bibkey: siddiquiwang2026concealment authors: Mohd Asad Siddiqui; Zizhu Wang year: 2026 title: “Operational Concealment of Measurement Incompatibility by Quantum Channels: Rank Loss versus Contraction” doi: 10.48550/arXiv.2607.11762 url: https://arxiv.org/abs/2607.11762v2 claim: “For channels with a common output space, equality of adjoint kernels (restricted to Hermitian operators) implies equality of the sets of concealed POVM pairs; whether equality of adjoint kernels is also necessary for concealment-equivalence is stated as open.” strata_touched:

• D5/S3/Quantum/QuantumChannels/ConcealmentKernelNecessityRefutation license: citation-only triage: anchor

Siddiqui and Wang, operational concealment of measurement incompatibility

M. A. Siddiqui and Z. Wang, Operational Concealment of Measurement Incompatibility by Quantum Channels: Rank Loss versus Contraction, arXiv:2607.11762v2 (quant-ph; v1 13 July 2026, v2 17 August 2026).

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DOI: 10.48550/arXiv.2607.11762. Preprint: https://arxiv.org/abs/2607.11762v2 (the abstract page lists v1 and v2). The TeX source final_v2.tex of the v2 e-print supplies the statements below.

Source statements

Standing assumptions (Remark, Section II): finite-dimensional Hilbert spaces, finite-outcome POVMs, exact tomography and tomographic completeness of the input family $\mathcal{T}$, that is ${\mathrm{span}}_{\mathbb{R}}(\mathcal{T})=\mathrm{Herm}({\mathcal{H}}_{\mathrm{in}})$. The adjoint kernel is restricted to Hermitian operators: $\ker ({\mathcal{E}}^{†})=\{X\in \mathrm{Herm}({\mathcal{H}}_{\mathrm{out}}):{\mathcal{E}}^{†}(X)=0\}$. Two POVMs are compatible if a joint POVM $\{J_{ab}\}$ has marginals ${\sum }_b{J}_{ab}=M_a$ and ${\sum }_a{J}_{ab}=N_b$.

Definition (Operational concealment):

Let $\mathcal{T}\subseteq \mathcal{S}({\mathcal{H}}_{\mathrm{in}})$ be a set of input states. POVMs $\{M_a\},\{N_b\}$ are concealed by a quantum channel $\mathcal{E}$ relative to $\mathcal{T}$ if there exist compatible POVMs $\{F_a\},\{G_b\}$ such that for all $\rho \in \mathcal{T}$, $\mathrm{Tr}[F_a\mathcal{E}(\rho )]=\mathrm{Tr}[M_a\mathcal{E}(\rho )]$ and $\mathrm{Tr}[G_b\mathcal{E}(\rho )]=\mathrm{Tr}[N_b\mathcal{E}(\rho )]$.

Section III: “For fixed finite outcome sets $\Omega$ and $\Lambda$, let ${\mathcal{C}}_{\mathcal{E}}^{\Omega ,\Lambda }$ denote the set of POVM pairs concealed by $\mathcal{E}$.” The theorem “Kernel invariance of concealment” proves ${\mathcal{C}}_{{\mathcal{E}}_1}={\mathcal{C}}_{{\mathcal{E}}_2}$ for channels with a common output space and $\ker ({\mathcal{E}}_1^{†})=\ker ({\mathcal{E}}_2^{†})$, and is followed by:

Equality of adjoint kernels is sufficient for concealment-equivalence: channels with different dynamical realizations but identical adjoint kernels are indistinguishable with respect to concealment of POVM pairs. […] Whether equality of adjoint kernels is also necessary for concealment-equivalence remains open.

The introduction uses complete dephasing as the basic example of concealment: after dephasing, the Pauli $X$ effects are operationally equivalent to $I/2$ while the Pauli $Z$ effects are unchanged.