Fig. 4
Space-time diagram of a protocol for disentangling the clocks from the mass. In configuration \({\mathrm{K}}_{{\mathrm{A}} \prec {\mathrm{B}}}\) the mass is at a distance r + L from a1, and at r + L + h from b1. In \({\mathrm{K}}_{{\mathrm{B}} \prec {\mathrm{A}}}\)—it is at r from a1 and at r + h from b1.The opposite holds for a2, b2. The initial mass superposition is swapped (after sufficient time to prepare the clocks in the correlated state) so that they finally show the same time. At the local time τa of a1 (at event A1) the agent applies \(U_{{\mathrm{A}}_{\mathrm{1}}}\) on S1. At the local time τb of b1 the agent applies \(U_{{\mathrm{B}}_{\mathrm{1}}}\) on S1. For the mass configuration \({\mathrm{K}}_{{\mathrm{A}} \prec {\mathrm{B}}}\) A1 is before B1 (orange-coloured events), while for \({\mathrm{K}}_{{\mathrm{B}} \prec {\mathrm{A}}}\) event B1 is before A1 (blue-coloured events). The opposite order holds for events A2, B2 occurring on the opposite side of the mass, where agents a2, b2 act on S2. Unitary operations should be applied in the future light cone of the event where the clocks get correlated and outside the future light cone of the event when the mass amplitudes are swapped, Bell measurements (at C1, C2) should be made when the clocks become disentangled (at future light-like events to when the mass amplitudes are brought together), and the measurement at event D should be space-like to C1, C2; dashed yellow lines represent the relevant light cones
