A novel entanglement criterion of two-qubit system

2020 ◽  
Vol 34 (05) ◽  
pp. 2050022
Author(s):  
Chao-Ying Zhao ◽  
Qi-Zhi Guo ◽  
Wei-Han Tan
Keyword(s):  
Density Matrix ◽  
Analytic Solution ◽  
Sufficient Conditions ◽  
High Dimensional ◽  
Dimensional System ◽  
Hard Problem ◽  
Qubit System ◽  
Separability Problem ◽  
Low Dimensional ◽  
Necessary And Sufficient

The “separability problem” in quantum information theory is a quite important and well-known hard problem. The low-dimensional system satisfies the PPT criterion. However, the high-dimensional system problem has been shown to be NP-hard problem. In general, it is very difficult to find the analytic solution of the density matrix for the high-dimensional system. Therefore, getting an analytic solution for two-qubit system is an interesting and useful problem. We propose a novel criterion for separability and entanglement-verification of two-qubit system. We expressed the density matrix by a sum of a principal density matrix and six separable density matrices. The necessary and sufficient conditions for the two-qubit system include that if the four involved coefficients [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and the principal density matrix [Formula: see text] are separable, then the two-qubit system is separable, otherwise the two-qubit system is entangled. Finally, our criterion results in a totally different conclusion compared to Horodecki’s criterion. We believe that the new criterion is more stringent than existing PPT methods.

A simple entanglement criterion of two-qubit system

2019 ◽  
Vol 33 (18) ◽  
pp. 1950197 ◽  
Author(s):  
Chao-Ying Zhao ◽  
Qi-Zhi Guo ◽  
Wei-Han Tan
Keyword(s):  
Density Matrix ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Density Matrices ◽  
Qubit System ◽  
Novel Method ◽  
Matrix Formula ◽  
Necessary And Sufficient ◽  
Entanglement Criterion

We present a novel method to judge the entanglement of two-qubit system, the density matrix of a two-qubit system can be constituted by the principle density matrix [Formula: see text] and the separable density matrices [Formula: see text]–[Formula: see text]. The necessary and sufficient conditions for the two-qubit system, the three involved coefficients p [Formula: see text] 0, p [Formula: see text] 0, p [Formula: see text] 0, and the principal density matrix [Formula: see text] being separable, then the two-qubit system is separable, otherwise the two-qubit system is entangled. Finally, the criterion for several known density matrices have been verified in this way.

Separability criteria based on the Bloch representation of density matrices

2007 ◽  
Vol 7 (7) ◽  
pp. 624-638
Author(s):  
J. de Vicente
Keyword(s):  
Sufficient Conditions ◽  
Quantum Systems ◽  
Necessary Condition ◽  
Sufficient Condition ◽  
Pure Product ◽  
Separability Problem ◽  
Arbitrary Dimensions ◽  
Necessary And Sufficient ◽  
Bloch Representation

We study the separability of bipartite quantum systems in arbitrary dimensions using the Bloch representation of their density matrix. This approach enables us to find an alternative characterization of the separability problem, from which we derive a necessary condition and sufficient conditions for separability. For a certain class of states the necessary condition and a sufficient condition turn out to be equivalent, therefore yielding a necessary and sufficient condition. The proofs of the sufficient conditions are constructive, thus providing decompositions in pure product states for the states that satisfy them. We provide examples that show the ability of these conditions to detect entanglement. In particular, the necessary condition is proved to be strong enough to detect bound entangled states.

scholarly journals Computer-controlled variable-structure systems

1992 ◽  
Vol 34 (1) ◽  
pp. 1-17 ◽  
Author(s):  
Xinghuo Yu ◽  
Renfrey B. Potts
Keyword(s):  
Computer Control ◽  
Sufficient Conditions ◽  
Variable Structure ◽  
Dimensional System ◽  
Sliding Modes ◽  
Variable Structure Systems ◽  
Computer Controlled ◽  
Two Dimensional System ◽  
Necessary And Sufficient ◽  
Theoretical Results

AbstractA theory is developed for the computer control of variable-structure systems, using periodic zero-order-hold sampling. A simple two-dimensional system is first analysed, and necessary and sufficient conditions for the occurrence of pseudo-sliding modes are discussed. The method is then applied to a discrete model of a cylindrical robot. The theoretical results are illustrated by computer simulations.

Information-Theoretic Characterization of Bifurcations in Nonlinear Dynamical Systems With Noise

2013 ◽  
Author(s):  
Navendu S. Patil ◽  
Joseph P. Cusumano
Keyword(s):  
Time Series ◽  
Time Series Data ◽  
Nonlinear Dynamical Systems ◽  
Period Doubling ◽  
Entropy Rate ◽  
Series Data ◽  
High Dimensional ◽  
Dimensional System ◽  
Dimensional Manifold ◽  
Low Dimensional

Detecting bifurcations in noisy and/or high-dimensional physical systems is an important problem in nonlinear dynamics. Near bifurcations, the dynamics of even a high dimensional system is typically dominated by its behavior on a low dimensional manifold. Since the system is sensitive to perturbations near bifurcations, they can be detected by looking at the apparent deterministic structure generated by the interaction between the noise and low-dimensional dynamics. We use minimal hidden Markov models built from the noisy time series to quantify this deterministic structure at the period-doubling bifurcations in the two-well forced Duffing oscillator perturbed by noise. The apparent randomness in the system is characterized using the entropy rate of the discrete stochastic process generated by partitioning time series data. We show that as the bifurcation parameter is varied, sharp changes in the statistical complexity and the entropy rate can be used to locate incipient bifurcations.

scholarly journals Spectral methods for nonlinear functionals and functional differential equations

2021 ◽  
Vol 8 (2) ◽  
Author(s):  
Daniele Venturi ◽  
Alec Dektor
Keyword(s):  
Differential Equations ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Approximate Solutions ◽  
Stability And Convergence ◽  
High Dimensional ◽  
Functional Derivatives ◽  
Nonlinear Functionals ◽  
Functional Differential ◽  
Necessary And Sufficient

AbstractWe present a rigorous convergence analysis for cylindrical approximations of nonlinear functionals, functional derivatives, and functional differential equations (FDEs). The purpose of this analysis is twofold: First, we prove that continuous nonlinear functionals, functional derivatives, and FDEs can be approximated uniformly on any compact subset of a real Banach space admitting a basis by high-dimensional multivariate functions and high-dimensional partial differential equations (PDEs), respectively. Second, we show that the convergence rate of such functional approximations can be exponential, depending on the regularity of the functional (in particular its Fréchet differentiability), and its domain. We also provide necessary and sufficient conditions for consistency, stability and convergence of cylindrical approximations to linear FDEs. These results open the possibility to utilize numerical techniques for high-dimensional systems such as deep neural networks and numerical tensor methods to approximate nonlinear functionals in terms of high-dimensional functions, and compute approximate solutions to FDEs by solving high-dimensional PDEs. Numerical examples are presented and discussed for prototype nonlinear functionals and for an initial value problem involving a linear FDE.

On the analytic solution of certain functional and difference equations

1983 ◽  
Vol 389 (1796) ◽  
pp. 1-13 ◽  
Keyword(s):  
Functional Equation ◽  
Analytic Solution ◽  
Sufficient Conditions ◽  
Irreducible Polynomial ◽  
Constant Solution ◽  
Simple Change ◽  
Change Of Variable ◽  
The Difference ◽  
Necessary And Sufficient ◽  
Complex Coefficients

Let P(u, v) be an irreducible polynomial with complex coefficients and let q ≥ 2 be an integer. We establish the necessary and sufficient conditions under which the functional equation P(f(z), f(z q )) = 0, (F) has a non-constant analytic solution that is either regular in a neighbour­hood of the point z = 0 or has a pole at this point (theorem 1). By a simple change of variable, the difference equation P ( F(Z) , F ( Z + 1)) = 0, (D) can be proved under the same restrictions to have a non-constant solution of the form F(Z) = Σ ∞ j=I f j e -jq z , which is regular in the strip Re Z ≥ X 0 , |Im Z | < π/2 ln q , if X 0 is sufficiently large (theorem 2).

scholarly journals Quantum Strassen’s theorem

2020 ◽  
Vol 23 (03) ◽  
pp. 2050020
Author(s):  
Shmuel Friedland ◽  
Jingtong Ge ◽  
Lihong Zhi
Keyword(s):  
Density Matrix ◽  
Hilbert Spaces ◽  
Sufficient Conditions ◽  
Trace Class ◽  
Product Spaces ◽  
Finite Product ◽  
Infinite Dimensional ◽  
Quantum Analog ◽  
Trace Class Operators ◽  
Necessary And Sufficient

Strassen’s theorem circa 1965 gives necessary and sufficient conditions on the existence of a probability measure on two product spaces with given support and two marginals. In the case where each product space is finite, Strassen’s theorem is reduced to a linear programming problem which can be solved using flow theory. A density matrix of bipartite quantum system is a quantum analog of a probability matrix on two finite product spaces. Partial traces of the density matrix are analogs of marginals. The support of the density matrix is its range. The analog of Strassen’s theorem in this case can be stated and solved using semidefinite programming. The aim of this paper is to give analogs of Strassen’s theorem to density trace class operators on a product of two separable Hilbert spaces, where at least one of the Hilbert spaces is infinite-dimensional.

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