On the Capacity of Vector Linear Computation over a Noiseless Quantum Multiple Access Channel with Entangled Transmitters

TL;DR

This work resolves the inverted 3-sum box problem for vector linear computation over a noiseless quantum MAC with entangled transmitters, proving that $N$-sum box protocols (together with time-sharing and TQC) are information-theoretically optimal for $K=3$. By deriving a complete polyhedral region $rak{D}^*$ of feasible per-transmitter download costs and establishing matching achievability and converse proofs, it demonstrates the centrality of the strong self-orthogonality constraint and the necessity of 3-way entanglement for general vector linear computations. The results extend known capacity concepts from the Σ-QMAC to the vector-linear LC-QMAC, and reveal both the power and the limits of entanglement-assisted coding in noiseless quantum networks. The findings have implications for quantum network function computation and motivate future work on $K>3$ transmitters and on noisy-channel extensions, where entanglement-assisted strategies may still yield fundamental capacity limits.

Abstract

Network function computation is an active topic in network coding, with much recent progress for linear (over a finite field) computations over broadcast (LCBC) and multiple access (LCMAC) channels. Over a quantum multiple access channel (QMAC) with quantum-entanglement shared among transmitters, the linear computation problem (LC-QMAC) is non-trivial even when the channel is noiseless, because of the challenge of optimally exploiting transmit-side entanglement through distributed coding. Given an arbitrary linear function of data streams defined in a finite field $\mathbb{F}_d$, the LC-QMAC problem seeks the optimal communication cost (minimum number of qudits that need to be sent by the transmitters to the receiver, per computation instance) over a noise-free QMAC, when the independent input data streams originate at the corresponding transmitters, who share quantum entanglement in advance. As our main result, we fully solve this problem for $K=3$ transmitters ($K\geq 4$ settings remain open). Coding schemes based on the $N$-sum box protocol (along with time-sharing and batch-processing) are shown to be information theoretically optimal in all cases.

Figures (5)

• Figure 1: LC-QMAC($\mathbb{F}_d, K, {\bm V}_1, {\bm V}_2, \cdots, {\bm V}_K$). $Q_1, Q_2,$$\cdots, Q_K$ are entangled quantum systems. Alice$_k$ encodes $W_k$ into $Q_k$, and Bob measures the joint system $Q_1Q_2\cdots Q_K$ to obtain the desired computation $F$.
• Figure 2: The $N$-sum box Allaix_N_sum_box is shown on the left as a black-box abstraction of a quantum protocol for classical distributed many-to-one linear computation. The actual quantum protocol is shown on the right (details can be found in Allaix_N_sum_box). $N$ qudits are initially prepared in a suitable stabilizer state $\ket{\psi}$ and distributed to $N$ transmitters, the classical inputs $(x_n,z_n)$ are applied by the $n^{th}$ transmitter to manipulate the $n^{th}$ qudit via conditional Pauli $X$ and $Z$ gates, all $N$ qudits are sent to a receiver (so the communication cost of the protocol is $N$ qudits), and a joint measurement at the receiver produces the linear function of the inputs, ${\bm y} = {\bm M}_x {\bm x}+{\bm M}_z {\bm z}$. Given any ${\bm M}_x, {\bm M}_z\in\mathbb{F}_d^{N\times N}$ there exists a stabilizer state $\ket{\psi}$ and a measurement that realizes this computation functionality, provided rank$[{\bm M}_x, {\bm M}_z]=N$ and ${\bm M}_x{\bm M}_z^\top = {\bm M}_z{\bm M}_x^\top$ (strong self-orthogonality).
• Figure 3: $\mathfrak{D}^*$ for Toy Example 3 is shown on the left. A coding scheme over $\mathbb{F}_3$ utilizing a $5$-sum box protocol is shown in the middle, achieving $2$ computations of the desired function $(A+B+C,D)$ at a communication cost of $N_1,N_2,N_3=1,1,3$ qudits from Alice$_1$, Alice$_2$, Alice$_3$, who have input streams $(A), (B), (C,D)$, respectively. A projection of $\mathfrak{D}^*$ into $2$ dimensions, by setting $\Delta_1=\Delta_2$, is shown on the right (blue region), along with the unentangled/classical feasible region (green, contained in blue, obtained directly from classical cut-set bounds). The black dot $(\Delta_1,\Delta_2,\Delta_3)=(0.5,0.5,1.5)$ per computation, corresponds to the scheme illustrated in the middle. The region that is outside the blue region, e.g., the red dot $(\Delta_1,\Delta_2,\Delta_3)=(0.5,0.5,1)$, is not feasible by any coding scheme, i.e., not even with other protocols that may not rely on the $N$-sum box, as shown by the information theoretic converse of Theorem \ref{['thm:new_bounds']}.
• Figure 4: A quantum coding scheme for the LC-QMAC. The output measured at the receiver, $Y ^{({\bm w})}$, must be equal to ${\bm V}_1w_1+{\bm V}_2 w_2+\cdots+{\bm V}_Kw_K$, for all realizations of $(w_1,w_2,\cdots, w_K)$.
• Figure 5: $\mathfrak{D}^*$ for Toy Example 6.

Theorems & Definitions (14)

• Lemma 1: $N$-sum box Allaix_N_sum_box
• Theorem 1: Communication bounds
• Theorem 2: Multiparty computation bounds
• Theorem 3
• Remark 1
• Remark 2
• Lemma 2: No-communication
• proof
• Lemma 3
• proof