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Bimodal behavior of measurement-dependent one-way quantum deficit for two-qubit X states

Mikhail A. Yurischev

Published Jun 9, 2017• 1 citation
OpenAlex

Abstract

It is shown that in some regions of X-state space the measurement-dependent one-way quantum deficit \Delta as a function of measurement angle \theta\in[0,\pi/2] exhibits a bimodal behavior inside the open interval (0,\pi/2), i.e., it has two interior extrema: one minimum and one maximum. Moreover, cases are found when the interior minimum of such a bimodal function \Delta(\theta) is less than that one at the endpoint with \theta=0 or \theta=\pi/2. This refutes the result of Ye et al. [Int. J. Theor. Phys. {\bf 55}, 2237 (2016)] which was intended for obtaining the one-way quantum deficit of general two-qubit X states.

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It is shown that in some regions of X-state space the measurement-dependent one-way quantum deficit \Delta as a function of measurement angle \theta\in[0,\pi/2] exhibits a bimodal behavior inside the open interval (0,\pi/2), i.e., it has two interior extrema: one minimum and one maximum. Moreover, cases are found when the interior minimum of such a bimodal function \Delta(\theta) is less than that one at the endpoint with \theta=0 or \theta=\pi/2. This refutes the result of Ye et al. [Int. J. Theor. Phys. {\bf 55}, 2237 (2016)] which was intended for obtaining the one-way quantum deficit of general two-qubit X states.
The one-way quantum work deficit, a measure of quantum correlation, exhibits the regions with the bimodal behavior of post-measurement entropy versus the measurement angle θ ε (0, π/2). Under certain conditions this can lead to the finite jumps Δν> 0 of optimal measurement angle ν from the endpoint 0 or π/2 to the interior minimum and vice versa. In turn, such sudden jumps lead to the fractures, i.e., non-analytical points on the dependencies of quantum correlation upon the state parameters. We examine a possibility to experimentally observe these phenomena on an example of three-component mixture consisting of one pair of Bell-diagonal states and a basis state orthogonal to the Bell ones. Observation of named effects, beyond their intrinsic interest, could demonstrate the experimental ability in controlling and manipulating the quantum states.
The work deficit, as introduced by Jonathan Oppenheim et al. [Phys. Rev. Lett. 89 (2002) 180402] is a good measure of the quantum correlations in a state and provides a new standpoint for understanding quantum non-locality. In this paper, we analytically evaluate the one-way information deficit (OWID) for the Bell-diagonal states and a class of two-qubit states and further give the geometry picture for OWID. The dynamic behavior of the OWID under decoherence channel is investigated and it is shown that the OWID of some classes of X states is more robust against the decoherence than the entanglement.
The work deficit, as introduced by Jonathan Oppenheim et al. [ Phys. Rev. Lett. 89 (2002) 180402 ] is a good measure of the quantum correlations in a state and provides a new standpoint for understanding quantum non-locality. In this paper, we analytically evaluate the one-way information deficit (OWID) for the Bell-diagonal states and a class of two-qubit states and further give the geometry picture for OWID. The dynamic behavior of the OWID under decoherence channel is investigated and it is shown that the OWID of some classes of X states is more robust against the decoherence than the entanglement.
So far, one-way information deficit (OWID) has been calculated explicitly only for Bell-diagonal states and the four-parameter family of X-states with additional assumptions and expressions for more general quantum states are not known. In this paper, we derive explicit expressions for OWID for a larger class of two-qubit states, namely, a five-parameter family of two-qubit states. The dynamic behavior of the OWID under decoherence channel is investigated and it is shown that the OWID is more robust against the decoherence than the entanglement.
We determine the least Greenberger-Horne-Zeilinger (GHZ) extractability compatible with a Mermin-Ardehali-Belinskii-Klyshko (MABK) Bell score for every n≥3n\geq3n≥3. Extractability is the largest squared overlap with a GHZ state obtainable by separate local quantum channels. Between the biseparable bound and the quantum maximum, its exact minimum is the affine interpolation from one half to one. This proves the bound conjectured for six or more parties. The result holds for arbitrary binary measurements and density operators on tensor products of finite- or infinite-dimensional local spaces. We also determine the exact minimum for qubits, which is strictly higher at every interior score. One qutrit and qubits elsewhere attain the unrestricted minimum with measurements fixed as the score varies and, at interior scores, realize the smallest product of ranks of the local reduced states among states attaining that minimum. For n≥4n\geq4n≥4, the attaining state with this rank product is unique at each interior score, up to local isometries and party permutations. On one qutrit and qubits elsewhere, for n≥4n\geq4n≥4 and scores a fixed fraction away from both endpoints, a sufficiently small excess of extractability above the minimum controls trace-norm distance to the attaining state of the same score, up to local unitaries. The dependence is of square-root order, with optimal exponent and explicit constants depending only on that fraction, uniformly in nnn. Equality and stability concern actual extractability; identical Bell statistics can accompany different extractabilities.