Pseudo-unitary freedom in the operator-sum representation

G. Alagar Ramanujam

Yong-Cheng Ou, Mark Byrd

A dynamical map is a map that takes one density operator to another. Such a map can be written in an operator-sum representation (OSR) by using a spectral decomposition. The method of construction applies to general maps, which need not be completely positive. The OSR is not unique; there is freedom to choose a different set of operators in the OSR, yet still obtain the same map. Here, we show that, whereas the freedom for completely positive maps is unitary, the freedom for maps that are not necessarily completely positive is pseudounitary.

Dennis Lima, Saif M. Al-Kuwari

Abstract Pseudo-unitary circuits are recurring in both S-matrix theory and analysis of No-Go theorems. We propose a matrix and diagrammatic representation for the operation that maps S-matrices to T-matrices and, consequently, a unitary group to a pseudo-unitary one. We call this operation ‘partial inversion’ and show its diagrammatic representation in terms of permutations. We find the expressions for the deformed metrics and deformed dot products that preserve physical constraints after partial inversion. Subsequently, we define a special set that allows for the simplification of expressions containing infinities in matrix inversion. Finally, we propose a renormalized-growth algorithm for the T-matrix as a possible application. The outcomes of our study expand the methodological toolbox needed to build a family of pseudo-unitary and inter-pseudo-unitary circuits with full diagrammatic representation in three dimensions, so that they can be used to exploit pseudo-unitary flexibilization of unitary No-Go theorems and renormalized circuits of large scattering lattices.

Lev Deych

Chandana Roy

Li Wang, Jinchuan Hou, Xiaofei Qi

Instead of unitary freedom for finite-dimensional cases, bi-contractive freedom in the operator-sum representation for quantum channels of infinite-dimensional systems is established. Specifically, if the channel sends every pure state to a finite rank state, then the isometric freedom feature holds. Then, a method of computing entanglement fidelity and a relation between quantum fidelity and entanglement fidelity for infinite-dimensional systems are obtained. In addition, upper and lower bounds of the quantum fidelity, and their connection to the trace distance, are also provided.