Solvable Model for the Quantum Measurement Process Armen E. Allahverdyan Roger Balian Theo M. Nieuwenhuizen Academia Sinica Taipei, June 26, 2004

Setup The model: system S + apparatus A S=spin-½ A = M + B = magnet + bath Classical measurement Statistical interpretation of QM Selection of collapse basis & collapse Registration of the Q-measurement Summary Post measurement & the Born rule The battlefield

Q-measurement is only contact of QM and experiment Interpretations of QM must be compatible with Q-meas. But no solvable models with enough relevant physics Interpretations Copenhagen: each system has its own wave function of QM multi-universe picture: no collapse but branching (Everett) mind-body problem: observation finishes measurement (Wigner) non-linear extensions of QM needed for collapse: GRW wave function is state of knowledge, state of belief consistent histories Bohmian, Nelsonian QM statistical interpretation of QM

The Hamiltonian Test system: spin ½, no dynamics during measurement: Magnet: N spins ½, with equal coupling J/4N^3 between all quartets Bath: standard harmonic oscillator bath: each component of each spin couples to its own set of harmonic oscillators System-Apparatus Apparatus=magnet+bath

Bath Hamiltonian

Initial density matrix Von Neuman eqn: Initial density matrix -> final density matrix Test system: arbitrary density matrix Magnet: N spins ½, starts as paramagnet (mixed state) Bath: Gibbs state (mixed state)

Classically: only eigenvalues show up: classical statistical physics Measure a spin s_z=+/- 1 with an apparatus of magnet and a bath Intermezzo: Classical measurement of classical Ising spin Dynamics Free energy F=U-TS: minima are stable states m = tanh h m

Statistical interpretation of QM Q-measurement describes ensemble of measurements on ensemble of systems Statistical interpretation: a density matrix (mixed or pure) describes an ensemble of systems Stern-Gerlach expt: ensemble of particles in upper beam described by |up> Copenhagen: the wavefunction is the most complete description of the system

Selection of collapse basis What selects collapse basis: The interaction Hamiltonian Trace out Apparatus (Magnet+Bath) Diagonal terms of r(t) conserved Off-diagonal terms endangered -> disappearence of Schrodinger cats

Fate of Schrodinger cats Consider off-diagonal terms of Cat hides itself after Bath suppresses its returns Initial step in collapse: effect of interaction Hamiltonian only (bath & spin-spin interactions not yet relevant)

Complete solution Mean field Ansatz: Solution: Result: decay of off-diagonal terms confirmed diagonal terms go exactly as in classical setup

Post-measurement state Density matrix: - maximal correlation between S and A - no cat-terms Born rule from classical interpretation

Summary Solution of measurement process in model of apparatus=magnet+bath Apparatus initially in metastable state (mixture) Collapse (vanishing Schrodinger-cats) is physical process, takes finite but short time Collapse basis determined by interaction Hamiltonian Measurement in two steps: Integration of quantum and classical measurements Born rule explained via classical interpretation of pointer readings Observation of outcomes of measurements is irrelevant Quantum Mechanics is a theory that describes statistics of outcomes of experiments Statistical interpretation: QM describes ensembles, not single systems Solution gives probabilities for outcomes of experiments: system in collapsed state + apparatus in pointer state

Bath Hamiltonian

Quantum mechanics Observables are operators in Hilbert space New parameter: Planck’s constant h Einstein: Statistical interpretation: Quantum state describes ensemble of systems Armen Allahverdyan, Roger Balian, Th.M. N. 2001; 2003: Exactly solvable models for quantum measurements. Ensemble of measurements on an ensemble of systems. explains: solid state, (bio-)chemistry high energy physics, early universe Interpretations: - Copenhagen: wave function = most complete description of the system - mind-body problem: mind needed for measurement - multi-universe picture: no collapse, system goes into new universe Max Born ( ) 1926: Quantum mechanics is a statistical theory John von Neumann ( ) 1932: Wave function collapses in measurement Classical measurement specifies quantum measurement. Collapse is fast; occurs through interaction with apparatus. All possible outcomes with Born probabilities.