CoherentClassicalCommunication

Aram Harrow (MIT)quant-ph/0307091

Outline

• What is coherent classical communication (CCC)?

• Where does CCC come from?• What is CCC good for? • Remote state preparation with CCC• Noisy CCC and applications

beyond qubits and cbits

Let {|xi}x=0,1 be a basis for C2.• [q!q]: |xiA!|xiB (qubit)• [c!c]: |xiA!|xiB|xiE (cbit)

• [qq]: |i=2-1/2x|xiA|xiB (ebit)• [cc]: 2-1/2x|xiA|xiB|xiE (rbit)

• [[c!c]]: |xiA!|xiA|xiB (coherent cbit)

(notation due to Devetak and Winter, quant-ph/0304196)

simple resource relations

Trivial relations:

[q!q] > [[c!c]] > [c!c] > [cc]

[q!q] > [[c!c]] > [qq] > [cc]

Teleportation (TP):

2[c!c] + [qq] > [q!q]

Super-dense coding (SDC):

[q!q] + [qq ] > 2[[c!c]] (coherent output!)

distributed unitary gatesTheorem:

If U is a unitary gate on HA HB such that

U + e [qq] > C![c!c] + CÃ[cÃc] (A)

then U + e [qq] > C![[c!c]] + CÃ[[cÃc]] (A).

Examples:CNOTAB|xiA|0iB=|xiA|xiB(HZa I)CNOTAB(Xa Zb)2-1/2x|xiA|xiB=|biA|aiB

Note:1. The proof requires careful accounting of ancillas.

2. It also holds for isometries (e.g. |xiA!|xiA|xiB)

Teleportation

H

X Z

2 [c!c] + 1 [qq] > 1 [q!q]

+ 2 [cc]

uniformly

random

Before measuring, the state is 2-1ab|ai|biAZaXb|iB.

Teleportation with coherent communication

H

X Z

2 [[c!c]] + 1 [qq] > 1 [q!q]+ 2 [qq]

coherent

classical

comm.

2-1ab|abiAZaXb|iB

2-1ab|abiA|abiBZaXb|iB

the power of coherent cbits

Teleportation with recycling:2 [[c!c]] + 1 [qq] > 1 [q!q]+ 2 [qq]2 [[c!c]] > 1 [q!q]+ 1 [qq] (C)

Super-dense coding:1 [q!q]+ 1 [qq] > 2 [[c!c]] (C)

Therefore:2 [[c!c]] = 1 [q!q]+ 1 [qq] (C)Teleportation and super-dense coding

are no longer irreversible.

Recycling in the remote CNOT

H

=

[c!c] + [cÃc] + [qq] > CNOT [Gottesman, quant-ph/9807006][[c!c]] + [[cÃc]] + [qq] > CNOT + 2 [qq][[c!c]] + [[cÃc]] > CNOT + [qq] (C)

the power of a CNOT

Making a remote CNOT coherent:[[c!c]] + [[cÃc]] > CNOT + [qq] (C)

Using a CNOT for bidirectional communication:(HZa I)CNOTAB(Xa Zb)2-1/2x|xiA|xiB=|biA|aiBCNOT + [qq] > [[c!c]] + [[cÃc]]

Combined: CNOT + [qq] = [[c!c]] + [[cÃc]] (C)2 CNOT = 2 [[c!c]] + 2 [[cÃc]] – 2 [qq]

= [q!q] + [qÃq] = SWAP (C)

Remote State Preparation

1 cbit + 1 ebit > 1 remote qubit

Given |di and a description of 2Cd, Alice can prepare in Bob’s lab with error by sending him log d + O(log (log d)/2) bits.

[Bennett, Hayden, Leung, Shor and Winter, quant-ph/0307100]

definitions of remote qubits

What does it mean for Alice to send Bob n remote qubits?

She can remotely prepare one of

RSP lemmaFor any d and any >0, there exists n=O(d log d/2) and a set of d x d unitary gates R1,…,Rn such that for any ,

Use this to define a POVM:

RSP protocol

k

Neumark’s theorem:any measurement can be made

unitary

k

UM

Entanglement recycling in RSP

UA

discard

coherent

classical

communication

of log n bits

Implications of recycled RSP

1 coherent cbit > 1 remote qubit (with catalysis)

Corollary 1: The remote state capacity of a unitary gate equals its classical capacity.

Corollary 2: Super-dense coding of quantum states (SDCQS)

1 qubit + 1 ebit > 2 remote qubits (with catalysis)

(Note: [Harrow, Hayden, Leung; quant-ph/0307221] have a direct proof of SDCQS.)

RSP of entangled states (eRSP)

Let E={pi,i} be an ensemble of bipartite pure states. Define S(E)=S(ipiTrAi), E(E)=ipiS(TrAi), (E)=S(E)-E(E).

eRSP: (E) [c!c] + S(E) [qq] > E (A) [BHLSW]

make it coherent: (E) [[c!c]] + E(E) [qq] > E (A)

use super-dense coding:

(E)/2 [q!q] + (E(E)+ (E)/2) [qq] > E (A)

Unitary gate capacities

Define Ce to be the forward classical capacity of U assisted by e ebits of entanglement per use, so that

1 use of U + e [qq] > Ce [c!c] (A)

(In [BHLS; quant-ph/0205057], this was proved for e=1.)

Solution:

Ce=supE {(UE) - (E) : E(E) - E(UE)6e}

Warmup: entanglement capacity

Define E(U) to be the largest number satisfying

U > E(U) [qq] (A).

Claim: E(U) = sup|i E(U|i) – E(|i)

Proof: [BHLS; quant-ph/0205057]

|i + U > U|i

> E(U|i) [qq] (concentration)

>|i + E(U|i)-E(|i) [qq] (dilution)

Thus: U > E(U|i)-E(|i) [qq] (A)

Coherent HSW coding

Lemma: Let E={pi,i} be an ensemble of bipartite pure states that Alice can prepare in superposition. Then

E > (E) [[c!c]] + E(E) [qq] (A)

Proof: Choose a good code on E n. Bob’s measurement obtains ¼n(E) bits of Alice’s message and determines the codeword with high probability, causing little disturbance. Thus, this measurement can be made coherent. Since Alice and Bob know the codeword, they can then do entanglement concentration to get ¼nE(E) ebits.

Protocol achieving Ce

E + U > UE

> (UE) [[c!c]] + E(UE) [qq] (coherent HSW)

> E + ((UE)-(E)) [[c!c]] + (E(UE)-E(E)) [qq]

(coherent RSP)

Thus, U + (E(E)-E(UE)) [qq] > ((UE)-(E)) [[c!c]] (A)

Quantum capacities of unitary gates

Define Qe(U) to be the largest number satisfying

U + e [qq] > Qe [q!q].

Using 2[[c!c] = 1[q!q] + 1[qq], we find

Summary

• 2 coherent cbits = 1 qubit + 1 ebit• 2 CNOT = SWAP (catalysis)• 1 qubit + 1 ebit > 2 remote qubits

(catalysis)• eSDCQS using /2 qubits and S-/2 ebits.

• Single-letter expressions for Ce and Qe.

• Remote state capacities and classical capacities are equal for unitary gates.

Noisy CCC[Devetak, Harrow, Winter; quant-

ph/0308044]• Two minute proofs of the hashing

inequality and the quantum channel capacity.

• Generalizations of these protocols to obtain the full trade-off curves for quantum channels assisted by a limited amount of entanglement and entanglement distillation with a limited amount of communication.

Noisy CCC: definitionsLet AB be a bipartite state and |iABE its purification.

I(A:B) = H(A) + H(B) – H(E)

I(A:E) = H(A) + H(E) – H(B)

Ic = H(B) – H(E) = ½ (I(A:B) – I(A:E))

If N is a noisy channel, then evaluate the above quantities on (I N)|i, where |i is a purification of Alice’s input .

{qq} = one copy of AB

{q!q} = one use of N

Noisy CCC: applications

Old results:

S(A) [qq] + {q!q} > I(A:B) [[c!c]] [BSST; q-ph/0106052]

{q!q} > Ic [q!q] [Shor; unpublished]

S(A) [q!q] + {qq} > I(A:B) [[c!c]] [HHHLT; q-ph/0106080]

I(A:E) [c!c] + {qq} > Ic [qq] [DW; q-ph/0306078]

New results:

I(A:E)/2 [qq] + {q!q} > I(A:B)/2 [q!q][father]

I(A:E)/2 [q!q] + {qq} > I(A:B)/2 [qq][mother]

A family of quantum protocols

fathermother

hashing inequality [DW]

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

CE [BSST]noisy SDC [HHHLT]

{q!q}>Ic[q!q]

[Shor]

TP

TP

TP

SDC

[q!q]

>[qq]

SDC

TP= teleportation

SDC = super-dense coding

A gate with asymmetric capacities?

x=0,…,d-1, U2Cd Cd

U |x0i = |xxiU |xxi = |x0iU |xyi = |xyi for xy0.

C1 = log d

C2 > (log d)/2