**Appendix B. Calculation of Feynman Diagrams with Explicit Unitarity and Reversibility**

The unitarity of the scattering matrix has not been explicitly visible in the main text. Reversibility that we have pointed out as crucial, is also not explicit. To remedy this we shall present a slightly more elaborate description of the whole process where the observed system *μ* is produced in its initial state


In this picture, the transition rate will instead be hidden and hence also the race to the final state. We therefore use the results that we have already obtained in the article, the transition rate (9) and the distribution (10) of the final states over the aggregated variable *Y*. The Born rule is also contained in (10).

As in the previous description, *A* starts in the initial state |0, *α<sup>A</sup>* but *μ* is produced by *B* at an early time −*T* in the state |*ψμ*. After *μA*-interaction around the time zero, *μ* is absorbed in an eigenstate |+*<sup>μ</sup>* or |−*<sup>μ</sup>* at the time +*T* by *D*<sup>+</sup> or *D*−, leaving *A* in the state |+, *β*+(*α*)*<sup>A</sup>* or |−, *β*−(*α*)*A*, respectively. We thus have one initial state |0, *αA*, a member of the ensemble of initial states, and three available final states, |0, *α<sup>A</sup>* (no change), |+, *β*+(*α*)*A*, and |−, *β*−(*α*)*A*; The system *μ* takes part only in intermediate states.

Schematic Feynman-diagram elements for the action of the source *B*, the transition matrix *M* in Equation (3) and the sinks *D*<sup>+</sup> and *D*<sup>−</sup> are shown in Figure A1, and the factors corresponding to them, *J*∗, *b*±*ψ*<sup>±</sup> and *F*±. We represent *μ* by a thin line and *A* by a thick line. As in Figure 1, the interaction between *μ* and *A* described by the transition matrix *M*, is represented by a shaded circle. Reversibility is included through the actions of the Hermitian or complex conjugates, *J*, *M*†, and *F*<sup>∗</sup> *j* .

**Figure A1.** Schematic Feynman-diagram elements for the action of the source *B*, the transition matrix *M* and the sinks *Dj* (*j* = ±) and their conjugates.

We use perturbation theory to compute the final-state density matrix,

$$S \mid 0, \mathfrak{a} \rangle\_{A \mid A} \nmid 0, \mathfrak{a} \mid S^{\dagger} \;. \tag{A8}$$

We use the method of Nakanishi [20] to calculate this bilinear quantity directly rather than the state vector *S* |0, *αA*, simply because it makes normalization easy.

The diagrams of perturbation theory are shown in Figure A2. The zero-order no-change term is only an *A*-line corresponding to a contribution equal to 1 (Figure A2a). Figure A2b shows the diagram corresponding to that of Figure 1 with the source *B* and one sink *Dj* (*j* = +, −). The inverse of this diagram is that of Figure A2c. The two taken together into one diagram represents a reduction of the no-change component due to transitions to the other states (Figure A2d). This can be repeated any number of times. All these diagrams leading back to the initial state (Figure A2e) contribute a geometrical series, representing the total no-change component of the final state,

$$\begin{split} 1 - \sum\_{j=+,-} J \, \psi\_{j}^{\*} b\_{j}^{\*} F\_{j}^{\*} F\_{j} b\_{j} \psi\_{j} I^{\*} + \left( \sum\_{j=+,-} J \, \psi\_{j}^{\*} b\_{j}^{\*} F\_{j}^{\*} F\_{j} b\_{j} \psi\_{j} I^{\*} \right)^{2} \pm \dots = \\ \frac{1}{1 + \left( |F\_{+}|^{2} |\psi\_{+}|^{2} |b\_{+}|^{2} + |F\_{-}|^{2} |\psi\_{-}|^{2} |b\_{-}|^{2} \right) |I|^{2}} = \frac{1}{1 + |J|^{2} |F|^{2} \left( |\psi\_{+}|^{2} e^{\Xi(Y - \frac{1}{2})} + |\psi\_{-}|^{2} e^{\Xi(-Y - \frac{1}{2})} \right)}. \end{split} \tag{A9}$$

**Figure A2.** Schematic Feynman diagrams: (**a**) zero-order diagram for no change (the I-like sign above the *A*-line symbolizes "no *μ*-system"); (**b**) lowest order diagram for transition to a state with *A* marked by *μ* in the *j* state (see Figure 1); (**c**) inverse diagram of b; (**d**) diagrams b and c combined to a no-change correction; (**e**) summation over d repeated any number of times, i.e., summation of no-change diagrams to all orders; (**f**) the full perturbation expansion of the diagonal elements of the final-state density matrix with *A* marked by *μ* in the state |*jμ*.

*Entropy* **2019**, *21*, 834

Here we have used the expressions for the amplitudes in Equation (7) and given equal strength *F* to the two sinks *D*<sup>+</sup> and *D*−. The total scattering probability, i.e., the probability of *A* being marked by *μ* is

$$\begin{split} 1 - \frac{1}{1 + |I|^2 |F|^2 \left( |\boldsymbol{\psi}\_{+}|^2 e^{\Xi(\boldsymbol{Y} - \frac{1}{2})} + |\boldsymbol{\psi}\_{-}|^2 e^{\Xi(-\boldsymbol{Y} - \frac{1}{2})} \right)} &= \\ I \, \mathbb{J} \boldsymbol{\psi}\_{+} \, ^\* e^{\frac{1}{2} \Xi(\boldsymbol{Y} - \frac{1}{2})} F^\* \, \frac{1}{1 + |I|^2 |F|^2 \left( |\boldsymbol{\psi}\_{+}|^2 e^{\Xi(\boldsymbol{Y} - \frac{1}{2})} + |\boldsymbol{\psi}\_{-}|^2 e^{\Xi(-\boldsymbol{Y} - \frac{1}{2})} \right)} \, F e^{\frac{1}{2} \Xi(\boldsymbol{Y} - \frac{1}{2})} \, \mathbb{J}^\* + \\ I \, \mathbb{J} \boldsymbol{\psi}\_{-} \, ^\* e^{\frac{1}{2} \Xi(-\boldsymbol{Y} - \frac{1}{2})} F^\* \, \frac{1}{1 + |I|^2 |F|^2 \left( |\boldsymbol{\psi}\_{+}|^2 e^{\Xi(\boldsymbol{Y} - \frac{1}{2})} + |\boldsymbol{\psi}\_{-}|^2 e^{\Xi(-\boldsymbol{Y} - \frac{1}{2})} \right)} \, F e^{\frac{1}{2} \Xi(-\boldsymbol{Y} - \frac{1}{2})} \, \mathbb{J}^\* \, \end{split} \tag{A10}$$

The two terms on the right side of (A10) are the probabilities for the final states |+, *β*+(*α*)*<sup>A</sup>* and |−, *β*−(*α*)*A*, corresponding to the diagrams of Figure A1f for the remaining diagonal elements of the density matrix. For large Ξ, the no-change contribution (A9) becomes negligible. The same is true for the non-diagonal elements of the density matrix. The diagonal terms for + and − in (A10) become

$$p\_{\pm} = \frac{|\psi\_{\pm}|^2 e^{\pm \Xi Y}}{|\psi\_{+}|^2 e^{\Xi Y} + |\psi\_{-}|^2 e^{-\Xi Y}} \, . \tag{A11}$$

For *Y* = +1, *p*<sup>+</sup> = 1 and the + channel takes everything and for *Y* = −1, *p*<sup>−</sup> = 1 and the − channel takes everything. The norm is preserved, i.e., *S* is unitary. Reversibility is also clearly visible: *J*∗, *M* and *F*<sup>±</sup> are active together with their conjugates that represent inverse processes.
