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arXiv:2609.14658

Exact Asymptotic Rates and an Exponential Strong Converse for quantum SMP and One-Way Communication

Daiki Suruga

quant-phSubmitted Sep 13, 2026• Announced Sep 15, 2026• 0 citations
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Abstract

Computing many instances jointly can reduce communication per instance. We ask whether such savings occur in the simultaneous-message-passing (SMP) model and how they depend on quantum messages and shared resources. For every finite total function $f$ and fixed error $0\le\varepsilon<1$, we prove that the optimal worst-case communication per instance for $f^n$ converges to \[ D^\parallel(f)=\log|\mathrm{Row}(f)|+\log|\mathrm{Col}(f)|, \] where $\mathrm{Row}(f)$ and $\mathrm{Col}(f)$ are the sets of distinct rows and columns of the table $(f(x,y))_{x,y}$. This holds for classical and quantum SMP without shared entanglement, with or without shared randomness. Thus, transmitting row and column indices is asymptotically optimal, even with bounded error, joint computation, and quantum messages. In particular, the exponential advantage of quantum fingerprinting for a single instance of equality without shared randomness disappears in the asymptotic rate. We also prove that the same threshold governs an exponential strong converse: every fixed worst-case communication rate below it forces the probability of computing all $n$ outputs correctly to decay as $2^{-Ω(n)}$, even on average under a fixed product input distribution. With shared randomness, the optimal expected communication rate is $(1-\varepsilon)D^\parallel(f)$. Entanglement shared between each sender and the referee halves both rates; arbitrary tripartite entanglement gives no further reduction. The results extend to a class of total relations and yield one-way characterizations with $D^\parallel(f)$ replaced by $\log|\mathrm{Row}(f)|$. Our proof reduces SMP to one-way communication and relates successful computation to sampling based on quantum min-entropy; we also give a simpler proof of the key lemma using the pretty-good measurement.

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Human-verified;AI-proven

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We study the relative power of public randomness and quantum communication in the multiparty Simultaneous Message Passing (SMP) model. By analyzing a total Boolean function and a relational problem, we establish polynomial and exponential separations between classical public-coin protocols and quantum protocols without shared randomness or prior entanglement. We first introduce XOR Equality, a total Boolean function in which each of kkk players receives an unrestricted string xj∈{0,1}nx_j\in\{0,1\}^nxj​∈{0,1}n, and the referee must decide whether x1⊕⋯⊕xk=0nx_1\oplus\cdots\oplus x_k=0^nx1​⊕⋯⊕xk​=0n. For k≥8k\ge8k≥8 and worst-case error at most 1/31/31/3, we prove quantum lower bounds of Ω(min⁡{n,k2})Ω(\min\{n,k^2\})Ω(min{n,k2}) on total communication and Ω(min⁡{n/k,k})Ω(\min\{n/k,k\})Ω(min{n/k,k}) on maximum message length. A classical public-coin protocol uses two bits per player with error at most 1/41/41/4. For n=k2n=k^2n=k2 and worst-case error at most 1/31/31/3, we prove that quantum protocols require maximum message length Ω(n)Ω(\sqrt n)Ω(n​) qubits, whereas two public-coin bits per player suffice. We then turn to relations, for which we obtain exponential separations. Specifically, we study Multiparty Index Coordination, a kkk-player generalization of the two-player relation studied by Gavinsky, Kempe, Regev, and de Wolf (STOC 2006). We prove an unambiguous quantum lower bound of Ω(n1−1/k)Ω(n^{1-1/k})Ω(n1−1/k) on maximum message length, matched by a classical private-coin protocol. For any fixed error ε∈(0,1/8)\varepsilon\in(0,1/8)ε∈(0,1/8), we also prove a bounded-error quantum lower bound of Ω(n(k−1)/(k+1))Ω(n^{(k-1)/(k+1)})Ω(n(k−1)/(k+1)) while we show that public-coin protocols solve the relation unambiguously with maximum message length O(log⁡n)O(\log n)O(logn), giving exponential separations in both regimes. The implicit constants are independent of kkk, and both quantum lower bounds become linear when k≥clog⁡nk\ge c\log nk≥clogn for any fixed c>0c>0c>0.
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In the private simultaneous message (PSM) setting, kkk players obtain inputs xi∈{0,1}nx_i\in\{0,1\}^nxi​∈{0,1}n and then each send messages to a referee, who should learn f(x1,...,xk)f(x_1,...,x_k)f(x1​,...,xk​) but no other information about (x1,...,xk)(x_1,...,x_k)(x1​,...,xk​). The PSM setting was introduced as a minimal model for secure multiparty computation and has connections to Boolean function complexity. In the quantum setting, PSM has been related to non-local quantum computation (NLQC). The communication and correlation cost of implementing PSM remains poorly understood. Here, we give new upper and lower bounds on the (quantum) PSM model. For lower bounds, we show: 1) Nečiporuk's measure lower bounds the entanglement required for kkk-player quantum PSM with perfect correctness. This leads to quadratic lower bounds for explicit functions. 2) The rank of the communication matrix of f(x1,x2)f(x_1,x_2)f(x1​,x2​) lower bounds 2-player quantum PSM with perfect privacy but imperfect correctness. This implies a previously unknown lower bound on classical PSM with imperfect correctness. When allowing quantum communication and shared entanglement, these are the first lower bounds on quantum PSM that make use of the privacy condition. For upper bounds, we show: 1) Letting sss be the size of a quantum circuit computing fff, dfd_fdf​ be the circuit depth, kkk the number of players, nnn the number of bits received by each player, and εεε a correctness parameter, we obtain PSMk∗(f)≤(kn+s)⋅log⁡O(df)(s/ε)\mathsf{PSM}_k^*(f) \leq (kn +s) \cdot \log^{O(d_f)}(s/ε)PSMk∗​(f)≤(kn+s)⋅logO(df​)(s/ε). 2) The square of the Fourier 1 norm of fff, ∥f^∥12\Vert \hat{f}\Vert_1^2∥f^​∥12​, upper bounds the classical PSM complexity, PSM(f)≤O(∥f^∥12)\mathsf{PSM}(f)\leq O(\Vert \hat{f} \Vert^2_1)PSM(f)≤O(∥f^​∥12​). In proving the first upper bound, we generalize existing TTT-depth based techniques for NLQC from 222 to k≥2k\geq 2k≥2 parties, and consider cases where the Clifford layers are restricted to having small light cones.
Buhrman, Cleve, Watrous, and de Wolf (PRL 2001) discovered the quantum fingerprinting protocol, which is the quantum SMP protocol with O(log⁡n)O(\log n)O(logn) qubits communication for the equality problem. In the protocol, Alice and Bob create some quantum fingerprints of their inputs, and the referee conducts the SWAP tests for the quantum fingerprints. Since Ω(n)Ω(\sqrt{n})Ω(n​) bits communication is required with the classical SMP scheme for the equality problem first shown by Newman and Szegedy (STOC 1996), there exists an exponential quantum advantage in the amount of communication. In this paper, we consider a setting in which the referee can do only incoherent measurements rather than coherent measurements including the SWAP tests. We first show that, in the case of one-way LOCC measurements, Ω(n)Ω(\sqrt{n})Ω(n​) qubits communication is required. To prove the result, we derive a new method to replace quantum messages by classical messages and consider a reduction to the optimal lower bound in the hybrid SMP model where one message is quantum and the other is classical, which was first shown by Klauck and Podder (MFCS 2014). Our method uses the result of Oszmaniec, Guerini, Wittek, and Acín (PRL 2017), who showed that general POVM measurements can be simulated by randomized projective measurements with small ancilla qubits, and Newman's theorem. We further investigate the setting of quantum SMP protocols with two-way LOCC measurements, and derive a lower bound against some restricted two-way LOCC measurements. To prove it, we revisit the technique to replace quantum messages by classical deterministic messages introduced by Aaronson (ToC 2005) and generalized by Gavinsky, Regev, and de Wolf (CJTCS 2008), and show that, using the deterministic message, the referee can simulate the two-way LOCC measurements.
Private simultaneous quantum messages (PSQM) provide a minimal model of noninteractive secure multiparty quantum computation, yet their communication complexity for concrete functions remains poorly understood. We show that familiar total functions already exhibit both genuine quantum advantages and strong structural limitations. For three-party equality on nnn-bit inputs, which determines whether the players' nnn-bit strings are identical, we construct an entanglement-assisted protocol with asymptotic communication (1.633…)n(1.633\ldots)n(1.633…)n qubits, whereas every perfectly correct protocol without prior entanglement requires at least 3n3n3n qubits. For three-bit majority, one shared GHZ state and one transmitted qubit per player suffice for perfect correctness and perfect privacy, reducing the communication from the optimal classical cost of six bits to three qubits. Three-bit AND behaves differently. No perfectly correct and perfectly private protocol exists when the players share an arbitrary three-qubit state, apply arbitrary local unitary encodings, and transmit one qubit each. However, increasing the local dimension overcomes this obstruction: one shared three-qutrit GHZ state and one transmitted qutrit per player yield a one-shot protocol with total communication 3log⁡233\log_2 33log2​3. Together with the six-bit classical lower bound, this gives a strict quantum communication advantage. More generally, for every k≥3k\geq 3k≥3, we construct a perfectly correct and perfectly private one-shot protocol for kkk-bit AND using one qudit of dimension at most kkk per player, and hence at most klog⁡2kk\log_2 kklog2​k qubits in total.
A quantum channel can transmit quantum states, classical messages, or classical messages that remain secret from its environment. We prove exponential strong converses for all three tasks over arbitrary finite-dimensional memoryless quantum channels, at their respective regularized capacities and allowing arbitrary entangled inputs across channel uses and joint decoding. Above quantum or classical capacity, entanglement fidelity or decoding success probability, respectively, decays exponentially with the number of channel uses. Above private capacity, fidelity to an ideal shared secret key decays exponentially; in particular, vanishing secrecy error forces decoding success to vanish. The proofs combine one-shot quantum blowing-up lemmas, which convert codes with low success into reliable codes with a controlled loss in code size, with a low-degree polynomial approximation of the channel's Stinespring image projector. The loss is governed by the projective tensor norm of the approximation across the receiver--environment bipartition. For memoryless channels, the polynomial construction gives exponential accuracy while keeping the rate loss arbitrarily small, turning the corresponding weak converses into strong converses.