Genuine multipartite nonlocality of entangled thermal states

We assess quantum nonlocality of multiparty entangled thermal states by studying, quantitatively, both tripartite and quadripartite states belonging to the Greenberger-Horne-Zeilinger, W, and linear cluster-state classes and showing violation of relevant Bell-like inequalities. We discuss the conditions for maximizing the degree of violation against the local thermal character of the states and the inefficiency of the detection apparatuses. We demonstrate that such classes of multipartite entangled states can be made to last quite significantly, notwithstanding adverse operating conditions. This opens up the possibility for coherent exploitation of multipartite quantum channels made out of entangled thermal states. Our study is accompanied by a detailed description of possible generation schemes for the states analyzed.

Figure 1

(a) Conditional scheme for generating GHZ-like ETS. Three displaced thermal states (labeled $j=1,2,3$) interact with a two-level ancilla prepared in state $|+\rangle {}_A=(|0\rangle {}_A+|1\rangle {}_A)/\sqrt{2}$. The interaction is ruled by the coupling Hamiltonian ${\mathrm{Ĥ}}_{\mathit{Aj}}=\hslash \Omega |1\rangle {}_A\langle 1|{\mathrm{â}}_j^{†}{\mathrm{â}}_j$ for a time equal to $\pi /\Omega$. The ancilla is finally projected onto $|\pm \rangle {}_A$ with ${}_A\langle +|-\rangle {}_A=0$. (b) Beam-splitter-based scheme for the generation of a tripartite GHZ-like ETS. A displaced thermal state as introduced in the body of the article undergoes a transformation that puts it into the form ${\rho }^{{\mathrm{th}}^{\prime }}(V,d)={\mathcal{N}}_{+}\int d^2\alpha P_{\alpha }^{\mathrm{th}}(V,d)|s_{\alpha }^{+}\rangle \langle s_{\alpha }^{+}|$, with $|s_{\alpha }^{+}\rangle$ defined as in Eq. (4) and ${\mathcal{N}}_{+}$ a normalization constant. This is then superimposed to two vacuum modes at beam splitters of transmittivity $t_1=\sqrt{2/3}$ and $t_2=1/\sqrt{2}$ [21]. (c) Scheme for Svetlichny test performed over a GHZ- or W-like ETS. Each mode of a state prepared off line is first appropriately rotated and then projected onto in-phase quadrature eigenstates by means of arbitrarily efficient homodyne detectors. The outcomes of the measurements are appropriately dichotomized (see the inset).

Figure 2

Violation of Svetlichny inequality by a tripartite GHZ-like ETS under effective rotations and inefficient homodyne detection (efficiency $\eta =0.1$). We show the Svetlichny function against displacement $d$ and variance $V$ of the local thermal distributions. The floor of the plot is given by the local realistic bound of $4$. For $d\gg \sqrt{V}$, the upper bound of $4\sqrt{2}$ is achieved, exactly as it would be for a pure GHZ state of three spin-$1/2$ particles under sharp measurements.

Figure 3

Comparison between the Svetlichny function associated with state ${\rho }_{\mathrm{GHZ},1}^{(3)}$ (solid lines) and ${\rho }_{\mathrm{GHZ},2}^{(3)}$ (dashed lines) for $V=5,10$ and $\eta =0.3$ (assumed to be the same for every detector).

Figure 4

Svetlichny function for a tripartite W-like ETS at various values of $V$. The local realistic bound is $4$ (shaded region of the plot). The maximum achieved violation of Svetlichny inequality, reached at large values of $d$, is quantitatively the same as in the spin-$1/2$ case.

Figure 5

Svetlichny function for a four-partite GHZ-like ETS. As before, we have used $\eta =0.1$. Local realistic theories enforce a bound of $8$, while the maximum achievable quantum mechanically is 8$\sqrt{2}$.

Figure 6

(a) Conditional scheme for the generation of a four-partite linear cluster state via local thermal states, conditional displacements, and postselection. The interpretation of the symbols is the same as in the caption of Fig. 1. Differently from the GHZ-like ETS case, here we need two ancillary two-level systems and their mutual interaction. In panel (b) we show an alternative way of generating a linear cluster ETS (see Ba an and Hoa in Ref. [34]). Two thermal states of annihilation operators ${\mathrm{â}}_j$ ($j=1,2$) enter a nonlinear cross-Kerr medium where the interaction ${\mathrm{Ĥ}}_{{\chi }^{(3)}}=\hslash \chi {\mathrm{â}}_1^{†}{\mathrm{â}}_1{\mathrm{â}}_2^{†}{\mathrm{â}}_2$ takes place for a time $\pi /\chi$. The output modes are then superimposed to two ancillary modes prepared in their vacuum state at 50:50 beam spitters. Local phase shifters (not shown) complete the generation process.

Figure 7

Violation of SASA inequality by a quadripartite clusterlike ETS under effective rotations and homodyne detection. The SASA function is plotted against $V$ and $d$. The local realistic bound is shown by the planeat $B_{\mathrm{SASA}}$ $=2$.

Figure 8

Violation of the WWZB inequality by a four-party linear cluster like ETS under effective rotations and homodyne detection. The local realistic bound is 4 and is indicated by the horizontal plane. For any $V$, a correspondingly large value of $d$ guarantees a violation of the WWZB inequality by a factor $\sqrt{2}$, in agreement with the spin-$1/2$ case.

Figure 9

Comparison between the SASA functions associated with states ${\rho }_{\mathrm{cluster}}^{(1)}$ (dashed line) and ${\rho }_{\mathrm{cluster}}^{(2)}$ (solid line) for $V=1$, 5, and 10.