Geometry of quantum correlations in space-time

The traditional formalism of nonrelativistic quantum theory allows the state of a quantum system to extend across space, but only restricts it to a single instant in time, leading to distinction between theoretical treatments of spatial and temporal quantum correlations. Here we unify the geometrical description of two-point quantum correlations in space-time. Our study presents the geometry of correlations between two sequential Pauli measurements on a single qubit undergoing an arbitrary quantum channel evolution together with two-qubit spatial correlations under a common framework. We establish a symmetric structure between quantum correlations in space and time. This symmetry is broken in the presence of nonunital channels, which further reveals a set of temporal correlations that are indistinguishable from correlations found in bipartite entangled states.

Figure 1

The surface enclosing the set of possible values of two-point temporal correlations in the real space of $\{〈{\sigma }_1{\sigma }_1〉,〈{\sigma }_2{\sigma }_2〉,〈{\sigma }_3{\sigma }_3〉\}$. The equation of the surface is derived in the Appendix [see Eq. (A4)].

Figure 2

Left A 3D visualization of the spatial and temporal tetrahedrons with the blue region representing ${\mathcal{T}}_s$, and the red region representing ${\mathcal{T}}_t$. Right: A perspective plot viewing from the $(-1,-1,-1)$ direction. The intersection of the spatial and temporal tetrahedra forms an octahedron that corresponds to the set of separable states. The purple hexagon is a projection of the octahedron overlap, while the blue and red triangles are projections of ${\mathcal{T}}_s$ and ${\mathcal{T}}_t$, respectively. The two tetrahedra are inscribed in the cube, which corresponds to the most general achievable region and is formed by a mixture of spatial and temporal correlations. It is clear that the cube is the largest possible set of space-time quantum correlations since $-1\le \langle {\sigma }_A{\sigma }_A\rangle \le 1$, and the set of possible correlation functions forms a convex set.

Figure 3

This figure present the types of correlations in two-point PDMs as 2D projections onto the planes of $\{〈{\sigma }_{A_1}{\sigma }_{B_1}〉,〈{\sigma }_{A_2}{\sigma }_{B_2}〉\}$ in (a), (b), and (c). The sets of correlations are shown in the first quadrant. The full 2D projection is generated in a symmetric manner about the origin. In (d) and (e) we give instances in the 3D spaces corresponding to this 2D projections. The red region highlights extra correlations attainable in a valid PDM compared to a valid density matrix. (a) Type a: $\left[{\sigma }_{A_1}\otimes {\sigma }_{B_1},{\sigma }_{A_2}\otimes {\sigma }_{B_2}\right]=0$. Temporal and spatial correlations both lie in the purple unit square. (b) Type b: $\{{\sigma }_{A_1}\otimes {\sigma }_{B_1},{\sigma }_{A_2}\otimes {\sigma }_{B_2}\}=0$, and one out of the four operators is ${\sigma }_0$. Spatial correlations lie in the purple quarter unit circle, while temporal correlations lie in the unit square. The red region is allowed by valid PDMs but not density matrices. (c) Type c: In all other cases, correlations are bounded by the purple quarter circle. (d) An example of 3D spaces corresponding to a combination of type a and type c 2D projections. (e) An example of 3D spaces corresponding to a combination of type b and type c 2D projections.