Quantifying Bell nonlocality with the trace distance

Measurements performed on distant parts of an entangled quantum state can generate correlations incompatible with classical theories respecting the assumption of local causality. This is the phenomenon known as quantum nonlocality that, apart from its fundamental role, can also be put to practical use in applications such as cryptography and distributed computing. Clearly, developing ways of quantifying nonlocality is an important primitive in this scenario. Here, we propose to quantify the nonlocality of a given probability distribution via its trace distance to the set of classical correlations. We show that this measure is a monotone under the free operations of a resource theory and, furthermore, that it can be computed efficiently with a linear program. We put our framework to use in a variety of relevant Bell scenarios also comparing the trace distance to other standard measures in the literature.

Figure 1

Bipartite Bell scenario where two parts, Alice and Bob, share a pair of correlated measurement devices, with inputs labeled by $x$ and $y$ and outputs labeled by $a$ and $b$, respectively.

Figure 2

Schematic drawing of a correlation $\mathbf{q}\in {\mathcal{C}}_{\mathrm{NS}}$ and $d=\mathrm{NL}\left(\mathbf{q}\right)$, the distance (with respect to the ${\ell }_1$ norm) from $\mathbf{q}$ to the closest local correlation ${\mathbf{p}}^{*}\in {\mathcal{C}}_{\mathrm{C}}$.

Figure 3

Value of $\mathrm{NL}\left(\mathbf{q}\right)$ as a function of the value of the CGLMP inequality $I_{\mathrm{CGLMP}}^d$ for $d=2,3,4,5$. The inset shows the maximum known quantum violation for each of these values of $d$, which in turn implies that $\mathrm{NL}\left(\mathbf{q}\right)=0.1035 (d=2), \mathrm{NL}\left(\mathbf{q}\right)=0.1143 (d=3), \mathrm{NL}\left(\mathbf{q}\right)=0.1215 (d=4)$, and $\mathrm{NL}\left(\mathbf{q}\right)=0.1269 (d=5)$.

Figure 4

Value of $\mathrm{NL}\left(\mathbf{q}\right)$ as a function of the value of $I_{nn22}$ for $n=2,3,4,5,6$. The inset shows the maximum known quantum violation (achieved with qubits) for $n=2,3$. Interestingly, even if the maximum NS violation of the inequality increases with the number of settings $n$, its trace distance to the set of local correlations decreases.

Figure 5

(a) Values of $\mathrm{NL}\left(\mathbf{q}\right)$ as a function of the violation of the 46 Sliwa classes of inequalities. (b) The values of $\mathrm{NL}\left(\mathbf{q}\right)$ for the maximal quantum [41] and nonsignalling violation of each one of these inequalities. The maximum value of $\mathrm{NL}\left(\mathbf{q}\right)=1/8$ is achieved by the maximum violation of the Mermin inequality [42] (corresponding to the second Sliwa inequality [40]).

Figure 6

$\mathrm{NL}\left(\mathbf{q}\right)$ as a function of the values of $M_N$ for $N=2,3,4,5,6$. We have used the straight line for odd cases and dashed lines for even cases. The circles correspond to $\mathrm{NL}\left(\mathbf{q}\right)$ evaluated assuming that each possible measurement setting has a probability $1/2^N$. The diamonds correspond to the case where only the measurement settings appearing in $M_N$ have a probability different from zero (happening for the $N$ odd case).

Figure 7

The black line shows $\mathrm{NL}\left(\mathbf{q}\right)$ for the distribution $\mathbf{q}=v{\mathbf{q}}_{\max }+(1-v)1/4$ as a function of $I_{3322}=2v-1$ considering that $p(x,y)=1/4\mathrm{iff}x,y\ne 2$ (in which case we basically recover $I_{2222}$). The red solid line shows $\mathrm{NL}\left(\mathbf{q}\right)$ as a function of $I_{3322}=2v-1$ considering the full distribution $\mathbf{q}$. The purple dashed line shows $\mathrm{NL}\left(\mathbf{q}\right)$ as a function of $I_{3322}=2v-1$ considering only the value of $I_{3322}$ (thus optimizing over all $\mathbf{q}$ compatible with it). Clearly, imposing the value of a given Bell inequality only provides a (in general, nontight) lower bound for $\mathrm{NL}\left(\mathbf{q}\right)$.

Figure 8

The behavior of the $I_{\mathrm{CGLMP}}^3$ (left) and $\mathrm{NL}\left(\mathbf{q}\right)$ (right) inequality as a function of $\gamma$ for the distribution $\mathbf{q}$ arising from measurements on the state $|\varphi \rangle =\gamma |00\rangle +\sqrt{1-2{\gamma }^2}|11\rangle +\gamma |22\rangle$. From the critical value of ${\gamma }_c>0.369$, we have that $I_{\mathrm{CGLMP}}^3>0$ and $\mathrm{NL}\left(\mathbf{q}\right)>0$. In both panels, the circle, square, and diamond correspond to the maximal entangled state (ME), the state that maximally violates (MV) the $I_{\mathrm{CGLMP}}^3$, and the optimal state (OS) that provides the maximal $\mathrm{KL}(\mathbf{q},\mathbf{p})$, respectively. As we can see, the maximal $I_{\mathrm{CGLMP}}^3$ violation is achieved by $\gamma =0.617$ and the maximal of $\mathrm{NL}\left(\mathbf{q}\right)$ is achieved for the same $\gamma$. This is opposed to the result found in [17], where the quantum distribution with maximum $\mathrm{KL}(\mathbf{q},\mathbf{p})$ does not violate $I_{\mathrm{CGLMP}}^3$ maximally. We also notice that according to the measure proposed in [21, 22], the maximal nonlocality is achieved exactly to the maximally entangled state. This shows how different measures can lead to very different orderings.

Figure 9

Plot of ${min}_{\mathbf{p}\in {\mathcal{C}}_{\mathrm{C}}}\mathrm{KL}(\mathbf{q},\mathbf{p})$ as a function of the CGLMP inequality for $d=3$. The black dashed curve represents the lower bound provided the Pinsker inequality, defined by $\mathrm{KL}(\mathbf{q},\mathbf{p})\ge 4C{\mathrm{NL}\left(\mathbf{q}\right)}^2$, where $C=\frac{1}{2}{log}_{2}e$ and $\mathrm{NL}\left(\mathbf{q}\right)=\frac{1}{2}{\parallel \mathbf{q}-\mathbf{p}\parallel }_{{\ell }_1}$. The red curve provides an upper bound for ${min}_{\mathbf{p}\in {\mathcal{C}}_{\mathrm{C}}}\mathrm{KL}(\mathbf{q},\mathbf{p})$ since it computes $KL({\mathbf{q}}^{*},{\mathbf{p}}^{*})$, where ${\mathbf{q}}^{*}$ and ${\mathbf{p}}^{*}$ are the solutions provided by the LP minimization of $\mathrm{NL}\left(\mathbf{q}\right)$ subjected to NS constraints and a given value of the CGLMP inequality. The blue curve is obtained by a brute force minimization of ${min}_{\mathbf{p}\in {\mathcal{C}}_{\mathrm{C}}}\mathrm{KL}({\mathbf{q}}^{*},\mathbf{p})$. The color points represent the three points found in [17] and discussed in the main text. Interestingly, for the green (circle) and pink (diamond) points, there are NS correlations (possibly postquantum) with the same value of the CGLMP inequality but providing a higher value of ${min}_{\mathbf{p}\in {\mathcal{C}}_{\mathrm{C}}}\mathrm{KL}(\mathbf{q},\mathbf{p})$.