Abstract-Let X and Y be finite non-empty sets and (X, Y ) a pair of random variables taking values in X × Y. We consider communication protocols between two parties, Alice and Bob, for generating X and Y . Alice is provided an x ∈ X generated according to the distribution of X, and is required to send a message to Bob in order to enable him to generate y ∈ Y, whose distribution is the same as that of Y |X=x. Both parties have access to a shared random string generated in advance. Let T (X : Y ) be the minimum (over all protocols) of the expected number of bits Alice needs to transmit to achieve this. We show thatWe also consider the worst-case communication required for this problem, where we seek to minimize the average number of bits Alice must transmit for the worst-case x ∈ X . We show that the communication required in this case is related to the capacity C(E) of the channel E, derived from (X, Y ), that maps x ∈ X to the distribution of Y |X=x. We show that the required communication T (E) satisfiesUsing the first result, we derive a direct sum theorem in communication complexity that substantially improves the previous such result shown by Jain, Radhakrishnan and Sen [In Proc. 30th International Colloquium of Automata, Languages and Programming (ICALP), ser. LNCS, vol. 2719LNCS, vol. . 2003.These results are obtained by employing a rejection sampling procedure that relates the relative entropy between two distributions to the communication complexity of generating one distribution from the other.

Compression of a message up to the information it carries is key to many tasks involved in classical and quantum information theory. Schumacher [B. Schumacher, Phys. Rev. A 51, 2738 (1995)PLRAAN1050-294710.1103/PhysRevA.51.2738] provided one of the first quantum compression schemes and several more general schemes have been developed ever since [M. Horodecki, J. Oppenheim, and A. Winter, Commun. Math. Phys. 269, 107 (2007); CMPHAY0010-361610.1007/s00220-006-0118-xI. Devetak and J. Yard, Phys. Rev. Lett. 100, 230501 (2008); PRLTAO0031-900710.1103/PhysRevLett.100.230501A. Abeyesinghe, I. Devetak, P. Hayden, and A. Winter, Proc. R. Soc. A 465, 2537 (2009)PRLAAZ1364-502110.1098/rspa.2009.0202]. However, the one-shot characterization of these quantum tasks is still under development, and often lacks a direct connection with analogous classical tasks. Here we show a new technique for the compression of quantum messages with the aid of entanglement. We devise a new tool that we call the convex split lemma, which is a coherent quantum analogue of the widely used rejection sampling procedure in classical communication protocols. As a consequence, we exhibit new explicit protocols with tight communication cost for quantum state merging, quantum state splitting, and quantum state redistribution (up to a certain optimization in the latter case). We also present a port-based teleportation scheme which uses a fewer number of ports in the presence of information about input.

In this work we introduce, both for classical communication complexity and query complexity, a modification of the partition bound introduced by Jain and Klauck [JK10]. We call it the public-coin partition bound. We show that (the logarithm to the base two of) its communication complexity and query complexity versions form, for all relations, a quadratically tight lower bound on the public-coin randomized communication complexity and randomized query complexity respectively.The partition bound introduced by Jain and Klauck [JK10] is known to be one of the strongest lower bound methods in classical communication complexity and query complexity. It is known to be stronger than most other lower bound methods, both in communication complexity and query complexity, except its relationship with the information complexity lower bound method in communication complexity is unknown. It is an interesting open question, in both these settings, as to how tight this lower bound method is. We are not aware, to the best of our knowledge, of any function or relation where this method is asymptotically weaker either for communication complexity or for query complexity.In this work we introduce, both for communication complexity and query complexity, a modification of the partition bound which we call the public-coin partition bound. Analogous to the partition bound, our new bound is also a linear-programming based lower bound method. We show that (the logarithm to the base two of) its communication and query complexity versions continue to form a lower bound on the public-coin communication complexity and randomized query complexity respectively. In addition we show that the square of (the logarithm to the base two of) its communication and query complexity versions form an upper bound on the public-coin communication complexity and randomized query complexity respectively. Also it is easily seen via their linear programs that our new bound is stronger than the partition bound for all relations, both in communication complexity and query complexity.