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arxiv: 2512.07822 · v4 · submitted 2025-12-08 · 🪐 quant-ph

Comparing quantum channels using Hermitian-preserving trace-preserving linear maps: A physically meaningful approach

Arindam Mitra , Jatin Ghai This is my paper

Pith reviewed 2026-05-17 00:08 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum channelsHPTP mapsquantum measurementspreorderphysical implementabilitydevice incompatibility
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The pith

If one quantum channel's output can be identified from the other's by a measurement that works for any unknown input, then the first is obtained by composing the second with an HPTP map.

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

The paper establishes that two quantum channels stand in a specific relation whenever the output of one can be uniquely recovered from the statistics produced by the other through a single quantum measurement that succeeds no matter which input state was fed in. When this holds, the identifiable channel equals the other channel followed by some Hermitian-preserving trace-preserving linear map, which need not be completely positive. This construction yields a preorder on channels and supplies a concrete quantifier, physical implementability, that measures how difficult it is to realize the first channel once the second has already been built. The relation also supplies a criterion for when two quantum devices are incompatible, illustrated by an explicit example.

Core claim

Given quantum channels Φ and Ψ, if there exists a measurement that distinguishes the output state of Φ from the output statistics of Ψ for every possible unknown input, then there is an HPTP linear map Λ such that Φ equals the composition Λ ∘ Ψ. This relation is a preorder on the set of quantum channels. The authors introduce the quantity of physical implementability to quantify the resources needed to realize the former channel given that the latter has already been implemented.

What carries the argument

Concatenation with a Hermitian-preserving trace-preserving (HPTP) linear map, which preserves Hermiticity and trace but drops the complete-positivity requirement of ordinary quantum channels.

If this is right

  • The relation defines a preorder that orders quantum channels according to this measurement-based distinguishability.
  • Physical implementability supplies a numerical measure of how much harder it is to realize one channel once the other is available.
  • The preorder yields a criterion for incompatibility of quantum devices.
  • Channels may be comparable under this relation even when neither can be obtained from the other by ordinary post-processing with a second channel.

Where Pith is reading between the lines

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

  • The same distinguishability condition could be used to induce resource orderings in other quantum information settings that currently rely only on complete positivity.
  • Experimental tests could begin with standard pairs such as the identity versus a depolarizing channel to check whether the predicted HPTP map appears.
  • The construction suggests a route to compare channel performance in continuous-variable systems where positivity constraints are harder to enforce.

Load-bearing premise

The distinguishing measurement must succeed for every possible unknown input state without any dependence on which state was chosen.

What would settle it

A concrete pair of quantum channels for which no HPTP map exists that maps one to the other, yet a single measurement still identifies one output from the statistics of the other for all inputs.

Figures

Figures reproduced from arXiv: 2512.07822 by Arindam Mitra, Jatin Ghai.

Figure 1
Figure 1. Figure 1: FIG. 1. For a given channel [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
read the original abstract

In quantum technologies, quantum channels are essential elements for the transmission of quantum states. The action of a quantum channel usually introduces noise in the quantum state and thereby reduces the information contained in it. These are mathematically described by completely positive trace-preserving linear maps that represent the generic evolution of quantum systems and are also special cases of Hermitian-preserving trace-preserving (HPTP) linear maps. Concatenating a quantum channel with another quantum channel makes it noisier and degrades its information and resource preservability. In this work, we demonstrate a physically meaningful way to compare a pair of quantum channels using Hermitian-preserving trace-preserving linear maps. More precisely, given a pair of quantum channels and an arbitrary unknown input state, we show that if the output state of one quantum channel from the pair can be uniquely identified from the output statistics of the other channel from the pair using some quantum measurement, then the former channel from the pair can be obtained from the latter channel by concatenating it with a Hermitian-preserving trace-preserving linear map that is not necessarily positive. In such cases, the former channel may not always be obtained from the latter through post-processing. This relation between these two channels is a preorder, and we try to study its characterization in this work. Furthermore, we try to characterize the difficulty of implementing the former channel given that the latter channel has already been implemented via a quantifier, namely, physical implementability. We also illustrate the implications of our results for the incompatibility of quantum devices through an example. In short, we try to provide valuable insights about the relevance of Hermitian-preserving trace-preserving linear maps in physically motivated settings.

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 paper claims that for a pair of quantum channels and an arbitrary unknown input state, if the output state of one channel can be uniquely identified from the output statistics of the other via some quantum measurement, then the former channel equals the latter composed with an HPTP (but not necessarily CP) linear map. This relation is asserted to be a preorder on channels; the authors introduce a 'physical implementability' quantifier to measure implementation difficulty and illustrate implications for device incompatibility with an example.

Significance. If the central equivalence holds with a uniform, input-independent HPTP map, the work supplies a physically motivated preorder on channels that properly extends post-processing and may aid noise and resource analysis in quantum technologies. The new quantifier could be a useful addition if it is shown to be well-defined and distinct from existing channel distances.

major comments (2)
  1. [Abstract] Abstract (main claim): the asserted equivalence requires that both the distinguishing measurement and the resulting HPTP map Λ be independent of the unknown input ρ so that Φ = Λ ∘ Ψ holds simultaneously for all ρ. The provided text supplies no derivation, explicit construction, or counter-example check confirming this uniformity; without it the claimed preorder is not guaranteed to be well-defined.
  2. [Central result] Central result (identification condition): the weakest assumption that a single measurement works for arbitrary unknown ρ is load-bearing. If the effective map or measurement must be adjusted per ρ, the relation fails to induce a preorder and the physical-implementability quantifier loses its intended meaning.
minor comments (2)
  1. The abstract repeatedly uses the phrase 'we try to' (study, characterize, provide); replace with direct statements of what is shown.
  2. Introduce the physical-implementability quantifier with an explicit formula rather than a descriptive phrase only.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments regarding the uniformity of the measurement and HPTP map with respect to the input state. We address each major comment below and will revise the manuscript to strengthen the presentation of our central result.

read point-by-point responses

  1. Referee: [Abstract] Abstract (main claim): the asserted equivalence requires that both the distinguishing measurement and the resulting HPTP map Λ be independent of the unknown input ρ so that Φ = Λ ∘ Ψ holds simultaneously for all ρ. The provided text supplies no derivation, explicit construction, or counter-example check confirming this uniformity; without it the claimed preorder is not guaranteed to be well-defined.

    Authors: We agree that input-independence of both the measurement and the HPTP map is necessary for the relation to induce a well-defined preorder on channels. Our central claim is that the identification condition, when required to hold for arbitrary unknown ρ, yields a single HPTP map Λ such that Φ = Λ ∘ Ψ for all ρ. While the current manuscript states this result, we acknowledge that an explicit derivation of the uniform map construction and a counter-example verification are not detailed in the provided text. We will add a dedicated subsection deriving the map from the measurement operator and confirming uniformity across all input states. revision: yes

  2. Referee: [Central result] Central result (identification condition): the weakest assumption that a single measurement works for arbitrary unknown ρ is load-bearing. If the effective map or measurement must be adjusted per ρ, the relation fails to induce a preorder and the physical-implementability quantifier loses its intended meaning.

    Authors: The referee is correct that the load-bearing assumption is the existence of a single measurement that works for arbitrary ρ. Our result shows that this uniform identification condition is equivalent to the existence of an input-independent HPTP map, thereby preserving the preorder structure and the meaning of the physical-implementability quantifier. To prevent any ambiguity, we will revise the central result section to explicitly state the uniformity requirement, provide the proof that the map does not depend on ρ, and discuss why ρ-dependent adjustments would invalidate the preorder. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained from identification condition

full rationale

The paper introduces a preorder on quantum channels directly from the premise that one channel's output statistics allow unique identification of the other's output state via a single measurement that works for arbitrary unknown input ρ. This condition is used to derive the existence of an HPTP map Λ such that one channel equals the other composed with Λ. No equations reduce the claimed implication to a fitted parameter or self-defined quantity by construction, and the physical-implementability quantifier is defined afterward as a derived measure of implementation difficulty under this preorder. The argument does not rely on load-bearing self-citations, imported uniqueness theorems, or ansatzes smuggled from prior work; the central relation follows from the stated identification assumption without presupposing the HPTP composition. The derivation remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim rests on standard quantum information axioms plus the new definition of the preorder via measurement distinguishability. No free parameters are introduced. The only invented entity is the 'physical implementability' quantifier.

axioms (2)
  • standard math Quantum channels are completely positive trace-preserving maps on finite-dimensional operator spaces.
    Invoked in the first paragraph when the authors state that channels are special cases of HPTP maps.
  • domain assumption The distinguishing measurement exists for an arbitrary unknown input state.
    The claim is conditioned on this identification being possible independently of the input.
invented entities (1)
  • physical implementability quantifier no independent evidence
    purpose: To measure the difficulty of implementing one channel given that the other has already been implemented.
    Introduced in the abstract as a new derived quantity; no independent evidence supplied.

pith-pipeline@v0.9.0 · 5591 in / 1542 out tokens · 30270 ms · 2026-05-17T00:08:23.885111+00:00 · methodology

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Reference graph

Works this paper leans on

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