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Emergence of Quantum Mechanics from Classical Statistics

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what is an atom ? quantum mechanics : isolated object quantum mechanics : isolated object quantum field theory : excitation of complicated vacuum quantum field theory : excitation of complicated vacuum classical statistics : sub-system of ensemble with infinitely many degrees of freedom classical statistics : sub-system of ensemble with infinitely many degrees of freedom

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quantum mechanics can be described by classical statistics !

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quantum mechanics from classical statistics probability amplitude probability amplitude entanglement entanglement interference interference superposition of states superposition of states fermions and bosons fermions and bosons unitary time evolution unitary time evolution transition amplitude transition amplitude non-commuting operators non-commuting operators

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probabilistic observables Holevo; Beltrametti,Bugajski

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classical ensemble, discrete observable Classical ensemble with probabilities Classical ensemble with probabilities one discrete observable A, values +1 or -1 one discrete observable A, values +1 or -1

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effective micro-states group states together σ labels effective micro-states, t σ labels sub-states in effective micro-states σ : probabilities to find A=1 : and A=-1: mean value in micro-state σ :

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expectation values only measurements +1 or -1 possible !

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probabilistic observables have a probability distribution of values in a microstate, classical observables a sharp value

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deterministic and probabilistic observables classical or deterministic observables describe atoms and environment classical or deterministic observables describe atoms and environment probabilities for infinitely many sub-states needed for computation of classical correlation functions probabilities for infinitely many sub-states needed for computation of classical correlation functions probabilistic observables can describe atom only probabilistic observables can describe atom only environment is integrated out environment is integrated out suitable system observables need only state of system for computation of expectation values and correlations suitable system observables need only state of system for computation of expectation values and correlations

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three probabilistic observables characterize by vector characterize by vector each A (k) can only take values ± 1, each A (k) can only take values ± 1, “orthogonal spins” “orthogonal spins” expectation values : expectation values :

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density matrix and pure states

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elements of density matrix probability weighted mean values of basis unit observables are sufficient to characterize the state of the system probability weighted mean values of basis unit observables are sufficient to characterize the state of the system ρ k = ± 1 sharp value for A (k) ρ k = ± 1 sharp value for A (k) in general: in general:

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purity How many observables can have sharp values ? How many observables can have sharp values ? depends on purity depends on 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 :

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purity for : at most M discrete observables can be sharp consider P ≤ 1 “ three spins, at most one sharp “

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density matrix define hermitean 2x2 matrix : define hermitean 2x2 matrix : properties of density matrix properties of density matrix

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M – state quantum mechanics density matrix for P ≤ M+1 : density matrix for P ≤ M+1 : choice of M depends on observables considered choice of M depends on observables considered restricted by maximal number of “commuting observables” restricted by maximal number of “commuting observables”

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quantum mechanics for isolated systems classical ensemble admits infinitely many observables (atom and its environment) classical ensemble admits infinitely many observables (atom and its environment) we want to describe isolated subsystem ( atom ) : finite number of independent observables we want to describe isolated subsystem ( atom ) : finite number of independent observables “isolated” situation : subset of the possible probability distributions “isolated” situation : subset of the possible probability distributions not all observables simultaneously sharp in this subset not all observables simultaneously sharp in this subset given purity : conserved by time evolution if subsystem is perfectly isolated given purity : conserved by time evolution if subsystem is perfectly isolated different M describe different subsystems ( atom or molecule ) different M describe different subsystems ( atom or molecule )

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density matrix for two quantum states hermitean 2x2 matrix : hermitean 2x2 matrix : P ≤ 1 P ≤ 1 “ three spins, at most one sharp “

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operators hermitean operators

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quantum law for expectation values

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operators do not commute at this stage : convenient way to express expectation values deeper reasons behind it …

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rotated spins correspond to rotated unit vector e k correspond to rotated unit vector e k new two-level observables new two-level observables expectation values given by expectation values given by only density matrix needed for computation of expectation values, only density matrix needed for computation of expectation values, not full classical probability distribution not full classical probability distribution

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pure states pure states show no dispersion with respect to one observable A pure states show no dispersion with respect to one observable A recall classical statistics definition recall classical statistics definition

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quantum pure states are classical pure states probability vanishing except for one micro-state probability vanishing except for one micro-state

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pure state density matrix elements ρ k are vectors on unit sphere elements ρ k are vectors on unit sphere can be obtained by unitary transformations can be obtained by unitary transformations SO(3) equivalent to SU(2) SO(3) equivalent to SU(2)

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wave function “root of pure state density matrix “ “root of pure state density matrix “ quantum law for expectation values quantum law for expectation values

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time evolution

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transition probability time evolution of probabilities ( fixed observables ) ( fixed observables ) induces transition probability matrix

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reduced transition probability induced evolution induced evolution reduced transition probability matrix reduced transition probability matrix

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evolution of elements of density matrix infinitesimal time variation infinitesimal time variation scaling + rotation scaling + rotation

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time evolution of density matrix Hamilton operator and scaling factor Hamilton operator and scaling factor Quantum evolution and the rest ? Quantum evolution and the rest ? λ=0 and pure state :

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quantum time evolution It is easy to construct explicit ensembles where λ = 0 λ = 0 quantum time evolution quantum time evolution

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evolution of purity change of purity attraction to randomness : decoherence attraction to purity : syncoherence

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classical statistics can describe decoherence and syncoherence ! unitary quantum evolution : special case

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pure state fixed point pure states are special : “ no state can be purer than pure “ “ no state can be purer than pure “ fixed point of evolution for approach to fixed point

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approach to pure state fixed point solution : syncoherence describes exponential approach to pure state if decay of mixed atom state to ground state

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purity conserving evolution : subsystem is well isolated

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two bit system and entanglement ensembles with P=3

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non-commuting operators 15 spin observables labeled by density matrix

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SU(4) - generators

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density matrix pure states : P=3 pure states : P=3

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entanglement three commuting observables three commuting observables L 1 : bit 1, L 2 : bit 2 L 3 : product of two bits L 1 : bit 1, L 2 : bit 2 L 3 : product of two bits expectation values of associated observables related to probabilities to measure the combinations (++), etc. expectation values of associated observables related to probabilities to measure the combinations (++), etc.

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“classical” entangled state pure state with maximal anti-correlation of two bits pure state with maximal anti-correlation of two bits bit 1: random, bit 2: random bit 1: random, bit 2: random if bit 1 = 1 necessarily bit 2 = -1, and vice versa if bit 1 = 1 necessarily bit 2 = -1, and vice versa

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classical state described by entangled density matrix

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entangled quantum state

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conditional correlations

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classical correlation pointwise multiplication of classical observables on the level of sub-states pointwise multiplication of classical observables on the level of sub-states not available on level of probabilistic observables not available on level of probabilistic observables definition depends on details of classical observables, while many different classical observables correspond to the same probabilistic observable definition depends on details of classical observables, while many different classical observables correspond to the same probabilistic observable classical correlation depends on probability distribution for the atom and its environment classical correlation depends on probability distribution for the atom and its environment needed : correlation that can be formulated in terms of probabilistic observables and density matrix !

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pointwise or conditional correlation ? Pointwise correlation appropriate if two measurements do not influence each other. Pointwise correlation appropriate if two measurements do not influence each other. Conditional correlation takes into account that system has been changed after first measurement. Conditional correlation takes into account that system has been changed after first measurement. Two measurements of same observable immediately after each other should yield the same value ! Two measurements of same observable immediately after each other should yield the same value !

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pointwise correlation pointwise product of observables does not describe A² =1: α=σα=σα=σα=σ

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conditional correlations probability to find value +1 for product probability to find value +1 for product of measurements of A and B of measurements of A and B … can be expressed in terms of expectation value of A in eigenstate of B probability to find A=1 after measurement of B=1

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conditional product conditional product of observables conditional product of observables conditional correlation conditional correlation does it commute ? does it commute ?

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conditional product and anticommutators conditional two point correlation commutes conditional two point correlation commutes =

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quantum correlation conditional correlation in classical statistics equals quantum correlation ! conditional correlation in classical statistics equals quantum correlation ! no contradiction to Bell’s inequalities or to Kochen-Specker Theorem no contradiction to Bell’s inequalities or to Kochen-Specker Theorem

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conditional three point correlation

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conditional three point correlation in quantum language conditional three point correlation is not commuting ! conditional three point correlation is not commuting !

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conditional correlations and quantum operators conditional correlations in classical statistics can be expressed in terms of operator products in quantum mechanics conditional correlations in classical statistics can be expressed in terms of operator products in quantum mechanics

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non – commutativity of operator product is closely related to conditional correlations !

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conclusion quantum statistics arises from classical statistics quantum statistics arises from classical statistics states, superposition, interference, entanglement, probability amplitudes states, superposition, interference, entanglement, probability amplitudes quantum evolution embedded in classical evolution quantum evolution embedded in classical evolution conditional correlations describe measurements both in quantum theory and classical statistics conditional correlations describe measurements both in quantum theory and classical statistics

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