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### Quantum Shannon Theory

Patrick Hayden (McGill)

http://www.cs.mcgill.ca/~patrick/QLogic2005.ppt

17 July 2005, Q-Logic Meets Q-Info

Overview

- Part I:
- What is Shannon theory?
- What does it have to do with quantum mechanics?
- Some quantum Shannon theory highlights

- Part II:
- Resource inequalities
- A skeleton key

Information (Shannon) 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

- Entropy: H(X) = - xp(x) log2p(x)
- Proportional to entropy of statistical physics
- Term suggested by von Neumann (more on him soon)
- Can arrive at definition axiomatically:
- H(X,Y) = H(X) + H(Y) for independent X, Y, etc.

- Operational point of view…

{0,1}n: 2n possible strings

2nH(X)typical strings

CompressionSource of independent copies of X

If X is binary:

0000100111010100010101100101

About nP(X=0) 0’s and nP(X=1) 1’s

X2 …

X1

Xn

Can compress n copies of X to

a binary string of length ~nH(X)

Uncertainty in X

when value of Y

is known

H(X|Y)

I(X;Y)

Information is that which reduces uncertainty

Quantifying informationH(X)

H(X,Y)

H(Y|X)

H(X|Y) = H(X,Y)-H(Y)

= EYH(X|Y=y)

I(X;Y) = H(X) – H(X|Y) = H(X)+H(Y)-H(X,Y)

m’

m

Decoding

Encoding

Shannon’s noisy coding theorem: In the limit of many uses, the optimal

rate at which Alice can send messages reliably to Bob through is

given by the formula

Sending information through noisy channelsStatistical model of a noisy channel:

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

- Major simplifying assumption:

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( trBAB)

Just quantum mechanics!

B n

(aka typical subspace)

dim(Effective supp of B n ) ~ 2nH(B)

CompressionSource 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

[Schumacher, Petz]

H(B)

Uncertainty in A

when value of B

is known?

H(A|B)

|iAB=|0iA|0iB+|1iA|1iB

Quantifying informationH(A)

H(AB)

H(B|A)

H(A|B) = H(AB)-H(B)

H(A|B) = 0 – 1 = -1

Conditional entropy can

be negative!

B = I/2

H(B)

Uncertainty in A

when value of B

is known?

H(A|B)

I(A;B)

Information is that which reduces uncertainty

Quantifying informationH(A)

H(AB)

H(B|A)

H(A|B) = H(AB)-H(B)

I(A;B) = H(A) – H(A|B) = H(A)+H(B)-H(AB)

¸ 0

I(A;B)

Data processing inequality(Strong subadditivity)Alice

Bob

time

U

I(A;B)

I(A;B)¸ I(A;B)

( state)

Decoding

(measurement)

m’

m

HSW noisy coding theorem: In the limit of many uses, the optimal

rate at which Alice can send messages reliably to Bob through is

given by the (regularization of the) formula

where

Sending classical information through noisy channelsPhysical model of a noisy channel:

(Trace-preserving, completely positive map)

( state)

Decoding

(measurement)

m’

m

2nH(B|A)

2nH(B|A)

2nH(B|A)

Sending classical information through noisy channelsB n

2nH(B)

X1,X2,…,Xn

(TPCP map)

Decoding

(TPCP map)

‘

|i2 Cd

LSD noisy coding theorem: In the limit of many uses, the optimal

rate at which Alice can reliably send qubits to Bob (1/n log d) through

is given by the (regularization of the) formula

Conditional

entropy!

where

Sending quantum information through noisy channelsPhysical model of a noisy channel:

(Trace-preserving, completely positive map)

Sets of size 2n(I(X;Z)+)

All x

Random 2n(I(X;Y)-) x

Entanglement and privacy: More than an analogyy=y1 y2 … yn

x = x1 x2 … xn

p(y,z|x)

z = z1 z2 … zn

How to send a private message from Alice to Bob?

Can send private messages at rate I(X;Y)-I(X;Z)

AC93

Sets of size 2n(I(X:E)+)

All x

Random 2n(I(X:A)-) x

Entanglement and privacy: More than an analogy|iBE = U n|xi

UA’->BE n

|xiA’

How to send a private message from Alice to Bob?

Can send private messages at rate I(X:A)-I(X:E)

D03

Sets of size 2n(I(X:E)+)

All x

Random 2n(I(X:A)-) x

H(E)=H(AB)

Entanglement and privacy: More than an analogyx px1/2|xiA|xiBE

UA’->BE n

x px1/2|xiA|xiA’

How to send a private message from Alice to Bob?

SW97

D03

Can send private messages at rate I(X:A)-I(X:E)=H(A)-H(E)

Notions of distinguishability

Basic requirement: quantum channels do not increase “distinguishability”

Fidelity

Trace distance

F(,)={Tr[(1/21/2)1/2]}2

T(,)=|-|1

F=0 for perfectly distinguishable

F=1 for identical

T=2 for perfectly distinguishable

T=0 for identical

F(,)=max |h|i|2

T(,)=2max|p(k=0|)-p(k=0|)|

where max is over POVMS {Mk}

F((),()) ¸ F(,)

T(,) ¸ T((,())

Statements made today hold for both measures

Conclusions: Part I

- Information theory can be generalized to analyze quantum information processing
- Yields a rich theory, surprising conceptual simplicity
- Operational approach to thinking about quantum mechanics:
- Compression, data transmission, superdense coding, subspace transmission, teleportation

Part I: Standard textbooks:

* Cover & Thomas, Elements of information theory.

* Nielsen & Chuang, Quantum computation and quantum information.

(and references therein)

Part II: Papers available at arxiv.org:

* Devetak, The private classical capacity and quantum capacity of a

quantum channel, quant-ph/0304127

* Devetak, Harrow & Winter, A family of quantum protocols,

quant-ph/0308044.

* Horodecki, Oppenheim & Winter, Quantum information can be

negative, quant-ph/0505062

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