‘No-signaling in time’ as an alternative

to the Leggett-Garg inequalities

University of Gdańsk, Poland

Sept 17th 2013

Johannes Kofler

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

Candidates for macro-superpositions

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)

Historical development

• Bell’s inequality & local realism

- well developed research field

- important for quantum information technologies

- experiments exist (photons, atoms, superconducting qubits, …)

• Leggett-Garg inequality & macroscopic realism

- gained momentum in last years

- experiments approach regime of macroscopic quantum superpositions

- candidates: superconducting devices, heavy molecules, quantum-optical systems in combination with atomic gases or massive objects

- community still divided into two groups

• This talk1

- local realism vs. macrorealism

- alternative to the Leggett-Garg inequality

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

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-classical correlations possible?

Local realism (e.g. classical physics) says

“no”(only classical correlations)

Bell test

has given experimental answer in favor of quantum mechanics

Macrorealism (e.g. classical physics, objective collapse models) says

“no”(only classical temporal correlations)

Leggett-Garg test

can/will give experimental answer,community still split

Practical relevance

qu. computation, qu. cryptography

Practical relevance

witnessing temporal qu. coherence

Local realism

• Realism is a worldview ”according to which external reality is assumed to exist and have definite properties, whether or not they are observed by someone.”1

• Locality demands that ”if two measurements are made at places remote from one another the [setting of one measurement device] does not influence the result obtained with the other.”2

• Joint assumption: Local realism (LR) or “local causality”:

1 J. F. Clauser and A. Shimony, Rep. Prog. Phys. 41, 1881 (1978)2 J. S. Bell, Physics (New York) 1, 195 (1964)

• Local realism restricts correlationsBell’s inequality (BI):

• Quantum mechanics (QM):

a

B = ±1A = ±1

b

No-signaling

• Causality demands the no-signaling (NS) condition: “Bob’s outcome statistics do not depend on space-like separated events on Alice’s side.”

• All local realistic theories are no-signaling but not the opposite (e.g. quantum mechanics, PR boxes):

• Violation of NS implies violation of LR, but all reasonable theories (including QM) fulfill NS:

Bell inequalities necessary

Macrorealism

• Macrorealism per se: ” A macroscopic object which has available to it two or more macroscopically distinct states is at any given time in a definite one of those states.”1

• Non-invasive measurability: “It is possible in principle to determine which of these states the system is in without any effect on the state itself or on the subsequent system dynamics.”1

• Joint assumption: Macrorealism (MR):

1 A. J. Leggett and A. Garg, Phys. Rev. Lett. 54, 857 (1985)

• Macrorealism restricts temporal correlationsLeggett-Garg inequality (LGI):

• Quantum mechanics (QM):

t1 t2 t3 t4

tA tBt0

t0

A B

Q Q Q Q ±1

LGI violation “macrorealism per se” and/or

“non-invasive measurability” fail/es

Derivation of the Leggett-Garg inequality

S := A1B1 + A2B1 + A2B2 – A1B2 2

K := Q1Q2 + Q2Q3 + Q3Q4 – Q1Q4 2

BI:

locality

=

=

LGI:

Q1Q2 + Q2Q3 + Q3Q4 – Q1Q4 = 2

Dichotomic quantity: Qi = 1

Temporal correlations t1 t2 t3 t4t0

Q Q Q Q ±1

KQM = 22

SQM = 22

non-invasiveness

Up to now: 1-to-1 correspondence between LR (BI) and MR (LGI)

No-signaling in time

• In analogy to NS:

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

• All macrorealistic theories fulfill NSIT but not the opposite:

• Key difference between NS and NSIT:

NS cannot be violated due to causality BI necessary

NSIT can be violated according to quantum mechanics

tA tBt0

A B

no need for LGI (the 1-to-1 correspondence breaks)

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

Violation of no-signaling in time

• NSIT demands “classical probability paths”:

tA tBt0

A B

• Advantages of NSIT compared to LGI:

- Only two measurement times (“simpler witness”)

- Usually violated for broader parameter regimes (“better witness”)

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

!

• Violation of NSIT due to quantum mechanical interference terms

tBt0

B

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:1

“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.”

• Step from stage 2 to 3:

logically important but straightforward via violation of NSIT

just need to show that QIMDS vanishes when prior measurement is made

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

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?

Bohmian mechanics

Space-like separation is of no help: non-local influence on hidden variable level

Realistic, non-local

Bohmian mechanics

Ideal negative measurements are of no help: wavefunction “collapse” changes subsequent evolution

Macrorealistic per se, invasive

? ?

–1 +1

–1 +1

Double slit experiment

t1

Picture: N. Bohr, in Quantum Theory and Measurement, eds. J. A. Wheeler and W. H. Zurek,Princeton University Press (1983)

t2

II Block lower slit at x = –d/2:

III Block upper slit at x = +d/2:

t0

x = d/2 x

fringes

no fringes

II,III: ideal negative measurements

NSIT is violated due to interference terms

LGI impossible to construct

I Both slits open:

t

x

Mach-Zehnder interferometer

LGI NSIT

violated in specific parameter regimes violated up to measure 0

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

The quantum-to-classical transition

J. K. and Č. Brukner, PRL 101, 090403 (2008)J. K. and Č. Brukner, PRL 99, 180403 (2007)

decoherence or or no-signaling in time LGI and NSIT: tools for witnessing temporal quantum coherence in complex systems

Conclusion

LGI NSIT

MR

QM

BI NS

LR

QM

BI necessary for LR tests

NS “useless”

LGI not essential for MR tests

alternative: NSIT