Quantum Fisher Information and the Curvature of Entanglement

TL;DR

This work establishes a time-resolved link between entanglement dynamics and quantum-metrological precision by introducing the curvature of entanglement (CoE) as the negative second derivative of concurrence with respect to the coupling in a two-qubit Hamiltonian. In pure, lossless evolution, CoE satisfies CoE ≤ F, with equality at times when the instantaneous concurrence is maximal, and the SLD becomes effectively separable, enabling simple product measurements to saturate the quantum Cramér-Rao bound. The authors further show that CoE = F coincides with maxima of entanglement and relate these points to fidelity measures via the Uhlmann fidelity, providing an operational interpretation of the bound. Extending to open dynamics with amplitude damping, the same coincidence properties persist for certain initial states, indicating that near-optimal sensitivity and easy readouts can be achieved even with loss. These results offer a practical perspective on when entanglement improves metrological performance and suggest avenues for extending the framework to larger systems and alternative noise models.

Abstract

We explore the relationship between quantum Fisher information (QFI) and the negative of the second derivative of concurrence with respect to the coupling between two qubits, referred to as the curvature of entanglement (CoE). We analyze in detail the pure-state lossless case for which general results can be inferred and we also consider a simple interaction Hamiltonian in the case of one form of loss applied to the qubits. For a two-qubit quantum probe used to estimate the coupling constant appearing in the interaction Hamiltonian we show, for certain initial conditions, that there are times such that CoE = QFI. These times can be associated with the concurrence, viewed as a function of the coupling parameter, being a maximum. We examine the time evolution of the concurrence of the eigenstates of the symmetric logarithmic derivative and show that, for several families of initially separable and initially entangled states, simple product measurements suffice to saturate the quantum Cramér-Rao bound when CoE = QFI, while otherwise, in general, entangled measurements are required giving an operational significance to the points in time when CoE = QFI.

Figures (4)

• Figure 1: (Color online) Entanglement and QFI dynamics for initial state with unequal superposition of Bell states and $\alpha = 0.3$ with $\eta_{xy}=1$. The upper panel shows the QFI normalised in units of $g$ as $g^2F$ (blue, dashed line) which acts as the upper bound for the CoE (red, solid line). The concurrence, $C$ (blue, dashed line) and the identical concurrences of the two entangled eigenstates of the SLD operators, $C_{\rm SLD}$ (purple line) are shown in the lower panel. The vertical dotted lines serve as a guide to the eye showing that the points where CoE = QFI coincides with the maxima of $C$ and the zeros of $C_{\rm SLD}$ at the points where $\sin(2gt\eta_{xy}) = 1$.
• Figure 2: (Color online) Entanglement and QFI dynamics for the initial state $|\Phi_{\alpha}^0\rangle$ with $\alpha = 0.3$ and $\eta=1$. The upper panel shows the QFI normalized in units of $g$ as $g^2F$ (blue, dashed line) which acts as the upper bound for the CoE (red, solid line). Also shown is the QFI corresponding to the the optimal initial state $|\Psi_{\rm opt}^0\rangle = |\Phi_1^0\rangle$. The concurrence, $C$ (blue, dashed line) and the three distinct values of concurrences of the three entangled eigenstates of the SLD operators, $C_{\rm SLD}^{(1)}$ (purple line), $C_{\rm SLD}^{(2)}$ (magenta line) and $C_{\rm SLD}^{(3)}$ (blue line) are shown in the lower panel. The vertical dotted lines serve as a guide to the eye showing that the points where CoE = QFI coincides with the maxima of $C$ when $\sin(2gt\eta_{xy}) = 1$. We see that at these points the SLD eigenstates do not become product states.
• Figure 3: (Color online) Entanglement and QFI dynamics for an initially separable state subject to dissipation. Upper panel: the QFI (dashed line) serves as the envelope for CoE (red solid line). CoE is not defined at the points where ($\sin(2gt)=0$). Lower panel: concurrence of the system state $\rho_s(t)$ (green dashed line) and the non-zero concurrence of two of the eigenvectors of SLD (purple solid line). Vertical dotted lines denotes points where CoE = QFI and $C_{\rm SLD}$ is zero. The decay rate is set to $\kappa /g =0.5$ in both panels.
• Figure 4: (Color online) Entanglement and QFI dynamics for an initially entangled state. Upper panel: The CoE (red solid line) touches the QFI (blue dashed line) at the maxima of the concurrence $C(g,t)$ shown in the lower panel (green dashed line). The lower panel also shows the concurrence of the eigenvectors of the SLD corresponding to non-zero eigenvalues (purple solid line). Vertical dotted lines denotes points where CoE = QFI and the four coincidence properties are satisfied. The decay rate is set to $\kappa /g =0.5$ and $\alpha = 0.25$ in both panels.