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
- actually a misnomer, it’s more the poor physicists’ problem… [↩]
arxiv, blackjack, entanglement, finite fields, games, mac, Manin, Mathieu, MUBs, puzzle, quantum, qubits, states, tetracode, wordpress
4 comments
Posted in games, geometry
Written on Thu, 28 February 2008 at 3:46 pm
Tags: arxiv, blackjack, entanglement, finite fields, games, mac, Manin, Mathieu, MUBs, puzzle, quantum, qubits, states, tetracode, wordpress
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February 28th, 2008 at 4:19 pm
[...] TrueGameHeadz BlogHeadz wrote an interesting post today onHere’s a quick excerpt 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 [...]
February 29th, 2008 at 12:58 am
Cool!!! And thanks. Note that we are associating the d=3 case with mass operators, from which one may derive actual rest masses. This is extremely controversial, physically speaking, because all good physicists ‘know’ that masses run in QFT renormalisation, and they don’t necessarily believe there is anything fundamental about the rest mass values. This applies also (sigh) to most physicists working in ‘quantum gravity’. My recent interest in codes stems from the urgent need to derive more than the 3 charged lepton masses, or the unknown neutrino masses, since the only truly convincing physical test is a set of numbers far beyond ‘coincidence’.
February 29th, 2008 at 1:39 am
Thanks for the link; I should make an interesting comment in return.
One of the mysteries of spin-1/2 quantum mechanics is why two eigenstates are sufficient to describe a state. Everyone knows how the math works, but physically, it is difficult to understand especially for those who wish their physics to have a classical interpretation.
In elementary particles, the electron is most naturally described as a combination of left and right handed particles. To get a left / right handed electron, you accelerate a regular electron in the direction / against the direction of its spin vector. How much do you accelerate it? To infinity, but not beyond. You have to get the electron to speed c. The left handed electron interacts weakly; the right handed one does not.
If one considers waves in 3 dimensions, say sound waves or classical light waves, to obtain sufficient degrees of freedom to describe an arbitrary direction wave requires that one begin with three degrees of freedom, say three waves travelling in perpendicular directions. This is one more than the quantum spin 1/2 case even though the electron is apparently built from components that do travel at speed c, and therefore should take 3 degrees of freedom, or maybe 6 (so that the waves can go either way, or you can account for position and momentum rather than just position), but certainly not 2.
When you replace the usual spin-1/2 basis states (spin up and spin down) with a Pauli MUB of 3 basis states, you end up with the classically expected 3 / 6 degrees of freedom. So MUBs are tied up with a more classical way of looking at quantum mechanics. From a preon point of view, this says that classical mechanics may be hiding underneath a quantum facade.
I guess I should admit that I’ve temporarily given up on the d=3 MUB case and am working on d=2 cases. Kea, I should have a post out soon with a new preon model that uses this idea.
February 29th, 2008 at 6:41 pm
[quote comment="5040"]
Thanks for the link; I should make an interesting comment in return.
[/quote]Carl, if only all people would react this way, I’d only write link-posts from now on…