bibkey: cha2026structural authors: Hyunho Cha, Jungwoo Lee year: 2026 title: “Structural perspectives from quantum states and measurements in optimal state discrimination” doi: 10.1007/s11128-026-05335-6 url: https://arxiv.org/abs/2507.05778v2 claim: “For an ensemble of N states with optimal minimum-error POVM, I_+ is the set of labels whose optimal effect is nonzero, and P^PGM_+ is the pretty-good-measurement score restricted to I_+ with the original weights. For equal priors the paper conjectures (|I_+| - 1)(P^PGM_+ - 1/N) <= (N - 1)(P^PGM - 1/N), the equiprobable form of ‘the restricted PGM bound is at most Renes’s PGM bound’.” strata_touched:

• D5/S3/Quantum/Measurement/EquiprobablePgmActiveSetRefutation license: citation-only triage: anchor

Structural perspectives from quantum states and measurements in optimal state discrimination

Hyunho Cha and Jungwoo Lee, Quantum Information Processing 25, 310 (2026); arXiv:2507.05778v2. Quotations are from the arXiv v2 source.

The optimal POVM (l. 232–243): maximize ${\sum }_i\text{tr}({\tilde{\sigma }}_i{E}_i)$ subject to $E_i⪰0$, ${\sum }_i{E}_i=I$; $\hat{E}(\mathcal{E})$ denotes the optimal POVM.

The active set and the restricted score (l. 429, 458–468):

Let ${\text{I}}_{+}(\mathcal{E})\equiv \{i|{\hat{E}}_i(\mathcal{E})\succ 0\}$ denote the set of indices of positive definite (optimal) measurement operators for $\mathcal{E}$. […] P_\mathcal{E}^{\text{PGM}+} = \sum_{i \in \text{I}_+(\mathcal{E})} \text{tr}(\Tilde{\sigma}_i E_i), where S \equiv \sum_{i \in \text{I}_+(\mathcal{E})} \Tilde{\sigma}_i and E_i \equiv S^{-1 / 2} \Tilde{\sigma}_i S^{-1 / 2} if $i\in {\text{I}}_{+}(\mathcal{E})$.

The journal version defines $I_{+}(\mathcal{E})=\{i:{\hat{E}}_i(\mathcal{E})\ne 0\}$, “nonzero (optimal) measurement operators” (seat-reported, Section 4, p. 7), and the arXiv example at l. 526 assigns ${\text{I}}_{+}=\{1,2\}$ to rank-one effects; the encoding uses nonzero effects.

The conjecture (l. 636–640; journal Section 4.1.3, Eq. (18)):

$(|{\text{I}}_{+}(\mathcal{E})|-1)\left(P_{\mathcal{E}}^{\text{PGM}+}-\frac{1}{N}\right)\le (N-1)\left(P_{\mathcal{E}}^{\text{PGM}}-\frac{1}{N}\right).$ Numerical attempts have failed to identify any exceptions to Eq. (equiprobable_pgm_bound_inequality). Consequently, we conjecture that it holds universally for equiprobable states.

The encoding states it for $N\ge 1$ positive definite states with equal priors and for some optimal POVM.

Verified locator

• DOI: https://doi.org/10.1007/s11128-026-05335-6 (Quantum Information Processing 25, 310 (2026); the conjecture is journal Eq. (18), Section 4.1.3, p. 12, as reported by the literature seat).
• URL: https://arxiv.org/abs/2507.05778v2 (source sn-article.tex, md5 b9a14a8589877a76996275e1e5230bde): the optimization (l. 232–243), the PGM (l. 254), the active set (l. 429), the restricted score (l. 458–468), the mirror-symmetric example (l. 526), the conjecture (l. 636–640).