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Quantum, classical &coarse-grained measurements
Johannes Kofler and Časlav Brukner
Faculty of PhysicsUniversity of Vienna, Austria
Institute for Quantum Optics and Quantum InformationAustrian Academy of Sciences
Young Researchers Conference
Perimeter Institute for Theoretical Physics
Waterloo, Canada, Dec. 3–7, 2007
Classical versus Quantum
Phase space
Continuity
Newton’s laws
Local Realism
Macrorealism
Determinism
- 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)
Hilbert space
Quantization, “Clicks”
Schrödinger + Projection
Violation of Local Realism
Violation of Macrorealism
Randomness
A. Peres, Quantum Theory: Concepts and Methods (Kluwer 1995)
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
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.”
t = 0
t
t1 t2
Q(t1) Q(t2)
A. J. Leggett and A. Garg, PRL 54, 857 (1985)
Dichotomic quantity: Q(t)
Temporal correlations
All macrorealistic theories fulfill the
Leggett–Garg inequality
t = 0
t
t1 t2 t3 t4
t
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.
The Leggett-Garg inequality
When is the Leggett-Garg inequality violated?
1/2
Rotating spin-1/2
Rotating classical spin
classical+1
–1
Evolution Observable
Violation of the Leggett-Garg inequality
precession around x
Classical evolution
for
Violation for arbitrary Hamiltonians
Initial state
State at later time t
Measurement
Survival probability
Leggett–Garg inequality
tt1 = 0 t2 t3
tt
Choose
can be violated for any E
classical limit
? ?!
Why don’t we see violations in everyday life?
- (Pre-measurement) Decoherence
- Coarse-grained measurements
Model system: Spin j, i.e. a qu(2j+1)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.
–j +j 1 2 3 4
Fuzzy measurements: any quantum state allows a classical description (i.e. hidden variable model).
This is macrorealism per se.
Probability for outcome m can be computed from an ensemble of classical spins with positive probability distribution:
J. Kofler and Č. Brukner, PRL 99, 180403 (2007)
Macrorealism per se
fuzzy measurement
Example: Rotation of spin j
classical limit
sharp parity measurement
Violation of Leggett-Garg inequality for arbitrarily large spins j
Classical physics of a rotated classical spin vector
J. Kofler and Č. Brukner, PRL 99, 180403 (2007)
Coarse-graining Coarse-graining
Neighbouring coarse-graining(many slots)
Sharp parity measurement(two slots)
Violation ofLeggett-Garg inequality
Classical Physics
1 3 5 7 ...
2 4 6 8 ...
Slot 1 (odd) Slot 2 (even)
Note:
Superposition versus Mixture
To see the quantumness of a spin j, you need to resolve j1/2 levels!
Albert Einstein and ...Charlie Chaplin
Non-invasive measurability
Fuzzy measurements only reduce previous ignorance about the spin mixture:
But for macrorealism we need more than that:
Depending on the outcome, measurement reduces state to
t = 0t
tti tj
Non-invasive measurability
J. Kofler and Č. Brukner, quant-ph/0706.0668
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”.
“Classical” Hamiltonians eq. is fulfilled (e.g. rotation)
“Non-classical” Hamiltonians eq. not fulfilled (e.g. osc. Schrödinger cat)
Given fuzzy measurements (or pre-measurement decoherence), it depends on the Hamiltonian whether macrorealism is satisfied.
J. Kofler and Č. Brukner, quant-ph/0706.0668
Non-classical Hamiltonians(no macrorealism despite of coarse-graining)
Hamiltonian:
- 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
Produces oscillating Schrödinger cat state:
Under fuzzy measurements it appears as a statistical mixture at every instance of time:
is not fulfilled
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
O(N) sequential steps1 single computation step
all N rotations can be done simultaneously
What are the key ingredients for anon-classical time evolution?
The initial state of the system
The Hamiltonian
The measurement observables
The candidates:
Answer: Sharp measurementsCoarse-grained measurements (or decoherence)
Any (non-trivial) Hamiltonian produces a non-classical time evolution
“Classical” Hamiltonians: classical time evolution
“Non-classical” Hamiltonians: violation of macrorealism
Quantum Physics Discrete Classical Physics(macrorealism)
Classical Physics(macrorealism)
fuzzy measurements
limit of infinite dimensionality
Macro Quantum Physics(no macrorealism)
macroscopic objects & non-classical Hamiltonians or
sharp measurements
macroscopic objects & classical Hamiltonians
Relation Quantum-Classical
1. Under sharp measurements every Hamiltonian leads to a non-classical time evolution.
2. Under coarse-grained measurements macroscopic realism (classical physics) emerges from quantum laws under classical Hamiltonians.
3. 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.
4. Non-classical Hamiltonians seem to be computationally complex.
5. Different coarse-grainings imply different macro-physics.
6. As resources are fundamentally limited in the universe and practically limited in any laboratory, does this imply a fundamental limit for observing quantum phenomena?
Conclusions and Outlook