Quantum Mechanics from Classical Statistics what is an
- Slides: 68
Quantum Mechanics from Classical Statistics
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 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 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 classes of probabilistic observables 2) incomplete statistics 3) correlations between measurements based on conditional probabilities
classical statistical implementation of quantum computer
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 of subsystem depends on three numbers n expectation value of qubit
classical probability distribution characterizes subsystem different δpe characterize environment
state of system independent of environment n ρj does not depend on precise choice of δpe
time evolution rotations of ρk example :
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
this realizes Hadamard gate
purity consider ensembles with P ≤ 1 purity conserved by time evolution
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 , e 1=e 2=0
quantum law for expectation values
pure state P=1 wave function unitary time evolution ρ2 = ρ
Hadamard gate
CNOT gate
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 copurity must vanish
suitable rotation of ρk yields transformation of the density matrix and realizes CNOT gate
classical probability distribution for 215 classical states
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 can be computed from state of subsystem
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 ! p=ps+δpe
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
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 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. Otherwise repetition of measurement could give a different result !
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
unique eigenstates for M=2:
eigenstates with A = 1 measurement preserves pure states if projection
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 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
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 same classical statistical setting n different time evolution , corresponding to different Hamiltonians n continuous interpolation between quantum and classical particle possible ! n
end ?
time evolution
transition probability time evolution of probabilities ( fixed observables ) induces 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 time variation n scaling + rotation
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 evolution
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
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 state if decay of mixed atom state to ground state
purity conserving evolution : subsystem is well isolated
two bit system and entanglement ensembles with P=3
non-commuting operators 15 spin observables labeled by density matrix
SU(4) - generators
density matrix n pure states : P=3
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: random , bit 2: random n if bit 1 = 1 necessarily bit 2 = -1 , and vice versa n
classical state described by entangled density matrix
entangled quantum state
end
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 expectation values
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