Faculty of Physics University of Vienna Austria Institute

Faculty of Physics University of Vienna, Austria Institute for Quantum Optics and Quantum Information Austrian Academy of Sciences Quantum violation of macroscopic realism and the transition to classical physics Johannes Kofler Ph. D Defense University of Vienna, Austria October 3 rd, 2008

List of publications Articles in refereed journals Submitted • J. Kofler and Č. Brukner Conditions for quantum violation of macroscopic realism Phys. Rev. Lett. 101, 090403 (2008) • T. Paterek, R. Prevedel, J. Kofler, P. Klimek, M. Aspelmeyer, A. Zeilinger, and Č. Brukner Mathemtical undecidability and quantum randomness • J. Kofler and Č. Brukner Classical world arising out of quantum physics under the restriction of coarse-grained measurements Phys. Rev. Lett. 99, 180403 (2007) • J. Kofler and Č. Brukner Entanglement distribution revealed by macroscopic observations Phys. Rev. A 74, 050304(R) (2006) • M. Lindenthal and J. Kofler Measuring the absolute photo detection efficiency using photon number correlations Appl. Opt. 45, 6059 (2006) • J. Kofler, V. Vedral, M. S. Kim, and Č. Brukner Entanglement between collective operators in a linear harmonic chain Phys. Rev. A 73, 052107 (2006) • J. Kofler, T. Paterek, and Č. Brukner Experimenter’s freedom in Bell's theorem and quantum cryptography Phys. Rev. A 73, 022104 (2006) Contributions in books • J. Kofler and Č. Brukner A coarse-grained Schrödinger cat Quantum Communication and Security, ed. M. Żukowski, S. Kilin, and J. Kowalik (IOS Press, 2007) Proceedings • R. Ursin et. al. Space-QUEST: Experiments with quantum entanglement in space 59 th International Astronautical Congress (2008) Articles in popular journals • A. Zeilinger and J. Kofler La dissolution du paradoxe Sciences et Avenir Hors-Série, No. 148, p. 54 (Oct. /Nov. 2006)

Classical versus Quantum Phase space Hilbert space Continuity Quantization, “clicks” Newton’s laws Schrödinger equation Definite states Superposition/Entanglement Determinism Randomness - When and how do physical systems stop to behave quantum mechanically and begin to behave classically? - What is the origin of quantum randomness? Isaac Newton Ludwig Boltzmann Albert Einstein Niels Bohr Erwin Schrödinger Werner Heisenberg

Double slit experiment With electrons! (or neutrons, molecules, photons, …) With cats? |cat left + |cat right ?

Why do we not see macroscopic superpositions? Two schools: - Decoherence uncontrollable interaction with environment; within quantum physics - Collapse models forcing superpositions to decay; altering quantum physics Alternative answer: - Coarse-grained measurements measurement resolution is limited; within quantum physics

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(t 1) Q(t 2) t t=0 t 1 t 2

The Leggett-Garg inequality Dichotomic quantity: t t=0 t Temporal correlations t 1 t 2 t 3 t 4 All macrorealistic theories fulfill the Leggett-Garg inequality Violation macrorealism per se or/and non-invasive measurability failes

When is the Leggett-Garg inequality violated? Rotating spin-½ Rotating classical spin precession around x measurement along z sign of z component Violation of the Leggett-Garg inequality Classical evolution classical limit ½

Why don’t we see violations in everyday life? Coarse-grained measurements Model system: Spin j macroscopic: j ~ 1020 Arbitrary state: - Measure Jz, outcomes: m = – j, –j+1, . . . , +j (2 j+1 levels) - Assume measurement resolution is much weaker than the intrinsic uncertainty such that neighbouring outcomes are bunched together into “slots” m. m = +j m = –j = 1 2 3 4

Macrorealism per se Probability for outcome m can be computed from an ensemble of classical spins with positive probability distribution: Coarse-grained measurements: any quantum state allows a classical description This is macrorealism per se. J. K. and Č. Brukner, PRL 99, 180403 (2007)

Example: Rotation of spin j j Coarse-grained measurement Sharp measurement of spin z-component –j +j 1 3 5 7. . . Q = – 1 –j +j 2 4 6 8. . . Q = +1 classical limit Fuzzy measurement Violation of Leggett-Garg inequality for arbitrarily large spins j Classical physics of a rotating classical spin vector J. K. and Č. Brukner, PRL 99, 180403 (2007)

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

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 Depending on the outcome, measurement reduces state to Fuzzy measurements only reduce previous ignorance about the spin mixture: For macrorealism we need more: Total ensemble without measurement should be the weighted mixture of the evolved subensembles after a measurement: Non-invasive measurability t=0 ti tj t t J. K. and Č. Brukner, PRL 101, 090403 (2008)

The sufficient condition for macrorealism is I. e. the statistical mixture has a classical time evolution, if no superpositions of macroscopically distinct states are produced. Given coarse-grained measurements, it depends on the Hamiltonian whether macrorealism is satisfied. “Classical” Hamiltonians “Non-classical” Hamiltonians eq. is fulfilled (e. g. rotation) eq. not fulfilled (e. g. osc. Schrödinger cat, next slide) J. K. and Č. Brukner, PRL 101, 090403 (2008)

Non-classical Hamiltonian (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: time

Non-classical Hamiltonians are complex Oscillating Schrödinger cat Rotation in real space “non-classical” rotation in Hilbert 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

Relation quantum-classical

The origin of quantum randomness Determinism (subjective randomness due to ignorance)? Objective randomness (no causal reason)?

Mathematical undecidability Axioms Proposition: true/false if it can be proved/disproved from the axioms “logically independent” or “mathematically undecidable” if neither the proposition nor its negation leads to an inconsistency (i) (ii) Euclid’s parallel postulate in neutral geometry “axiom of choice” in Zermelo-Fraenkel set theory intuitively: independent proposition contains new information Information-theoretical formulation of undecidability (Chaitin 1982): “If a theorem contains more information than a given set of axioms, then it is impossible for theorem to be derived from the axioms. ”

Logical complementarity Consider (Boolean) bit-to-bit function f(a) = b (with a = 0, 1 and b = 0, 1) (A) f(0) = 0 (B) f(1) = 0 (C) f(0) + f(1) = 0 logically complementary Given any single 1 -bit axiom, i. e. (A) or (B) or (C), the two other propositions are undecidable. Physical “black box” can encode the Boolean function: f(a) = 0 f(0) = 1 f(1) = 1 Example qubit a=1 a=0 f(1) = 1 f(0) = 0

Mathematical undecidability and quantum randomness x Preparation Black box Measurement z Information gain f(0) (A) f(1) (B) f(0) + f(1) (C) x y However: x Random outcomes! (B) is undecidable within axiom (A)

Experimental test of mathematical undecidability: 1 qubit “A qubit carries only one bit of information” (Holevo 1973, Zeilinger 1999) T. Paterek, R. Prevedel, J. K. , P. Klimek, M. Aspelmeyer, A. Zeilinger, Č. Brukner, submitted (2008)

Generalization to many qubits N qubits, N Boolean functions f 1, …, f. N Black box: New feature: Partial undecidability T. Paterek, R. Prevedel, J. K. , P. Klimek, M. Aspelmeyer, A. Zeilinger, Č. Brukner, submitted (2008)

Conclusions Quantum-to-classical transition under coarse-grained measurements Quantum randomness: a manifestation of mathematical undecidability

Thank you!

Appendix

Violation for arbitrary Hamiltonians t Initial state t t State at later time t t 1 = 0 Measurement t 2 ! Survival probability Leggett-Garg inequality classical limit Choose can be violated for any E t 3 ? ?

Continuous monitoring by an environment Exponential decay of survival probability - Leggett-Garg inequality is fulfilled (despite the non-classical Hamiltonian) - However: Decoherence cannot account for a continuous spatiotemporal description of the spin system in terms of classical laws of motion. - Classical physics: differential equations for observable quantitites (real space) - Quantum mechanics: differential equation for state vector (Hilbert space)

Experimental setups