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arXiv:2304.02960 · Open Systems & Information Dynamics

Random Lindblad Operators Obeying Detailed Balance

Wojciech Tarnowski, Dariusz Chruściński, Sergey Denisov, Karol Życzkowski

Published Jun 1, 2023• 3 citations
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Abstract

We introduce different ensembles of random Lindblad operators [Formula: see text], which satisfy quantum detailed balance condition with respect to given stationary state [Formula: see text] of size [Formula: see text], and investigate their spectral properties. Such operators are known as ‘Davies generators’ and their eigenvalues are real; however, their spectral densities depend on [Formula: see text]. We propose different structured ensembles of random matrices, which allow us to tackle the problem analytically in the extreme cases of Davies generators corresponding to random [Formula: see text] with a nondegenerate spectrum or the maximally mixed stationary state, [Formula: see text]. Interestingly, in the latter case the density can be reasonably well approximated by integrating out the imaginary component of the spectral density characteristic to the ensemble of random unconstrained Lindblad operators. The case of asymptotic states with partially degenerated spectra is also addressed. Finally, we demonstrate that similar universal properties hold for the detailed balance-obeying Kolmogorov generators obtained by applying superdecoherence to an ensemble of random Davies generators. In this way we construct an ensemble of random classical generators with imposed detailed balance condition.

DOI
10.1142/s1230161223500075

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We introduce different ensembles of random Lindblad operators [Formula: see text], which satisfy quantum detailed balance condition with respect to given stationary state [Formula: see text] of size [Formula: see text], and investigate their spectral properties. Such operators are known as ‘Davies generators’ and their eigenvalues are real; however, their spectral densities depend on [Formula: see text]. We propose different structured ensembles of random matrices, which allow us to tackle the problem analytically in the extreme cases of Davies generators corresponding to random [Formula: see text] with a nondegenerate spectrum or the maximally mixed stationary state, [Formula: see text]. Interestingly, in the latter case the density can be reasonably well approximated by integrating out the imaginary component of the spectral density characteristic to the ensemble of random unconstrained Lindblad operators. The case of asymptotic states with partially degenerated spectra is also addressed. Finally, we demonstrate that similar universal properties hold for the detailed balance-obeying Kolmogorov generators obtained by applying superdecoherence to an ensemble of random Davies generators. In this way we construct an ensemble of random classical generators with imposed detailed balance condition.
This paper deals with the asymptotic behavior of a quantum dynamical semigroup [Formula: see text] acting on the algebra of all linear bounded operators on a given Hilbert space. In practice, all these semigroups have a generator which can be written in a well-known form named after Lindblad and Davies. If the semigroup has a faithful normal stationary state ρ, necessary and sufficient conditions are derived for the w*-convergence of [Formula: see text] to [Formula: see text], where [Formula: see text] is the conditional expectation of an element X onto the subalgebra of fixed points. Our main results are expressed in terms of the Lindblad–Davies generator .
Continuous-time Markovian evolution appears to be manifestly different in classical and quantum worlds. We consider ensembles of random generators of N-dimensional Markovian evolution, quantum and classical ones, and evaluate their universal spectral properties. We then show how the two types of generators can be related by superdecoherence. In analogy with the mechanism of decoherence, which transforms a quantum state into a classical one, superdecoherence can be used to transform a Lindblad operator (generator of quantum evolution) into a Kolmogorov operator (generator of classical evolution). We inspect spectra of random Lindblad operators undergoing superdecoherence and demonstrate that, in the limit of complete superdecoherence, the resulting operators exhibit spectral density typical to random Kolmogorov operators. By gradually increasing strength of superdecoherence, we observe a sharp quantum-to-classical transition. Furthermore, we define an inverse procedure of supercoherification that is a generalization of the scheme used to construct a quantum state out of a classical one. Finally, we study microscopic correlation between neighboring eigenvalues through the complex spacing ratios and observe the horseshoe distribution, emblematic of the Ginibre universality class, for both types of random generators. Remarkably, it survives both superdecoherence and supercoherification.
Generators of continuous-time quantum Markovian evolution, known as Lindblad operators, play a key role in the theory of open quantum systems. The question of how to define and characterize random Lindblad operators has attracted considerable attention in recent years. Purely dissipative random Lindbladians have been extensively studied using toolboxes of free probability and non-Hermitian random matrix theories, which helped to uncover a characteristic lemon-like spectral support. In this work, we focus on random Lindbladians with a Hamiltonian component. We analyze three ensembles of increasing complexity, designed to illustrate the joint effects of dissipative and Hamiltonian dynamics. While the first two allow for an exact analytic treatment with standard probabilistic and scalar-valued free-probability methods, the most general case requires the more general framework of operator-valued free probability. We explain how the spectral support changes with the relative Hamiltonian strength and the rank of the dissipative component and characterize the effect of Hamiltonian-induced spectral squeezing.
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).
We study various methods to generate ensembles of random density matrices of a fixed size N, obtained by partial trace of pure states on composite systems. Structured ensembles of random pure states, invariant with respect to local unitary transformations are introduced. To analyze statistical properties of quantum entanglement in bi-partite systems we analyze the distribution of Schmidt coefficients of random pure states. Such a distribution is derived in the case of a superposition of k random maximally entangled states. For another ensemble, obtained by performing selective measurements in a maximally entangled basis on a multi-partite system, we show that this distribution is given by the Fuss-Catalan law and find the average entanglement entropy. A more general class of structured ensembles proposed, containing also the case of Bures, forms an extension of the standard ensemble of structureless random pure states, described asymptotically, as N → ∞, by the Marchenko-Pastur distribution.
We give an explicit entropy production formula for a class of quantum Markov semigroups, arising in the weak coupling limit of a system coupled with reservoirs, whose generators [Formula: see text] are sums of other generators [Formula: see text] associated with positive Bohr frequencies [Formula: see text] of the system. As a consequence, we show that any such semigroup satisfies the quantum detailed balance condition with respect to an invariant state if and only if all semigroups generated by each [Formula: see text] do so with respect to the same invariant state.