Interpreting a relation which uses the equal a priori probability principle

Question

Currently, I am going through The role of quantum information in thermodynamics—a topical review by Goold et al. I am stuck with Equation $1$ of the paper. It is given below.

The equal a priori probability principle would describe the equilibrium state as $$ \varepsilon_R = \frac{1_R}{d_R} $$

So, what does this equation mean? I understand that $d_R$ is just a dimensionless quantity as it is the number of dimensions of the system $R$. But what about the other quantities in this equation? Any help?

Answer 1

$\epsilon_R$ is a density matrix which describes a mixed state, where every orthogonal direction in the Hilbert space $\mathcal{H_R}$ (in some particular basis) is weighted with equal probability.

$1_R$ is the unit $d_R$ x $d_R$ matrix (where $d_R$ is the dimensionality of $H_R$). Just 1's on the diagonal and 0's everywhere else. But to be properly normalized, density matrices must have their trace equal to 1 (so that the probability of being in any state adds up to 100%). Since ${\rm tr}(1_R) $ = $d_R$, it needs to be divided by $d_R$ to ensure ${\rm tr}(\epsilon_R) = 1$.

The physical interpretation of a density matrix like this is that there is a maximal amount of uncertainty about what state the system is in. It could be in literally any possible state, all with equal probability. You could think of this as an "a priori" state of a system before you gain any information about it. Or you could think of it as the equilibrium state of maximum entropy (S = $k\log d_R$)