introduction to quantum shannon theory patrick hayden (mcgill university) 12 february 2007, birs...
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Introduction to Quantum Shannon Theory
Patrick Hayden (McGill University)
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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X1X2 …Xn
Compression
Source of independent copies of X
{0,1}n: 2n possible strings
~2nH(X) typical strings
If X is binary:0000100111010100010101100101About nP(X=0) 0’s and nP(X=1) 1’s
Can compress n copies of X toa binary string of length ~nH(X)
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H(Y)
Quantifying information
H(X)
H(Y|X)
Information is that which reduces uncertainty
I(X;Y)H(X|Y)
Uncertainty in Xwhen value of Yis known
H(X|Y) = EY H(X|Y=y) = H(X,Y)-H(Y)
I(X;Y) = H(X) – H(X|Y) = H(X)+H(Y)-H(X,Y)
H(X,Y)
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Sending information through noisy channels
Statistical model of a noisy channel: ´
mEncoding Decoding
m’
Shannon’s noisy coding theorem: In the limit of many uses, the optimalrate at which Alice can send messages reliably to Bob through is given by the formula
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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
What’s missing (for a quantum mechanic)? Features from linear structure:
Entanglement 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
|+AB=|0iA|0iB+|1iA|1iB
1 ebit
1 qubit
| span{ |0, |1}
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j 2 {0,1,2,3}
Brilliant Triviality # 1: Superdense coding
j
Time
1 qubit
1 ebit
BW92
j: 2 bits
Entanglement allows one qubit to carry two bits of classical data
|+
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|
Brilliant Triviality # 2: Teleportation
j
Time1 qubit
2 bits (j)
1 ebit
BBCJPW93
Reality:Fiction:
Two classical bits and one ebit can be used send one qubit
|+
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Quantifying uncertainty
Let = x p(x) |xihx| 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 :
B n
dim(Support of B n ) ~ 2nH(B)
Can compress n copies of B toa system of ~nH(B) qubits whilepreserving correlations with A
No statistical assumptions:Just quantum mechanics!
A A A
B B B
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H(B)
Quantifying information
H(A)
H(B|A)H(A|B)
Uncertainty in Awhen value of Bis known?
H(A|B) = H(AB)-H(B)
|iAB=|0iA|0iB+|1iA|1iB
B = I/2
H(A|B) = 0 – 1 = -1
Conditional entropy canbe negative!
H(AB)
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H(B)
Quantifying information
H(A)
H(B|A)
Information is that which reduces uncertainty
I(A;B)H(A|B)
Uncertainty in Awhen value of Bis known?
H(A|B) = H(AB)-H(B)
I(A;B) = H(A) – H(A|B) = H(A)+H(B)-H(AB)¸ 0
H(AB)
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Sending classical information
through noisy channels
Physical model of a noisy channel:(Trace-preserving, completely positive map)
m Encoding( state)
Decoding(measurement)
m’
HSW noisy coding theorem: In the limit of many uses, the optimalrate at which Alice can send messages reliably to Bob through is given by the (regularization of the) formula
where
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Sending quantum information
through noisy channels
Physical model of a noisy channel:(Trace-preserving, completely positive map)
|i 2 CdEncoding(TPCP map)
Decoding(TPCP map)
‘
LSD noisy coding theorem: In the limit of many uses, the optimalrate at which Alice can reliably send qubits to Bob (1/n log d) through is given by the (regularization of the) formula
whereConditionalentropy!
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The family paradigmMany problems in quantum Shannon theory are all
versions of the same problem: protocols transform into each other
Mother
Entanglement distillation
Superdense coding with noisy states
Teleporting over noisy states
Devetak, Harrow, Winter [2003]
Father
Entanglement-assistedclassical capacity
Quantum capacity
TP
TP
SD
SD
Stupid
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Further unificationFully quantum Slepian-Wolf
Mother
Entanglement distillation
Superdense coding with noisy states
Teleporting over noisy states
Abeyesinghe, Devetak, Hayden, Winter [2006]
Father
Entanglement-assistedclassical capacity
Quantum capacity
Distributed compressionQuantum multiple access capacities
Channel simulation
Time-reversalSpecia
l case
Schmidt symmetry
TP
TP
SD
SD
Stupid
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The art of forgetting
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The art of forgetting
AB1B2B3
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?
TRASH
In this case, by discarding 2 particles, Bob succeeded ineliminating all correlations with Alice’s particle
AB2B3AB2
= A B2
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Purification and correlation
|ABCDi
TrBD ABCD = A C
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.
Purifications are essentially unique.(Up to local transformations
of the purifying space.)
A σC
=(idAC UBD)|ABi|CDi|ABi|CDi = (idAC UBD-1)|ABCDi
B D
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The benefits of forgetting:
Applied theology
AB1B2B3
TRASH
AB2B3AB2
= A B2
|AB1B2B3CiPurification
Charlie’s Magical Bucket
O’ Particles
Watch again:
All purifications equivalent up to a local transformation in Charlie’s lab.
Charlie holds uncorrelated purifications of bothAlice’s particle and Bob’s remaining particles.
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The benefits of forgetting:
Applied theology
TRASH
|AB1B2B3Ci
TRASH
|AC1i|
B2C2C3i
Before After
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-Wolf:How much does Bob need to
send?
TRASH
Before
|ABCi n
Uncertainty: von Neumann entropy
H(A) = H(A) = - tr[ A log A ]
Correlation: mutual informationI(A;B) = H(A) + H(B) – H(AB)
0 if and only if AB = A B
I(A;B)= m for m pairs of correlated bits 2m for m ebits (maximal)
Initial mutual information: n I(A;B)Final mutual information:
Each qubit Bob discards has the potentialto eliminate at most 2 bits of correlation
Bob should (ideally) send around nI(A;B)/2 qubits to Charlie.
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How does Bob choose whichqubits?
TRASH
Before
|ABCi n
?
At random!
(According to the unitarily invariant measureon the typical subspace of B n.)
Bob can ignore the correlation structure of his state!
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Final accounting
TRASH
|AC1i|
B2C2C3i
AfterInvestment: Bob sends Charlie ~n[I(A;B)]/2 qubits
Rewards: 1) Charlie holds Alice’s purification 2) B and C establish ~n[I(B;C)]/2 ebits
OK – but what good is it?
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Entanglement distillation
(BC) n
Bob and Charlie share many copies of a noisy entangled stateand would like to convert them to ebits.
Only local operations and classicalcommunication are allowed.Forgetting protocol good but uses quantumcommunication
Implement quantum communication using teleportation:Transmit 1 qubit using 2 cbits and 1 ebit.
Net rate of ebit production:I(B;C)/2 – I(A;B)/2 = H(C)-H(BC)
Optimal
[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