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bibkey: baekjonha2026maximalcoherence authors: K. Baek; Y. Jo; H. Nha year: 2026 title: “Maximal coherence of quantum measurement and the resource theory of sharpness” doi: null url: “https://arxiv.org/abs/2607.05847v1” claim: ‘Section III.C asks whether maximal distance-based measurement coherence is monotone under general CP unital maps for non-dichotomic POVMs.’ strata_touched:

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Source: https://arxiv.org/abs/2607.05847v1

Definitions

Section II.A, p. 2:

A quantum measurement on a $d$-dimensional Hilbert space with $n$ outcomes is described by a POVM, i.e., a tuple $\mathsf{A}=\{A_x{\}}_{x=1}^n$ of positive semidefinite operators satisfying the completeness relation ${\sum }_{x=1}^n{A}_x=I$.

In the same spirit, a quantum measurement is called incoherent when all its elements are diagonal in the reference basis:

The defining expression is $A_x={\sum }_{i=1}^d{\lambda }_{i|x}|i⟩⟨i|$. The set of these POVMs is $\mathcal{I}(d,n)$. Section II.B, pp. 3–4, defines trivial POVMs as $\{p_{x}I{\}}_{x=1}^n$ with ${\sum }_x{p}_x=1$, denoted $\mathcal{T}(d,n)$.

Section II.A, p. 2, states:

A natural family of coherence monotones for measurements arises from a distance $D$ between POVMs as

The coherence of Eq. (4), p. 2, and the maximal coherence of Section III.A, p. 4, are

$$C_D(\mathsf{A})=\underset{\mathsf{B}\in \mathcal{I}(d,n)}{\min }D(\mathsf{A},\mathsf{B}),C_D^{\max }(\mathsf{A})=\underset{U}{\max }{C}_D(U^{†}\mathsf{A}U).$$

Section III.A, p. 4, states:

Coherence is intrinsically basis dependent: rotating the reference basis by a unitary $U$ alters the off-diagonal structure of every POVM element. We define the maximal coherence of measurement of $\mathsf{A}$ as the basis-optimized value of any coherence monotone $C_D$,

The maximum is over all unitaries. Lean uses sInf and sSup of the corresponding sets of real distances, with the spectral norm from Matrix.Norms.L2Operator. The sharpness in Section II.B, p. 4, is $S_D(\mathsf{A})={\min }_{\mathsf{B}\in \mathcal{T}(d,n)}D(\mathsf{A},\mathsf{B})$.

The element-additive distance of Eq. (32), p. 5, at $p=\infty$, is $D_{\infty }(\mathsf{A},\mathsf{B})=\frac{1}{2}{\sum }_x\Vert A_x-B_x{\Vert }_{\infty }$. Section IV.A, p. 7, adopts it for the tetrahedral SIC qubit POVM. Its contractivity under CP unital maps is the $p=\infty$ case of Theorem 5 of Tendick et al., arXiv:2205.08546.

Fuzzifying operations, Eq. (12), p. 4, act as $A_x\mapsto \alpha {\mathcal{E}}^{†}(A_x)+(1-\alpha )q_x{I}_B$, with $\alpha \in [0,1]$ and $q$ a probability distribution. The source says:

By construction, fuzzifying operations preserve the number of outcomes $n$ but may change the Hilbert space dimension; in particular, dimension-preserving fuzzifying operations arise when ${\mathcal{H}}_A={\mathcal{H}}_B$.

The CP unital dual has the finite-dimensional Kraus form $\Phi (X)={\sum }_j{K}_j^{†}X{K}_j$, ${\sum }_j{K}_j^{†}{K}_j=I$.

Question

Section III.C, p. 7, states verbatim:

For non-dichotomic POVMs, the equivalence with sharpness is generally unavailable, and monotonicity must be considered at the channel level: it holds under mixed-unitary channels (Theorem 5) and under unitarily covariant channels (Corollary 6), but whether $C_D^{\max }$ remains monotonic under general CP unital maps in this regime is left as an open problem.

Table I, p. 7, marks the non-dichotomic POVM row under “any fuzzifyng operations” as “Open”. The source’s spelling is retained.

Refutation and scope

The four tetrahedral SIC effects are $A_i=(I+{\vec{n}}_i\cdot \vec{\sigma })/4$, with ${\vec{n}}_i=(1,1,1),(1,-1,-1),(-1,1,-1),(-1,-1,1)$ divided by $\sqrt{3}$. Adjoining an identity qubit is $J(X)=X\otimes I_2$, a CP unital dual of discarding that qubit. Coordinates use finProdFinEquiv, $(i,j)\mapsto 2i+j$, identifying ${\mathbb{C}}^2\otimes {\mathbb{C}}^2$ with Fin 4 in the order $|00⟩,|01⟩,|10⟩,|11⟩$.

The module proves $C_{D_{\infty }}^{\max }({\mathsf{M}}_{\mathrm{SIC}})\le \sqrt{2/3}/2$ and $C_{D_{\infty }}^{\max }(J({\mathsf{M}}_{\mathrm{SIC}}))\ge 1/2$. Bell-diagonal effects have scalar normalized partial trace; applying the Rayleigh bound with both signs gives the lower bound. The qubit upper bound uses dephasing and the simultaneous tetrahedral off-diagonal square sum. This refutes general monotonicity with four outcomes and a dimension-changing operation. Dichotomic, mixed-unitary and covariant results retain their stated hypotheses. Diamond distance and dimension-preserving failure remain open.

Numerical reading

The independent numerical check is check.py, SHA256 b0d07f99a84ffde3e5a0285f79184a2e0dfe8cf0dcf4ad2c8c151bc7d971d96c. Running python3 check.py exits 0 and reports bad= 0. The sampled dephasing maximum is 0.408247, below the proved upper bound 0.408248. Sampling is computed evidence; the exact upper and lower bounds are checked in Lean.