Bell and Leggett-Garg inequalitiesin tests of local and macroscopic realism

University of Valencia, Spain25 June 2013

Johannes Kofler

Max Planck Institute of Quantum Optics (MPQ)Garching / Munich, Germany

Outlook

• Quantum entanglement vs. local realism

- Bell’s inequality

- Loopholes

- Entanglement swapping & teleportation

• Macroscopic quantum superpositions vs. macrorealism

- Leggett-Garg inequality

- Quantum-to-classical transition

- Witnessing non-classical evolutions in complex systems

• Conclusion and outlook

Local realism

• Realism: properties of physical objects exist independent of whether or not they are observed by anyone

• Locality: no physical influence can propagate faster than the speed of light

External world

Passive observers

Classical world view:

Bell’s inequality

Realism

*J. S. Bell, Phys. 1, 195 (1964); J. F. Clauser et al., PRL 23, 880 (1969)

a1,a2

B = ±1A = ±1

b1,b2

A1 (B1+B2) + A2 (B1–B2) = ±2

Local realism: A = A(a,,b,B)B = B(b,,a,A)

outcomes

settings

variables

S := A1B1 + A1B2 + A2B1 – A2B2 2 Bell’s inequality*

Quantum mechanics:

SQM = 22 2.83

First experimental violation: 1972Since then: tests with photons, atoms, superconducting qubits, …

using entangled quantum states, e.g.

Locality

|AB = (|HVAB + |VHAB) / 2

Alice Bob

Quantum entanglement

Entangled state:

|AB = (|AB + |AB) / 2= (|AB + |AB) / 2

BobAlice

locally: random

/: /: /: /: /: /: /: /:

/: /: /: /: /: /: /: /:

globally: perfect correlations

basis: result basis: result

Top picture: http://en.wikipedia.org/wiki/File:SPDC_figure.png

A1

A2B1

B2

Entanglement and knowledge

Erwin Schrödinger

“Total knowledge of a composite system does not necessarily include maximal knowledge of all its parts, not even when these are fully separated from each other and at the moment are not influencing each other at all.” (1935)

Loopholes

Why important?- Quantum foundations- Security of entanglement-based quantum cryptography

Three main loopholes:

• Locality loopholehidden communication between the partiesclosing: hard for atoms, achieved for photons (19821,19982)

• Freedom of choicesettings are correlated with hidden variables closing: hard for atoms, achieved for photons (20103)

• Fair samplingmeasured ensemble is not representativeclosing: achieved for atoms (20014) and photons (20135)

1 A. Aspect et al., PRL 49, 1804 (1982)2 G. Weihs et al., PRL 81, 5039 (1998)3 T. Scheidl et al., PNAS 107, 10908 (2010)

4 M. A. Rowe et al., Nature 409, 791 (2001)5 M. Giustina et al., Nature 497, 227 (2013)

Loopholes: maintain local realism despite Sexp > 2

E

Locality: A is space-like sep. from b and BB is space-like sep. from a and A

T. Scheidl, R. Ursin, J. K., T. Herbst, L. Ratschbacher, X. Ma, S. Ramelow, T. Jennewein, A. Zeilinger, PNAS 107, 10908 (2010)

Ensuring locality & freedom of choice

b,B

E,A

a

Tenerife

La Palma

Freedom of choice: a and b are randoma and b are space-like sep. from E

E

p(a,b|) = p(a,b)

p(A,B|a,b,) = p(A|a,) p(B|b,)

La Palma Tenerife

Sexp = 2.37 0.02

Ensuring fair sampling

Solution:

• very good detectors• Eberhard inequality*

- undetected (“u”) events in derivation- required detection efficiency only 2/3

0)()(),(),(),(),( 1122122111 --- Bo

Aooooooooo SSCCCCJ

From Topics in Applied Physics 99, 63-150 (2005)

* P. H. Eberhard, PRA 47, 747 (1993)

+1–1 Source

+1–1

local realism

Problem: detection efficiency could depend on settingsA = A(), B = B() Superconducting transition

edge sensors

First fair sampling of photons

M. Giustina, A. Mech, S. Ramelow, B. Wittmann, J. K., Jörn Beyer, A. Lita, B. Calkins, T. Gerrits, S. W. Nam, R. Ursin, A. Zeilinger, Nature 497, 227 (2013)

0)()(),(),(),(),( 1122122111 --- Bo

Aooooooooo SSCCCCJ

Detection efficiency 75%Violation by 70 standard deviations

local realism

quantum violation of local realism with fair sampling

Photon: only system for which all loop-holes are closed (not yet simultaneously)

Large distances

* M. Žukowski et al., PRL 71, 4287 (1993)

Bell-state measurement (BSM): Entanglement swapping

How to distribute entanglement over large distances?- qu. cryptography between Vienna and Paris- distributed quantum computation

Two answers:

- glass fibers & quantum repeaters- no fibers: free space

Quantum repeaters use entanglement swapping*

Delayed-choice entanglement swapping

Later measurement on photons 2 & 3 decides whether 1 & 4 were separable or entangled

Naïve class. interpretation would require influences into the past

X. Ma, S. Zotter, J. K., R. Ursin, T. Jennewein, Č. Brukner, A. Zeilinger, Nature Phys. 8, 479 (2012)

Temporal order does not matter in qu. mechanics

Quantum teleportation

Towards a world-wide “quantum internet”

X. Ma, T. Herbst, T. Scheidl, D. Wang, S. Kropatschek, W. Naylor, A. Mech, B. Wittmann, J. K., E. Anisimova, V. Makarov, T. Jennewein, R. Ursin, A. Zeilinger, Nature 489, 269 (2012)

The next step

ISS (350 to 400 km altitude)

Contents

• Quantum entanglement vs. local realism

- Bell’s inequality

- Loopholes

- Entanglement swapping & teleportation

• Macroscopic quantum superpositions vs. macrorealism

- Leggett-Garg inequality

- Quantum-to-classical transition

- Witnessing non-classical evolutions in complex systems

• Conclusion

The double slit experiment

Picture: http://www.blacklightpower.com/theory/DoubleSlit.shtml

Particles Waves Quanta

Superposition:| = |left + |right

With photons, electrons, neutrons, molecules etc.

With cats?

|cat left + |cat right ?

When and how do physical systems stop to behave quantum mechanically and begin to behave classically (“measurement problem”)?

Macroscopic superpositions

6910 AMU*

* S. Gerlich et al., Nature Comm. 2, 263 (2011)

Quantum mechanics says“yes”(if you manage to defy decoherence)

Are macroscopic superpositions possible?

Local realism vs. macrorealism

Quantum mechanics says“yes”(use entanglement)

Are “non-local” correlations possible?

Local realism (e.g. classical physics) says“no”(only classical correlations)

Bell testhas given experimental answer in favor of quantum mechanics

Macrorealism (e.g. classical physics, objective collapse models) says“no”(only classical temporal correlations)

Leggett-Garg testcan/will give experimental answer,community still split

Practical relevancequ. computation, qu. cryptography

Practical relevancewitnessing temporal qu. coherence

Macrorealism

• Macrorealism per se: given a set of macroscopically distinct states, a macroscopic object is at any given time in a

definite one of these states

• Non-invasive measurability: measurements reveal the state without any effect on the state itself or on the subsequent dynamics

• Leggett-Garg inequality (LGI)

A. J. Leggett and A. Garg, PRL 54, 857 (1985)

• Quantum mechanics:

t1 t2 t3 t4t0

Q Q Q Q ±1

S := A1B1 + A1B2 + A2B1 – A2B2 2

K := Q1Q2 + Q2Q3 + Q3Q4 – Q1Q4 2

Bell:

KQM = 22 2.83

locality

non-invasiveness=

=

time

½

Rotating spin ½ particle (eg. electron)

Rotating classical spin vector (eg. gyroscope)

K > 2: violation of Leggett-Garg inequality

K 2: no violation, classical time evolution

classical limit

Precession around an axis(via magnetic field or external force)

Measurments along different axis

Quantum vs. classical

22

classical limit

Sharp measurement of spin z-component

Violation of Leggett-Garg inequality for arbitrarily large spins j

Classical physics of a rotating classical spin vector

J. K. and Č. Brukner, PRL 99, 180403 (2007)

Spin j

1 3 5 7 ...

2 4 6 8 ...Q = +1

Q = –1–j +j –j +j

Coarse-grained measurement or decoherence

Sharp vs. coarse-grained measurements

macroscopically distinct states

Sharp measurements

Coarse-grained measurements or decoherence

Superposition vs. mixture

To see quantumness: need to resolve j1/2 levels & protect system from environment

J. K. and Č. Brukner, PRL 101, 090403 (2008)

Oscillating Schrödinger cat“non-classical” rotation in Hilbert space

Rotation in real space“classical”

N sequential steps per t1 single computation step per tall N rotations can be done simultaneously

Non-classical evolutions are complex

J. K. and Č. Brukner, PRL 101, 090403 (2008)

N elemen-tary spins ½

time time

“+” “+”

t t t t

Relation quantum-classical

Macroscopic candidates

Heavy molecules1

(position)

Nanomechanics4

(position, momentum)

Superconducting devices2

(current)

Atomic gases3

(spin)

1 S. Gerlich et al., Nature Comm. 2, 263 (2011) 3 B. Julsgaard et al., Nature 413, 400 (2001)2 M. W. Johnson et al., Nature 473, 194 (2011) 4 G. Cole et al., Nature Comm. 2, 231 (2011)

Alternative to Leggett-Garg inequality

• No-signaling in time (NSIT): “A measurement does not change the outcome statistics of a later measurement.”*

• MR NSITViolation of NSIT witnesses non-classical time evolution

• Advantages of NSIT compared to LGI:- Only two measurement times (simpler witness)- Violated for broader parameter regime (better witness)

• LGI and NSIT are tools for witnessing temporal quantum coherence in complex systems (not necessarily having macroscopic superpositions)

• Does quantum coherence give biological systems an evolutionary advantage?

tA tBt0

A B

* J. K. and Č. Brukner, PRA 87, 052115 (2013)

Candidates for quantum biology

Photosynthesis:Light harvesting in the FMO complex

M. Sarovar et al., Nature Phys. 6, 462 (2010)

Avian compass

electronic excitation (by sunlight) in antenna is transferred to reaction centerevidence for efficiency increase due to quantum coherent transport

radical pair mechanism proposedreaction products depend on earth magnetic field

N. Lambert et al., Nature Phys. 9, 10 (2013)

Conclusion and outlook

• Local realism- world view radically different from quantum mechanics- violated experimentally (Bell tests) by qu. entanglement- all loopholes are closed, but not yet simultaneously- loopholes relevant for qu. cryptography- long distance distribution of entanglement

• Macrorealism- related to the measurement problem (Schrödinger’s cat)- quantum mechanics predicts violation- quantum-to-classical transition- Leggett-Garg inequality (LGI) not yet violated for macroscopic objects; several candidates- no-signaling in time (NSIT) as an alternative- LGI and NSIT: tools for witnessing quantum time evolution in mesoscopic systems including biological organisms

Acknowledgments

Anton Zeilinger

Maximilan EbnerMarissa GiustinaThomas Herbst

Thomas JenneweinMichael Keller

Mateusz KotyrbaXiao-song Ma

Caslav Brukner

Alexandra MechSven RamelowThomas ScheidlMandip SinghRupert Ursin

Bernhard WittmannStefan Zotter

Ignacio Cirac

Appendix

Einstein vs. Bohr

Albert Einstein(1879–1955)

Niels Bohr(1885–1962)

What is nature? What can be saidabout nature?

Interpretations

Copenhagen interpretation quantum state (wave function) only describes probabilitiesobjects do not possess all properties prior to and independent of measurements (violating realism)individual events are irreducibly random

Bohmian mechanics quantum state is a real physical object and leads to an additional “force”particles move deterministically on trajectoriesposition is a hidden variable & there is a non-local influence (violating locality)individual events are only subjectively random

Many-worlds interpretation all possibilities are realizedparallel worlds

Entanglement from Bose-Einstein condensates

J. K., M. Singh, M. Ebner, M. Keller, M. Kotyrba, A. Zeilinger, PRA 86, 032115 (2012)

First entanglement of massive particles in external degree of freedom (momentum)

Picture: A. Perrin et al., PRL 99, 150405 (2007)

Ideal negative measurements

Taking only those results where no interaction with the object took place

How to enforce non-invasiveness?

Locality vs. non-invasiveness

Space-like separation

Special relativity guarantees impossibility of physical influence

How to enforce locality?

? ?

–1 +1

–1 +1

Stages towards violation of MR

• Quantum interference between macroscopically distinct states (QIMDS)does not necessarily establish the truth of quantum mechanics (QM)

• Leggett’s three stages of experiments:*

“Stage 1. One conducts circumstantial tests to check whether the relevant macroscopic variable appears to be obeying the prescriptions of QM.

Stage 2. One looks for direct evidence for QIMDS, in contexts where it does not (necessarily) exclude macrorealism.

Stage 3. One conducts an experiment which is explicitly designed so that if the results specified by QM are observed, macrorealism is thereby excluded.”

• However: step from stage 2 to 3 is straightforward via violation of NSIT

* A. J. Leggett, J. Phys.: Cond. Mat. 14, R415 (2002)