The fact that eigenvalues of PT-symmetric Hamiltonians H can be real for some values of a parameter and complex for others is explained by showing that the matrix elements of H, and hence the secular equation, are real, not only for PT but also for any antiunitary operator A satisfying A 2k = 1 with k odd. The argument is illustrated by a 2 × 2 matrix Hamiltonian, and two examples of the generalization are given.
We propose a general method for introducing extensive characteristics of quantum entanglement. The method relies on polynomials of nilpotent raising operators that create entangled states acting on a reference vacuum state. By introducing the notion of tanglemeter, the logarithm of the state vector represented in a special canonical form and expressed via polynomials of nilpotent variables, we show how this description provides a simple criterion for entanglement as well as a universal method for constructing the invariants characterizing entanglement. We compare the existing measures and classes of entanglement with those emerging from our approach. We derive the equation of motion for the tanglemeter and, in representative examples of up to four-qubit systems, show how the known classes appear in a natural way within our framework. We extend our approach to qutrits and higher-dimensional systems, and make contact with the recently introduced idea of generalized entanglement. Possible future developments and applications of the method are discussed.
According to Hudson's theorem, any pure quantum state with a positive Wigner function is necessarily a Gaussian state. Here, we make a step toward the extension of this theorem to mixed quantum states by finding upper and lower bounds on the degree of non-Gaussianity of states with positive Wigner functions. The bounds are expressed in the form of parametric functions relating the degree of non-Gaussianity of a state, its purity, and the purity of the Gaussian state characterized by the same covariance matrix. Although our bounds are not tight, they permit us to visualize the set of states with positive Wigner functions. The Wigner representation of quantum states ͓1͔, which is realized by joint quasiprobability distributions of canonically conjugate variables in phase space, has a specific property which differentiates it from a true probability distribution: it can attain negative values. Among pure states, it was proven by Hudson ͓2͔ ͑and later generalized to multimode quantum systems by Soto and Claverie ͓3͔͒ that the only states which have non-negative Wigner functions are Gaussian states ͓4͔. The question that naturally arises ͓2͔ is whether this theorem can be extended to mixed states, among which not only Gaussian states may possess a positive Wigner function. A logical extension of the theorem would be a complete characterization of the convex set of states with positive Wigner function. Although this question can be approached by using the notion of Wigner spectrum ͓5͔, a simple and operational extension of Hudson's theorem has not yet been achieved due to the mathematical complications which emerge when dealing with states with positive Wigner functions ͓5͔.Motivated by the increasing interest for non-Gaussian states in continuous-variable quantum information theory ͑see, e.g., ͓6͔͒ and the need for a better understanding of the de-Gaussification procedures for mixed states ͑see, e.g., ͓7͔͒, we attempt here an exploration of the set of states with positive Wigner functions using Gaussian states as a reference. More precisely, we consider the subset of such states that have the same covariance matrix as a reference Gaussian state. We obtain a partial solution to the problem by analytically deriving necessary conditions ͑bounds͒ on a measure of non-Gaussianity for a state to have a positive Wigner function. This set of conditions bounds a region in a threedimensional space with coordinates being the purity of the state, the purity of the corresponding Gaussian state, and the non-Gaussianity. As intuitively expected, the maximum degree of non-Gaussianity increases with a decrease in the purity of both the state and its Gaussian corresponding state.Before deriving the main results of this paper, let us recall a convenient representation of the trace of the product of two one-mode quantum states, and Ј, in terms of the Wigner representation ͓8͔,where W is the Wigner function of the state . For example, the purity of a state, ͓͔ =Tr͑ 2 ͒, may be calculated with the help of this formula. For a state with a G...
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