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Quantum Information Processing

Measurement-induced geometric measures of correlations based on the Hellinger distance and their local decoherence

Zbigniew Walczak

Published Sep 3, 2019• 11 citations
Publisher

Abstract

We apply the modified Brodutch and Modi method of constructing geometric measures of correlations to obtain analytical expressions for measurement-induced geometric classical and quantum correlations based on the Hellinger distance for two-qubit Bell diagonal states. Moreover, we study dynamics of the classical and quantum correlations for these states under decoherence induced by the action of bit flip, bit-phase flip, phase flip, depolarizing and generalized amplitude damping local quantum channels and identify all possible types of the decoherence dynamics. In particular, we show that these correlations exhibit freezing, sudden change and sudden transition between classical and quantum decoherence phenomena.

DOI
10.1007/s11128-019-2437-9

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We apply the modified Brodutch and Modi method of constructing geometric measures of correlations to obtain analytical expressions for measurement-induced geometric classical and quantum correlations based on the Hellinger distance for two-qubit Bell diagonal states. Moreover, we study dynamics of the classical and quantum correlations for these states under decoherence induced by the action of bit flip, bit-phase flip, phase flip, depolarizing and generalized amplitude damping local quantum channels and identify all possible types of the decoherence dynamics. In particular, we show that these correlations exhibit freezing, sudden change and sudden transition between classical and quantum decoherence phenomena.
Abstract The classification and quantification of quantum correlations are the important issues in quantum information theory. Quantum relative entropy and Rényi relative entropy are both non-negative, and can therefore be regarded as generalized distances between quantum states. We first introduced the method proposed by Brodutch and Modi for constructing measurements of classical and quantum correlations. Subsequently, we use this method to construct measurement-induced geometric classical and quantum correlations based on quantum relative entropy and Rényi− 1 2 relative entropy. For two-qubit Bell diagonal states, the analytic expressions for these geometric classical correlations and quantum correlations are obtained.
We investigate the square-norm distance correlation dynamics of the Bell-diagonal states under different local decoherence channels, including phase flip, bit flip, and bit-phase flip channels by employing the geometric discord (GD) and its modified geometric discord (MGD), as the measures of the square-norm distance correlations. Moreover, an explicit comparison between them is made in detail. The results show that there is no distinct dominant relative ordering between them. Furthermore, we obtain that the GD just gradually deceases to zero, while MGD initially has a large freezing interval, and then suddenly changes in evolution. The longer the freezing interval, the less the MGD is. Interestingly, it is shown that the dynamic behaviors of the two geometric discords under the three noisy environments for the Werner-type initial states are the same.
We investigate the dynamics of correlations for two-parameter qubit-qutrit states under various local decoherence channels including depahsing, phase-flip, bit- and trit-flip, bit- and trit-phase-flip, and depolarizing channels. We find that, under certain conditions, the classical correlations may not be affected by the noise or decay monotonically. The quantum correlations measured by measurement-induced disturbance (MID) show three types of dynamical behaviors: (i) monotonic decay to zero, (ii) monotonic decay to a nonzero steady value, (iii) increase from zero and then decrease to zero in a monotonic way. Consequently, we find that, differing from the dynamics of entanglement, the present classical and quantum correlations do not reveal sudden death behavior.
We provide analytical expressions for classical and total trace-norm (Schatten 1-norm) geometric correlations in the case of two-qubit XXX states. As an application, we consider the open-system dynamical behavior of such correlations under phase and generalized amplitude damping evolutions. Then, we show that geometric classical correlations can characterize the emergence of the pointer basis of an apparatus subject to decoherence in either Markovian or non-Markovian regimes. In particular, as a non-Markovian effect, we obtain a time delay for the information to be retrieved from the apparatus by a classical observer. Moreover, we show that the set of initial XXX states exhibiting sudden transitions in the geometric classical correlation has nonzero measure.