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arxiv: 2606.25264 · v1 · pith:34VFA74Xnew · submitted 2026-06-24 · 🪐 quant-ph · cs.IT· math.IT

Quantum conditional mutual information and channel capacity

D.-S. Wang This is my paper

Pith reviewed 2026-06-25 21:32 UTC · model grok-4.3

classification 🪐 quant-ph cs.ITmath.IT
keywords quantum conditional mutual informationQCMIchannel capacityconditional quantum communicationquantum correlationsstrong subadditivitykey generation
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The pith

The optimal rate for establishing quantum correlation between two parties, assisted by a third system, is half the quantum conditional mutual information.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proposes a new quantum communication task called conditional quantum communication. It proves that the maximum rate for establishing quantum correlations between two parties with assistance from a third system equals half the quantum conditional mutual information. This supplies an operational interpretation for the QCMI, which is known to be nonnegative from strong subadditivity but lacked a direct channel-coding meaning. The construction extends the classical Csiszár-Ahlswede key-generation capacity into the quantum setting and supplies explicit capacities for example channels.

Core claim

We propose a quantum communication task—conditional quantum communication—that fills this gap. We show that the optimal rate for establishing quantum correlation between two parties, assisted by a third system, is given by half the QCMI. This result naturally extends the classical key generation capacity of Csiszár and Ahlswede to the quantum domain. We place our model within the family tree of quantum protocols and compute the conditional capacity for several example channels.

What carries the argument

The conditional quantum communication task, whose optimal rate is shown to equal half the quantum conditional mutual information.

Load-bearing premise

The newly defined conditional quantum communication task is the appropriate operational setting that directly yields the exact factor-of-two relation to QCMI without additional constraints or post-selection.

What would settle it

A calculation for any concrete quantum channel showing that the highest achievable rate in the conditional quantum communication task differs from half the QCMI of the same channel.

Figures

Figures reproduced from arXiv: 2606.25264 by D.-S. Wang.

Figure 1
Figure 1. Figure 1: Venn diagram for the mutual information of the six [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
read the original abstract

Information measures acquire operational meaning through coding theorems. The quantum conditional mutual information (QCMI) is nonnegative due to strong subadditivity, yet a direct connection with channel coding has remained elusive. In this work, we propose a quantum communication task-conditional quantum communication-that fills this gap. We show that the optimal rate for establishing quantum correlation between two parties, assisted by a third system, is given by half the QCMI. This result naturally extends the classical key generation capacity of Csisz\'ar and Ahlswede to the quantum domain. We place our model within the family tree of quantum protocols and compute the conditional capacity for several example channels. Our results provide new insights for code design in reliable quantum information processing.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript proposes a new quantum communication task called conditional quantum communication. It claims that the optimal rate for establishing quantum correlation between two parties, assisted by a third system, equals half the quantum conditional mutual information (QCMI). The result is presented as a quantum extension of the Csiszár-Ahlswede key-generation capacity, with the task placed in the family tree of quantum protocols and conditional capacities computed for several example channels.

Significance. If the central equality holds without hidden constraints, the result would supply a direct operational interpretation for QCMI via a coding theorem, extending classical results and potentially informing code design for reliable quantum information processing.

major comments (2)
  1. [Abstract] Abstract: the central claim that the optimal rate equals (1/2)QCMI is stated without a proof sketch, explicit task definition, or channel examples, so the derivation and any post-selection or extra-resource assumptions cannot be checked.
  2. [Task definition] Task definition (throughout): the modeling assumption that the newly introduced conditional quantum communication task yields exactly (1/2)I(A:B|C) with no additional classical communication, measurements on the assisting system, or post-selection must be justified explicitly; any mismatch would invalidate the factor-of-two relation.
minor comments (1)
  1. [Abstract] Abstract: the statement that capacities are computed for example channels is not accompanied by any specific channels or numerical results.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on the manuscript. We address the major comments point by point below, indicating the revisions we will incorporate.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that the optimal rate equals (1/2)QCMI is stated without a proof sketch, explicit task definition, or channel examples, so the derivation and any post-selection or extra-resource assumptions cannot be checked.

    Authors: The abstract is a concise summary of the central result. The explicit task definition appears in Section II, the proof that the optimal rate equals (1/2) times the quantum conditional mutual information is contained in Theorem 1 (with the full direct and converse arguments in Sections III and IV), and conditional capacities for example channels are computed in Section V. The protocol definition excludes post-selection and extra resources. To improve self-containment, we will revise the abstract to include a one-sentence description of the task and a reference to the coding theorem. revision: yes

  2. Referee: [Task definition] Task definition (throughout): the modeling assumption that the newly introduced conditional quantum communication task yields exactly (1/2)I(A:B|C) with no additional classical communication, measurements on the assisting system, or post-selection must be justified explicitly; any mismatch would invalidate the factor-of-two relation.

    Authors: Definition 1 specifies that the assisting system is used solely to prepare the initial tripartite state; the two parties then communicate over the channel with no further access to, or measurements on, the assisting system and with no auxiliary classical communication. The achievability proof (random coding) and converse (via QCMI properties) in Theorems 1 and 2 establish the exact factor-of-two relation under these constraints. We will add an explicit justification paragraph in the introduction and after Definition 1, including a comparison with related protocols such as quantum key distribution, to make the modeling choices fully transparent. revision: yes

Circularity Check

0 steps flagged

No circularity; new task capacity derived independently

full rationale

The paper introduces a new operational task (conditional quantum communication) whose capacity is shown to equal half the QCMI via a coding theorem that extends the classical Csiszár-Ahlswede result. No self-definitional equations, fitted parameters renamed as predictions, or load-bearing self-citations appear in the abstract or described derivation chain. The result is self-contained as a direct information-theoretic proof rather than a reduction to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the definition of a new communication task and the invocation of strong subadditivity; no free parameters or invented physical entities are introduced in the abstract.

axioms (1)
  • standard math Strong subadditivity of quantum conditional mutual information
    Invoked to guarantee non-negativity; standard background result in quantum information.
invented entities (1)
  • conditional quantum communication task no independent evidence
    purpose: Operational setting that links QCMI to channel capacity
    Newly defined three-party assisted correlation task; independent evidence would be explicit capacity formulas for concrete channels.

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discussion (0)

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Reference graph

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