*3.3. Time Evolution of Entropy*

Let us mention that we limit our analysis here only to weak coupling. In this regime, the dynamics of a quantum system are reversible and can be formulated in terms of the density matrix evolution. This time evolution depends on the the time-dependent Hamiltonian *H* (*t*) = *HA* + *HB* + *HAB* as follows:

$$\frac{d\rho}{dt} = \frac{\dot{\imath}}{\hbar} \left[ H(t), \rho(t) \right]. \tag{11}$$

We transform the basis to the interaction frame by using defining a unitary operator with the non-interaction part of the Hamiltonian *U*(*t*) = exp [−*<sup>i</sup>*(*HA* + *HB*)*t*]. The density matrix transforms as *R* (*t*) = *U* (*t*) *ρ* (*t*) *U*† (*t*), thereby not changing its entropy, neither in parts nor in total. In the new basis, Equation (11) becomes

$$\frac{dR}{dt} = \frac{i}{\hbar} \left[ \mathcal{U}^{\dagger}(t) H\_{AB}(t) \mathcal{U}(t), \mathcal{R}(t). \right] \tag{12}$$

Let us refer to the interaction Hamiltonian *HAB* in the new basis as *HI*, i.e., *HI* ≡ *U*† (*t*) *HAB* (*t*) *U* (*t*). The solution to the time evolution Equation (12) can be written as

$$R(t) = R\_0 + R^{(1)}(t) + \mathcal{O}(2) \tag{13}$$

with

$$R\_0 \equiv R(0) \tag{14}$$

$$\mathcal{R}^{(1)}(t) \equiv \frac{i}{\hbar} \int\_0^t ds \left[ H\_I(\mathbf{s}), R\_0 \right] \tag{1st order} \tag{15}$$

This solution can (repeatedly) be inserted back into the right side of Equation (12), declaring its cycle of internal interaction:

$$\frac{d\mathcal{R}\left(t\right)}{dt} = \Lambda^{(1)} + \Lambda^{(2)} + \mathcal{O}(3) \tag{16}$$

with

$$\Delta^{(1)} \equiv \frac{i}{\hbar} \left[ H\_I \left( t \right), \mathcal{R}\_0 \right],\tag{1st order} \tag{17}$$

$$\Delta^{(2)} \equiv -\frac{1}{\hbar^2} \int\_0^t ds \left[ H\_I \left( t \right), \left[ H\_I \left( s \right), R\_0 \right] \right]. \tag{2nd order} \tag{18}$$

In order to find the time evolution of the Renyi and von Neumann entropies, we first notice that the unitary transformation *<sup>U</sup>*(*t*), defining the basis change, also transforms any power of the density matrix, i.e.,

$$\mathcal{R}\left(t\right)^{M} = \mathcal{U}(t)\left(\rho\left(t\right)^{M}\right)\mathcal{U}^{\dagger}(t). \tag{19}$$

Now, all we need to do is to generalize the evolution of density matrix to the powers of density matrix (*R* (*t*))*<sup>M</sup>*. We follow the terminology of Nazarov in [31] and name each copy of replica *R* (*t*) in the matrix (*R* (*t*))*<sup>M</sup>* a 'world', thus (*R* (*t*))*<sup>M</sup>* is the generalized density matrix of *M* worlds:

$$\begin{split} \frac{d}{dt} \left( \boldsymbol{R} \left( \boldsymbol{t} \right)^{M} \right) &= \quad \left[ \frac{d}{dt} \boldsymbol{R} \left( \boldsymbol{t} \right) \right] \left( \boldsymbol{R} \left( \boldsymbol{t} \right) \right)^{M-1} + \boldsymbol{R} \left( \boldsymbol{t} \right) \left[ \frac{d}{dt} \boldsymbol{R} \left( \boldsymbol{t} \right) \right] \left( \boldsymbol{R} \left( \boldsymbol{t} \right) \right)^{M-2} \\ &+ \dots + \left( \boldsymbol{R} \left( \boldsymbol{t} \right) \right)^{M-2} \left[ \frac{d}{dt} \boldsymbol{R} \left( \boldsymbol{t} \right) \right] \boldsymbol{R} \left( \boldsymbol{t} \right) + \left( \boldsymbol{R} \left( \boldsymbol{t} \right) \right)^{M-1} \left[ \frac{d}{dt} \boldsymbol{R} \left( \boldsymbol{t} \right) \right]. \end{split}$$

By substituting the solutions of Equations (13) to (18), and limiting the result to second order, we find the the following time evolution of the *M*-world density matrix:

$$\begin{split} \frac{d}{dt} \begin{Bmatrix} \mathbf{R} \begin{pmatrix} \mathbf{r} \end{pmatrix}^{M} \end{pmatrix} &= \begin{array}{c} \boldsymbol{\Lambda}^{(2)} \mathbf{R}\_{0}^{M-1} + \boldsymbol{R}\_{0} \boldsymbol{\Lambda}^{(2)} \mathbf{R}\_{0}^{M-2} + \cdots + \boldsymbol{R}\_{0}^{M-1} \boldsymbol{\Lambda}^{(2)} \\ \\ + \boldsymbol{\Lambda}^{(1)} \left\{ \boldsymbol{R}\_{0}^{(1)} \mathbf{R}\_{0}^{M-2} + \boldsymbol{R}\_{0} \boldsymbol{R}^{(1)} \boldsymbol{R}\_{0}^{M-3} + \cdots + \boldsymbol{R}\_{0}^{M-2} \boldsymbol{R}^{(1)} \right\} \end{array} \\ &+ \boldsymbol{R}\_{0} \boldsymbol{\Lambda}^{(1)} \left\{ \boldsymbol{R}^{(1)} \boldsymbol{R}\_{0}^{M-3} + \boldsymbol{R}\_{0} \boldsymbol{R}^{(1)} \boldsymbol{R}\_{0}^{M-4} + \cdots + \boldsymbol{R}\_{0}^{M-3} \boldsymbol{R}^{(1)} \right\} \\ &+ \boldsymbol{R}\_{0}^{2} \boldsymbol{\Lambda}^{(1)} \left\{ \boldsymbol{R}^{(1)} \boldsymbol{R}\_{0}^{M-4} + \boldsymbol{R}\_{0} \boldsymbol{R}^{(1)} \boldsymbol{R}\_{0}^{M-5} + \cdots + \boldsymbol{R}\_{0}^{M-4} \boldsymbol{R}^{(1)} \right\} \\ &+ \cdots \\ &+ \left\{ \boldsymbol{R}^{(1)} \boldsymbol{R}\_{0}^{M-2} + \boldsymbol{R}\_{0} \boldsymbol{R}^{(1)} \boldsymbol{R}\_{0}^{M-3} + \cdots + \boldsymbol{R}\_{0}^{M-2} \boldsymbol{R}^{(1)} \right\} \boldsymbol{\Lambda}^{(1)}. \end{split}$$

This is how the *M*-world density matrix evolves in time. The first line in Equation (20) denotes the case where the 2nd order perturbation takes place in one world while the *M* − 1 remaining worlds are left non-interacting. All these remaining terms have in common that they don't contain a 2nd order term occurring in a single replica. Instead, these terms contain two 1st order interactions, each acting in a single replica, which together combine to give a 2nd order perturbation term. These new terms have recently been found [34].

If you decide to consider higher perturbative orders, say up to *k*-th order with *k* ≤ *M*, there will be terms like *RM*−<sup>1</sup> 0 Δ(*k*) in the expansions that have *k* interactions taking place in one replica, leaving *M* − 1 replicas noninteracting as well as terms having *k* first-order configurations combining to give a *k*th order interaction term, such as *RM*−*<sup>k</sup>* 0 Δ(1)*<sup>k</sup>*. In the case *k* > *M*, some of the lowest-order interactions will obviously become excluded from the summations.

Let us show the time evolution pictorially using the following diagrams, in which the evolution of (*R*(*t*))*<sup>M</sup>* is shown by *M* parallel lines, each one denoting the time evolution of one world, starting in the past at the bottom and arriving at the present time on the top. In the following diagrams, we show five time-slices by horizontal dashed lines. Blue dots denote the interaction *HI*(*t*) and our diagrams are limited to the 2nd order only. Curly photon-like lines connect the two interactions and represent the correlation function.

The first line of Equation (20) contains all terms that have two interactions in a single world. These two interactions within the same world are called 'self-replica interactions'. They can be illustrated pictorially by the following diagrams in Figure 1 from left to right:

**Figure 1.** Diagrammatic representation of terms in the first line in Equation (20).

The following diagram in Figure 2 illustrates the typical term (*<sup>R</sup>*0) 2 <sup>Δ</sup>(1)*R*0*R*(1) (*<sup>R</sup>*0) *M*−4 from Equation (20) and pictorially shows the contribution of two first order interactions in two different worlds that together evolve the generalized density matrix of *M* worlds in the second order.

**Figure 2.** A typical diagram with two first order interactions acting on two different worlds.

A typical higher order digram limited to two-correlation interactions can diagrammatically be shown as below in Figure 3.

**Figure 3.** A typical higher order diagram.
