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arXiv:2512.03935 · Journal of Physics A Mathematical and Theoretical

Thermodynamics of an Open $\mathcal{PT-}$Symmetric Quantum System

Baibhab Bose, Devvrat Tiwari, Subhashish Banerjee

quant-phcond-mat.stat-mechSubmitted Dec 3, 2025• Announced Dec 4, 2025• 0 citations
v1 · Dec 3, 2025v2 · Aug 31, 2026
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Abstract

For a subclass of a general $\mathcal{PT}-$symmetric Hamiltonian obeying anti-commutation relation with its conjugate, a Hermitian basis is found that spans the bi-orthonormal energy eigenvectors. Using the modified projectors constructed from these eigenvectors, the generalized density matrix of the $\mathcal{PT}-$symmetric evolution is calculated, and subsequently, ergotropy for a closed system is obtained. The $\mathcal{PT}-$symmetric system, in an open system scenario, is studied to understand ergotropy under different regimes of non-Hermiticity of the Hamiltonian. The consistency of the three laws of thermodynamics for the $\mathcal{PT}-$symmetric system in an open system scenario is also analyzed.

Journal reference
J. Phys. A: Math. Theor. 59, 345302 (2026)
DOI
10.1088/1751-8121/ae98c7
Comments
10 pages, 4 figures

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Thermodynamics of an Open PT−\mathcal{PT-}PT−Symmetric Quantum System
For a subclass of a general PT−\mathcal{PT}-PT−symmetric Hamiltonian obeying anti-commutation relation with its conjugate, a Hermitian basis is found that spans the bi-orthonormal energy eigenvectors. Using the modified projectors constructed from these eigenvectors, the generalized density matrix of the PT−\mathcal{PT}-PT−symmetric evolution is calculated, and subsequently, ergotropy for a closed system is obtained. The PT−\mathcal{PT}-PT−symmetric system, in an open system scenario, is studied to understand ergotropy under different regimes of non-Hermiticity of the Hamiltonian. The consistency of the three laws of thermodynamics for the PT−\mathcal{PT}-PT−symmetric system in an open system scenario is also analyzed.
Anti-PT\mathcal{PT}PT symmetry for a non-Hermitian Hamiltonian
Abstract Anti-PT\mathcal{PT}PT symmetry, (PT)H=−H(PT)(\mathcal{PT})H=-H(\mathcal{PT})(PT)H=−H(PT), is a plausible variant of PT\mathcal{PT}PT symmetry. Of particular interest is the situation when all the eigenstates of an anti-PT\mathcal{PT}PT-symmetric non-Hermitian Hamiltonian HHH are also eigenstates of the PT\mathcal{PT}PT operator; then, the quasi-energies are purely imaginary, which implies that the Hermitian conjugate H+=−HH^{+}=-HH+=−H, and thus they are connected via the relation (PT)H=H+PT(\mathcal{PT})H=H^{+}\mathcal{PT}(PT)H=H+PT, similar to the quasi-Hermiticity relation. Therefore, the eigenfunctions of the anti-PT\mathcal{PT}PT-symmetric HHH form a complete orthonormal set with positive definite norms, and moreover the time evolution is unitary.
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Abstract The theory of a two-level η -Hermitian Hamiltonian with P T symmetry is reviewed and extended to include open system dynamics. A first-principles derivation of the generalized Gorini-Kossakowski-Sudarshan-Lindblad master equation appropriate for a P T − symmetric Hamiltonian is presented. Inspired by a simple light–matter interaction open system model, information theoretic quantities like a non-Markovian witness and fidelity are calculated for the P T − symmetric Hamiltonian, and the results are compared with their corresponding two-level Hermitian counterparts. The nature of entanglement between two P T − symmetric and Hermitian open quantum systems is calculated, and the contrast observed.
Non-Hermitian systems with parity-time (PT\mathcal{PT}PT) symmetry and anti-PT\mathcal{PT}PT symmetry give rise to exceptional points (EPs) with important properties due to the coalescence of eigenvectors. Because of the limitations on the Hermiticity of the Hamiltonian, these kinds of non-Hermitian systems have recently attracted extensive attention and emerged as a new way to engineer the response of open quantum systems. We propose an efficient and easily computable tool, based on the Hilbert-Schmidt speed (HSS), which does not require the diagonalization of the evolved density matrix, to characterize non-Markovian effects in (anti-)PT ⁣−\mathcal{PT}\!-PT−symmetric systems. Our proposed witness of non-Markovianity can also detect exactly the EPs and hence the critical behavior of the non-Hermitian systems, especially high-dimensional qudits. Our theoretical predictions are completely consistent with results extracted from recent experiments studying the non-Markovianity and criticality in (anti-)PT ⁣−\mathcal{PT}\!-PT−symmetric systems.
Quantum thermodynamics has emerged as a central field for understanding how energy conversion processes occur in microscopic systems. In these systems, effects such as coherence, entanglement, and non-Markovianity play key roles. In this thesis, we explore different ways to describe quantum thermodynamics, using two main approaches: one based on entropy and the other on ergotropy. First, we introduce a generalized approach to quantify non-Markovianity through the breakdown of monotonicity in thermodynamic functions. In this context, the entropy-based heat flow serves as a practical tool to witness and measure quantum memory in unital maps that do not reverse the sign of the internal energy. Next, we analyze the dynamics of ergotropy in open qubits under both Markovian and non-Markovian evolutions. We identify phenomena such as freezing and sudden death of ergotropy, and we establish an analytical relation between the change in ergotropy and the environment-induced work . This provides a clear physical interpretation for the additional term in the first law in the entropy-based formulation. Finally, we propose an ergotropy-based thermodynamic formulation, in which heat is reinterpreted in terms of the change of the passive state associated with the density operator governing the quantum dynamics. This approach allows one to measure non-Markovianity of unital maps more generally and accurately than the entropy-based heat flow. This advantage comes from the direct link between heat and von Neumann entropy, a property ensured by invariance under passive transformations. Moreover, the out-of-equilibrium temperature naturally remains non-negative, similarly to equilibrium thermodynamics.