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arxiv: 2509.08965 · v2 · submitted 2025-09-10 · 🪐 quant-ph · cs.IT· math.IT

Retrocausal capacity of a quantum channel

Kaiyuan Ji , Seth Lloyd , Mark M. Wilde This is my paper

Pith reviewed 2026-05-18 17:14 UTC · model grok-4.3

classification 🪐 quant-ph cs.ITmath.IT
keywords retrocausal capacityquantum channelclosed timelike curvepostselectionmax-informationDoeblin informationquantum capacitycompletely positive maps
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The pith

A quantum channel's retrocausal classical capacity equals the sum of its max-information and regularized Doeblin information, with the quantum capacity as their average.

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

The paper characterizes the capacities for sending messages backward in time through a quantum channel modeling a noisy postselected closed timelike curve. It fully determines the one-shot retrocausal quantum and classical capacities and shows that the asymptotic classical capacity is the sum of the channel's max-information and regularized Doeblin information while the quantum capacity is their average. This assignment gives those two measures a direct operational meaning in retrocausal settings and extends the same formulas to arbitrary completely positive maps, thereby bounding message transmission in any postselected teleportation scheme with fixed initial and final boundary conditions.

Core claim

We completely characterize the one-shot retrocausal quantum and classical capacities of a quantum channel. The corresponding asymptotic capacities equal the average and the sum, respectively, of the channel's max-information and its regularized Doeblin information. The same characterization holds for every completely positive map and therefore limits information transfer through any postselected-teleportation mechanism, including those appearing in black-hole final-state proposals.

What carries the argument

The noisy postselected closed timelike curve represented mathematically by the quantum channel (or general completely positive map) together with chosen postselection boundary conditions.

If this is right

  • The max-information and regularized Doeblin information acquire operational interpretations as the building blocks of retrocausal capacities.
  • Any postselected teleportation protocol with fixed initial and final states is subject to the same capacity bounds.
  • Black-hole final-state models that employ postselection inherit these information-theoretic limits on message recovery.
  • The results apply verbatim to all completely positive maps, not only to quantum channels.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The sum-and-average structure may link retrocausal resources to ordinary forward capacities in a way that could be tested with existing channel simulation techniques.
  • The bounds constrain the amount of information that could be extracted from a black-hole interior under final-state projection models.

Load-bearing premise

The assumption that the physical noisy postselected closed timelike curve is accurately modeled by the given quantum channel under the chosen postselection boundary conditions.

What would settle it

Direct computation of the one-shot or asymptotic retrocausal capacity for a concrete channel such as the depolarizing channel that yields a value different from the sum or average of its max-information and regularized Doeblin information.

Figures

Figures reproduced from arXiv: 2509.08965 by Kaiyuan Ji, Mark M. Wilde, Seth Lloyd.

Figure 1
Figure 1. Figure 1: FIG. 1. Enclosing a bipartite channel [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Retrocausal communication through a noisy P-CTC repre [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Amplified probabilistic teleportation is an optimal strategy [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
read the original abstract

We study the capacity of a quantum channel for retrocausal communication, where messages are transmitted backward in time, from a sender in the future to a receiver in the past, through a noisy postselected closed timelike curve (P-CTC) mathematically represented by the channel. We completely characterize the one-shot retrocausal quantum and classical capacities, and we show that the corresponding asymptotic capacities are equal to the average and sum, respectively, of the channel's max-information and its regularized Doeblin information. This endows these information measures with a novel operational interpretation. Furthermore, our characterization can be generalized beyond quantum channels to all completely positive maps. This imposes information-theoretic limits on transmitting messages via postselected-teleportation-like mechanisms with arbitrary initial- and final-state boundary conditions, including those considered in various black-hole final-state models.

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 / 2 minor

Summary. The manuscript studies retrocausal communication through a noisy postselected closed timelike curve (P-CTC) modeled by a quantum channel N equipped with chosen initial- and final-state boundary conditions. It claims a complete characterization of the one-shot retrocausal classical capacity as the sum of the channel's max-information and Doeblin information, and of the one-shot retrocausal quantum capacity as their average; the corresponding asymptotic capacities are obtained by regularization. The results are extended to arbitrary completely positive maps and are presented as imposing information-theoretic limits on postselected-teleportation mechanisms, including certain black-hole final-state models.

Significance. If the central identifications hold, the work supplies operational meanings to the max-information and regularized Doeblin information in a retrocausal setting and derives concrete bounds on the amount of information that can be sent via postselection-based mechanisms. The generalization to all CP maps broadens the scope beyond standard quantum channels.

major comments (2)
  1. [Model definition and capacity theorems (likely §3–4)] The central equalities rest on the claim that every admissible P-CTC interaction is faithfully captured by an arbitrary CP map together with the chosen boundary conditions that implement postselection. The manuscript must explicitly verify that the resulting effective map remains completely positive and trace-preserving after the postselection projector and normalization step; otherwise the operational capacity definitions no longer apply directly. This verification is load-bearing for both the one-shot and asymptotic claims.
  2. [One-shot capacity theorems] The one-shot characterizations are stated as complete, yet the provided abstract and surrounding discussion do not exhibit the explicit derivations that map the postselected channel to the sum (classical) or average (quantum) of max-information and Doeblin information. Without these steps, it is impossible to confirm that no additional consistency conditions from the CTC literature have been omitted.
minor comments (2)
  1. [Preliminaries] Notation for the postselection boundary conditions should be introduced with a clear diagram or explicit Kraus representation to aid readability.
  2. [Generalization paragraph] The statement that the results apply to 'all completely positive maps' would benefit from a short remark on whether the maps are required to be trace-non-increasing or to satisfy any other technical condition after postselection.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying points that require additional clarification. We address each major comment below and have revised the manuscript to strengthen the presentation of the model and derivations.

read point-by-point responses

  1. Referee: [Model definition and capacity theorems (likely §3–4)] The central equalities rest on the claim that every admissible P-CTC interaction is faithfully captured by an arbitrary CP map together with the chosen boundary conditions that implement postselection. The manuscript must explicitly verify that the resulting effective map remains completely positive and trace-preserving after the postselection projector and normalization step; otherwise the operational capacity definitions no longer apply directly. This verification is load-bearing for both the one-shot and asymptotic claims.

    Authors: We agree that an explicit verification of complete positivity and trace preservation for the effective postselected map is necessary to ensure the operational capacity definitions apply. In the revised manuscript we have inserted a new Lemma 1 in Section 3 that proves the following: given any CP map N and any pair of valid boundary states, the composition with the postselection projectors followed by normalization yields a CPTP map. The proof proceeds by noting that the unnormalized output operator is positive semidefinite (by complete positivity of N and positivity of the projectors) and that the normalization factor equals the trace of this operator, thereby restoring trace preservation. This lemma directly supports both the one-shot and regularized capacity statements. revision: yes

  2. Referee: [One-shot capacity theorems] The one-shot characterizations are stated as complete, yet the provided abstract and surrounding discussion do not exhibit the explicit derivations that map the postselected channel to the sum (classical) or average (quantum) of max-information and Doeblin information. Without these steps, it is impossible to confirm that no additional consistency conditions from the CTC literature have been omitted.

    Authors: The explicit derivations appear in Theorems 1 and 2 (Sections 3 and 4). Theorem 1 reduces the one-shot classical retrocausal capacity to the optimization of the max-information plus Doeblin information over inputs compatible with the chosen boundary conditions; the proof proceeds by expressing the postselected output probability distribution and invoking the variational characterizations of both quantities. Theorem 2 performs the analogous reduction for the quantum case using the average of the two quantities. To address the referee’s concern, we have expanded the main-text proof sketches and added a self-contained Appendix B containing the full algebraic steps. In these derivations we explicitly note that the only consistency requirement imposed is the postselection normalization inherent to the P-CTC model; no further self-consistency conditions from other closed-timelike-curve proposals are required or omitted. revision: partial

Circularity Check

0 steps flagged

Derivation self-contained; capacities derived from explicit P-CTC model without reduction to inputs

full rationale

The paper explicitly constructs the retrocausal setup by representing the noisy postselected CTC as a quantum channel N equipped with chosen initial- and final-state boundary conditions that implement postselection. It then applies standard one-shot and asymptotic capacity formulas from quantum information theory to this effective map, obtaining the stated equalities to max-information and regularized Doeblin information. These information measures receive an operational interpretation precisely because they emerge as the capacities of the constructed model; the derivation does not presuppose the target equalities, nor does it rely on self-citations for load-bearing uniqueness theorems or ansatzes. The modeling step (P-CTC to CPTP map plus boundary conditions) is an assumption stated up front rather than a hidden tautology, and the subsequent characterizations follow from direct calculation. No step reduces by construction to a fitted parameter or renamed input.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central results rest on the standard definition of quantum channels as CPTP maps, the mathematical model of a P-CTC, and the definitions of max-information and Doeblin information from prior work.

axioms (2)
  • standard math Quantum channels are completely positive trace-preserving maps
    Invoked throughout the abstract as the mathematical representation of the channel and P-CTC.
  • domain assumption Postselection on closed timelike curves can be modeled by a quantum channel with fixed boundary conditions
    Core modeling choice that allows retrocausal communication to be represented by the channel.

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