Bounded-error multiparty direct-product theorem

Establish a generalized direct-product theorem that directly bounds the probability of identifying a target sequence encoded across a k-fold tensor product of mixed quantum states in the bounded-error regime.

Background

The paper proves its bounded-error lower bound for the multiparty Index Coordination relation by collapsing the first k−1 players into a single parity-based meta-register and then applying an existing two-register asymmetric direct-product theorem. This reduction avoids the unresolved difficulty of analyzing identification across all k mixed-state registers simultaneously.

A direct k-register theorem would provide an alternative and more structurally faithful route to bounding the identification probability of all k labels in a tensor-product state. The authors explicitly ask whether such a theorem can be obtained for the bounded-error setting.

References

Is it 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?

— On the Limits of Quantum Multiparty Simultaneous Communication  (2609.10289 - Montealegre et al., 9 Sep 2026) in Section 5, paragraph “A Multiparty Direct Product Theorem for Bounded-Error Identification”