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Laser Physics Letters

A note on hidden quantum discord in two- and three-qubit systems under local filtering operations

Chunyan Yang, Youwang Xiao, Jing Wang, Ming Li, Lei Li, Shu‐Qian Shen

Published Jun 9, 2022• 0 citations
Publisher

Abstract

Abstract One of the primary goal of quantum information theory is to understand and quantify the various correlations that exist in quantum systems. Quantum correlation is broader than quantum entanglement, and quantum discord has always attracted more attention. In this paper, we investigate the hidden properties of quantum discord. In particular, for a class of two-qubit states, we prove the existence of hidden quantum discord under local filtering operations and give sufficient conditions for the absence of hidden quantum discord. For a class of three-qubit states, we investigate hidden geometric discord and analyze its dynamic behavior under several conditions. Furthermore, by comparing two- and three-qubit systems, we discover the inconsistency of hidden discord for ρ = 1 2 N ( I + ∑ j = 1 3 c j σ j ⊗ … ⊗ σ j ) ( c 3 = 1 , N = 2 , 3 ) .

DOI
10.1088/1612-202x/ac6fc4

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•May 4, 2012•0 citations
Abstract One of the primary goal of quantum information theory is to understand and quantify the various correlations that exist in quantum systems. Quantum correlation is broader than quantum entanglement, and quantum discord has always attracted more attention. In this paper, we investigate the hidden properties of quantum discord. In particular, for a class of two-qubit states, we prove the existence of hidden quantum discord under local filtering operations and give sufficient conditions for the absence of hidden quantum discord. For a class of three-qubit states, we investigate hidden geometric discord and analyze its dynamic behavior under several conditions. Furthermore, by comparing two- and three-qubit systems, we discover the inconsistency of hidden discord for ρ = 1 2 N ( I + ∑ j = 1 3 c j σ j ⊗ … ⊗ σ j ) ( c 3 = 1 , N = 2 , 3 ) .
We evaluate analytically the quantum discord for a large family of multiqubit states. It is interesting to note that the quantum discord of three qubits and five qubits is the same, as is the quantum discord of two qubits and six qubits. We discover that the quantum discord of this family of states can be classified into three categories. The level surfaces of the quantum discord in the three categories are shown through images. Furthermore, we investigated the dynamic behavior of quantum discord under decoherence. For the odd partite systems, we prove the frozen phenomenon of quantum discord does not exist under the phase flip channel, while it can be found in the even partite systems.
Quantum discord characterizes "nonclassicality" of correlations in quantum mechanics. It has been proposed as the key resource present in certain quantum communication tasks and quantum computational models without containing much entanglement. We obtain a necessary and sufficient condition for the existence of nonzero quantum discord for any dimensional bipartite states. This condition is easily experimentally implementable. Based on this, we propose a geometrical way of quantifying quantum discord. For two qubits this results in a closed form of expression for discord. We apply our results to the model of deterministic quantum computation with one qubit, showing that quantum discord is unlikely to be the reason behind its speedup.
The counterintuitive effect of non-unique ordering of two-qubit states with quantum entanglement measures was discovered over ten years ago. More precisely, it was shown by Monte Carlo simulations that there exist states for which the entanglement of formation and the negativity do not impose the same ordering of states, i.e. E F (ρ AB )⩽(⩾)E F (ρ' AB ) is not equivalent to N (ρ AB )⩽(⩾) N (ρ' AB ). Recently, it was discovered that quantum discord and the geometric quantum discord do not necessarily imply the same ordering of two-qubit X -states, which means that the lack of the unique ordering of states with quantum entanglement measures goes beyond entanglement. Inspired by this observation, we study the problem of the states ordering with quantum discords, considering two-qubit Bell-diagonal states for analytical simplicity. In particular, we identify some classes of states for which the states ordering with quantum discords is preserved as long as the states belong to the same class and give a few illustrative examples.
Open quantum systems have attracted great attention, since inevitable coupling between quantum systems and their environment greatly affects the features of interest of these systems. Quantum discord, is a measure of the total nonclassical correlation in a quantum system that includes, but is not exclusive to, the distinct property of quantum entanglement. Quantum discord can exist in separated quantum states and plays an important role in many fundamental physics problems and practical quantum information tasks. There have been numerous investigations on quantum discord and its counterpart classical correlation. This short review focuses on highlighting the system–environment dynamics of two-qubit quantum discord and the influence of initial system–environment correlations on the dynamics of open quantum systems. The external control effect on the dynamics of open quantum systems are involved. Several related experimental works are discussed.
Quantum discord quantifies quantum correlations beyond entanglement and assumes nonzero values, which are notoriously hard to compute, for almost all quantum states. Here we provide computable tight bounds for the quantum discord for qubit-qudit states. In the case of two qubits our lower and upper bounds coincide for a 7-parameter family of filtered XXX-states, whose quantum discords can therefore be evaluated analytically. An application to the accessible information of the binary qubit channel is also presented. For the qubit-qudit state output by the circuit of deterministic computation with one qubit, nontrivial lower and upper bounds that respect the zero-discord conditions are obtained for its quantum discord.
Open quantum systems have attracted great attention, since inevitable coupling between quantum systems and their environment greatly affects the features of interest of these systems. Quantum discord, is a measure of the total nonclassical correlation in a quantum system that includes, but is not exclusive to, the distinct property of quantum entanglement. Quantum discord can exist in separated quantum states and plays an important role in many fundamental physics problems and practical quantum information tasks. There have been numerous investigations on quantum discord and its counterpart classical correlation. This short review focuses on highlighting the system-environment dynamics of two-qubit quantum discord and the influence of initial system-environment correlations on the dynamics of open quantum systems. The external control effect on the dynamics of open quantum systems are involved. Several related experimental works are discussed.