Bidirectional classical-communication cost of bipartite channel simulation
- Fields
- Topics
Problem
What is the asymptotic classical-communication cost of simulating a bipartite quantum channel with bidirectional classical communication and non-signalling assistance? Let \(\mathcal N:\mathcal L(A_0\otimes B_0)\to\mathcal L(A_1\otimes B_1)\) be a quantum channel on finite-dimensional systems, with Alice holding \(A_0,A_1\) and Bob holding \(B_0,B_1\). A bidirectional simulation protocol consists of a bipartite channel shared in advance that can signal in neither direction (a non-signalling correlation, which includes every shared entangled state), one classical message with \(m_\to\) values from Alice to Bob, one classical message with \(m_\leftarrow\) values from Bob to Alice, and local operations; it uses \(m:=m_\to m_\leftarrow\) classical values in total. For \(\varepsilon\in[0,1)\) define the one-shot cost
where \(\|\cdot\|_\diamond\) is the diamond norm. The asymptotic exact and vanishing-error costs are
Let \(\mathrm{NS}\) denote the set of bipartite channels from \(A_0B_0\) to \(A_1B_1\) that are non-signalling in both directions, and define the max-relative entropy of bidirectional communication
where \(J\) denotes the Choi operator. Determine \(S_{\leftrightarrow,0}(\mathcal N)\) and \(S_{\leftrightarrow}(\mathcal N)\) in Eq. (2) for a general bipartite channel. In particular, is either cost equal to the regularization \(\lim_{n\to\infty}\frac1n \mathfrak D^{\leftrightarrow}_{\max}(\mathcal N^{\otimes n})\) of Eq. (3), and does a single-letter formula exist?
Source
The question is implicit in Zhu, Zhao, and Wang, who determine the asymptotic exact one-way cost of a bipartite channel but obtain only one-shot semidefinite programs and converse bounds for the bidirectional cost [ZZW25].
Progress
For a point-to-point channel \(\mathcal M:\mathcal L(A)\to\mathcal L(B)\) and tensor-power input sources, the entanglement-assisted quantum reverse Shannon theorem makes simulation and coding reversible: the asymptotic classical-communication cost of simulating \(\mathcal M^{\otimes n}\) with free entanglement equals the entanglement-assisted classical capacity
\begin{equation} C_E(\mathcal M) =\max_{\phi_{RA}}I(R;B)_{(\operatorname{id}_R\otimes\mathcal M)(\phi)}. \tag{4} \end{equation}Equation (4) concerns a single sender and receiver and does not address interactive bipartite processes [BDHSW14].
For one-way simulation of a bipartite channel, Zhu, Zhao, and Wang bound the one-shot \(\varepsilon\)-error cost above and below by a smooth max-relative entropy of one-way classical communication and prove that the asymptotic exact one-way cost equals the additive quantity
\begin{equation} \mathfrak D^{\to}_{\max}(\mathcal N) :=\min_{\mathcal E\in\mathrm{NS}_\to}D_{\max}(\mathcal N\|\mathcal E), \tag{5} \end{equation}where \(\mathrm{NS}_\to\) is the set of bipartite channels that cannot signal from Alice to Bob. Equation (5) settles the one-way problem but has no proven bidirectional analogue [ZZW25].
In the bidirectional setting the same work gives a semidefinite program for the one-shot exact cost \(S^{(1)}_{\leftrightarrow,0}(\mathcal N)\) of Eq. (1) in terms of non-signalling bipartite superchannels, introduces a bipartite conditional min-entropy of the channel as an efficiently computable lower bound on the asymptotic exact cost, and shows that \(\mathfrak D^{\leftrightarrow}_{\max}\) in Eq. (3) is subadditive under tensor products. Additivity of \(\mathfrak D^{\leftrightarrow}_{\max}\) and an achievability theorem matching either converse are not known [ZZW25].
Comment
The point-to-point reverse Shannon theorem does not determine the two-directional resource trade-off of an interactive bipartite process. For bidirectional simulation only converses and one-shot semidefinite programs are available: it is unknown whether the asymptotic costs in Eq. (2) equal a regularized or single-letter bidirectional max-relative entropy, and no coding theorem attains the known lower bounds.