CLASSICAL MARKOV CHAINS AS SHADOWS OF QUANTUM STATE UNITARY DYNAMICS
A MATHEMATICAL ALLEGORY
Profº Eduardo Jordão Neves - IME
Dia: 15/03, 6ª feira, 14h. Café às 15h., na sala 265 A (Chefia do MAP).
Local: Instituto de Matemática e Estatística (IME-USP), Sala Gilioli.
Link para transmissão pelo IPTV:
https://iptv.usp.br/portal/transmission.action?idItem=40251
Abstract:
The standard approach to describe the dynamics of classical Markov chains is based on Kolmogorov’s equation. We present a different approach, playing around ideas and results from quantum mechanics and quantum computing. In a nutshell, we relate the original Markov chain dynamics, which preserves l1-norm and lives in a probability simplex, to a dynamical system on larger spaces, on which l2-norm is preserved - being orthogonal or unitary transformations. It brings about beautiful geometrical structures, like Hopf fibrations. We suggest a (mathematically-irrelevant) parallel between the approach and a classical Greek allegory, pointing out that the evolution of a N-state Markov chain are somehow like shadows in a Mathematical Plato’s Cave of some Unitary Dynamics described by the Lie algebra su(N). The mathematical setting starts representing real-valued probability vectors as Probability Amplitude vectors in the Hilbert space C N . We move deep into the realm of quantum mechanics as we relate the classical stochastic evolution to dynamics of a quantum spin systems. The strategy, standard in quantum mechanics, takes quantum spin dynamics into rotations on a unit-radius sphere in R N2−1 , the so called N-level Bloch sphere. Inside this sphere, within a (N − 1)-dimensional Euclidean subspace, we find the projected stochastic dynamics. The two-state Markov Chain is associated to the dynamics of a single quantum bit, or qubit, the elementary unit of quantum information in quantum computation. While the full su(N) may be required for a complicated N-state Markov chain, we may require much less. To illustrate, we present some examples. In particular, the asymmetric random walk in the discrete circle and the classical Ehrenfest Urn Model. For the N state Ehrenfest model, N ≥ 2, we show that su(2) is always enough, that is, the dynamics is ”just” a simple rotation in the 2-level Bloch sphere, the usual sphere S 2 ⊂ R 3 . To prove this, we need some less well known quantum mechanics tools, which allows the description of a N-state Markov chain dynamics through the collective movement of (N − 1) points - known as the Majorana’s stars - in the Bloch sphere S 2 .
