slug: roofeh-karimipour-2024-werner-holevo-lindblad-embedding
bibkey: roofehkarimipour2024wernerholevo
doi: 10.1103/PhysRevA.109.052620
url: https://arxiv.org/abs/2310.15353v7
triage: theorem
motivation_gids:
  - D5/S3/Quantum/QuantumChannels/NoisyWernerHolevoLindbladEmbedding.result

The noisy qutrit Werner–Holevo channel is a GKSL exponential only at zero noise on $[0,2/3]$

Problem

Shayan Roofeh and Vahid Karimipour, The noisy Werner-Holevo channel and its properties, Phys. Rev. A 109, 052620 (2024), arXiv:2310.15353v7, subsection “Spectral properties of the channel,” ask:

The question remains whether, within the range of $0\le x\le \frac{2}{3}$, the channel can be described as the exponential of Lindbladian dynamics. We will explore this question in our future work.

This is verbatim from ModifiedLSChannel-12.tex, line 249, in the downloaded v7 source, md5 bcff52a86196a2e1a0054a34c48a8117. The channel formula is at line 153; its spectral multipliers and determinant are at lines 237–247. The literature note records these locators.

The module defines, for every real $x$, the complex-linear map on $M_3(\mathbb{C})$

${\Lambda }_x(X)=(1-x)X-\frac{x}{2}{X}^{\mathsf{T}}+\frac{x}{2}\mathrm{tr}(X)I_3.$

This is wernerHolevo; the source uses the same formula with $0\le x\le 1$. IsGKSLGenerator G means that there exist $m\in \mathbb{N}$, a Hermitian $H\in M_3(\mathbb{C})$ and matrices $V_k\in M_3(\mathbb{C})$ indexed by $k\in \mathrm{Fin}m$ such that, for every $X\in M_3(\mathbb{C})$,

$G(X)=-i(HX-XH)+{\sum }_k\left(V_{k}X{V}_k^{†}-\frac{1}{2}(V_k^{†}{V}_{k}X+X{V}_k^{†}{V}_k)\right).$

The empty jump family is allowed. generatorExp G is the ordinary exponential of the $9\times 9$ complex matrix of $G$ in the matrix-unit basis, transported back to a linear map. Equality is equality on the entire complex matrix space.

Issue #21274 fixes the statement: for every $x\in [0,2/3]$, there exists such a $G$ with $\mathrm{generatorExp}(G)={\Lambda }_x$ if and only if $x=0$. This statement is proved.

Motivation

The question distinguishes a noise channel specified by its input-output action from a finite-time exponential of a time-independent Lindblad generator. A positive determinant alone leaves the interior $0<x<2/3$ undecided: the two nontrivial spectral multipliers are positive there. The settlement supplies a stronger obstruction involving their relative decay rates. It also clarifies the relation to the source’s small-angle random-rotation approximation at TeX lines 212–216.

Gap

The bounded literature check in #21274 records:

• the v7 source retains the explicit question; the quotation and source hash are independently reproduced here;
• the issue reports OpenAlex work W4396887671 with six citations and five distinct citing works from OpenCitations: arXiv:2405.11216v3, 2506.13397v3, 2411.19644v2, 2402.07700v5, and Luo et al., DOI 10.1007/s10773-025-06093-4;
• the issue reports full-text readings of the four arXiv papers, only an abstract reading of Luo et al., and readings of related arXiv:2401.16312v6, 2408.05733v5, 2610.03706v1, 2609.40355v2, 2603.12130v1 and 2603.07761v1, without finding a settlement;
• the preregistration records no existing repository issue, module or Problems record for this question.

These citation counts, secondary-paper readings and repository-search readings are attributed to #21274, not independently reproduced here. The search outcome is not-found-in-searched-scope, not an exhaustive novelty claim. Conditional complete positivity and the determinant obstruction are known ingredients recorded in the issue; the target is the answer for this specified family and parameter interval.

Route

All linear maps and matrices below are over $\mathbb{C}$ on $M_3(\mathbb{C})$. Define the module’s two projections by

$P(X)=\frac{1}{2}(X+X^{\mathsf{T}})-\frac{1}{3}\mathrm{tr}(X)I_3,Q(X)=\frac{1}{2}(X-X^{\mathsf{T}}),$

and put $s(G)=\mathrm{Re}\mathrm{Tr}(G\circ P)$ and $a(G)=\mathrm{Re}\mathrm{Tr}(G\circ Q)$. Here $\mathrm{Tr}$ is the endomorphism trace; $\mathrm{tr}$ is the matrix trace. These are symmetricProjection, antisymmetricProjection, symmetricTrace and antisymmetricTrace.

1. Zero noise. zero_gksl supplies the zero generator with $H=0$ and $m=0$. generatorExp_zero and wernerHolevo_zero identify its exponential and ${\Lambda }_0$ with the identity map.

2. The singular endpoint. generatorExp_injective applies to every complex-linear $G$, with no GKSL hypothesis. endpoint_kernel gives the nonzero matrix $E_{00}-E_{11}$ annihilated by ${\Lambda }_{2/3}$. Thus endpoint_not_exp excludes $\mathrm{generatorExp}(G)={\Lambda }_{2/3}$ for every $G$.

3. A GKSL trace inequality. For every $H\in M_3(\mathbb{C})$, sector_trace_hamiltonian gives $s({\mathcal{H}}_H)-5a({\mathcal{H}}_H)=0$, where ${\mathcal{H}}_H(X)=-i(HX-XH)$; this identity does not require $H$ to be Hermitian. For every $V\in M_3(\mathbb{C})$, sector_trace_dissipator gives

   $s({\mathcal{D}}_V)-5a({\mathcal{D}}_V)=2{\sum }_{i<j}|V_{ii}-V_{jj}{|}^2+3{\sum }_{i<j}|V_{ij}+V_{ji}{|}^2,$

   where ${\mathcal{D}}_V(X)=VX{V}^{†}-\frac{1}{2}(V^{†}VX+X{V}^{†}V)$ and $0\le i<j\le 2$. Summing these identities proves gksl_trace_inequality: for every $G$ satisfying IsGKSLGenerator G, $s(G)-5a(G)\ge 0$. No unitality or commutation hypothesis is needed for this inequality.

4. The required commutation. generatorExp_commute applies to every $G$. gksl_trace_zero gives $\mathrm{tr}(G(X))=0$ for every $X$ whenever IsGKSLGenerator G. For a real $x\ne 0$, a $G$ satisfying IsGKSLGenerator G and $\mathrm{generatorExp}(G)={\Lambda }_x$, generator_unital proves $G(I_3)=0$, generator_transpose proves $G(X^{\mathsf{T}})=G(X{)}^{\mathsf{T}}$ for every $X$, and generator_projection_commute proves that $G$ commutes with both $P$ and $Q$. These lemmas do not infer generator unitality merely from exponential unitality without their other hypotheses.

5. Branch-independent sector traces. For complex-linear $G,R$, projected_log_trace assumes $R^2=R$, $GR=RG$, a real $d>0$, and $\mathrm{generatorExp}(G)R=dR$. It proves $\mathrm{Re}\mathrm{Tr}(GR)=\mathrm{Re}\mathrm{Tr}(R)\log d$. On the range of $R$, every eigenvalue $\mu$ of $G$ satisfies $e^{\mu }=d$, hence $\mathrm{Re}\mu =\log d$; summing characteristic roots supplies the trace without choosing a logarithm branch. symmetricProjection_idempotent, antisymmetricProjection_idempotent, symmetricProjection_trace and antisymmetricProjection_trace give idempotence and traces $5$ and $3$. For every real $x$, wernerHolevo_symmetricProjection and wernerHolevo_antisymmetricProjection give ${\Lambda }_{x}P=(1-3x/2)P$ and ${\Lambda }_{x}Q=(1-x/2)Q$. Consequently, spectral_traces, with all its hypotheses $0<x<2/3$, IsGKSLGenerator G and $\mathrm{generatorExp}(G)={\Lambda }_x$, gives

   $s(G)=5\log (1-3x/2),a(G)=3\log (1-x/2).$

6. The interior contradiction. For a real $x$ with $0<x<2/3$, strict_log_inequality uses strict Bernoulli and logarithm monotonicity to prove $5\log (1-3x/2)<15\log (1-x/2)$. Together with spectral_traces and gksl_trace_inequality, interior_not_exp excludes $\mathrm{generatorExp}(G)={\Lambda }_x$ for every $G$ satisfying IsGKSLGenerator G under those same strict interval hypotheses. result combines this with the two endpoints.

Falsifier

The formalization would miss the question if it excluded only a particular logarithm, assumed a symmetric generator without justification, or changed the channel or generator class. Its definitions use the source’s ordinary transpose, the full complex matrix space and arbitrary finite GKSL jump families with a Hermitian Hamiltonian. Its interior obstruction applies to every such generator, including logarithms with nonzero imaginary branches; the needed projection commutation is derived with the hypotheses stated above.

The theorem concerns an exact exponential of one time-independent generator. It does not assert nonexistence of a time-dependent evolution, a product of different generator exponentials, an approximation, or a limiting realization. Complete positivity of ${\Lambda }_x$ as a channel is source context, not a separate conclusion of result.

Evidence

The canonical source is D5/S3/Quantum/QuantumChannels/NoisyWernerHolevoLindbladEmbedding.lean (459 lines).

• Proof. It proves result : claim.
• Axioms. It uses only the standard axioms propext, Classical.choice and Quot.sound; no sorry, native_decide, or new axiom. The axiom closure is read from #print axioms on the imported result; the source scan for those forbidden constructs has no matches.

Frozen state: Golden/Frozen/state/D5/S3/Quantum/QuantumChannels/NoisyWernerHolevoLindbladEmbedding.lean.json, statement sha256:e8aafc82c498f30e7909e12fec4d9e2957d5e577e3bb8b2dcb74d1bdb8624844.

Declaration statements:

• result: sha256:ba8cd229ed7fa12e768f0e1783787fbe516cf6ad26cc3490575357d47ae1635f;
• claim: sha256:3b03c90ddc64af18d11c11c1f88a45118bbd716020ff816e431bad9f9c6ecee2.

Triage

Tier 1: an explicit open question of a 2024 paper left for future work, preregistered in issue #21274 before any Lean. theorem; resolution proved.

The module is admitted under the open-problem-resolution basis of CLAUDE.md §3.2. There is no digestion atom.

What the settlement shows

Proved by result: for every real $x\in [0,2/3]$, existence of a complex-linear $G$ on $M_3(\mathbb{C})$ satisfying IsGKSLGenerator G and $\mathrm{generatorExp}(G)=\mathrm{wernerHolevo}(x)$ is equivalent to $x=0$.

Argued, not formalized:

• Decisive mechanism. Under the full hypotheses of spectral_traces, namely $0<x<2/3$, IsGKSLGenerator G and $\mathrm{generatorExp}(G)={\Lambda }_x$, that lemma and gksl_trace_inequality force $b\ge c^3$. The polynomial difference $c^3-b=x^2(6-x)/8$ is positive for $0<x<2/3$, since $x^2>0$ and $6-x>0$. The obstruction is therefore a relation between two positive decay multipliers, stronger than a determinant sign test, and excludes every logarithm branch.
• A neighbouring necessary condition. Let $P_0(X)=\mathrm{tr}(X)I_3/3$ and ${\Phi }_{b,c}=P_0+bP+cQ$, where $b,c>0$ and $1,b,c$ are pairwise distinct. If $G$ satisfies IsGKSLGenerator G and $\mathrm{generatorExp}(G)={\Phi }_{b,c}$, commutation with its exponential (generatorExp_commute) makes $G$ commute with $P$ and $Q$: these are polynomials in ${\Phi }_{b,c}$, for example $P=({\Phi }_{b,c}-\mathrm{id})({\Phi }_{b,c}-c\mathrm{id})/((b-1)(b-c))$. The projections are idempotent with traces $5$ and $3$, and the exponential acts on them by $b$ and $c$. These are all the hypotheses needed to apply projected_log_trace separately to $P$ and $Q$. Thus $s(G)=5\log b$ and $a(G)=3\log c$, and gksl_trace_inequality gives $b\ge c^3$. This argument gives a necessary condition, not a classification of all such channels; coincident multipliers require a separate argument.
• Sharpness of this neighbouring obstruction. Let $E_{ij}$ be the matrix units and $W_{ij}=E_{ij}-E_{ji}$ for $0\le i<j\le 2$; sums over $W_{ij}$ range over these three pairs. The skew-jump identities admit an entrywise verification: the diagonal entries of $\sum W_{ij}X{W}_{ij}^{†}$ are $\mathrm{tr}(X)-X_{ii}$ and its off-diagonal entries are $-X_{ji}$. With $\sum W_{ij}^{†}{W}_{ij}=2I_3$, their dissipator sum is $L_0(X)=\mathrm{tr}(X)I_3-X^{\mathsf{T}}-2X=-3P(X)-Q(X)$. For any $\gamma \ge 0$, $\gamma L_0$ has a finite GKSL representation with $H=0$ and jumps $\sqrt{\gamma }{W}_{ij}$. Their definitions give $P_0+P+Q=\mathrm{id}$, $P_0^2=P_0$, and zero products between distinct projections; $P$ and $Q$ are idempotent as recorded in the Route. Thus its exponential is $P_0+e^{-3\gamma }P+e^{-\gamma }Q$. Hence $b=c^3$ is attained by this neighbouring family. This construction and its exponential are not declarations of the settlement module.
• Small noise and temporal scope. Algebraically ${\Lambda }_x=\mathrm{id}+(x/2)L_0$, whereas the genuine exponential $e^{(x/2)L_0}$ has multipliers $e^{-3x/2}$ and $e^{-x/2}$. Their Taylor expansions agree with the channel multipliers to first order at zero; the discrepancy starts at second order. Exact non-embedding at every positive $x\le 2/3$ therefore permits an infinitesimal Lindblad approximation. Also, for a time-independent finite GKSL generator $L$ and any $t\ge 0$, $tL$ has the same form using $tH$ and $\sqrt{t}{V}_k$, so the time-one convention includes arbitrary nonnegative finite times. Neither statement settles time-dependent embeddability.
• Effect on the source. The question at TeX line 249 receives a negative answer for every $0<x\le 2/3$, with the identity as its sole affirmative case. The source explicitly leaves this question to future work, rather than asserting an embedding theorem there. Its separate determinant observation on $2/3<x\le 1$ is outside result. Its random-rotation discussion reaches the noisy-channel formula through the small-angle approximation at lines 212–216; an approximation does not entail an exact exponential representation. The settlement neither verifies nor refutes the source’s separate capacity and degradability calculations, or its questions about higher-spin and higher-dimensional generalizations at lines 911–913. Those assertions do not acquire a new proof from this settlement.

ASSUMED-UNVERIFIED

The bounded literature check in #21274 does not establish exhaustive worldwide novelty or priority. Its OpenAlex and OpenCitations counts, secondary-paper readings, abstract-only Luo et al. reading, assertion that v7 is the latest version, and repository-search results are not independently reproduced here. The v7 TeX quotation, locator and md5 are reproduced; the DOI and journal metadata are recorded in the Library note and preregistration, without an independent Crossref query here.

Everything labelled “Argued, not formalized” lies outside the compiled statement of result. In particular, the generic two-multiplier necessary condition, its sharpness construction, the polynomial identity as a named theorem, the Taylor comparison, and absorption of a time parameter are not formalized in this module. No sufficiency classification for general ${\Phi }_{b,c}$, time-dependent embedding result, higher-dimensional extension, or conclusion about the other open questions of the source is claimed; these remain open in this dossier.