Unifying framework for spatial and temporal quantum correlations

Measurements on a single quantum system at different times reveal rich nonclassical correlations similar to those observed in spatially separated multipartite systems. Here we introduce a theory framework that unifies the description of temporal, spatial, and spatiotemporal resources for quantum correlations. We identify and experimentally demonstrate simple cases where an exact mapping between the domains is possible. We then identify correlation resources in arbitrary situations, where not all spatial quantum states correspond to a process and not all temporal measurements have a spatial analog. These results provide a starting point for the systematic exploration of multipoint temporal correlations as a powerful resource for quantum information processing.

Figure 1

Spatial vs temporal quantum correlations. Spatial quantum correlations are revealed by local measurements $E^A$ and $E^B$ on spatially separated quantum systems, whereas temporal correlations are observed between subsequent measurements $M^{A_I{A}_O}$ and $M^{B_I{B}_O}$ on the same system. The solid arrows indicate that every spatial measurement has a temporal analog, while the converse is not true in general (dashed arrow). Since temporal measurements must specify the postmeasurement state, we compare them to spatial measurements of systems of double the dimension of the temporal counterpart (see the double solid lines). The resource (red shading) generating the observed spatial correlations is the initial quantum state $\rho$, whereas in the temporal case it is the quantum process $W$. Solid (dashed) arrows indicate that every $W$ can be mapped to a $\rho$ but not vice versa.

Figure 2

Spatial, temporal, and spatiotemporal correlations. (a) In the temporal scenario, Alice and Bob measure the same quantum system at different times. (b) In the traditional spatial scenario, Alice and Bob perform local measurements on parts of a shared quantum system. (c) In a general scenario, there may be multiple parties—Alice, Bob, Charlie, etc.—who perform temporally and spatially separate measurements in a quantum network. (d) A simple experiment with two interacting quantum systems already exhibits all three forms of correlations. Pairs of single photons are generated via spontaneous parametric downconversion in a femtosecond-pumped $\beta$-barium-borate (BBO) crystal. The polarization state of the system photon is prepared by Alice using a half waveplate (HWP), quarter waveplate (QWP), and Glan-Taylor polarizer (GT). The ancilla photon is entangled nondeterministically to the system via two-photon interference in a partially polarizing beam splitter (PPBS) [17]. Finally Bob and Charlie measure the states of system and ancilla, using a HWP, QWP, and GT, before detecting their photons with avalanche photodiodes (APD). In this experiment, Alice and Bob observe temporal correlations on the system qubit, Bob and Charlie observe spatial correlations between the final measurements, and taking into account all three parties we observe spatiotemporal correlations (dotted line).

Figure 3

Spatiotemporal GHZ state. (a) Scheme for generating a spatiotemporal GHZ state and (b) the reconstructed density matrix for $\kappa =1$. (c) Violation of the Svetlichny inequality for a range of $\kappa$ witnesses genuine tripartite entanglement. Error bars for $S_{svet}$ represent $3\sigma$-equivalent statistical confidence regions obtained from ${10}^5$ Monte Carlo resampled data points under Poissonian counting statistics; error bars for $\kappa$ correspond to $3\sigma$ confidence intervals estimated from independent classical calibration. The solid line is the theory prediction for perfect states and measurements. The dashed line is the theory prediction taking into account imperfections in the state preparation.