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the King’s problem on MUBs

Published February 28, 2008 by lievenlb

MUBs (for Mutually Unbiased Bases) are quite popular at the moment. Kea is running a mini-series Mutual Unbias as is Carl Brannen. Further, the Perimeter Institute has a good website for its seminars where they offer streaming video (I like their MacromediaFlash format giving video and slides/blackboard shots simultaneously, in distinct windows) including a talk on MUBs (as well as an old talk by Wootters).

So what are MUBs to mathematicians? Recall that a d-state quantum system is just the vectorspace $\mathbb{C}^d $ equipped with the usual Hermitian inproduct $\vec{v}.\vec{w} = \sum \overline{v_i} w_i $. An observable $E $ is a choice of orthonormal basis ${ \vec{e_i} } $ consisting of eigenvectors of the self-adjoint matrix $E $. $E $ together with another observable $F $ (with orthonormal basis ${ \vec{f_j} } $) are said to be mutally unbiased if the norms of all inproducts $\vec{f_j}.\vec{e_i} $ are equal to $1/\sqrt{d} $. This definition extends to a collection of pairwise mutually unbiased observables. In a d-state quantum system there can be at most d+1 mutually unbiased bases and such a collection of observables is then called a MUB of the system. Using properties of finite fields one has shown that MUBs exists whenever d is a prime-power. On the other hand, existence of a MUB for d=6 still seems to be open…

The King’s Problem (( actually a misnomer, it’s more the poor physicists’ problem… )) is the following : A physicist is trapped on an island ruled by a mean
king who promises to set her free if she can give him the answer to the following puzzle. The
physicist is asked to prepare a d−state quantum system in any state of her choosing and give it
to the king, who measures one of several mutually unbiased observables on it. Following this, the physicist is allowed to make a control measurement
on the system, as well as any other systems it may have been coupled to in the preparation
phase. The king then reveals which observable he measured and the physicist is required
to predict correctly all the eigenvalues he found.

The Solution to the King’s problem in prime power dimension by P. K. Aravind, say for $d=p^k $, consists in taking a system of k object qupits (when $p=2l+1 $ one qupit is a spin l particle) which she will give to the King together with k ancilla qupits that she retains in her possession. These 2k qupits are diligently entangled and prepared is a well chosen state. The final step in finding a suitable state is the solution to a pure combinatorial problem :

She must use the numbers 1 to d to form $d^2 $ ordered sets of d+1 numbers each, with repetitions of numbers within a set allowed, such that any two sets have exactly one identical number in the same place in both. Here’s an example of 16 such strings for d=4 :

11432, 12341, 13214, 14123, 21324, 22413, 23142, 24231, 31243, 32134, 33421, 34312, 41111, 42222, 43333, 44444

Here again, finite fields are used in the solution. When $d=p^k $, identify the elements of $\mathbb{F}_{p^k} $ with the numbers from 1 to d in some fixed way. Then, the $d^2 $ of number-strings are found as follows : let $k_0,k_1 \in \mathbb{F}_{p^k} $ and take as the first 2 numbers the ones corresponding to these field-elements. The remaning d-2 numbers in the string are those corresponding to the field element $k_m $ (with $2 \leq m \leq d $) determined from $k_0,k_1 $ by the equation

$k_m = l_{m} * k_0+k_1 $

where $l_i $ is the field-element corresponding to the integer i ($l_1 $ corresponds to the zero element). It is easy to see that these $d^2 $ strings satisfy the conditions of the combinatorial problem. Indeed, any two of its digits determine $k_0,k_1 $ (and hence the whole string) as it follows from
$k_m = l_m k_0 + k_1 $ and $k_r = l_r k_0 + k_1 $ that $k_0 = \frac{k_m-k_r}{l_m-l_r} $.

In the special case when d=3 (that is, one spin 1 particle is given to the King), we recover the tetracode : the nine codewords

0000, 0+++, 0—, +0+-, ++-0, +-0+, -0-+, -+0-, –+0

encode the strings (with +=1,-=2,0=3)

3333, 3111, 3222, 1312, 1123, 1231, 2321, 2132, 2213

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Writing & Blogging

Published February 27, 2008 by lievenlb

Terry Tao is reworking some of his better blogposts into a book, to be published by the AMS (here’s a preliminary version of the book “What’s New?”)

After some thought, I decided not to transcribe all of my posts from last year (there are 93 of them!), but instead to restrict attention to those articles which (a) have significant mathematical content, (b) are not announcements of material that will be published elsewhere, and (c) are not primarily based on a talk given by someone else. As it turns out, this still leaves about 33 articles from 2007, leading to a decent-sized book of a couple hundred pages in length.

If you have a blog and want to turn it into a LaTeX-book, there’s no need to transcribe or copy every single post, thanks to the WPTeX tool. Note that this is NOT a WP-plugin, but a (simple at that) php-program which turns all posts into a bookcontent.tex file. This file can then be edited further into a proper book.

Unfortunately, the present version chokes on LaTeXrender-code (which is easy enough to solve doing a global ‘find-and-replace’ of the tex-tags by dollar-signs) but worse, on Markdown-code… But then, someone fluent in php-regex will have no problems extending the libs/functions.php file (I hope…).

At the moment I’m considering turning the Mathieu-games-posts into a booklet. A possible title might be Mathieumatical Games. Rereading them (and other posts) I regret to be such an impatient blogger. Often I’m interested in something and start writing posts about it without knowing where or when I’ll land. This makes my posts a lot harder to get through than they might have been, if I would blog only after having digested the material myself… Typical recent examples are the tori-crypto-posts and the Bost-Connes algebra posts.

So, I still have a lot to learn from other bloggers I admire, such as Jennifer Ouellette who maintains the Coctail Party Physics blog. At the moment, Jennifer is resident blogger-journalist at the Kavli Institute where she is running a “Journal Club” workshop giving ideas on how to write better about science.

But the KITP is also committed to fostering scientific communication. That’s where I come in. Each Friday through April 26th, I’ll be presiding over a “Journal Club” meeting focusing on some aspect of communicating science.

Her most recent talk was entitled To Blog or Not to Blog? That is the Question and you can find the slides as well as a QuickTime movie of her talk. They even plan to set up a blog for the participants of the workshop. I will surely follow the rest of her course with keen interest!

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iTouch as network sniffer

Published February 22, 2008 by lievenlb

In the iTouch warwalking post I was considering trying to gain access to closed networks for innocent purposes such as checking mail, rather than stealing secret passwords from people allowing you free access to their wireless network, but still, I should have thought of the following possibility

Here’s a walk-through :

  • type the following command into your iTouch Terminal.app (assuming you’ve installed the BSD subsystem) :

tcpdump -v -s 65535 -w log.txt

  • once you’ve collected enough packets, cancel the command (ctrl c), AFPd the file from the iTouch to your Mac and open it with Wireshark (this is the most convenient way to install binaries under Leopard as well as an updated version of X11. For other platforms, or source code, see here)

  • do whatever black magic you feel you have to perform using Wireshark (the new name for Ethereal) or other password crackers

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