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arXiv:2404.05473 · Journal of Physics A Mathematical and Theoretical

Modeling the effects of perturbations and steepest entropy ascent on the time evolution of entanglement

César Damián, Robert Tyler Holladay, Michael R. von Spakovsky, Adriana Saldaña‐Robles

Published Apr 8, 2025• 1 citation
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Abstract

Abstract This work presents an analysis of the evolution of perturbed Bell diagonal states using the equation of motion of steepest-entropy-ascent quantum thermodynamics (SEAQT), the Lindblad equation, and various measures of entanglement loss. The Bell diagonal states considered are those generated by the quantum optics study conducted by Liu et al (2016 Phys. Rev. A 94 062107). First, a brief derivation is provided, showing that Bell diagonal states are stationary but not stable equilibrium states within the SEAQT formalism. This highlights the need for perturbation methods to study the evolution of nearby states with, for example, similar values of the energy and entropy. In contrast, under the Lindblad equation of motion, only some Bell diagonal states remain stationary. Although SEAQT is a non-standard, nonlinear framework, it exhibits several unique features that other approaches do not. In particular, not only does it satisfy the postulates of quantum mechanics, but it, as well, is fully compatible with the second law of thermodynamics, providing a physically consistent description of non-equilibrium quantum systems. The perturbation methods used include a weighted-average method for perturbing bipartite system states and a general bipartite method based on a set of unitary operations constrained to maintain constant system energy and entropy. Sets of density operators are randomly generated using each method, and the resulting time-dependent characteristics of the entanglement are analyzed within the SEAQT and Lindblad frameworks. The findings reveal that the evolutions associated with constrained perturbations accurately predict the loss of non-locality and align well with measured concurrence. Additionally, within the SEAQT framework, the deep connection between the thermodynamic states of the evolution of the system and the loss of non-locality is quantitatively demonstrated.

DOI
10.1088/1751-8121/adca75

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Abstract This work presents an analysis of the evolution of perturbed Bell diagonal states using the equation of motion of steepest-entropy-ascent quantum thermodynamics (SEAQT), the Lindblad equation, and various measures of entanglement loss. The Bell diagonal states considered are those generated by the quantum optics study conducted by Liu et al (2016 Phys. Rev. A 94 062107). First, a brief derivation is provided, showing that Bell diagonal states are stationary but not stable equilibrium states within the SEAQT formalism. This highlights the need for perturbation methods to study the evolution of nearby states with, for example, similar values of the energy and entropy. In contrast, under the Lindblad equation of motion, only some Bell diagonal states remain stationary. Although SEAQT is a non-standard, nonlinear framework, it exhibits several unique features that other approaches do not. In particular, not only does it satisfy the postulates of quantum mechanics, but it, as well, is fully compatible with the second law of thermodynamics, providing a physically consistent description of non-equilibrium quantum systems. The perturbation methods used include a weighted-average method for perturbing bipartite system states and a general bipartite method based on a set of unitary operations constrained to maintain constant system energy and entropy. Sets of density operators are randomly generated using each method, and the resulting time-dependent characteristics of the entanglement are analyzed within the SEAQT and Lindblad frameworks. The findings reveal that the evolutions associated with constrained perturbations accurately predict the loss of non-locality and align well with measured concurrence. Additionally, within the SEAQT framework, the deep connection between the thermodynamic states of the evolution of the system and the loss of non-locality is quantitatively demonstrated.
Quantum information and quantum computing (QIQC) systems, relying on the phenomena of superposition and entanglement, offer the potential for vast improvements in certain computations. A practical QC realization requires maintaining the stored information for time-scales long enough to implement algorithms. One primary cause of information loss is decoherence, i.e., the loss of coherence between two energy levels in a quantum system. This work attributes decoherence to dissipation occurring as the system evolves and uses steepest-entropy-ascent quantum thermodynamics (SEAQT) to predict the evolution of system state. SEAQT asserts that, at any instant of time, the system state evolves such that the rate of system entropy change is maximized while conserving system energy. With this principle, the SEAQT equation of motion is applicable to systems in any state, near or far from stable equilibrium, making SEAQT particularly well suited for predicting the dissipation occurring as quantum algorithms are implemented. In the present research, the dynamics of qubits (quantum-bits) using the SEAQT framework are first examined during common quantum gates (combinations of which form algorithms). This is then extended to modeling a system of multiple qubits implementing Shor's algorithm on a nuclear-magnetic-resonance (NMR) QC. Additionally, the SEAQT framework is used to predict experimentally observed dissipation occurring in a two-qubit NMR QC undergoing a so called ``quenching'' process. In addition, several methods for perturbing the density or so-called ``state'' operator used by the SEAQT equation of motion subject to an arbitrary set of expectation value constraints are presented. These are then used as the basis for randomly generating states used in analyzing the dynamics of entangled, non-interacting systems within SEAQT. Finally, a reservoir interaction model is developed for general quantum systems where each system locally experiences a heat interaction with an external reservoir. This model is then used as the basis for developing a decoherence control scheme, which effectively transfers entropy out of the QIQC system as it is generated, thus, reducing the decoherence. Reservoir interactions are modeled for single qubits and the control scheme is employed in modeling an NMR QC and shown to eliminate nearly all of the noise caused by decoherence/dissipation.
A formal development of the hypoequilibrium (HE) state concept within the Steepest-Entropy-Ascent Quantum Thermodynamics (SEAQT) framework is presented, emphasizing its rigorous mathematical formulation. Using a general decomposition of the Hilbert space, HE states are defined in operator language and the reduced evolution of the associated intensive parameters for the regime where the dissipative dynamics commutes with the Hamiltonian is derived. It is proved that the MMM-th order HE family (where MMM is the number of spectral sectors) constitutes an invariant manifold under the SEAQT equation of motion, ensuring that states initially representing a ``mixture of canonicals'' maintain this structure throughout their evolution. Furthermore, a formal connection is established between the HE ansatz and the rate-controlled constrained equilibrium (RCCE) method, identifying HE variables as constraint potentials. Finally, the model is extended to non-Hamiltonian SEAQT (NH-SEAQT) interactions to describe thermodynamically consistent energy and entropy exchanges between subsystems and heat baths. This work provides the formal foundation for reduced-order modeling of far-from-equilibrium relaxation and transport processes, and supports a methodology previously applied across various physical and chemical systems.
The steepest-entropy-ascent quantum thermodynamic (SEAQT) framework is used to model the decoherence that occurs during the state evolution of two different microscopic composite systems. The test cases are a $\text{two}\ensuremath{-}\text{spin}\ensuremath{-}\frac{1}{2}\ensuremath{-}\text{particle}$ composite system and a particle-photon field composite system like that experimentally studied in cavity quantum electrodynamics. The first system is used to study the characteristics of the nonlinear equation of motion of the SEAQT framework when modeling the state evolution of a microscopic composite system with particular interest in the phenomenon of decoherence. The second system is used to compare the numerical predictions of the SEAQT framework with experimental cavity quantum electrodynamic data available in the literature. For the two different numerical cases presented, the time evolution of the density operator of the composite system as well as that of the reduced operators belonging to the two constituents is traced from an initial nonequilibrium state of the composite along its relaxation towards stable equilibrium. Results show for both cases how the initial entanglement and coherence is dissipated during the state relaxation towards a state of stable equilibrium.
The equation of motion of steepest-entropy-ascent quantum thermodynamics (SEA-QT) was first postulated in the early 1980s with the intent of modeling the non-linear dynamic behavior encountered in nature, which the unitary (linear) dynamics of the Schrödinger-von Neumann equation cannot. The SEA-QT equation is used here to model the decoherence phenomenon between two distinguishable and indivisible elementary constituents of type spin–½ (e.g., quantum bits or qubits). The resulting set of non-linear, first-order differential equations is solved with a fourth-order-Runge-Kutta routine provided by Matlab®. The time evolution of the state of the composite system as well as that of the reduced and locally-perceived states of the two constituents are traced from an initial non-equilibrium state of the composite along its relaxation towards stable equilibrium at constant system energy. An entangled and generally coherent, initial non-equilibrium state of the composite quantum system is prepared using a heuristic approach, which consists of randomly and homogeneously generating an initial point on the Bloch sphere for each of the constituents and then using a weighted average of their projections to arrive at an initial state for the composite. Results show how the initial entanglement and coherence between the two spin–½ constituents are reduced during relaxation towards a state of stable equilibrium. When the two particles are non-interacting, the initial coherence is lost once stable equilibrium is reached. When they are interacting, the coherence in the final stable equilibrium state is only that due to the interaction.
Quantum systems driven by time-dependent Hamiltonians are considered here within the framework of steepest-entropy-ascent quantum thermodynamics (SEAQT) and used to study the thermodynamic characteristics of such systems. In doing so, a generalization of the SEAQT framework valid for all such systems is provided, leading to the development of an ab initio physically relevant expression for the intrarelaxation time, an important element of this framework and one that had as of yet not been uniquely determined as an integral part of the theory. The resulting expression for the relaxation time is valid as well for time-independent Hamiltonians as a special case and makes the description provided by the SEAQT framework more robust at the fundamental level. In addition, the SEAQT framework is used to help resolve a fundamental issue of thermodynamics in the quantum domain, namely, that concerning the unique definition of process-dependent work and heat functions. The developments presented lead to the conclusion that this framework is not just an alternative approach to thermodynamics in the quantum domain but instead one that uniquely sheds new light on various fundamental but as of yet not completely resolved questions of thermodynamics.
The Steepest Entropy Ascent (SEA) ansatz, recently recognized as the fourth law of thermodynamics, governs the irreversible evolution of a system from a non-equilibrium state toward a unique maximum-entropy equilibrium. SEA builds upon the second law to unify mechanics and thermodynamics. Due to its nonlinear nature, exact solutions to the SEA equation of motion are scarce. To address this, the Fixed Lagrange Multiplier (FLM) method is developed as an approximate analytical tool, applicable to both two-level and higher-dimensional quantum systems. Using quantum walks, a universal computation model, the study applies FLM to analyze and solve the SEA dynamics for single-component N−N-N−level systems. The approximate FLM solutions show strong agreement with full numerical simulations, particularly in regions of maximum entropy production consistent with SEA predictions. To extend SEA analysis to composite systems, especially two-qubit systems, the work provides a general framework for N−N-N−level Bloch vector parametrization. It includes analytical roots for N=3N=3N=3 and a complete parametrization for N=4N=4N=4, along with a method for analytically computing operator traces in this representation. Finally, the study examines the no-signaling condition in nonlinear quantum theories. While nonlinearity often implies faster-than-light signaling, the SEA framework inherently respects no-signaling. The equation of motion for both separable and entangled (e.g., Bell-diagonal) composites confirms that SEA maintains locality and provides a robust foundation for modeling decoherence in both open and closed quantum systems. (Abridged for ArXiv)