Contextuality in Phase Space

We present a general framework for contextuality tests in phase space using displacement operators. First, we derive a general condition that a single-mode displacement operator should fulfill in order to construct Peres-Mermin square and similar scenarios. This approach offers a straightforward scheme for experimental implementations of the tests via modular variable measurements. In addition to the continuous variable case, our condition can also be applied to finite-dimensional systems in discrete phase space, using Heisenberg-Weyl operators. This approach, therefore, offers a unified picture of contextuality with a geometric flavor.

Figure 1

Pictorial representation of three displacement amplitudes ${\alpha }_i$ which span a triangle of area $\mathcal{A}=\pi /4$ and, therefore, satisfy the condition in Eq. (6) used for constructing a PM square in phase space.

Figure 2

Illustration of the displacement amplitudes ${\overrightarrow{A}}_{jk}$ used for the approximate construction of the PM square for a single mode. See the text for further details.

Figure 3

A quantum circuit composed of a sequence of three Ramsey measurements used for measuring the correlations between modular variables.