Figure 2: (color online) Complementarity between the predictive and retrodictive approaches of quantum physics - the successful test of a proposition in one of these approaches gives the state of the other one, with which we can make predictions about measurement results or preparation choices. and the proposition operators are given by: Their resolution of the Hilbert space (3) is in fact equivalent to the maximization of Von Neumann entropy S [ˆ ρ ] = − Tr { ˆ ρ log ˆ ρ } by the mixture ˆ ρ $^{[?]}$. This one is then maximally mixed and probes all the responses of the apparatus. Finally, we propagate these retrodicted states backward in time contrary to prepared states, as pictured on Fig. 1. Indeed, when we take the Heisenberg picture for the POVM elements in the predictive approach, we retrieve the pre-measurement state (6) in the Schr¨ odinger picture with the time-evolution operator ˆ U ( − t ) = ˆ U $^{†}$( t ). The complementarity between the predictive and retrodictive approaches is illustrated on Fig. 2. ˆ Θ$_{m}$ = D P $_{m}$ˆ ρ$_{m}$. (7) 7