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Journal of Physics A Mathematical and Theoretical

Local channels preserving the states without measurement-induced nonlocality

Yu Jing Guo, Jinchuan Hou

Published Jul 24, 2013• 10 citations
Publisher

Abstract

Measurement-induced nonlocality is a measure of nonlocality introduced recently by Luo and Fu. We present here sufficient and necessary conditions for a quantum state for which this quantity is equal to zero. Furthermore it is shown that for such a state ?ab with any local channel acting on Ha cannot create measurement-induced nonlocality if and only if either it is a completely contractive channel or it is a nontrivial isotropic channel. For the qubit case this property is an additional characteristic of the completely contractive channel or the commutativity-preserving unital channel.

DOI
10.1088/1751-8113/46/32/325301

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Measurement-induced nonlocality is a measure of nonlocality introduced recently by Luo and Fu. We present here sufficient and necessary conditions for a quantum state for which this quantity is equal to zero. Furthermore it is shown that for such a state ?ab with any local channel acting on Ha cannot create measurement-induced nonlocality if and only if either it is a completely contractive channel or it is a nontrivial isotropic channel. For the qubit case this property is an additional characteristic of the completely contractive channel or the commutativity-preserving unital channel.
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 quantify the measurement-induced nonlocality [Luo and Fu, Phys. Rev. Lett. 106, 120401 (2011)] from the perspective of the relative entropy. This quantification leads to an operational interpretation for the measurement-induced nonlocality, namely, it is the maximal entropy increase after the locally invariant measurements. The relative entropy of nonlocality is upper bounded by the entropy of the measured subsystem. We establish a relationship between the relative entropy of nonlocality and the geometric nonlocality based on the Hilbert-Schmidt norm, and show that it is equal to the maximal distillable entanglement. Several trade-off relations are obtained for tripartite pure states. We also give explicit expressions for the relative entropy of nonlocality for Bell-diagonal states.
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.
A novel resource, i.e. measurement-induced nonlocality, was proposed by Luo and Fu [ Phys. Rev. Lett. 106 (2011) 120401 ]. It can directly reflect the usefulness of quantum nonlocality in quantum protocols. We investigate the dynamics of measurement-induced nonlocality by exactly solving a model which consists of two independent atoms each subject to a zero-temperature muti-mode cavity. We find that the dynamics of measurement-induced nonlocality is discontinuity under some special conditions. The reason for this phenomenon is the constraint of the von Neumann measurements which do not disturb the reduced density operator. We also find the sudden death phenomenon and the sudden birth phenomenon.
Measurement-Induced NonLocality was introduced by Luo and Fu (Phys. Rev. Lett. \textbf{106}, 120401,(2011)) as a measure of nonlocality in a bipartite state. In this paper we will discuss monogamy property of measurement-induced nonlocality for some three- and four-qubit classes of states. Unlike discord, we find quite surprising results in this situation. Both the GHZ and W states satisfy monogamy relations in the three-qubit case, however, in general there are violations of monogamy relations in both the GHZ-class and W-class states. In case of four-qubit system, monogamy holds for most of the states in the generic class. Four qubit GHZ does not satisfy monogamy relation, but W-state does. We provide several numerical results including counterexamples regarding monogamy nature of measurement induced nonlocality. We will also extend our results of generalized W-class to n-qubit.