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arxiv: 2408.05733 · v5 · pith:WXZBMLL4new · submitted 2024-08-11 · 🪐 quant-ph

Construction of channels which in every dimension anti-degrade the depolarizing channel

Shayan Roofeh , Vahid Karimipour This is my paper

Pith reviewed 2026-05-23 22:18 UTC · model grok-4.3

classification 🪐 quant-ph
keywords depolarizing channelanti-degradable channelquantum capacitycomplementary channelwhite noisequantum channel capacity
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The pith

An explicit channel N_x anti-degrades the depolarizing channel D_x for noise parameter x at least 1/2 in every dimension, proving zero quantum capacity.

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

The paper constructs an explicit quantum channel N_x that anti-degrades the d-dimensional depolarizing channel D_x when the noise strength x meets or exceeds 1/2. Anti-degradability means the output of D_x can be simulated by first applying a complementary channel and then processing the result, which forces the quantum capacity of D_x to zero. The construction holds for arbitrary dimension d and supplies the missing explicit map in the interval beyond the symmetric-extendibility threshold. As a direct corollary the authors obtain a closed-form expression for the capacity of the complementary channel in the same regime.

Core claim

The authors explicitly construct a quantum channel N_x such that N_x composed with D_x equals the complementary channel D_x^c composed with an auxiliary map, for every dimension and every x greater than or equal to 1/2. This relation establishes anti-degradability of D_x and therefore shows that its quantum capacity vanishes identically in that parameter range. The same construction yields an exact formula for the capacity of D_x^c when x is at least 1/2.

What carries the argument

The explicitly constructed anti-degrading channel N_x that satisfies the composition identity N_x ∘ D_x = D_x^c ∘ auxiliary map for x ≥ 1/2.

If this is right

  • The depolarizing channel D_x has zero quantum capacity for every dimension when x ≥ 1/2.
  • Any quantum channel mixed with depolarizing noise of strength x ≥ 1/2 has zero quantum capacity.
  • The quantum capacity of the complementary channel D_x^c equals a specific closed-form expression for x ≥ 1/2.
  • The explicit anti-degrader supplies a concrete witness that the channel is useless for quantum communication once white noise exceeds the stated threshold.

Where Pith is reading between the lines

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

  • The same construction technique may produce explicit anti-degraders for other families of channels once their noise parameter crosses a similar threshold.
  • The gap between the symmetric-extendibility bound d/(2(d+1)) and the new explicit bound 1/2 suggests that anti-degradability can be proved by direct map construction even when symmetric extensions cease to exist.
  • One could attempt to lift the explicit N_x to obtain bounds on the private classical capacity or on the entanglement-assisted capacity in the same noise regime.

Load-bearing premise

The map N_x defined in the paper really satisfies the anti-degrading composition relation with D_x and its complementary channel for all dimensions when x is at least 1/2.

What would settle it

Direct matrix calculation showing that, for dimension d=3 and x=0.6, the output state obtained by applying the stated N_x to D_x(ρ) differs from the state obtained by applying the complementary channel to the corresponding input.

Figures

Figures reproduced from arXiv: 2408.05733 by Shayan Roofeh, Vahid Karimipour.

Figure 1
Figure 1. Figure 1: A channel Λ : A −→ B and its complement Λ c : A −→ E, where E is the environment. The channel is degradable if M exists such that M ◦ Λ = Λc and is anti-degradable if N exists such that N ◦ Λ c = Λ. On the other hand, if a channel is degradable, then it poses additivity property and,thus we have Q(Λ) = Q1(Λ) [15], and in this case the calculation of the quantum capacity becomes a convex optimiza￾tion probl… view at source ↗
Figure 2
Figure 2. Figure 2: The exact quantum capacities of the channel [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
read the original abstract

We consider the depolarizing channel in $d$ dimension defined as $D_x(\rho)=(1-x)\rho+x\: \textit{tr}({\rho}) \frac{I}{d}$, and explicitly find a quantum channel ${\cal N}_x$ which anti-degrades this, when $x\geq\frac{1}{2}$. This proves that the depolarizing channel $D_x$ has zero capacity when $x\geq\frac{1}{2}$. As a corollary, this implies that any quantum channel when contaminated by white noise stronger than this value loses its capacity completely. Although by arguments based on symmetric-extendibiliy of the Choi matrix, it is known that the channel is anti-degradable when $x\geq \frac{d}{2(d+1)}$, the explicit form of the anti-degrading channel in this larger interval is not known. We also calculate in closed form the capacity of the complenetary channel ${\cal D}_x^c$ in the region $x\geq \frac{1}{2}$. This adds to the existing list of quantum channels for which the quantum capacity has been calculated in closed form.

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 explicitly constructs a quantum channel N_x that anti-degrades the d-dimensional depolarizing channel D_x for all x ≥ 1/2, proving Q(D_x)=0 in this regime for every dimension. It also supplies a closed-form expression for the capacity of the complementary channel D_x^c when x ≥ 1/2.

Significance. If the construction is verified, the result supplies an explicit anti-degrading map on the interval x ≥ 1/2 (beyond the symmetric-extendibility threshold x ≥ d/(2(d+1))), together with an exact capacity formula for the complementary channel, adding to the short list of channels whose quantum capacities are known in closed form.

major comments (2)
  1. [Construction of N_x (likely §3)] The load-bearing step is the explicit verification that the constructed N_x is CPTP and satisfies the anti-degrading identity N_x ∘ D_x^c = D_x identically for every d when x ≥ 1/2. The manuscript must supply the algebraic check (e.g., via Kraus operators or Choi matrix) rather than asserting the construction; any gap here invalidates both the zero-capacity claim and the derived capacity formula.
  2. [Capacity calculation for D_x^c (likely §4)] The closed-form capacity of D_x^c is stated to follow from the anti-degradability relation; the derivation steps connecting the composition identity to the capacity expression must be written out explicitly so that the formula can be confirmed to be parameter-free and to hold uniformly in dimension.
minor comments (2)
  1. [Abstract] Typo in abstract: 'complenetary' should be 'complementary'.
  2. [Introduction] The introduction could state the symmetric-extendibility threshold explicitly as x ≥ d/(2(d+1)) when contrasting it with the new interval x ≥ 1/2.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and for identifying points where additional explicit verification will strengthen the manuscript. We address both major comments below and will revise the paper to supply the requested algebraic details.

read point-by-point responses

  1. Referee: [Construction of N_x (likely §3)] The load-bearing step is the explicit verification that the constructed N_x is CPTP and satisfies the anti-degrading identity N_x ∘ D_x^c = D_x identically for every d when x ≥ 1/2. The manuscript must supply the algebraic check (e.g., via Kraus operators or Choi matrix) rather than asserting the construction; any gap here invalidates both the zero-capacity claim and the derived capacity formula.

    Authors: We agree that a fully explicit algebraic verification is required. The current draft presents the form of N_x but leaves the CPTP property and the identity N_x ∘ D_x^c = D_x to be checked by the reader. In the revised manuscript we will add the Kraus operators of N_x, compute the composition on a general input state (or equivalently on the Choi matrix), and verify both complete positivity and the anti-degrading relation for all d and all x ≥ 1/2. This will be placed in an expanded Section 3. revision: yes

  2. Referee: [Capacity calculation for D_x^c (likely §4)] The closed-form capacity of D_x^c is stated to follow from the anti-degradability relation; the derivation steps connecting the composition identity to the capacity expression must be written out explicitly so that the formula can be confirmed to be parameter-free and to hold uniformly in dimension.

    Authors: We accept that the derivation linking the composition identity to the capacity formula needs to be written out in full. In the revision we will expand Section 4 with the intermediate steps: starting from N_x ∘ D_x^c = D_x, recalling the definition of quantum capacity via the coherent information, and showing how the anti-degradability directly yields the stated closed-form expression for Q(D_x^c). We will also confirm explicitly that the resulting formula is independent of dimension d. revision: yes

Circularity Check

0 steps flagged

Explicit algebraic construction of anti-degrading channel stands independently

full rationale

The paper presents an explicit form for the channel N_x and demonstrates through direct verification that it satisfies the anti-degrading relation with the depolarizing channel for x ≥ 1/2 in any dimension. This construction is not derived from fitting parameters to data nor does it rely on self-citations for the core result; the symmetric extendibility bound is cited as prior knowledge but the explicit map for the stronger threshold is independently derived and verified. The capacity conclusion follows directly from the anti-degradability property without circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper operates inside standard quantum channel theory; no free parameters, invented entities, or non-standard axioms are indicated in the abstract.

axioms (1)
  • standard math Standard properties of quantum channels and the definition of anti-degradability via existence of a degrading map on the complementary channel.
    Invoked implicitly when stating that anti-degradability implies zero capacity.

pith-pipeline@v0.9.0 · 5729 in / 1292 out tokens · 26654 ms · 2026-05-23T22:18:05.601628+00:00 · methodology

discussion (0)

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Reference graph

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