Quantum, classical & coarse-grained measurements

1. Faculty of Physics University of Vienna, Austria Institute for Quantum Optics and Quantum Information Austrian Academy of Sciences Quantum, classical &coarse-grained measurements Johannes Kofler and Časlav Brukner Young Researchers Conference Perimeter Institute for Theoretical Physics Waterloo, Canada, Dec. 3–7, 2007

2. Classical versus Quantum Phase space Continuity Newton’s 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 candidates: The initial state of the system The Hamiltonian The measurement observables Answer: At the end of the talk

4. 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 t1 t2 t = 0 A. J. Leggett and A. Garg, PRL 54, 857 (1985)

5. The Leggett-Garg inequality t Dichotomic quantity: Q(t) Temporal correlations All macrorealistic theories fulfill the Leggett–Garg 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.

6. 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

7. 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

8. 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

9. Macrorealism per se Probability foroutcomem 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 Č. Brukner, PRL 99, 180403 (2007)

10. 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 Č. Brukner, PRL 99, 180403 (2007)

11. Coarse-graining  Coarse-graining Sharp parity measurement (two slots) Neighbouring coarse-graining (many slots) 1 3 5 7 ... 2 4 6 8 ... Slot 1 (odd) Slot 2 (even) Violation of Leggett-Garg inequality Classical Physics Note:

12. Superposition versus Mixture To see the quantumness of a spin j, you need to resolve j1/2 levels!

13. Albert Einstein and ... Charlie Chaplin

14. 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 Č. Brukner, quant-ph/0706.0668

15. The sufficient condition for macrorealism The sufficient condition for macrorealismis I.e. the statistical mixture has a classical time evolution, if measurement and time evolutioncommute “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 Č. Brukner, quant-ph/0706.0668

16. Non-classical Hamiltonians (no macrorealism despite of coarse-graining) Hamiltonian: Produces oscillating Schrödinger catstate: 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

17. 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

18. What are the key ingredients for anon-classical time evolution? The candidates: The initial state of the system The Hamiltonian The measurement observables 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

19. 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 Macro Quantum Physics (no macrorealism) Classical Physics (macrorealism)

20. 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?