Statistical Mechanics of Random Mixed State Ensembles with Fixed Energy

Abstract

Mixed state ensembles such as the Bures-Hall and Hilbert-Schmidt measure are probability distributions that characterise the statistical properties of random density matrices and can be used to determine the typical features of mixed quantum states. Here we extend this framework by considering the properties of random states with fixed average energy, and the ensemble-averaged density matrix is derived under this additional physical constraint. This gives rise to a type of microcanonical ensemble for random mixed states and we connect its properties to a statistical mechanical entropy and temperature. Our results are illustrated using a variety of simple spin systems, and we find that they can exhibit exotic features such as phase transitions in the absence of energetic interactions and finite relative energy fluctuations in the thermodynamic limit.

Figures (5)

• Figure 1: Volume entropy $S(E)$ for a two-level system with spacing $\epsilon$, plotted as a function of energy $E$. We compare the BH ensemble's entropy (blue) with the HS ensemble (orange).
• Figure 2: Average energy $E(T)$ as a function of temperature $T$ for the BH microcanonical ensemble with a linear spectrum.
• Figure 3: Energy fluctuations $\delta E$ as a function of average energy $E$ for the BH microcanonical ensemble with a linear spectrum.
• Figure 4: Comparison of energy fluctuations $\delta E$ of the HS (blue), BH (orange) and pure-state (red) microcanonical density matrix with a linear spectrum at different dimensions $d$.
• Figure 5: Plots of the average magnetisation per spin of the Curie-Weiss model for the microcanonical HS density matrix as a function of temperature and for different system sizes. Plot (a) is weakly coupled $J=0.2B$ while plot (b) is strongly coupled, $J=20B$.