This paper proposes a revised definition for the entanglement cost of a quantum channel $\mathcal{N}$. 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 $\mathcal{N}$, such that the most general discriminator cannot distinguish the $n$ calls to $\mathcal{N}$ 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)] 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 '07 (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's entanglement cost [Berta et al., IEEE Trans. Inf. Theory 59, 6779 (2013)], 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.

• Received 2 August 2018

DOI:https://doi.org/10.1103/PhysRevA.98.042338