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arXiv:2610.15955

Quantum simultaneous decoding via pretty-good measurements of tilted states

Mark M. Wilde

quant-phcs.ITSubmitted Oct 11, 2026• Announced Oct 13, 2026• 0 citations

Abstract

A multiple-access channel transmits messages from several senders to one receiver. For quantum outputs, the receiver must distinguish states depending on all messages, while tests for different decoding errors may not commute. In this paper, we give a self-contained account of simultaneous decoding, in which one joint measurement recovers the messages at the established achievable rates using independent codebooks. The proof combines Sen's tilted-subspace method with Cheng's analysis of the pretty-good measurement, a standard measurement for distinguishing quantum states. Auxiliary labels close to a common reference state separate the geometric directions associated with different errors. A Gram-matrix bound then allows separately chosen binary tests to certify the performance of one measurement. The labels are removed from the completed decoder, leading to a deterministic packing theorem that applies to finite state ensembles, including unequal priors. We develop the two-sender case first, with worked examples and figures, before treating any fixed number of senders. Sharper finite-error guarantees for the same decoder are developed separately, with sharp bounds for two specified steps of the analysis rather than a claim of globally optimal decoding. Applications recover known achievable rates for unassisted and entanglement-assisted classical communication, distributed classical compression with quantum side information, and coherent-state communication through a beamsplitter or passive multiport device. The common proof makes the geometric mechanism and its quantitative consequences available without repeating the construction for each application.

Comments: 92 pages, 18 figures

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A multiple-access channel transmits messages from several senders to one receiver. For quantum outputs, the receiver must distinguish states depending on all messages, while tests for different decoding errors may not commute. In this paper, we give a self-contained account of simultaneous decoding, in which one joint measurement recovers the messages at the established achievable rates using independent codebooks. The proof combines Sen's tilted-subspace method with Cheng's analysis of the pretty-good measurement, a standard measurement for distinguishing quantum states. Auxiliary labels close to a common reference state separate the geometric directions associated with different errors. A Gram-matrix bound then allows separately chosen binary tests to certify the performance of one measurement. The labels are removed from the completed decoder, leading to a deterministic packing theorem that applies to finite state ensembles, including unequal priors. We develop the two-sender case first, with worked examples and figures, before treating any fixed number of senders. Sharper finite-error guarantees for the same decoder are developed separately, with sharp bounds for two specified steps of the analysis rather than a claim of globally optimal decoding. Applications recover known achievable rates for unassisted and entanglement-assisted classical communication, distributed classical compression with quantum side information, and coherent-state communication through a beamsplitter or passive multiport device. The common proof makes the geometric mechanism and its quantitative consequences available without repeating the construction for each application.

In this work, we prove a novel one-shot ‘multi-sender’ decoupling theorem generalising Dupuis' seminal single sender decoupling theorem. We start off with a multipartite quantum state, say on

A_{1}A_{2}R

, where

A_{1}A_{2}R

are treated as the two ‘sender’ systems and

R

is the reference system. We apply independent Haar random unitaries in tensor product on

A_{1}

and

A_{2}

and then send the resulting systems through a quantum channel. We want the channel output

B

to be almost in tensor with the untouched reference

R

. Our main result shows that this is indeed the case if suitable entropic conditions are met. An immediate application of our main result is to obtain a one-shot simultaneous decoder for sending quantum information over a

k

-sender entanglement unassisted quantum multiple access channel (QMAC). The rate region achieved by this decoder is the natural one-shot quantum analogue of the pentagonal classical rate region. Assuming a simultaneous smoothing conjecture, this one-shot rate region approaches the optimal rate region of Yard et al. [20] in the asymptotic iid limit. Our work is the first one to obtain a non-trivial simultaneous decoder for the QMAC with limited entanglement assistance in both one-shot and asymptotic iid settings; previous works used unlimited entanglement assistance.

In this work, we prove a novel one-shot multi-sender decoupling theorem generalising Dupuis result. We start off with a multipartite quantum state, say on A1 A2 R, where A1, A2 are treated as the two sender systems and R is the reference system. We apply independent Haar random unitaries in tensor product on A1 and A2 and then send the resulting systems through a quantum channel. We want the channel output B to be almost in tensor with the untouched reference R. Our main result shows that this is indeed the case if suitable entropic conditions are met. An immediate application of our main result is to obtain a one-shot simultaneous decoder for sending quantum information over a k-sender entanglement unassisted quantum multiple access channel (QMAC). The rate region achieved by this decoder is the natural one-shot quantum analogue of the pentagonal classical rate region. Assuming a simultaneous smoothing conjecture, this one-shot rate region approaches the optimal rate region of Yard, Dein the asymptotic iid limit. Our work is the first one to obtain a non-trivial simultaneous decoder for the QMAC with limited entanglement assistance in both one-shot and asymptotic iid settings; previous works used unlimited entanglement assistance.

Quantum communication requires a receiver to distinguish several, generally mixed, signal states. A useful theoretical question is whether errors on smaller signal alphabets can certify the error of a decoder for the full alphabet. We study this question for the pretty good measurement and present it as a use case for the programme of extracting reusable mathematical instruments from a large collection of research manuscripts. A constrained spectral optimization mechanism identified in that collection leads to the sharp pairwise coefficient (1 + sqrt(1 - 2/N))^2 for all equally likely pure ensembles. A separate matrix analysis proves the sharp coefficient (1 + sqrt(1 - k/N))^2 for every subset size when the average state is maximally mixed on its support, allowing arbitrary mixed signals. For general mixed ensembles, the sharp leave-one coefficient (1 + 1/sqrt(N))^2 holds whenever the normalized full error is at least 1/N; consequently it holds for all equally likely ensembles in dimension at most N - 1. We give a noisy qutrit decoding example, an unrestricted bound for nearly full subsets, and an exact polynomial certificate for a further four-signal family. The account separates the extracted mechanisms, their classical antecedents, and the additional arguments needed for the quantum problem. These results provide conditional error certificates, while the unrestricted sharp mixed-state subset problem and improvements to operational communication rates remain open.

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Abstract Recently, a coding technique called position-based coding has been used to establish achievability statements for various kinds of classical communication protocols that use quantum channels. In the present paper, we apply this technique in the entanglement-assisted setting in order to establish lower bounds for error exponents, lower bounds on the second-order coding rate, and one-shot lower bounds. We also demonstrate that position-based coding can be a powerful tool for analyzing other communication settings. In particular, we reduce the quantum simultaneous decoding conjecture for entanglement-assisted or unassisted communication over a quantum multiple access channel to open questions in multiple quantum hypothesis testing. We then determine achievable rate regions for entanglement-assisted or unassisted classical communication over a quantum multiple-access channel, when using a particular quantum simultaneous decoder. The achievable rate regions given in this latter case are generally suboptimal, involving differences of Rényi-two entropies and conditional quantum entropies.

We introduce and analyze an information theoretical task that we call the quantum multiple-access one-time pad. Here, a number of senders initially share a correlated quantum state with a receiver and an eavesdropper. Each sender performs a local operation to encode a classical message and sends their system to the receiver, who subsequently performs a measurement to decode the messages. The receiver will be able to decode the messages almost perfectly, while the eavesdropper must not be able to extract information about the messages even if they have access to the quantum systems transmitted. We consider a “conditional” scenario in which a portion of the receiver’s side information is also accessible to the eavesdropper. We investigate the maximum amount of classical information that can be encoded by each of the senders. We derive a single-letter characterization for the achievable rate region in an asymptotic limit of infinitely many copies and vanishingly small error.