Fig. 3: Temporal circuit for a general tripartite unitary process.
From: Existence of processes violating causal inequalities on time-delocalised subsystems
UA and UB are applied either on the time-local target system \({T}_{1}^{({\prime} )}\) or \({T}_{2}^{({\prime} )}\) (and the ancillary systems), depending coherently on the state of the two-dimensional control systems \({Q}_{1}^{({\prime} )}\) and \({Q}_{2}^{({\prime} )}\). These two applications of the coherently controlled operations UA and UB are surrounded by circuit operations \({\omega }_{1}({U}_{C}):{{{{{{\mathcal{H}}}}}}}^{p{C}_{I}^{{\prime} }{P}_{O}}\to {{{{{{\mathcal{H}}}}}}}^{{\bar{T}}_{1}{E}_{1}{\bar{Q}}_{1}},\, {\omega }_{2}^{\circ }({U}_{C}):{{{{{{\mathcal{H}}}}}}}^{{\bar{T}}_{1}^{{\prime} }{E}_{1}}\to {{{{{{\mathcal{H}}}}}}}^{{\bar{T}}_{2}{E}_{2}},\, {\omega }_{2}^{\bullet }({U}_{C}):{{{{{{\mathcal{H}}}}}}}^{{\bar{{T}}_{1}^{{\prime} }{E}_{1}}}\to {{{{{{\mathcal{H}}}}}}}^{{\bar{{T}}_{2}{E}_{2}}}\) (these two also being coherently controlled), and \({\omega }_{3}({U}_{C}):{{{{{{\mathcal{H}}}}}}}^{{\bar{{T}}_{2}^{{\prime} }{E}_{2}{\bar{Q}}_{2}^{{\prime} }}}\to {{{{{{\mathcal{H}}}}}}}^{f{C}_{O}^{{\prime} }{F}_{I}}\), which can (together with the therein introduced ancillary systems E1, E2) in general all depend on UC, the third party’s (Charlie’s) operation. The boxes \({\mathbb{I}}\) stand for identity channels that relate the systems with and without the bars. The ancillary system p is prepared in the state \({\left|0\right\rangle }^{p}\) in the beginning, and the final ancillary system f is discarded. (Note that, with a slight abuse of notation, we use the ground symbol for this discarding of f, which is commonly used for mixed circuits where the boxes represent CP maps, rather than for circuits consisting of pure operations, as we have here. The system f however always ends up in the state \({\left|0\right\rangle }^{f}\) (see Supplementary Note 3A), so that taking the partial trace over f is equivalent to projecting onto \({\left|0\right\rangle }^{f}\), and does not introduce any decoherence or loss of purity. The coherently controlled applications of UA and UB, as well as of \({\omega }_{2}^{\circ }({U}_{C})\) and \({\omega }_{2}^{\bullet }({U}_{C})\), are displayed with a slight shift for graphical clarity, but they can be taken to act at the same time.
