Krawtchouk matrices from classical and quantum random walks

Abstract

Krawtchouk's polynomials occur classically as orthogonal polynomials with respect to the binomial distribution. They may be also expressed in the form of matrices, that emerge as arrays of the values that the polynomials take. The algebraic properties of these matrices provide a very interesting and accessible example in the approach to probability theory known as quantum probability. First it is noted how the Krawtchouk matrices are connected to the classical symmetric Bernoulli random walk. And we show how to derive Krawtchouk matrices in the quantum probability context via tensor powers of the elementary Hadamard matrix. Then connections with the classical situation are shown by calculating expectation values in the quantum case.

Recent Advances in Applied Probability

“…For more properties we refer the reader to [Feinsilver, 2001]. For more properties we refer the reader to [Feinsilver, 2001].…”

“…Background note. For some further development of this idea, see [Feinsilver, 2001]. The idea of setting them in a matrix form appeared in the 1985 work of N. Bose [Bose, 1985] on digital filtering in the context of the Cayley transform on the complex plane.…”

“…Krawtchouk matrices interpreted as operators give rise to two new interpretations in the context of both classical and quantum random walks [Feinsilver, 2001]. Krawtchouk matrices interpreted as operators give rise to two new interpretations in the context of both classical and quantum random walks [Feinsilver, 2001].…”

“…Now we proceed to the interpretation leading to a symmetric Bernoulli quantum random walk ( [Feinsilver, 2001]). For this interpretation, the Hilbert space of states is represented by the tensor power of the original 2-dimensional space V, that is, by the -dimensional Hilbert space .…”

Bilinear Forms and Fierz Identities for Real Spin Representations

“…From manipulations of a rank four spinorial tensor introduced in [1], we are able to find a general class of identities which, upon specializing from four spinors to two spinors and one spinor in signatures (1,3) and (10,1), yield some well-known Fierz identities. We will see, surprisingly, that the identities we construct are partly encoded in certain involutory real matrices that resemble the Krawtchouk matrices [4] [5]. …”