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arxiv: 2506.06089 · v1 · submitted 2025-06-06 · 🪐 quant-ph

Optimizing entanglement distribution via noisy quantum channels

Piotr Masajada , Marco Fellous-Asiani , Alexander Streltsov This is my paper

Pith reviewed 2026-05-19 10:53 UTC · model grok-4.3

classification 🪐 quant-ph
keywords entanglement distributionnoisy quantum channelsmidpoint strategysemidefinite programmingnegativityqubit channelsamplitude dampingdepolarizing noise
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The pith

Placing the entanglement source at the channel midpoint maximizes distributable entanglement for qubit systems.

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

The paper compares two placements for an entanglement source sending pairs through noisy channels to distant parties: one at the midpoint and one at an endpoint. For specific families of qubit channels it proves analytically that the midpoint placement yields more entanglement at the output. Extensive numerics lead to the conjecture that midpoint placement is optimal for every qubit channel. In the midpoint setup the authors formulate semidefinite programs that compute the maximum negativity achievable and show that, for combinations of amplitude damping and depolarizing noise, only weakly entangled input states succeed while stronger inputs destroy the output entanglement.

Core claim

For certain families of qubit channels the midpoint strategy is optimal; numerical evidence suggests the same holds for all qubit channels. Semidefinite programming applied to the midpoint configuration quantifies the largest negativity that can be distributed, and this quantity is reliably captured in many relevant cases. For several amplitude-damping plus depolarizing models, successful distribution occurs only when the input state is weakly entangled.

What carries the argument

The midpoint configuration of the entanglement source, optimized and quantified via semidefinite programming on the negativity.

Load-bearing premise

That semidefinite programming applied to the midpoint configuration gives the true maximum amount of distributable entanglement when negativity is the chosen measure.

What would settle it

An explicit qubit channel (or family) for which either analytical calculation or reliable numerics shows strictly higher output negativity when the source sits at one end rather than at the midpoint.

Figures

Figures reproduced from arXiv: 2506.06089 by Alexander Streltsov, Marco Fellous-Asiani, Piotr Masajada.

Figure 1
Figure 1. Figure 1: Distribution of entanglement in a different settings. between the parties and placing it at one end of the channel. When the source is located at the midpoint (Fig. 1b), one par￾ticle travels through the left segment of the channel, and the other through the right. Denoting the action of the left and right segments by Λ1 and Λ2, respectively, the effective action of the full channel is given by: Λmidway = … view at source ↗
Figure 2
Figure 2. Figure 2: Plot of the maximal negativity of the depolarizing - [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Initial geometric entanglement c of the input state |φ⟩ (defined in (27)) that maximizes the negativity of the out￾put state ρ Depol,AD f (given in (28)). The vertical axis represents the depolarizing parameter ps , while the horizontal axis cor￾responds to the amplitude damping parameter γ. For strong depolarization (i.e., as ps approaches 2 3 ), input states with very low initial entanglement yield the m… view at source ↗
Figure 4
Figure 4. Figure 4: Schematic plot of the entanglement of the output of [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Schematic plot illustrating the entanglement prop [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
read the original abstract

Entanglement distribution is a crucial problem in quantum information science, owing to the essential role that entanglement plays in enabling advanced quantum protocols, including quantum teleportation and quantum cryptography. We investigate strategies for distributing quantum entanglement between two distant parties through noisy quantum channels. Specifically, we compare two configurations: one where the entanglement source is placed at the midpoint of the communication line, and another where it is located at one end. For certain families of qubit channels we show analytically that the midpoint strategy is optimal. Based on extensive numerical analysis, we conjecture that this strategy is generally optimal for all qubit channels. Focusing on the midpoint configuration, we develop semidefinite programming (SDP) techniques to assess whether entanglement can be successfully distributed through the network, and to quantify the amount of entanglement that can be distributed in the process. In many relevant cases the SDP formulation reliably captures the maximal amount of entanglement which can be distributed, if entanglement is quantified using the negativity. We analyze several channel models and demonstrate that, for various combinations of amplitude damping and depolarizing noise, entanglement distribution is only possible with weakly entangled input states. Excessive entanglement in the input state can hinder the channel's ability to establish entanglement. Our findings have implications for optimizing entanglement distribution in realistic quantum communication networks.

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

1 major / 2 minor

Summary. The paper investigates entanglement distribution over noisy qubit channels by comparing a midpoint source placement to an end-point placement. It analytically proves optimality of the midpoint strategy for certain qubit channel families and conjectures general optimality for all qubit channels on the basis of numerical evidence. Focusing on the midpoint configuration, the authors formulate semidefinite programs to decide whether entanglement can be distributed and to quantify the maximum distributable negativity; they apply the method to amplitude-damping and depolarizing channels and report that only weakly entangled input states succeed for some noise combinations.

Significance. If the midpoint optimality conjecture is confirmed and the SDP bounds are tight, the work supplies both analytical results and a practical computational tool for network design. The explicit demonstration that excessive input entanglement can prevent distribution under realistic noise models is a useful, counter-intuitive observation with direct implications for quantum communication protocols.

major comments (1)
  1. [§4] §4 (SDP formulation and negativity optimization): the claim that the SDP 'reliably captures the maximal amount of entanglement' (abstract and §4) is load-bearing for all quantitative results and for the numerical support of the general conjecture. Because the underlying optimization over input states and channel outputs is non-convex, the SDP is necessarily a relaxation; no proof of zero duality gap, no a-posteriori bound on the relaxation error, and no systematic comparison against exact methods for the tested amplitude-damping + depolarizing pairs are provided. If the gap is nonzero for any of the reported instances, the stated maximal negativities become only upper bounds, weakening both the quantitative claims and the optimality conjecture.
minor comments (2)
  1. [Abstract] The abstract states that analytical optimality holds 'for certain families of qubit channels' but does not name the families; listing them explicitly would improve readability.
  2. [Numerical results section] Numerical evidence for the general conjecture is described as 'extensive' without reporting the number of channel instances, the sampling method, or error-control procedures; adding these details would strengthen the supporting material.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their detailed and constructive report on our manuscript arXiv:2506.06089. We address the major comment regarding the SDP formulation point by point below. We believe the proposed revisions will improve the rigor of our quantitative claims without altering the core analytical results.

read point-by-point responses

  1. Referee: §4 (SDP formulation and negativity optimization): the claim that the SDP 'reliably captures the maximal amount of entanglement' (abstract and §4) is load-bearing for all quantitative results and for the numerical support of the general conjecture. Because the underlying optimization over input states and channel outputs is non-convex, the SDP is necessarily a relaxation; no proof of zero duality gap, no a-posteriori bound on the relaxation error, and no systematic comparison against exact methods for the tested amplitude-damping + depolarizing pairs are provided. If the gap is nonzero for any of the reported instances, the stated maximal negativities become only upper bounds, weakening both the quantitative claims and the optimality conjecture.

    Authors: We thank the referee for highlighting this important technical point. The optimization over input states is indeed non-convex, and the SDP in §4 is a convex relaxation. We acknowledge that a general proof of zero duality gap is not provided. However, for the specific qubit channels and negativity measure considered, the SDP solutions we obtain correspond to valid physical states (verified by reconstructing the input density operator from the SDP output), and the reported negativities match independent numerical checks we performed via direct parameterization of the input state for representative amplitude-damping and depolarizing pairs. To address the concern, we will revise the manuscript to (i) explicitly describe the SDP as a relaxation that furnishes an upper bound, (ii) add a discussion of conditions under which tightness is expected for negativity, and (iii) include a systematic comparison table against exact optimization methods for the tested noise combinations. These changes will clarify the scope of the quantitative results while preserving the analytical optimality proofs for the midpoint strategy and the conjecture based on the numerical evidence. revision: partial

Circularity Check

0 steps flagged

No significant circularity; optimality claims and SDP bounds rest on standard channel analysis and numerics

full rationale

The paper shows analytical optimality of the midpoint strategy for specific qubit channel families using direct comparison of configurations, then conjectures generality from numerical optimization over input states and channel parameters. The SDP is formulated to upper-bound negativity after two independent channels and is described as reliably capturing the maximum in relevant cases based on those numerics; no equation reduces a prediction to a fitted parameter by construction, nor does any central claim rely on a self-citation chain that itself assumes the target result. The derivation remains self-contained against external benchmarks such as known negativity formulas and standard SDP relaxations for entanglement measures.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so the ledger records typical background assumptions of quantum information rather than paper-specific free parameters or invented entities.

axioms (1)
  • domain assumption Standard properties of memoryless qubit quantum channels and the definition of negativity as an entanglement monotone
    The comparison of configurations and the SDP formulation presuppose the usual framework of completely positive trace-preserving maps on qubits.

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