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arXiv:1811.11784 · Journal of Physics A Mathematical and Theoretical

Mathematical models of Markovian dephasing

Franco Fagnola, John Edward Gough, Hendra I Nurdin, Lorenza Viola

Published Aug 6, 2019• 10 citations
arXivHTMLPDF

Abstract

Abstract We develop a notion of dephasing under the action of a quantum Markov semigroup in terms of convergence of operators to a block-diagonal form determined by irreducible invariant subspaces. If the latter are all one-dimensional, we say the dephasing is maximal . With this definition, we show that a key necessary requirement on the Lindblad generator is bistochasticity, and focus on characterizing whether a maximally dephasing evolution may be described in terms of a unitary dilation with only classical noise, as opposed to a genuine non-commutative Hudson–Parthasarathy dilation. To this end, we make use of a seminal result of Kümmerer and Maassen on the class of commutative dilations of quantum Markov semigroups. In particular, we introduce an intrinsic quantity constructed from the generator, the Hamiltonian obstruction , which vanishes if and only if the latter admits a self-adjoint representation and quantifies the hindrance to having a classical diffusive noise model.

DOI
10.1088/1751-8121/ab38ec

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Abstract We develop a notion of dephasing under the action of a quantum Markov semigroup in terms of convergence of operators to a block-diagonal form determined by irreducible invariant subspaces. If the latter are all one-dimensional, we say the dephasing is maximal . With this definition, we show that a key necessary requirement on the Lindblad generator is bistochasticity, and focus on characterizing whether a maximally dephasing evolution may be described in terms of a unitary dilation with only classical noise, as opposed to a genuine non-commutative Hudson–Parthasarathy dilation. To this end, we make use of a seminal result of Kümmerer and Maassen on the class of commutative dilations of quantum Markov semigroups. In particular, we introduce an intrinsic quantity constructed from the generator, the Hamiltonian obstruction , which vanishes if and only if the latter admits a self-adjoint representation and quantifies the hindrance to having a classical diffusive noise model.
We present, for the first time, the result (from 2008) that (normal, strongly continuous) Markov semigroups on B(G) (G a separable Hilbert space) admit a Hudson-Parthasarathy dilation (that is, a dilation to a cocycle perturbation of a noise) if and only if the Markov semigroup is spatial (that is, if it dominates an elementary CP-semigroup).The proof is by general abstract nonsense (taken from Arveson's classification of E 0 -semigroups on B(H) by Arveson systems up to cocycle conjugacy) and not, as usual, by constructing the cocycle as a solution of a quantum stochastic differential equation in the sense of Hudson and Parthasarathy.All other results that make similar statements (especially, [Mem.Amer.Math.Soc.240 (2016), vi+126 pages, arXiv:0901.1798]) for more general C * -algebras) have been proved later by suitable adaptations of the methods exposed here.(They use Hilbert module techniques, which we carefully avoid here in order to make the result available without any appeal to Hilbert modules.)
We study a class of generic quantum Markov semigroups on the algebra of all bounded operators on a Hilbert space h arising from the stochastic limit of a discrete system with generic Hamiltonian HS, acting on h, interacting with a Gaussian, gauge invariant, reservoir. The selfadjoint operator HS determines a privileged orthonormal basis of h. These semigroups leave invariant diagonal and off-diagonal bounded operators with respect to this basis. The action on diagonal operators describes a classical Markov jump process. We construct generic semigroups from their formal generators by the minimal semigroup method and discuss their conservativity (uniqueness). When the semigroup is irreducible we prove uniqueness of the equilibrium state and show that, starting from an arbitrary initial state, the semigroup converges towards this state. We also prove that the exponential speed of convergence of the quantum Markov semigroup coincides with the exponential speed of convergence of the classical (diagonal) semigroup towards its unique invariant measure. The exponential speed is computed or estimated in some examples.
In recent years, digraph induced generators of quantum dynamical semigroups have been introduced and studied, particularly in the context of unique relaxation and invariance. We define the class of pair block diagonal generators, which allows for additional interaction coefficients but preserves the main structural properties. Namely, when the basis of the underlying Hilbert space is given by the eigenbasis of the Hamiltonian (for example the generic semigroups), then the action of the semigroup leaves invariant the diagonal and off-diagonal matrix spaces. In this case, we explicitly compute all invariant states of the semigroup. In order to define this class we provide a characterization of when the Gorini- Kossakowski-Sudarshan-Lindblad (GKSL) equation defines a proper generator when arbitrary Lindblad operators are allowed (in particular, they do not need to be trace- less as demanded by the GKSL Theorem). Moreover, we consider the converse con- struction to show that every generator naturally gives rise to a digraph, and that under certain assumptions the properties of this digraph can be exploited to gain knowledge of both the number and the structure of the invariant states of the corre- sponding semigroup. We also consider more general constructions on the von Neumann algebra of all bounded linear operators on a Hilbert space, perhaps infinite dimensional. In partic- ular, we prove that for every semigroup of Schwarz maps on such an algebra which has a subinvariant faithful normal state there exists an associated semigroup of con- tractions on the space of Hilbert-Schmidt operators of the Hilbert space. Moreover, we show that if the original semigroup is weak∗ continuous then the associated semigroup is strongly continuous. We introduce the notion of the extended generator of a semigroup on the bounded operators of a Hilbert space with respect to an or- thonormal basis of the Hilbert space. We describe this form of the generator of a quantum Markov semigroup on the von Neumann algebra of all bounded linear op- erators on a Hilbert space which has an invariant faithful normal state under the assumption that the generator of the associated semigroup has compact resolvent, or under the assumption that the minimal unitary dilation of the associated semigroup of contractions is compact.
Abstract The dynamics of Markovian open quantum systems are described by Lindblad master equations, generating a quantum dynamical semigroup. An important concept for such systems is (Davies) irreducibility, i.e. the question whether there exist non-trivial invariant subspaces. Steady states of irreducible systems are unique and faithful, i.e. they have full rank. In the 1970s, Frigerio showed that a system is irreducible if the Lindblad operators span a self-adjoint set with trivial commutant. We discuss a more general and powerful algebraic criterion, showing that a system is irreducible if and only if the multiplicative algebra generated by the Lindblad operators L a and the operator i H + ∑ a L a † L a , involving the Hamiltonian H , is the entire operator space. Examples for two-level systems, show that a change of Hamiltonian terms as well as the addition or removal of dissipators can render a reducible system irreducible and vice versa. Examples for many-body systems show that a large class of spin chains can be rendered irreducible by dissipators on just one or two sites. Additionally, we discuss the decisive differences between (Davies) reducibility and Evans reducibility for quantum channels and dynamical semigroups which has lead to some confusion in the recent physics literature, especially, in the context of boundary-driven systems. We give a criterion for quantum reducibility in terms of associated classical Markov processes and, lastly, discuss the relation of the main result to the stabilization of pure states and argue that systems with local Lindblad operators cannot stabilize pure Fermi-sea states.
We present different characterizations of the notion of irreducibility for Quantum Markov Semigroups (QMSs) and investigate its relationship with other relevant features of the dynamics, such as primitivity, positivity improvement and relaxation; in particular, we show that irreducibility, primitivity and relaxation towards a faithful invariant density are equivalent when the semigroup admits an invariant density. Moreover, in the case of uniformly continuous QMSs, we present several useful ways of checking irreducibility in terms of the operators appearing in the generator in GKLS form. Our exposition is as much self-contained as possible, we present some well known results with elementary proofs (collecting all the relevant literature) and we derive new ones. We study both finite and infinite dimensional evolutions and we remark that many results only require the QMS to be made of Schwarz maps.