Extracting many-body quantum resources within one-body reduced density matrix functional theory

Quantum Fisher information (QFI) is a central concept in quantum sciences used to quantify the ultimate precision limit of parameter estimation, detect quantum phase transitions, witness genuine multipartite entanglement, or probe nonlocality. Despite this widespread range of applications, computing the QFI value of quantum many-body systems is, in general, a very demanding task. Here we combine ideas from functional theories and quantum information to develop a functional framework for the QFI of fermionic and bosonic ground states. By relying upon the constrained-search approach, we demonstrate that the QFI matrix terms can universally be determined by the one-body reduced density matrix (1-RDM), thus avoiding the use of exponentially large wave functions. Furthermore, we show that QFI functionals can be determined from the universal 1-RDM functional by calculating its derivatives with respect to the coupling strengths, thus becoming the generating functional of the QFI. We showcase our approach with the Bose-Hubbard model and present exact analytical and numerical QFI functionals. Our results provide the first connection between the one-body reduced density matrix functional theory and the quantum Fisher information.

Figure 1

Universal functionals of QFIM for the repulsive Bose-Hubbard model (for all $U>0$) for $N=2$. The limit ${\mathcal{M}}_{\alpha \beta }=2$ is indicated as a disk in gray.

Figure 2

Universal functionals of QFIM for the attractive Bose-Hubbard model (for all $U<0$) for $N=2$. The limit ${\mathcal{M}}_{\alpha \beta }=2$ is indicated as a disk in gray.

Figure 3

Representation of $\mathit{\gamma }$, parametrized with the angles $\theta$ and $\phi$, inside the Bloch sphere of radius $N/2$. The color code represents the value of ${\mathcal{M}}_{zz}$ close to BEC for $N=1000$ and $\delta =0.1$ [see Eq. (15)]. The white line marks the standard quantum limit ${\mathcal{M}}_{zz}=N$.