Quantum Information Theory

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Quantum Information Theory

• Graduate Course - Spring 2005
• Lecture 5 2/21/05
• Marco Lanzagorta Jeff Tollaksen
• George Mason University

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Von Neumann Entropy unifies aspects of QI

• Transmission of classical info over quantum
  channels (i.e. what is the max info in bits that
  can be obtained using best measmts)
• measure of success Prob of error - 0
• Tradeoff between gaining info about a quantum
  state and disturbance of the state
• Quantifying entanglement E(?AB?)S(?A)
• Transmission of quantum info over quantum
  channels (compressibility of an ensemble of pure
  quantum states, i.e. min of qubits per letter
  of message needed to encode information)
• measure of success Fidelity - 1

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Basic properties of the von Neumann entropy
Classically, information is a function of prob
distribs quantum information is a function of
density matrices

• von Neumann entropy S(?)?-log ??? tr ? log ?

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Basic properties of the von Neumann entropy

• Suppose we have an ensemble of pure states ?
  ?iqi?k???k
• H(q) S(?)
• Only equal if the ??k are orthogonal, then
  quantum source reduces to a classical source
  all signal states can be distinguished

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Transmission of quantum info over quantum
channels-quantum analog of noiseless coding
theorem
How many qubits are needed to store the output of
the source, so the output can be reliably
recovered?
In this case each letter of the message is drawn
from an ensemble of pure states ?k? (e.g.
polarization states of a photon that are not
necessarily orthogonal) with probability pi Thus
each letter is described by a density matrix
The entire message is given by
How redundant is this message?
Smaller HS
?n

nR qubits
?n
n uses
compress
decompress
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The typical subspace
Devise quantum code to compress w/o loss of
fidelity develop notion of typical subspace
rather than typical sequence
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Quantum analog of noiseless coding theorem

• Use diagonal form
• A given sequence has
• The typical subspace has size
• Therefore, we can Schumacher compress n?nS(?)

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• Von Neumann entropy is of qubits of quantum
  info carried per letter of the message
• Use large blocks of N independent inputs, w/ N
  large
• (N large so we can use properties of typical
  subspace)
• Use a coding scheme with NS(?) qubits that are
• Faithful for states in the typical subspace
• No good otherwise
• Can always compress unless ? ½I (cant compress
  random qubits)
• Schumachers protocol
• Project onto typical subspace
• If successful projection, then encode
• If not successful, then do nothing (according to
  the law of large s, this prob?0 as n??)
• It can then be shown that the average fidelity? 1

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Quantum data compression 1 letter example

• Suppose letters are single qubits taken from an
  ensemble
• Density matrix of each letter
• Eigenstates of the density matrix are qubits in n
  axis
• Which overlap w/ initial state

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Quantum data compression 1 letter example

• On the other hand
• Thus, if we dont know whether up z or up x was
  sent, the best guess we can make is 0 which
  has max fidelity
• For arbitrary input ?? this is 0.8535
• Thus by diagonalizing the density matrix we can
  decompose HS of one qubit into a likely 1-d
  subspace and an unlikely 1-d subspace

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Quantum data compression 3 letter example

• Suppose Alice needs to send 3 letters but can
  only afford 2 qubits
• She could send 2 of the 3 qubits and ask Bob to
  guess 0 for the third with F.8535 overall.
  Is there a better procedure?
• Yes, in the 1 letter example we saw that by
  diagonalizing the density matrix we can decompose
  HS of one qubit into a likely 1-d subspace and an
  unlikely 1-d subspace
• Similarly we can de-compose HS of 3 qubits into
  likely and unlikely subspaces and encode only the
  most likely

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Quantum data compression 3 letter example

• If the signal state is
• (where each of the qubits is either ?z? or
  ?x?)
• Then the de-composition of HS of 3 qubits into
  likely and unlikely subspaces is given by
• Likely space spanned by

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Quantum data compression 3 letter example

• If we make a fuzzy measurement that projects the
  signal onto the likely subspace then, probability
  of likely
• And probability of projecting onto un-likely
  sub-space
• E.g., Alice could apply a U that rotates the 4
  high prob basis states to ??0? and low-prob
  to ??1?
• Then Alice performs a measurement on the 3rd
  qubit. If the outcome is 0? then Alices input
  state has been projected onto the likely subspace

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Quantum data compression 3 letter example

• She then sends the remaining 2 unmeasured qubits
  to Bob
• When Bob receives this compressed 2-qubit state,
  he decompresses it by appending a 0? and
  applying U-1
• If Alices measurement of the third qubit gives
  1? then the best she can do is send to Bob the
  state that he will decompress as the most likely
  state 0?0?0?
• Thus, if Alice encodes 3-qubit signal state ??
  and sends 2 qubits to Bob who decodes them, then
  Bob obtains state

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Quantum data compression 3 letter example

• The fidelity of this procedure
• This is better than the procedure of sending 2 of
  the 3 qubits with perfect fidelity (total
  fidelity was 0.8535)
• With longer messages, the fidelity improves, the
  eigenvalues of the diagonalized density matrix
  are
• Von Neumann entropy of 1-qubit ensemble is S(?)
  H(cos2p/8).6009, thus we can shorten message by
  .6009

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Quantum data compression

• If Alice just sent classical information
  orthogonal quantum states then Bob could follow
  the previous procedure to re-construct Alices
  initial state and achieve high-fidelity
  compression to H(X) bits per letter
• But if states are drawn from non-orthogonal pure
  states, then this compression is not optimal
  classical information about preparation of state
  is redundant because non-orthogonal states cannot
  be perfectly distinguished
• Hence Schumacher coding can achieve optimal
  compression to S(?) qubits per letter but at a
  price
• Bob received message from Alice, but he doesnt
  know what it is
• Bob cant make a measurement that will determine
  Alices message correctly because the state would
  be disturbed

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Quantum data compression
decompression
compression
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Outline of Schumachers data compression
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Schumacher compression
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Fidelity

• Begin with and try to preserve it
• Wind up with
• Did we success?
• Bad criterion
• Good criterion subject to a strong
  observational test
• This gives the fidelity

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The idea of the proof is similar to Shannons
proof.
Two known proofs
One is a complicated kludge, done from
first principles.
The other proof is an elegant easy proof
that relies on other deep theorems.
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Some remarks about Schumacher compression
The states do not have to be orthogonal, in fact
it works better if they are not!
Classical messages require H S bits
Can also transmit entangled states with F?1 by
using S qubits.
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Entropy and Thermodynamics

• 2 approaches to quantum statistical physics
• Evolution of closed system and perform coarse
  graining to obtain thermodynamic variables
• Evolution of open system quantum system in
  contact with an environment track only system,
  dont monitor environ.
• If system A and environment E are initially
  uncorrelated
• And then the entropy is additive
• If we let the open system evolve via UAE, then

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Entropy and Thermodynamics

• Since unitary evolution preserves S
• Applying sub-additivity ?
• If A and E are un-correlated then these are equal
• Hence entropy of world cannot decrease
• Info initially encoded in system (i.e. ability to
  distinguish one state from another) is LOST and
  ends up encoded in entanglement between Sys-env
  (in principle recoverable but not in practice)

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Entropy and Thermodynamics

• Sys-env interactions leads to well-defined global
  properties while local properties become unc
• Simple derivation of decoherence (w/o ref to H)
• Specify of states of sys (gas g, HS dim ng)
• Specify of states of env (container c HS dim
  nc)
• Ensemble average of von Neumann entropy
  approaches maximum as nc increases

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Entropy and Thermodynamics

• Even though we dont know actual pure state of
  the whole almost all states have same local
  properties in terms of objective uncertainties
  described by finite local entropy
• This is generic for any bi-partite system and
  gives rise to thermodynamic behavior
• Highly entangled states are typical and so are
  uncertain local properties

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Thermodynamics of computation

• All computers produce waste heathow much waste
  heat is necessary?
• Landauer 1961 the only operations that are
  thermodynamically irrev are the logically irrev
• Bennet 1982 only inherently irrev operation is
  erasure of info
• Landauers principle thermo cost of erasure
• ? Qkt ln 2 per bit
• if answer has 40 bits, then min energy needed is
  40 kT

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Thermodynamics of computation

• If we could discrimate non-orthogonal states then
  we could violate 2nd law of thermodynamics
• Could make a closed loop in which non-orthog
  states are made orthog by perfect
  distinguishability then used to do work and
  finally return to same state
• No heat is dissipated but work is done

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Encoding classical information into quantum states
Alice prepares ?kQ probability pk
Bob measures decoding observable B and tries to
infer k
Mutual information H(KB)H(K)-H(KB) How much
classical info can a Quantum system carry?
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Encoding classical information into quantum states

• Potential problems
• Alice could use non-orthogonal states that cannot
  be perfectly distinguished
• Noise
• Bob can choose different decoding observables

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Encoding classical information into quantum states

• Alice prepares state ?kQ with probability pk
  ?QSpk ?kQ
• What is most efficient way to distinguish them?
  Define
• S(Spk ?kQ ) - S pkS(?kQ )??Q
• Entropy of avg signal average signal entropy
• ?Q measures distinguishability of the signal
  states ?kQ
• ?Q0 (by convexity of S)
• ?Q0 iff ?kQ all are the same
• ?QH(p) w/ equality iff the signals are
  orthogonal
• If signals are pure states, then ?QS(?Q)

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Encoding classical information into quantum states

• No measurement can provide more than ?Q bits of
  info about the preparation of Q
• S(Spk ?kQ ) - S pkS(?kQ ) ? ?Q
• Entropy of avg signal average signal entropy
• Note if ?kQ ?k???k then ?Q SQ
• This quantity depends not just on ? but on how it
  is realized as a ensemble of mixed states
• It is also similar to mutual info which tells us
  how much on average the shannon entropy of Y is
  reduced once we learn X, similarily ?Q tells us
  how much on avg the Von Neumann entropy of an
  ensemble is reduced when prep is known
• By suitable choice of code and decoding
  observable, the alphabet can be used to send up
  to ?Q bits of information per letter with
  arbitrarily low prob of error

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Encoding classical information into quantum states

• Alice prepares state ?kRQ with probability pk
  then
• ?Q ?RQ property of partial tr
• Distinguishability is not increased by discarding
  subsystem
• Distinguishability cannot be increased by
  dynamics
• Mutual info between input K and measurement A
• Holevo
• H(AK) ?A

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Accessible information what is the max info in
bits that can be obtained using best measurements

• The close analyogy between Holevo info and
  classical mutual info suggests that ?Q is related
  to amount of classical info that can be stored in
  a quantum source
• What is classical info content that can be
  extracted?

• Let Imax H(AK) (maximize over all decoding
  observables)
• I ?A , for example
• I H(AK)0.278 bits (can make I ?A iff ?kQ
  commute

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Accessible information example

• Suppose papbpc1/3, then ?A 1
• Best measurement gives I0.585 bits, so use 2
  copies
• 9 possible states aa?, ab?, etc.
• Use only aa?, bb?, cc?, entropy is ?Q1Q2
  SQ1Q21.369 bits
• This is 0.685 bits/system
• Improved H(AK) by using blocks of systems, only
  some signal states, and a good decoding obserable

Material in this talk used w/ permission from M.
Nielsen
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Error correction overview

• Digitization of errors project back onto state
  w/ no errors
• Measure errors without measuring data
• Errors are local but the quantum information is
  stored in a non-local way
• Info is stored in correlations amongst several
  qubits
• Assumption errors affecting qubit are
  un-correlated
• If they are correlated, then this coding will not
  improve the reliability
• Non-local information is robust to local
  disturbances