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Physical Review A

Norm-based measurement of quantum correlation

Yu-Chun Wu, Guang‐Can Guo

Published Jun 3, 2011• 16 citations
Publisher

Abstract

In this paper we derived a necessary and sufficient condition for classical correlated states and proposed a norm-based measurement $Q$ of quantum correlation. Using the max norm of operators, we gave the expression of the quantum correlation measurement $Q$ and investigated the dynamics of $Q$ in Markovian and non-Markovian cases, respectively. $Q$ decays exponentially and vanishes only asymptotically in the Markovian case and causes periodical death and rebirth in the non-Markovian case. In the pure state, the quantum correlation $Q$ is always larger than the entanglement, which was different from other known measurements. In addition, we showed that locally broadcastable and broadcastable are equivalent and reproved the density of quantum correlated states.

DOI
10.1103/physreva.83.062301

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In this paper we derived a necessary and sufficient condition for classical correlated states and proposed a norm-based measurement QQQ of quantum correlation. Using the max norm of operators, we gave the expression of the quantum correlation measurement QQQ and investigated the dynamics of QQQ in Markovian and non-Markovian cases, respectively. QQQ decays exponentially and vanishes only asymptotically in the Markovian case and causes periodical death and rebirth in the non-Markovian case. In the pure state, the quantum correlation QQQ is always larger than the entanglement, which was different from other known measurements. In addition, we showed that locally broadcastable and broadcastable are equivalent and reproved the density of quantum correlated states.
In order to measure the quantum correlation of a bipartite state quickly, an easy method is to construct a test matrix through the commutations among the blocks of its density matrix. Then, the Frobenius norm of the test matrix can be used to measure the quantum correlation. In this paper, we apply the measurement by Frobenius norm ([Formula: see text] to the dynamics evolution of the non-Markovian quantum system and compare it with the typical quantum discord ([Formula: see text] proposed by Ollivier and Zurek. The research results show that [Formula: see text] can indeed measure the quantum correlation of a bipartite state as same as [Formula: see text]. Further studies find that there are still differences between the two measurements: in some regions, when [Formula: see text] is zero, [Formula: see text] is not zero. It indicates that [Formula: see text] is more detailed than [Formula: see text] to measure quantum correlation of a bipartite state.
We address the problem of quantifying the non-markovian character of quantum time evolutions of general systems in contact with an environment. We introduce two different measures of non-markovianity that exploit the specific traits of quantum correlations and are suitable for opposite experimental contexts. When complete tomographic knowledge about the evolution is available, our measure provides a necessary and sufficient condition to quantify strictly the non-markovianity. In the opposite case, when no information whatsoever is available, we propose a sufficient condition for non-markovianity. Remarkably, no optimization procedure underlies our derivation, which greatly enhances the practical relevance of the proposed criteria.
Abstract We propose an alternative non-Markovianity measure of open quantum systems, which is based on max-information. The max-information is a one-shot generalization of the mutual information derived from the max-relative entropy. This operational interpretation provides a new site for viewing the non-Markovian process. As applications, some typical noisy channels have been investigated: for phase damping and amplitude damping channels, non-Markovianity measure based on max-information is equivalent to the three previous non-Markovianity measures, i.e. the measures based on the distinguishability, entanglement of environment and the mutual information. For the random unitary channel, the proposed measurement in the witness of Markovianity is equivalent to the measures based on quantum trace distance and thus overlaps partly with the measurement based on the mutual information. It shows that max-information may be a better candidate for correlation-based measurement of non-Markovian than the mutual information.
The problem of recognizing (non-)Markovianity of a quantum dynamics is revisited through analyzing quantum correlations. We argue that instantaneouslyvanishing quantum discord provides a necessary and sufficient condition for Markovianity of a quantum map. This is used to introduce a measure of non-Markovianity. This measure, however, requires demanding knowledge about the system and the environment. By using a quantum correlation monogamy property and an ancillary system, we propose a simplified measure with fewer requirements. Non-Markovianity is thereby decided by quantum state tomography of the system and the ancilla.
We theoretically study the non-Markovian disentanglement dynamics of a two-qubit system coupled to nonequilibrium environments with nonstationary and non-Markovian random telegraph noise statistical properties. The reduced density matrix of the two-qubit system can be expressed as the Kraus representation in terms of the tensor products of the single qubit Kraus operators. We derive the relation between the entanglement and nonlocality of the two-qubit system which are both closely associated with the decoherence function. We identify the threshold values of the decoherence function to ensure the existences of the concurrence and nonlocal quantum correlations for an arbitrary evolution time when the two-qubit system is initially prepared in the composite Bell states and the Werner states, respectively. It is shown that the environmental nonequilibrium feature can suppress the disentanglement dynamics and reduce the entanglement revivals in non-Markovian dynamics regime. In addition, the environmental nonequilibrium feature can enhance the nonlocality of the two-qubit system. Moreover, the entanglement sudden death and rebirth phenomena and the transition between quantum and classical nonlocalities closely depend on the parameters of the initial states and the environmental parameters in nonequilibrium environments.
We analyze the dynamics of the entanglement in two independent non-Markovian channels. In particular, we focus on the entanglement dynamics as a function of the initial states and the channel parameters like the temperature and the ratio rrr between ω0ω_0ω0​ the characteristic frequency of the quantum system of interest, and ωcω_cωc​ the cut-off frequency of Ohmic reservoir. We give a stationary analysis of the concurrence and find that the dynamic of non-markovian entanglement concurrence Cρ(t)\mathcal{C}_ρ(t)Cρ​(t) at temperature kBT=0k_BT=0kB​T=0 is different from the kBT>0k_BT>0kB​T>0 case. We find that "entanglement sudden death" (ESD) depends on the initial state when kBT=0k_BT=0kB​T=0, otherwise the concurrence always disappear at finite time when kBT>0k_BT>0kB​T>0, which means that ESD must happen. The main result of this paper is that the non-Markovian entanglement dynamic is fundamentally different from the Markovian one. In the Markovian channel, entanglement decays exponentially and vanishes only asymptotically, but in the non-Markovian channel the concurrence Cρ(t)\mathcal{C}_ρ(t)Cρ​(t) oscillates, especially in the high temperature case. Then an open-loop controller adjusted by the temperature is proposed to control the entanglement and prolong the ESD time.