Iterative optimization in quantum metrology and entanglement theory using semidefinite programming

TL;DR

The paper tackles the problem of achieving quantum-enhanced metrology in bipartite systems by optimizing over local Hamiltonians, introducing the metrological gain $g(\varrho)$ as a state-dependent benchmark relative to the best separable performance. It develops an efficient iterative see-saw method that recasts the quantum Fisher information as a bilinear form, enabling fast convergence via semidefinite programming, and pairs this with a moment-based SDP relaxation to certify global optimality on small systems. The authors demonstrate the approach on various states, including isotropic two-qudit states and bound-entangled PPT states, and extend the framework to a broader class of quadratic-optimization problems in quantum information, such as maximizing Wigner-Yanase skew information and CCNR-related trace norms. The work provides a practical, scalable toolkit for designing metrologically valuable quantum states and dynamics under experimentally affordable local Hamiltonians, with implications for resource theories and experimental quantum sensing. The methods offer a path toward reliably identifying quantum advantages in metrology and related informational tasks, while connecting optimization, entanglement structure, and measurement precision in a unified framework.

Abstract

We discuss efficient methods to optimize the metrological performance over local Hamiltonians in a bipartite quantum system. For a given quantum state, our methods find the best local Hamiltonian for which the state outperforms separable states the most from the point of view of quantum metrology. We show that this problem can be reduced to maximizing the quantum Fisher information over a certain set of Hamiltonians. We present the quantum Fisher information in a bilinear form and maximize it by an iterative see-saw (ISS) method, in which each step is based on semidefinite programming. We also solve the problem with the method of moments that works very well for smaller systems. Our approach is one of the efficient methods that can be applied for an optimization of the unitary dynamics in quantum metrology, the other methods being, for example, machine learning, variational quantum circuits, or neural networks. The advantage of our method is the fast and robust convergence due to the simple mathematical structure of the approach. We also consider a number of other problems in quantum information theory that can be solved in a similar manner. For instance, we determine the bound entangled quantum states that maximally violate the Computable Cross Norm-Realignment (CCNR) criterion.

Figures (5)

• Figure 1: Scheme of the metrological process considered here. We start by preparing a given bipartite probe state $\varrho$ that undergoes the unitary dynamics $U_{\theta}=e^{-i\mathcal{H}\theta}$. The output state is $\varrho_{\theta}=U_{\theta}\varrho U_{\theta}^\dagger$, where $\theta$ is the parameter that we try to estimate. It is assumed that the Hamiltonian ${\mathcal{H}}$ that generates the unitary dynamics is local (i. e., ${\mathcal{H}}\in{\mathcal{L}}$). A Hamiltonian ${\mathcal{H}}$ on a bipartite system is said to be local if it can be written as ${\mathcal{H}}=H_1\otimes \openone_2+\openone_1\otimes H_2,$ where $H_1$ ($H_2$) is a Hamiltonian on the first (second) subsystem. Such Hamiltonians do not contain interaction terms between the two parties. In this setting, for a given probe state $\varrho$, we look for the optimal local Hamiltonian such that the output state provides the best possible metrological precision in estimating the parameter $\theta.$ We will discuss in the text, how to quantify the metrological performance of a quantum state with a given local Hamiltonian.
• Figure 2: The convex set of local Hamiltonians fulfilling Eq. \ref{['eq:cnHconst']}. $\mathcal{H}_p$ is a "mixture" of $\mathcal{H}'$ and $\mathcal{H}"$ as given by Eq. \ref{['eq:Hp']}.
• Figure 3: Based on the formula given in Eq. \ref{['eq:seesaw']}, we can optimize the quantum Fisher information over local Hamiltonians for a fixed probe state with an iterative see-saw (ISS) method.
• Figure 4: Plot of $g(x,\tilde{x})$ as defined in Eq. \ref{['eq:gxy']} for $q=1.$ In the insets, curves for $x_0=0$ and $\tilde{x}_0=0.5$ are shown on the sides, while the middle inset is for $\tilde{x}=x.$ Red lines in the main figure correspond to the three insets. By inspection we can see that $g(x,\tilde{x})$ is not concave.
• Figure 5: Plot of $h(x,y)$ as defined in Eq. \ref{['eq:expr2']}. In the insets, curves for $x_0=0$ and $y_0=0.5$ are shown on the sides, while the middle inset is for $y=x.$ Red lines in the main figure correspond to the three insets. By inspection, we can see that $h(x,y)$ is not concave.