Recently, apparent nonphysical implications of non-Hermitian quantum mechanics (NHQM) have been discussed in the literature. In particular, the apparent violation of the no-signaling theorem, discrimination of nonorthogonal states, and the increase of quantum entanglement by local operations were reported, and therefore NHQM was not considered as a fundamental theory. Here we show that these and other no-go principles (including the no-cloning and no-deleting theorems) of conventional quantum mechanics still hold in finite-dimensional non-Hermitian quantum systems, including parity-time symmetric and pseudo-Hermitian cases, if its formalism is properly applied. We have developed a modified formulation of NHQM based on the geometry of Hilbert spaces which is consistent with the conventional quantum mechanics for Hermitian systems. Using this formulation the validity of these principles can be shown in a simple and uniform approach.
Recently, apparent nonphysical implications of non-Hermitian quantum mechanics (NHQM) have been discussed in the literature. In particular, the apparent violation of the no-signaling theorem, discrimination of nonorthogonal states, and the increase of quantum entanglement by local operations were reported, and therefore NHQM was not considered as a fundamental theory. Here we show that these and other no-go principles (including the no-cloning and no-deleting theorems) of conventional quantum mechanics still hold in finite-dimensional non-Hermitian quantum systems, including parity-time symmetric and pseudo-Hermitian cases, if its formalism is properly applied. We have developed a modified formulation of NHQM based on the geometry of Hilbert spaces which is consistent with the conventional quantum mechanics for Hermitian systems. Using this formulation the validity of these principles can be shown in a simple and uniform approach.
A diagonalizable non-Hermitian Hamiltonian having a real spectrum may be used to define a unitary quantum system, if one modifies the inner product of the Hilbert space properly. We give a comprehensive and essentially self-contained review of the basic ideas and techniques responsible for the recent developments in this subject. We provide a critical assessment of the role of the geometry of the Hilbert space in conventional quantum mechanics to reveal the basic physical principle motivating our study. We then offer a survey of the necessary mathematical tools, present their utility in establishing a lucid and precise formulation of a unitary quantum theory based on a non-Hermitian Hamiltonian, and elaborate on a number of relevant issues of fundamental importance. In particular, we discuss the role of the antilinear symmetries such as [Formula: see text], the true meaning and significance of the so-called charge operators [Formula: see text] and the [Formula: see text]-inner products, the nature of the physical observables, the equivalent description of such models using ordinary Hermitian quantum mechanics, the pertaining duality between local-non-Hermitian versus nonlocal-Hermitian descriptions of their dynamics, the corresponding classical systems, the pseudo-Hermitian canonical quantization scheme, various methods of calculating the (pseudo-) metric operators, subtleties of dealing with time-dependent quasi-Hermitian Hamiltonians and the path-integral formulation of the theory, and the structure of the state space and its ramifications for the quantum Brachistochrone problem. We also explore some concrete physical applications and manifestations of the abstract concepts and tools that have been developed in the course of this investigation. These include applications in nuclear physics, condensed matter physics, relativistic quantum mechanics and quantum field theory, quantum cosmology, electromagnetic wave propagation, open quantum systems, magnetohydrodynamics, quantum chaos and biophysics.
A diagonalizable non-Hermitian Hamiltonian having a real spectrum may be used to define a unitary quantum system, if one modifies the inner product of the Hilbert space properly. We give a comprehensive and essentially self-contained review of the basic ideas and techniques responsible for the recent developments in this subject. We provide a critical assessment of the role of the geometry of the Hilbert space in conventional quantum mechanics to reveal the basic physical principle motivating our study. We then offer a survey of the necessary mathematical tools and elaborate on a number of relevant issues of fundamental importance. In particular, we discuss the role of the antilinear symmetries such as PT, the true meaning and significance of the charge operators C and the CPT-inner products, the nature of the physical observables, the equivalent description of such models using ordinary Hermitian quantum mechanics, the pertaining duality between local-non-Hermitian versus nonlocal-Hermitian descriptions of their dynamics, the corresponding classical systems, the pseudo-Hermitian canonical quantization scheme, various methods of calculating the (pseudo-) metric operators, subtleties of dealing with time-dependent quasi-Hermitian Hamiltonians and the path-integral formulation of the theory, and the structure of the state space and its ramifications for the quantum Brachistochrone problem. We also explore some concrete physical applications of the abstract concepts and tools that have been developed in the course of this investigation. These include applications in nuclear physics, condensed matter physics, relativistic quantum mechanics and quantum field theory, quantum cosmology, electromagnetic wave propagation, open quantum systems, magnetohydrodynamics, quantum chaos, and biophysics.
Abstract P T -symmetric quantum mechanics, the extension of conventional quantum mechanics to the non-Hermitian Hamiltonian invariant under the combined parity ( P ) and time reversal ( T ) symmetry, has been successfully applied to a variety of fields—such as solid state physics, mathematical physics, optics, quantum field theory. Recently, the extension of P T -symmetrical theory to entangled quantum systems has been challenged, in that P T formulation within the conventional Hilbert space violates the no-signaling principle. Here, we revisit the derivation of non-signaling principle in the framework of C P T inner product prescription. Our results preserve the no-signaling principle for a two-qubit system, reaffirm the invariance of the entanglement, and reproduce the Clauser–Horne–Shimony–Holt (CHSH) inequality. We conclude that P T -symmetric quantum mechanics satisfies the requirements for a fundamental theory and provides a consistent description of quantum systems.
Non-Hermitian quantum mechanics (NHQM) is an important alternative to the standard (Hermitian) formalism of quantum mechanics, enabling the solution of otherwise difficult problems. The first book to present this theory, it is useful to advanced graduate students and researchers in physics, chemistry and engineering. NHQM provides powerful numerical and analytical tools for the study of resonance phenomena - perhaps one of the most striking events in nature. It is especially useful for problems whose solutions cause extreme difficulties within the structure of a conventional Hermitian framework. NHQM has applications in a variety of fields, including optics, where the refractive index is complex; quantum field theory, where the parity-time (PT) symmetry properties of the Hamiltonian are investigated; and atomic and molecular physics and electrical engineering, where complex potentials are introduced to simplify numerical calculations.
As a fundamental and challenging problem in quantum information processing, quantum state discrimination (QSD) has recently been revisited from the perspective of non-Hermitian (NH) physics. The existing work about QSD in NH systems is mostly limited to PT-symmetric or pseudo-Hermitian systems with real spectra, while generic NH Hamiltonians possess complex spectra. To explore the underlying physics of quantum state discrimination in NH systems, we first demonstrate that the unambiguous discrimination of non-orthogonal quantum states can be realized with P-pseudo-Hermitian and PT-symmetric Hamiltonians in the broken phase and the required evolution time can be arbitrarily small. We further extend the regime to generic non-Hermitian Hamiltonians and demonstrate the feasibility of unambiguous discrimination. Under the same energy constraint, we establish the criteria for constructing P-pseudo-Hermitian Hamiltonians that enable unambiguous discrimination of quantum states with smaller angular separation or within shorter evolution time than any fixed PT-symmetric system. Furthermore, we also discuss the potential impact of exceptional points (EPs) on QSD. Our results suggest that the non-orthogonal eigenstates of NH Hamiltonians, rather than PT symmetry or pseudo-Hermitian symmetry, are fundamental to unambiguous discrimination and information flow in NH systems.
The basic structure of quantum mechanics was delineated in the early days of the theory and has not been modified since.One of the fundamental assumptions used in formulating the theory is that operators are represented by Hermitian matrices.In recent years it has been shown that quantum mechanics can be formulated consistently without making this assumption, using instead a combination of the parity (P) and time-reversal (T) operators and a number of other requirements related to P and T .Only the case of even T has been analyzed in the literature; here we generalize the principles to include odd time-reversal.We use this generalization to construct a non-Hermitian version of the Dirac equation, and in doing so discover a new type of particle not allowed within the (Hermitian) Standard Model.Finally we present a potential application of the ideas of non-Hermitian quantum mechanics to the unsolved problems of quantum magnetism and high temperature superconductivity.