emergence of quantum mechanics from classical statistics
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Emergence of Quantum Emergence of Quantum MechanicsMechanics
from Classical Statisticsfrom Classical Statistics
what is an atom ?what is an atom ?
quantum mechanics : isolated objectquantum mechanics : isolated object quantum field theory : excitation of quantum field theory : excitation of
complicated vacuumcomplicated vacuum classical statistics : sub-system of classical statistics : sub-system of
ensemble with infinitely many ensemble with infinitely many degrees of freedomdegrees of freedom
quantum mechanics quantum mechanics can be described by can be described by classical statistics !classical statistics !
quantum mechanics from quantum mechanics from classical statisticsclassical statistics
probability amplitudeprobability amplitude entanglemententanglement interferenceinterference superposition of statessuperposition of states fermions and bosonsfermions and bosons unitary time evolutionunitary time evolution transition amplitudetransition amplitude non-commuting operatorsnon-commuting operators
probabilistic observablesprobabilistic observables
Holevo; Beltrametti,BugajskiHolevo; Beltrametti,Bugajski
classical ensemble , classical ensemble , discrete observablediscrete observable
Classical ensemble with probabilitiesClassical ensemble with probabilities
one discrete observable A , values one discrete observable A , values +1 or -1+1 or -1
effective micro-stateseffective micro-states
group states togethergroup states together
σσ labels effective labels effective micro-statesmicro-states , t , tσσ labels labels sub-statessub-states
in effective micro-states in effective micro-states σσ : :probabilities to find A=1 : and A=-probabilities to find A=1 : and A=-
1:1:mean value in micro-state mean value in micro-state σσ : :
expectation valuesexpectation values
only measurements +1 or -1 possible !only measurements +1 or -1 possible !
probabilistic observables probabilistic observables have a probability have a probability
distribution of values in a distribution of values in a microstate ,microstate ,
classical observables a classical observables a sharp valuesharp value
deterministic and deterministic and probabilistic observablesprobabilistic observables
classical or deterministic observablesclassical or deterministic observables describe atoms and environmentdescribe atoms and environment
probabilities for infinitely many sub-states needed probabilities for infinitely many sub-states needed for computation of classical correlation functionsfor computation of classical correlation functions
probabilistic observablesprobabilistic observables can describe atom can describe atom only only
environment is integrated outenvironment is integrated out suitable system observables need only state of suitable system observables need only state of
system for computation of expectation values and system for computation of expectation values and correlationscorrelations
three probabilistic three probabilistic observablesobservables
characterize by vectorcharacterize by vector
each Aeach A(k)(k) can only take values ± 1 , can only take values ± 1 ,
““orthogonal spins”orthogonal spins” expectation values : expectation values :
elements of density elements of density matrixmatrix
probability weighted mean values of probability weighted mean values of basis unit observables are sufficient basis unit observables are sufficient to characterize the state of the systemto characterize the state of the system
ρρkk = ± 1 sharp value for A = ± 1 sharp value for A(k)(k)
in general:in general:
puritypurity
How many observables can have sharp How many observables can have sharp values ?values ?
depends ondepends on purity purity
P=1 : one sharp observable ok P=1 : one sharp observable ok
for two observables with sharp values :for two observables with sharp values :
puritypurity
for for ::
at most M discrete observables can be at most M discrete observables can be sharpsharp
consider P ≤ 1consider P ≤ 1
“ “ three spins , at most one sharp “three spins , at most one sharp “
density matrixdensity matrix
define hermitean 2x2 matrix :define hermitean 2x2 matrix :
properties of density matrix properties of density matrix
M – state quantum M – state quantum mechanicsmechanics
density matrix for P ≤ M+1 :density matrix for P ≤ M+1 :
choice of M depends on observables choice of M depends on observables consideredconsidered
restricted by maximal number of restricted by maximal number of “commuting observables”“commuting observables”
quantum mechanics forquantum mechanics forisolated systemsisolated systems
classical ensemble admits infinitely many classical ensemble admits infinitely many observables (observables (atom and its environmentatom and its environment))
we want to describe isolated subsystem we want to describe isolated subsystem ( ( atomatom ) : finite number of independent ) : finite number of independent observablesobservables
““isolated” situation : subset of the possible isolated” situation : subset of the possible probability distributionsprobability distributions
not all observables simultaneously sharp in not all observables simultaneously sharp in this subsetthis subset
given purity : conserved by time evolution given purity : conserved by time evolution if subsystem is perfectly isolatedif subsystem is perfectly isolated
different M describe different subsystems different M describe different subsystems ( ( atom or moleculeatom or molecule ) )
density matrix for density matrix for two quantum statestwo quantum states
hermitean 2x2 matrix :hermitean 2x2 matrix :
P ≤ 1P ≤ 1
“ “ three spins , at most one three spins , at most one sharp “sharp “
operators do not operators do not commutecommute
at this stage : convenient way to at this stage : convenient way to express expectation valuesexpress expectation values
deeper reasons behind it …deeper reasons behind it …
rotated spinsrotated spins
correspond to rotated unit vector ecorrespond to rotated unit vector ekk
new two-level observablesnew two-level observables expectation values given byexpectation values given by
only density matrix needed for only density matrix needed for computation of expectation values , computation of expectation values ,
not full classical probability not full classical probability distribution distribution
pure statespure states
pure states show no dispersion with pure states show no dispersion with respect to one observable Arespect to one observable A
recall classical statistics definitionrecall classical statistics definition
quantum pure states are quantum pure states are classical pure statesclassical pure states
probability vanishing except for one probability vanishing except for one micro-statemicro-state
pure state density matrixpure state density matrix
elements elements ρρkk are vectors on unit are vectors on unit spheresphere
can be obtained by unitary can be obtained by unitary transformationstransformations
SO(3) equivalent to SU(2)SO(3) equivalent to SU(2)
wave functionwave function
““root of pure state density matrix “root of pure state density matrix “
quantum law for expectation valuesquantum law for expectation values
transition probabilitytransition probability
time evolution of probabilities time evolution of probabilities
( fixed ( fixed observables )observables )
induces transition probability matrixinduces transition probability matrix
reduced transition reduced transition probabilityprobability
induced evolutioninduced evolution
reduced transition probability matrixreduced transition probability matrix
evolution of elements of evolution of elements of density matrixdensity matrix
infinitesimal time variationinfinitesimal time variation
scaling + rotationscaling + rotation
time evolution of density time evolution of density matrixmatrix
Hamilton operator and scaling factorHamilton operator and scaling factor
Quantum evolution and the rest ?Quantum evolution and the rest ?
λλ=0 and pure state :=0 and pure state :
quantum time evolutionquantum time evolution
It is easy to construct explicit It is easy to construct explicit ensembles whereensembles where
λλ = 0 = 0
quantum time evolutionquantum time evolution
evolution of purityevolution of purity
change of puritychange of purity
attraction to randomness :attraction to randomness :decoherencedecoherence
attraction to purity :attraction to purity :syncoherencesyncoherence
classical statistics can classical statistics can describe describe
decoherence and decoherence and syncoherence !syncoherence !
unitary quantum evolution : unitary quantum evolution : special casespecial case
pure state fixed pointpure state fixed point
pure states are special :pure states are special :
“ “ no state can be purer than pure “no state can be purer than pure “
fixed point of evolution forfixed point of evolution for
approach to fixed pointapproach to fixed point
approach to pure state approach to pure state fixed pointfixed point
solution :solution :
syncoherence describes exponential syncoherence describes exponential approach to pure state ifapproach to pure state if
decay of mixed atom state to ground decay of mixed atom state to ground statestate
purity conserving evolution :purity conserving evolution :subsystem is well isolatedsubsystem is well isolated
non-commuting non-commuting operatorsoperators
15 spin observables labeled by15 spin observables labeled by
density matrixdensity matrix
entanglemententanglement three commuting observablesthree commuting observables
LL11 : bit 1 , L : bit 1 , L22 : bit 2 L : bit 2 L33 : product of : product of two bitstwo bits
expectation values of associated expectation values of associated observables related to probabilities to observables related to probabilities to measure the combinations (++) , etc.measure the combinations (++) , etc.
““classical” entangled classical” entangled statestate
pure state with maximal anti-pure state with maximal anti-correlation of two bitscorrelation of two bits
bit 1: random , bit 2: randombit 1: random , bit 2: random if bit 1 = 1 necessarily bit 2 = -1 , if bit 1 = 1 necessarily bit 2 = -1 ,
and vice versaand vice versa
classical state described by classical state described by entangled density matrixentangled density matrix
classical correlationclassical correlation pointwise multiplication of classical pointwise multiplication of classical
observables on the level of sub-statesobservables on the level of sub-states not available on level of probabilistic not available on level of probabilistic
observablesobservables
definition depends on details of classical definition depends on details of classical observables , while many different observables , while many different classical observables correspond to the classical observables correspond to the same probabilistic observablesame probabilistic observable
classical correlation depends on classical correlation depends on probability distribution for the atom and probability distribution for the atom and its environmentits environment
needed : correlation that can be formulated needed : correlation that can be formulated in terms of probabilistic observables and in terms of probabilistic observables and density matrix !density matrix !
pointwise or conditional pointwise or conditional correlation ?correlation ?
Pointwise correlation appropriate if two Pointwise correlation appropriate if two measurements do not influence each othermeasurements do not influence each other..
Conditional correlation takes into account that Conditional correlation takes into account that system has been changed after first system has been changed after first measurement.measurement.
Two measurements of same observable Two measurements of same observable immediately after each other should yield the immediately after each other should yield the same value !same value !
pointwise correlationpointwise correlation
pointwise product of observablespointwise product of observables
does not describe A² =1:does not describe A² =1:
αα==σσ
conditional correlationsconditional correlations
probability to find value +1 for probability to find value +1 for product product
of measurements of A and Bof measurements of A and B
… … can be expressed in can be expressed in terms of expectation valueterms of expectation valueof A in eigenstate of Bof A in eigenstate of B
probability to find A=1 probability to find A=1 after measurement of B=1after measurement of B=1
conditional productconditional product
conditional product of observablesconditional product of observables
conditional correlationconditional correlation
does it commute ?does it commute ?
conditional product and conditional product and anticommutatorsanticommutators
conditional two point correlation conditional two point correlation commutescommutes
==
quantum correlationquantum correlation
conditional correlation in classical conditional correlation in classical statistics equals quantum correlation statistics equals quantum correlation !!
no contradiction to Bell’s no contradiction to Bell’s inequalities or to Kochen-Specker inequalities or to Kochen-Specker TheoremTheorem
conditional three point conditional three point correlation in quantum correlation in quantum
languagelanguage conditional three point conditional three point
correlation is not commuting !correlation is not commuting !
conditional correlations conditional correlations and quantum operatorsand quantum operators
conditional correlations in classical conditional correlations in classical statistics can be expressed in terms statistics can be expressed in terms of operator products in quantum of operator products in quantum mechanicsmechanics
non – commutativitynon – commutativityof operator productof operator productis closely related tois closely related to
conditional correlations !conditional correlations !
conclusionconclusion
quantum statistics arises from classical quantum statistics arises from classical statisticsstatistics
states, superposition , interference , states, superposition , interference , entanglement , probability amplitudesentanglement , probability amplitudes
quantum evolution embedded in quantum evolution embedded in classical evolutionclassical evolution
conditional correlations describe conditional correlations describe measurements both in quantum theory measurements both in quantum theory and classical statisticsand classical statistics
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