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Let $\mathcal{B} = \{B_1, \dots, B_k\}$ be a set of Mutually Unbiased Bases (MUB) in $\mathbb{C}^n$, i.e. each $B_i$ is an orthonormal basis and for $v \in B_i, w \in B_j, i \neq j $ we have $|\langle v\vert w\rangle| = \frac{1}{\sqrt{n}}$. We are interested in discriminating between arbitrary vectors from $\mathcal{B}$. Is the optimal (worst case or average with uniform prior) POVM measurement identified explicitly anywhere in the literature (e.g. using Holevo criterion), at least for some specific constructions of MUBs?

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It seems that this problem in full-generality is though. These two references might be helpful to you.

  1. Here [1] the pure-state discrimination of MUBs is studied in a cryptographic set-up. The optimality of different measurement schemes is rigorously discussed. It also includes a good bunch of useful references about distinguishability of pure quantum states.

  2. For particular choices of pure-state ensembles the Pretty Good Measurement is proven to be optimal in this task. This [2] is a nice exposition on this topic, although not focused on MUBs.

If you are interested in more restricted scenarios that the ones considered above, regard that there are some factors that influence the complexity of this problem. The following two are considered in several references:

  • The choice of quantum states to distinguish (in this case the choice of MUBs). This issue is important [3] to find efficient implementations of optimal POVMs.
  • The particular probabilities $p_{i,j}$ of receiving the $i$th state of the $j$th basis $B_k$ as an input [1][2] (in your notation).

Also, in cryptographic applications the next two seem to be relevant [1]:

  • If you are using this states to encode some information the particular functions used to encode and decode this information.
  • Other: ability to store qubits between measurements, some given knowledge about the bases used.

Hope it helps.

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