Dynamics in a model for quantum measurements and insights in the Quantum Measurement Problem

Armen Allahverdyan, Yerevan Roger Balian, Saclay

Foundations of Probability and Physics -6 Linneus University, Vaxjo, 16-6, 2011

    Theo M. NieuwenhuizenCCPP, NYU &ITP, Univ. of Amsterdam Europhysics Letters 2003

arXiv 2004,Vaxjo 2005, 2006Beyond the Quantum 2007arXiv 2008

Opus Magnum, in progress155 pages, Physics Reports

SetupStatistical interpretation of QM

Problems & paradoxes + the big questions

The model: system S + apparatus A spin-½ A = M + B = magnet + bath

Selection of collapse basis & fate of Schrodinger cats

Registration of the Q-measurement & classical measurement

Post measurement & the Born rule

The Q measurement problem elucidated

Summary

Statistical interpretation of QMDensity matrix describes ensemble of identically prepared

systems

Pure state is limiting case; same meaning (no special role)

Ensemble can be real: (many particles: bundle at LHC one trapped ion in photon field, repeated excitation) or virtual: (as in classical statistical physics)

QM = tool for calculating averages from density matrix ρ

=> QM = about what we can measure, not about what is epistomology ontology

Quantum measurement theory describes ensemble of measurements on ensemble of identically prepared systems

Problems & paradoxesEinstein, Bohr, de Broglie, von Neumann, Wigner,

Bohm, Bell, Balian, van Kampen ...

Collapse = non-unitary

Small part of apparatus described by QM why not measurement process?

Preferred basis paradox: on which basis will reduction take place?

When does collapse happen ? How long does it take? What happens to Schrödinger cats?

The biggest of them all:The quantum measurement problem: QM = statistical theory, but in experiments we see individual outcomesSolvable within QM ?? => probably not …or??

Requirements for Q measurement modelsR1: simulate as much as possible real experiments;

R2: ensure unbiased, robust, permanent registration by macroscopic pointer

R3: to amplify quantum signal: involve an apparatus initially in a metastable state and evolving towards one of the stable states under the influence of S; the transition of A, instead of occurring spontaneously, is triggered by S;

R4: include a bath where the free energy may be dumped;

R5: be solvable so as to provide a complete scenario of the joint evolution of S + A and to exhibit the characteristic times;

R6: conserve the tested observable;

R7: lead to a final state devoid of “Schrodinger cats”;

R8: satisfy Born’s rule for the registered results;

R9: produce, for ideal measurements or preparations, the required diagonal correlations between the tested system and the indication of the pointer,

R10: be su ciently flexible to allow discussing imperfect measurements.ffi

Curie-Weiss model for quantum measurements

System S: s = spin-½. For ideal measurement:

Apparatus = Magnet + Bath (A = M + B).

M = N spins-½ σ(n). Pointer = magnetization

B = phonon bath: harmonic oscillators

Spin-spin interactions in magnet M

• J2 term: Curie-Weiss magnet

• J4 term: allows first order phase transition

• Extreme case of first order transition: J2 = 0, J4 = J

Bath HamiltonianStandard harmonic oscillator bath:

x,y,z components of the spins of M couple to their own harmonic oscillators

Bath characterized bycorrelator K(t-t’)

Weak M-B coupling: γ << 1 allows to work at lowest order in γ

Initial density matrix

Tested system S: arbitrary density matrix uncorrelated with A

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

Bath: Gibbs state (mixed state) β = 1/T

Apparatus A starts in mixed state, (large system cannot be prepared in pure state)

Apparatus starts in mixed state => system may end up in mixed state No unitarity paradox

Selection of collapse basis

What selects collapse basis? The interaction Hamiltonian !

Liouville-von Neumann eqn

The 4 sectors decoupleideality of the measurement

a=z: Diagonal terms of r(t) conserved ->= rii(t) = rii(0 ): Born probabilities

a=x,y: Off-diagonal terms evolve => disappearence of Schrodinger cats

Motion of the spin

Fate of Schrodinger cat termsShort times: spin-spin couplings in magnet are irrelevant

x,y components of spin n of M moves in the magnetic field g of spin S

Larmor procession damping by zero-point flucts of bath

Schrodinger cat:

dephasing decoherence as in NMR comes in later

Recurrences eliminated by: decoherence by bath N B >> 1 or even by a spread in g’s

Creation of multi-particle correlations: weak, dying out

Schrodinger cat terms die quickly in cascade of weak multi-spin correlations

CCR: Continuous Cat Reduction

• Reduction is ongoing (not once-and-for-all), like the repeated Stosszahlansatz in the Boltzmann equation

• Through resonant excitation of bath modes

( is energy needed to flip spin of M)

Cat suppression factor

time

Magnetization m involves only eigenvalues , as in classical statistical physics

In sector with sz =1: analogy to classical measurement of classical Ising spin sz =1

Dynamics

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

m

Registration of the measurementSolve Q-dynamics of diagonal elements to second order in the coupling to bath

Measure a spin with a “classical” apparatus of magnet and a bath

Free energy landscape: classical Curie-Weiss model

mm

At g=0:

Low T:paramagnetic initial state; stable ferromagnetic statescan act as measuring apparatus

High T: paramagnet is stable

Perform measurement: turn on coupling field h=g sz

m m

Bath is needed to dump the free energy

Procedure:Turn off coupling g after minimum is reachedMagnet goes to ferromagnetic minimum at g=0. Stays there a time exp(N)

(Result is stable and may be read off, or not)

Real solution: QM widens the distribution P(m;t)

• Finally: sharp peak at m=+mF in sector sz=+1

• m=−mF in sector sz=−1

Post-measurement state

Magnet ends up in up/down ferromagnetic stateSign of magnetization maximally correlated with sign spin SProbabilities satisfy Born rule

No Schrodinger cat termsPhysical disappearance: sums of many oscillating terms vanish despite of mathematical survival: individual terms have fixed amplitude

On the Quantum Measurement Problem

The macroscopic magnet itself is well understood. Initial paramagnet relaxes to stable or ferromagnetEnsemble theory describes breaking of ergodicityIndividual and ferromaget is stable, Poincare time is very large

In Q measurement this transition is triggered by coupling to the spin SLong life time of FM state ensures uniqueness of outcome for m=±mF

and for the measured spin sz = ± 1 that is fully correlated with it

No survival of Schrodinger cat terms, that could spoil this view.

One may select individual outcomes with magnetization (possible because ferromagnetic state longlived)This unambiguous subensemble is a pure ensemble of spins of S

Summary: the measurement problem elucidated

• Collapse basis determined by interaction Hamiltonian

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

cascade of small multi-particle correlations ongoing reduction

• Registration : some classical features ( Bohr) macroscopic pointer: irreversible dynamics, entropy increase

• Born rule results from the dynamics

• Observation of pointer is irrelevant for outcomes ( Wigner) (results may be read off, or processed automatically)

• Statistical interpretation of QM

• Individual outcomes due to irreversible phase transition in magnet long lived pointer indication, no cat terms

How to teach QM

• Copenhagen interpretation• Statistical interpretation: minimalistic, simple

On the interpretation of QM

• Solution gives probabilities for outcomes of experiments: • system in collapsed state + apparatus in pointer state

• Density matrix describes my knowledge about the ensemble of identically prepared systems

• Should be updated after measurement

Statistical interpretation: QM describes ensembles, not single systemsQ-measurement theory = about ensembles of measurements

Q-measurement theory describes the statistics of outcomes of an ensemble of measurements on an 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>

Statistical interpretation of QM

Einstein again wrote on it even in 1955Kemble : in his book 1937Ballentine: Rev Mod Phys 1975van Kampen: Am J Phys 1988Balian: Am J Phys 1989

Quantum Mechanics is a theory

that describesthe statistics of outcomes of experiments

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

The Quantum measurement problem

• Why do we see individual outcomes in individual measurements?

• Perhaps because there is an underlying reality

in which the events occur (“world”, “nature”, “life”)?

• Should there exist a theory for that?Hidden-variables theory, sub-quantum mechanics

Other interpretations• Copenhagen: each system has its own wavefunction,

that has the most complete info in a statistical sense.There is an underlying wavefunction of system + apparatus

• Ghirardi-Rimini-Weber GRW: collapse is a physical process, modification of QM needed to achieve the collapses.

• Multi-universe: at each measurement one of the branches is chosen, other ones in other universes.

• Wigner’s friend: the observer “finishes” the measurement

• Decoherence picture: pointer is a mode of the bath(The air measures my height)

The Born ruleas an identification at the macroscopic level

In practice: describes statistics of pointer states

determine relative frequencies (von Mises)

Post-measurement state of tested system is a mixed state,