bibkey: baebera2022mubbound
authors: J. Bae; A. Bera; D. Chruściński; B. C. Hiesmayr; D. McNulty
year: 2022
title: "How many mutually unbiased bases are needed to detect bound entangled states?"
doi: 10.1088/1751-8121/acaa16
url: https://arxiv.org/abs/2108.01109
claim: "Equation (4) defines the MUB entanglement witness; Conjecture 1 predicts that the minimal number of MUBs for bound entanglement detection in dimension d = 2^r, r > 1, is d/2 + 1."
strata_touched:
  - D5/S3/Quantum/Entanglement/FourMUBBoundEntanglementRefutation
license: citation-only
triage: anchor

MUB witnesses and the power-of-two resource conjecture

J. Bae, A. Bera, D. Chruściński, B. C. Hiesmayr and D. McNulty, How many mutually unbiased bases are needed to detect bound entangled states?, Journal of Physics A: Mathematical and Theoretical 55 (2022), 505303. Page numbers refer to arXiv:2108.01109v2.

Verified locator

DOI: https://doi.org/10.1088/1751-8121/acaa16

Source: https://arxiv.org/abs/2108.01109

Version: https://arxiv.org/pdf/2108.01109v2

Source statements

Section I, p. 2:

Generalising to larger collections of measurements, we say a set of bases is mutually unbiased if all bases are pairwise unbiased.

Section VIII, p. 8:

When d = 2^r, we predict that d/2 + 1 MUBs are both necessary and sufficient for the witness W^Γ(M_m, s) to be non-decomposable.

Conjecture 1. If d = 2^r and r > 1, the minimal number of MUBs required for bound entanglement detection equals m = d/2 + 1 = 2^{r−1} + 1.

Section II, Eq. (4), p. 3:

$$W({\mathcal{M}}_m,s)=\frac{d+m-1}{d}1\otimes 1-B({\mathcal{M}}_m,s).$$

The computational-basis contribution to $B$ pairs $\ell$ with $\ell +s$ modulo $d$. Each remaining basis contributes the sum of equal-column rank-one tensor projectors. Partial transposition conjugates the second projector, giving $W^{\Gamma }({\mathcal{M}}_m,s)$.

Scope

The Lean claim expresses the necessity half: no family with fewer than $2^{r-1}+1$ bases detects a positive-partial-transpose density matrix through this witness. The refutation uses $r=3$, four bases and shift $s=1$. It does not assert that four is the minimum. Theorem 1’s sufficiency for $m>d/2+1$ is a separate source result.