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

<ol>
<li><p><a href="http://arxiv.org/abs/quant-ph/0608014" rel="nofollow">Here [1]</a> the pure-state discrimination of <strong>MUBs</strong> is studied in a cryptographic set-up. The <strong>optimality</strong> of different  measurement schemes is rigorously discussed. It also includes a good bunch of useful references about distinguishability of pure quantum states.</p></li>
<li><p>For particular choices of pure-state ensembles the <strong>Pretty Good Measurement</strong>  is proven to be optimal in this task. <a href="http://authors.library.caltech.edu/3842/1/MOCpra06.pdf" rel="nofollow">This [2]</a> is a nice exposition on this topic, although not focused on MUBs.</p></li>
</ol>

<p>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:</p>

<ul>
<li>The choice of quantum states to distinguish (in this case the choice of MUBs). This issue is <a href="http://arxiv.org/abs/quant-ph/0504083" rel="nofollow">important [3]</a> to find efficient implementations of optimal POVMs.</li>
<li>The particular probabilities $p_{i,j}$ of receiving the $i$th state of the $j$th basis $B_k$ as an input <a href="http://arxiv.org/abs/quant-ph/0608014" rel="nofollow">[1]</a><a href="http://authors.library.caltech.edu/3842/1/MOCpra06.pdf" rel="nofollow">[2]</a> (in your notation).</li>
</ul>

<p>Also, in cryptographic applications the next two seem to be relevant <a href="http://arxiv.org/abs/quant-ph/0608014" rel="nofollow">[1]</a>:</p>

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

<p>Hope it helps.</p>

CC BY-SA 3.0
2012-02-05T14:47:37.877Z
2012-02-05T17:50:58.910Z
Juan Bermejo Vega
<p>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?</p>

CC BY-SA 3.0
2012-01-31T14:27:07.580Z
@MarkS.Everitt IMHO we should refrain from forcing to accept answers, see a newly opened thread: http://meta.theoreticalphysics.stackexchange.com/questions/230/accepting-answers-and-accept-rates.
Piotr Migdal
CC BY-SA 3.0
2012-02-03T03:27:57.597Z
You may want to have a look at http://arxiv.org/abs/0907.3704 and references therein.
Marco
<reference-request><quantum-information>
Optimal measurement for MUBs
Marcin Kotowski
# Optimal measurement for MUBs

(Asked by: username_0 on 2012-01-31)

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?

**Tags:** reference-request, quantum-information

> **Comments:**
> **username_1**: @MarkS.Everitt IMHO we should refrain from forcing to accept answers, see a newly opened thread: http://meta.theoreticalphysics.stackexchange.com/questions/230/accepting-answers-and-accept-rates.  
> **username_2**: You may want to have a look at http://arxiv.org/abs/0907.3704 and references therein.  

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## 1 answers

### Answer by username_3 on 2012-02-05. Score: 4

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

1. [Here [1]](http://arxiv.org/abs/quant-ph/0608014) 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]](http://authors.library.caltech.edu/3842/1/MOCpra06.pdf) 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]](http://arxiv.org/abs/quant-ph/0504083) 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]](http://arxiv.org/abs/quant-ph/0608014)[[2]](http://authors.library.caltech.edu/3842/1/MOCpra06.pdf) (in your notation).

Also, in cryptographic applications the next two seem to be relevant [[1]](http://arxiv.org/abs/quant-ph/0608014):

* 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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