Separability of Mixed Quantum States: Linear Contractions and Permutation Criteria

2006 ◽  
Vol 13 (01) ◽  
pp. 103-111 ◽  
Author(s):  
Michał Horodecki ◽  
Paweł Horodecki ◽  
Ryszard Horodecki
Keyword(s):  
Density Matrix ◽  
Main Idea ◽  
Necessary And Sufficient Condition ◽  
Separability Criterion ◽  
Positive Maps ◽  
Partial Transposition ◽  
Special Cases ◽  
Partial Transpose ◽  
Necessary And Sufficient

Recently, a powerful separability criterion was introduced by O. Rudolf in [5] and by K. Chen et al. in [6] — basing on realignment of elements of density matrix. Composing the main idea behind the above criterion and the necessary and sufficient condition in terms of positive maps, we provide a characterization of separable states by means of linear contractions. The latter need not be positive maps. We extend the idea to multipartite systems, and find that, somewhat suprisingly, partial realigment (unlike partial transposition) can detect genuinely tri-parite entanglement. We generalize it by introducing a family of so called permutation separability criteria for multipartite states. Namely, any permutation of indices of density matrix written in product basis leads to a separability criterion. Partial transpose and realignment criterion are special cases of permutation criteria.

On representation of Poisson mixtures as Poisson sums and a characterization of the gamma distribution

1977 ◽  
Vol 82 (2) ◽  
pp. 297-300 ◽  
Author(s):  
A. V. Godambe
Keyword(s):  
Gamma Distribution ◽  
Exponential Type ◽  
Similar Condition ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Poisson Mixture ◽  
Mixing Distribution ◽  
Poisson Mixtures ◽  
Necessary And Sufficient

AbstractA necessary and sufficient condition for a Poisson mixture with an exponential type mixing distribution to be equivalently represented as a Poisson sum is obtained. The problem of deriving a similar condition under any mixing distribution on (0, ∞) is discussed. Finally, a characterization of the gamma distribution is obtained.

A characterization of the Poisson process

1974 ◽  
Vol 11 (1) ◽  
pp. 72-85 ◽  
Author(s):  
S. M. Samuels
Keyword(s):  
Poisson Process ◽  
Renewal Process ◽  
Poisson Processes ◽  
Renewal Processes ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Complete Proof ◽  
Necessary And Sufficient

Theorem: A necessary and sufficient condition for the superposition of two ordinary renewal processes to again be a renewal process is that they be Poisson processes.A complete proof of this theorem is given; also it is shown how the theorem follows from the corresponding one for the superposition of two stationary renewal processes.

scholarly journals A Critical Review of Works Pertinent to the Einstein-Bohr Debate and Bell’s Theorem

Symmetry ◽  
2022 ◽  
Vol 14 (1) ◽  
pp. 163
Author(s):  
Karl Hess
Keyword(s):  
Quantum Theory ◽  
Physical Reality ◽  
Point Of View ◽  
Statistical Sampling ◽  
Necessary And Sufficient Condition ◽  
Special Cases ◽  
General Physical ◽  
Necessary And Sufficient ◽  
Physical Features ◽  
Chsh Inequalities

This review is related to the Einstein-Bohr debate and to Einstein–Podolsky–Rosen’s (EPR) and Bohm’s (EPRB) Gedanken-experiments as well as their realization in actual experiments. I examine a significant number of papers, from my minority point of view and conclude that the well-known theorems of Bell and Clauser, Horne, Shimony and Holt (CHSH) deal with mathematical abstractions that have only a tenuous relation to quantum theory and the actual EPRB experiments. It is also shown that, therefore, Bell-CHSH cannot be used to assess the nature of quantum entanglement, nor can physical features of entanglement be used to prove Bell-CHSH. Their proofs are, among other factors, based on a statistical sampling argument that is invalid for general physical entities and processes and only applicable for finite “populations”; not for elements of physical reality that are linked, for example, to a time-like continuum. Bell-CHSH have, furthermore, neglected the subtleties of the theorem of Vorob’ev that includes their theorems as special cases. Vorob’ev found that certain combinatorial-topological cyclicities of classical random variables form a necessary and sufficient condition for the constraints that are now known as Bell-CHSH inequalities. These constraints, however, must not be linked to the observables of quantum theory nor to the actual EPRB experiments for a variety of reasons, including the existence of continuum-related variables and appropriate considerations of symmetry.

On weak convergence within the hnbue family of life distributions

1984 ◽  
Vol 21 (03) ◽  
pp. 654-660 ◽  
Author(s):  
Sujit K. Basu ◽  
Manish C. Bhattacharjee
Keyword(s):  
Weak Convergence ◽  
Exponential Distribution ◽  
Limiting Distribution ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Moment Sequence ◽  
Positive Order ◽  
Life Distributions ◽  
Necessary And Sufficient

We show that the HNBUE family of life distributions is closed under weak convergence and that weak convergence within this family is equivalent to convergence of each moment sequence of positive order to the corresponding moment of the limiting distribution. A necessary and sufficient condition for weak convergence to the exponential distribution is given, based on a new characterization of exponentials within the HNBUE family of life distributions.

A characterization of the exponential distribution

1972 ◽  
Vol 9 (02) ◽  
pp. 457-461 ◽  
Author(s):  
M. Ahsanullah ◽  
M. Rahman
Keyword(s):  
Probability Distribution ◽  
Order Statistics ◽  
Lebesgue Measure ◽  
Random Variable ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Positive Random Variable ◽  
Absolutely Continuous ◽  
Necessary And Sufficient

A necessary and sufficient condition based on order statistics that a positive random variable having an absolutely continuous probability distribution (with respect to Lebesgue measure) will be exponential is given.

FOURTH MOMENT STRUCTURE OF THE GARCH(p,q) PROCESS

1999 ◽  
Vol 15 (6) ◽  
pp. 824-846 ◽  
Author(s):  
Changli He ◽  
Timo Teräsvirta
Keyword(s):  
Statistical Theory ◽  
Autocorrelation Function ◽  
Fourth Moment ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Special Cases ◽  
The Moment ◽  
Necessary And Sufficient

In this paper, a necessary and sufficient condition for the existence of the unconditional fourth moment of the GARCH(p,q) process is given and also an expression for the moment itself. Furthermore, the autocorrelation function of the centered and squared observations of this process is derived. The statistical theory is further illustrated by a few special cases such as the GARCH(2,2) process and the ARCH(q) process.

On the general bilinear time series model

1988 ◽  
Vol 25 (3) ◽  
pp. 553-564 ◽  
Author(s):  
Jian Liu ◽  
Peter J. Brockwell
Keyword(s):  
Time Series ◽  
Stationary Solution ◽  
Time Series Model ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Special Cases ◽  
Non Linear ◽  
Necessary And Sufficient ◽  
Special Case

A sufficient condition is derived for the existence of a strictly stationary solution of the general bilinear time series equations. The condition is shown to reduce to the conditions of Pham and Tran (1981) and Bhaskara Rao et al. (1983) in the special cases which they consider. Under the condition specified, a solution is constructed which is shown to be causal, stationary and ergodic. It is moreover the unique causal solution and the unique stationary solution of the defining equations. In the special case when the defining equations contain no non-linear terms, our condition reduces to the well-known necessary and sufficient condition for existence of a causal stationary solution.

A dynamical characterization of diagonal-preserving -isomorphisms of graph -algebras

2017 ◽  
Vol 38 (7) ◽  
pp. 2401-2421 ◽  
Author(s):  
SARA E. ARKLINT ◽  
SØREN EILERS ◽  
EFREN RUIZ
Keyword(s):  
Directed Graphs ◽  
Sufficient Condition ◽  
Orbit Equivalence ◽  
Necessary And Sufficient Condition ◽  
Graph Algebras ◽  
Boundary Path ◽  
Necessary And Sufficient

We characterize when there exists a diagonal-preserving $\ast$-isomorphism between two graph $C^{\ast }$-algebras in terms of the dynamics of the boundary path spaces. In particular, we refine the notion of ‘orbit equivalence’ between the boundary path spaces of the directed graphs $E$ and $F$ and show that this is a necessary and sufficient condition for the existence of a diagonal-preserving $\ast$-isomorphism between the graph $C^{\ast }$-algebras $C^{\ast }(E)$ and $C^{\ast }(F)$.

scholarly journals QKSpaces on the Unit Circle

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Jizhen Zhou
Keyword(s):  
Sobolev Space ◽  
Unit Circle ◽  
Measurable Function ◽  
Weight Functions ◽  
Sufficient Condition ◽  
General Criterion ◽  
Necessary And Sufficient Condition ◽  
New Space ◽  
Necessary And Sufficient

We introduce a new spaceQK(∂D)of Lebesgue measurable functions on the unit circle connecting closely with the Sobolev space. We obtain a necessary and sufficient condition onKsuch thatQK(∂D)=BMO(∂D), as well as a general criterion on weight functionsK1andK2,K1≤K2, such thatQK1(∂D)QK2(∂D). We also prove that a measurable function belongs toQK(∂D)if and only if it is Möbius bounded in the Sobolev spaceLK2(∂D). Finally, we obtain a dyadic characterization of functions inQK(∂D)spaces in terms of dyadic arcs on the unit circle.

A CHARACTERIZATION OF KNOTS IN A SPATIAL GRAPH

2000 ◽  
Vol 09 (08) ◽  
pp. 1069-1084 ◽  
Author(s):  
KAZUKO ONDA
Keyword(s):  
Finite Graph ◽  
Restriction Map ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Spatial Graph ◽  
Necessary And Sufficient

For a finite graph G, let Γ(G) be the set of all cycles of G. Suppose that for each γ∈Γ(G), an embedding ϕγ:γ→S3 is given. A set {ϕγ|γ∈Γ(G)} of embeddings is said to be realizable if there is an embedding f:G→S3 such that the restriction map f|γ is ambient isotopic to ϕγ for any γ∈Γ(G). In this paper on seven specified graphs G, we give a necessary and sufficient condition for the set {ϕγ|γ∈Γ(G)} to be realizable by using the second coefficients of Conway polynomials of knots.

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