Quantum communication from Alice to Bob

Quantum communication from Alice to Bob

Andreas Winter, Bristol

quant-ph/0308044quant-ph/0308044Aram Harrow, MIT

Igor Devetak, USC

outline

• Introduction– basic concepts and resource inequalities – historical overview of quantum information

theory

• A family of protocols– Rederive and connect old protocols– Prove new protocols (parents)

• Optimal trade-off curves

the setting•two parties: Alice and Bob

•one-way communication from Alice to Bob

•we want asymptotic communication rates

AliceBob

(noisy) classical communication

(noisy) quantum communication

(noisy) shared entanglement

or classical correlations

cbit [c!c] 1 noiseless bit channel

ebit [qq] the state (|0iA|0iB + |1iA|1iB)/p2

qubit [q!q] 1 noiseless qubit channel

noisystate

{qq} noisy bipartite quantum state AB

noisychannel

{q!q}N noisy cptp map: N:HA’!HB

Information processing resources may be:

• classical / quantum c / q

• noisy / noiseless (unit) { } / [ ]

• dynamic / static ! / ¢

examples of bipartite resources

Church of the larger Hilbert space

static AB ) purification |iABE s.t. AB = trEABE.

If AB=i pi |iihi|AB, then |iABE=i ppi |iiE |iiAB.

|iAA’

UN

A

A0 B

E

|iABE

Channel N:HA’!HB ) isometric extension UN:HA’!HB HE s.t. N() = trEUN().

Use a test source |iAA’ and define |iABE = (IA UN)|iAA’

information theoretic quantities

von Neumann entropy: H(A) = -tr [A log A]

mutual information: I(A:B) = H(A) + H(B) – H(AB)

coherent information: Ic(AiB) = H(B) – H(AB) = -H(A|B)

conditional mutual information:

I(A:B|X) = H(A|X) + H(B|X) – H(AB|X)

= I(A:BX) – I(B:X)

resource inequalitiesExample: classical noisy channel coding [Shannon]

{c!c}N > I (A:B)p [c!c]Meaning there exists an asymptotic and approximate protocol transforming the LHS into the RHS.

For any >0 and any R<I(A:B) and for sufficiently large n there exist encoding and decoding maps E: {0,1}nR ! Xn and D: Xn ! {0,1} nR such that for any input x2{0,1}nR

(D ¢ N n ¢ E)|xi ¼ |xi

The capacity is given by maxp I(A:B)p, where p is a distribution on AB resulting from B = N(A).

resource inequalitiesExample: quantum channel coding

{q!q}N > Ic(AiB) [q!q]Meaning there exists an asymptotic and approximate protocol transforming the LHS into the RHS.

For any >0 and any R<Ic(AiB) and for sufficiently large n there exist encoding and decoding maps E: H2

nR ! HA’ n and D: HB

n ! H2 nR such that for any input |

i2H2 n,

(D ¢ N n ¢ E)|i ¼ |i

The capacity is given by limn!1 (1/n) max Ic(AiB), where the maximization is over all arising from N

n.

the history of quantum information theory, part

one

first generation: semi-classical

Characterized by:

•Results depend only on average density matrix

•Protocols can be analyzed by looking at one party’s stateExamples:

Schumacher compression: [PRA 51, 2738 (1995)]

RA + S() [q!q] > RB

entanglement concentration/dilution: [BBPS, quant-ph/9511030]

= S(A) [qq]

remote state preparation: [BDSSTW, quant-ph/0006044]

S(B) [c!c] + S(B) [qq] > EAB = {pi, |iiB}

first generation techniques

semi-classical reductions:

Schmidt decomposition: |iAB=i ppi |aiiA|biiB matrix diagonalization: = i pi |viihvi|

Typical sequences: p a probability distribution

p-typical sequences i1,…,in have |#{ij = x} – npx| < n for all x

# of p-typical sequences is ¼exp(n(S(p)+))

each has probability exp(-n(S(p) ))Typical projectors and subspaces: a state with spectrum p

= I typical |vIihvI| projects onto a typical subspace

where I=i1,…,in is a typical sequence and |vIi=|vi1i…|vini

2nd generation: CQ ensembles

HSW theorem: [H, IEEE IT 44, 269 (1998); SW, PRA 56, 131 (1997)]

E = i pi |iihi|A iB {c!q} > [c!c]

I(A:B) = S(ipii) - iS() = (E)

Entanglement assisted channel capacity: [BSST, quant-ph/0106052]

{q!q} + H(A) ebits > I(A:B) [c!c]

RSP of entangled states: [BHLSW, quant-ph/0307100]

H(A) [qq] + I(A:B) [c!c] > E = i pi |iihi|X |iihi|AB

Measurement compression: [Winter, quant-ph/0109050]

I(X:R)[c!c] + H(X|R) [cc] > T:A! AEXAXB on |iAR

2nd generation techniques

conditionally typical subspaces: E = i pi |iihi|A iB

Compressing B requires S(B) qubits, but if you know (or have) A then you need

S(B|A) = S(AB) – S(A) = i pi S(i) qubits.

The difference is S(B)-S(B|A) = S(A)+S(B)-S(AB) = I(A:B) = .

operator Chernoff bounds: [AW, quant-ph/0012127]

X1,…,Xn i.i.d. Hermitian matrices s.t. 06Xi6I and =EXi>I

3rd generation: fully quantum

•quantum channel capacity: {q!q} > Ic(AiB)

•super-dense coding of quantum states

•double and triple-tradeoff curves:

N > R [c!c] + Q[q!q] + E[qq]

•unification of different protocols

•entanglement distillation using limited quantum or classical communication

3rd generation techniques

derandomization: If the output state is pure, [cc] inputs are unnecessary.

piggybacking: Time-sharing protocol Px with probability px allows an extra output of I(X:B) [c!c]. [DS, quant-ph/0311131]

coherent classical communication: [H, quant-ph/0307091]

Modify protocols to obtain [[c!c]]: |xiA!|xiA|xiB

use coherent TP and SD to get 2 [[c!c]] = [q!q] + [qq].

main result #1: parent protocols

father: {q!q} + ½ I(A:E) [qq] > ½ I(A:B) [q!q]

mother: {qq} + ½ I(A:E) [q!q] > ½ I(A:B) [qq]

Basic protocols combine with parents to get children.

(TP) 2[c!c] + [qq] > {q!q}

(SD) [q!q] + [qq] > 2[c!c]

(QE) [q!q] > [qq]

the family tree

{q!q} + ½ I(A:E) [qq] > ½ I(A:B) [q!q]

{qq} + ½ I(A:E) [q!q] > ½ I(A:B) [qq]

{q!q} + H(A) [qq] > I(A:B) [c!c]

BSST, [IEEE IT 48, 2002], E-assisted cap.

{q!q} > Ic(AiB) [q!q]

L/S/D, quantum channel cap.

{qq} + H(A) [q!q] > I(A:B) [c!c]

H3LT, [QIC 1, 2001], noisy SD

{qq} + I(A:B) [c!c] > Ic(AiB) [q!q]

DHW, noisy TP

SD

QE

TP SD

TP

{qq} + I(A:E) [c!c] > Ic(AiB) [q!q]

DW, entanglement distillation

TP

(TP) 2[c!c] + [qq] > {q!q}

(SD) [q!q] + [qq] > 2[c!c]

(QE) [q!q] > [qq]

coherent classical communication

rule I:

X + C [c!c] > Y ) X + C/2 ([q!q] – [qq]) > Y

rule O:

X > Y + C [c!c] ) X > Y + C/2 ([q!q] + [qq])Whenever the classical message in the original protocol is almost uniformly distributed and is almost decoupled from the remaining quantum state of Alice, Bob and Eve.

based on PRL 92, 097902 (2004)

generating the parents

{q!q} + ½ I(A:E) [qq] > ½ I(A:B) [q!q]

{qq} + ½ I(A:E) [q!q] > ½ I(A:B) [qq]

{q!q} + H(A) [qq] > I(A:B) [c!c]

BSST, [IEEE IT 48, 2002], E-assisted cap.

{q!q} > Ic(AiB) [q!q]

L/S/D, quantum channel cap.

{qq} + H(A) [q!q] > I(A:B) [c!c]

H3LT, [QIC 1, 2001], noisy SD

{qq} + I(A:B) [c!c] > Ic(AiB) [q!q]

DHW, noisy TP

SD

QE

TP SD

TP

{qq} + I(A:E) [c!c] > Ic(AiB) [q!q]

DW, entanglement distillation

TP

O OI

I(A:B)/2

[BSST; quant-ph/0106052]

H(A)+I(A:B)

main result #2: tradeoff curves

Q: q

ub

its sen

t per u

se o

f ch

an

nel

E: ebits allowed per use of channel

Ic(A>B)

[L/S/D]

qubit > ebit bound

45o

example: quantum channel capacity with limited entanglement

father trade-off curve

Q: q

ub

its sen

t per u

se o

f ch

an

nel

E: ebits allowed per use of channel

Ic(AiB)

[L/S/D]

45o

I(A:E)/2 = I(A:B)/2 - Ic(AiB)

I(A:B)/2

father

mother trade-off curve

{qq} + ½ I(A:E) [q!q] > ½ I(A:B)[qq]

preprocessing instrument T:A!AE’X

{qq} + ½ I(A:EE’|X) [q!q] + H(X)[c!c] > ½ I(A:B|X)[qq]

H(X) [c!c]measurement

compressionI(X:BE) [c!c] + H(X|BE) [cc]

I(X:BE) [c!c]derandomization

½ I(X:BE) ([q!q] – [qq])rule I

{qq} + ½ (I(A:EE’|X) + I(X:BE)) [q!q] > ½ (I(A:B|X) + I(X:BE)) [qq]

converse proof techniques

Holevo bound/data processing inequality:

X Q Y: I(X:Y) 6 I(X:Q)

Fano/Fannes inequality: error on n qubits makes entropy change by O((n+log(1/)).

unnamed identity that shows up everywhere:

I(X:AB) = H(A) + Ic(AiBX) – I(A:B) + I(X:B)

quantum data processing inequality: [quant-ph/9604022]

RQ RQ’E1 RQ’’E1E2: H(R)=Ic(RiQ)>Ic(RiQ’) > Ic(RiQ’’)

what’s left

• In quant-ph/0308044, we prove similar tradeoff curves for the rest of the resource inequalities in the family.

• Remaining open questions include

– Finding single-letter formulae (i.e. additivity)

– Reducing the optimizations over instruments

– Addressing two-way communication

– Multiple noisy resources

– Reverse coding theorems