Stability of quantum statistical ensembles with respect to local measurements

We introduce a stability criterion for quantum statistical ensembles describing macroscopic systems. An ensemble is called “stable” when a small number of local measurements cannot significantly modify the probability distribution of the total energy of the system. We apply this criterion to lattices of spins-1/2, thereby showing that the canonical ensemble is nearly stable, whereas statistical ensembles with much broader energy distributions are not stable. In the context of the foundations of quantum statistical physics, this result justifies the use of statistical ensembles with narrow energy distributions such as canonical or microcanonical ensembles.

Figure 1

Schematic representation of the evolution of a two-peak energy distribution $g_n\left(E\right)$ (solid red lines) governed by Eq. (10): $g_0\left(E\right)=1/2[\delta (E-E_1)+\delta (E-E_2)]$; $g_1\left(E\right)\cong {\left[{\mathcal{P}}_1\right]}_{\text{diag}}\left(E\right)g_0\left(E\right)$; and $g_2\left(E\right)\cong {\left[{\mathcal{P}}_2\right]}_{\text{diag}}\left(E\right)g_1\left(E\right)$. Here ${\left[{\mathcal{P}}_1\right]}_{\text{diag}}\left(E\right)$ and ${\left[{\mathcal{P}}_2\right]}_{\text{diag}}\left(E\right)$ (dashed blue lines) correspond to two single-spin measurements with respective outcomes ${\vartheta }_1=\pi$ and ${\vartheta }_2=\pi$ substituted in Eq. (13).

Figure 2

Averaged ensemble stability measure $\overline{\mathrm{\Delta }G}\left(n\right)$ as a function of the number of measurements $n$ for a two-peak initial distribution $g_n\left(E\right)$. Points represent exact numerically computed results as explained in Appendix pp5. Lines correspond to the approximated expression (15) with $\lambda =0.3\frac{{(E_1-E_2)}^2}{{(E_{max}-E_{min})}^2}$. Different colors represent different pairs of values for $\left(E_1,E_2\right)$: blue (circles) ($-0.9$,0.9), yellow (squares) ($-0.9$,0.0), and green (rhombi) $\left(-0.9,-0.6\right)$ in units where $E_{min}=-1$ and $E_{max}=1$.

Figure 3

The energy distribution $g_n\left(E\right)$ is shown after each even-numbered measurement as indicated in the legend. The system size is $N_s=24$.

Figure 4

The quantity $\frac{E_{\text{av},n}-E_{\text{av},0}}{E_{max}-E_{min}}$ is shown for the interacting spin system introduced in Sec. 5. The energy distribution corresponds to the Gibbs distribution with the temperature $T=0.1$. Different system sizes were chosen as indicated in the legend. The plotted values have been averaged over many iterations. The points are connected by lines in order to guide the eye.

Figure 5

Schematic representation of the energy intervals $\text{bin}\left(E\right),\overline{\text{bin}\left(E\right)},X_{-},{\overline{X}}_{-},X_{+}$, and ${\overline{X}}_{+}$, introduced in the text. The characteristic size of the intervals ${\mathrm{\Delta }}_e$ and ${\mathrm{\Delta }}_{\mathcal{O}}$ are indicated above.