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Nonlocality in pure and mixed n-qubit X states

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Abstract

Nonlocality for general multiqubit X states is studied in detail. Pure and mixed states are analyzed as far as their maximum amount of nonlocality is concerned, and analytic results are obtained for important families of these states. The particular form of nonzero diagonal and antidiagonal matrix elements makes the corresponding study easy enough to obtain exact results. We also provide a numerical recipe to randomly generate an important family of X states endowed with a given degree of mixture.

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Acknowledgments

J. Batle acknowledges fruitful discussions with J. Rosselló, Maria del Mar Batle and Regina Batle. J. Batle is grateful to the villages of sa Pobla and Santa Margalida for their warm hospitality. R. O. acknowledges support from High Impact Research MoE Grant UM.C/625/1/HIR/MoE/CHAN/04 from the Ministry of Education Malaysia.

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Appendix: Generation of two-qubits states with a fixed value of the participation ratio R

Appendix: Generation of two-qubits states with a fixed value of the participation ratio R

Here, we describe a numerical recipe to randomly generate two-qubit states, according to a definite measure and with a given, fixed value of R. Suppose that the states \(\rho \) are generated according to the product measure \(\nu = \mu \times \mathcal{L}_{N-1}\), where \(\mu \) is the Haar measure on the group of unitary matrices \(\mathcal{U}(N)\) and the standard normalized Lebesgue measure \(\mathcal{L}_{N-1}\) on \(\mathcal{R}^{N-1}\) provides a reasonable computation of the simplex of eigenvalues of \(\rho \). In this case, the numerical procedure we are about to explain owes its efficiency to the following geometrical picture which is valid only if the states are supposed to be distributed according to measure \(\nu \)). We shall identify the simplex \(\Delta \) with a regular tetrahedron of side length 1, in \(\mathcal{R}^3\), centered at the origin. Let \(\mathbf{r}_i\) stand for the vector positions of the tetrahedron’s vertices. The tetrahedron is oriented in such a way that the vector \(\mathbf{r}_4\) points toward the positive z-axis and the vector \(\mathbf{r_2}\) is contained in the (xz)-semiplane corresponding to positive x-values. The positions of the tetrahedron’s vertices correspond to the vectors

$$\begin{aligned} \mathbf {r_1}= & {} \left( -\frac{1}{2\sqrt{3}},-\frac{1}{2},-\frac{1}{4}\sqrt{\frac{2}{3}}\right) \nonumber \\ \mathbf {r_2}= & {} \left( \frac{1}{\sqrt{3}},0,-\frac{1}{4}\sqrt{\frac{2}{3}}\right) \nonumber \\ \mathbf {r_3}= & {} \left( -\frac{1}{2\sqrt{3}},\frac{1}{2},-\frac{1}{4}\sqrt{\frac{2}{3}}\right) \nonumber \\ \mathbf {r_4}= & {} \left( 0,0,\frac{3}{4}\sqrt{\frac{2}{3}}\right) . \end{aligned}$$
(27)

The mapping connecting the points of the simplex \(\Delta \) (with coordinates \((\lambda _1,\ldots , \lambda _4)\)) with the points \(\mathbf r\) within tetrahedron is given by the equations

$$\begin{aligned} \lambda _i \,= & {} \, 2(\mathbf{r}\cdot \mathbf{r}_i ) \, + \, \frac{1}{4} \,\,\,\, i=1, \dots , 4, \nonumber \\ \mathbf{r} \,= & {} \, \sum _{i=1}^4 \lambda _i \mathbf{r}_i \end{aligned}$$
(28)

The degree of mixture is characterized by the quantity \(R^{-1} \equiv Tr(\rho ^2) = \sum _i \lambda _i^2\). This quantity is related to the distance \(r=\mid \mathbf{r} \mid \) to the center of the tetrahedron \(T_{\Delta }\) by

$$\begin{aligned} r^2 \, = \, -\frac{1}{8} \, + \, \frac{1}{2} \sum _{i=1}^4 \lambda _i^2. \end{aligned}$$
(29)

Thus, the states with a given degree of mixture lie on the surface of a sphere \(\Sigma _r\) of radius r concentric with the tetrahedron \(T_{\Delta }\). To choose a given R is tantamount to define a given radius of the sphere. Three different possible regions exist:

  • region I: \(r \in [0, h_1]\) (\(R \in [4,3]\)), where \(h_1 \equiv h_c={1 \over 4 }\sqrt{2 \over {3}}\) is the radius of a sphere tangent to the faces of the tetrahedron \(T_{\Delta }\). In this case, the sphere \(\Sigma _r\) lies completely within the tetrahedron \(T_{\Delta }\). Therefore, we only need to generate at random points over its surface. The Cartesian coordinates for the sphere are given by

    $$\begin{aligned} x_1= & {} r \, \sin \theta \, \cos \phi \nonumber \\ x_2= & {} r \, \sin \theta \, \sin \phi \nonumber \\ x_3= & {} r \, \cos \theta , \end{aligned}$$
    (30)

    Denoting rand_u() a random number uniformly distributed between 0 an 1, the random numbers \(\phi =2\pi \) rand_u() and \(\theta =\arccos (2\) rand_u() \(-1)\) (its probability distribution being \(P(\theta )=\frac{1}{2}\sin (\theta )\)) define an arbitrary state \(\rho \) on the surface inside \(T_{\Delta }\). The angle \(\theta \) is defined between the center of the tetrahedron (the origin) and the vector \(\mathbf{r_4}\), and any point aligned with the origin. Substitution of \(\mathbf{r}=(x_1,x_2,x_3)\) in (28) provides us with the eigenvalues \(\{\lambda _i\}\) of \(\rho \), with the desired R as prescribed by the relationship (29). With the subsequent application of the unitary matrices U, we obtain a random state \(\rho = U D(\Delta ) U^{\dag }\) distributed according to the usual measure \(\nu = \mu \times \mathcal{L}_{N-1}\).

  • region II: \(r \in [h_1, h_2]\) (\(R \in [3,2]\)), where \(h_2 \equiv \sqrt{h^{2}_{c}+(\frac{D}{2})^2}={\sqrt{2}\over 4}\) denotes the radius of a sphere which is tangent to the sides of the tetrahedron \(T_{\Delta }\). Contrary to the previous case, part of the surface of the sphere lies outside the tetrahedron. This fact means that we are able to still generate the states \(\rho \) as before, provided we reject those ones with negative weights \(\lambda _i\).

  • region III: \(r \in [h_2, h_3]\) (\(R \in [2,1]\)), where \(h_3 \equiv \sqrt{h^{2}_{c}+D^2}={\sqrt{6}\over 4}\) is the radius of a sphere passing through the vertices of \(T_{\Delta }\). The generation of states is a bit more involved in this case. Again \(\phi =2\pi \) rand_u(), but the available angles \(\theta \) now range from \(\theta _c(r)\) to \(\pi \). It can be shown that \(w\equiv \cos (\theta _c)\) results from solving the equation \(3r^2 w^2 - \sqrt{\frac{3}{2}}r w + \frac{3}{8}-2r^2 = 0\). Thus, \(\theta (r)=\arccos (w(r))\), with \(w(r)=\cos \theta _c(r) + (1-\cos \theta _c(r))\) rand_u(). Some states may be unacceptable (\(\lambda _i<0\)) still, but the vast majority are accepted.

Combining these three previous regions, we are able to generate arbitrary mixed states \(\rho \) endowed with a given participation ratio R.

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Batle, J., Ooi, C.H.R., Farouk, A. et al. Nonlocality in pure and mixed n-qubit X states. Quantum Inf Process 15, 1553–1567 (2016). https://doi.org/10.1007/s11128-015-1216-5

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