bibkey: nandi2025genuine authors: S. Nandi year: 2025 title: “Genuine multipartite entanglement detection with mutually unbiased bases (MUBs)” doi: 10.48550/arXiv.2509.24045 url: https://arxiv.org/abs/2509.24045v1 claim: “The correlation I2 = C(a,b) + C(a’,b’), summed over equal outcomes of local measurements in two mutually unbiased bases, is conjectured to be monotonically non-increasing under LOCC operations.” strata_touched:

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Nandi, genuine multipartite entanglement detection with MUBs

S. Nandi, Genuine multipartite entanglement detection with mutually unbiased bases (MUBs), arXiv:2509.24045v1 (28 September 2025, quant-ph); published in EPL 154 (2026), DOI 10.1209/0295-5075/ae6f48.

Verified locator

DOI: 10.48550/arXiv.2509.24045. Primary version: https://arxiv.org/abs/2509.24045v1 (the only arXiv version). The TeX source main.tex of v1 supplies §II (Eqs. (pAB), (cAB), the definition of $I_2$, the Conjecture and Eq. (10)).

Source statements (§II)

Mutual predictability: “We denote the joint probability that the outcome of $a$ is $i$ and the outcome of $b$ is $j$ by $P_{a,b}(i,j)$ given by (P_{a,b}(i,j)=\langle i_a|\otimes\langle j_b|\rho|i_a\rangle\otimes|j_b\rangle)”, “(C_{a,b}=\sum_{i=0}^{d-1}P_{a,b}(i,i))”, and, for observables “({a^\prime, b^\prime}\in B_2) which is mutually unbiased to (B_1)”, “(C_{a^\prime,b^\prime}=\sum_{i=0}^{d-1}P_{a^\prime,b^\prime}(i,i))” and “(I_{2}=C_{a^\prime,b^\prime}+C_{a,b})”.

Conjecture: “The quantity (I_2) is monotonically non-increasing under LOCC operations.” The test that follows considers “POVMs ({E_1, E_2}) acting locally on a particular subsystem of a bipartite density matrix (\rho)” with “(E_1^\dagger E_1+E_2^\dagger E_2=\mathbb{I})” and the inequality “(I_2(\rho)-p_1I_2(\rho_1)-(1-p)I_2(\rho_2)\ge0)” (Eq. (10)), where “(\rho_k=\frac{(E_k\otimes I).\rho.(E_k\otimes I)^\dagger}{p_k}), (p_k=Tr[\rho (E_k\otimes I)(E_k\otimes I)^\dagger])”.

Scope

The paper’s numerical check of Eq. (10) (its Fig. 1) uses the Bell state only. For pure bipartite states it evaluates $I_2$ in the Schmidt basis.