5 research outputs found
On Hidden States in Quantum Random Walks
It was recently pointed out that identifiability of quantum random walks and
hidden Markov processes underlie the same principles. This analogy immediately
raises questions on the existence of hidden states also in quantum random walks
and their relationship with earlier debates on hidden states in quantum
mechanics. The overarching insight was that not only hidden Markov processes,
but also quantum random walks are finitary processes. Since finitary processes
enjoy nice asymptotic properties, this also encourages to further investigate
the asymptotic properties of quantum random walks. Here, answers to all these
questions are given. Quantum random walks, hidden Markov processes and finitary
processes are put into a unifying model context. In this context, quantum
random walks are seen to not only enjoy nice ergodic properties in general, but
also intuitive quantum-style asymptotic properties. It is also pointed out how
hidden states arising from our framework relate to hidden states in earlier,
prominent treatments on topics such as the EPR paradoxon or Bell's
inequalities.Comment: 26 page
Generic identification of binary-valued hidden Markov processes
The generic identification problem is to decide whether a stochastic process
is a hidden Markov process and if yes to infer its parameters for all
but a subset of parametrizations that form a lower-dimensional subvariety in
parameter space. Partial answers so far available depend on extra assumptions
on the processes, which are usually centered around stationarity. Here we
present a general solution for binary-valued hidden Markov processes. Our
approach is rooted in algebraic statistics hence it is geometric in nature. We
find that the algebraic varieties associated with the probability distributions
of binary-valued hidden Markov processes are zero sets of determinantal
equations which draws a connection to well-studied objects from algebra. As a
consequence, our solution allows for algorithmic implementation based on
elementary (linear) algebraic routines.Comment: 28 page
Efficient tests for equivalence of hidden Markov processes and quantum random walks
While two hidden Markov process (HMP) resp.~quantum random walk (QRW)
parametrizations can differ from one another, the stochastic processes
arising from the