Quantum violation of macroscopic realism and the transition to classical physics

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Quantum violation of macroscopic realism and the
transition to classical physics
Faculty of Physics University of Vienna, Austria
Institute for Quantum Optics and Quantum
Information Austrian Academy of Sciences
Johannes Kofler
PhD Defense University of Vienna, Austria October
3rd, 2008
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List of publications

• Articles in refereed journals
• J. Kofler and C. Brukner
• Conditions for quantum violation of macroscopic
  realism
• Phys. Rev. Lett. 101, 090403 (2008)
• J. Kofler and C. Brukner
• Classical world arising out of quantum physics
  under the restriction of coarse-grained
  measurements
• Phys. Rev. Lett. 99, 180403 (2007)
• J. Kofler and C. Brukner
• Entanglement distribution revealed by
  macroscopic observations
• Phys. Rev. A 74, 050304(R) (2006)
• M. Lindenthal and J. Kofler
• Measuring the absolute photo detection
  efficiency using photon number correlations
• Appl. Opt. 45, 6059 (2006)
• J. Kofler, V. Vedral, M. S. Kim, and C. Brukner

• Submitted
• T. Paterek, R. Prevedel, J. Kofler, P. Klimek, M.
  Aspelmeyer, A. Zeilinger, and C. Brukner
• Mathemtical undecidability and quantum
  randomness
• Contributions in books
• J. Kofler and C. Brukner
• A coarse-grained Schrödinger cat
• Quantum Communication and Security, ed. M.
  Zukowski, S. Kilin, and J. Kowalik (IOS Press,
  2007)
• Proceedings
• R. Ursin et. al.
• Space-QUEST Experiments with quantum
  entanglement in space
• 59th International Astronautical Congress (2008)
• Articles in popular journals

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Classical versus Quantum
Phase space Continuity Newtons laws Definite
states Determinism
Hilbert space Quantization, clicks Schrödinger
equation Superposition/Entanglement Randomness

• When and how do physical systems stop to behave
  quantum mechanically and begin to behave
  classically?
• What is the origin of quantum randomness?

Isaac Newton
Ludwig Boltzmann
Albert Einstein
Niels Bohr
Erwin Schrödinger
Werner Heisenberg
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Double slit experiment
With electrons! (or neutrons, molecules, photons,
)
With cats?
cat left? cat right? ?
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Why do we not seemacroscopic superpositions?

• Two schools
• Decoherence
• uncontrollable interaction with environment
  within quantum physics
• Collapse models
• forcing superpositions to decay altering
  quantum physics

• Alternative answer
• Coarse-grained measurements
• measurement resolution is limited within
  quantum physics

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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
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The Leggett-Garg inequality
?t
Dichotomic quantity Temporal correlations
t 0
t
t1
t2
t3
t4
All macrorealistic theories fulfill
the Leggett-Garg inequality
Violation ? macrorealism per se or/and
non-invasive measurability failes
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When is the Leggett-Garg inequality violated?
Rotating spin-½
Rotating classical spin
precession around x sign of z component
precession around x measurement along z
½
Violation of the Leggett-Garg inequality
Classical evolution
classical limit
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Why dont we see violations in everyday life?
Coarse-grained measurements
Model system Spin j macroscopic j 1020
Arbitrary state

• Measure Jz, outcomes m j, j1, ..., j
  (2j1 levels)

• Assume measurement resolution is much weaker than
  the intrinsic uncertainty such that
  neighbouring outcomes are bunched together
  into slots m.

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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.
J. K. and C. Brukner, PRL 99, 180403 (2007)
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Example Rotation of spin j
j
Coarse-grained measurement
Sharp measurement of spin z-component
j
j
Q 1
1 3 5 7 ...
j
j
Q 1
2 4 6 8 ...
classical limit
Fuzzy measurement
Classical physics of a rotating classical spin
vector
Violation of Leggett-Garg inequality for
arbitrarily large spins j
J. K. and C. Brukner, PRL 99, 180403 (2007)
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Coarse-graining ? Coarse-graining
Neighbouring coarse-graining (many slots)
Sharp parity measurement (two slots)
1 3 5 7 ...
2 4 6 8 ...
Slot 1 (odd)
Slot 2 (even)
Violation of Leggett-Garg inequality
Classical physics
Note
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Superposition versus Mixture
To see the quantumness of a spin j, you need to
resolve j1/2 levels!
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Albert Einstein and ...
Charlie Chaplin
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Non-invasive measurability
Depending on the outcome, measurement reduces
state to
Fuzzy measurements only reduce previous ignorance
about the spin mixture
For macrorealism we need more Total ensemble
without measurement should be the weighted
mixture of the evolved subensembles after a
measurement
Non-invasive measurability
t 0
t
tj
ti
t
J. K. and C. Brukner, PRL 101, 090403 (2008)
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The sufficient condition for macrorealism
The sufficient condition for macrorealism is
I.e. the statistical mixture has a classical time
evolution, if no superpositions of
macroscopically distinct states are produced.
Given coarse-grained measurements, it depends on
the Hamiltonian whether macrorealism is
satisfied.
Classical Hamiltonians eq. is fulfilled (e.g.
rotation) Non-classical Hamiltonians eq. not
fulfilled (e.g. osc. Schrödinger cat, next
slide)
J. K. and C. Brukner, PRL 101, 090403 (2008)
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Non-classical Hamiltonian (no macrorealism
despite of coarse-graining)
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
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Non-classical Hamiltonians are complex
Oscillating Schrödinger cat non-classical
rotation in Hilbert space
Rotation in real space classical
Complexity is estimated by number of sequential
local operations and two-qubit manipulations
Simulate a small time interval ?t
1 single computation step all N rotations can be
done simultaneously
O(N) sequential steps
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Relation quantum-classical
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The origin of quantum randomness
Determinism (subjective randomness due to
ignorance)? Objective randomness (no causal
reason)?
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Mathematical undecidability
Axioms Proposition true/false if it can be
proved/disproved from the axioms logically
independent or mathematically undecidable if
neither the proposition nor its negation leads to
an inconsistency (i) Euclids parallel
postulate in neutral geometry (ii) axiom of
choice in Zermelo-Fraenkel set
theory intuitively independent proposition
contains new information
Information-theoretical formulation of
undecidability (Chaitin 1982)
If a theorem contains more information than a
given set of axioms, then it is impossible for
the theorem to be derived from the axioms.
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Logical complementarity
Consider (Boolean) bit-to-bit function f(a) b
(with a 0,1 and b 0,1) (A) f(0)
0 (B) f(1) 0 (C) f(0) f(1) 0
logically complementary
Given any single 1-bit axiom, i.e. (A) or (B) or
(C), the two other propositions are undecidable.
Physical black box can encode the Boolean
function
f(a) 0
f(0) 1
f(1) 1
Example
qubit
a 1
a 0
f(0) 0
f(1) 1
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Mathematical undecidability and quantum randomness
x
Preparation Black box
Measurement Information gain
f(0)
(A)
z
f(1)
(B)
x
f(0) f(1)
(C)
y
However
Random outcomes! (B) is undecidable within axiom
(A)
x
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Experimental test of mathematical
undecidability1 qubit
A qubit carries only one bit of information
(Holevo 1973, Zeilinger 1999)
T. Paterek, R. Prevedel, J. K., P. Klimek, M.
Aspelmeyer, A. Zeilinger, C. Brukner, submitted
(2008)
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Generalization to many qubits
N qubits, N Boolean functions f1,,fN Black
box New feature Partial undecidability
T. Paterek, R. Prevedel, J. K., P. Klimek, M.
Aspelmeyer, A. Zeilinger, C. Brukner, submitted
(2008)
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Conclusions

• Quantum-to-classical transition under
  coarse-grained measurements

Quantum randomness a manifestation of
mathematical undecidability
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Thank you!
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Appendix
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Violation for arbitrary Hamiltonians
?t
?t
Initial state
t
State at later time t
t1 0
t2
t3
Measurement
?
?
!
Survival probability
Leggett-Garg inequality
classical limit
Choose
?
can be violated for any ?E
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Continuous 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)

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Experimental setups