On circle preserving quadratic operators
type: Item
instance of: scholarly article
title: On circle preserving quadratic operators
zbMATH Open document ID: 1370.46044
author: Farruh Mukhamedov
published in: Bulletin of the Malaysian Mathematical Sciences Society. Second Series
publication date: 2017-04-20T00:00:00Z
review text: Let \(U : M_2(\mathbb C) \otimes M_2(\mathbb C) \to M_2(\mathbb C) \otimes M_2(\mathbb C)\) be the linear operator such that \(U(x \otimes y) = y \otimes x\) for all \(x, y \in M_2(\mathbb C)\). A linear operator \(\Delta : M_2(\mathbb C) \to M_2(\mathbb C) \otimes M_2(\mathbb C)\) is said to be a quantum quadratic operator (q.q.o.)\ if it is unital and positive, i.e., \(\Delta x \geq 0\) whenever \(x \geq 0\). The operator \(\Delta\) is a quasi-quantum quadratic operator (quasi q.q.o.)\ if is is unital, \(*\)-preserving and \(V_{\Delta}(\varphi)\in M_2(\mathbb C)\), where \(\varphi \in M_2(\mathbb C)_+^*\). Also, it is symmetric if one has \(U \Delta=\Delta\). A quasi q.q.o.\ \(\Delta\) is called \(q\)-pure if for any pure state \(\varphi\), the state \(V _{\Delta}(\varphi)\) is also pure. In the present paper, the author shows that any \(q\)-pure anti-symmetric quasi q.q.o.\ is not positive. He considers quadratic operators, which preserve the unit circle, and shows that the corresponding quasi q.q.o.\ cannot be positive. He also classifies \(q\)-pure circle preserving quadratic operators into three disjoint classes. Moreover, he proves that a quasi q.q.o.\ corresponding to the first class is block positive.
reviewed by: Maryam Amyari
Mathematics Subject Classification ID: 46L55
Mathematics Subject Classification ID: 47B48
zbMATH DE Number: 6706530
zbMATH Keywords: quasi-quantum quadratic operator
zbMATH Keywords: positive operator
zbMATH Keywords: pure
zbMATH Keywords: symmetric
zbMATH Keywords: anti-symmetric
zbMATH Keywords: circle
MaRDI profile type: Publication
full work available at URL: s40840-015-0240-z
OpenAlex ID: W2192508837
cites work: On Classical and Quantum Liftings
cites work: Geometry of Quantum States
cites work: Quantum-correlation breaking channels, quantum conditional probability and Perron-Frobenius theory
cites work: A class of symmetric Bell diagonal entanglement witnesses -- a geometric perspective
cites work: Geometry of quantum states: new construction of positive maps
cites work: Exposed Positive Maps in M4(ℂ)
cites work: Ergodic properties of discrete quadratic stochastic processes defined on von Neumann algebras
cites work: QUADRATIC STOCHASTIC OPERATORS AND PROCESSES: RESULTS AND OPEN PROBLEMS
cites work: Quantum entanglement
cites work: Separability of mixed states: necessary and sufficient conditions.
cites work: Q3724143
cites work: On a characterization of positive maps
cites work: On expansion of quantum quadratic stochastic processes into fibrewise Markov processes defined on von Neumann algebras
cites work: On Kadison-Schwarz type quantum quadratic operators on \(\mathbb M_2(\mathbb C)\)
cites work: On pure quasi-quantum quadratic operators of \(\mathbb M_2(\mathbb C)\)
cites work: Quantum quadratic operators and processes
cites work: On quantum quadratic operators of \(\mathbb M_2(\mathbb C)\) and their dynamics
cites work: Q2706552
cites work: An analysis of completely-positive trace-preserving maps on \({\mathcal M}_{2}\)
cites work: Positive Linear Maps of Operator Algebras
DOI: 10.1007/S40840-015-0240-Z
description: scientific article; zbMATH DE number 6706530