33 research outputs found
Non-commutative Markov chains and multi-analytic operators
We study a model of repeated interaction between quantum systems which can be
thought of as a non-commutative Markov chain. It is shown that there exists an
outgoing Cuntz scattering system associated to this model which induces an
input-output formalism with a transfer function corresponding to a
multi-analytic operator, in the sense of multivariate operator theory. Finally
we show that observability for this system is closely related to the scattering
theory of non-commutative Markov chains.Comment: 19 page
Weak Markov Processes as Linear Systems
A noncommutative Fornasini-Marchesini system (a multi-variable version of a
linear system) can be realized within a weak Markov process (a model for
quantum evolution). For a discrete time parameter the resulting structure is
worked out systematically and some quantum mechanical interpretations are
given. We introduce subprocesses and quotient processes and then the notion of
a -extension for processes which leads to a complete classification of
all the ways in which processes can be built from subprocesses and quotient
processes. We show that within a -extension we have a cascade of
noncommutative Fornasini-Marchesini systems. We study observability in this
setting and as an application we gain new insights into stationary Markov
chains where observability for the system is closely related to asymptotic
completeness in a scattering theory for the chain.Comment: Expanded version v2 (43 pages) with substantial additions and
improvements compared to v1. More details and examples, in particular in
sections 3, 4 and 7. Also changes in terminology, compare Def. 3.1, 4.2, 6.4,
page 33. To appear in the journal: Mathematics of Control, Signals, and
Systems (MCSS
Characteristic Functions for Ergodic Tuples
Motivated by a result on weak Markov dilations, we define a notion of
characteristic function for ergodic and coisometric row contractions with a
one-dimensional invariant subspace for the adjoints. This extends a definition
given by G. Popescu. We prove that our characteristic function is a complete
unitary invariant for such tuples and show how it can be computed.Comment: 22 pages, changes made after referee's comments, to appear in
Integral Equations and Operator Theor
Noncommutative Independence from the Braid Group B∞
We introduce `braidability' as a new symmetry for (infinite) sequences of
noncommutative random variables related to representations of the braid group
. It provides an extension of exchangeability which is tied to the
symmetric group . Our key result is that braidability implies
spreadability and thus conditional independence, according to the
noncommutative extended de Finetti theorem (of C. K\"{o}stler). This endows the
braid groups with a new intrinsic (quantum) probabilistic interpretation.
We underline this interpretation by a braided extension of the Hewitt-Savage
Zero-One Law.
Furthermore we use the concept of product representations of endomorphisms
(of R. Gohm) with respect to certain Galois type towers of fixed point algebras
to show that braidability produces triangular towers of commuting squares and
noncommutative Bernoulli shifts. As a specific case we study the left regular
representation of and the irreducible subfactor with infinite Jones
index in the non-hyperfinite -factor related to it. Our
investigations reveal a new presentation of the braid group , the
`square root of free generator presentation' . These new
generators give rise to braidability while the squares of them yield a free
family. Hence our results provide another facet of the strong connection
between subfactors and free probability theory and we speculate about
braidability as an extension of (amalgamated) freeness on the combinatorial
level.Comment: minor changes, added 3.3-3.6, version to be published in
Comm.Math.Phys. (47 pages