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arxiv: 2606.31243 · v1 · pith:I2TKPSHBnew · submitted 2026-06-30 · 🪐 quant-ph

Absorption capacity of separable noise: Bell-mixing thresholds on separability and teleportation

Xuan Du Trinh This is my paper

Pith reviewed 2026-07-01 05:47 UTC · model grok-4.3

classification 🪐 quant-ph
keywords entanglementquantum teleportationseparable statesBell statesnoise absorptionmixing thresholdsfidelity
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The pith

Separable two-qubit noise absorbs a definite amount of Bell state before entanglement or teleportation advantage emerges.

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

The paper defines absorption capacities that quantify how much a separable noise state can mix with a fixed Bell reference before crossing two operational thresholds. The entanglement absorption capacity C_abs is the largest amount of Bell reference absorbable while the partial transpose stays positive. The fidelity absorption capacity C_F is the largest amount absorbable while the maximum teleportation fidelity stays at or below the classical bound of 2/3. Closed-form expressions are obtained for product noise states, where C_abs depends only on local marginal purities, and for separable complex X noise states, where both capacities are explicit in every Bell frame. The two thresholds are recovered from the same Möbius map applied to each capacity.

Core claim

For separable two-qubit noise σ the entanglement absorption capacity C_abs(σ) is the largest coefficient such that λ Φ⁺ + (1-λ) σ remains positive under partial transpose, and the fidelity absorption capacity C_F(σ) is the largest coefficient such that the mixture keeps maximal teleportation fidelity ≤ 2/3; both thresholds are recovered from the map λ = C/(1+C).

What carries the argument

The Bell-mixing line ρ_λ = λ Φ⁺ + (1-λ) σ together with the partial-transpose positivity test and the maximum teleportation fidelity, packaged as the two absorption capacities C_abs and C_F.

Load-bearing premise

The noise state σ is separable so that partial-transpose positivity exactly locates the entanglement threshold for these two-qubit states.

What would settle it

For a concrete product noise state with known marginal purities, compute the formula for C_abs, then numerically locate the smallest λ where the partial transpose of the mixture acquires a negative eigenvalue and check whether that λ equals C_abs/(1+C_abs).

Figures

Figures reproduced from arXiv: 2606.31243 by Xuan Du Trinh.

Figure 1
Figure 1. Figure 1: Two complementary views of the product-law threshold (absorption capacity [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Bell-mixing thresholds for separable X noise evolving under local amplitude damping and pure de￾phasing. Each panel shows the separability threshold λ∗ and the teleportation threshold λF as functions of the channel strength. Noise state A starts with equal middle populations b = c, and the channels preserve this along the whole evolution, so the two thresholds coincide by (43). For noise state B the middle… view at source ↗
Figure 3
Figure 3. Figure 3: Closed-form separability and teleportation thresholds for a separable [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗
read the original abstract

We study Bell-mixing lines $\rho_\lambda=\lambda\Phi^+ +(1-\lambda)\sigma$, where $\Phi^+$ is a fixed Bell reference and $\sigma$ is a separable two-qubit noise state. Along this line there are two operational crossings: the state becomes entangled, and it reaches quantum teleportation advantage over classical strategies. We package these crossings as capacities of the noise state. The entanglement absorption capacity $C_{\rm abs}(\sigma)$ is the largest amount of Bell reference that $\sigma$ can absorb while the partial transpose remains positive. The fidelity absorption capacity $C_F(\sigma)$ is the largest amount of Bell reference that $\sigma$ can absorb while keeping the maximal teleportation fidelity at or below the classical bound $2/3$. The thresholds corresponding to the two crossing points are obtained from the same M\"obius map, $\lambda_* = C_{\rm abs}/(1+C_{\rm abs})$ and $\lambda_F = C_F/(1+C_F)$. We derive closed-form capacities and thresholds for product noise states and separable complex $X$ noise states. For product noise, $C_{\rm abs}$ depends only on local marginal purities, while $C_F$ also depends on orientation relative to the maximally entangled reference. For $X$ noise states, both capacities are explicit in all four Bell frames. We also study three extensions: arbitrary pure-state references, the evolution of $X$ noise states and their capacities under local amplitude-damping and dephasing channels, and decomposition certificates that give lower bounds on the capacities, hence on the thresholds, for general separable noise.

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

0 major / 2 minor

Summary. The paper defines entanglement absorption capacity C_abs(σ) and fidelity absorption capacity C_F(σ) for separable two-qubit noise σ along Bell-mixing lines ρ_λ = λ Φ^+ + (1-λ) σ. These capacities quantify the maximum λ such that the mixture remains PPT-positive or has teleportation fidelity ≤ 2/3. Closed-form expressions are derived for product noise (C_abs depending only on local purities) and separable complex X noise (explicit in Bell frames) via the Möbius map λ_* = C/(1+C); extensions cover pure references, local channel evolution of X states, and decomposition-based lower bounds.

Significance. If the closed-form derivations hold, the work supplies explicit, computable thresholds for noise tolerance in entanglement and teleportation that depend only on measurable marginals or Bell-frame coefficients for the treated classes. The reliance on PPT=separability for two qubits and the standard 2/3 fidelity bound is correctly applied; the parameter-free character for product states and the four-frame expressions for X states are concrete strengths.

minor comments (2)
  1. [product noise section] § on product states: the dependence of C_F on orientation is stated but the explicit formula relating the reference Bell state to the local bases of the product state could be written out for direct verification.
  2. [decomposition certificates] The decomposition certificates for general separable σ are presented as lower bounds; a brief remark on how tight they are for the X and product cases already treated would clarify their added value.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript and the recommendation to accept.

Circularity Check

0 steps flagged

No significant circularity; derivations are self-contained

full rationale

The paper defines C_abs(σ) and C_F(σ) operationally from the crossing points of the Bell-mixing line ρ_λ = λ Φ⁺ + (1-λ)σ where PPT positivity or teleportation fidelity hits its bound. The Möbius reparametrization λ = C/(1+C) is applied uniformly to convert those independently computed λ_* thresholds into capacities; this is a definitional packaging step, not a reduction of any result to its inputs. Closed-form expressions for product states (depending on marginal purities) and separable X-states (explicit in Bell bases) follow from direct application of the PPT criterion and fidelity formula to the two-qubit density matrix, with no fitted parameters, self-citations, or ansatzes invoked as load-bearing premises. Standard external facts (PPT ⇔ separability for 2 qubits; classical fidelity bound 2/3) are used correctly and do not create internal circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Builds on standard quantum information concepts; no free parameters or invented entities are introduced in the abstract. The capacities are derived quantities from existing thresholds.

axioms (2)
  • standard math For two-qubit states, positive partial transpose is equivalent to separability
    Used to determine the entanglement threshold along the Bell-mixing line.
  • domain assumption The classical bound for maximal teleportation fidelity is 2/3
    Used to define the fidelity absorption capacity threshold.

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