Quantum Mechanics from Classical Statistics what is an

  • Slides: 68
Download presentation
Quantum Mechanics from Classical Statistics

Quantum Mechanics from Classical Statistics

what is an atom ? quantum mechanics : isolated object n quantum field theory

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

quantum mechanics can be described by classical statistics !

quantum mechanics can be described by classical statistics !

quantum mechanics from classical statistics n n n n n probability amplitude entanglement interference

quantum mechanics from classical statistics n n n n n probability amplitude entanglement interference superposition of states fermions and bosons unitary time evolution transition amplitude non-commuting operators violation of Bell’s inequalities

statistical picture of the world basic theory is not deterministic n basic theory makes

statistical picture of the world basic theory is not deterministic n basic theory makes only statements about probabilities for sequences of events and establishes correlations n probabilism is fundamental , not determinism ! n quantum mechanics from classical statistics not a deterministic hidden variable theory

essence of quantum mechanics description of appropriate subsystems of classical statistical ensembles 1) equivalence

essence of quantum mechanics description of appropriate subsystems of classical statistical ensembles 1) equivalence classes of probabilistic observables 2) incomplete statistics 3) correlations between measurements based on conditional probabilities

classical statistical implementation of quantum computer

classical statistical implementation of quantum computer

classical ensemble , discrete observable n Classical ensemble with probabilities qubit : one discrete

classical ensemble , discrete observable n Classical ensemble with probabilities qubit : one discrete observable A , values +1 or -1 probabilities to find A=1 : w+ and A=-1: wn

classical ensemble for one qubit n classical states labeled by eight states n state

classical ensemble for one qubit n classical states labeled by eight states n state of subsystem depends on three numbers n expectation value of qubit

classical probability distribution characterizes subsystem different δpe characterize environment

classical probability distribution characterizes subsystem different δpe characterize environment

state of system independent of environment n ρj does not depend on precise choice

state of system independent of environment n ρj does not depend on precise choice of δpe

time evolution rotations of ρk example :

time evolution rotations of ρk example :

time evolution of classical probability n evolution of ps according to evolution of ρk

time evolution of classical probability n evolution of ps according to evolution of ρk n evolution of δpe arbitrary , consistent with constraints

state after finite rotation

state after finite rotation

this realizes Hadamard gate

this realizes Hadamard gate

purity consider ensembles with P ≤ 1 purity conserved by time evolution

purity consider ensembles with P ≤ 1 purity conserved by time evolution

density matrix n define hermitean 2 x 2 matrix : n properties of density

density matrix n define hermitean 2 x 2 matrix : n properties of density matrix

operators if observable obeys associate hermitean operators in our case : e 3=1 ,

operators if observable obeys associate hermitean operators in our case : e 3=1 , e 1=e 2=0

quantum law for expectation values

quantum law for expectation values

pure state P=1 wave function unitary time evolution ρ2 = ρ

pure state P=1 wave function unitary time evolution ρ2 = ρ

Hadamard gate

Hadamard gate

CNOT gate

CNOT gate

Four state quantum system - two qubits k=1, …, 15 P ≤ 3 normalized

Four state quantum system - two qubits k=1, …, 15 P ≤ 3 normalized SU(4) – generators :

four – state quantum system P≤ 3 pure state : P = 3 and

four – state quantum system P≤ 3 pure state : P = 3 and copurity must vanish

suitable rotation of ρk yields transformation of the density matrix and realizes CNOT gate

suitable rotation of ρk yields transformation of the density matrix and realizes CNOT gate

classical probability distribution for 215 classical states

classical probability distribution for 215 classical states

probabilistic observables for a given state of the subsystem , specified by {ρk} :

probabilistic observables for a given state of the subsystem , specified by {ρk} : The possible measurement values +1 and -1 of the discrete two - level observables are found with probabilities w+(ρk) and w-(ρk). In a quantum state the observables have a probabilistic distribution of values , rather than a fixed value as for classical states.

probabilistic quantum observable spectrum { γα } probability that γα is measured : wα

probabilistic quantum observable spectrum { γα } probability that γα is measured : wα can be computed from state of subsystem

non – commuting quantum operators for two qubits : n all Lk represent two

non – commuting quantum operators for two qubits : n all Lk represent two – level observables n they do not commute n n the laws of quantum mechanics for expectation values are realized uncertainty relation etc.

incomplete statistics joint probabilities depend on environment and are not available for subsystem !

incomplete statistics joint probabilities depend on environment and are not available for subsystem ! p=ps+δpe

quantum mechanics from classical statistics n n n n n probability amplitude ☺ entanglement

quantum mechanics from classical statistics n n n n n probability amplitude ☺ entanglement interference superposition of states fermions and bosons unitary time evolution ☺ transition amplitude non-commuting operators ☺ violation of Bell’s inequalities

conditional correlations

conditional correlations

classical correlation n n point wise multiplication of classical observables on the level of

classical correlation n n point wise multiplication of classical observables on the level of classical states classical correlation depends on probability distribution for the atom and its environment 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 needed : correlation that can be formulated in terms of probabilistic observables and density matrix !

conditional probability to find value +1 for product of measurements of A and B

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

measurement correlation After measurement A=+1 the system must be in eigenstate with this eigenvalue.

measurement correlation After measurement A=+1 the system must be in eigenstate with this eigenvalue. Otherwise repetition of measurement could give a different result !

measurement changes state in all statistical systems ! quantum and classical eliminates possibilities that

measurement changes state in all statistical systems ! quantum and classical eliminates possibilities that are not realized

physics makes statements about possible sequences of events and their probabilities

physics makes statements about possible sequences of events and their probabilities

unique eigenstates for M=2:

unique eigenstates for M=2:

eigenstates with A = 1 measurement preserves pure states if projection

eigenstates with A = 1 measurement preserves pure states if projection

measurement correlation equals quantum correlation probability to measure A=1 and B=1 :

measurement correlation equals quantum correlation probability to measure A=1 and B=1 :

probability that A and B have both the value +1 in classical ensemble not

probability that A and B have both the value +1 in classical ensemble not a property of the subsystem probability to measure A and B both +1 can be computed from the subsystem

sequence of three measurements and quantum commutator two measurements commute , not three

sequence of three measurements and quantum commutator two measurements commute , not three

conclusion quantum statistics arises from classical statistics states, superposition , interference , entanglement ,

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

quantum particle from classical statistics quantum and classical particles can be described within the

quantum particle from classical statistics quantum and classical particles can be described within the same classical statistical setting n different time evolution , corresponding to different Hamiltonians n continuous interpolation between quantum and classical particle possible ! n

end ?

end ?

time evolution

time evolution

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

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

reduced transition probability n induced evolution n reduced transition probability matrix

reduced transition probability n induced evolution n reduced transition probability matrix

evolution of elements of density matrix in two – state quantum system n infinitesimal

evolution of elements of density matrix in two – state quantum system n infinitesimal time variation n scaling + rotation

time evolution of density matrix n Hamilton operator and scaling factor n Quantum evolution

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

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

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

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

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

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

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

pure state fixed point pure states are special : “ no state can be

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

approach to pure state fixed point solution : syncoherence describes exponential approach to pure

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

purity conserving evolution : subsystem is well isolated

purity conserving evolution : subsystem is well isolated

two bit system and entanglement ensembles with P=3

two bit system and entanglement ensembles with P=3

non-commuting operators 15 spin observables labeled by density matrix

non-commuting operators 15 spin observables labeled by density matrix

SU(4) - generators

SU(4) - generators

density matrix n pure states : P=3

density matrix n pure states : P=3

entanglement n three commuting observables L 1 : bit 1 , L 2 :

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

“classical” entangled state n pure state with maximal anti-correlation of two bits bit 1:

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

classical state described by entangled density matrix

classical state described by entangled density matrix

entangled quantum state

entangled quantum state

end

end

pure state density matrix elements ρk are vectors on unit sphere n can be

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

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

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