*3.2. Replica Trick*

Calculating the full reduced density matrix for a general system is the subject of active research. Here, we use a different method that is reminiscent of the 'replica trick' in disorder systems. The trick has been introduced in the context of quantum field theory by Wilczek [32] and Cardy [33] and later in the context of quantum transport by Nazarov [31]. The key point is that, if we can evaluate Tr*ρ<sup>M</sup>* for any *M* ≥ 1, we are able to evaluate the von Neumann entropy using the following relation:

$$S^{(A)} = \lim\_{M \to 1} \frac{d}{dM} S\_M^{(A)} = \lim\_{M \to 1} \frac{d}{dM} \text{Tr}\_A \left( \rho\_A \right)^M. \tag{10}$$

One can see that there is no need to take the logarithm of Tr*A* (*ρA*) *M*. This is only a mathematical simplification in the vicinity of *M* → 1, i.e., when we want to reproduce von Neumann entropy by analytically continuing the derivative of the Renyi entropy. Otherwise, the presence of the logarithm is essential for the definition of the Renyi entropy. It might be useful to further comment that the Renyi entropy without the logarithm has many names such as Tsallis entropy or power entropy, etc. However, the presence of the logarithm is necessary for what we call the Renyi entropy. Otherwise, we would have lim *M* →1 *Trρ<sup>M</sup>* = 1, which, in this important limit, cannot be a true measure of information.

However, calculating Tr*A* (*ρA*) *M* for a real or complex number *M* is a hopeless task. The 'replica trick' does the following: compute Tr*A* (*ρA*) *M* only for integer *M* and then analytically continue it to a general real or even complex number.
