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 :

density matrix and density matrix and pure statespure states

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 “

operatorsoperators

hermitean operatorshermitean operators

quantum law for quantum law for expectation valuesexpectation values

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

time evolutiontime evolution

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

two bit system andtwo bit system andentanglemententanglement

ensembles with P=3ensembles with P=3

non-commuting non-commuting operatorsoperators

15 spin observables labeled by15 spin observables labeled by

density matrixdensity matrix

SU(4) - generatorsSU(4) - generators

density matrixdensity matrix

pure states : P=3pure states : P=3

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

entangled quantum stateentangled quantum state

conditional correlationsconditional correlations

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 correlationcorrelation

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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