Nonlinear Channels of Werner States

Abstract

We study the nonlinear positive map of the density matrix of two-qubit Werner states, called the nonlinear channel. The map ρ → Φ(ρ) is realized by the rational function Φ. We discuss the influence of the map on the entanglement properties of the transformed density matrix. We investigate the violation of the Bell inequality (CHSH inequality) for the two-qubit state Φ(ρ). The nonlinear channels under discussion create the entangled state from a separable Werner state. We study the quantum spin tomograms of the states.

Sensitivity to Initial Noise in Measurement-Induced Nonlinear Quantum Dynamics

“…Repeating the above procedure can lead to entanglement distillation [3][4][5]. The same scheme acting pairwise arXiv:1808.08151v1 [quant-ph] 24 Aug 2018 on an ensemble of identically prepared systems in a pure quantum state may result in a postselected ensemble in a nonlinearly transformed pure state, providing a possible physical realization of nonlinear quantum channels [6,7]. Iterating such nonlinear quantum state transformations may result in strong dependence on the initial conditions and in complex chaos [8].…”

Maps of Matrices and Portrait Maps of the Density Operators of Composite and Noncomposite Systems

“…In quantum information theory, the completely positive maps are called the quantum channels [6]. If the function Φ(ρ) does not describe the linear transform of the density matrix, the map is called the nonlinear map or the nonlinear quantum channel [7]. In the probability representation of quantum states [8][9][10][11][12], the examples of nonlinear positive maps were given in [13] on the example of unitary tomograms of qudit states.…”

“…The common properties of the probability vectors within the classical and quantum frameworks were studied in [17]. Recently, it was observed that the properties of quantum entanglement and other aspects of quantum correlations [7,[18][19][20][21][22] existing in composite quantum systems, e.g., in the form of entropic inequalities [20,[23][24][25][26][27], exist also in a single qudit system.…”

Characterization of the nonlinear qubit map using the probability parametrization

“…Moreover, it is interesting to investigate a model of a nonlinear map of density matrices ρ → Φ(ρ) for which the matrix Φ(ρ) has the properties of density matrices. In the papers [18,19] the nonlinear map ρ → Φ n (ρ) = ρ n /Trρ n with positive integer n was introduced. It can be easily seen that the map Φ n preserves the properties of density matrices, namely hermiticity, non-negativity and trace.…”

“…The quantum relative entropy is the quantum-mechanical analog of the classical relative entropy (18). Indeed, let p(x) and q(x) be the eigenvalues of ρ and σ, respectively.…”