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Faculty of Physics,
University of Vienna, Austria
Institute for Quantum Optics and Quantum Information
Austrian Academy of Sciences
Macrorealism,
the freedom-of-choice loophole,
and an EPR-type BEC experiment
Johannes Kofler
Max Planck Institute for Quantum Optics
Garching, July 12th 2011
Double slit experiment
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?
Macroscopic superpositions
Two schools:
- Decoherence
uncontrollable interaction with environment;
within quantum physics
- Objective collapse models (GRW, Penrose, etc.)
forcing superpositions to decay;
altering quantum physics
Alternative answer:
- Coarse-grained measurements
measurement resolution is limited;
within quantum physics
A. Peres, Quantum Theory: Concepts and Methods, Kluver (2002)
Macrorealism
Leggett and Garg (1985):
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.”
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.”
Q(t1)
Q(t2)
t
t=0
t1
t2
The Leggett-Garg inequality
Dichotomic quantity:
t
t=0
t
Temporal correlations
t1
t2
t3
t4
All macrorealistic theories fulfill the
Leggett-Garg inequality
Violation  macrorealism per se or/and non-invasive measurability failes
Violation of the inequality
Rotating spin ½ particle
(eg. electron)
½
Rotating classical spin
vector (e.g. torque)
Precession around axis with frequency 
(through manetic field or external force)
Measurement along orthogonal axis
K > 2: Violation of the
Leggett-Garg inequality
K  2: Classical time
evolution, no violation
classical limit
Violation for arbitrary Hamiltonians
t
Initial state
t
t
State at later time t
t1 = 0 t2
Measurement
!
t3
?
?
Survival probability
Leggett-Garg inequality
classical limit
Choose

can be violated for any E
J. Kofler and Č. Brukner, PRL 101, 090403 (2008)
Why no violation in everyday life?
Coarse-grained measurements
Model system: Spin j
macroscopic: j ~ 1020
Arbitrary state:
- Measure Jz, outcomes: m = – j, –j+1, ..., +j (2j+1 levels)
- Assume measurement resolution is much weaker than the intrinsic
uncertainty such that
neighbouring outcomes are bunched
together into “slots” m.
m = –j
m=
m = +j
1
2
3
4
Example: Rotation of spin j
Sharp measurement
of spin z-component
Coarse-grained measurement
–j
1 3 5 7 ... Q = –1
–j
+j
+j
2 4 6 8 ... Q = +1
classical limit
Fuzzy measurement
Violation of Leggett-Garg inequality
for arbitrarily large spins j
Classical physics of a rotating
classical spin vector
J. Kofler and Č. Brukner, PRL 99, 180403 (2007)
Coarse-graining  Coarse-graining
Sharp parity measurement
Neighbouring coarse-graining
(two slots)
(many slots)
1 3 5 7 ...
2 4 6 8 ...
Slot 1 (odd)
Slot 2 (even)
Violation of
Leggett-Garg inequality
Note:
Classical physics
Superposition vs. mixture
To see the quantumness of a spin j, you need to resolve j1/2 levels
Non-classical Hamiltonians
Hamiltonian:
Produces oscillating Schrödinger cat state:
Under fuzzy measurements it appears as a
statistical mixture at every instance of time:
But the time evolution of this mixture cannot
be understood classically:
time
J. Kofler and Č. Brukner, PRL 101, 090403 (2008)
Non-classical Hamiltonians are complex
Oscillating Schrödinger cat
Rotation in real space
“non-classical” rotation in Hilbert space
“classical”
Complexity is estimated by number of sequential
local operations and two-qubit manipulations
Simulate a small time interval t
O(N) sequential steps
1 single computation step
all N rotations can be done simultaneously
Monitoring by an environment
Exponential decay of survival probability
- Leggett-Garg inequality is fulfilled (despite the non-classical Hamiltonian)
- However: Decoherence cannot account for a continuous spatiotemporal
description of the spin system in terms of classical laws of motion.
- Classical physics: differential equations for observable quantitites (real space)
- Quantum mechanics: differential equation for state vector (Hilbert space)
Relation quantum-classical
A brief history of hidden variables
Quantum mechanics and realism
1927 Kopenhagen interpretation
(Bohr, Heisenberg)
1932 von Neumann’s (wrong) proof of
non-possibility of hidden variables
1935 Einstein-Podolsky-Rosen paradox
1952 De Broglie-Bohm (nonlocal)
hidden variable theory
1964 Bell’s theorem on local hidden
variables
1972 First successful Bell test
(Freedman & Clauser)
Bohr and Einstein, 1925
Bell’s assumptions
λ
Realism:
[J. F. Clauser & A. Shimony, Rep. Prog. Phys. 41, 1881 (1978)]
Hidden variables λ determine outcome probabilities: p(A,B|a,b,λ)
[J. S. Bell, Physics 1,
195 (1964)]
Locality: (OI) Outcome Independence: p(A|a,b,B,λ) = p(A|a,b,λ) & vice versa
(SI) Setting Independence:
p(A|a,b,λ) = p(A|a,λ)
& vice versa
[J. S. Bell, Speakable and Unspeakable in
Quantum Mechanics, p. 243 (2004)]
Freedom of Choice:
(FC) p(a,b|λ) = p(a,b)  p(λ|a,b) = p(λ)
Bell’s theorem
Realism + Locality + Freedom of Choice  Bell‘s Inequality
CHSH form: |E(a1,b2) + E(a2,b1) + E(a2,b1) - E(a2,b2)|  2
The original Bell paper (1964) implicitly assumes freedom of choice:
explicitly: A(a,b,B,λ)
locality (outcome and setting independence)
freedom of choice
implicitly: (λ|a,b) A(a,λ) B(b,λ) – (λ|a,c) A(a,λ) B(c,λ)
Loopholes
Locality loophole:
There may be a communication from the setting or outcome on one side to the
outcome on the other side
Closed by Aspect et al., PRL 49, 1804 (1982) & Weihs et al., PRL 81, 5039 (1998)
Fair-sampling loophole:
The measured events stem from an unrepresentative subensemble
Closed by Rowe et al., Nature 409, 791 (2001)
Freedom-of-choice loophole:
The setting choices may be correlated with the hidden variables
Closed by Scheidl et al., PNAS 107, 10908 (2010) [this talk]
Geography
Space-time diagram
t
l
l
l
l
l
B
l
l
A
l
b
a
x
E
La Palma
Tenerife
Locality:
Freedom of choice:
A is space-like separated from B (OI) and b (SI)
a and b are random and
B is space-like separated from A (OI) and a (SI)
space-like separated from E
Geographic details
Tenerife
La Palma
144 km free-space link
144 km free-space link
NOT
Source
6 km fiber channel
Alice
OGS
1.2 km RF link
QRNG
Bob
QRNG
Experimental results
Coincidence rate detected: 8 Hz
Measurement time: 2400 s
Number of total detected coincidences: 19200
Polarizer settings a, b
0°, 22.5°
0, 67.5°
45°, 22.5°
45°, 67.5°
Correlation E(a,b)
0.62 ± 0.01 0.63 ± 0.01 0.55 ± 0.01 –0.57 ± 0.01
Obtained Bell value Sexp
2.37 ± 0.02
T. Scheidl, R. Ursin, J. Kofler, S. Ramelow, X. Ma, T. Herbst, L. Ratschbacher, A. Fedrizzi,
N. Langford, T. Jennewein, and A. Zeilinger, PNAS 107, 19708 (2010)
Important remarks
•
In a fully deterministic world, neither the locality nor the freedom-ofchoice loophole can be closed:
Setting choices would be predetermined and could not be space-like
separated from the outcome at the other side (locality) or the particle
pair emission (freedom-of-choice).
•
Thus, we need to assume stochastic local realism:
There, setting choices can be created randomly at specific points in
space-time.
•
We have to consistently argue within local realism:
The QRNG is the best candidate for producing stochastic settings.
•
Practical importance:
freedom of choice can be seen as a resource for device-independent
cryptography and randomness generation/amplification
Acknowledgments
Thomas Scheidl
Rupert Ursin
Lothar Ratschbacher Alessandro Fedrizzi
Sven Ramelow
Xiao-Song Ma
Thomas Herbst
Nathan Langford
Thomas Jennewein
Anton Zeilinger
Colliding BECs
Cigar-shaped BEC of metastable He4 (high
internal energy)
Three laser beams kick the atoms:
Recoil velocity:
Two freely falling species are produced and
undergo s-wave scattering
Momentum-entangled particle pairs are
produced, lying on a shell in velocity space:
A. Perrin, H. Chang, V. Krachmalnicoff, M. Schellekens, D. Boiron,
A. Aspect, and C. I. Westbrook, PRL 99, 150405 (2007)
Proposal: The double double slit
If the condensate is too small, there is a
product of one-particle interference patterns:
If the condensate is sufficiently large, one
obtains two-particle interference (conditional
interference fringes):
Experimental conditions
(I)
Sufficiently large source size Sx to achieve well defined momentum correlation
(px  Sx–1) and wash out the single-particle interference pattern:
(II) Sufficiently small source to not wash out the two-particle interference pattern:
(III) Resolution of interference fringes:
(IV) Ability to identify pairs, i.e. coincidences:
In preparation (2011)
Two-particle interference
In preparation (2011)
Acknowledgments
Michael Keller
Maximilian Ebner
Mateusz Kotyrba
Mandip Singh
Anton Zeilinger
Summary
•
Coarse-grained measurements are a way to
understand the quantum-to-classical
transition (complementary to decoherence)
•
We simultaneously closed the locality
and the freedom-of-choice loophole; a
loophole-free Bell test is still missing
•
Proposal: A BEC double double slit
experiment can show EPR-type
entanglement of massive particles
Thank you for your attention!
Appendix
Macrorealism per se
Probability for outcome m can be computed
from an ensemble of classical spins with
positive probability distribution:
Coarse-grained measurements:
any quantum state allows a
classical description
This is macrorealism per se.
Experimental setup