Since ρ ρ A happens to be precisely equal to the density matrix of a single quantum harmonic oscillator of frequency ω ω ≡ ≡ ω ω + ω ω − − at thermal equilibrium with temperature T (such that ω ω / k B T = cosh − − 1 ⁡ ⁡ ( 8 ω ω + ω ω − − + ( ω ω + − − ω ω − − ) 2 ( ω ω + − − ω ω − − ) 2 ) where k B is the Boltzmann constant), the eigenvalues of ρ ρ A are λ λ n = ( 1 − − e − − ω ω / k B T ) e − − n ω ω / k B T for nonnegative integers n . The Von Neumann Entropy is thus