One can obtain a Hilbert space from a wide variety of physical situations. Suppose we are to use a Hilbert space to model elementary particles, for instance. One the one hand, we’d like to have our model cover as many particles as possible. But to do this, we may have to assume a symmetry that is not exactly correct. For example, to get a model that includes isospin, we have a natural inclination to assume that isospin is a perfect symmetry when, in reality, it cannot be. Assuming perfect symmetry as an approximation leaves us with the need to later correct for this by including a symmetry breaking.
Recently, we’ve been discussing using Mutual Unbiased Bases (MUBs) to model the elementary particles. If we want to include very disparate objects as states in different MUB bases, we need to also include a symmetry breaking mechanism.
When elementary particle theory is built from symmetry alone, one finds that there are a nearly infinite number of ways of breaking symmetry. The freedom to do this makes it difficult to find our way. In the case of MUBs, however, we can write down some very natural ways to break symmetry if we treat the Hilbert space as the vector space of a Clifford algebra like the Dirac algebra. In this post we will show how the natural symmetry breaking in the Dirac algebra can lead to structures that are reminiscent of the structure of the elementary particles.
Mutually Unbiased Bases are sets of bases for a Hilbert space that are “unbiased:” the transition probabilities between any two states from different bases are equal. For a Hilbert space of dimension 3 (i.e. qutrits), the transition probability is 1/3. The operator space of a Hilbert space of dimension n is n^2, in this case the operator space has 9 dimensions. Each base consists of 3 quantum states. It turns out that a base uses up 3-1 = 2 degrees of freedom of the operator space, and the scalar part of the operator space is shared by all. So for a 3-dimensional Hilbert space, there are at most four mutually unbiased bases. In this post I will derive a set of four such bases.
I’ve been watching Perimeter Institute lectures again, and one that dips into the stuff I’m working came up. In addition to the lecture, there is an acrobat file that contains the slides and photos of the blackboard. This is 124 pages long and it is these pages that I will reference.
Applications of the generalized Pauli group in quantum information
Speaker(s): Thomas Durt – Vrije Universiteit Brussel
Abstract: It is known that finite fields with d elements exist only when d is a prime or a prime power. When the dimension d of a finite dimensional Hilbert space is a prime power, we can associate to each basis state of the Hilbert space an element of a finite or Galois field, and construct a finite group of unitary transformations, the generalised Pauli group or discrete Heisenberg-Weyl group. Its elements can be expressed, in terms of the elements of a Galois field. This group presents numerous applications in Quantum Information Science e.g. tomography, dense coding, teleportation, error correction and so on. The aim of our talk is to give a general survey of these properties and to present recently obtained results in connection with three problems: -the so-called ”Mean King’s problem” in prime power dimension, -discrete Wigner distributions, -and quantum tomography . Finally we shall discuss a limitation of the possible dimensions in which the so-called epistemic interpretation can be consistently formulated, in relation with the existence of finite affine planes, Euler’s conjecture and the 36 officers problem.
Date: 10/10/2007 – 4:00 pm
This has been discussed a little over at Kea’s blog and at Matti’s blog as well, and I figure I should write something up here as well, since I am also applying “Mutually Unbiased Bases” (MUBs); what I call snuarks are the Pauli algebra example of a set of MUBs.
I’ve added an explicit calculation in density matrix formalism for a maximal MUB for the Dirac algebra, and I’ve converted the solution into spinor form. This is a wonderful illustration of how much more powerful the density matrix formalism is than the spinor formalism.
I’m continuing to watch the Perimeter Institute lectures and watching the latest one, Dynamic Time: The “Missing Link” in the Search for a Unified Theory? by Avshalom Elitzur, has influenced me to type up my version of the ontology of quantum mechanics.
As Elitzur states, the central conundrum is to ask the question, “what is the past, the present and the future”. The intuitive answer to this is that only the present is real. The future is just our expectations, and the past is just our memory. The standard physics answer to the question is that physics must be written as a theory on space time. The past and future exist just as surely as left and right, or up and down, and the notion of “now” is either a figment of our imagination, or an arbitrary choice of the time coordinate origin.
My view on this is in the middle. I believe that the past and future do exist, but that the present should be treated as a preferred time. In this, I part ways with relativity which denies preferred things. And analogously, I part ways with gauge theory, which denies a preferred gauge. As a matter of philosophy, I believe in the existence of a unique (i.e. exactly 1) world which must therefore have a unique representation in mathematics. If we are unable to distinguish between two different representations, it is only because of the inadequacy of our understanding and not due to the world being several different things at once.
These ideas are difficult to explain because the word “time” is overloaded in our language. To get the point across, let us consider the 2-slit experiment with a single quantum particle, and examine the target region from two different perspectives. This experiment is simpler than the “2-hole experiment” because the symmetry in the direction of the slit reduces one of the dimensions. Instead of three dimensions of space and one time dimension, the 2-slit experiment has only two space dimensions. Furthermore, we concentrate our attention on the target (which is the only part of the experiment which we can “measure” anyway). This reduces the problem to one spatial dimension. We look at only a single time slice, so our pictures are only 1+0 dimensional.
The wave function shows interference. It might look like this:
I’ve been listening to Perimeter Institute lectures and came upon one by Roderich Tumulka that was interesting enough that I looked up his papers. They are voluminous. One that applies more or less directly to the area I play in is this one: On the Role of Density Matrices in Bohmian Mechanics quant-ph/0311127, by Detlef Dürr, Sheldon Goldstein, Roderich Tumulka, and Nino Zanghì and published as Foundations of Physics 35 (2005) 449-467.
A density matrix is relevant in yet another way: in a modified version of Bohmian mechanics in which the particles are guided not by a wave function but by a density matrix. Let us call this W-Bohmian mechanics. Whereas in the conventional version of Bohmian mechanics the wave function (of the universe) is something real, as an objective component of the state of the universe at a given time, in W-Bohmian mechanics instead of a wave function (of the universe) we may have only a density matrix. This density matrix does not arise in any way from an analysis of the theory, but is built into the fundamental postulates of W-Bohmian mechanics. It is a fundamental density matrix, , in contrast to the four other density matrices we have discussed, which were derived objects, derived from and Q.
This is one of only a very few references I’ve seen in the literature to the possibility that the density matrix is fundamental. Since this is one of my assumptions in my view of elementary particles, it is gratifying to see it in print.
They approach the subject very thinly. One can find how one defines a density matrix from a state vector in most quantum mechanics text books, but as far as I know, it is only in my book on density matrices that one can find a demonstration of how one defines a state vector from the density matrix. As part of my NYR to be more professional about physics, I’m rewriting the book to remove the inside jokes, make the cover art actually look like a physics book instead of a heavy equipment operation guide, etc. So if, like some of my friends, you think a physics book with a picture of a ditch witch on the front is cool, you’d better order a copy soon because it is not going to be available for very much longer (and I’m not going to buy you one).
The new book will be in paperback format 8.5″ x 11″ (Misner Thorne & Wheeler inspired my layout for both this and the paperback), and will be cheaper at around $15 to $20.