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arXiv:2512.03935 · Journal of Physics A Mathematical and Theoretical

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

Baibhab Bose, Devvrat Tiwari, Subhashish Banerjee

quant-phcond-mat.stat-mechSubmitted Dec 3, 2025• Announced Dec 4, 2025• 0 citations• Revised Aug 31, 2026 (v2)
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
16 pages, 6 figures

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Thermodynamics of an Open Two-Level 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.
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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.
The Hamiltonian H specifies the energy levels and time evolution of a quantum theory. A standard axiom of quantum mechanics requires that H be Hermitian because Hermiticity guarantees that the energy spectrum is real and that time evolution is unitary (probability-preserving). This paper describes an alternative formulation of quantum mechanics in which the mathematical axiom of Hermiticity (transpose + complex conjugate) is replaced by the physically transparent condition of space-time reflection (PT) symmetry. If H has an unbroken PT symmetry, then the spectrum is real. Examples of PT-symmetric non-Hermitian quantum-mechanical Hamiltonians are H=p^2+ix^3 and H=p^2-x^4. Amazingly, the energy levels of these Hamiltonians are all real and positive! In general, if H has an unbroken PT symmetry, then it has another symmetry represented by a linear operator C. Using C, one can construct a time-independent inner product with a positive-definite norm. Thus, PT-symmetric Hamiltonians describe a new class of complex quantum theories having positive probabilities and unitary time evolution. The Lee Model is an example of a PT-symmetric Hamiltonian. The renormalized Lee-model Hamiltonian has a negative-norm "ghost" state because renormalization causes the Hamiltonian to become non-Hermitian. For the past 50 years there have been many attempts to find a physical interpretation for the ghost, but all such attempts failed. Our interpretation of the ghost is simply that the non-Hermitian Lee Model Hamiltonian is PT-symmetric. The C operator for the Lee Model is calculated exactly and in closed form and the ghost is shown to be a physical state having a positive norm. The ideas of PT symmetry are illustrated by using many quantum-mechanical and quantum-field-theoretic models.
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