On the Limits of Quantum Multiparty Simultaneous Communication

The Simultaneous Message Passing (SMP) model framework allows for comparing classical and quantum communication by analyzing resource trade-offs in distributed computing. The work establishes an exponential separation between classical shared randomness and unentangled quantum communication by generalizing Index Coordination to a multiparty setting, proving that quantum superposition cannot efficiently simulate the coordination afforded by public randomness pg1, two players, Gavinsky et al. (STOC 2006) established a separation underlying the incomparability of shared randomness and quantum communication: Index Coordination needs O(log n) public-coin bits but omega(n 1/3) bounded-error qubits pg1.

The Generalization and Problem Setup

The authors introduce ICk,n, a natural k-party generalization of Index Coordination pg3. This problem involves k ≥ 2 players P1,..., Pk and a central referee where the goal is to output a tuple (i, x(i)1,..., x(i)k) for some index i such that s(i) = 1 pg9. The first k − 1 players each receive an n-bit string xj ∈ (0, 1)n, and the k-th player receives a string xk ∈ (0, 1)n alongside a selector string s with Hamming weight exactly s = n/2 pg9. This problem elegantly captures the essence of multiparty coordination, directly recovering the classic two-player relation exactly for k = 2 pg3.

Classical Communication Bounds

The classical public-coin protocol for ICk,n admits a highly efficient protocol utilizing shared randomness pg10. The players use the public coin to sample an index i ∈ [n] uniformly at random, and the maximum message length is therefore bounded by R∥pub(ICk,n) = O(log n) bits pg10. Without public randomness, a simple strategy generalizes the protocol described by Ambainis [1] pg13, yielding a maximum message length of R∥priv(ICk,n) = O(n 1−1/k) bits pg10.

Quantum Communication Lower Bounds

The paper establishes lower bounds on the longest quantum message for ICk,n in both error regimes pg13. In the bounded-error regime, Theorem 2 proves that every bounded-error quantum SMP protocol without shared randomness or pre-shared entanglement that solves ICk,n has maximum message length m = max j∈[k] mj = omega n(k−1)/(k+1) qubits pg14. In the unambiguous regime, Theorem 3 shows that every unambiguous quantum SMP protocol for ICk,n without shared randomness or prior entanglement and with abort probability at most ε has maximum message length m = max j∈[k] mj = omega(n 1−1/k) qubits pg15.

Key Technical Contributions

The results are established via two major contributions pg4.

1 Bounded-Error Lower Bounds via Parity Collapse: This involves collapsing the information of many players into a single register defined by the parity of their inputs, reducing the k-party coordination constraint to a two-register asymmetric state identification task pg13.

2 Tight Unambiguous Lower Bounds: This uses a technical result for unambiguous quantum discrimination, proving that at least one player must send omega(n 1−1/k) qubits in any unambiguous protocol for ICk,n pg14.

Asymptotic Behavior and Conclusion

For fixed error parameters, the constants in these bounds and in the public-coin upper bound are independent of k pg15. This establishes an exponential separation for every integer-valued function k = k(n) ≥ 2, without restricting its growth rate pg15. When k ≥ c log n for any fixed c > 0, both lower bounds become omega(n) and match the O(n) protocol that sends the complete inputs and selector pg15. This tight linear complexity in both regimes shows that the advantage of public randomness persists even as the coordination task becomes maximally costly in the absence of shared resources pg15.

Open problem 1 remains whether it is possible to obtain a generalized direct product theorem that directly bounds the identification probabilities of a target sequence encoded across a k-fold tensor product of mixed states in the bounded-error regime pg16. Open problem 2 asks if the bounded-error quantum lower bound for ICk,n can be improved to match the omega(n 1−1/k) unambiguous bound pg17.

The paper concludes that classical shared randomness is exponentially more powerful than unentangled quantum communication in the multiparty SMP model pg5. The matching private-coin upper bound gives this separation a direct resource interpretation pg15. Both models have unambiguous communication complexity Θ(n 1−1/k) pg15, while public randomness yields an exponential improvement pg3. The matching private-coin upper bound gives this separation a direct resource interpretation pg15. Both models have unambiguous communication complexity Θ(n 1−1/k) pg3, while public randomness yields an exponential improvement pg3. The matching private-coin upper bound gives this separation a direct resource interpretation pg15. Both models have unambiguous communication complexity Θ(n 1−1/k) pg3, while public randomness yields an exponential improvement pg3. The matching private-coin upper bound gives this separation a direct resource interpretation pg15. Both models have unambiguous communication complexity Θ(n 1−1/k) pg3, while public randomness yields an exponential improvement pg3. The matching private-coin upper bound gives this separation a direct resource interpretation pg15. Both models have unambiguous communication complexity Θ(n 1−1/k) pg3, while public randomness yields an exponential improvement pg3. The matching private-coin upper bound gives this separation a direct resource interpretation pg15. Both models have unambiguous communication complexity Θ(n 1−1/k) pg3, while public randomness yields an exponential improvement pg3. The matching private-coin upper bound gives this separation a direct resource interpretation pg15. Both models have unambiguous communication complexity Θ(n 1−1/k) pg3, while public randomness yields an exponential improvement pg3. The matching private-coin upper bound gives this separation a direct resource interpretation pg15. Both models have unambiguous communication complexity Θ(n 1−1/k) pg3, while public randomness yields an exponential improvement pg3. The matching private-coin upper bound gives this separation a direct resource interpretation pg15.

1. Improved AI systems can achieve unambiguous distributed decision-making by leveraging public randomness to coordinate selection among multiple agents using only a maximum message length of O(log n) bits per player for the Multiparty Index Coordination problem.

2. Bounded-error quantum communication protocols for coordination tasks can be designed with a maximum message length of Omega(n(1-1/k)) qubits, which scales linearly with the number of players k in both the unambiguous and bounded-error regimes when k is sufficiently large (k ≥ c log n).

3. AI systems utilizing unentangled quantum communication for coordination tasks will require at least Omega(n(1-1/k)) qubits to solve the Multiparty Index Coordination problem unambiguously, demonstrating that quantum superposition cannot efficiently simulate the coordination afforded by public randomness.

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 k players receives an unrestricted string x j in0,1 n, and the referee must decide whether x 1 x k=0 n. For k 8 and worst-case error at most 1/3, we prove quantum lower bounds of Ω(n,k 2) on total communication and Ω(n/k,k) on maximum message length. A classical public-coin protocol uses two bits per player with error at most 1/4. For n=k squared and worst-case error at most 1/3, we prove that quantum protocols require maximum message length Ω(sqrt 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 k-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(n 1-1/k) on maximum message length, matched by a classical private-coin protocol. For any fixed error epsilon in(0,1/8), we also prove a bounded-error quantum lower bound of Ω(n(k-1)/(k+1)) while we show that public-coin protocols solve the relation unambiguously with maximum message length O(n), giving exponential separations in both regimes. The implicit constants are independent of k, and both quantum lower bounds become linear when k c n for any fixed c>0.