Quantcast
  • Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.

News

Please welcome our new moderators!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback

Attributions

(propose a free ad)

Site Statistics

122 submissions , 103 unreviewed
3,497 questions , 1,172 unanswered
4,544 answers , 19,342 comments
1,470 users with positive rep
408 active unimported users
More ...

Optimal measurement for MUBs

+ 9 like - 0 dislike
15 views

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?

This post has been migrated from (A51.SE)
asked Jan 31, 2012 in Theoretical Physics by Marcin Kotowski (405 points) [ no revision ]
retagged Mar 18, 2014 by dimension10
You may want to have a look at http://arxiv.org/abs/0907.3704 and references therein.

This post has been migrated from (A51.SE)
@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.

This post has been migrated from (A51.SE)
At the moment, beneath your name and badge, it says "0% accept rate" in red. This is likely to deter people who might otherwise be inclined to answer your question. Please consider the other questions that you have asked, and whether any of the answers to them may be accepted.

This post has been migrated from (A51.SE)

1 Answer

+ 3 like - 0 dislike

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.

This post has been migrated from (A51.SE)
answered Feb 5, 2012 by jbvega (285 points) [ no revision ]

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
p$\hbar$ysicsOverfl$\varnothing$w
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
To avoid this verification in future, please log in or register.




user contributions licensed under cc by-sa 3.0 with attribution required

Your rights
...