**3. Mathematical Model of the** *μA***-interaction**

We shall now model the amplitudes describing the *μA*-interaction, leading to the final state (3), and how it depends on the initial state |0, *αA*.

To have a generic model, we think of our system *A* as consisting of *N* independent subsystems, each interacting with *μ*, resulting in amplitudes that are products of *N* factors [15,16]. Since only factors resulting in differences between the two amplitudes are important, we assume

$$\left|b\_{\rangle}(\boldsymbol{a})\right|^{2} = \prod\_{n=1}^{N} \left(1 + j\kappa\_{\mathbb{I}}(\boldsymbol{a})\right), \quad \text{with } j = + \text{ or } - \,, \tag{5}$$

where *κn*(*α*)<sup>∗</sup> = *κn*(*α*). Small deviations from unity in the factors are characterized by a zero mean, *κn* <sup>=</sup> 0, while the independence between the subsystems is expressed by *κnκn* <sup>=</sup> *<sup>δ</sup>nnχ*2, and 0 < *χ* << 1. We have followed the convention, used in stochastic dynamics (as for instance in Ref. [5]), to calculate to second order in *κ<sup>n</sup>* and then replace *κnκn* by its mean *δnnχ*2. This model reflects the unbiased character of the measurement device, and it guarantees that on average the squared moduli of the amplitudes are identical and equal to unity, |*bj*(*α*)| <sup>2</sup> <sup>=</sup> 1.

As an illustration, in Appendix C, we describe a situation where *μ* is a fast charged particle emerging from some process and then interacting electrically with the system *A*. Here A consists of a chain of small cylinders of ionizable material along the track where *μ* is passing in one of its state components. We show how the amplitude factorizes in this case.

We have chosen to keep each factor in the model close to unity in order to illustrate that even very small variations in how the subsystems interact with the system *μ* may result in one of the channels (one of the amplitudes), dominating over the other one, depending on the microstate *α* of the device. This makes it necessary to have a very large number *N* of subsystems. The critical assumptions are (i) the unbiased character of the device, i.e., not favoring any of the channels; and (ii) the independence of the subsystems of the device. The statistics of the interaction is treated in the following section.
