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Theoretical Physics of Quantum Measurements. Armen E. Allahverdyan, Yerevan Roger Balian, Saclay . Europhysics Letters 2003 Beyond the Quantum 2007 Opus Magnum, in progress. Theo M. Nieuwenhuizen. AG Grenzen der Quantummechanik Die Junge Akademie Berlin, 28-4-2008. Setup.

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theoretical physics of quantum measurements

Theoretical Physics ofQuantum Measurements

Armen E. Allahverdyan, Yerevan

Roger Balian, Saclay

Europhysics Letters 2003Beyond the Quantum 2007

Opus Magnum, in progress

Theo M. Nieuwenhuizen

AG Grenzen der Quantummechanik

Die Junge Akademie

Berlin, 28-4-2008

setup
Setup

Statistical interpretation of QM

The model: system S + apparatus A

spin-½ A = M + B = magnet + bath

Selection of collapse basis & fate of Schrodinger catsRegistration of the Q-measurement & classical measurement

Post measurement & the Born rule

Summary

On the interpretation of Quantum mechanics

slide3

Statistical interpretation of QM

Einstein again wrote on it even in 1955Kemble 1937Ballentine 1975van Kampen 1988Balian 1989

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>

Q-measurement theory describes an ensemble of measurements on an ensemble of systems

the model for this talk

ABN Europhys Lett 61, 452 (2003); cond-mat/0408316

The model for this talk

Tested system: spin ½, no dynamics during measurement:

Apparatus=magnet+bath

Magnet: N spins ½, with equal coupling between all quartets (Curie-Weiss type mean field model)

System-Apparatus interaction: spin-spin coupling

slide5

Bath Hamiltonian

Standard weak coupling to a harmonic oscillator bath:

each component of each spin couples to its own set of harmonic oscillators

slide6

Initial density matrix

Tested system: arbitrary density matrix uncorrelated with apparatus

Apparatus in mixed state,product of magnet and bath

Magnet: N spins ½, starts as paramagnet (mixed state)

Bath: Gibbs state (mixed state)

slide7

Selection of collapse basis

What selects collapse basis? The interaction Hamiltonian

Trace out Apparatus (Magnet+Bath) in von Neumann eqn

Diagonal terms of r(t) conserved -> Born probabilities

Off-diagonal terms endangered -> disappearence of Schrodinger cats

slide8

Fate of Schrodinger cats

Consider off-diagonal terms of

Initial step in collapse: effect of interaction Hamiltonian only

(spin-spin &bath interactions not yet relevant)

Dephasing

Decoherence

Cat hides itself after

Bath suppresses its returns after

Returns can also be suppressed by randomness in the g’s

slide9

Registration of the measurement:Solve Q-dynamics of diagonal elements to second order in the coupling to bath

In sector with s_z=1: analogous to classical measurement of classical Ising spin s_z=1

Apparatus: only eigenvalues show up: classical statistical physics

Measure a spin with an apparatus of magnet and a bath

m

Dynamics

Free energy F=U-TS: minima are stable states of free energy

free energy landscape classical curie weiss model
Free energy landscape: classical Curie-Weiss model

At g=0:

m

m

High T: paramagnet is stable

Low T: can act as

measuring apparatus

during measurement turn on coupling field g s z
During measurement: turn on coupling field g s_z

m

m

Bath is needed to dump the released energy (Coleman-Hepp miss it)

Turn off coupling g after minimum is reached

Magnet goes to g=0 minimum and stays there a time exp(N)(Result is stable and may be read off, or not)

post measurement state
Post-measurement state

Density matrix: - maximal correlation between S and A

- no Schrodinger cat-terms

Bath back to Gibbs state

Magnet to ferromagnetic state

explanation of the born rule via identification at the macroscopic level
Explanation of the Born rulevia identification at the macroscopic level

In practice: describes statistics of pointer states

Probabilities = relative frequencies (von Mises)

Post-measurement state of tested system

also has frequency interpretation

simultaneously measuring two non commuting variables
Simultaneously measuring two non-commuting variables
  • Measure s_x and s_z by coupling spin to 2 apparati
  • Simplification if apparati identical
  • Solve early time dynamics of the pointer variables
  • Probabilities set already then:
  • Reduction factor
summary new or confirmed insights
Collapse basis determined by interaction Hamiltonian

Measurement in two steps: cats die & registration of the result very fast is much slower

Registration=Integration of quantum and classical measurements (Bohr) pointer = macroscopic: QM with some classical features

Born rule explained via identification of use of Q predictionsat the macroscopic level (interpretation of pointer readings)

Observation of outcomes of measurements is irrelevant ( Wigner)

Statistical interpretation of QM

Extension to measuring non-commuting variables

Summary: New or confirmed insights

ABN: EPL 2003; in: Beyond the Quantum 2007; Opus Magnum 2108

on the interpretation of qm
On the interpretation of QM

Statistical interpretation: QM describes ensembles, not single systems

Q-measurement theory = about ensembles of measurements

  • Solution gives probabilities

for outcomes of experiments:

  • system in collapsed state + apparatus in pointer state
quantum mechanics is a theory
Quantum Mechanics is a theory

that describes the statistics of outcomes of experiments

It cannot and should not describe individual experiments(otherwise than in a probablistic sense)