Transcript ppt
‘No-signaling in time’ as an alternative
to the Leggett-Garg inequalities
Johannes Kofler
University of Gdańsk, Poland
Sept 17th 2013
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”)?
Candidates for macro-superpositions
1
Heavy molecules1
Superconducting devices2
(position)
(current)
Atomic gases3
Nanomechanics4
(spin)
(position, momentum)
S. Gerlich et al., Nature Comm. 2, 263 (2011)
3 B. Julsgaard et al., Nature 413, 400 (2001)
2
4
M. W. Johnson et al., Nature 473, 194 (2011)
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)
Local realism vs. macrorealism
Are non-classical correlations
possible?
Are macroscopic superpositions
possible?
Quantum mechanics says
Quantum mechanics says
“yes”
(use entanglement)
“yes”
(if you manage to defy decoherence)
Local realism (e.g. classical
physics) says
Macrorealism (e.g. classical physics,
objective collapse models) says
“no”
(only classical correlations)
“no”
(only classical temporal correlations)
Bell test
Leggett-Garg test
has given experimental answer
in favor of quantum mechanics
can/will give experimental answer,
community still split
Practical relevance
Practical relevance
qu. computation, qu. cryptography
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”:
• Local realism restricts correlations
Bell’s inequality (BI):
• Quantum mechanics (QM):
1
2
J. F. Clauser and A. Shimony, Rep. Prog. Phys. 41, 1881 (1978)
J. S. Bell, Physics (New York) 1, 195 (1964)
A = ±1
B = ±1
a
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):
t0
• Macrorealism restricts temporal correlations
Leggett-Garg inequality (LGI):
t0
• Quantum mechanics (QM):
1
A. J. Leggett and A. Garg, Phys. Rev. Lett. 54, 857 (1985)
A
B
tA
tB
Q
Q
Q Q ±1
t1
t2
t3
t4
Derivation of the Leggett-Garg inequality
Dichotomic quantity: Qi = 1
Temporal correlations
t0
Q
Q
Q Q ±1
t1
t2
t3
t4
Q1Q2 + Q2Q3 + Q3Q4 – Q1Q4 = 2
LGI:
K := Q1Q2 + Q2Q3 + Q3Q4 – Q1Q4 2
KQM = 22
=
non-invasiveness
BI:
S := A1B1 + A2B1 + A2B2 – A1B2 2
SQM = 22
=
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)
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.”
t0
• 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, 052115 (2013)
A
B
tA
tB
Violation of no-signaling in time
• NSIT demands “classical probability paths”:
!
B
t0
tB
t0
A
B
tA
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, 052115 (2013)
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)
Locality vs. non-invasiveness
How to enforce locality?
How to enforce non-invasiveness?
Space-like separation
Ideal negative measurements
Special relativity guarantees
impossibility of physical influence
Taking only those results where no
interaction with the object took place
?
?
–1
+1
–1
+1
Bohmian mechanics
Bohmian mechanics
Space-like separation is of no
help: non-local influence on
hidden variable level
Ideal negative measurements are of
no help: wavefunction “collapse”
changes subsequent evolution
Realistic, non-local
Macrorealistic per se, invasive
Double slit experiment
x = d/2
x
t1
t2
t0
t
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)
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
decoherence or
J. K. and Č. Brukner, PRL 99, 180403 (2007)
J. K. and Č. Brukner, PRL 101, 090403 (2008)
or no-signaling in time
LGI and NSIT: tools for
witnessing temporal
quantum coherence in
complex systems
Conclusion
LR
BI
MR
NS
LGI
NSIT
QM
QM
BI necessary for LR tests
LGI not essential for MR tests
NS “useless”
alternative: NSIT