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? Who has a poster related to coherent communication that you should see after this talk?

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

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: |x i A ! |x i A |x i 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 > 1 cbit (  ) CNOT + ebit > 1 cbit (  ) + 1 cbit (  ) cobit: |x i A ! |x i A |x i B

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. 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

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/ ]

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/ ].) Corollary 2: The remote state capacity of a unitary gate equals its classical capacity. cobit: |x i A ! |x i A |x i B

Noisy coherent communication [“A family of quantum protocols.” Devetak, Harrow, Winter; quant-ph/ ] 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/ I. Devetak, A.W. Harrow and A. Winter. “A family of quantum protocols.” quant- ph/ Also, see the poster this afternoon by Igor Devetak!