Semi-Device-Independent Certification of Causal Nonseparability with Trusted Quantum Inputs

While the standard formulation of quantum theory assumes a fixed background causal structure, one can relax this assumption within the so-called process matrix framework. Remarkably, some processes, termed causally nonseparable, are incompatible with a definite causal order. We explore a form of certification of causal nonseparability in a semi-device-independent scenario where the involved parties receive trusted quantum inputs, but whose operations are otherwise uncharacterized. Defining the notion of causally nonseparable distributed measurements, we show that certain causally nonseparable processes that cannot violate any causal inequality, including the canonical example of the quantum switch, can generate noncausal correlations in such a scenario. Moreover, by imposing some further natural structure to the untrusted operations, we show that all bipartite causally nonseparable process matrices can be certified with trusted quantum inputs.

Figure 1

SDI-QI scenario (main): A process matrix $W^{AB}$ connects two parties who receive quantum inputs ${\rho }_x^{\tilde{A}}$ and ${\rho }_y^{\tilde{B}}$, respectively. They each perform a joint operation [$(M_a^{\tilde{A}{A}_I{A}_O}{)}_a$ and $(M_b^{\tilde{B}{B}_I{B}_O}{)}_b$, respectively], and produce the classical outcomes $a$ and $b$. The purple box shows the D-POVM $(E_{a,b}^{\tilde{A}\tilde{B}}{)}_{a,b}$ induced by these instruments and the process matrix. Inset: In the MDCI scenario (see later), additional structure is assumed on the quantum instruments (shown here for Alice). The quantum input is a bipartite state in ${\mathcal{H}}^{{\tilde{A}}_I{\tilde{A}}_O}$, a measurement is performed jointly on ${\mathcal{H}}^{{\tilde{A}}_I{A}_I}$ and a channel sends ${\mathcal{H}}^{{\tilde{A}}_O}$ to the process matrix through ${\mathcal{H}}^{A_O}$.