**Appendix A. Scattering Matrix** *S* **and Transition Matrix** *M*

The unitary (i.e., probability preserving) scattering operator *S*, takes an initial state |*i* into a final state *S* |*i*. If a certain final state | *f* is of interest to us then we calculate the scattering-matrix element *f* | *S* |*i*. When dealing with particle scattering, it is convenient to do this in momentum space. Eigenstates of momentum are plane waves, i.e., states that occupy all space and cannot be normalized.

We shall be interested in final states | *f* that are different from the initial state |*i*, so that | *f* and |*i* are orthogonal, i.e., *f* |*i* = 0, and we can replace *S* by *S* − 1.

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

We use here the Quantum Electrodynamics book by Jauch and Rohrlich as our reference [17], to emphasize the development that had taken place between the physics of the 1930s and the quantum field theory of the 1950s.

To consider energy and momentum conservation, it is usual to write (Ref. [17], Equation (8)–(29))

$$
\langle f \vert \left(\mathcal{S} - 1\right) \vert i \rangle = \delta(P\_f - P\_i) \left\langle f \vert \left.M \vert i \right\rangle \; , \tag{A1}
$$

where *δ*(*Pf* − *Pi*) is the 4-dimensional delta function over energy-momentum and *M* is the transition matrix.

Usually the probability for a transition into the final state | *f*, given the initial state |*i*, would be the squared modulus of (A1) but the square of a delta function does not make sense. Then one imposes a very large but finite length *L* in space and requires normalization for the wave-functions in the volume *L*3, and, similarly, one imposes a time *T* for the whole process. Energy-momentum conservation is nearly exact for large *L* and *T*. One delta function in the squared modified (A1) becomes replaced by (2*π*)−4*L*3*T*. When normalization conventions are taken into account, the result becomes independent of *L* and proportional to *T*. After this we divide by *T* to get the transition probability per unit time (see Ref. [17], Equation (8)–(40)),

$$(2\pi)^{-1}\delta(P\_f - P\_i)|\left|^2. \tag{A2}$$

Then requesting the states |*i* and | *f* to have the same energy and momentum, we can interpret

$$(2\pi)^{-1} |\left< f \right| M \left| i \right> \right|^2 = (2\pi)^{-1} \text{Tr}[\left| f \right> \left< f \right| M \rho^{(0)} M^\dagger] \,. \tag{A3}$$

as the transition probability per unit time, induced by *M*, from an initial state described by the density operator

$$\rho^{(0)} = |i\rangle\langle i|\tag{A4}$$

to a final state described by the projection operator | *f f* |. We thus find that the transition probability-rate matrix obtained from the initial state (A4) is (2*π*)−<sup>1</sup> times

$$R = M\rho^{(0)}M^\dagger.\tag{A5}$$

Thus, (2*π*)−1*R* is the total transition rate times the density operator for the final state. Since the trace of a density operator is unity,

$$(2\pi)^{-1}w = (2\pi)^{-1}\text{Tr}\,R\tag{A6}$$

is the total transition rate. The normalized final-state density matrix is then

$$\rho^{(f)} = \frac{1}{w} R = \frac{M\rho^{(0)}M^{\dagger}}{\text{Tr}[M\rho^{(0)}M^{\dagger}]} \,. \tag{A7}$$

Let us consider the systems *μ* and *A*. *M* makes *A* entangled with *μ* without changing the state of *μ*. Still the transition amplitudes can differ between + and −. This can distort the entanglement and induce changes in the relative proportions of + and − in the final state (Equation (2)). Thus, the proportions are no longer fixed by the von Neumann dilemma; the dilemma does not arise in the scattering theory that we are considering.
