Introduction to Quantum Shannon Theory Patrick Hayden (McGill University) 12 February 2007, BIRS...

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

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

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…

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)

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

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

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

Basic resources

|+AB=|0iA|0iB+|1iA|1iB

1 ebit

1 qubit

| span{ |0, |1}

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)

…

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

H(B)


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)

H(B)


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)

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

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!

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

Further unificationFully quantum Slepian-Wolf

Mother


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

The art of forgetting

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

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

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.

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!

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.

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!

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?


(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]

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

Introduction to Quantum Shannon Theory Patrick Hayden (McGill University) 12 February 2007, BIRS...

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