slug: siddiqui-wang-2026-concealment-kernel-necessity
bibkey: siddiquiwang2026concealment
doi: 10.48550/arXiv.2607.11762
url: https://arxiv.org/abs/2607.11762v2
triage: theorem
motivation_gids:
  - D5/S3/Quantum/QuantumChannels/ConcealmentKernelNecessityRefutation.result

Concealment-equivalent channels with different adjoint kernels

Problem

M. A. Siddiqui and Z. Wang, Operational Concealment of Measurement Incompatibility by Quantum Channels: Rank Loss versus Contraction, arXiv:2607.11762v2, Section III, prove that two channels with a common output space and equal adjoint kernels conceal the same POVM pairs, and state:

Whether equality of adjoint kernels is also necessary for concealment-equivalence remains open.

The definitions are quoted in the literature note.

• A pair of POVMs is concealed by a channel $\mathcal{E}$ relative to an input set $\mathcal{T}$ if compatible POVMs reproduce its statistics on every output $\mathcal{E}(\rho )$, $\rho \in \mathcal{T}$.
• The adjoint kernel is $\ker ({\mathcal{E}}^{†})$ on Hermitian operators.

Issue #14233 reads the affirmative answer with the hypothesis in its strongest form. For all finite dimensions, all trace-preserving Kraus channels ${\mathcal{E}}_1,{\mathcal{E}}_2$ with common input and output spaces, and every tomographically complete $\mathcal{T}$: if the two channels conceal the same pairs for every pair of finite outcome sets, then their adjoint kernels coincide.

Motivation

The source shows that the adjoint kernel decides whether concealment is possible at all, and that kernel equality is a sufficient invariant for the whole concealment set. A necessary-and-sufficient answer would make the kernel a complete invariant of concealment-equivalence.

Gap

Issue #14233 records the literature check before any Lean:

• arXiv lists v1 and v2, and the sentence stands in v2;
• arXiv searches find only the source;
• Semantic Scholar lists no citing work;
• Zenodo has no record.

not-found-in-searched-scope.

Route

On ${\mathbb{C}}^2$ take the complete dephasing $D(A)=\mathrm{diag}(A_{00},A_{11})$ and the complete depolarization $R(A)=\mathrm{Tr}(A)I/2$, and let $\mathcal{T}$ be all density matrices.

• Both conceal every pair of POVMs, for all finite outcome sets. For $D$, the diagonal parts of the effects are compatible, through the joint POVM of products of diagonal entries. For $R$, the trace multiples of $I/2$ are compatible, through the joint POVM of products of traces. Both reproduce every output statistic.
• The kernels differ. $R^{†}(Z)=0$, while $D^{†}(Z)=Z\ne 0$.

Falsifier

The refutation depends on the reading of concealment-equivalence as equality of the concealment sets of two channels with common input and output spaces, as in the source’s theorem. A different notion (for example one restricted to incompatible pairs) would not change the answer, since both channels conceal every pair.

Evidence

The canonical source is D5/S3/Quantum/QuantumChannels/ConcealmentKernelNecessityRefutation.lean.

• It has public IsPOVM, Compatible, Concealed, TomographicallyComplete, AdjointKernel, claim and result : ¬ claim.
• It reuses the frozen Kraus carrier MatrixMap.of_kraus and its transplanted dual MatrixMap.dual, the density predicate and the Hermitian space of the frozen quantum-channel modules, and the frozen dephasing channel of D5/S3/QuantumChannels/CumulantRenyiDataProcessingRefutation, whose dephase_apply and pinching_kraus are made public at their origin, and the frozen Pauli carrier Pauli/pauliMatrix of D5/S3/Quantum/Information/StabilizerPairLocalUnitaryInequivalence for the depolarizing Kraus operators.
• The axiom closure of result is exactly propext, Classical.choice and Quot.sound. There is no sorry, no native_decide and no new axiom.

The module statement is sha256:9773b9f136826c316e78f3ccfff398b112fcb6093dda26799ff4966ea4ea1254, the result statement sha256:e7fb999e85afe613f8e15e9453d76265fd17396c7288166ee78261eeda0a54d8 and the claim statement sha256:3971ed2f6455f7a415446ea94a6c56e8ea7851e42d01f47e20b12103ae81e551. The Freeze event is sha256:08263f215d6184737b059eba1b394d15439ea823b9649309966243efd476d295. Its project-level prerequisites are PositiveFilterTransposeRefutation, StabilizerPairLocalUnitaryInequivalence and the re-pinned CumulantRenyiDataProcessingRefutation (Freeze event sha256:0f6f4fff2e03748a2cf4315cd065d1e5b08b1a0e399a4e6ea6ec0584353b73f0).

Triage

Tier 1 open question of a July 2026 paper, preregistered in issue #14233 before any Lean. theorem; resolution refuted.

The private helpers (the diagonal joint POVM, and concealment by channels with diagonal or scalar outputs, for arbitrary finite outcome types) lie on the proof path of result. Utility is kind=certified-instance; basis=refutes. There is no digestion atom.

What the refutation shows

Proved by result: the affirmative answer fails in dimension $2$. Complete dephasing and complete depolarization conceal exactly the same pairs (all of them) for every pair of finite outcome sets, but their adjoint kernels differ: $Z$ lies in the kernel of $R^{†}$ ($R^{†}(Z)=0$) and not in the kernel of $D^{†}$ ($D^{†}(Z)=Z\ne 0$).

Proved inside the proof, for general data: a channel whose outputs are all diagonal, or all multiples of the identity, conceals every pair of POVMs with arbitrary finite outcome sets.

Computed, not formalized: on the Pauli basis, $D^{†}$ maps $I,X,Y,Z$ to $I,0,0,Z$ and $R^{†}$ maps them to $I,0,0,0$ (exact SymPy check, issue #14233). By linearity the complete kernels are ${\mathrm{span}}_{\mathbb{R}}\{X,Y\}$ for $D^{†}$ and ${\mathrm{span}}_{\mathbb{R}}\{X,Y,Z\}$ for $R^{†}$.

Argued, not formalized.

• Mechanism. When every output of a channel commutes with a fixed basis, any measurement is simulated on the outputs by its diagonal part. Diagonal effects are always compatible, through the product joint POVM. Such a channel conceals everything, whatever its kernel, so the concealment set forgets which observable directions are hidden.
• Family. The same argument applies in every dimension $d$: any two channels whose outputs are diagonal conceal all pairs. Their kernels all contain the off-diagonal Hermitian operators and may differ on the diagonal part. On ${\mathbb{C}}^d$ the kernel of the complete dephasing has codimension $d$, and that of the complete depolarization has codimension $1$.
• What survives. The counterexample uses channels whose concealment set is everything. Whether the kernel is a complete invariant among channels with a noncommutative output range, or whether the concealment sets determine the kernel up to the commutative part of the output algebra, is not addressed.

Open. The two refinements just stated are not settled here.

Effect on the paper. Kernel equality is sufficient but not necessary for concealment-equivalence. The kernel is therefore not a complete invariant of the concealment sets. The paper’s other results (the existence criterion and kernel monotonicity) are unaffected.

ASSUMED-UNVERIFIED

The bounded literature check does not establish exhaustive worldwide novelty, priority, or the absence of an independent answer.