Entanglement negativity bounds for fermionic Gaussian states

The entanglement negativity is a versatile measure of entanglement that has numerous applications in quantum information and in condensed matter theory. It can not only efficiently be computed in the Hilbert space dimension, but for noninteracting bosonic systems, one can compute the negativity efficiently in the number of modes. However, such an efficient computation does not carry over to the fermionic realm, the ultimate reason for this being that the partial transpose of a fermionic Gaussian state is no longer Gaussian. To provide a remedy for this state of affairs, in this work, we introduce efficiently computable and rigorous upper and lower bounds to the negativity, making use of techniques of semidefinite programming, building upon the Lagrangian formulation of fermionic linear optics, and exploiting suitable products of Gaussian operators. We discuss examples in quantum many-body theory and hint at applications in the study of topological properties at finite temperature.

Figure 1

Logarithmic negativity bounds vs exact results in the ground state, as a function of the dimerization $\delta$, with $N=8$ and $\ell =2$.

Figure 2

Logarithmic negativity bounds vs exact results for thermal states, as a function of the dimerization $\delta$, and for various values of $\beta$. The symbols represent the exact data, while the solid lines with matching colors show the corresponding bounds.

Figure 3

Single-particle spectra ${\varepsilon }_k^{\times }$ for various $\ell$.

Figure 4

Single-particle spectra ${\varepsilon }_k^{\times }$ for ${\ell }_1+{\ell }_2=200$ and various ${\ell }_1$.

Figure 5

Upper bound against CFT scaling variable with ${\ell }_1+{\ell }_2=200$ fixed and varying ${\ell }_1$. For comparison, the solid line shows the equal-segment result (93), with ${\ell }_1={\ell }_2=\ell$.

Figure 6

Thermal single-particle spectra for $\ell =100$ and various $\beta$.

Figure 7

Upper bound for thermal states with various $\beta$ against $\ell$. The inset shows the data against CFT scaling variable.