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arXiv:2510.07628 · Physical Review Research

Generating entangled steady states in multistable open quantum systems via initial state control

Diego Fallas Padilla, Raphael Kaubruegger, Adrianna Gillman, Stephen Becker, Ana María Rey

Published Feb 8, 2026• 0 citations
arXivHTMLPDF

Abstract

Entanglement underpins the power of quantum technologies, yet it is fragile and typically destroyed by dissipation. Paradoxically, the same dissipation, when carefully engineered, can drive a system toward robust entangled steady states. However, this engineering task is nontrivial, as dissipative many-body systems are complex, particularly when they support multiple steady states. Here, we derive analytic expressions that predict how the steady state of a system evolving under a Lindblad equation depends on the initial state, without requiring integration of the dynamics. These results extend Refs. [V. V. Albert and L. Jiang, ; V. V. Albert , ], showing that while the steady-state manifold is determined by the Liouvillian kernel, the weights within it depend on both the Liouvillian and the initial state. We identify a special class of Liouvillians for which the steady state depends only on the initial overlap with the kernel. Our framework provides analytical insight and a computationally efficient tool for predicting steady states in open quantum systems. As an application, we propose schemes to generate metrologically useful entangled steady states in spin ensembles via balanced collective decay.

DOI
10.1103/s8kc-crf3

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Entanglement underpins the power of quantum technologies, yet it is fragile and typically destroyed by dissipation. Paradoxically, the same dissipation, when carefully engineered, can drive a system toward robust entangled steady states. However, this engineering task is nontrivial, as dissipative many-body systems are complex, particularly when they support multiple steady states. Here, we derive analytic expressions that predict how the steady state of a system evolving under a Lindblad equation depends on the initial state, without requiring integration of the dynamics. These results extend Refs. [V. V. Albert and L. Jiang, ; V. V. Albert , ], showing that while the steady-state manifold is determined by the Liouvillian kernel, the weights within it depend on both the Liouvillian and the initial state. We identify a special class of Liouvillians for which the steady state depends only on the initial overlap with the kernel. Our framework provides analytical insight and a computationally efficient tool for predicting steady states in open quantum systems. As an application, we propose schemes to generate metrologically useful entangled steady states in spin ensembles via balanced collective decay.
Highly multipartite entangled states play an important role in various quantum computing tasks. We investigate the dissipative generation of a complex entanglement structure as in a cluster state through engineered Markovian dynamics in the spin systems coupled via Ising interactions. Using the Lindblad master equation, we design a projection based dissipative channel that drives the system toward a unique pure steady state corresponding to the desired cluster state. This is done by removing the contribution of the orthogonal states. By explicitly constructing the Liouvillian superoperator in the full 2N2^N2N-dimensional Hilbert space, we compute the steady-state density matrix, the Liouvillian spectral gap, entanglement witness and the fidelity with respect to the ideal cluster state. The results demonstrate that the cluster state emerges as the steady state when the engineered Liouvillian dissipation dominates over the local Ising interaction between spins. Moreover, we find that the fidelity and Liouvillian spectral gap is relatively insensitive to the system size once the saturation dissipation has been achieved that scales linearly with the qubit number. This analysis illustrates a physically realizable path towards steady-state entanglement generation in the spin systems using engineered dissipation.
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Realistic quantum systems are affected by environmental loss, which is often seen as detrimental for applications in quantum technologies. Alternatively, weak coupling to an environment can aid in stabilizing highly entangled and mixed states, but determining optimal system-environment parameters can be challenging. Here, we describe a technique to optimize parameters for generating desired nonequilibrium steady states (NESSs) in driven-dissipative quantum systems governed by the Lindblad equation. We apply this approach to predict highly entangled and mixed NESSs in Ising, Kitaev, and Dicke models in several quantum phases.
Open quantum systems, where quantum dynamics is influenced by (non-)Markovian environments, have attracted increasing attention due to their rich physical properties. Understanding phenomena such as non-equilibrium phases, anomalous thermalization, and dissipative state preparation requires the knowledge of the low-lying eigenvalues and eigenstates of the Lindbladian. In this thesis, we introduce a framework that systematically computes not only steady states but also low-lying excited states in large, dissipative, interacting quantum many-body systems. The framework is based on tensor-network methods and utilizes recent advances in complex-time Krylov spaces. Extending these ideas to the challenging non-Hermitian eigenvalue problem ubiquitous in open quantum systems enables simulations at unprecedented Hilbert space dimensions. We employ the framework to analyze a dissipative state preparation protocol for Bose-Einstein condensates (BECs) on optical lattices. Analytical and numerical studies allow us to characterize the steady state and perform a finite-size scaling of the dissipative gap, uncovering strong evidence for non-linear hydrodynamics described by the Kardar-Parisi-Zhang universality class. Furthermore, we show analytically that the dissipative state preparation can be substantially accelerated via the quantum Mpemba effect. Our approach exploits weak symmetries to analytically identify a class of simple, experimentally realizable states that converge exponentially faster to the steady state than typical random initializations. Overall, this work establishes a versatile framework for the spectral analysis of generic open quantum many-body systems, enabling the analysis of complex collective phenomena and additionally paving the way toward faster dissipative preparation of highly entangled states in analog quantum simulators.
Strong dynamical symmetries provide an algebraic description of persistent oscillations in open quantum systems governed by the Lindblad equation. For finite-dimensional systems with a full-rank stationary state, such symmetries are necessary and sufficient for non-decaying oscillatory modes to appear in the Liouvillian's spectrum. These modes can be directly constructed by acting on stationary states with the identified symmetry operators. Here we establish an asymptotic counterpart of this algebra for collective spins. Taking a single spin SSS in a magnetic field, subject to dissipation through linear jump operators, we identify the exact conditions on the field and dissipative couplings under which non-decaying oscillatory modes emerge as S→∞S \rightarrow \inftyS→∞. We then write down a simple, closed form for the corresponding `symmetry' operator which can be applied to the steady state to construct these modes, which are eigenvectors of the Liouvillian, with imaginary eigenvalues, up to O(1/S)O(1/S)O(1/S) corrections. A second operator, an emergent conserved quantity, completes the algebra, and its slow relaxation determines how quantum noise selects the stationary state. The construction can be directly extended to a pair of interacting spins sharing a common reservoir, with the resulting quantum algebra making the entanglement during a quench analytically accessible.
We investigate dissipation-driven topological phase transitions in one-dimensional quantum open systems governed by the Lindblad equation with linear dissipation operators, which ensure the density matrix retains its Gaussian form throughout the dynamics. By employing the modular Hamiltonian framework, we rigorously demonstrate that the Z2\rm{Z}_2Z2​ topological invariant characterizing steady states in one-dimensional class D systems is exclusively dependent on the dissipation operators, rather than the system Hamiltonian. Through a sudden quench protocol where the system evolves from the steady state of one Lindbladian to another, we reveal that topological transitions can occur at analytically predictable critical times, even when the initial and final steady states share identical topological indices. These transitions are shown, both analytically and numerically, to depend solely on dissipation parameters. Entanglement spectrum analysis demonstrates bulk-edge correspondence in non-equilibrium density matrices via coexisting single-particle gap closures (periodic boundaries) and topologically protected zero modes (open boundaries), directly underpinning the detection of dissipation-induced topology in quantum simulators.