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arXiv:2405.10245

A Graph-Theoretical Framework to Analyse Zero Discord Quantum States

Anoopa Joshi, Parvinder Singh, Kumar, Atul

Published May 16, 2024• 0 citations
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Abstract

This article comprehensively explores matrices and their prerequisites for achieving positive semidefiniteness. The study delves into a series of theorems concerning pure quantum states in the context of weighted graphs. The main objective of this study is to establish a graph-theoretic framework for the study of quantum discord and to identify the necessary and sufficient conditions for zero quantum discord states using unitary operators. This research aims to advance the understanding of quantum discord and its implications for quantum information theory with a graph-theoretic framework.

DOI
10.48550/arxiv.2405.10245

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This article comprehensively explores matrices and their prerequisites for achieving positive semidefiniteness. The study delves into a series of theorems concerning pure quantum states in the context of weighted graphs. The main objective of this study is to establish a graph-theoretic framework for the study of quantum discord and to identify the necessary and sufficient conditions for zero quantum discord states using unitary operators. This research aims to advance the understanding of quantum discord and its implications for quantum information theory with a graph-theoretic framework.
This work is at the interface of graph theory and quantum mechanics. Quantum correlations epitomize the usefulness of quantum mechanics. Quantum discord is an interesting facet of bipartite quantum correlations. Earlier, it was shown that every combinatorial graph corresponds to quantum states whose characteristics are reflected in the structure of the underlined graph. A number of combinatorial relations between quantum discord and simple graphs were studied. To extend the scope of these studies, we need to generalize the earlier concepts applicable to simple graphs to weighted graphs, corresponding to a diverse class of quantum states. To this effect, we determine the class of quantum states whose density matrix representation can be derived from graph Laplacian matrices associated with a weighted directed graph and call them graph Laplacian quantum states. We find the graph theoretic conditions for zero and nonzero quantum discord for these states. We apply these results on some important pure two qubit states, as well as a number of mixed quantum states, such as the Werner, Isotropic, and [Formula: see text]-states. We also consider graph Laplacian states corresponding to simple graphs as a special case.
Quantum discord is usually referred to as a measure for quantum correlations. In the search of the fundamental resource to gain a quantum advantage in quantum information applications, quantum discord is considered a promising candidate. In this paper we present an alternative view on quantum correlations in terms of the rank of the correlation matrix as introduced by Daki\ifmmode \acute{c}\else \'{c}\fi{}, Vedral, and Brukner [Phys. Rev. Lett. 105, 190502 (2010)]. According to our analysis, information about the quantum discord does not necessarily determine the amount of quantum correlations but rather the quantumness of the state. Nonzero quantum discord is only a necessary but not a sufficient condition for correlations above the classically achievable limit. This becomes clear when we consider states of nonzero discord, which can be created from zero discord states only by a single local operation. We further show that the set of these states has measure zero.
In this paper we determine the class of quantum states whose density matrix representation can be derived from graph Laplacian matrices associated with a weighted directed graph and we call them graph Laplacian quantum states. Then we obtain structural properties of these graphs such that the corresponding graph Laplacian states have zero quantum discord by investigating structural properties of clustered graphs which provide a family of commuting normal matrices formed by the blocks of its Laplacian matrices. We apply these results on some important mixed quantum states, such as the Werner, Isotropic, and XXX-states.
Quantum discord is a measure of the quantumness of correlations. After reviewing its different versions and properties, we apply it to the questions of quantum information processing. First we show that changes in discord in the processed unentangled states indicate the need for entanglement in the distributed implementation of quantum gates. On the other hand, it was shown that zero system-environment discord is a necessary and sufficient condition for applicability of the standard completely positive description of the system's evolution. We demonstrate that this result does not translate into useful quantum process tomography. Depending on the details of the preparation procedure only absence of any initial correlations may guarantees consistency of the process tomography.
Quantum discord is a measure of the quantumness of correlations. After reviewing its different versions and properties, we apply it to the questions of quantum information processing. First we show that changes in discord in the processed unentangled states indicate the need for entanglement in the distributed implementation of quantum gates. On the other hand, it was shown that zero system-environment discord is a necessary and sufficient condition for applicability of the standard completely positive description of the system's evolution. We demonstrate that this result does not translate into useful quantum process tomography. Depending on the details of the preparation procedure only absence of any initial correlations may guarantees consistency of the process tomography.