Quantum Entanglement Assistance Improves the Capacity and Activates the Zero-Error Capacity of Classical Channels with Causal CSIT

Abstract

For classical point-to-point channels, it has been shown by Bennett et al. that quantum entanglement assistance cannot improve their capacity, and by Cubitt et al. that entanglement assistance cannot activate (increase from zero to non-zero) their zero-error capacity. In contrast, we show that for classical point-to-point channels with causal CSIT (channel state information at the transmitter), quantum entanglement assistance can in some cases improve their capacity, and in some cases activate their zero-error capacity.

Figures (8)

• Figure 1: General entanglement-assisted coding scheme with causal CSIT. A message $W$ is encoded into symbols $X_1,\dots,X_n$ that are sent by the transmitter over $n$ uses of a memoryless channel with state $\mathsf{N}_{Y\mid XS}$. The state sequence $S_1,\dots,S_n\stackrel{\hbox{\tiny i.i.d.}}{\sim} \mathsf{P}_S$ is revealed causally to the transmitter. The encoding of $X_i$ over the $i^{th}$ channel use is based on a quantum instrument depicted as $\mathcal{E}^{(i)}$ in the figure which can depend (the dependencies are not shown in the figure) on $W, S_1,\dots, S_i, X_1,\dots,X_{i-1}$, acting on the transmitter's side $(A)$ of an (entangled) quantum system $AB$ shared with the receiver in advance. The decoding at the receiver is based on a POVM that depends on the entire channel output sequence $Y_1,\dots,Y_n$, acting on $B$. See Section \ref{['sec:EAcoding']} for a formal description.
• Figure 2: A strategy that converts each use (say, the $i^{th}$ use) of a channel $\mathsf{N}$ with state ($S_i$) and entanglement assistance ($A_iB_i$), informally $(A_i,S_i,X_i)\stackrel{\mathsf{N}}{\longrightarrow} (B_i,Y_i)$, into an effective classical channel $\mathsf{N}'$ with input $X_i'$ and output $Y_i'$, that has no state and no entanglement assistance, informally $X_i'\stackrel{\mathsf{N}'}{\longrightarrow} Y_i'$.
• Figure 3: Two examples of graph channels with state. The left-hand side shows $\mathcal{C}_5$ and the right-hand side shows $\mathcal{K}_4$. For each graph channel with state, each vertex (corresponding to an output symbol) is labeled by a value $y\in \mathcal{Y}$ (without loss of generality we let $\mathcal{Y} = \{0,1,\cdots, m-1\}$), and each edge (corresponding to a channel state) is labeled by a tuple $s=(s_0,s_1)$, where $s_0, s_1\in \mathcal{Y}$ denote the two endpoints of the edge $s$. Given the channel state $S=s$ and input $X\in \{0,1\}$, the receiver observes the output $Y=s_X$.
• Figure 4: Plots of the classical capacity $C^{\mathrm{ C} }(\mathcal{K}_8,p)$ (Eq. \ref{['eq:CC_Km']}), achievable rate by entanglement assistance $R^{\mathrm{ EA} }(\mathcal{K}_8,p)$ (Eq. \ref{['eq:RQ_Km']}), and the gain factor $R^{\mathrm{ EA} }(\mathcal{K}_8,p)/C^{\mathrm{ C} }(\mathcal{K}_8,p)$, for the graph channel with state associated with $\mathcal{K}_8$, as functions of $p$.
• Figure 5: A channel with state defined from the B-KS set \ref{['eq:BKS']}: For each $\mathcal{A}_i, i\in \{1,2,3\}$ (and similarly $\mathcal{B}_j, j\in \{1,2,3\}$), there are $4$ channel inputs (dark circles in the figure) associated with the $4$ vectors in $\mathcal{A}_i$ ($\mathcal{B}_j$). The channel thus contains $24$ inputs in total. The channel state $S\in \{(i,j)\colon i\in \{1,2,3\}, j\in \{1,2,3\}\}$, such that when the state is $S=(i,j)$, the transmitter is restricted to picking an input from the subsets of inputs associated with $\mathcal{A}_i \cup \mathcal{B}_j$ (See Remark \ref{['rem:input_constraint']}). Two inputs are connected by an edge if and only if their associated vectors are orthogonal (not all such edges are drawn in the figure above). The edges correspond to the outputs of the channel. Conditioned on the state $S=(i,j)$, when the transmitter chooses an input $x$ from the allowed subset, the output of the channel is an edge $y$ drawn randomly from the edges incident to $x$. The graph can be also viewed as the confusability graph of the underlying channel without the state-dependent input constraints.

Theorems & Definitions (17)

• Remark 1
• Remark 2
• Definition 1: Graph channel with state
• Remark 3
• Theorem 1: Cyclic graph $\mathcal{C}_5$
• Definition 2: Noisy version
• Theorem 2: Noisy complete graph $\mathcal{K}_m$
• Corollary 1
• Theorem 3
• Definition 3: B-KS set BPQS