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Dynamics in a model for quantum
measurements and insights in the
Quantum Measurement Problem
Theo M. Nieuwenhuizen
CCPP, NYU &
ITP, Univ. of Amsterdam
Armen Allahverdyan, Yerevan
Roger Balian, Saclay
Europhysics Letters 2003
arXiv 2004,Vaxjo 2005, 2006
Beyond the Quantum 2007
arXiv 2008
Opus Magnum, in progress
155 pages, Physics Reports
Foundations of Probability and Physics -6
Linneus University, Vaxjo, 16-6, 2011
Setup
Statistical 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 QM
Density matrix
Pure state
describes ensemble of identically prepared
systems
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 & paradoxes
Einstein, 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 outcomes
Solvable within QM ?? => probably not …or??
Requirements for Q measurement models
R1: 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 sufficiently flexible to allow discussing imperfect measurements.
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 Hamiltonian
Standard harmonic oscillator bath:
x,y,z components of the spins of M couple to their own harmonic oscillators
Bath characterized by
correlator 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
Apparatus A starts in mixed state,
(large system cannot be prepared in pure state)
Magnet M: N spins ½, starts as paramagnet (mixed state)
Bath: Gibbs state (mixed state)
β = 1/T
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 decouple
ideality of the measurement
Motion of the spin
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
Fate of Schrodinger cat terms
Short 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
as in NMR
decoherence
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
Cat suppression factor
time
• 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)
Registration of the measurement
Solve Q-dynamics of diagonal elements to second order in the coupling to bath
In sector with sz =1: analogy to classical measurement of classical Ising spin sz =1
Magnetization m involves only eigenvalues
Measure a spin
, as in classical statistical physics
with a “classical” apparatus of magnet and a bath
Dynamics
Free energy F=U-TS: minima are stable states of free energy
m
Free energy landscape: classical Curie-Weiss model
At g=0:
m
High T: paramagnet is stable
m
Low T:
paramagnetic initial state;
stable ferromagnetic states
can act as measuring apparatus
Perform measurement: turn on coupling
m
Bath is needed to dump the free energy
field h=g sz
m
Procedure:
Turn off coupling g after minimum is reached
Magnet 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 state
Sign of magnetization maximally correlated with sign spin S
Probabilities satisfy Born rule
No Schrodinger cat terms
Physical 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  ferromagnet
Ensemble theory describes breaking of ergodicity
Individual  and  ferromaget is stable, Poincare time is very large
In Q measurement this transition is triggered by coupling to the spin S
Long 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
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
Density matrix describes my knowledge about the ensemble of
identically prepared systems
Should be updated after measurement
Statistical interpretation of QM
Einstein again wrote on it even in 1955
Kemble : in his book 1937
Ballentine: Rev Mod Phys 1975
van Kampen: Am J Phys 1988
Balian: Am J Phys 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 the statistics of outcomes
of an ensemble of measurements
on an ensemble of systems
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)
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 rule
as 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,