Mutually equi-biased bases

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 $q=(q_0,\cdots,q_{d-1})$. 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 the light of this, we define the notion of \emph{mutually equi-biased bases} (MEBs) such that within each basis the states are equi-biased with respect to the states of the other basis and that the bases are mutually equi-biased with respect to each other. For $d=2,3$, we derive a complete set of $d+1$ MEBs. The mutual equi-biasedness 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.