bibkey: chitambargeorgedoolittlejunge2023communication
authors: Eric Chitambar; Ian George; Brian Doolittle; Marius Junge
year: 2023
title: "The Communication Value of a Quantum Channel"
doi: 10.1109/TIT.2022.3218540
url: https://arxiv.org/abs/2109.11144v1
claim: "The paper defines the communication value cv of a quantum channel as the largest unnormalized guessing score of the classical channels it induces, its entanglement-assisted version cv*, characterizes both by conic programs over the separable and positive cones, and conjectures cv*(N) <= d_B cv(N) for every channel."
strata_touched:
  - D5/S3/Quantum/QuantumChannels/EntanglementAssistedCommunicationValue
license: citation-only
triage: anchor

The Communication Value of a Quantum Channel

E. Chitambar, I. George, B. Doolittle and M. Junge, arXiv:2109.11144v1 (2021-09-23); IEEE Trans. Inf. Theory 69(3), 1660–1679 (2023). Quotations are from the v1 TeX source.

The communication value of a classical channel (Eq:Cv):

\cv(\mbf{P})=\sum_{y\in[n’]}\max_{x\in[n]}P(y|x).

The communication value of a quantum channel (Defn:cv):

The $[n]\to [n^{\prime }]$ communication value (cv) of a quantum channel $\text{mc}N\in \text{cptp}(A\to B)$ is \cv^{n\to n’}(\mc{N})=\max_{\substack{{\Pi_{y}}{y=1}^{n’}\{\rho_x}{x=1}^n}}{\cv(\mbf{P});|; P(y|x)=\tr[\Pi_y\mc{N}(\rho_x)]}, and the cv value of $\text{mc}N$ is defined as \cv(\mc{N})=\sup_{n,n\in\mbb{N}}\cv^{n\to n’}(\mc{N}).

The entanglement-assisted protocol (Eq:Ea-cv):

P(y|x)=\tr\left[\Pi_y^{BB’}\left(\mc{N}^{A\to B}\circ\mc{E}_x^{A’\to A}\otimes\id^{B’}[\varphi^{A’B’}]\right)\right].

The entanglement-assisted communication value:

The entanglement-assisted communication value (ea cv) of a quantum channel $\text{mc}N\in \text{cptp}(A\to B)$, denoted ${\text{cv}}^{\ast }(\text{mN})$, is \sup_{\varphi^{A’B’}}\max{\cv(\mbf{P});|;\text{$P(y|x)$ given by Eq. \eqref{Eq:Ea-cv}}} , , where the supremum is taken over all entangled states ${\phi }^{A^{\prime }{B}^{\prime }}$ (and all dimensions $A^{\prime }{B}^{\prime }$), while the maximization considers all $n,n^{\prime }\in \text{mbb}N$ along with arbitrary states $\{{\rho }_x{\}}_{x\in [n]}$ and POVMs $\{{\Pi }_y{\}}_{y\in [n^{\prime }]}$.

The conjecture:

\noindent\textit{Conjecture}: For any channel $\text{mc}N\in \text{cptp}(A\to B)$, \cv^*(\mc{N})\leq d_B\cdot \cv(\mc{N}).

Verified locator

Source: https://arxiv.org/abs/2109.11144v1

• URL: https://arxiv.org/abs/2109.11144v1 (source arxiv_draft_v1.tex, md5 7008e4461225de9fc07e55cf4b3ffcce):
  • the classical communication value, l. 128–131;
  • the communication value of a quantum channel, l. 143–153;
  • the entanglement-assisted protocol, l. 1353–1356;
  • the entanglement-assisted communication value, l. 1358–1364;
  • the conjecture, l. 1462–1465.
• DOI: 10.1109/TIT.2022.3218540 (https://doi.org/10.1109/TIT.2022.3218540).