property of states of composite systems, so we mustintroduce the notion of composition of systems. We shall dene this in away which applies to classical and quantum systems alike. If A and B arethe algebras of observables of the subsystems, the algebra of observables ofthe composition is dened to be the tensor product A B. In the nite-dimensional case, which is our main concern, this is dened as the space oflinear combinations of elements that can be written as A B with A Aand B B, such that A B is linear in A and linear in B. The algebraicoperations are dened by (A B) = A B, and (A1 B1)(A2 B2) =(A1A2) (B1B2). Thus 1I = 1IA1IB. Since positivity is dened in terms of astar operation (adjoint) and a product, these denitions also determine thestates and eects of the composite system.

Let us explore how this unies the more common denitions in the clas-sical and quantum cases. For two classical factors C(X) C(Y ), a basis isformed by the elements ex ey, so the general element can be expanded as

f =x,y

f(x, y)ex ey ,


and each element can be identied with a function on the Cartesian productX Y . Hence C(X) C(Y ) = C(X Y ). Similarly, in the purely quantumcase, we can expand in matrix units and obtain quantities with four indices:(AB), = AB . In a basis-free way, i.e. when A,B are consideredas operators on Hilbert spaces HA,HB, this is dened by the equation

(AB)( ) = (A) (B) ,where HA and HB, and the tensor product of the Hilbert spaces isformed in the usual way. Hence B(HA) B(HB) = B(HA HB).

But the denition of a composition by a tensor product of algebras of ob-servables also determines how a quantumclassical hybrid must be described.Such systems occur frequently in quantum information theory, whenever acombination of classical and quantum information is given. We shall approachhybrids in two equivalent ways, which are also useful more generally. Supposewe know only that the rst subsystem is classical and make no assumptionsabout the nature of the second, i.e. we want to characterize tensor prod-ucts of the form C(X) B. Then every element can be expanded in theform B =

x ex Bx, where now Bx B. Clearly, the elements Bx de-

termine B, and hence we can identify the tensor product with the space(sometimes denoted by C(X ;B)) of B-valued functions on X with pointwisealgebraic operations. Similarly, suppose that we know only that B = Md isthe algebra of d d matrices. Then, expanding in matrix units, we nd thatA =

A e with A A. That is, we can identify A Md with

the space (sometimes denoted byMd(A)) of dd matrices with entries fromA. By using the relation ee = e , one can readily verify that theproduct in A Md indeed corresponds to the usual matrix multiplicationin Md(A), with due care given to the order of factors in products with ele-ments from A, if A happens to be noncommutative. The adjoint is given by(A) = (A). Hence a hybrid algebra C(X) Md can be viewed eitheras the algebra of C(X)-valued d d matrices or as the space of Md-valuedfunctions on X .

The physical interpretation of a composite system AB in terms of statesand eects is straightforward. When F A and G B are eects, so is FG,and this is interpreted as the joint measurement of F on the rst subsystemand of G on the second subsystem, where the yes outcome is taken as botheects give yes. In particular, F1IB corresponds to measuring F on the rstsystem, completely ignoring the second. Thus, for any state on A B wedene the restriction A of to A by A(A) = (A1IB). In the classical case,the probability density for A is obtained by integrating out the B variables.In the quantum case, it corresponds to the partial trace of density matriceswith respect to HB. In general, it is not possible to reconstruct the state from the restrictions A and B, which is another way of saying that alsodescrib

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