LOEM Strategy in Quantum Multiparameter Estimation

• Local Operation with Entangling Measurements (LOEM) Strategy is a quantum protocol that uses classically correlated orthogonal pure states and global entangling measurements to enable multiparameter estimation with optimal precision.
• It circumvents measurement incompatibility by engineering orthogonal state ensembles that satisfy the weak commutativity condition, ensuring decoupled parameter variances and QCRB saturation.
• Experimental implementations on photonic platforms validate its effectiveness, demonstrating Heisenberg scaling through iterative parameter encoding and iterated enhancement of the quantum Fisher information.

A Local Operation with Entangling Measurements (LOEM) strategy is a quantum protocol for multiparameter estimation that combines locally preparable classically correlated orthogonal pure states with optimal entangling (global) measurements to attain the ultimate precision specified by the multiparameter quantum Cramér–Rao bound (QCRB). The LOEM approach achieves this by engineering an ensemble of orthogonal states, each independently carrying the encoded parameters, such that the measurement incompatibility barrier (typified by noncommuting optimal measurements for different parameters) is removed under the action of an entangling measurement across these orthogonal states. This framework is especially significant for practical quantum metrology, where attaining the QCRB for multiple parameters simultaneously is a central challenge, and is supported by experimental validation on photonic platforms (Mi et al., 12 Sep 2025).

1. The Multiparameter Quantum Cramér–Rao Bound and Measurement Incompatibility

In quantum multiparameter estimation, a vector of parameters $\boldsymbol{x} = (x_1, x_2, ..., x_m)$ is encoded in a quantum state via $$|\psi_{\boldsymbol{x}}\rangle = U(\boldsymbol{x}) |\psi_0\rangle$$. The QCRB establishes a fundamental lower bound on the covariance matrix $V_{\boldsymbol{x}}$ of any unbiased estimator as

$V_{\boldsymbol{x}} \geq (F)^{-1} \geq (Q)^{-1}$

where $F$ is the Fisher information matrix (FIM) for a measurement, and $Q$ is the quantum Fisher information matrix (QFIM) of the state.

The QFIM for a set of pure states is given by

$$Q_{ij} = 4 \operatorname{Re} \left( \langle \partial_{x_i} \psi_{\boldsymbol{x}} | \partial_{x_j} \psi_{\boldsymbol{x}} \rangle - \langle \partial_{x_i} \psi_{\boldsymbol{x}} | \psi_{\boldsymbol{x}} \rangle \langle \psi_{\boldsymbol{x}} | \partial_{x_j} \psi_{\boldsymbol{x}} \rangle \right)$$

Saturating the QCRB in multiparameter settings is generically obstructed by measurement incompatibility: different parameters can correspond to noncommuting optimal measurements, formalized by the Uhlmann curvature matrix

$$\mathcal{U}_{ij} = \frac{i}{4} \langle \psi_{\boldsymbol{x}} | [L_i, L_j] | \psi_{\boldsymbol{x}} \rangle$$

where $L_i$ denotes the symmetric logarithmic derivative (SLD) for $x_i$. The weak commutativity condition (WCC), $\mathcal{U}_{ij} = 0$ for all $i,j$, is necessary and sufficient for the QCRB to be saturable.

2. LOEM State Preparation: Orthogonality and Classical Correlation

The LOEM protocol circumvents the measurement incompatibility problem by replacing the usual parallel state encoding with a set of $d$ mutually orthogonal pure states, each prepared with local unitary operations:

$$|\psi_m\rangle = U_m |\psi_0\rangle, \quad m = 0,1,\ldots,d-1$$

Each $|\psi_m\rangle$ is orthogonal to the others, resulting in a classically correlated set. Applying the parameter-encoding unitary $U(\boldsymbol{x})$ to each state, the composite state is

$$|\Psi_{\boldsymbol{x}}\rangle = \bigotimes_{m=0}^{d-1} U(\boldsymbol{x}) |\psi_m\rangle$$

This naturally leads to a global state $|\Psi_{\boldsymbol{x}}\rangle$ in a $d^d$-dimensional Hilbert space with the property that imaginary parts of overlaps of the encoded states' derivatives vanish:

$$\operatorname{Im}\langle \partial_{x_i} \Psi_{\boldsymbol{x}} | \partial_{x_j} \Psi_{\boldsymbol{x}} \rangle = 0 \quad \forall i,j$$

corresponding to the WCC and thus making the QCRB accessible.

This protocol exploits the fact that the information about all parameters is redundantly encoded in each orthogonal component, rendering them suitable for joint analysis via collective measurement.

3. Entangling Measurement Design and Attainment of the QCRB

Optimal measurements for multiparameter estimation generally require global projections across the complete $d^d$-dimensional space. The LOEM scheme designs entangling measurements, typically projective in a basis that spans the full tensor-product space of the orthogonal preparations.

For the two-parameter ($\theta$, $\phi$) case in a qubit system, suitable orthogonal pure states can be

$$U(\theta, \phi) |0\rangle \otimes U(\theta, \phi) |1\rangle = |n, -n\rangle,$$

referred to as the "antiparallel state." This state achieves a diagonal QFIM,

$Q(|n, -n\rangle) = \begin{bmatrix}2 & 0\0 & 2\sin^2\theta\end{bmatrix} = 2 Q(|n\rangle),$

ensuring the variances in the estimation of $\theta$ and $\phi$ are decoupled, which is essential for attaining the quantum-optimal bound.

Iterative (or sequential) enhancement, where parameter encoding is amplified as $U(N\theta, N\phi)$ (for $N$ rounds), further yields QFIM scaling as $2N^2$ on the diagonal, thus achieving Heisenberg scaling ($1/N^2$ precision improvement).

4. Experimental Realization: Photonic Implementation

An experimental validation of the LOEM strategy was conducted on a quantum photonic system:

• State Preparation: Orthogonal states were generated using the path and polarization degrees of freedom of single photons from a $1560\,\mathrm{nm}$ SPDC source, with spatial modes representing logical qubits.
• Parameter Encoding: $\theta$ and $\phi$ were encoded using half-wave plates (HWPs) and phase shifters to realize the desired $U(\theta,\phi)$ operations.
• Entangling Measurement: Measurement was performed in an entangled basis, including projective measurements onto $|01\rangle$, $|10\rangle$, and $(|00\rangle\pm|11\rangle)/\sqrt{2}$, implemented with cascaded wave plates, polarizing beam splitters, and single-photon detectors.
• Iterative Enhancement: The experiment included iterated encoding, where $U(N\theta,N\phi)$ is realized to observe the quadratic reduction in mean squared error (MSE) with number of rounds $N$, confirming Heisenberg scaling.

The experiment confirmed that the achievable classical Fisher information for both $\theta$ and $\phi$ equals the QFIM, indicating QCRB saturation for the LOEM protocol and yielding variances matching the theoretical lower bounds.

5. Heisenberg Scaling Through Iterative Interactions

The LOEM strategy readily incorporates iterative (sequential) enhancement for further precision gains. In each round, the parameter-encoding unitary is exponentiated, e.g., $U(N\theta, N\phi)$ for $N$ rounds, producing the state

$$|\Psi_\mathbf{x}^{(N)}\rangle = U(N\theta, N\phi) |0\rangle \otimes U(N\theta, N\phi) |1\rangle.$$

The QFIM for this state has diagonal entries $2N^2$ (for parameter $\theta$) and $2N^2\sin^2 N\theta$ (for $\phi$), so that the mean squared error for each parameter estimation, with $M$ experimental runs, satisfies $M \cdot \mathrm{MSE} \sim 1/N^2$. The experiment confirmed that the measured MSE scales as $1/N^2$, in full agreement with the theoretical Heisenberg limit.

6. Context and Implications for Multiparameter Quantum Metrology

LOEM protocols offer a configuration that both circumvents the typical quantum measurement incompatibility in multiparameter estimation (through satisfying the WCC by construction) and provides a physically implementable path to saturating the QCRB. The principal implications include:

• Decoupling of Parameter Variances: The resulting QFIM being diagonal greatly simplifies joint estimation tasks.
• Feasibility of Entangling Measurements: Entangling measurements are engineered via global projectors on the orthogonal ensemble, rather than requiring joint optimal measurements on $N$ identical copies.
• Practical Applicability: The method is robust to imperfections, and can be realized in scalable photonic architectures, providing a pathway for high-efficiency quantum sensors, imaging systems, and complex sensor networks.

7. Comparison to Related Approaches and Prospects

The LOEM strategy is distinct from approaches using entangled probe states or adaptive local measurements alone, as it relies on the orthogonality of the prepared states and the use of global entangling measurements. The scheme stands out in:

• Avoiding Probe Entanglement Preparation: Only local unitary operations are required for state engineering.
• Harnessing "Classical" Correlation: Orthogonality ensures mutual exclusivity, facilitating global projective measurement design.
• Generic Applicability: The scheme generalizes beyond two-parameter, two-qubit cases, and is, in principle, extensible to higher-dimensional and multipartite settings.

Future directions involve extending LOEM architectures to multi-qubit and high-dimensional quantum systems, as well as exploring realizations in platforms with collective measurement access, furthering practical quantum metrology and quantum-enhanced sensing (Mi et al., 12 Sep 2025).