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arXiv:2202.10989 · Quantum Information Processing

Antilinear superoperator, quantum geometric invariance, and antilinear symmetry for higher-dimensional quantum systems

Lu Wei, Zhian Jia, Dagomir Kaszlikowski, Sheng Tan

Published Jul 29, 2024• 0 citations
arXivHTMLPDF
DOI
10.1007/s11128-024-04499-3

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Abstract Currently there is much interest in Hamiltonians that are not Hermitian but instead possess an antilinear PT symmetry. Here we seek to put such PT symmetric theories into as general a context as possible. After providing a brief overview of the PT symmetry program, we show that having an antilinear symmetry that acts on a well-defined Hilbert space is the most general condition that one can impose on a quantum theory for which one can have a well-defined inner product that is time independent, have a Hamiltonian that is self-adjoint, and have energy eigenvalues that are all real. For each of these properties Hermiticity is only a sufficient condition but not a necessary one, with Hermiticity thus being the special case in which the Hamiltonian has both antilinearity and Hermiticity. As well as being the necessary condition for the reality of energy eigenvalues, antilinearity in addition allows for the physically interesting cases of manifestly non-Hermitian but nonetheless self-adjoint Hamiltonians that have energy eigenvalues that appear in complex conjugate pairs, or that are Jordan block and cannot be diagonalized at all. We show that one can extend these ideas to quantum field theory, with the dual requirements of the existence of time independent inner products and invariance under complex Lorentz transformations forcing the antilinear symmetry to uniquely be CPT . We thus extend the CPT theorem to non-Hermitian Hamiltonians. For theories that are separately charge conjugation invariant, PT symmetry then follows, with the case for the physical relevance of the PT -symmetry program thus being advanced. While CPT symmetry can be defined at the classical level for every classical path in a path integral quantization procedure, in contrast, in such a path integral there is no reference at all to the Hermiticity of the Hamiltonian or the quantum Hilbert space on which it acts, as they are strictly quantum-mechanical concepts that can only be defined after the path integral quantization has been performed and the quantum Hilbert space has been constructed. CPT symmetry thus goes beyond Hermiticity and has primacy over it, with our work raising the question of how Hermiticity ever comes into quantum theory at all. To this end we show that whether or not a CPT -invariant theory has a Hamiltonian that is Hermitian is a property of the solutions to the theory and not of the Hamiltonian itself. Hermiticity thus never needs to be postulated at all.
We present a full symmetry classification of fermion matter in and out of thermal equilibrium.Our approach starts from first principles, the ten different classes of linear and antilinear state transformations in fermionic Fock spaces, and symmetries defined via invariance properties of the dynamical equation for the density matrix.The object of classification is then the generators of reversible dynamics, dissipation and fluctuations, featuring' in the generally irreversible and interacting dynamical equations.A sharp distinction between the symmetries of equilibrium and out-of-equilibrium dynamics, respectively, arises from the different role played by "time" in these two cases: In unitary quantum mechanics as well as in "microreversible" thermal equilibrium, antilinear transformations combined with an inversion of time define time-reversal symmetry.However, out of equilibrium an inversion of time becomes meaningless, while antilinear transformations in Fock space remain physically significant, and hence must be considered in autonomy.The practical consequence of this dichotomy is a novel realization of antilinear symmetries (six out of the ten fundamental classes) in nonequilibrium quantum dynamics that is fundamentally different from the established rules of thermal equilibrium.At large times, the dynamical generators thus symmetry classified determine the steady-state nonequilibrium distributions for arbitrary interacting systems.To illustrate this principle, we consider the fixation of a symmetry protected topological phase in a system of interacting lattice fermions.More generally, we consider the practically important class of mean field interacting systems, represented by Gaussian states.This class is naturally described in the language of non-Hermitian matrices, which allows us to compare to previous classification schemes in the literature.
We present a full symmetry classification of fermion matter in and out of thermal equilibrium. Our approach starts from first principles, the ten different classes of linear and antilinear state transformations in fermionic Fock spaces, and symmetries defined via invariance properties of the dynamical equation for the density matrix. The object of classification is then the generators of reversible dynamics, dissipation and fluctuations, featuring' in the generally irreversible and interacting dynamical equations. A sharp distinction between the symmetries of equilibrium and out-of-equilibrium dynamics, respectively, arises from the different role played by time in these two cases: In unitary quantum mechanics as well as in microreversible thermal equilibrium, antilinear transformations combined with an inversion of time define time-reversal symmetry. However, out of equilibrium an inversion of time becomes meaningless, while antilinear transformations in Fock space remain physically significant, and hence must be considered in autonomy. The practical consequence of this dichotomy is a novel realization of antilinear symmetries (six out of the ten fundamental classes) in nonequilibrium quantum dynamics that is fundamentally different from the established rules of thermal equilibrium. At large times, the dynamical generators thus symmetry classified determine the steady-state nonequilibrium distributions for arbitrary interacting systems. To illustrate this principle, we consider the fixation of a symmetry protected topological phase in a system of interacting lattice fermions. More generally, we consider the practically important class of mean field interacting systems, represented by Gaussian states. This class is naturally described in the language of non-Hermitian matrices, which allows us to compare to previous classification schemes in the literature.