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arxiv: 2606.16916 · v2 · pith:5VKKEDJGnew · submitted 2026-06-15 · 🪐 quant-ph

Analytic Benchmarks for Coherence-to-Entanglement Conversion under Post-Gate Noise in CNOT-Based Protocols

Asad Ali , H. Kuniyil , M.I. Hussain , M.T. Rahim , Abdallah Slaoui , Saif Al-Kuwari This is my paper

Pith reviewed 2026-07-03 23:24 UTC · model grok-4.3

classification 🪐 quant-ph
keywords coherence-to-entanglement conversionCNOT gatepost-gate noisenegativityX-statesentanglement sudden deathphase dampingamplitude damping
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The pith

CNOT conversion from coherence to entanglement obeys noiseless negativity exactly equal to half the l1 coherence for arbitrary mixed inputs.

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

The paper derives closed-form expressions for how entanglement generated by a CNOT from a coherent qubit and incoherent ancilla degrades under four post-gate noise channels. It proves that without noise the negativity is always half the l1-norm of coherence regardless of input mixedness. For channels that keep the output in X form a master relation accounts for entanglement loss through the combined effect of coherence decay and shifts in the partial-transpose eigenvalues. Exact survival fractions and sudden-death thresholds are obtained for each channel and mapped to physical parameters T1 and Tphi.

Core claim

We prove the noiseless law N0 = C_l1 / 2, valid for arbitrary mixed inputs, and obtain exact negativities, survival fractions, and entanglement-sudden-death thresholds. For all X-state-preserving channels, a master relation shows that entanglement loss results from the competition between coherence suppression and partial-transpose spectral shifts.

What carries the argument

The master relation for X-state-preserving channels that expresses entanglement loss as the competition between coherence suppression and partial-transpose spectral shifts.

If this is right

  • Phase damping produces survival fraction eta=1-p with no finite-noise sudden death.
  • Global depolarizing noise yields coherence-dependent sudden death.
  • Amplitude damping adds an excited-population penalty and triggers sudden death only for theta greater than pi/4.
  • Local depolarization is most destructive at equal noise strength on both qubits.
  • Initial survival slopes equal -1, -3/2, -2 and -3 for the four channels and serve as noise fingerprints.

Where Pith is reading between the lines

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

  • The closed forms enable direct inference of effective noise parameters from measured entanglement without simulation.
  • Because concurrence equals twice the negativity for these states, all robustness orderings carry over to concurrence.
  • The same master-relation approach may apply to other entangling gates that produce X-states from single-qubit coherence.

Load-bearing premise

Post-gate noise channels preserve the X-state form of the two-qubit density matrix.

What would settle it

An experiment that applies known phase-damping noise of strength p after the CNOT and measures negativity deviating from the predicted factor (1-p) times the ideal value.

Figures

Figures reproduced from arXiv: 2606.16916 by Abdallah Slaoui, Asad Ali, H. Kuniyil, M.I. Hussain, M.T. Rahim, Saif Al-Kuwari.

Figure 1
Figure 1. Figure 1: FIG. 1. Geometry of the unified master relation ( [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Ideal (noiseless) coherence-to-entanglement propor [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Negativity [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Two-dimensional negativity landscapes [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Consolidated head-to-head comparison of all four channels at [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. (a) Entanglement-sudden-death thresholds versus input angle [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗
read the original abstract

Coherence-to-entanglement conversion transforms single-qubit superposition into a practical two-qubit resource, but noise limits this process in near-term quantum hardware. We derive closed-form benchmarks for a minimal CNOT primitive in which a coherent qubit and an incoherent ancilla generate entanglement before undergoing phase damping, global depolarizing, amplitude damping, or independent local depolarizing noise. Using the $\ell_1$-norm of coherence and negativity, we prove the noiseless law $\mathcal{N}_0=C_{\ell_1}/2$, valid for arbitrary mixed inputs, and obtain exact negativities, survival fractions, and entanglement-sudden-death thresholds. For all $X$-state-preserving channels, a master relation shows that entanglement loss results from the competition between coherence suppression and partial-transpose spectral shifts. Phase damping yields $\eta=1-p$ without finite-noise sudden death; global depolarization gives coherence-dependent sudden death; amplitude damping adds an excited-population penalty and sudden death only for $\theta>\pi/4$; while local depolarization is most destructive at equal depolarizing strength. The initial survival slopes, $-1$, $-3/2$, $-2$, and $-3$, act as compact noise fingerprints. Since concurrence satisfies $C=2\mathcal{N}$ for the generated states, all robustness rankings remain unchanged. Mapping channel parameters to $T_1$, $T_\varphi$, and average gate fidelity connects the theory to hardware-level performance.

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 manuscript derives closed-form analytic benchmarks for coherence-to-entanglement conversion via a CNOT gate acting on a coherent qubit and diagonal ancilla, followed by one of four post-gate noise channels (phase damping, global depolarizing, amplitude damping, independent local depolarizing). It proves the noiseless relation N_0 = C_{l1}/2 valid for arbitrary mixed inputs to the protocol, obtains exact negativities, survival fractions and entanglement-sudden-death thresholds, and presents a master relation for all X-state-preserving channels in which entanglement loss arises from the competition between coherence suppression and partial-transpose spectral shifts. Initial survival slopes (-1, -3/2, -2, -3) are identified as noise fingerprints, and concurrence is noted to satisfy C = 2N for the generated states, leaving robustness rankings unchanged. Channel parameters are mapped to T1, T_φ and average gate fidelity.

Significance. If the derivations hold, the work supplies exact, parameter-free expressions that function as hardware-independent benchmarks for near-term CNOT-based entanglement generation. The noiseless law and master relation are falsifiable predictions that follow directly from channel action on X-states without fitted parameters or self-referential definitions; the explicit ranking of the four channels by destructiveness and the hardware mapping add practical utility. These features allow direct experimental comparison and could serve as reference standards in the field.

minor comments (2)
  1. [Abstract / §1] The abstract states the noiseless law holds for 'arbitrary mixed inputs'; the introduction should explicitly restate the precise input class (coherent qubit ⊗ diagonal ancilla) to avoid any ambiguity about the scope.
  2. [§3] Notation for the survival fraction η and the partial-transpose eigenvalues in the master relation should be cross-referenced to the definitions in the X-state section for immediate readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, the accurate summary of our results, and the recommendation to accept. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The central results (noiseless law N0 = C_l1/2 and master relation for X-state-preserving channels) are obtained by direct computation: the post-CNOT state is an X-state whose negativity equals half the input l1 coherence by explicit matrix construction, and the listed channels (phase damping, depolarizing, amplitude damping) are shown to map X-states to X-states by their action on off-diagonal and diagonal elements. No parameter is fitted then renamed as prediction, no self-citation supplies a load-bearing uniqueness theorem, and no ansatz is smuggled in. The derivation chain is self-contained against the stated assumptions and does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

All derivations rest on standard quantum channel formalism and the definition of l1-coherence and negativity; no free parameters, ad-hoc axioms, or new entities are introduced in the abstract.

axioms (2)
  • standard math Standard quantum mechanics: density-matrix evolution under completely positive trace-preserving maps
    Invoked for applying phase damping, depolarizing, and amplitude damping channels to the post-CNOT state.
  • domain assumption X-state preservation under the considered noise channels
    Required to obtain the master relation and closed-form negativity expressions.

pith-pipeline@v0.9.1-grok · 5817 in / 1380 out tokens · 28546 ms · 2026-07-03T23:24:02.610638+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

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    We show that the four channels produce quali- tatively distinct degradation patterns: (i) multi- plicative suppression with no sudden death (phase damping); (ii) coherence-dependent sudden death for all nonzero inputs, with more coherent inputs more robust (global depolarizing); (iii) population- transfer correction causing sudden death only for θ > π/4, ...

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    We show that all four channels are gov- erned by a single master relationN= max[0, f(noise) 1 2 sin 2θ−g(noise)], reducing the four-channel comparison to a per-channel pair of coherence-suppression and spectrum-shift func- arXiv:2606.16916v2 [quant-ph] 1 Jul 2026 2 tions (Table I), and derive the analytic sudden- death thresholds as explicit functions of ...

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    We establish that all results aremeasure- independent: every state in the protocol satisfies C= 2Nexactly (concurrence equals twice the neg- ativity), so all thresholds and orderings are identi- cal under either measure, and the noiseless identity N0 = 1 2 Cℓ1 holds for arbitrary mixed single-qubit inputs, not only pure ones

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    Whether this ratio isoptimal—that is, whether 1 2 is the largest output negativity obtain- able per unit input coherence under any incoherent two- qubit operation, or whether a different entangling gate or measurement-assisted protocol could exceed it—is not addressed here and remains open. Establishing 1 2 as a tight bound, or exhibiting a protocol that ...

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