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