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Izvestiya Mathematics

On translation-covariant quantum Markov equations

Alexander Semenovich Holevo

Published Apr 30, 1995• 26 citations
Publisher

Abstract

The structure of quantum Markov control equations with unbounded generators and covariant with respect to 1) irreducible representation of the Weyl CCR on and 2) representation of the group of , is completely described via non-commutative Levy-Khinchin-type formulae. The existence and uniqueness of solutions for such equations is briefly discussed.

DOI
10.1070/im1995v059n02abeh000019

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The structure of quantum Markov control equations with unbounded generators and covariant with respect to 1) irreducible representation of the Weyl CCR on and 2) representation of the group of , is completely described via non-commutative Levy-Khinchin-type formulae. The existence and uniqueness of solutions for such equations is briefly discussed.
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In this thesis I analyze the structure of decoherence-free subalgebra N(T) of a uniformly continuous covariant Quantum Markov Semigroup (QMS) with respect to a representation ? of a compact group G on a Hilbert space h. In particular, I obtain that, when ? is irreducible, N(T) is isomorphic to (?(k) ? 1m)^d for suitable Hilbert spaces k and m, and an integer d related to the connected components of G. I extend this result when ? is reducible and N(T) is atomic by the decomposition of h due to the Peter?Weyl theorem. The thesis is then concluded by exploring the possibility of extending these results beyond the direct sum by using the concept of direct integral.
Every Markov-regular quantum Levy process on a multiplier C*-bialgebra is shown to be equivalent to one governed by a quantum stochastic differential equation, and the generating functionals of norm-continuous convolution semigroups on a multiplier C*-bialgebra are then completely characterised. These results are achieved by extending the theory of quantum Levy processes on a compact quantum group, and more generally quantum stochastic convolution cocycles on a C*-bialgebra, to locally compact quantum groups and multiplier C*-bialgebras. Strict extension results obtained by Kustermans, together with automatic strictness properties developed here, are exploited to obtain existence and uniqueness for coalgebraic quantum stochastic differential equations in this setting. Then, working in the universal enveloping von Neumann bialgebra, we characterise the stochastic generators of Markov-regular, *-homomorphic (respectively completely positive and contractive), quantum stochastic convolution cocycles.
A classical theorem of Stone and von Neumann says that the Schrödinger representation is, up to unitary equivalences, the only irreducible representation of the Heisenberg group on the Hilbert space of square-integrable functions on configuration space. Using the Wigner-Moyal transform we construct an irreducible representation of the Heisenberg group on a certain Hilbert space of square-integrable functions defined on phase space. This allows us to extend the usual Weyl calculus into a phase-space calculus and leads us to a quantum mechanics in phase space, equivalent to standard quantum mechanics. We also briefly discuss the extension of metaplectic operators to phase space and the probabilistic interpretation of the solutions of the phase space Schrödinger equation
Let [Formula: see text] be a quantum Markov semigroup on [Formula: see text] with a faithful normal invariant state ρ. The decoherence-free subalgebra [Formula: see text] of [Formula: see text] is the biggest subalgebra of [Formula: see text] where the completely positive maps [Formula: see text] act as homomorphisms. When [Formula: see text] is the minimal semigroup whose generator is represented in a generalised GKSL form [Formula: see text], with possibly unbounded H, Lℓ, we show that [Formula: see text] coincides with the generalised commutator of [Formula: see text] under some natural regularity conditions. As a corollary we derive simple sufficient algebraic conditions for convergence towards a steady state based on multiple commutators of H and Lℓ. We give examples of quantum Markov semigroups [Formula: see text], with h infinite-dimensional, having a non-trivial decoherence-free subalgebra.
Quantum Markovian master equations with generally unbounded generators, having physically relevant symmetries, such as Weyl, Galilean or boost covariance, are characterized. It is proven in particular that a fully Galilean covariant zero spin Markovian evolution reduces to the free motion perturbed by a covariant stochastic process with independent stationary increments in the classical phase space. A general form of the boost covariant Markovian master equation is discussed and a formal dilation to the Langevin equation driven by quantum Boson noises is described.