Full expectation-value statistics for randomly sampled pure states in high-dimensional quantum systems

We explore how the expectation values $\langle \psi \left|A\right|\psi \rangle$ of a largely arbitrary observable $A$ are distributed when normalized vectors $|\psi \rangle$ are randomly sampled from a high-dimensional Hilbert space. Our analytical results predict that the distribution exhibits a very narrow peak of approximately Gaussian shape, while the tails significantly deviate from a Gaussian behavior. In the important special case that the eigenvalues of $A$ satisfy Wigner's semicircle law, the expectation-value distribution for asymptotically large dimensions is explicitly obtained in terms of a large deviation function, which exhibits two symmetric nonanalyticities akin to critical points in thermodynamics.

Figure 1

Solid: The function $Y\left(a\right)$ from (35) by numerically evaluating $y\left(a\right)$ according to (7) for $N=2000$ (blue), $N=10000$ (red), and $N=50000$ (brown). In each case, the $a_n$ in (7) are the eigenvalues of an $N\times N$ matrix, randomly sampled from a GOE [15], and properly rescaled so that (2) and (31) are fulfilled. Bold black line: theoretical large $N$ limit according to (26), (33), and (34). Dashed: The corresponding functions $p_N\left(a\right)$ from (5). The theoretical large $N$ predictions are $p_N\left(a\right)=0$ for $a\in [0,1/2]$ and $p_N\left(1\right)=1$ but are not shown in the plot. (a) and (b): Same data displayed on linear and logarithmic scales. The fluctuations for different samples of the random matrices turned out to be quite small (not shown). Some remnants are still visible close to $a=1/2$ as apparent “irregularities” in the $N$ dependence of the curves.

Figure 2

Same as in Fig. 1 except that the eigenvalues $a_n$ were randomly generated via Wigner distributed differences $a_{n+1}-a_n$ [15] and that the bold line was now obtained by numerically solving (37). Very similar results were also found for Poisson distributed differences $a_{n+1}-a_n$ (not shown).