Quantum Mechanics from Classical Statistics

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Quantum Mechanicsfrom Classical Statistics
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what is an atom ?

• quantum mechanics isolated object
• quantum field theory excitation of complicated
  vacuum
• 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
• entanglement
• interference
• superposition of states
• fermions and bosons
• unitary time evolution
• transition amplitude
• non-commuting operators
• violation of Bells inequalities

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statistical picture of the world

• basic theory is not deterministic
• basic theory makes only statements about
  probabilities for sequences of events and
  establishes correlations
• probabilism is fundamental , not determinism !

quantum mechanics from classical statistics not
a deterministic hidden variable theory
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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
• 4) unitary time evolution for isolated
  subsystems

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classical statistical implementation of quantum
computer
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classical ensemble , discrete observable

• Classical ensemble with probabilities
• qubit
• one discrete observable A , values 1 or -1
• probabilities to find A1 w and A-1 w-

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classical ensemble for one qubit

• classical states labeled by
• state of subsystem depends on three numbers
• expectation value of qubit

eight states
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classical probability distribution
characterizes subsystem
different dpe characterize environment
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state of system independent of environment

• ?j does not depend on precise choice of dpe

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time evolution

• rotations of ?k

example
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time evolution of classical probability

• evolution of ps according to evolution of ?k
• evolution of dpe arbitrary , consistent with
  constraints

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state after finite rotation
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this realizes Hadamard gate
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purity
consider ensembles with P 1
purity conserved by time evolution
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density matrix

• define hermitean 2x2 matrix
• properties of density matrix

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operators

• if observable obeys
• associate hermitean operators

in our case e31 , e1e20
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quantum law for expectation values
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pure state
P 1 ?2 ?
wave function
unitary time evolution
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Hadamard gate
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CNOT gate
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Four state quantum system- two qubits -
k1, ,15 P 3
normalized SU(4) generators
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four state quantum system
P 3
pure state P 3 and copurity
must vanish
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suitable rotation of ?k
yields transformation of the density matrix
and realizes CNOT gate
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classical probability distributionfor 215
classical states
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probabilistic observables

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

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probabilistic quantum observable
spectrum ?a probability that ?a is measured
wa can be computed from state of subsystem
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non commuting quantum operators

• for two qubits
• all Lk represent two level observables
• they do not commute
• the laws of quantum mechanics for expectation
  values are realized
• uncertainty relation etc.

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incomplete statistics
joint probabilities depend on environment and are
not available for subsystem !
ppsdpe
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quantum mechanics from classical statistics

• probability amplitude ?
• entanglement
• interference
• superposition of states
• fermions and bosons
• unitary time evolution ?
• transition amplitude
• non-commuting operators ?
• violation of Bells inequalities

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conditional correlations
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classical correlation

• 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 !
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conditional probability

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

probability to find A1 after measurement of B1
can be expressed in terms of expectation
value of A in eigenstate of B
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measurement correlation
After measurement A1 the system must be in
eigenstate with this eigenvalue. Otherwise
repetition of measurement could give a different
result !
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measurement changes statein all statistical
systems !quantum and classicaleliminates
possibilities that are not realized
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physics makes statements about
possiblesequences of events and their
probabilities
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unique eigenstates for M2
M 2
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eigenstates with A 1
measurement preserves pure states if projection
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measurement correlation equals quantum correlation
probability to measure A1 and B1
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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
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sequence of three measurements and quantum
commutator
two measurements commute , not three
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conclusion

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

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quantum particle from classical statistics

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

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end ?
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time evolution
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transition probability

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

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reduced transition probability

• induced evolution
• reduced transition probability matrix

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evolution of elements of density matrix in two
state quantum system

• infinitesimal time variation
• scaling rotation

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time evolution of density matrix

• Hamilton operator and scaling factor
• Quantum evolution and the rest ?

?0 and pure state
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quantum time evolution

• It is easy to construct explicit ensembles where
• ? 0
• 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
• 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 andentanglement
ensembles with P3
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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 P3

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entanglement

• three commuting observables
• L1 bit 1 , L2 bit 2 L3 product of two
  bits
• 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
• bit 1 random , bit 2 random
• 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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end
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pure state density matrix

• elements ?k are vectors on unit sphere
• can be obtained by unitary transformations
• SO(3) equivalent to SU(2)

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wave function

• root of pure state density matrix
• quantum law for expectation values