Fig. 5

a The process by which the quantum transducer generates future statistics as outlined in Fig. 3. b A more detailed breakdown of. Upon input x at time t, the transducer first applies the selection operator S _x . The subsequent operator A can be decomposed into two operators, a linear mapping \(B:\left|{s}_{i}^{x}\right\rangle \left\langle {s}_{i}^{x}\right|\to {\sum }_{y,k}{T}_{ik}^{y|x}\left|y\right\rangle \left|{\tau }_{k}\right\rangle \left\langle y\right|\left\langle {\tau }_{k}\right|\), and a decompression operator U that rotates each \(\left|{\tau }_{k}\right\rangle\) into |s _k 〉 (always possible when suitable ancillary systems in states |0〉 are supplied). Σ is emitted as output, while \({\mathcal{W}}\) is retained as the subsequent causal state at time t. This circuit makes it clear that \(\left|{\tau }_{k}\right\rangle\) also make perfectly valid causal states