Introduction to Quantum Shannon Theory

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Introduction to Quantum Shannon Theory

• Patrick Hayden (McGill University)

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12 February 2007, BIRS Quantum Structures Workshop
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Overview

• What is Shannon theory?
• Why quantum Shannon theory?
• Highlights
• The brilliant trivialities
• Basic capacity theorems
• The grand unified theory

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Information theory

• A practical question
• How to best make use of a given communications
  resource?
• A mathematico-epistemological question
• How to quantify uncertainty and information?
• Shannon
• Solved the first by considering the second.
• A mathematical theory of communication 1948

The
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Quantifying uncertainty

• Shannon entropy H(X) - ?x p(x) log2 p(x)
• Term suggested by von Neumann (more on him
  later)
• Can arrive at definition axiomatically
• H(X,Y) H(X) H(Y) for independent X, Y, etc.
• Operational point of view

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Compression
Source of independent copies of X
If X is binary 0000100111010100010101100101 About
nP(X0) 0s and nP(X1) 1s
X1
X2
Xn
Can compress n copies of X to a binary string of
length nH(X)
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Quantifying information
H(X)
H(X,Y)
H(YX)
H(XY) EY H(XYy) H(X,Y)-H(Y)
I(XY) H(X) H(XY) H(X)H(Y)-H(X,Y)
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Sending information through noisy channels
Statistical model of a noisy channel
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Shannon theory provides

• Practically speaking
• A holy grail for error-correcting codes
• Conceptually speaking
• A operationally-motivated way of thinking about
  correlations
• Whats missing (for a quantum mechanic)?
• Features from linear structureEntanglement and
  non-orthogonality

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Quantum Shannon Theory provides

• General theory of interconvertibility between
  different types of communications resources
  qubits, cbits, ebits, cobits, sbits
• Relies on a
• Major simplifying assumption
• Computation is free
• Minor simplifying assumption
• Noise and data have regular structure

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Basic resources
?? ? span 0?, 1?
1 qubit
??AB0iA0iB1iA1iB
1 ebit
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Brilliant Triviality 1 Superdense coding
j 2 0,1,2,3
Time
?j
1 qubit
1 ebit
??
j 2 bits
Entanglement allows one qubit to carry two bits
of classical data
BW92
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Brilliant Triviality 2 Teleportation
Two classical bits and one ebit can be used send
one qubit
Time
1 qubit
??
2 bits (j)
1 ebit
??
??
?j
Reality
Fiction
BBCJPW93
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Quantifying uncertainty

• Let ? ?x p(x) ?xih?x be a density operator
• von Neumann entropy H(?) - tr ? log ?
• Equal to Shannon entropy of ? eigenvalues
• Analog of a joint random variable
• ?AB describes a composite system A B
• H(A)? H(?A) H( trB ?AB)

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Compression
Source of independent copies of ?AB
?
?
?
A
A
A
B
B
B
Can compress n copies of B to a system of nH(B)
qubits while preserving correlations with A
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Quantifying information
H(A)?
H(AB)?
H(BA)?
H(AB) H(AB)-H(B)
H(AB)? 0 1 -1
Conditional entropy can be negative!
?B I/2
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Quantifying information
H(A)?
H(AB)?
H(BA)?
H(AB) H(AB)-H(B)
I(AB) H(A) H(AB) H(A)H(B)-H(AB)
0
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Sending classical information through noisy
channels
Physical model of a noisy channel (Trace-preservi
ng, completely positive map)
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Sending quantum information through noisy
channels
Physical model of a noisy channel (Trace-preservi
ng, completely positive map)
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The family paradigm
Many problems in quantum Shannon theory are all
versions of the same problem protocols
transform into each other
Father
Mother
TP
Stupid
TP
Entanglement distillation
Quantum capacity
SD
SD
Teleporting over noisy states
Entanglement-assisted classical capacity
Superdense coding with noisy states
Devetak, Harrow, Winter 2003
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Further unification
Fully quantum Slepian-Wolf
Special case
Schmidt symmetry
Time-reversal
Channel simulation
Father
Mother
TP
Stupid
TP
Entanglement distillation
Quantum capacity
SD
SD
Teleporting over noisy states
Entanglement-assisted classical capacity
Superdense coding with noisy states
Quantum multiple access capacities
Distributed compression
Abeyesinghe, Devetak, Hayden, Winter 2006
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The art of forgetting
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The art of forgetting
TRASH
?AB1B2B3
?AB2B3
?AB2 ?A ?B2
How can Bob unilaterally destroy his correlation
with Alice?
What is the minimal number of particles he must
discard before the remaining state is
uncorrelated?
In this case, by discarding 2 particles, Bob
succeeded in eliminating all correlations with
Alices particle
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Purification and correlation
B
D
Purification When faced with a mixed state, we
can always imagine that the state describes part
of a larger system on which the state is pure.
?ABCDi
?ABi?CDi (idACUBD-1)?ABCDi
(idAC UBD)?ABi?CDi
Purifications are essentially unique. (Up to
local transformations of the purifying space.)
TrBD ?ABCD ?A ?C
?A sC
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The benefits of forgettingApplied theology
Watch again
?AB2 ?A ?B2
?AB1B2B3
?AB2B3
?AB1B2B3Ci
Charlies Magical Bucket O Particles
Purification
TRASH
All purifications equivalent up to a local
transformation in Charlies lab.
Charlie holds uncorrelated purifications of
both Alices particle and Bobs remaining
particles.
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The benefits of forgettingApplied theology
Before
After
TRASH
TRASH
?AB1B2B3Ci
?AC1i?B2C2C3i
Alice never did anything ) Her marginal state ?A
?A is unchanged
Originally, her purification is held by both Bob
and Charlie. Afterwards, entirely by Charlie.
Bob transferred his Alice entanglement to Charlie
and distilled entanglement with Charlie, just by
discarding particles!
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Fully quantum Slepian-WolfHow much does Bob
need to send?
Uncertainty von Neumann entropy
Before
H(A)? H(?A) - tr ?A log ?A
Correlation mutual information
I(AB)? H(A)? H(B)? H(AB)?
TRASH
0 if and only if ?AB ?A
?B I(AB)? m for m pairs of correlated bits
2m for m ebits (maximal)
?ABCi n
Initial mutual information n I(AB)?
Final mutual information ?
Each qubit Bob discards has the potential to
eliminate at most 2 bits of correlation
Bob should (ideally) send around nI(AB)?/2
qubits to Charlie.
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How does Bob choose which qubits?
?
Before
TRASH
?ABCi n
(According to the unitarily invariant measure on
the typical subspace of Bn.)
Bob can ignore the correlation structure of his
state!
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Final accounting
After
Investment Bob sends Charlie nI(AB)?/2
qubits
Rewards 1) Charlie holds Alices purification
2) B and C establish nI(BC)?/2 ebits
TRASH
?AC1i?B2C2C3i
OK but what good is it?
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Entanglement distillation
(?BC) n
Bob and Charlie share many copies of a noisy
entangled state and would like to convert them to
ebits.
Only local operations and classical communication
are allowed. Forgetting protocol good but uses
quantum communication
Implement quantum communication using
teleportation Transmit 1 qubit using 2 cbits and
1 ebit.
Optimal
Net rate of ebit production I(BC)/2 I(AB)/2
H(C)-H(BC)
Devetak/Winter 03
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Conclusions

• Information theory can be generalized to analyze
  quantum information processing
• Yields a rich theory, surprising conceptual
  simplicity
• Compression, data transmission, superdense
  coding, teleportation, subspace transmission
• Capacity zoo, using noisy entanglement, channel
  simulation all are closely related
• Operational approach to thinking about quantum
  mechanics