An effective iterative method to build the Naimark extension of rank-n POVMs

2017 ◽  
Vol 15 (04) ◽  
pp. 1750029 ◽  
Author(s):  
Nicola Dalla Pozza ◽  
Matteo G. A. Paris
Keyword(s):  
Hilbert Space ◽  
Iterative Method ◽  
Projective Measurement ◽  
Probability Operator

We revisit the problem of finding the Naimark extension of a probability operator-valued measure (POVM), i.e. its implementation as a projective measurement in a larger Hilbert space. In particular, we suggest an iterative method to build the projective measurement from the sole requirements of orthogonality and positivity. Our method improves existing ones, as it may be employed also to extend POVMs containing elements with rank larger than one. It is also more effective in terms of computational steps.

scholarly journals When do Projections Commute?

1980 ◽  
Vol 35 (4) ◽  
pp. 437-441 ◽  
Author(s):  
W. Rehder
Keyword(s):  
Hilbert Space ◽  
Conditional Probability ◽  
Mechanical Systems ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Quantum Mechanical ◽  
Probability Operator ◽  
Quantum Mechanical Systems ◽  
Necessary And Sufficient ◽  
New Criteria

Abstract Necessary and sufficient conditions for commutativity of two projections in Hilbert space are given through properties of so-called conditional connectives which are derived from the conditional probability operator PQP. This approach unifies most of the known proofs, provides a few new criteria, and permits certain suggestive interpretations for compound properties of quantum-mechanical systems.

scholarly journals Explicit Scheme for Fixed Point Problem for Nonexpansive Semigroup and Split Equilibrium Problem in Hilbert Space

2014 ◽  
Vol 2014 ◽  
pp. 1-12
Author(s):  
Pei Zhou ◽  
Gou-Jie Zhao
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Iterative Method ◽  
Equilibrium Problems ◽  
Fixed Point Problem ◽  
Explicit Scheme ◽  
Common Element ◽  
Nonexpansive Semigroup ◽  
Point Problem ◽  
Split Equilibrium Problem

We establish an iterative method for finding a common element of the set of fixed points of nonexpansive semigroup and the set of split equilibrium problems. Under suitable conditions, some strong convergence theorems are proved. Our works improve previous results for nonexpansive semigroup.

scholarly journals A New Iterative Method for Equilibrium Problems and Fixed Point Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Abdul Latif ◽  
Mohammad Eslamian
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Variational Inequality ◽  
Iterative Method ◽  
Equilibrium Problems ◽  
Variational Inequality Problem ◽  
Real Hilbert Space ◽  
Common Element ◽  
Continuous Mappings ◽  
New Iterative Method

Introducing a new iterative method, we study the existence of a common element of the set of solutions of equilibrium problems for a family of monotone, Lipschitz-type continuous mappings and the sets of fixed points of two nonexpansive semigroups in a real Hilbert space. We establish strong convergence theorems of the new iterative method for the solution of the variational inequality problem which is the optimality condition for the minimization problem. Our results improve and generalize the corresponding recent results of Anh (2012), Cianciaruso et al. (2010), and many others.

Application of g-frames in conjugate gradient

2016 ◽  
Vol 7 (3) ◽  
Author(s):  
Hassan Jamali ◽  
Neda Momeni
Keyword(s):  
Hilbert Space ◽  
Iterative Method ◽  
Operator Equation ◽  
Conjugate Gradient ◽  
Separable Hilbert Space

AbstractThis paper proposes an iterative method for solving an operator equation on a separable Hilbert space

scholarly journals Cyclic Iterative Method for Strictly Pseudononspreading in Hilbert Space

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Bin-Chao Deng ◽  
Tong Chen ◽  
Zhi-Fang Li
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Variational Inequality ◽  
Iterative Method ◽  
Strong Convergence ◽  
Common Fixed Point ◽  
Convex Subset ◽  
Real Hilbert Space ◽  
Cyclic Algorithm ◽  
The Common

Let{Ti}i=1NbeNstrictly pseudononspreading mappings defined on closed convex subsetCof a real Hilbert spaceH. Consider the problem of finding a common fixed point of these mappings and introduce cyclic algorithms based on general viscosity iteration method for solving this problem. We will prove the strong convergence of these cyclic algorithm. Moreover, the common fixed point is the solution of the variational inequality〈(γf-μB)x*,v-x*〉≤0,∀v∈⋂i=1NFix(Ti).

scholarly journals Modified inertial double Mann type iterative algorithm for a bivariate weakly nonexpansive operator

2020 ◽  
Vol 36 (1) ◽  
pp. 127-139
Author(s):  
ANANTACHAI PADCHAROEN ◽  
KAMONRAT SOMBUT
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Iterative Method ◽  
Iterative Algorithm ◽  
Convex Subset ◽  
Fixed Point Theory ◽  
Coupled Fixed Point ◽  
Point Theory ◽  
Nonexpansive Operator ◽  
The One

"We introduce a modified inertial double Mann type iterative method to approximate coupled solutions of a bivariate nonexpansive operator T : C x C→ C, where C is a nonempty closed and convex subset of a Hilbert space. The one theorem and complement important old and recent results in coupled fixed point theory. Some appropriate examples to illustrate our results and their generalization are also given. "

scholarly journals Continuous method of second order with constant coefficients for monotone equations in Hilbert space

2020 ◽  
Vol 22 (4) ◽  
pp. 449-455
Author(s):  
Irina P. Ryazantseva
Keyword(s):  
Hilbert Space ◽  
Iterative Method ◽  
Sufficient Conditions ◽  
Variable Coefficients ◽  
Second Order ◽  
Order Difference ◽  
Monotone Equations ◽  
Sufficient Conditions For Convergence ◽  
Second Order Difference Equation ◽  
Constant Coefficients

Convergence of an implicit second-order iterative method with constant coefficients for nonlinear monotone equations in Hilbert space is investigated. For non-negative solutions of a second-order difference numerical inequality, a top-down estimate is established. This estimate is used to prove the convergence of the iterative method under study. The convergence of the iterative method is established under the assumption that the operator of the equation on a Hilbert space is monotone and satisfies the Lipschitz condition. Sufficient conditions for convergence of proposed method also include some relations connecting parameters that determine the specified properties of the operator in the equation to be solved and coefficients of the second-order difference equation that defines the method to be studied. The parametric support of the proposed method is confirmed by an example. The proposed second-order method with constant coefficients has a better upper estimate of the convergence rate compared to the same method with variable coefficients that was studied earlier.

scholarly journals A New Iterative Algorithm for Pseudomonotone Equilibrium Problem and a Finite Family of Demicontractive Mappings

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
F. U. Ogbuisi ◽  
F. O. Isiogugu
Keyword(s):  
Hilbert Space ◽  
Iterative Method ◽  
Equilibrium Problem ◽  
Numerical Experiments ◽  
Real Hilbert Space ◽  
Convex Combination ◽  
Convergence Result ◽  
Finite Family ◽  
Demicontractive Mappings ◽  
New Iterative Method

In this paper, we introduce a new iterative method in a real Hilbert space for approximating a point in the solution set of a pseudomonotone equilibrium problem which is a common fixed point of a finite family of demicontractive mappings. Our result does not require that we impose the condition that the sum of the control sequences used in the finite convex combination is equal to 1. Furthermore, we state and prove a strong convergence result and give some numerical experiments to demonstrate the efficiency and applicability of our iterative method.

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