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arXiv:1206.2159 · Journal of Physics A Mathematical and Theoretical

Necessary and sufficient conditions for the local creation of quantum discord

Yu Guo, Jinchuan Hou

Published Mar 28, 2013• 23 citations
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Abstract

Quantum discord (QD) is one of the main quantum correlations in quantum information theory. In this paper, we show that a local channel cannot create QD for zero QD states of size d ⩾ 3 if and only if either it is a completely decohering channel or it is a nontrivial isotropic channel. For the qubit case this property is an additional characteristic of the completely decohering channel or the commutativity-preserving unital channel. In particular, the exact forms of the completely decohering channel and the commutativity-preserving unital qubit channel are proposed. Consequently, our results confirm and improve the conjecture proposed by Hu et al for the case of d ⩾ 3 and improve the result proposed by Streltsov et al for the qubit case.

DOI
10.1088/1751-8113/46/15/155301

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Quantum discord (QD) is one of the main quantum correlations in quantum information theory. In this paper, we show that a local channel cannot create QD for zero QD states of size d ⩾ 3 if and only if either it is a completely decohering channel or it is a nontrivial isotropic channel. For the qubit case this property is an additional characteristic of the completely decohering channel or the commutativity-preserving unital channel. In particular, the exact forms of the completely decohering channel and the commutativity-preserving unital qubit channel are proposed. Consequently, our results confirm and improve the conjecture proposed by Hu et al for the case of d ⩾ 3 and improve the result proposed by Streltsov et al for the qubit case.
Quantum correlation can be created by a local operation from some initially classical states. We prove that the necessary and sufficient condition for a local trace-preserving channel to create quantum correlation is that it is not a commutativity-preserving channel. This condition is valid for arbitrary finite dimension systems. We also derive the explicit form of commutativity-preserving channels. For a qubit, a commutativity-preserving channel is either a completely decohering channel or a mixing channel. For a three-dimensional system (qutrit), a commutativity-preserving channel is either a completely decohering channel or an isotropic channel.
We identify and characterize all the local quantum channels that preserves the set of classical states, i.e., does not create any quantum correlations. At first we show that the quantum correlations cannot be created from without if and only if the local quantum channel preserves the commutativity, i.e., the images of any two commuting states also commute. And then we provide an operational necessary and sufficient criterion for a known quantum channel to preserve the commutativity as well as a single observable to witness an arbitrary unknown commutativity-preserving channel. All the distance-based measures for quantum correlations, e.g., the geometric measure, are non-increasing while the quantum discord defined by von Neumann measurements can be increasing or decreasing under local commutativity-preserving channels.
We prove the condition for local trace-preserving channels to create quantum correlation from initially classical states. For two-qubit states, the necessary and sufficient condition for a channel that cannot create quantum correlation in any initially classical state is that it is either a completely decohering channel or a mixing channel. For states in higher dimensions, we show that even a mixing channel can create quantum correlation from a classically correlated state. Our results reveal that mixedness is important for creation of quantum correlation in high-dimension quantum systems.
We show that two qubits, initially in a fully classical state, can develop significant quantum correlations as measured by the quantum discord (QD) under the action of a local memoryless noise (specifically we consider the case of a Markovian amplitude-damping channel). This is analytically proven after deriving in a compact form the QD for the class of separable states involved in such a process. We provide a picture in the Bloch sphere that unambiguously highlights the physical mechanism behind the effect regardless of the specific measure of quantum correlations adopted.
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.
Through comparisons with an entanglement breaking (EB) channel, we characterize the local quantum channels that destroy quantum correlations, especially for quantum discord, and give a detailed description of a discord-breaking (DB) channel. We prove that the quantum discord can be broken if and only if the local quantum channel is a commutativity-creating (CoC) channel. Then we show which kind of channels is a CoC channel, and address the relation between EB and DB channels. Furthermore we apply our results to three typical noise channels.