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Theoretical Physics of Quantum Measurements

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

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  1. 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

  2. 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

  3. 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

  4. 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

  5. Bath Hamiltonian Standard weak coupling to a harmonic oscillator bath: each component of each spin couples to its own set of harmonic oscillators

  6. 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)

  7. 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

  8. 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

  9. 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

  10. Free energy landscape: classical Curie-Weiss model At g=0: m m High T: paramagnet is stable Low T: can act as measuring apparatus

  11. 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)

  12. Post-measurement state Density matrix: - maximal correlation between S and A - no Schrodinger cat-terms Bath back to Gibbs state Magnet to ferromagnetic state

  13. 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

  14. 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

  15. 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

  16. 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

  17. 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)

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