**3. Quantum**

### *3.1. Von Neumann and Renyi Entropy*

Let us now consider that a large system *A* with many degrees of freedom interacts with a small quantum system *q*. This can be thought of as the two share some degrees of freedom. The two exchange some energy via those shared degrees of freedom. Quantumness indicates that *q* carries a discrete energy spectrum and can be found in superposition between energy levels. Let *ρ* be the density matrix of the compound system. The partial density matrix of *A* is defined by tracing out the system *q* from *ρ*, i.e., *ρA* = Tr*qρ*. The von Neumann entropy for system *A* in the Boltzmann constant unit is defined as

$$S^{(A)} = -\text{Tr}\_A \rho\_A \ln \rho\_A \tag{5}$$

and the generalization of entropy in quantum theory will naturally give rise to defining the following quantum Renyi entropy for system *A*:

$$S\_M^{(A)} = -\ln \text{Tr}\_A \left(\rho\_A\right)^M. \tag{6}$$

The density matrix of the isolated compound system evolves between the times *t* and *t* > *t* using a unitary transformation that depends on the time difference *U* (*t* − *<sup>t</sup>*). Therefore, one can evaluate Tr*A* (*ρA*)*<sup>M</sup>* using the unitary transformation to trace it back to the time *t*; i.e.,

$$\begin{split} \left(\text{Tr}\left(\boldsymbol{\rho}\left(t\right)\right)^{M} \right)^{M} &=& \text{Tr}\left\{ \left(\boldsymbol{\mathcal{U}}\left(t - t'\right)\boldsymbol{\rho}\left(t'\right)\boldsymbol{\mathcal{U}}^{\dagger}\left(t - t'\right)\right)^{M} \right\} =& \text{Tr}\left\{ \left(\boldsymbol{\mathcal{U}}\left(t - t'\right)\boldsymbol{\rho}\left(t'\right)^{M}\boldsymbol{\mathcal{U}}^{\dagger}\left(t - t'\right)\right) \right\} \\ &=& \text{Tr}\boldsymbol{\rho}\left(t'\right)^{M}. \end{split}$$

After taking the logarithm from both sides, one finds that the Renyi entropy remains unchanged between the two times *t* and *t*. In other words, in a closed system, similar to energy and charge, Renyi entropy is a conserved quantity:

$$\frac{dS\_M}{dt} = 0.\tag{7}$$

Let us consider for now that there is no interaction between *A* and *q*. One can expect naturally that partial entropies are conserved as the result of no interaction because each subsystem can evolve with an independent unitary operator:

$$\frac{dS\_M^{(A)}}{dt} = \frac{dS\_M^{(q)}}{dt} = 0.\tag{8}$$

Interesting physical systems interact. Therefore, let us now consider that *A* and *q* interact. Consider that the total Hamiltonian is *H* = *HA* + *Hq* + *HAq*. For interacting systems, there is an important difference between conserved physical and information quantities. For physical quantities, the conservation holds in the whole system as well as in each subsystem. As far as Renyi entropies are concerned, there is a conservation law for the total Renyi entropy ln *S*(*<sup>A</sup>*+*q*) *M* ; however, this quantity is only approximately equal to the sum ln *S*(*A*) *M* + ln *S*(*q*) *M* , up to the terms proportional to the volume of the system. Therefore, no exact conservation law can be expected for the extensive quantity summation: ln *S*(*A*) *M* + ln *S*(*q*) *M* [30]. The reason is that, although the evolution of the entire system is governed by a unitary operator, the subsystem evolves non-unitarily. In the limit of weak coupling |*HAq*|/|*HA* + *Hq*| 1, the entropy of entire system can only be approximated with the sum of two partial entropies, thus the sum of partial entropies can only approximately satisfy a conservation, i.e., *dS*(*A*) *M* /*dt* + *dS*(*q*) *M* /*dt* ≈ 0. Outside of the validity of the weak coupling approximation, we must expect that, although the total entropy conserves, the interacting parts have entropy flows different from each other:

$$\frac{dS\_M^{(A)}}{dt} \neq -\frac{dS\_M^{(q)}}{dt}.\tag{9}$$

This makes the conservation of Renyi entropy different from the conservation of physical quantities. The root for the difference is in fact in the nonlinear dependence on the density matrix, namely 'non-observability' of entropy [31].
