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Mutually equibiased bases

Seyed Javad Akhtarshenas, Saman Karimi, Mahdi Salehi

TL;DR

The paper introduces mutually equibiased bases (MEBs) as a relaxed generalization of mutually unbiased bases (MUBs), where inter-basis measurement statistics are fixed by a distribution q up to permutation. It develops the mathematical framework using unistochastic matrices, a q parameterization via an anti-diagonal N, and explicit constructions for d=2 and d=3, including a one-parameter family of four MEBs in dimension 3 with μ in [1/3,1/2]. It then derives an entropic uncertainty relation for MEBs and constructs positive maps and entanglement witnesses based on MEBs, illustrating entanglement detection in isotropic states and exposing a tradeoff between the number of MEBs and detection strength depending on μ. The results demonstrate nontrivial constraints on q and reveal that not all bases in an MEB set contribute equally to entanglement detection, offering new insights into measurement incompatibility and quantum state discrimination in higher dimensions.

Abstract

In the framework of mutually unbiased bases (MUBs), a measurement in one basis gives \emph{no information} about the outcomes of measurements in another basis. Here, we relax the no-information condition by allowing the $d$ outcomes to be predicted according to a predefined probability distribution $q=(q_0,\ldots,q_{d-1})$. The notion of mutual unbiasedness, however, is preserved by requiring that the extracted information is the same for any preparation and any measurement; regardless of which state from which basis is chosen to prepare the system, the outcomes of measuring the system with respect to the other basis generate the same probability distribution. In light of this, we define the notion of \emph{mutually equibiased bases} (MEBs) such that within each basis the states are equibiased with respect to the states of the other basis and that the bases are mutually equibiased with respect to each other. For $d=2,3$, we derive a set of $d+1$ MEBs. The mutual equibiasedness imposes nontrivial constraints on the distribution $q$, leading for $d=3$ to the restriction $1/3\leμ\le 1/2$ where $μ=\sum_{k=0}^{2}q_k^2$. To capture the incompatibility of the measurements in MEBs, we derive an inequality for the probabilities of projective measurements in a qudit system, which yields an associated entropic uncertainty inequality. Finally, we construct a class of positive maps and their associated entanglement witnesses based on MEBs. While an entanglement witness constructed from MUBs is generally finer than one based on MEBs when both use the same number of bases, for certain values of the index $μ$, employing a larger set of MEBs can yield a finer witness. We illustrate this behavior using isotropic states of a $3\times 3$ system. Our results reveal that not all bases in a set of $L$ MEBs can contribute to the entanglement detection. A constraint, dependent on the probability ...

Mutually equibiased bases

TL;DR

The paper introduces mutually equibiased bases (MEBs) as a relaxed generalization of mutually unbiased bases (MUBs), where inter-basis measurement statistics are fixed by a distribution q up to permutation. It develops the mathematical framework using unistochastic matrices, a q parameterization via an anti-diagonal N, and explicit constructions for d=2 and d=3, including a one-parameter family of four MEBs in dimension 3 with μ in [1/3,1/2]. It then derives an entropic uncertainty relation for MEBs and constructs positive maps and entanglement witnesses based on MEBs, illustrating entanglement detection in isotropic states and exposing a tradeoff between the number of MEBs and detection strength depending on μ. The results demonstrate nontrivial constraints on q and reveal that not all bases in an MEB set contribute equally to entanglement detection, offering new insights into measurement incompatibility and quantum state discrimination in higher dimensions.

Abstract

In the framework of mutually unbiased bases (MUBs), a measurement in one basis gives \emph{no information} about the outcomes of measurements in another basis. Here, we relax the no-information condition by allowing the outcomes to be predicted according to a predefined probability distribution . The notion of mutual unbiasedness, however, is preserved by requiring that the extracted information is the same for any preparation and any measurement; regardless of which state from which basis is chosen to prepare the system, the outcomes of measuring the system with respect to the other basis generate the same probability distribution. In light of this, we define the notion of \emph{mutually equibiased bases} (MEBs) such that within each basis the states are equibiased with respect to the states of the other basis and that the bases are mutually equibiased with respect to each other. For , we derive a set of MEBs. The mutual equibiasedness imposes nontrivial constraints on the distribution , leading for to the restriction where . To capture the incompatibility of the measurements in MEBs, we derive an inequality for the probabilities of projective measurements in a qudit system, which yields an associated entropic uncertainty inequality. Finally, we construct a class of positive maps and their associated entanglement witnesses based on MEBs. While an entanglement witness constructed from MUBs is generally finer than one based on MEBs when both use the same number of bases, for certain values of the index , employing a larger set of MEBs can yield a finer witness. We illustrate this behavior using isotropic states of a system. Our results reveal that not all bases in a set of MEBs can contribute to the entanglement detection. A constraint, dependent on the probability ...

Paper Structure

This paper contains 14 sections, 5 theorems, 70 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Mutually equibiased bases (MEBs)
  3. Preliminaries
  4. Bistochastic and unistochastic matrices
  5. Parametrization of the distribution $q=(q_0,\ldots,q_{d-1})$
  6. Mutually equibiased bases for $d=2$
  7. Mutually equibiased bases for $d=3$
  8. Example: $d=3$ and $\mu=1/2$
  9. Entropic uncertainty relation for MEBs
  10. Inequality for the probabilities of projective measurements in MEBs of a qudit system
  11. Entanglement detection
  12. Conclusion
  13. Proof of Proposition \ref{['Proposition-d3']}
  14. Derivation of the eigenvalues of $G$

Key Result

Proposition 2

Starting from the standard basis $\{|e_{k}^{(1)}\rangle\}_{k=0}^{2}$, the unitary matrices $\mathcal{U}$, $\mathcal{V}$, and $\mathcal{W}$ generate a one-parameter class of four MEBs. Here $\mathcal{V}=\mathcal{D}_{\alpha}\mathcal{U}$ and $\mathcal{W}=\mathcal{D}_{\beta}\mathcal{U}$, where $\mathcal where for $\mu\in[1/3,1/2]$. Furthermore, the probabilities $(q_0,q_1,q_2)$ are different permutat

Figures (5)

  • Figure 1: Bloch vector representation for MEB, when $d=2$. Geometrically, equibiasedness means equal angle, $\vartheta=\cos^{-1}[\delta]$ between Bloch vectors of different bases.
  • Figure 2: Geometrical representation of the probability distribution $q=(q_0,q_1,q_2)$, for which a set of four MEBs exist. The figure is plotted in the $\delta_2\delta_1$ plane, where independent parameters $\delta_1$ and $\delta_2$ are defined in Eq. \ref{['delta1delta2']}. The curves represent the allowed values of the distribution $q=(q^\mu_{+}, q^\mu_0, q^\mu_{-})$, or any permutation of its probabilities, when $\alpha_1=0$ and $\alpha_2=\cos^{-1}[-\mu/(1-\mu)]$. The circles correspond to regions with constant index $\mu$. Corresponding to each $\mu$, only six points, i.e., the distribution $q$ and its permutations, are allowed for a given $\mu$.
  • Figure 3: Plot of $(L\gamma)^{-1}$ in terms of index $\mu$ for $d=3$ and different values of $L=2,3,4$. For equal $L$'s, the MUB-based witness (dashed lines) always captures a wider range of states relative to the MEB-based witness (solid curves). However, if a MEB-based witness employs a larger set of measurements, it can surpass the MUB-based witness in detecting entanglement, particularly within low values of the index $\mu$. As discussed in the text, for $L=4$, the plot is restricted to $\mu\in[1/3,11/27)$, due to the imposed limit on $\mu$.
  • Figure 4: Contour plot of the left-hand side of Eq. \ref{['AlphaCondition']} in $\alpha_1\alpha_2$ plane for $\alpha_1,\alpha_2\in(-\pi,+\pi)$. (a) The inner boundary (green ellipse) corresponds to $\mu=1/3$, (b) the outer boundary (red hexagon) corresponds to $\mu=1/2$, and (c) two little pink contours are plotted for $\mu\approx5/9$. The exact value for $\mu=5/9$ correspond to two points at $(2\pi/3,-2\pi/3)$ and $(-2\pi/3,2\pi/3)$. The $\alpha_1\alpha_2$ plane is divided by dashed grid lines to make it easier to identify the quadrant of the angles $\alpha_1$ and $\alpha_2$.
  • Figure 5: Geometric representation of the probability distribution $q=(q_0,q_1,q_2)$ and the construction of the feasible region. (a) Equilateral triangle associated with the probability distribution $q=(q_0,q_1,q_2)$, with its unistochastic subset. All plots are in the $\delta_2\delta_1$ plane. Each panel corresponds to a constraint applied in succession. See Remark \ref{['Remark-FeasibleRegion']} for details.

Theorems & Definitions (12)

  • Definition 1
  • Proposition 2
  • Lemma 3
  • proof
  • Theorem 4
  • proof
  • Theorem 5
  • proof
  • Proposition 6
  • proof
  • ...and 2 more