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arXiv:1907.04548 · Physical review. E

Steepest entropy ascent solution for a continuous-time quantum walker

Rohit Kishan Ray

Published Aug 15, 2022• 4 citations
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Abstract

We consider the steepest entropy ascent (SEA) ansatz to describe the nonlinear thermodynamic evolution of a quantum system. Recently this principle has been dubbed the fourth law of thermodynamics [Beretta, Phil. Trans. R. Soc. A 378, 20190168 (2020)10.1098/rsta.2019.0168]. A unique global equilibrium state exists in this context, and any other state is driven by the maximum entropy generation principle towards this equilibrium. We study the SEA evolution of a continuous-time quantum walker (CTQW) on a cycle graph with N nodes. SEA solutions are difficult to find analytically. We provide an approximate scheme to find a general single-particle evolution equation governed by the SEA principle, whose solution produces dissipation dynamics. We call this scheme the fixed Lagrange's multiplier (FLM) method. In the Bloch sphere representation, we find trajectories traced out by the Bloch vector within the sphere itself. We have discussed these trajectories under various initial conditions for the case of a qubit. A similar dissipative motion is also observed in the case of CTQW, where probability amplitudes have been used to characterize decoherence. Our FLM scheme shows good agreement with numerical results. As we report, in CTQW, a strong delocalization exists for low system relaxation time.

DOI
10.1103/physreve.106.024115

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We consider the steepest entropy ascent (SEA) ansatz to describe the nonlinear thermodynamic evolution of a quantum system. Recently this principle has been dubbed the fourth law of thermodynamics [Beretta, Phil. Trans. R. Soc. A 378, 20190168 (2020)10.1098/rsta.2019.0168]. A unique global equilibrium state exists in this context, and any other state is driven by the maximum entropy generation principle towards this equilibrium. We study the SEA evolution of a continuous-time quantum walker (CTQW) on a cycle graph with N nodes. SEA solutions are difficult to find analytically. We provide an approximate scheme to find a general single-particle evolution equation governed by the SEA principle, whose solution produces dissipation dynamics. We call this scheme the fixed Lagrange's multiplier (FLM) method. In the Bloch sphere representation, we find trajectories traced out by the Bloch vector within the sphere itself. We have discussed these trajectories under various initial conditions for the case of a qubit. A similar dissipative motion is also observed in the case of CTQW, where probability amplitudes have been used to characterize decoherence. Our FLM scheme shows good agreement with numerical results. As we report, in CTQW, a strong delocalization exists for low system relaxation time.
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)
Quantum walks play a crucial role in quantum algorithms and computational problems. Many-body quantum walks can reveal and exploit quantum correlations that are unavailable for single-walker cases. Studying quantum walks under noise and dissipation, particularly in multiwalker systems, has significant implications. In this context, we use a thermodynamically consistent formalism of dissipation modeling, namely the steepest entropy ascent (SEA) formalism. We analyze two spinless fermionic continuous-time walkers on a one-dimensional graph with tunable Hubbard and extended Hubbard-like interactions. By contrasting SEA-driven dynamics with unitary evolution, we systematically investigate how interaction strengths modulate thermalization and entropy production. Our findings highlight the relevance of the SEA formalism in modeling nonlinear dissipation in many-body quantum systems and its implications for quantum thermalization.
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.
The steepest entropy ascent (SEA) dynamical principle provides a general framework for modeling the dynamics of nonequilibrium (NE) phenomena at any level of description, including the atomistic one. It has recently been shown to provide a precise implementation and meaning to the maximum entropy production principle and to encompass many well-established theories of nonequilibrium thermodynamics into a single unifying geometrical framework. Its original formulation in the framework of quantum thermodynamics (QT) assumes the simplest and most natural Fisher-Rao metric to geometrize from a dynamical standpoint the manifold of density operators, which represent the thermodynamic NE states of the system. This simplest SEAQT formulation is used here to develop a general mathematical framework for modeling the NE time evolution of the quantum state of a chemically reactive mixture at an atomistic level. The method is illustrated for a simple two-reaction kinetic scheme of the overall reaction F+H_{2}⇔HF+F in an isolated tank of fixed volume. However, the general formalism is developed for a reactive system subject to multiple reaction mechanisms. To explicitly implement the SEAQT nonlinear law of evolution for the density operator, both the energy and the particle number eigenvalue problems are set up and solved analytically under the dilute gas approximation. The system-level energy and particle number eigenvalues and eigenstates are used in the SEAQT equation of motion to determine the time evolution of the density operator, thus effectively describing the overall kinetics of the reacting system as it relaxes toward stable chemical equilibrium. The predicted time evolution in the near-equilibrium limit is compared to the reaction rates given by a standard detailed kinetic model so as to extract the single time constant needed by the present SEA model.
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.