Identifying nonconvexity in the sets of limited-dimension quantum correlations

Quantum theory is known to be nonlocal in the sense that separated parties can perform measurements on a shared quantum state to obtain correlated probability distributions, which cannot be achieved if the parties share only classical randomness. Here we find that the set of distributions compatible with sharing quantum states subject to some sufficiently restricted dimension is neither convex nor a superset of the classical distributions. We examine the relationship between quantum distributions associated with a dimensional constraint and classical distributions associated with limited shared randomness. We prove that quantum correlations are convex for certain finite dimension in certain Bell scenarios and that they sometimes offer a dimensional advantage in realizing local distributions. We also consider if there exist Bell scenarios where the set of quantum correlations is never convex with finite dimensionality.

Figure 1

Bell scenarios may be viewed schematically as shown above, where $n$ parties ($n=2$ shown in this example) make synchronized measurements on systems prepared by a common source. Each party has $m$ input (measurement setting) choices and $v$ output (measurement result) possibilities. For $n=2$, the parties are conventionally named Alice and Bob. Inputting $x$ and $y$ will return outputs $a$ and $b$ for Alice and Bob, respectively, with probability $p\left[ab\right|xy]$.

Figure 2

Various causal structures associated with two space-like separated parties who make synchronized measurements on systems prepared by a common source, as in Fig. 1. (a) Sharing a multipartite quantum state. (b) Sharing only classical randomness. (c) Sharing both resources. The joint distribution on the observed outcomes differs depending on the nature of the shared resource. If the preparation is quantum, such as in (a), then $p\left[ab\right|xy]=Tr\left[\rho {\widehat{A}}_{a|x}\otimes {\widehat{B}}_{b|y}\right]$, per Eq. (1). If the preparation is classical, such as in (b), then $p\left[ab\right|xy]={\sum }_{\lambda }p\left[\lambda \right]p\left[a\right|x\lambda ]p\left[b\right|y\lambda ]$, per Eq. (3). If the shared system is both classical and quantum, such as in (c), then $p\left[ab\right|xy]={\sum }_{\lambda }p\left[\lambda \right]Tr\left[{\rho }^{\left(\lambda \right)}{\widehat{A}}_{a|x}^{\left(\lambda \right)}\otimes {\widehat{B}}_{b|y}^{\left(\lambda \right)}\right]$, per Eq. (5). Hybrid preparations can equivalently be considered classical-quantum (CQ) states [40, 44].

Figure 3

Three examples of the nonconvexity of the set $Q_2$ are shown above, in terms of the boxes defined in Table 1. (a) If $\tilde{P}=c_1{P}_1+c_3{P}_3+c_4{P}_4,$ then $\tilde{P}\in {\mathcal{L}}_{\infty }$ if and only if $c_4\le 1-c_3,$ and $\tilde{P}\in {\mathcal{Q}}_2$ if and only if $\left\{c_1\text{or}{c}_3\text{or}{c}_4\right\}=0$. (b) If $\tilde{P}=c_0{P}_0+c_1{P}_1+c_{3:4}{P}_{3:4},$ then $\tilde{P}\in {\mathcal{L}}_{\infty }$ if and only if $c_{3:4}\le 1-c_1,$ and $\tilde{P}\in {\mathcal{Q}}_2$ if and only if $c_{3:4}\le {\left(1-c_1\right)}^{3/2}.$ (c) If $\tilde{P}=c_0{P}_0+c_1{P}_1+c_{\text{TB}}{P}_{\text{TB}},$ then $\tilde{P}\in {\mathcal{L}}_{\infty }$ if and only if $c_{\text{TB}}\le 2^{-½}\left(1-c_1\right),$ and $\tilde{P}\in {\mathcal{Q}}_2$ if and only if $c_{\text{TB}}\le {\left(1-c_1\right)}^{5/4}.$ Each triangle represents the set of possible convex combination of its vertex boxes, where $c_i$ represents the weight of $P_i$ in the mixture and ${\sum }_i{c}_i=1$. To emphasize the notion of convex combination we have plotted the regions as equilateral triangles; nevertheless any pair of edges should be thought of as two independent axes such that the third weight is fixed by normalization. Although in each triangle all the vertex boxes are achievable with shared-qubits preparations, only a fraction of possible convex combinations are also achievable, as indicated by the shaded region. Indeed, in (a) only the zero-area edges of the triangle are qubit-achievable. The local polytope ${\mathcal{L}}_{\infty }$ contains the entire triangular regions of (a) and (b) since the vertex boxes of those triangle are all local, and in (c) only the region below the dashed red line is consistent with classical shared randomness. The essential feature of all three triangles is that the shaded subset representing ${\mathcal{Q}}_2$ is not convex and that ${\mathcal{L}}_{\infty }⊈{\mathcal{Q}}_2$. The ${\mathcal{Q}}_2$ region in (c) was obtained via computational maximization; ${\left(1-c_1\right)}^{5/4}$ was noticed to precisely coincide with the numeric boundary. The given boundaries of ${\mathcal{Q}}_2$ in (a) and (b) were recovered analytically. Consistent with Ref. [41], the qubit-accessible region shrinks if one restricts to projective measurements. The edges where $c_1\ne 0$ in (a), for example, are not in ${\mathcal{Q}}_2$-PVM, and in (c) we find that $\tilde{P}\in {\mathcal{Q}}_2$-PVM only if $c_{\text{TB}}\lesssim {\left(1-c_1\right)}^{3/2}$. Thus the use of POVMs can be certified in a device-independent way given a dimension promise, akin the the results of Ref. [55].