In a recent paper (arXiv:1905.07348v1), Dyson scheme to deal with density matrix of non-Hermitian Hamiltonians has been used to investigate the entanglement of states of a PT-symmetric bosonic system. They found that von Neumann entropy can show a different behavior in the broken and unbroken regime. We show that their results can be recast in terms of an abstract model of pseudo-Hermitian random matrices. It is found however that, although the formalism is practically the same, the entanglement is not of Fock states but of Bell states.
We provide the first solution of a time-dependent metric operator for the non-Hermitian Jaynes-Cummings Hamiltonian. We use this solution to calculate the entanglement between two identical isolated such Hamiltonians. The presence of a non-Hermitian interaction term leads to a spontaneously broken $\mathcal{PT}$-symmetric regime which manifests itself in the exotic time-evolution of entanglement. When the symmetry is broken, oscillatory modes transition into decay. As such that there is a drastic difference in behaviour between the broken and unbroken regimes.
We derive several entanglement conditions employing non-hermitian operators. We start with two conditions that were derived previously for field mode operators, and use them to derive conditions that can be used to show the existence of field-atom entanglement and entanglement between groups of atoms. The original conditions can be strengthened by making them invariant under certain sets of local unitary transformations, such as Gaussian operations. We then apply these conditions to several examples, such as the Dicke model. We conclude with a short discussion of how local uncertainty relation…
In this paper the entanglement of multi-qubit fermionic pseudo Hermitian coherent states (FPHCS) described by anticommutative Grassmann numbers is studied. The pseudo-Hermitian versions of the well known maximally entangled pure states such as Bell and GHZ, W and biseparable states are introduced through integrating over the tensor products of FPHCSs with suitable choice of Grassmannian weight functions. Meanwhile as an illustration, the method is applied to tensor product of 2 and 3 qubit pseudo Hermitian systems. Then the measures of concurrence and average entropy are applied to quantify th…
We investigate the connection between pseudo-Hermitian and Hermitian descriptions for a lattice, which consists of a set of isomorphic pseudo-Hermitian clusters. We show that such non-Hermitian systems can act as Hermitian systems. This is made possible by considering the dynamics of a state involving the superposition of a single eigenmode of each isomorphic clusters. It still holds when multiple eigenmodes are involved when additional restriction on the state is imposed. This Hermitian dynamics is demonstrated for the case of an exactly solvable PT-symmetric ladder system.
We study the quantum entanglement and separability of Hermitian and pseudo-Hermitian systems of identical bosonic or fermionic particles with point interactions. The separability conditions are investigated in detail.
Entanglement is one of the key feature of quantum world that has no classical counterpart. This arises due to the linear superposition principle and the tensor product structure of the Hilbert space when we deal with multiparticle systems. In this paper, we will introduce the notion of entanglement for quantum systems that are governed by non-Hermitian yet PT-symmetric Hamiltonians. We will show that maximally entangled states in usual quantum theory behave like non-maximally entangled states in PT-symmetric quantum theory. Furthermore, we will show how to create entanglement between two PTQub…
We generalize a recently proposed approach for the construction of pseudo-Hermitian Hamiltonians with real spectra. Present technique is based on a simple and straightforward similarity transformation of the coordinate and momentum.
A complexified von Roos Hamiltonian is considered and a Hermitian first-order intertwining differential operator is used to obtain the related position dependent mass $η$-weak-pseudo-Hermitian Hamiltonians. Using a Liouvillean-type change of variables, the $η$-weak-pseudo-Hermitian von Roos Hamiltonians H(x) are mapped into the traditional Schrodinger Hamiltonian form H(q), where exact isospectral correspondence between H(x) and H(q) is obtained. Under a user-friendly position dependent mass settings, it is observed that for each exactly-solvable $η$-weak-pseudo-Hermitian reference-Hamiltonian…
We study the time evolution of quantum systems with a time-dependent non-Hermitian Hamiltonian given by a linear combination of SU(1,1) and SU(2) generators.With a time-dependent metric, the pseudo-Hermitian invariant operator is constructed in the same manner as for both the SU(1,1) and SU(2) systems. The exact common solutions of the Schrödinger equations for both the SU(1,1) and SU(2) systems are obtained in terms of eigenstates of the pseudo-Hermitian invariant operator.
We investigate some questions on the construction of $η$ operators for pseudo-Hermitian Hamiltonians. We give a sufficient condition which can be exploited to systematically generate a sequence of $η$ operators starting from a known one, thereby proving the non-uniqueness of $η$ for a particular pseudo-Hermitian Hamiltonian. We also study perturbed Hamiltonians for which $η$'s corresponding to the original Hamiltonian still work.
We consider a class of (possibly nondiagonalizable) pseudo-Hermitian operators with discrete spectrum, showing that in no case (unless they are diagonalizable and have a real spectrum) they are Hermitian with respect to a semidefinite inner product, and that the pseudo-Hermiticity property is equivalent to the existence of an antilinear involutory symmetry. Moreover, we show that a typical degeneracy of the real eigenvalues (which reduces to the well known Kramers degeneracy in the Hermitian case) occurs whenever a fermionic (possibly nondiagonalizable) pseudo-Hermitian Hamiltonian admits an a…
We extend the definition of generalized parity $P$, charge-conjugation $C$ and time-reversal $T$ operators to nondiagonalizable pseudo-Hermitian Hamiltonians, and we use these generalized operators to describe the full set of symmetries of a pseudo-Hermitian Hamiltonian according to a fourfold classification. In particular we show that $TP$ and $CTP$ are the generators of the antiunitary symmetries; moreover, a necessary and sufficient condition is provided for a pseudo-Hermitian Hamiltonian $H$ to admit a $P$-reflecting symmetry which generates the $P$-pseudounitary and the $P$-pseudoantiunit…
We introduce creation and annihilation operators of pseudo-Hermitian fermions for two-level systems described by pseudo-Hermitian Hamiltonian with real eigenvalues. This allows the generalization of the fermionic coherent states approach to such systems. Pseudo-fermionic coherent states are constructed as eigenstates of two pseudo-fermion annihilation operators. These coherent states form a bi-normal and bi-overcomplete system, and their evolution governed by the pseudo-Hermitian Hamiltonian is temporally stable. In terms of the introduced pseudo-fermion operators the two-level system' Hamilto…
Attention has been brought to the possibility that statistical fluctuation properties of several complex spectra, or, well-known number sequences may display strong signatures that the Hamiltonian yielding them as eigenvalues is PT-symmetric (Pseudo-Hermitian). We find that the random matrix theory of pseudo-Hermitian Hamiltonians gives rise to new universalities of level-spacing distributions other than those of GOE, GUE and GSE of Wigner and Dyson. We call the new proposals as Gaussian Pseudo-Orthogonal Ensemble and Gaussian Pseudo-Unitary Ensemble. We are also led to speculate that the enig…
We study the pseudo-Hermitian systems with general spin-coupling point interactions and give a systematic description of the corresponding boundary conditions for PT-symmetric systems. The corresponding integrability for both bosonic and fermionic many-body systems with PT-symmetric contact interactions is investigated.
We consider a generalization of recently proposed non-Hermitian model for resonant cavities coupled by a chiral mirror by taking into account number non-conservation and nonlinear interactions. We analyze non-Hermitian quantum dynamics of populations and entanglement of the cavity modes. We find that an interplay of initial coherence and non-Hermitian coupling leads to a counterintuitive population transfer. While an initially coherent cavity mode is depleted, the other empty cavity can be populated more or less than the initially filled one. Moreover, presence of nonlinearity yields populatio…
We formulate a systematic algorithm for constructing a whole class of Hermitian position-dependent-mass Hamiltonians which, to lowest order of perturbation theory, allow a description in terms of PT-symmetric Hamiltonians. The method is applied to the Hermitian analogue of the PT-symmetric cubic anharmonic oscillator. A new example is provided by a Hamiltonian (approximately) equivalent to a PT-symmetric extension of the one-parameter trigonometric Poschl-Teller potential.
Quantum sensing exploits fundamental features of quantum system to achieve highly efficient measurement of physical quantities. Here, we propose a strategy to realize a single-qubit pseudo-Hermitian sensor from a dilated two-qubit Hermitian system. The pseudo-Hermitian sensor exhibits divergent susceptibility in dynamical evolution that does not necessarily involve exceptional point. We demonstrate its potential advantages to overcome noises that cannot be averaged out by repetitive measurements. The proposal is feasible with the state-of-art experimental capability in a variety of qubit syste…
A class of non-Hermitian d-dimensional Hamiltonias with position dependent mass and their $η$-pseudo-Hermiticity generators is presented. Illustrative examples are given in 1D, 2D, and 3D for different position dependent mass settings.
We show that non-Hermitian dynamics generate substantial entanglement in many-body systems. We consider the non-Hermitian Lipkin-Meshkov-Glick model and show that its phase transition occurs with maximum multiparticle entanglement: there is full N-particle entanglement at the transition, in contrast to the Hermitian case. The non-Hermitian model also exhibits more spin squeezing than the Hermitian model, showing that non-Hermitian dynamics are useful for quantum metrology. Experimental implementations with trapped ions and cavity QED are discussed.
We give an explicit characterization of the most general quasi-Hermitian operator H, the associated metric operators η_+, and η_+-pseudo-Hermitian operators acting in two-dimensional complex Euclidean space C^2. These operators represent the physical observables of a model whose Hamiltonian and Hilbert space are respectively H and C^2 endowed with the inner product defined by η_+. Our calculations allows for a direct demonstration of the fact that the choice of an irreducible family of observables fixes the metric operator up to a multiplicative factor.
We investigate in this paper time-dependent non-Hermitian Hamiltonians, which consist respectively of SU(1,1) and SU(2) generators. The former Hamiltonian is PT symmetric but the latter one is not. A time-dependent non-unitary operator is proposed to construct the non-Hermitian invariant, which is verified as pseudo-Hermitian with real eigenvalues. The exact solutions are obtained in terms of the eigenstates of the pseudo-Hermitian invariant operator for both the SU(1,1)and SU(2)systems in a unified manner. Then, we derive the LR phase, which can be separated to the dynamic phase and the geome…
Recently some authors have broadened the scope of canonical quantum mechanics by replacing the conventional Hermiticity condition on the Hamiltonian by a weaker requirement through the introduction of the notion of pseudo-Hermiticity. In the present study we investigate eigenvalues, transmission and reflection from complex optical potentials enjoying the property of pseudo-Hermiticity.
We prove that any symmetric Hamiltonian that is a quadratic function of the coordinates and momenta has a pseudo-Hermitian adjoint or regular matrix representation. The eigenvalues of the latter matrix are the natural frequencies of the Hamiltonian operator. When all the eigenvalues of the matrix are real, then the spectrum of the symmetric Hamiltonian is real and the operator is Hermitian. As illustrative examples we choose the quadratic Hamiltonians that model a pair of coupled resonators with balanced gain and loss, the electromagnetic self-force on an oscillating charged particle and an ac…
In contrast to classical systems, actual implementation of non-Hermitian Hamiltonian dynamics for quantum systems is a challenge because the processes of energy gain and dissipation are based on the underlying Hermitian system-environment dynamics that is trace preserving. Recently, a scheme for engineering non-Hermitian Hamiltonians as a result of repetitive measurements on an anicillary qubit has been proposed. The induced conditional dynamics of the main system is described by the effective non-Hermitian Hamiltonian arisng from the procedure. In this paper we demonstrate the effectiveness o…
In study of pseudo(quasi)-hermitian operators, the key role is played by the positive-definite metric operator. It enables physical interpretation of the considered systems. In the article, we study the pseudo-hermitian systems with constant number of particles in equilibrium. We show that the explicit knowledge of the metric operator is not essential for study of thermodynamic properties of the system. We introduce a simple example where the physically relevant quantities are derived without explicit calculation of either metric operator or spectrum of the Hamiltonian.
The concept of energy-dependent forces in quantum mechanics is re-analysed. We suggest a simplification of their study via the representation of each self-adjoint and energy-dependent Hamiltonian H=H(E) with real spectrum by an auxiliary non-Hermitian operator K which remains energy-independent. Practical merits of such an approach to the Schroedinger equations with energy-dependent potentials are illustrated using their quasi-exactly solvable example.
We investigate a two-level spin system based anti-parity-time (anti-$\mathcal{PT}$)-symmetric qubit and study its decoherence as well as entanglement entropy properties. We compare our findings with that of the corresponding $\mathcal{PT}$-symmetric and Hermitian qubits. First we consider the time-dependent Dyson map to find the exact analytical dynamics for a general non-Hermitian qubit system coupled with a bath, then we specialize it to the case of the anti-$\mathcal{PT}$-symmetric qubit. We find that the decoherence function for the anti-$\mathcal{PT}$-symmetric qubit decays slower than th…
Non-Hermiticity and dephasing, collaborating in an unusual wave packet dynamics, realizes unconventional entanglement evolution in a disordered, interacting and asymmetric (non-reciprocal) quantum medium. Taking the Hatano-Nelson model as a concrete example, we first consider how wave packet spreads in a non-Hermitian disordered system for demonstraing that it is very different from the Hermitian case. Interestingly, a cascade like wave packet spreading as in the Hermitian case is suppressed in the clean limit and at weak disorder, while it revives in the vicinity of the localization-delocaliz…
Hamiltonian Mechanics works for conserved systems and Quantum Mechanics is given in Hamiltonian language. It is considered that complexifying the quantum Hamiltonian, a balanced loss and gain model can be created. The usual mathematics of density operator formalism and entanglement is extrapolated to such systems and the consequences are studied. Namely, a complete formalism using Density operators is created for real eigenvalue regime of these Non- Hermitian systems and correct forms of Von-Neumann and Entanglement Entropy are created. The consequences are studied in this regime and depicted …
We consider a couple of examples to study the pseudo-Hermitian interaction in relativistic quantum mechanics. Rasbha interaction, commonly used to study the spin Hall effect, is considered with imaginary coupling. The corresponding Dirac Hamiltonian is shown to be parity pseudo-Hermitian. In the other example we consider parity pseudo-Hermitian scalar interaction with arbitrary parameter in Dirac theory. In both the cases we show that the energy spectrum is real and all the other features of non-relativistic pseudo-Hermitian formulation are present. Using the spectral method the positive defin…
We demonstrate how to discriminate two non-orthogonal, entangled quantum state which are slightly different from each other by using pseudo-Hermitian system. The positive definite metric operator which makes the pseudo-Hermitian systems fully consistent quantum theory is used for such a state discrimination. We further show that non-orthogonal states can evolve through a suitably constructed pseudo-Hermitian Hamiltonian to orthogonal states. Such evolution ceases at exceptional points of the pseudo-Hermitian system.
Usually duality process keeps energy spectrum invariant. In this paper, we provide a duality, which keeps entanglement spectrum invariant, in order to diagnose quantum entanglement of non-Hermitian non-interacting fermionic systems. We limit our attention to non-Hermitian systems with a complete set of biorthonormal eigenvectors and an entirely real energy spectrum. The original system has a reduced density matrix $ρ_\mathrm{o}$ and the real space is partitioned via a projecting operator $\mathcal{R}_{\mathrm o}$. After dualization, we obtain a new reduced density matrix $ρ_{\mathrm{d}}$ and a…
