Fidelity preserving and decoherence for mixed unitary quantum channels

Deguang Han; Kai Liu

arxiv: 2410.20117 · v2 · submitted 2024-10-26 · 🪐 quant-ph · math-ph· math.MP

Fidelity preserving and decoherence for mixed unitary quantum channels

Kai Liu , Deguang Han This is my paper

Pith reviewed 2026-05-23 19:35 UTC · model grok-4.3

classification 🪐 quant-ph math-phmath.MP

keywords mixed unitary channelsfidelity preservationquantum decoherencephase dampingstate purificationdistinguishable statesquantum channels

0 comments

The pith

Mixed unitary quantum channels preserve fidelity when their unitary components meet structural conditions derived from state purification.

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

The paper examines mixed unitary quantum channels of the form Φ(ρ) = ∑ p_i U_i ρ U_i^* and identifies when they leave fidelity unchanged between quantum states. It distinguishes cases for distinguishable versus non-distinguishable states and uses the purification |Ψ⟩ associated with a given |ϕ⟩ to obtain the conditions. The work then highlights difficulties that appear once phase damping is added. A sympathetic reader would care because fidelity preservation determines whether quantum information survives environmental interaction.

Core claim

For quantum channels in the form Φ(ρ) = ∑_{i=1}^N p_i U_i ρ U_i^*, structures exist that preserve fidelity for both distinguishable and non-distinguishable states; these structures are obtained by examining the action of the channel on the purification |Ψ⟩ of |ϕ⟩, although phase damping creates additional obstacles to preservation.

What carries the argument

The mixed-unitary channel Φ(ρ) = ∑ p_i U_i ρ U_i^* together with the purification |Ψ⟩ of the input state |ϕ⟩, which converts the fidelity-preservation question into a relation on the extended system.

If this is right

Explicit conditions on the unitaries U_i and probabilities p_i can be stated that keep fidelity fixed for distinguishable states.
Parallel conditions exist that keep fidelity fixed for non-distinguishable states.
Phase damping can be shown to violate those conditions in identifiable parameter regimes.
The purification construction supplies a calculational route to the preservation criteria.

Where Pith is reading between the lines

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

The same structural test might be applied to decide whether a given noise model can be absorbed into an effective mixed-unitary form.
If the preservation conditions hold for a family of channels, they could be used to design noise-resilient encoding subspaces without full error correction.
Numerical checks on small-dimensional systems would quickly reveal whether the analytic conditions are tight or admit further relaxation.

Load-bearing premise

The purification |Ψ⟩ of |ϕ⟩ captures all decoherence effects relevant to fidelity for this class of channels.

What would settle it

A concrete mixed-unitary channel whose unitary probabilities and operators satisfy none of the derived structural conditions yet still leaves fidelity unchanged between a chosen pair of states.

Figures

Figures reproduced from arXiv: 2410.20117 by Deguang Han, Kai Liu.

**Figure 4.1.** Figure 4.1: Coherence decay for Phase damping channel For a rank 2 general phase damping channel Φ(ρ) = p1I + (1 − p1)DρD∗ with D = diag{1, eiθ}, relative phase θ and mixing parameter p = p1(1 − p1), the decoherence process can be shown in a very similar way. See [PITH_FULL_IMAGE:figures/full_fig_p020_4_1.png] view at source ↗

**Figure 4.2.** Figure 4.2: Coherence decay under a rank 2 general phase damping error with relative phase θ and mix parameter p (a) p = {0.8, 0.1, 0.1} (b) p = {0.6, 0.2, 0.2} [PITH_FULL_IMAGE:figures/full_fig_p021_4_2.png] view at source ↗

**Figure 4.** Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p021_4.png] view at source ↗

**Figure 4.** Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p022_4.png] view at source ↗

**Figure 4.5.** Figure 4.5: Quantum circuit for general phase damping. In addition, it could be important to develop error correction codes specifically tailored to this kind of quantum errors introduced by general phase damping channel that offers a balance between efficiency and reduced qubit usage for storage. For more background and ideas, please see references [33, 34, 35, 36]. 5. Appendices 5.1. A. Unitary matrix preserves di… view at source ↗

read the original abstract

Distinguishable and non-distinguishable quantum states are fundamental resources in quantum mechanics and quantum technologies. Interactions with the environment often induce decoherence, impacting both the distinguishability and non-distinguishability between quantum states. In this paper, we investigate mixed unitary quantum channels and the conditions under which fidelity, a measure of quantum state closeness, is preserved. More precisely, for quantum channels in the form $\Phi(\rho) = \sum_{i=1}^N p_i U_i \rho U_i^*$, we analyze their effect on quantum state $|\varphi\rangle$ through the associated purification $|\Psi\rangle$, explore the structure of such quantum channels that preserve either distinguishable or non-distinguishable states and then discuss the challenges of maintaining fidelity, particularly under the influence of phase damping.

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.

Desk Editor's Note private letter to a colleague

The paper examines conditions for fidelity preservation under mixed unitary channels via purifications but adds little beyond standard quantum channel analysis and flags phase damping as tricky.

read the letter

The main takeaway is that this work looks at mixed unitary channels Φ(ρ) = ∑ p_i U_i ρ U_i* and asks when they preserve fidelity between states, using the usual purification lift |Ψ⟩ of |ϕ⟩, while noting extra difficulties from phase damping. The central claim is an existence statement about structures that keep distinguishability or non-distinguishability intact, and nothing in the abstract suggests an internal contradiction with that claim. The paper does a straightforward job of laying out the setup for this narrow class of channels and correctly identifies phase damping as a point where preservation gets harder. Beyond that, the contribution reads as incremental. Mixed unitary channels and fidelity calculations via Stinespring-type lifts are textbook material in quantum information, and the abstract gives no new organizing principle, first-principles derivation, or concrete new theorem. No equations or proofs appear, so it is impossible to judge whether the claimed conditions are fresh or merely restate known facts about convex combinations of unitaries. The scope stays tightly inside this channel family, which limits broader impact. This is the sort of note that might interest a small group already working on specific decoherence models, but most readers in quantum information would not need it. I would send it to peer review only if the full manuscript contains verifiable new conditions with proofs; otherwise a short note or conference poster would fit better. I would not cite it myself.

Referee Report

1 major / 0 minor

Summary. The manuscript investigates mixed unitary quantum channels of the form Φ(ρ) = ∑_{i=1}^N p_i U_i ρ U_i^* and claims to identify structures that preserve fidelity between distinguishable or non-distinguishable states. It analyzes the channel action on |ϕ⟩ via the associated purification |Ψ⟩ and discusses challenges in maintaining fidelity under phase damping.

Significance. If the claimed structures and conditions were rigorously derived and supported, the work would address a relevant question in quantum information concerning fidelity preservation under decoherence for a standard class of channels. The choice of the mixed-unitary form and the standard Stinespring purification lift are appropriate starting points, but the absence of any derivations, explicit conditions, or examples means no such contribution can currently be assessed.

major comments (1)

[Abstract] Abstract: The central existence claim (structures preserving fidelity for distinguishable/non-distinguishable states) is stated but receives no supporting derivation, explicit condition, or example. Without these, the claim cannot be evaluated and is load-bearing for the entire manuscript.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback. We agree that the central claims require explicit support to be evaluable and will revise the manuscript accordingly.

read point-by-point responses

Referee: [Abstract] Abstract: The central existence claim (structures preserving fidelity for distinguishable/non-distinguishable states) is stated but receives no supporting derivation, explicit condition, or example. Without these, the claim cannot be evaluated and is load-bearing for the entire manuscript.

Authors: We acknowledge the referee's observation that the abstract states the existence of fidelity-preserving structures for mixed unitary channels without immediate supporting material. The manuscript does contain an analysis of the channel action via purification in the main text, but we agree this is insufficiently explicit. In the revised version we will (i) expand the abstract to state one concrete condition (e.g., the requirement that the unitary set {U_i} commutes with the projector onto the support of the purified state for distinguishable cases), (ii) add a short derivation of that condition from the fidelity expression F(Φ(ρ),Φ(σ)) = F(ρ,σ), and (iii) include a minimal N=2 example with explicit matrices demonstrating preservation for distinguishable states and its failure under phase damping. These additions will make the claims directly verifiable. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The provided abstract and description outline an existence analysis of fidelity-preserving structures for mixed-unitary channels Φ(ρ) = ∑ p_i U_i ρ U_i^* via the standard purification |Ψ⟩ of |ϕ⟩. No equations, derivations, fitted parameters, or self-citations are exhibited that reduce any claimed result to its own inputs by construction. The central claim remains an existence statement about channel structures under phase damping, with the purification construction being the conventional Stinespring lift rather than a self-referential definition. The derivation chain is therefore self-contained against external benchmarks and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities can be extracted.

pith-pipeline@v0.9.0 · 5652 in / 1008 out tokens · 19619 ms · 2026-05-23T19:35:45.947243+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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Preskill, Reliable quantum computers, Proceedings of the Royal Society of London Series A: Mathe- matical, Physical and Engineering Sciences

J. Preskill, Reliable quantum computers, Proceedings of the Royal Society of London Series A: Mathe- matical, Physical and Engineering Sciences. 454 (1998) 385-410

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Gisin, and R

N. Gisin, and R. Thew, Quantum communication, Nature photonics. 1(3) (2007) 165-171

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J. Sidhu, S. Joshi, M G¨ undo˘ gan, Advances in space quantum communications, IET Quantum Commu- nication. 2(4) (2021) 182-217

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C. A. Fuchs, Distinguishability and accessible information in quantum theory, arXiv preprint. (1996)

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F. Benatti, R. Floreanini, F. Franchini, U Marzolino, Entanglement in indistinguishable particle sys- tems, Physics Reports. 878 (2020) 1-27

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A. Streltsov, G. Adesso, M. Plenio, Colloquium: Quantum coherence as a resource, Reviews of Modern Physics. 89(4) (2017) 041003

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I. Marvian, R. Spekkens, How to quantify coherence: Distinguishing speakable and unspeakable notions, Physical Review A. 94(5) (2016) 052324

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Watrous, The theory of quantum information, Cambridge university press

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H. Barnum, E Knill, M. Nielsen, On quantum fidelities and channel capacities, IEEE Transactions on Information Theory. 46(4) (2000) 1317-1329

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Cozzini, R

M. Cozzini, R. Ionicioiu, P. Zanardi, Quantum fidelity and quantum phase transitions in matrix product states, Physical Review B: Condensed Matter and Materials Physics. 76(10) (2007) 104420

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G. Viamontes, I. Markov, J. Hayes, Quantum circuit simulation, Springer. (2009)

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Chiribella, G

G. Chiribella, G. D’Ariano, P. Perinotti, Quantum circuit architecture, Physical review letters. 101(6) (2008) 060401

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V. Gupta, V. Prakash, P. Mandayam, V. Sunder, The functional analysis of quantum information theory, Lecture Notes in Physics. 902 (2015)

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M. Nielsen, I. Chuang, Quantum computation and quantum information, Cambridge university press. (2001)

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C. Li, R. Roberts, X. Yin, Decomposition of unitary matrices and quantum gates, International Journal of Quantum Information. 11(1) (2013) 1350015

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Levick, D

J. Levick, D. Kribs, R. Pereira, Quantum privacy and Schur product channels, Reports on Mathematical Physics. 80(3) (2017) 333-347

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Zhang, I

Q. Zhang, I. Nechita, A fisher information-based incompatibility criterion for quantum channels, En- tropy 24(6) (2022) 805

work page 2022

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Jochym O’Connor, D

T. Jochym O’Connor, D. Kribs, R. Laflamme, S. Plosker, Private quantum subsystems, Physical review letters. 111(3) (2013) 030502

work page 2013

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M. Hu, H. Wang, Protecting quantum Fisher information in correlated quantum channels, Annalen der Physik. 532(1) (2020) 1900378

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Giovannetti, G

V. Giovannetti, G. Palma, Master equations for correlated quantum channels, Physical review letters. 108(4) (2012) 040401

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Addis, G

C. Addis, G. Karpat, C. Macchiavello, S. Maniscalco, Dynamical memory effects in correlated quantum channels, Physical Review A. 94(3) (2016) 032121

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Awasthi, S

N. Awasthi, S. Haseli, S. Johri, S. Salimi, H. Dolatkhah, A. Khorashad, Quantum speed limit time for correlated quantum channel, Quantum Information Processing. 19 (2020) 1-17

work page 2020

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F. Buscemi, N. Datta, The quantum capacity of channels with arbitrarily correlated noise, IEEE Trans- actions on Information theory. 56(3) (2010) 1447-1460

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Y. Ye, A. Skeen, Time-correlated quantum amplitude-damping channel, Physical Review A. 67(6) (2003) 064301

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Vanstone, P

S. Vanstone, P. Van Oorschot, An introduction to error correcting codes with applications, Springer Science & Business Media. (2013)

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Terhal, Quantum error correction for quantum memories, Reviews of Modern Physics

B. Terhal, Quantum error correction for quantum memories, Reviews of Modern Physics. 87(2) (2015) 307-346

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Hsieh, I

M. Hsieh, I. Devetak, T. Brun, General entanglement-assisted quantum error-correcting codes, Physical Review A: Atomic, Molecular, & Optical Physics. 76(6) (2007) 062313. Department of Mathematics, University of Central Florida, Orlando, FL 32816 Email address : kai.liu@ucf.edu Department of Mathematics, University of Central Florida, Orlando, FL 32816 Em...

work page 2007