Quantum, classical

1
Quantum, classical coarse-grained measurements
Faculty of Physics University of Vienna, Austria
Institute for Quantum Optics and Quantum
Information Austrian Academy of Sciences

• Johannes Kofler and Caslav Brukner

Young Researchers Conference Perimeter Institute
for Theoretical Physics Waterloo, Canada, Dec.
37, 2007
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Classical versus Quantum
Phase space Continuity Newtons laws Local
Realism Macrorealism Determinism
Hilbert space Quantization, Clicks Schrödinger
Projection Violation of Local Realism Violation
of Macrorealism Randomness

• Does this mean that the classical world is
  substantially different from the quantum world?
• When and how do physical systems stop to behave
  quantumly and begin to behave classically?
• Quantum-to-classical transition without
  environment (i.e. no decoherence) and within
  quantum physics (i.e. no collapse models)

A. Peres, Quantum Theory Concepts and Methods
(Kluwer 1995)
3
What are the key ingredients for anon-classical
time evolution?
The initial state of the system The
Hamiltonian The measurement observables
The candidates
Answer
At the end of the talk
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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
A. J. Leggett and A. Garg, PRL 54, 857 (1985)
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The Leggett-Garg inequality
?t
Dichotomic quantity Q(t) Temporal
correlations All macrorealistic theories
fulfill the LeggettGarg inequality
t 0
t
t1
t2
t3
t4
Violation ? at least one of the two postulates
fails (macrorealism per se or/and non-invasive
measurability). Tool for showing quantumness in
the macroscopic domain.
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When is the Leggett-Garg inequality violated?
Rotating spin-1/2
Evolution Observable
1/2

for
Violation of the Leggett-Garg inequality
Rotating classical spin
precession around x
1
classical
Classical evolution
1
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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
LeggettGarg inequality
classical limit
Choose
?
can be violated for any ?E
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Why dont we see violations in everyday life?
- (Pre-measurement) Decoherence - Coarse-grained
measurements
Model system Spin j, i.e. a qu(2j1)it
Arbitrary state
Assume measurement resolution is much weaker than
the intrinsic uncertainty such that
neighbouring outcomes in a Jz measurement
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
Fuzzy measurements any quantum state allows a
classical description (i.e. hidden variable
model). This is macrorealism per se.
J. Kofler and C. Brukner, PRL 99, 180403 (2007)
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Example Rotation of spin j
sharp parity measurement
classical limit
fuzzy measurement
Classical physics of a rotated classical spin
vector
Violation of Leggett-Garg inequality for
arbitrarily large spins j
J. Kofler 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
But for macrorealism we need more than that
Non-invasive measurability
t 0
t
tj
ti
t
J. Kofler and C. Brukner, quant-ph/0706.0668
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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 measurement and time evolution
commute on the coarse-grained level.
Given fuzzy measurements (or pre-measurement
decoherence), 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)
J. Kofler and C. Brukner, quant-ph/0706.0668
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Non-classical Hamiltonians (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 - Cosine-law between
macroscopically distinct states - Coarse-graining
(even to northern and southern hemi-sphere) does
not help as j and j are well separated
is not fulfilled
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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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What are the key ingredients for anon-classical
time evolution?
The initial state of the system The
Hamiltonian The measurement observables
The candidates
Coarse-grained measurements (or decoherence)
Answer
Sharp measurements
Any (non-trivial) Hamiltonian produces a
non-classical time evolution
Classical Hamiltonians classical time
evolution Non-classical Hamiltonians violation
of macrorealism
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Relation Quantum-Classical
fuzzy measurements
Discrete Classical Physics (macrorealism)
Quantum Physics
macroscopic objects classical Hamiltonians
macroscopic objects non-classical Hamiltonians
or sharp measurements
limit of infinite dimensionality
Classical Physics (macrorealism)
Macro Quantum Physics (no macrorealism)
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Conclusions and Outlook

• Under sharp measurements every Hamiltonian leads
  to a non-classical time evolution.
• Under coarse-grained measurements macroscopic
  realism (classical physics) emerges from quantum
  laws under classical Hamiltonians.
• Under non-classical Hamiltonians and fuzzy
  measurements a quantum state can be described by
  a classical mixture at any instant of time but
  the time evolution of this mixture cannot be
  understood classically.
• Non-classical Hamiltonians seem to be
  computationally complex.
• Different coarse-grainings imply different
  macro-physics.
• As resources are fundamentally limited in the
  universe and practically limited in any
  laboratory, does this imply a fundamental limit
  for observing quantum phenomena?