Universal Spectra of Random Lindblad Operators

Денисов, С. В.; Laptyeva, T. V.; Tarnowski, Wojciech; Chruściński, Dariusz; Życzkowski, Karol

doi:10.1103/physrevlett.123.140403

Cited by 100 publications

(98 citation statements)

References 63 publications

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“…c)-e) Higher order Pauli strings are progressively inserted and the eigenvalue clusters begin to merge until f) the full basis of traceless matrices {Ln} is obtained once six-body Pauli strings are included. The support of the single eigenvalue cluster matches the result for purely random Liouvillians (gray curve)[14].…”

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confidence: 75%

Hierarchy of Relaxation Timescales in Local Random Liouvillians

2020

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To characterize the generic behavior of open quantum systems, we consider random, purely dissipative Liouvillians with a notion of locality. We find that the positivity of the map implies a sharp separation of the relaxation timescales according to the locality of observables. Specifically, we analyze a spin-1/2 system of size with up to n-body Lindblad operators, which are n-local in the complexity-theory sense. Without locality (n = ), the complex Liouvillian spectrum densely covers a "lemon"-shaped support, in agreement with recent findings [Phys. Rev. Lett. 123, 140403; arXiv:1905.02155]. However, for local Liouvillians (n < ), we find that the spectrum is composed of several dense clusters with random matrix spacing statistics, each featuring a lemon-shaped support wherein all eigenvectors correspond to n-body decay modes. This implies a hierarchy of relaxation timescales of n-body observables, which we verify to be robust in the thermodynamic limit.Introduction. For unitary quantum many-body dynamics, the characterization of generic features common to the vast majority of systems is well developed, in the form of an effective random matrix theory [1-6]. It is for example a crucial ingredient to our understanding of thermalization in unitary quantum systems, manifest in the eigenstate thermalization hypothesis (ETH) [7][8][9][10][11][12][13].For open quantum many-body systems analogous organizing principles are yet missing. Only very recently the first developments in this direction appeared with the investigation of spectral features of a purely random Liouvillian [14-18], describing generic properties of trace preserving positive quantum maps. The purely random Liouvillians considered in [14,15,17] constitute the least structured models aiming at describing generic features of open quantum many-body systems. A main result of these recent studies is that the spectrum of purely random Liouvillians is densely covering a lemon-shaped support whose form is universal, differentiating random Liouvillians from completely random Ginibre matrices with circular spectrum [1,[19][20][21].How close is a completely random Liouvillian to a physical system? In this Letter, we make a step towards answering this question by adding the minimal amount of structure to a purely random Liouvillian, namely a notion of locality. We consider a dissipative spin system of size with Lindblad operators given by Pauli strings classified by their length n ≤ , the latter being a measure of their n-locality in a complexity-theory sense [22].In this language, purely random Liouvillians are completely nonlocal and structureless, since they include the full operator basis including Pauli strings of all lengths 1 ≤ n ≤ .Here, we restrict the maximal number of non-identity operators in the Lindblad operators to n max and examine

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confidence: 75%

Hierarchy of Relaxation Timescales in Local Random Liouvillians

2020

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“…Note Added -While this paper was in preparation, the preprint [114] appeared which considered the spectrum of pure dissipators with the maximal N 2 − 1 number of jump operators. They similarly found a spectral gap, as well as an explicit expression for the limiting large N distribution of eigenvalues.…”

Section: Resultsmentioning

confidence: 99%

Random Lindblad dynamics

Can

2019

J. Phys. A: Math. Theor.

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We study the mixing behavior of random Lindblad generators with no symmetries, using the dynamical map or propagator of the dissipative evolution. In particular, we determine the longtime behavior of a dissipative form factor, which is the trace of the propagator, and use this as a diagnostic for the existence or absence of a spectral gap in the distribution of eigenvalues of the Lindblad generator. We find that simple generators with a single jump operator are slowly mixing, and relax algebraically in time, due to the closing of the spectral gap in the thermodynamic limit. Introducing additional jump operators or a Hamiltonian opens up a spectral gap which remains finite in the thermodynamic limit, leading to exponential relaxation and thus rapid mixing. We use the method of moments and introduce a novel diagrammatic expansion to determine exactly the form factor to leading order in Hilbert space dimension N . We also present numerical support for our main results.

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“…(7), with a 'dense', randomly generated, rate matrix A, is a heavy computational task already for N = 100 -when performed on a single node, without accounting for a sparse structure of the matrices. By taking explicitly into account the sparsity and implementing trivial parallelization, it was possible to sample over a large ensemble (with more than 10 3 realizations) of random Lindbladian generators for N = 100 and thus explore universal spectral features of the ensemble [61].…”

Section: Discussionmentioning

confidence: 99%

Unfolding a quantum master equation into a system of real-valued equations: Computationally effective expansion over the basis of SU(N) generators

et al. 2019

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Dynamics of an open N -state quantum system is typically modeled with a Markovian master equation describing the evolution of the system's density operator. By using generators of SU (N ) group as a basis, the density operator can be transformed into a real-valued 'Bloch vector'. The Lindbladian, a super-operator which serves a generator of the evolution, can be expanded over the same basis and recast in the form of a real matrix. Together, these expansions result is a nonhomogeneous system of N 2 − 1 real-valued linear differential equations for the Bloch vector. Now one can, e.g., implement a high-performance parallel simplex algorithm to find a solution of this system which guarantees exact preservation of the norm and Hermiticity of the density matrix. However, when performed in a straightforward way, the expansion turns to be an operation of the time complexity O(N 10 ). The complexity can be reduced when the number of dissipative operators is independent of N , which is often the case for physically meaningful models. Here we present an algorithm to transform quantum master equation into a system of real-valued differential equations and propagate it forward in time. By using a scalable model, we evaluate computational efficiency of the algorithm and demonstrate that it is possible to handle the model system with N = 10 3 states on a single node of a computer cluster.

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Universal Spectra of Random Lindblad Operators

Cited by 100 publications

References 63 publications

Hierarchy of Relaxation Timescales in Local Random Liouvillians

Hierarchy of Relaxation Timescales in Local Random Liouvillians

Random Lindblad dynamics

Unfolding a quantum master equation into a system of real-valued equations: Computationally effective expansion over the basis of SU(N) generators

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