Coherent Communication of Classical Messages Aram Harrow MIT

Coherent Communication of Classical Messages Aram Harrow (MIT) quant-ph/0307091

outline • What is coherent communication? • Why should you care about it? • Where can you obtain it? • How can you use it? • Who has a poster related to coherent communication that you should see after this talk?

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

cobit: |xi. A!|xi. A|xi. B why? motivation #1: What is the power of sending a classical message using a bipartite unitary gate or isometry? motivation #2: Why are quantum resource transformations irreversible?

cobit: |xi. A!|xi. A|xi. B sources of cobits Super-dense coding: 1 qubit + 1 ebit > 2 coherent bits Distributed unitary gates: If U is a unitary gate and U > C cbits, then U > C coherent bits. Example: CNOT > 1 cbit ( ) CNOT + ebit > 1 cbit ( ) + 1 cbit ( )

Teleportation uniformly H random X Z 2 cbits + 1 ebit > 1 qubit + 2 rbits Before measuring, the state is 2 -1åab|ai|bi. AZa. Xb|yi. B.

cobit: |xi. A!|xi. A|xi. B Teleportation with coherent communication 2 -1åab|abi. A|abi. BZa. Xb|yi. B H coherent comm. 2 -1åab|abi. AZa. Xb|yi. B X Z 2 cobits +1 ebit > 1 qubit + 2 ebits

cobit: |xi. A!|xi. A|xi. B Simple consequences • 2 coherent bits = 1 qubit + 1 ebit (C) (using entanglement catalytically) • Teleportation and super-dense coding are no longer irreversible.

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

Remote State Preparation 1 cbit + 1 ebit > 1 remote qubit (A) Given |Fdi and a description of |yi 2 Cd, Alice can prepare |yi in Bob’s lab with error e by sending him log d + O(log d)/e 2) cbits. [Bennett, Hayden, Leung, Shor and Winter, quant-ph/0307100]

cobit: |xi. A!|xi. A|xi. 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 (with catalysis) (Independent direct proof in [Harrow, Hayden, Leung; quant-ph/0307221]. ) Corollary 2: The remote state capacity of a unitary gate equals its classical capacity.

Noisy coherent communication [“A family of quantum protocols. ” 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.

References A. W. Harrow. “Coherent Communication of Classical Messages” quant-ph/0307091 I. Devetak, A. W. Harrow and A. Winter. “A family of quantum protocols. ” quant-ph/0308044 Also, see the poster this afternoon by Igor Devetak!