Coherent Classical Communication Aram Harrow, MIT Quantum Computing Graduate Research Fellow Objective Objective ApproachStatus Determine tradeoff curves for quantum channel coding and entanglement distillaton assisted by entanglement or classical or quantum communication. Find efficient methods to calculate and achieve these capacities. Relate different quantum protocols. Reconsider the role of classical communication in quantum information theory. Found classical and quantum capacities of unitary gates and quantum channels assisted by arbitrary amounts of entanglement. Proved super-dense coding of quantum states. Derived several new quantum protocols and related several old ones. H XZ coherent classical comm.

Research Overview A. Harrow, MIT A.W. Harrow and H.-K. Lo. “A tight lower bound on the classical communication cost of entanglement dilution.” quant-ph/ , accepted IEEE-IT. C.H. Bennett, A.W. Harrow, D.W. Leung and J.A. Smolin. “On the capacities of bipartite Hamiltonians and unitary gates.” quant- ph/ , accepted IEEE-IT. A.W. Harrow and M.A. Nielsen. “Robustness of gates in the presence of noise,” quant-ph/ , accepted PRA. A.W. Harrow. “Coherent Classical Communication.” quant- ph/ I. Devetak, A.W. Harrow and A. Winter. “A family of quantum protocols.” quant-ph/ A.W. Harrow, P. Hayden and D.W. Leung. “Superdense coding of quantum states.” quant-ph/ D. Bacon, A.W. Harrow and I.L. Chuang. “Efficient circuits for Clebsch-Gordon transformations.” in preparation C.H. Bennett, A.W. Harrow and S. Lloyd. “Universal compression via gentle tomography.” in preparation

Inequivalent resources? One bit of quantum communication (1 qubit) can be used to send one classical bit (1 cbit) or generate one EPR pair (1 ebit). Conversely, teleportation uses two cbits and 1 ebit to send 1 qubit. Although the above protocols are optimal, combining them uses three qubits to send one qubit. Similar inefficiencies exist throughout quantum information theory. Are they necessary?

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 cbit:|x i A ! |x i A |x i B ebit: |  i =2 -1/2  x |x i A |x i B 1 qubit > 1 coherent cbit > 1 cbit 1 qubit > 1 coherent cbit > 1 ebit

sources of CCC Super-dense coding: 1 qubit + 1 ebit > 2 coherent cbits Distributed unitary gates: If U is a unitary gate and U > C cbits, then U > C coherent cbits. Example: CNOT > 1 cbit (  )CNOT > 1 cbit (  ) CNOT + ebit > 1 cbit (  ) + 1 cbit (  )

uses of CCC Entanglement recycling using CCC: Suppose X + C cbits > Y and the classical message sent is independent of the output state. Then X + C coherent cbits > Y + C ebits

Example: teleportation with coherent communication H XZ 2 coherent cbits + 1 ebit > 1 qubit + 2 ebits coherent classical comm.

Simple consequences 2 coherent cbits = 1 qubit + 1 ebit (using entanglement catalytically) Teleportation and super-dense coding are no longer irreversible.

There’s more! Super-dense coding of quantum states: 1 qubit + 1 ebit > 2 remote qubits (asymptotically) Single-letter expression for capacity of a unitary interaction to communicate classical or quantum data assisted by any amount of entanglement. 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 C.H. Bennett, A.W. Harrow, D.W. Leung and J.A. Smolin. “On the capacities of bipartite Hamiltonians and unitary gates.” quant- ph/ , accepted IEEE-IT. A.W. Harrow. “Coherent Classical Communication.” quant-ph/ I. Devetak, A.W. Harrow and A. Winter. “A family of quantum protocols.” quant- ph/