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Quantum Shannon Theory

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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

- 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

- 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

- 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

Source 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)

H(Y)

Uncertainty in X

when value of Y

is known

H(X|Y)

I(X;Y)

Information is that which reduces uncertainty

H(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

Statistical model of a noisy channel:

- 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

- 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:

- 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)

No statistical assumptions:

Just quantum mechanics!

B n

(aka typical subspace)

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

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

[Schumacher, Petz]

H(B)

Uncertainty in A

when value of B

is known?

H(A|B)

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

H(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

H(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)

Alice

Bob

time

U

I(A;B)

I(A;B)¸ I(A;B)

Encoding

( 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

Physical model of a noisy channel:

(Trace-preserving, completely positive map)

Encoding

( state)

Decoding

(measurement)

m’

m

2nH(B|A)

2nH(B|A)

2nH(B|A)

B n

2nH(B)

X1,X2,…,Xn

Encoding

(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

Physical 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

y=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

|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)

x 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)

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

- 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

Some references:

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