Institute for Experimental Physics University of Vienna Institute

Institute for Experimental Physics University of Vienna Institute for Quantum Optics and Quantum Information Austrian Academy of Sciences Classical World because of Quantum Physics Johannes Kofler and Časlav Brukner University of Leeds, United Kingdom September 2006

Classical versus Quantum Phase space Hilbert space Continuity Events, ”Clicks” Newton’s laws Schrödinger + Projection Local Realism Violation of Local Realism Macrorealism Violation of Macrorealism Determinism 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?

Macrorealism [Leggett–Garg (1985)] Macrorealism per se Non-invasive measurability “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. ” “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

Quantity: Q t t=0 t Temporal correlations t 1 t 2 t 3 t 4 All macrorealistic theories fulfill the Leggett–Garg inequality Violation no objective properties prior to and independent of measurements

When is Macrorealism violated? Spin-1/2 Evolution Observable 1/2 for Violation of macrorealism Classical Spin precession around x +1 Macrorealism – 1 classical

Violation of Macrorealism for macroscopically large spins? Spin-j precession in magnetic field (totally mixed state!) j Parity of eigenvalue m of Jz measurement classical limit Violation of macrorealism for arbitrarily large spins j Shown for local realism [Mermin, Peres]

The quantum-to-classical transition Coherent spin state (t = 0): exact measurement fuzzy measurement & limit of large spins This is (continuous and non-invasive) classical physics of a rotated classical spin vector!

Transition to Classicality: General state Quantum General density matrix: Classical Probability to detect in a slot: f can be negative! Probability for result m: h is non-negative! Hamilton operator: Hamilton function: Classical limit: Ensemble of classical spins with probability distribution h

Relation Quantum-Classical inaccurate measurements Quantum Physics macroscopic objects limit of large spins Macro Quantum Physics (no macrorealism) ? Discrete Classical Physics (macrorealism) limit of large spins Classical Physics (macrorealism)

Is there a fundamental limit for observability of quantum phenomena? Algorithmic Complexity of Quantum States [Mora and Briegel]: The length of the shortest string that encodes the preparation procedure for a given state with a given precision ε. Most of the states of N qubits are of complexity 2 N random amplitudes Number of bits needed to realize the state: Number of bits in the Universe [Lloyd]: Limit [Davis]: Every state is effectively a mixture over all pure states that are not distinguishable within the precision ε

Conclusions 1. 2. 3. Classical physics emerges from quantum laws under the restriction of coarse-grained measurements, not alone through the limit of large quantum numbers. Conceptually different from decoherence. Not dynamical, puts the stress on observability and works also for fully isolated systems. As the resources in the world are limited, there is a fundamental limit for observability of quantum phenomena (even if there is no such limit for the validity of quantum theory itself). quant-ph/0609079