1. University of Gdańsk, Poland
‘No-signaling in time’ as an alternative to the Leggett-Garg inequalities
Johannes Kofler
University of Gdańsk, Poland
Sept 17th 2013

2. With photons, electrons, neutrons, molecules etc.
Macroscopic superpositions
With photons, electrons, neutrons, molecules etc.
With cats?
|cat left + |cat right ?
6910 AMU
When and how do physical systems stop to behave quantum mechanically and begin to behave classically (“measurement problem”)?

3. Superconducting devices2
Candidates for macro-superpositions
Heavy molecules1
(position)
Superconducting devices2
(current)
Atomic gases3
(spin)
Nanomechanics4
(position, momentum)
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)

4. 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, (2013)

5. Local realism vs. macrorealism
Are non-classical correlations possible?
Are macroscopic superpositions possible?
Quantum mechanics says
“yes”
(use entanglement)
Quantum mechanics says
“yes”
(if you manage to defy decoherence)
Local realism (e.g. classical physics) says
“no”
(only classical correlations)
Macrorealism (e.g. classical physics, objective collapse models) says
“no”
(only classical temporal correlations)
Bell test
has given experimental answer in favor of quantum mechanics
Leggett-Garg test
can/will give experimental answer,
community still split
Practical relevance
qu. computation, qu. cryptography
Practical relevance
witnessing temporal qu. coherence

6. 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”:
A = ±1
B = ±1 a b
Local realism restricts correlations
Bell’s inequality (BI):
Quantum mechanics (QM):
1 J. F. Clauser and A. Shimony, Rep. Prog. Phys. 41, 1881 (1978)
2 J. S. Bell, Physics (New York) 1, 195 (1964)

7. 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

8. 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): A B t0 tA tB
Macrorealism restricts temporal correlations
Leggett-Garg inequality (LGI): Q Q Q Q ±1 t0 t1 t2 t3 t4
Quantum mechanics (QM):
1 A. J. Leggett and A. Garg, Phys. Rev. Lett. 54, 857 (1985)

9. Derivation of the Leggett-Garg inequality
Dichotomic quantity: Qi = 1
Temporal correlations Q Q Q Q ±1 t0 t1 t2 t3 t4
Q1Q2 + Q2Q3 + Q3Q4 – Q1Q4 = 2
LGI:
K := Q1Q2 + Q2Q3 + Q3Q4 – Q1Q4  2
KQM = 22 =
non-invasiveness
S := A1B1 + A2B1 + A2B2 – A1B2  2
BI:
SQM = 22 =
locality
LGI violation  “macrorealism per se” and/or
“non-invasive measurability” fail/es
Up to now: 1-to-1 correspondence between LR (BI) and MR (LGI)

10. 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.” A B t0 tA tB
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
 no need for LGI (the 1-to-1 correspondence breaks)
J. K. and Č. Brukner, PRA 87, (2013)

11. Violation of no-signaling in time
NSIT demands “classical probability paths”: ! A B B t0 tA tB t0 tB
Violation of NSIT due to quantum mechanical interference terms
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, (2013)

12. 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)

13. Locality vs. non-invasiveness
How to enforce locality?
How to enforce non-invasiveness?
Space-like separation
Special relativity guarantees impossibility of physical influence
Ideal negative measurements
Taking only those results where no interaction with the object took place –1 +1 ? ? –1 +1
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

14. Double slit experiment
x = d/2 x t t0 t1 t2 x 
NSIT is violated due to interference terms
LGI impossible to construct
I Both slits open:
fringes
II Block lower slit at x = –d/2:
III Block upper slit at x = +d/2:
no fringes
II,III: ideal negative measurements
Picture: N. Bohr, in Quantum Theory and Measurement, eds. J. A. Wheeler and W. H. Zurek,
Princeton University Press (1983)

15. violated in specific parameter regimes
Mach-Zehnder interferometer
LGI
NSIT
violated in specific parameter regimes
violated up to measure 0
J. K. and Č. Brukner, PRA 87, (2013)

16. or no-signaling in time
The quantum-to-classical transition
decoherence or
or no-signaling in time
LGI and NSIT: tools for witnessing temporal quantum coherence in complex systems
J. K. and Č. Brukner, PRL 99, (2007)
J. K. and Č. Brukner, PRL 101, (2008)

17. Conclusion LR MR BI NS LGI NSIT QM QM BI necessary for LR tests
NS “useless”
LGI not essential for MR tests
alternative: NSIT