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Infinite Dimensional Analysis Quantum Probability and Related Topics

GENERIC QUANTUM MARKOV SEMIGROUPS, CYCLE DECOMPOSITION AND DEVIATION FROM EQUILIBRIUM

Franco Fagnola, Veronica Umanità

Published Aug 23, 2012• 17 citations
Publisher

Abstract

A generic quantum Markov semigroup [Formula: see text] of a d-level quantum open system with a faithful normal invariant state ρ admits a dual semigroup [Formula: see text] with respect to the scalar product induced by ρ. We show that the difference of the generators [Formula: see text] can be written as the sum of a derivation 2i[H, ⋅] and a weighted difference of automorphisms [Formula: see text] where [Formula: see text] is a family of cycles on the d levels of the system, wc are positive weights and [Formula: see text] are unitaries. This formula allows us to represent the deviation from equilibrium (in a "small" time interval) as the superposition of cycles of the system where the difference between the forward and backward evolution is written as the difference of a reversible evolution and its time reversal. Moreover, it generalises cycle decomposition of Markov jump processes. We also find a similar formula with partial isometries instead of unitaries.

DOI
10.1142/s0219025712500166

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A generic quantum Markov semigroup [Formula: see text] of a d-level quantum open system with a faithful normal invariant state ρ admits a dual semigroup [Formula: see text] with respect to the scalar product induced by ρ. We show that the difference of the generators [Formula: see text] can be written as the sum of a derivation 2i[H, ⋅] and a weighted difference of automorphisms [Formula: see text] where [Formula: see text] is a family of cycles on the d levels of the system, wc are positive weights and [Formula: see text] are unitaries. This formula allows us to represent the deviation from equilibrium (in a "small" time interval) as the superposition of cycles of the system where the difference between the forward and backward evolution is written as the difference of a reversible evolution and its time reversal. Moreover, it generalises cycle decomposition of Markov jump processes. We also find a similar formula with partial isometries instead of unitaries.
For a quantum Markov semigroup [Formula: see text] on the algebra [Formula: see text] with a faithful invariant state ρ, we can define an adjoint [Formula: see text] with respect to the scalar product determined by ρ. In this paper, we solve the open problems of characterizing adjoints [Formula: see text] that are also a quantum Markov semigroup and satisfy the detailed balance condition in terms of the operators H, Lk in the Gorini–Kossakowski–Sudarshan–Lindblad representation [Formula: see text] of the generator of [Formula: see text]. We study the adjoint semigroup with respect to both scalar products 〈a, b〉 = tr (ρa*b) and 〈a, b〉 = tr (ρ1/2a*ρ1/2 b).
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.
An invariant state of a quantum Markov semigroup is an equilibrium state if it satisfies a quantum detailed balance condition. In this paper, we introduce a notion of entropy production for faithful normal invariant states of a quantum Markov semigroup on B(h) as a numerical index measuring "how much far" they are from equilibrium. The entropy production is defined as the derivative of the relative entropy of the one-step forward and backward evolution in analogy with the classical probabilistic concept. We prove an explicit trace formula expressing the entropy production in terms of the completely positive part of the generator of a norm continuous quantum Markov semigroup showing that it turns out to be zero if and only if a standard quantum detailed balance condition holds.
We study the structure of generic quantum Markov semigroups, arising from the stochastic limit of a discrete system with generic Hamiltonian interacting with a Gaussian gauge invariant reservoir. We show that they can be essentially written as the sum of their irreducible components determined by closed classes of states of the associated classical Markov jump process. Each irreducible component turns out to be recurrent, transient or have an invariant state if and only if its classical (diagonal) restriction is recurrent, transient or has an invariant state, respectively. We classify invariant states and study convergence towards invariant states as time goes to infinity.
Quantum Bernoulli noises (QBNs) refer to the annihilation and creation operators acting on the space [Formula: see text] of square integrable Bernoulli functionals, which satisfy the canonical anti-commutation relation (CAR) in equal time. In this paper, we consider the Markov evolution of an open quantum system interacting with QBNs. Let [Formula: see text] be the system space of an open quantum system interacting with QBNs. Then [Formula: see text] just describes the coupled quantum system. In the framework of [Formula: see text], we first construct a quantum Markov semigroup that respects the interactions between the system and QBNs, and then we prove that under some mild conditions the semigroup has faithful invariant states. To support our main results mentioned above, we prove several technical propositions and theorems about operators defined in [Formula: see text]. Some other results are also obtained.
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.