Coherent Communication of Classical Messages Aram Harrow (MIT) quant-ph/

outline What is coherent communication? Why should you care about it? Where can you obtain it? How can you use it?

beyond qubits and cbits Let {|x i } x=0,1 be a basis for C 2. qubit:|x i A ! |x i B cbit:|x i A ! |x i B |x i E coherent bit (cobit):|x i A ! |x i A |x i B ebit: |  i =2 -1/2  x |x i A |x i B 1 qubit > 1 cobit > 1 cbit 1 qubit > 1 cobit > 1 ebit

motivation

motivation #1 irreversibility in q. info. theory 1 qubit > 1 ebit and 1 qubit > 1 cbit are both individually optimal, but the best way to generate qubits is 2 cbits + 1 ebit ¸ 1 qubit. Super-dense coding and teleportation appear to be dual, but composing them wastes entanglement. cobit: |x i A ! |x i A |x i B Claim: Irreversibility comes from the transformation “1 cobit > 1 cbit.”

motivation #2 communication with unitary gates Suppose Alice can send Bob n cbits using a unitary interaction (i.e. U|x i A |0 i B ¼ |x i B |  x i AB for x 2 {0,1} n ). This must be more powerful than an arbitrary noisy interaction, because it implies the ability to create n ebits. But what exactly is its power? cobit: |x i A ! |x i A |x i B Claim: Unitary interactions send cobits while general interactions send cbits.

I(A:B)/2 = (H(A) + H(B) - H(AB))/2 [BSST; quant-ph/ ] H(A)+I(A:B) motivation #3: quantum channel capacity with limited entanglement Q : qubits sent per use of channel E: ebits allowed per use of channel I c =H(B) - H(AB) [L/S/D] qubit > ebit bound 45 o

motivation #3 quantum channel capacity with limited entanglement Q : qubits sent per use of channel E : ebits allowed per use of channel I c =H(B) - H(AB) [LSD] 45 o I(A:E)/2 = (H(A) + H(AB) - H(B))/2 I(A:B)/2 = (H(A) + H(B) - H(AB))/2 “father protocol” [DHW; quant-ph/ ]

sources of cobits Super-dense coding: 1 qubit + 1 ebit > 2 coherent bits cobit: |x i A ! |x i A |x i B ZX H

sources of cobits cobit: |x i A ! |x i A |x i B Distributed unitary gates: If U is a unitary gate or isometry and U > C cbits, then U > C coherent bits. Examples: CNOT > cobit( ! )CNOT > cobit( Ã ) CNOT + ebit > cobit( ! ) + cobit( Ã ) Z YH

noisy sources of cobits [Devetak, Harrow, Winter; quant-ph/ ] cobit: |x i A ! |x i A |x i B General rule: If N can send a classical message that is nearly independent of the residual state (of Alice, Bob and environment), then that message can be made coherent. The environment always sees a maximally mixed state! N + H(A) ebits > I(A:B) cobits D UxUx N Example: entanglement-assisted classical capacity [BSST] N + H(A) ebits > I(A:B) cbits ¼ |x i B |  x i BE

Teleportation H XZ 2 cbits + 1 ebit > 1 qubit + 2 rbits uniformly random Before measuring, the state is  ab |a i |b i A Z a X b |  i B.

Teleportation with coherent communication H XZ 2 cobits +1 ebit > 1 qubit + 2 ebits coherent comm  ab |ab i A Z a X b |  i B 2 -1  ab |ab i A |ab i B Z a X b |  i B cobit: |x i A ! |x i A |x i B

Simple consequences 2 coherent bits = 1 qubit + 1 ebit (C) (using entanglement catalytically) Teleportation and super-dense coding are no longer irreversible. N + H(A) ebits > I(A:B) cobits = I(A:B)/2 (qubit + ebit) H(A) - I(A:B)/2 = (H(A) + H(E) - H(B))/2 = I(A:E)/2 father: N + I(A:E)/2 ebits > I(A:B)/2 qubits cobit: |x i A ! |x i A |x i B

general rule for using cobits cobit: |x i A ! |x i A |x i B Suppose X + C cbits > Y and the classical message sent is independent of the output state. Then X + C coherent bits > Y + C ebits Simultaneous communication and entanglement generation

Recycling in the remote CNOT H = cbit ( ! )+ cbit( Ã ) + ebit > CNOT [Gottesman, quant-ph/ ] cobit ( ! )+ cobit( Ã ) + ebit > CNOT + 2 ebits cobit ( ! )+ cobit( Ã ) = CNOT + ebit (C) SWAP = qubit( ! ) + qubit( Ã ) = 2 CNOT (C) cobit: |x i A ! |x i A |x i B

Remote State Preparation 1 cbit + 1 ebit > 1 remote qubit (A) Given |  d i and a description of |  i2 C d, Alice can prepare |  i in Bob’s lab with error  by sending him log d + O(log (log d)/  2 ) cbits. [Bennett, Hayden, Leung, Shor and Winter, quant- ph/ ]

RSP protocol k

Neumark’s theorem: any measurement can be made unitary k UAUA

Entanglement recycling in RSP UAUA discard coherent communication of log n bits cobit: |x i A ! |x i A |x i B

Coherent RSP 1 cobit + 1 ebit > 1 remote qubit + 1 ebit 1 cobit > 1 remote qubit (C) Corollary 1: Super-dense coding of quantum states 1 qubit + 1 ebit > 2 remote qubits (C) (Independent direct proof in [Harrow, Hayden, Leung; quant- ph/ ].) Corollary 2: The remote state capacity of a unitary gate equals its classical capacity. cobit: |x i A ! |x i A |x i B

RSP of entangled states (eRSP) Let E ={p i,|  i i AB } be an ensemble of bipartite pure states. Define S( E )=S(  i p i Tr A  i ), E( E )=  i p i S(Tr A  i ),  ( E )=S( E )-E( E ). eRSP:  ( E ) cbits + S( E ) ebits > E (A) [BHLSW] make it coherent:  ( E ) cobits + E( E ) ebits > E (A)

Unitary gate capacities Define C e to be the forward classical capacity of U assisted by e ebits of entanglement per use, so that 1 use of U + e ebits > C e cbits (A) Solution: C e =sup E {  (U E ) -  ( E ) : E( E ) - E(U E ) 6 e } (In [BHLS; quant-ph/ ], this was proved for e= 1.)

Warmup: entanglement capacity Define E(U) to be the largest number satisfying U > E(U) ebits (A). Claim: E(U) = sup |  i E(U|  i ) – E(|  i ) Proof: [BHLS; quant-ph/ ] |  i + U > U|  i > E(U|  i ) ebits (concentration) > |  i + E(U|  i )-E(|  i ) ebits (dilution) Thus: U > E(U|  i )-E(|  i ) ebits (A)

Coherent HSW coding Lemma: Let E ={p i,  i } be an ensemble of bipartite pure states that Alice can prepare in superposition. Then E >  ( E ) cobits + E( E ) ebits (A) Proof: Choose a good code on E ­ n. Bob’s measurement obtains ¼ n  ( 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 ¼ nE( E ) ebits.

Protocol achieving C e E + U > U E >  (U E ) cobits + E(U E ) ebits(coherent HSW) > E + (  (U E )-  ( E )) cobits + (E(U E )-E( E )) ebits (coherent RSP) Thus, U + (E( E )-E(U E )) ebits > (  (U E )-  ( E )) cobits (A)

Quantum capacities of unitary gates Define Q e (U) to be the largest number satisfying U + e ebits > Q e qubits. Using 2 cobits = 1qubit + 1ebit, we find

References A.W. Harrow. “Coherent Communication of Classical Messages” quant-ph/ I. Devetak, A.W. Harrow and A. Winter. “A family of quantum protocols.” quant- ph/

Conclusions Whenever a classical message is independent of the residual Alice-Bob-environment state, it can be made coherent. 2 coherent bits = 1 qubit + 1 ebit consequences include: super-dense coding of [entangled] quantum states, SWAP=2 CNOT, classical and quantum capacities of unitary gates, tradeoff curves for quantum channel capacities and the asymptotic equivalence of ensembles of bipartite pure states.