Structure of temporal correlations of a qubit

Abstract

In quantum mechanics, spatial correlations arising from measurements at separated particles are well studied. This is not the case, however, for the temporal correlations arising from a single quantum system subjected to a sequence of generalized measurements. We first characterize the polytope of temporal quantum correlations coming from the most general measurements. We then show that if the dimension of the quantum system is bounded, only a subset of the most general correlations can be realized and identify the correlations in the simplest scenario that can not be reached by twodimensional systems. This leads to a temporal inequality for a dimension test, and we discuss a possible implementation using nitrogen-vacancy centers in diamond.

Genuine temporal correlations can certify the quantum dimension

“…In this paper we propose a dimension witness using true multipoint temporal correlations which does not allow for an interpretation in the P&M scheme and shows a potentially larger ratio of violation than the dimension witnesses obtained in [13]. We report the implementation of this witness on a single trapped 171 Yb + ion and certify the manipulation of a three-dimensional system in the experiment.…”

“…A scenario similar to ours in which however no relation among the measurements performed at different times is assumed has been discussed in [34]. But dimension bounds have been derived only based on the correlations obtained for measurement sequences of length two [13,34]. Measurement sequences of length two may still be interpreted in the P&M scenario.…”

“…In contrast to that, for the temporal scenario it is possible to obtain all correlations that do not exhibit signaling with respect to the past within quantum theory, if one does not restrict the dimension of the quantum system and the type of measurements [12,13]. If the dimension of the system is restricted, however, some correlations cannot be realized [13]. This allows one to exploit temporal correlations for testing the dimension of a quantum system.…”

“…Recently, dimension witnesses based on sequences of general measurements have been introduced [13]. In this scenario sequences of measurements are performed on a quantum state  in (see figure 1).…”

“…We report the implementation of this witness on a single trapped 171 Yb + ion and certify the manipulation of a three-dimensional system in the experiment. For reference, the dimension witnesses of [13] have been experimentally implemented. This shows that dimension tests based on temporal correlations are of practical relevance and can be used to certify lower bounds on the dimension with current state technology.…”