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 equipped with the usual Hermitian inproduct . An observable is a choice of orthonormal basis consisting of eigenvectors of the self-adjoint matrix . together with another observable (with orthonormal basis ) are said to be mutally unbiased if the norms of all inproducts are equal to . 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 Problem¹ 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 , consists in taking a system of k object qupits (when 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 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 , identify the elements of with the numbers from 1 to d in some fixed way. Then, the of number-strings are found as follows : let 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 (with ) determined from by the equation

where is the field-element corresponding to the integer i ( corresponds to the zero element). It is easy to see that these strings satisfy the conditions of the combinatorial problem. Indeed, any two of its digits determine (and hence the whole string) as it follows from and that .

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

1. actually a misnomer, it’s more the poor physicists’ problem… [↩]

Via the n-category cafe (and just now also the Arcadian functor ) I learned that Benjamin Mann of DARPA has constructed a list of 23 challenges for mathematics for this century.

DARPA is the “Defense Advanced Research Projects Agency” and is an agency of the United States Department of Defense ‘responsible for the development of new technology for use by the military’.

Bejamin Mann is someone in their subdivision DSO, that is, the “Defense Sciences Office” that ‘vigorously pursues the most promising technologies within a broad spectrum of the science and engineering research communities and develops those technologies into important, radically new military capabilities’.

I’m not the greatest fan of the US military, but the proposed list of 23 mathematical challenges is actually quite original and interesting.

What follows is my personal selection of what I consider the top 5 challenges from the list (please disagree) :

1. The Mathematics of Quantum Computing, Algorithms, and Entanglement (DARPA 15) : “In the last century we learned how quantum phenomena shape our world. In the coming century we need to develop the mathematics required to control the quantum world.”

2. Settle the Riemann Hypothesis (DARPA 19) : “The Holy Grail of number theory.”

3. Geometric Langlands and Quantum Physics (DARPA 17) : “How does the Langlands program, which originated in number theory and representation theory, explain the fundamental symmetries of physics? And vice versa?”

4. The Geometry of Genome Space (DARPA 15) : “What notion of distance is needed to incorporate biological utility?”

5. Algorithmic Origami and Biology (DARPA 10) : “Build a stronger mathematical theory for isometric and rigid embedding that can give insight into protein folding.”

All of this will have to wait a bit, for now

HAPPY & HEALTHY 2008