On Environment-Assisted Capacities of Quantum Channels
2007, v.13, №2, 297-314
Following initial work by Gregoratti and Werner [M. Gregoratti and R.F. Werner, Quantum lost and found, J. Mod. Optics, 2003, v. 50, N 6&7, 913-933] and [M. Gregoratti and R.F. Werner, On quantum error correction by classical feedback in discrete time, J. Math. Phys., 2004, v. 45, N 7, 2600-2612, quant-ph/0403092] and Hayden and King [P. Hayden and C. King, Correcting quantum channels by measuring the environment, Quantum Information and Computation, 2005, v. 5, N 2, 156-160, quant-ph/0409026], we study the problem of the capacity of a quantum channel assisted by a "friendly (channel) environment" that can locally measure and communicate classical messages to the receiver. Previous work [J.A. Smolin, F. Verstraete and A. Winter, Entanglement of assistance and multipartite state distillation, Phys. Rev. A, 2005, v. 72, 052317, quant-ph/0505038] has yielded a capacity formula for the quantum capacity under this kind of help from the environment. Here we study the problem of the environment-assisted classical capacity, which exhibits a somewhat richer structure (at least, it seems to be the harder problem). There are several, presumably inequivalent, models of the permitted local operations and classical communications between receiver and environment: one-way, arbitrary, separable and PPT POVMs. In all these models, the task of decoding a message amounts to discriminating a set of possibly entangled states between the two receivers, by a class of operations under some sort of locality constraint. After introducing the operational capacities outlined above, we show that a lower bound on the environment-assisted classical capacity is always half the logarithm of the input space dimension. Then we develop a few techniques to prove the existence of channels which meet this lower bound up to terms of much smaller order, even when PPT decoding measurements are allowed (assuming a certain superadditivity conjecture).
Keywords: entanglement of assistance,quantum error correction,feedbackcontrol,LOCC discrimination,PPT discrimination,additivity conjecture