Quantum Estimation with State Symmetry-Induced Optimal Measurements

Abstract

A central challenge in quantum metrology is identifying optimal measurements that saturate the quantum Cramer-Rao bound under realistic constraints, e.g., local measurements. We show that symmetries of the probe state provide a general principle for identifying optimal measurement strategies. Building on this idea, we demonstrate that when a parameter is encoded in the real coefficients of a fixed-basis expansion, the optimal measurement reduces to projection in that basis, with an application to critical metrology. Under local-measurement constraints, we show that local state symmetries provide a systematic route to constructing optimal local measurements. We illustrate this framework using graph states, explicitly constructing optimal local measurements from their local symmetries. Furthermore, weak and strong connection rules are introduced to generate broader classes of graph states that achieve Heisenberg-scaling precision using local measurements. By relaxing the number of stabilizer generators, graph states are extended to a stabilizer-code subspace. Analytical and numerical results show that coherent states in these subspaces offer multiple metrological advantages: high precision, partial noise resilience, local-measurement accessibility, and built-in error correction. These findings advance the theory of optimal measurements in quantum metrology and underscore the central role of state symmetry.

Figures (6)

• Figure 1: Schematic illustration of protocols for estimating a global phase parameter $\theta$ in an $N$-qubit system. (a) Global unitary encoding $U_\theta=\exp(-i\theta H/2)$ generated by an arbitrary Hamiltonian $H$, followed by collective measurements $\{M_x\}_{x\in \mathcal{X}}$. (b) Local unitary encoding $U_\theta=\exp(-i\theta H/2)$ generated by a linear Hamiltonian $H=\sum_j H_j$, followed by local measurements $\{M_{\vec{x}}=\bigotimes_{j=1}^N M_{x_j}\}_{\vec{x}\in\mathcal{X}}$.
• Figure 2: Examples of graphs used in this work: (a) an example graph with highlighted vertex types, (b) the complete graph $\mathbb G_C$, (c) the star graph $\mathbb G_S$, and (d) the complete bipartite graph $\mathbb G_{m,n}$.
• Figure 3: (a) A W-type state obtained by weakly connecting a complete graph and a star graph; (b) A S-type state obtained from a two-vertex parent graph $a\!\leftrightarrow\! b$ with both vertices being replaced by edgeless subgraphs; (c) A S-type state obtained from a three-vertex parent graph $b\!\leftrightarrow\! a\!\leftrightarrow\! c$, with vertices replaced by complete bipartite, star, and complete subgraphs, respectively. The vertices belonging to the subset $\alpha$ that determines the state symmetry \ref{['S_alpha_1']} are marked by red circles.
• Figure 4: Comparison of Protocols 1 and 2. Both protocols share the same state symmetry $S$ guaranteed by the stabilizer generators $\{K_j^{(s)}\}$. Accordingly, by Theorem 3 [or Eq. \ref{['qubit']}], the same Hamiltonian and local measurements ${M_x}$ can be used to construct the estimation protocol and to saturate the QCRB. Protocol 2 differs from Protocol 1 in that it removes some of stabilizer generators irrelevant to the state symmetry, $K_{N-m+2}^{(o)},\cdots,K_{N}^{(o)}$, thereby enlarging the probe space from a single graph state $|\mathbb{G}\rangle$ to a relaxed-stabilizer subspace $\mathcal{R}_m$. Here, the superscripts $(s)$ and $(o)$ denote stabilizer generators that are respectively related and unrelated to the state-symmetry operator $S$.
• Figure 5: QFI of different probe states \ref{['rss_qubit']} with $r_x=1$, $r_y=1$, or $r_z=1$ (GHZ) under $(X,Z)$ or $(Y,Z)$ noise. The QFI is normalized as $\widetilde{\mathcal{F}}_Q=\mathcal{F}_Q/[(1-2p)^2 N_1^2]$, according to Eq. \ref{['rss_XZ']}. In all figures, $N_1=N_2$ and the noise probabilities are taken as $p=0.1$ for $X$-noise and $q=0.15$ for $Z$-noise.