Abstract
The exact solutions of the super quantum discord are derived for general two qubit X states in terms of a one-variable function. Several exact solutions of the super quantum discord are given for the general X state over nontrivial regions of a seven-dimensional manifold.
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Ollivier, H., Zurek, W.H.: Quantum discord: a measure of the quantumness of correlations. Phys. Rev. Lett. 88, 017901 (2001)
Henderson, L., Vedral, V.: Classical, quantum and total correlations. J. Phys. A Math. Gen. 34, 6899 (2001)
Luo, S.: Quantum discord for two-qubit systems. Phys. Rev. A 77, 042303 (2008)
Dakíc, B., Vedral, V., Brukner, Č.: Necessary and sufficient condition for nonzero quantum discord. Phys. Rev. Lett. 105, 190502 (2010)
Ali, M., Rau, A.R.P., Alber, G.: Quantum discord for two-qubit X states. Phys. Rev. A 81, 042105 (2010)
Girolami, D., Adesso, G.: Quantum discord for general two-qubit states: analytical progress. Phys. Rev. A 83, 052108 (2011)
Galve, F., Giorgi, G., Zambrini, R.: Orthogonal measurements are almost sufficient for quantum discord of two qubits. Europhys. Lett. 96, 40005 (2011)
Chen, Q., Zhang, C., Yu, S., Yi, X.-X., Oh, C.-H.: Quantum discord of two-qubit X states. Phys. Rev. A 84, 042313 (2011)
Fanchini, F.F., Werlang, T., Brasil, C.A., Arruda, L.G.E., Caldeira, A.O.: Non-Markovian dynamics of quantum discord. Phys. Rev. A 81, 052107 (2010)
Huang, Y.: Quantum discord for two-qubit X states: analytical formula with very small worst-case error. Phys. Rev. A 88, 014302 (2013)
Streltsov, A.: Quantum Correlations Beyond Entanglement. Springer Briefs in Physics. Springer International Publishing (2015)
Aharonov, Y., Albert, D.Z., Vaidman, L.: How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100. Phys. Rev. Lett. 60, 1351 (1988)
Singh, U., Pati, A.K.: Super quantum discord with weak measurements. Ann. Phys. 343, 141 (2014)
Wang, Y.K., Ma, T., Fan, H., Fei, S.M., Wang, Z.X.: Super-quantum correlation and geometry for Bell-diagonal states with weak measurements. Quantum Inf. Process. 13, 283 (2014)
Li, B., Chen, L., Fan, H.: Non-zero total correlation means non-zero quantum correlation. Phys. Lett. A 378, 1249 (2014)
Rfifi, S., Siyouri, F., Baz, M.E.I., Hassouni, Y.: Super quantum discord and other correlations for tripartite GHZ-type squeezed states. J. Korean Phys. Soc. 67, 311 (2015)
Li, T., Ma, T., Wang, Y.K., Fei, S.M., Wang, Z.X.: Super quantum discord for X-type states. Int. J. Theor. Phys. 54, 680 (2015)
Oreshkov, O., Brun, T.A.: Weak measurements are universal. Phys. Rev. Lett. 95, 110409 (2005)
Partovi, M.H.: Irreversibility, reduction, and entropy increase in quantum measurements. Phys. Lett. A 137, 445 (1989)
Nielsen, M.A., Chuang, I.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2010)
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The research is partially supported by NSFC Grant (Nos. 11271138, 11531004) and Simons Foundation (No. 198129).
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Appendix
Appendix
Proof of Lemma 3.1
First we notice that \(G(\theta , z)\) is a strictly decreasing function of \(\theta \):
Therefore, there are no interior critical points and extremal points must lie on the boundary of the domain. Since \(\frac{\partial G}{\partial \theta }<0\), we further conclude that \(\min G\) takes place at the largest value of \(\theta \) for some \(z\in [0, 1]\). As \(z_1^2+z_2^2+z^2=1\), we have that
where \(b^2=\frac{|c_1|^2+|c_2|^2+\sqrt{(|c_1|^2-|c_2|^2)^2+4|c_1\times c_2|^2}}{2}\).
For each fixed z, the maximum value \(b^2+(c_3^2-b^2)z^2\) can be achieved by appropriate \(z_1, z_2\). Therefore, \(\min G(z, \theta )=\min _{z\in [0,1]}G(z, b^2+(c_3^2-b^2)z^2)\), which is explicitly given in (3.14). \(\square \)
Proof of Theorem 3.2
We compute the derivative of F(z).
where \(H_{\pm }^{\prime }(z)=H_\pm ^{-1}(\pm rc_3\tanh x+(c_3^2-b^2)z\tanh ^2x)\) and \(\displaystyle A_{\pm }=\frac{H_\pm }{1\pm sz\tanh x}\in [0, 1]\). \(\square \)
Case (a) Since
Then the first term of \(F'(z)\le 0\) iff \(-s\tanh x(A_+^2-A_-^2)\geqslant 0\), which holds if \(s\tanh x\geqslant 0\) and \(rc_3\tanh x\leqslant 0\) or \(s\tanh x\leqslant 0\) and \(rc_3\tanh x\geqslant 0\).
Note that \(\displaystyle g(x)=\frac{1}{x}\ln \frac{1+x}{1-x}\) is a strictly increasing function on (0, 1), as
Therefore, \(A_+\geqslant A_-\) iff
(i) If \(s\tanh x\geqslant 0,rc_3\tanh x\leqslant 0\) and \(c_3^2-b^2\geqslant 0\), it follows from (5.2) that \(A_+\leqslant A_-\), then (5.3) implies that
(ii) If \(s\tanh x\geqslant 0,rc_3\tanh x\leqslant 0\) and \(src_3\le c_3^2-b^2\leqslant 0\) , we have that
Case (b) Is treated in two subcases, (i) Suppose that \(s\tanh x\leqslant 0, rc_3\tanh x\geqslant 0\), and \(c_3^2-b^2\geqslant 0\), (ii) Suppose that \(s\tanh x\le 0,rc_3\tanh x\geqslant 0\) and \(src_3\le c_3^2-b^2\le 0\). We can solve the problem by the same method as Case (a).
Case (c) If \(s=r=0\), then
Therefore, \(\min F(z)\) is F(0) according to \(c_3^2\le b^2\) or not. Thus, the minimum of F(z) on \(z\in [0,1]\) is F(0).
Case (d) If \(s=rc_3, b^2=c_3^2\), and \(r^2+c_3^2\tanh ^x\pm rc_3\tanh x\ge 1\), it follows from (5.2) that the first term \(F'(z)\ge 0\).
Let \(k(z)=\frac{rc_3\tanh x}{H(z)}\log _2\frac{1+sz\tanh x z+H(z)}{1+sz\tanh x z -H(z)}\), where \(H(z)=\sqrt{r^2+b^2\tanh ^2x+2sz\tanh x }\). Then
As a function of z, we have that \(H'(z)=\frac{s\tanh x}{H(z)}, H''(z)=-\frac{s^2\tanh ^2x}{H^3(z)}\) and
the inequality holds because
Similarly \(k'(-z)\le 0\), thus \(-\frac{1}{4}(k'(z)+k'(-z))\ge 0\), which implies that \(F'(z)\ge 0\).
Therefore, the minimum of F(z) on \(z\in [0,1]\) is F(0).
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Jing, N., Yu, B. Super quantum discord for general two qubit X states. Quantum Inf Process 16, 99 (2017). https://doi.org/10.1007/s11128-017-1547-5
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DOI: https://doi.org/10.1007/s11128-017-1547-5