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Journal of Physics A Mathematical and Theoretical

Fixed points of commutative super-operators

Long Long, Shifang Zhang

Published Feb 10, 2011• 13 citations
Publisher

Abstract

In this paper we study fixed points of commutative super-operators. First, under some circumstances, we discuss the question of necessary and sufficient conditions for the set of fixed points of a super-operator to be non-trivial, then we establish a structure lemma to completely characterize this set. Using this lemma, we show that if a commutative super-operator is also a generalized Lüders operator, then its fixed points are precisely those operators intertwining the two sequences of operation elements associated with it. Specifically, the main results of Weihua and Junde ( 2010 J. Phys. A: Math. Theor. 43 395206 ) are special cases of our results.

DOI
10.1088/1751-8113/44/9/095201

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In this paper we study fixed points of commutative super-operators. First, under some circumstances, we discuss the question of necessary and sufficient conditions for the set of fixed points of a super-operator to be non-trivial, then we establish a structure lemma to completely characterize this set. Using this lemma, we show that if a commutative super-operator is also a generalized Lüders operator, then its fixed points are precisely those operators intertwining the two sequences of operation elements associated with it. Specifically, the main results of Weihua and Junde ( 2010 J. Phys. A: Math. Theor. 43 395206 ) are special cases of our results.
Quantum operations frequently occur in quantum measurement theory, quantum probability, quantum computation, and quantum information theory. If an operator A is invariant under a quantum operation φ, we call A a φ-fixed point. Physically, the φ-fixed points are the operators that are not disturbed by the action of φ. Our main purpose is to answer the following question. If A is a φ-fixed point, is A compatible with the operation elements of φ? We shall show in general that the answer is no and we shall give some sufficient conditions under which the answer is yes. Our results will follow from some general theorems concerning completely positive maps and injectivity of operator systems and von Neumann algebras.
We formulate a general framework for the study of operator systems arising from discrete groups. We study in detail the operator system of the free group on nnn generators, as well as the operator systems of the free products of finitely many copies of the two-element group Z2\mathbb Z_2Z2​. We examine various tensor products of group operator systems, including the minimal, the maximal, and the commuting tensor products. We introduce a new tensor product in the category of operator systems and formulate necessary and sufficient conditions for its equality to the commuting tensor product in the case of group operator systems. We express various sets of quantum correlations studied in the theoretical physics literature in terms of different tensor products of operator systems of discrete groups. We thus recover earlier results of Tsirelson and formulate a new approach for the study of quantum correlation boxes.
This paper is a continuation of [J. Phys. A: Math. Theor. 55, 405203 (2022)], in which a set of matrix elements of local operators was computed for the XXZ spin-1/2 open chain with a particular case of unparallel boundary fields. Here, we extend these results to the more general case in which both fields are non-longitudinal and related by one constraint, allowing for a partial description of the spectrum by usual Bethe equations. More precisely, the complete spectrum and eigenstates can be characterized within the Separation of Variables (SoV) framework. One uses here the fact that, under the constraint, a part of this SoV spectrum can be described via solutions of a usual, homogeneous, TQ-equation, with corresponding transfer matrix eigenstates coinciding with generalized Bethe states. We explain how to generically compute the action of a basis of local operators on such kind of states, and this under the most general boundary condition on the last site of the chain. As a result, we can compute the matrix elements of some of these basis elements in any eigenstate described by the homogenous TQ-equation. Assuming, following a conjecture of Nepomechie and Ravanini, that the ground state itself can be described in this framework, we obtain multiple integral representations for these matrix elements in the half-infinite chain limit, generalizing those previously obtained in the case of longitudinal boundary fields and in the case of the special boundary conditions considered in [J. Phys. A: Math. Theor. 55, 405203 (2022)].