The equiprobable pretty-good-measurement comparison of Cha and Lee fails

Abstract

Cha and Lee (arXiv:2507.05778, Quantum Information Processing 25, 310) compare two upper bounds on the optimal success probability of minimum-error state discrimination, one built from the pretty good measurement on all states and one built from it on the active set I_+ of labels with nonzero optimal effects. For equal priors they conjecture (|I_+| - 1)(P^PGM_+ - 1/N) <= (N - 1)(P^PGM - 1/N). Three positive definite qubit states with rational entries violate it.

Definition 1.1 (The success probability).

$$\forall d,N:\mathbb{N},\forall rho,E:\mathrm{Fin}\left(N\right)\to \mathrm{Matrix}\left(\mathrm{Fin}\left(d\right),\mathrm{Fin}\left(d\right),\mathbb{C}\right),\mathrm{success}\left(rho,E\right)=\sum _{i\in \mathrm{Fin}\left(N\right)}\mathrm{Re}\left(\mathrm{tr}\left(\frac{1}{N}\cdot rho\left(i\right)\cdot E\left(i\right)\right)\right)$$

Formalization. D5/S3/Quantum/Measurement/EquiprobablePgmActiveSetRefutation.success (✓ std3).

Citation. Hyunho Cha, Jungwoo Lee (2026). 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.

Commentary.

For the equiprobable ensemble sigma_i = rho_i / N, the success probability of the POVM E is sum_i tr(sigma_i E_i); the real part reads this real trace.

Definition 1.2 (The pretty good measurement on a set of labels).

$$\forall d,N:\mathbb{N},\forall rho:\mathrm{Fin}\left(N\right)\to \mathrm{Matrix}\left(\mathrm{Fin}\left(d\right),\mathrm{Fin}\left(d\right),\mathbb{C}\right),\forall A:\mathrm{Finset}\left(\mathrm{Fin}\left(N\right)\right),\mathrm{pgmScore}\left(rho,A\right)=\sum _{i\in A}\mathrm{Re}\left(\mathrm{tr}\left(\frac{1}{N}\cdot rho\left(i\right)\cdot (\sum _{j\in A}\frac{1}{N}\cdot rho\left(j\right){)}^{-\frac{1}{2}}\cdot \frac{1}{N}\cdot rho\left(i\right)\cdot (\sum _{j\in A}\frac{1}{N}\cdot rho\left(j\right){)}^{-\frac{1}{2}}\right)\right)$$

Formalization. D5/S3/Quantum/Measurement/EquiprobablePgmActiveSetRefutation.pgmScore (✓ std3).

Citation. Hyunho Cha, Jungwoo Lee (2026). 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.

Commentary.

For a set A of labels, S_A = sum_{j in A} sigma_j and the pretty good measurement has effects S_A^{-1/2} sigma_i S_A^{-1/2} for i in A; its score keeps the original weights sigma_i = rho_i / N. S_A^{-1/2} is the continuous-functional-calculus power. With A all labels this is P^PGM; with A = I_+ it is P^PGM_+.

Definition 1.3 (The conjectured comparison).

$$(claim)\iff (\forall k,N:\mathbb{N},\forall rho:\mathrm{Fin}\left(N\right)\to \mathrm{Matrix}\left(\mathrm{Fin}\left(k+1\right),\mathrm{Fin}\left(k+1\right),\mathbb{C}\right),(0<N)\Rightarrow (((\forall i:\mathrm{Fin}\left(N\right),\mathrm{PosDef}\left(rho\left(i\right)\right))\wedge (\forall i:\mathrm{Fin}\left(N\right),\mathrm{tr}\left(rho\left(i\right)\right)=1))\Rightarrow (\exists E:\mathrm{Fin}\left(N\right)\to \mathrm{Matrix}\left(\mathrm{Fin}\left(k+1\right),\mathrm{Fin}\left(k+1\right),\mathbb{C}\right),((\mathrm{finitePOVM}\left(E\right))\wedge (\forall F:\mathrm{Fin}\left(N\right)\to \mathrm{Matrix}\left(\mathrm{Fin}\left(k+1\right),\mathrm{Fin}\left(k+1\right),\mathbb{C}\right),(\mathrm{finitePOVM}\left(F\right))\Rightarrow (\mathrm{success}\left(rho,F\right)\le \mathrm{success}\left(rho,E\right))))\wedge (\forall A:\mathrm{Finset}\left(\mathrm{Fin}\left(N\right)\right),(A=\{i\mid E\left(i\right)\ne 0\})\Rightarrow ((|A|-1)\cdot (\mathrm{pgmScore}\left(rho,A\right)-\frac{1}{N})\le (N-1)\cdot (\mathrm{pgmScore}\left(rho,\mathrm{univ}\left(\mathrm{Fin}\left(N\right)\right)\right)-\frac{1}{N}))))))$$

Formalization. D5/S3/Quantum/Measurement/EquiprobablePgmActiveSetRefutation.claim (✓ std3).

Citation. Hyunho Cha, Jungwoo Lee (2026). 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.

Commentary.

Journal Eq. (18) for equal priors, with I_+ the set of labels whose optimal effect is nonzero. POVMs are the frozen finitePOVM: positive semidefinite effects on C^(k+1) that sum to the identity. The encoding asks for some optimal POVM, the weakest reading of the optimal POVM, and restricts to N >= 1 positive definite states, for which every nonempty S_A is invertible.

Definition 1.4 (The three states).

$$((rho\left(0\right)=\mathrm{mat}\left(\frac{400}{401},\frac{18}{401},\frac{18}{401},\frac{1}{401}\right))\wedge (rho\left(1\right)=\mathrm{mat}\left(\frac{400}{401},-\frac{18}{401},-\frac{18}{401},\frac{1}{401}\right)))\wedge (rho\left(2\right)=\mathrm{mat}\left(\frac{47963}{48922},0,0,\frac{959}{48922}\right))$$

Formalization. D5/S3/Quantum/Measurement/EquiprobablePgmActiveSetRefutation.rho (✓ std3).

Source. Repository-derived.

Acknowledgement. Hyunho Cha, Jungwoo Lee (2026). 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.

Commentary.

Three positive definite qubit density matrices with rational entries; mat(a, b, c, e) is the 2 x 2 matrix with rows (a, b) and (c, e).

Theorem 1.5 (The comparison fails for three qubit states).

$$\neg claim$$

Proof. Machine-checked in Lean as D5/S3/Quantum/Measurement/EquiprobablePgmActiveSetRefutation.result (✓ std3). ∎

Resolves. Problems/cha-2026-equiprobable-pgm-active-set (refuted) by D5/S3/Quantum/Measurement/EquiprobablePgmActiveSetRefutation.result.

Source. Repository-derived.

Acknowledgement. Hyunho Cha, Jungwoo Lee (2026). 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.

Commentary.

Take N = 3, d = 2 and sigma_i = rho_i / 3. The matrix Gamma = diag(418, 19)/1203 has Gamma - sigma_1 = (6/401)(1, -1)(1, -1)^T, Gamma - sigma_2 = (6/401)(1, 1)(1, 1)^T and Gamma - sigma_3 = diag(1011, 453)/48922, all positive semidefinite. Hence every POVM satisfies sum_i tr(sigma_i E_i) = tr(Gamma) - sum_i tr((Gamma - sigma_i) E_i) <= tr(Gamma) = 437/1203, and the projectors (1/2)(1, 1)(1, 1)^T, (1/2)(1, -1)(1, -1)^T with E_3 = 0 attain it. For an optimal POVM every term tr((Gamma - sigma_i) E_i) vanishes. Since Gamma - sigma_3 is at least (453/48922) I, E_3 = 0; and E_1 = 0 or E_2 = 0 would leave tr(Gamma - sigma_j) = 12/401. So I_+ = {1, 2}. Then S = diag(121, 1)/122 and S_+ = diag(800, 2)/1203, whose inverse square roots are diagonal; P^PGM = 20192347/58370763 and P^PGM_+ = 2167/6015, and (2 - 1)(P^PGM_+ - 1/3) - (3 - 1)(P^PGM - 1/3) = 168714/97284605 > 0.

References

• Truth anchor: D5/S3/Quantum/Measurement/EquiprobablePgmActiveSetRefutation.claim
• Truth anchor: D5/S3/Quantum/Measurement/EquiprobablePgmActiveSetRefutation.pgmScore
• Truth anchor: D5/S3/Quantum/Measurement/EquiprobablePgmActiveSetRefutation.result
• Truth anchor: D5/S3/Quantum/Measurement/EquiprobablePgmActiveSetRefutation.rho
• Truth anchor: D5/S3/Quantum/Measurement/EquiprobablePgmActiveSetRefutation.success
• Dependency: D5/S3/Estimation/TransmissivityTwoPointProbeRefutation