Entropy in the Quantum World

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Entropy in theQuantum World

• Panagiotis Aleiferis
• EECS 598, Fall 2001

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Outline

• Entropy in the classic world
• Theoretical background
• Density matrix
• Properties of the density matrix
• The reduced density matrix
• Shannons entropy
• Entropy in the quantum world
• Definition and basic properties
• Some useful theorems
• Applications
• Entropy as a measure of entanglement
• References

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Entropy in the classic world

• Murphys Laws

If something can go wrong, it will!
The more we complicate the plan, the greater the
chance of failure.
Nothing is ever so bad, that it can't get worse.
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Why does heat always flow from warm to cold?
1st law of thermodynamics 2nd law of
thermodynamics There is some degradation of the
total energy U in the system, some non-useful
heat, in any thermodynamic process.
?W
?Q
?U
Rudolf Clausius (1822 - 1988)
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The more disordered the energy, the less useful
it can be!
When energy is degraded, the atoms become more
disordered, the entropy increases! At
equilibrium, the system will be in its most
probable state and the entropy will be maximum.
Ludwig Boltzmann (1844 - 1906)
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All possible microstates of 4 coins

Three heads, one tails
Two heads, two tails
One heads, three tails
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• Boltzmann statistics 5 dipoles in external field

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• General Relations of Boltzmann statistics
• For a system in equilibrium at temperature T
• Statistical entropy

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Theoretical Background

• The density matrix ?
• In most cases we do NOT completely know the exact
  state of the system. We can estimate the
  probabilities Pi that the system is in the states
  ?igt.
• Our system is in an ensemble of pure states
  Pi,?igt.

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Define
tr(?)1
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• Properties of the density matrix
• tr(?)1
• ? is a positive operator
• (positive, means is real,
  non-negative, )
• if a unitary operator U is applied, the density
  matrix transforms as
• ? corresponds to a pure state, if and only if
• ? corresponds to a mixed state, if and only if

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• if we choose the energy eigenfunctions for our
  basis set, then H and ? are both diagonal, i.e.
• in any other representation ? may or may not be
  diagonal, but generally it will be symmetric,
  i.e.
• Detailed balance is essential so that
  equilibrium is maintained (i.e. probabilities do
  NOT explicitly depend on time).

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• The reduced density matrix
• What happens if we want to describe a subsystem
  of the composite system?
• Divide our system AB into parts A, B.
• Reduced density matrix for the subsystem A
• where trB partial trace over subsystem B

trace over subspace of system B
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Shannons entropy

• Definition
• How much information we gain, on average, when we
  learn the value of a random variable X?
• OR equivalently,
• What is the uncertainty, on average, about X
  before we learn its value?
• If p1, p2, ,pn the probability distribution of
  the n possible values of X

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• By definition 0log20 0
• (events with zero probability do not
  contribute to entropy.)
• Entropy H(X) depends only on the respective
  probabilities of the individual events Xi !
• Why is the entropy defined this way?
• It gives the minimal physical resources
  required to store information so that at a later
  time the information can be reconstructed.
• - Shannons noiseless coding theorem.

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• Example of Shannons noiseless coding theorem
• Code 4 symbols 1, 2, 3, 4 with
  probabilities 1/2, 1/4, 1/8, 1/8.
• Code without compression
• But, what happens if we use this code
  instead?
• Average string length for the second code
• Note
  !!!

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• Joint and Conditional Entropy
• A pair (X,Y) of random variables.
• Joint entropy of X and Y
• Entropy of X conditional on knowing Y
• Mutual Information
• How much do X, Y have in common?
• Mutual information of X and Y

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H(X)
H(Y)
H(XY)
H(YX)
H(YX)

• , equality when Y
  f(X)
• Subadditivity
  ,
• equality when X, Y are independent variables.

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Entropy in the quantum world

• Von Neumanns entropy
• Probability distributions replaced by the density
  matrix ?. Von Neumanns definition
• If ?i are the eigenvalues of ?, use the
  equivalent definition

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• Basic properties of Von Neumanns entropy
• , equality if and only if in
  pure state.
• In a d-dimensional Hilbert space
  ,
• the equality if and only if in a completely
  mixed state, i.e.
• If system AB in a pure state, then

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• Triangle inequality and subadditivity
• with
• Both these inequalities hold for Shannons
  entropy H.

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• Strong subadditivity
• First inequality also holds for Shannons
  entropy H, since
• BUT, for Von Neumanns entropy it is possible
  that
• However, somehow nature conspires so that
  both of these inequalities are NOT true
  simultaneously!

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Applications

• Entropy as a measure of entanglement
• Entropy is a measure of the uncertainty about a
  quantum system before we make a measurement of
  its state.
• For a d-dimensional Hilbert space

Pure state
Completely mixed state
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• Example
• Consider two 4-qbit systems with initial
  states
• Which one is more entangled ?

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• Partial measurement randomizes the initially pure
  states.
• The entropy of the resulting mixed states
  measures the amount of this randomization!
• The larger the entropy, the more randomized the
  state after the measurement is, the more
  entangled the initial state was!
• We have to go through evaluating the density
  matrix of the randomized states

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• System 1
• Trace over (any) 1 qbit
• Trace over (any) 2 qbits

Pure state
?1,20 , ?3,41/2
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• Trace over (any) 3 qbits
• Summary
• 1. initially
• 2. measure (any) 1 qbit
• 3. measure (any) 2 qbits
• 4. measure (any) 3 qbits

?1,21/2
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• System 2
• Trace over (any) 1 qbit

diagonal
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• Trace over (any) 2 qbits
• Trace over (any) 3 qbits

?10, ?2,31/6, ?42/3
?1,21/2
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Summary 1. initially 2. measure (any) 1 qbit 3.
measure (any) 2 qbits 4. measure (any) 3
qbits Therefore, ?2 is more entangled than ?1.
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• Ludwin Boltzmann, who spent much of his life
  studying statistical mechanics, died in 1906, by
  his own hand. Paul Ehrenfest, carrying on the
  work, died similarly in 1933. Now it is our turn
  to study statistical mechanics.
• - States of Matter, D. Goodstein

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References

• Quantum Computation and Quantum Information,
• Nielsen Chuang, Cambridge Univ. Press,
  2000
• Quantum Mechanics,
• Eugen Merzbacher, Wiley, 1998
• Lecture notes by C. Monroe (PHYS 644, Univ. of
  Michigan)
• coursetools.ummu.umich.edu/2001/fall/physics
  /644/001.nsf
• Lecture notes by J. Preskill (PHYS 219, Caltech)
• www.theory.caltech.edu/people/preskill/ph229