Inequality on tensor product in trace Let $\rho$ and $\sigma$ be density operators and $N$ an integer. We assume that $\sigma>0$. I want to show that the inequality is true:
$$N \cdot \text{Tr}[\rho\log\sigma]- \text{Tr}[\rho^{\otimes N}\log{(\sigma}^{\otimes N})]\leq 0$$
 A: Actually the equality holds and this is one of the cases where establishing the notation takes longer than the actual proof.
To start, let $H$ denote a finite-dimensional complex Hilbert space and define for $\mathbb N \ni N\geq 2$ the Hilbert space $$H_N:=H^{\otimes N}:=\underbrace{H\otimes H\otimes \ldots\otimes H}_{N-\mathrm{factors}} \tag{1}$$ and for an operator $A$ on $H$ we define $$A_N:= A^{\otimes N}:=\underbrace{A\otimes A\otimes \ldots\otimes A}_{N-\mathrm{factors}} \tag{2}$$ as an operator on $H_N$. We want to show that
$$ N\, \mathrm{Tr}_{H} \rho \log \sigma  = \mathrm{Tr}_{H_N} \rho_N\log \sigma_N\tag{3} \quad , $$
for $\rho$ and $\sigma$ density matrices, i.e. positive semi-definite operators of unit trace, on $H$, where we additionally assume that $\sigma$ is positive definite. Note that $\rho_N$ and $\sigma_N$ are density matrices on $H_N$ and that $\sigma_N$ is positive definite, too.
To prove equation $(3)$, we make use of the following equality:
$$\log \sigma_N= \sum\limits_{k=1}^N \underbrace{\mathbb I_H\otimes \mathbb I_H\otimes \ldots\otimes \mathbb I_H\otimes\underbrace{ \log\sigma}_{k\mathrm{-th\, slot}}\otimes \mathbb I_H \otimes  \mathbb I_H\otimes \ldots \otimes\mathbb I_H }_{N-\mathrm{factors}}\tag{4} \quad . $$
There are several ways to prove equation $(4)$ and it is not hard at all. For example by an induction proof or by showing that the action of the LHS and RHS as operators are equal on an orthonormal basis of $H_N$ (which you can choose in a clever way).
Equation $(4)$ in turn immediately yields
$$ \rho_N \log\sigma_N =\sum\limits_{k=1}^N \underbrace{\rho\otimes\rho\otimes\ldots\otimes \rho \otimes \underbrace{\rho\log\sigma}_{k-\mathrm{th\,slot}}\otimes\rho\otimes \rho\otimes\ldots\otimes\rho}_{N\mathrm{-factors}}  \quad .  \tag{5}$$
Finally, by applying the trace operation $\mathrm{Tr}_{H_N}$ on $(5)$, using the very basic properties of the trace on tensor product spaces and the fact that $\rho$ is of unit trace, we arrive at the desired result.
