Quantum Information Theory

Patrick Hayden (McGill)

4 August 2005, Canadian Quantum Information Summer School

Overview

Part I: What is information theory?

Entropy, compression, noisy coding and beyond What does it have to do with quantum mechanics? Noise in the quantum mechanical formalism

Density operators, the partial trace, quantum operations Some quantum information 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) = - x p(x) log2 p(x) Proportional to entropy of statistical physics 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)

Typicality in more detail

Let xn = x1,x2,…,xn with xj 2 X We say that xn is -typical with respect to p(x)

if For all a 2 X with p(a)>0, |1/n N(a|xn) – p(a) | < / |X| For all a 2 X with p(a) = 0, N(a|xn)=0.

For >0, the probability that a random string Xn is -typical goes to 1.

If xn is -typical, 2-n[H(X)+]· p(xn) · 2-n[H(X)-]

The number of -typical strings is bounded above by 2n[H(X)+

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

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

H(X,Y)

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

Data processing inequality

Alice Bob

time),( YX

I(X;Y) ¸ I(Z;Y)

I(X;Y)

X Y

p(z|x)Z Y

I(Z;Y)

Optimality in Shannon’s theorem

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

Assume there exists a code with rate R and perfect decoding. Let M be the random variable corresponding to the uniform distribution over messages.

Xn Yn

nR = H(M) = I(M;M’) · I(M;Yn) · I(Xn;Yn) · j=1n I(Xj,Yj) · n¢maxp(x) I(X;Y)

M has nR bits of entropy

Perfect decoding: M=M’Data processing

Some fiddlingTerm by term

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

Before we get going: Some unavoidable formalism

We need quantum generalizations of: Probability distributions (density operators) Marginal distributions (partial trace) Noisy channels (quantum operations)

Mixing quantum states: The density operator

23

4 1

Draw |xi with probability p(x) Perform a measurement {|0i,|1i}:

Probability of outcome j:

qj = x p(x) |hj|xi |2

= x p(x) tr[|jih j|xihx|] = tr[ |jih j| ],

i

xxxp )(where

Outcome probability is linear in

Properties of the density operator

is Hermitian: y = [x p(x) |xihx|]y = x p(x) [|xihx|]y =

is positive semidefinite: h||i = x p(x) h|xihx|i¸ 0

tr[] = 1: tr[] = x p(x) tr[|xihx|] = x p(x) = 1

Ensemble ambiguity: I/2 = ½[|0ih 0| + |1ih 1|] = ½[|+ih+| + |-ih-|]

The density operator: examples

Which of the following are density operators?

The partial trace

Suppose that AB is a density operator on A B

Alice measures {Mk} on A Outcome probability is

qk = tr[ (Mk IB) AB] Define A = trB[AB] = j Bhj|AB|jiB.

Then qk = tr[ Mk A ] A describes outcome statistics for all

possible experiments by Alice alone

{Mk}

i

Purification

Suppose that A is a density operator on A

Diagonalize A = i i |iihi|

Let |i = i i

1/2 |iiA|iiB Note that A = trB[] |i is a purification of Symmetry:

=

and

have same

non-zero eigenvalues

Quantum (noisy) channels:Analogs of p(y|x)

What reasonable constraints might such a channel :A! B satisfy?

1) Take density operators to density operators2) Convex linearity: a mixture of input states should be mapped to

a corresponding mixture of output states

All such maps can, in principle, be realized physically

Must be interpreted very strictly

Require that ( IC)(AC) always be a density operator too

Doesn’t come for free! Let T be the transpose map on A.If |i = |00iAC + |11iAC, then (T IC)(|ih|) has negative eigenvalues

The resulting set of transformations on density operators are known astrace-preserving, completely positive maps

Quantum channels: examples

Adjoining ancilla: |0ih0| Unitary transformations: UUy

Partial trace: AB trB[AB] That’s it! All channels can be built out of

these operations:

U

|0i

Further examples

The depolarizing channel:

(1-p) + p I/2 The dephasing channel

j hj||ji

|0i

Equivalent to measuring {|ji} then forgetting the outcome

One last thing you should see...

What happens if a measurement is preceded by a general quantum operation?

Leads to more general types of measurements: Positive Operator-Valued Measures (forevermore POVM)

{Mk} such that Mk ¸ 0, k Mk = 1 Probability of outcome k is tr[Mk ]

POVM’s: What are they good for?

Try to distinguish |0i=|0i and |1i = |+i = (|0i+|1i)/21/2

States are non-orthogonal, so projective measurements won’t work.

Let N = 1/(1+1/21/2).

Exercise: M0 = N |1ih1|, M1 = N |-ih-|, M2 = I – M0 – M1 is a POVM

Note: * Outcome 0 implies |1i* Outcome 1 implies |0i* Outcome 2 is inconclusive

Instead of imperfect distinguishability all of the time, the POVM provides perfect distinguishability some of the time.

Notions of distinguishability

Basic requirement: quantum channels do not increase “distinguishability”

Fidelity Trace distance

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

T(,)=|-|1F(,)=[Tr(1/21/2)]2

F=0 for perfectly distinguishableF=1 for identical

T=2 for perfectly distinguishableT=0 for identical

T(,)=2max|p(k=0|)-p(k=0|)|where max is over measurements {Mk}

F((),()) ¸ F(,) T(,) ¸ T((,())

Statements made today hold for both measures

Back to information theory!

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)

Quantifying uncertainty: examples

H(|ih|) = 0 H(I/2) = 1 H( ) = H() + H() H(I/2n) = n H(p © (1-p)) =

H(p,1-p) + pH() + (1-p)H()

…

Compression

Source of independent copies of :

B n

dim(Effective supp 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(aka typical subspace)

[Schumacher, Petz]

The typical subspace

Diagonalize = x p(x) |exihex|

Then n = xn p(xn) |exn ihexn| The -typical projector t is the projector

onto the span of the |exn ihexn| such that xn is typical

tr[ n t] ! 1 as n ! 1

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)

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)

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 classical information

through noisy channels

m Encoding( state)

Decoding(measurement)

m’

B n

2nH(B)

X1,X2,…,Xn

2nH(B|A)

2nH(B|A)

2nH(B|A)

Sending classical information

through noisy channels

m Encoding( state)

Decoding(measurement)

m’

B n

2nH(B)

X1,X2,…,Xn

2nH(B|A)

2nH(B|A)

2nH(B|A)

Distinguish using well-chosen POVM

Data processing inequality(Strong subadditivity)

Alice Bob

timeAB

U

I(A;B)

I(A;B)

I(A;B) ¸ I(A;B)

Optimality in the HSW theorem

Assume there exists a code with rate R with perfect decoding. Let M be the random variable corresponding to the uniform distribution over messages.

nR = H(M) = I(M;M’) · I(A;B)

M has nR bits of entropy

Perfect decoding: M=M’Data processing

m Encoding( state)

Decoding(measurement)

m’

where

m

Sending quantum information

through noisy channels

Physical model of a noisy channel:(Trace-preserving, completely positive map)

|i 2 Cd Encoding(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

whereConditional

entropy!

All x

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

Entanglement and privacy: More than an analogy

p(y,z|x)x = x1 x2 … xn

y=y1 y2 … yn

z = z1 z2 … zn

How to send a private message from Alice to Bob?

AC93Can send private messages at rate I(X;Y)-I(X;Z)

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

All x

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

Entanglement and privacy: More than an analogy

UA’->BE n|xiA’

|iBE = U n|xi

How to send a private message from Alice to Bob?

D03Can send private messages at rate I(X:A)-I(X:E)

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

All x

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

Entanglement and privacy: More than an analogy

UA’->BE nx px

1/2|xiA|xiA’x px

1/2|xiA|xiBE

How to send a private message from Alice to Bob?

SW97D03Can send private messages at rate I(X:A)-I(X:E)=H(A)-H(E)

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

H(E)=H(AB)

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

Some references:

Part I: Standard textbooks:* Cover & Thomas, Elements of information theory.* Nielsen & Chuang, Quantum computation and quantum information. (and references therein)* Devetak, The private classical capacity and quantum capacity of a quantum channel, quant-ph/0304127

Part II: Papers available at arxiv.org:* Devetak, Harrow & Winter, A family of quantum protocols,

quant-ph/0308044.* Horodecki, Oppenheim & Winter, Quantum information can be

negative, quant-ph/0505062