Quantum, classical & coarse-grained measurements Johannes Kofler and Časlav Brukner Faculty of Physics University of Vienna, Austria Institute for Quantum Optics and Quantum Information Austrian 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 a non-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 t1t1 t2t2 Q(t1)Q(t1)Q(t2)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 t1t1 t2t2 t3t3 t4t4 t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 t t 1 = 0t2t2 t3t3 tt tt 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 J z measurement are bunched together into “slots” m. –j–j+j

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, (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, (2007)

Coarse-graining  Coarse-graining Neighbouring coarse-graining (many slots) Sharp parity measurement (two slots) Violation of Leggett-Garg inequality Classical Physics Slot 1 (odd)Slot 2 (even) Note:

Superposition versus Mixture To see the quantumness of a spin j, you need to resolve j 1/2 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 = 0 t t titi tjtj Non-invasive measurability J. Kofler and Č. Brukner, quant-ph/

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” Hamiltonianseq. is fulfilled (e.g. rotation) “Non-classical” Hamiltonianseq. 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/

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 steps 1 single computation step all N rotations can be done simultaneously

What are the key ingredients for a non-classical time evolution? The initial state of the system The Hamiltonian The measurement observables The candidates: Answer:Sharp measurements Coarse-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