Entanglement cost and quantum channel simulation

This paper proposes a revised definition for the entanglement cost of a quantum channel N. In particular, it is defined here to be the smallest rate at which entanglement is required, in addition to free classical communication, in order to simulate n calls to N, such that the most general discriminator cannot distinguish the n calls to N from the simulation. The most general discriminator is one who tests the channels in a sequential manner, one after the other, and this discriminator is known as a quantum tester [Chiribella et al., Phys. Rev. Lett. 101, 060401 (2008)PRLTAO0031-900710.1103/PhysRevLett.101.060401] or one who is implementing a quantum costrategy [Gutoski and Watrous, in Proceedings of the Thirty-Ninth Annual ACM Symposium on Theory of Computing STOC \u2707 (ACM Press, New York, 2007), pp. 565-574]. As such, the proposed revised definition of entanglement cost of a quantum channel leads to a rate that cannot be smaller than the previous notion of a channel\u27s entanglement cost [Berta et al., IEEE Trans. Inf. Theory 59, 6779 (2013)IETTAW0018-944810.1109/TIT.2013.2268533], in which the discriminator is limited to distinguishing parallel uses of the channel from the simulation. Under this revised notion, I prove that the entanglement cost of certain teleportation-simulable channels is equal to the entanglement cost of their underlying resource states. Then I find single-letter formulas for the entanglement cost of some fundamental channel models, including dephasing, erasure, three-dimensional Werner-Holevo channels, and epolarizing channels (complements of depolarizing channels), as well as single-mode pure-loss and pure-amplifier bosonic Gaussian channels. These examples demonstrate that the resource theory of entanglement for quantum channels is not reversible. Finally, I discuss how to generalize the basic notions to arbitrary resource theories