Fraction of isospectral states exhibiting quantum correlations

For several types of correlations (mixed-state entanglement in systems of distinguishable particles, particle entanglement in systems of indistinguishable bosons and fermions, and non-Gaussian correlations in fermionic systems) we estimate the fraction of noncorrelated states among the density matrices with the same spectra. We prove that for the purity exceeding some critical value (depending on the considered problem) fraction of noncorrelated states tends to zero exponentially fast with the dimension of the relevant Hilbert space. As a consequence, a state randomly chosen from the set of density matrices possessing the same spectra is asymptotically a correlated one. To prove this we developed a systematic framework for detection of correlations via nonlinear witnesses.

Figure 1

Illustration of the geometry of correlated states in the space of density matrices $\mathcal{D}\left(\mathcal{H}\right)$. The space $\mathcal{D}\left(\mathcal{H}\right)$ is located inside the region bounded by the dashed line. For simplicity of presentation we identified the boundary of $\mathcal{D}\left(\mathcal{H}\right)$ with space of pure states ${\mathcal{D}}_1\left(\mathcal{H}\right)$. The class of pure states $\mathcal{M}$ is given by the thick solid segments laying on ${\mathcal{D}}_1\left(\mathcal{H}\right)$. The class of noncorrelated states, ${\mathcal{M}}^c$, is marked by a gray region. The set of isospectral density matrices $\Omega$ is marked by the solid loop.