Operator representation of spatiotemporal quantum correlations

While quantum correlations between two spacelike-separated systems are fully encoded by the bipartite density operator associated with the joint system, there does not exist an analogous operator representing general quantum correlations across space and time. This is in stark contrast to the case of classical random variables, which make no distinction between spacelike and timelike correlations. Despite this, we show that spatiotemporal correlations between light-touch observables (i.e., observables whose eigenvalues are all equal in magnitude) admit a unique operator representation for arbitrary timelike-separated quantum systems. A special case of our result reproduces generalized Pauli observables and pseudo-density matrices, which have, up until now, only been defined for multi-qubit systems. In the case of qutrit systems, we use our results to illustrate an intriguing connection between light-touch observables and symmetric, informationally complete, positive operator-valued measures (SIC-POVMs).