Non-Hermitian Quantum Metrology

• Non-Hermitian quantum metrology is the study of quantum parameter estimation under nonunitary dynamics, incorporating losses, gain, and decoherence.
• It generalizes quantum Fisher information to achieve Heisenberg scaling, optimizing measurement strategies through generalized uncertainty relations.
• Experimental protocols using single-photon interferometry confirm that both PT-symmetric and generic non-Hermitian Hamiltonians can reach quantum-limited precision.

Non-Hermitian quantum metrology is the paper and implementation of quantum parameter estimation protocols where the probe evolution or measurement process is governed by non-Hermitian dynamics. In such scenarios, the Hamiltonian or the effective evolution operator is not necessarily Hermitian, and thus the standard machinery of quantum Cramér–Rao theory and symmetric logarithmic derivatives (SLDs) must be generalized to nonunitary maps, often accommodating loss, gain, and decoherence processes intrinsic to realistic measurement environments. This field has seen substantial advances in foundational mathematical formalism, operational criteria for optimal measurements, experimental verification of ultimate quantum limits, and the connection to critical phenomena and exceptional points.

1. Quantum Fisher Information in Non-Hermitian Dynamics

A central quantity in quantum metrology is the quantum Fisher information (QFI), which sets the ultimate bound for estimation precision via the quantum Cramér–Rao inequality. For non-Hermitian (and thus nonunitary) evolution, the QFI must consistently accommodate non-normalization of the quantum state and generalizes beyond the standard Hermitian settings. The following unified expression applies to a parameter $\alpha$ encoded in a (normalized) pure state $|\varphi_\alpha\rangle$ generated by a non-Hermitian Hamiltonian $H_\alpha$ (Yu et al., 16 Sep 2025, Yu et al., 2022, Wei et al., 25 May 2025): $$\mathcal{F}_\alpha = 4\left(\langle \hat{h}^\dagger_\alpha \hat{h}_\alpha \rangle_\alpha - \langle \hat{h}^\dagger_\alpha \rangle_\alpha \langle \hat{h}_\alpha \rangle_\alpha\right)$$ where $$\hat{h}_\alpha = i (\partial_\alpha U_\alpha) U_\alpha^{-1}$$ and $U_\alpha = \exp(-i H_\alpha t)$. When $H_\alpha$ is Hermitian and the evolution is unitary, this reduces to the standard result $\mathcal{F}_\alpha = 4 t^2{\rm Var}(G)$ for $H=sG$ (multiplicative parameter encoding).

In scenarios where loss/gain is present, this formalism remains consistent provided the output state is appropriately normalized. For more complex, non-multiplicative encoding, this QFI formula retains full generality and governs the attainable precision even with significant non-Hermitian effects.

2. Heisenberg Limit: Scaling of Precision and Its Attainability

A key finding is that non-Hermitian dynamics allow the realization of the Heisenberg scaling in parameter estimation: the mean squared error in the estimator scales as $1/t^2$, so the standard deviation $\sigma(\hat{\theta}) \sim 1/t$ in the optimal case. When the base Hamiltonian is of multiplicative form ($H = s G$), the QFI is manifestly

$$\mathcal{F}_s = 4t^2 \left(\langle G^\dagger G \rangle_s - \langle G^\dagger \rangle_s \langle G \rangle_s\right)$$

demonstrating explicit quadratic scaling with evolution time. For more general non-Hermitian Hamiltonians (including those with no particular symmetry such as parity-time ($\mathcal{PT}$) symmetry), the QFI formula yields the same $t^2$ scaling (modulo corrections from decoherence, loss, or normalization factors), validating that the ultimate Heisenberg scaling can be achieved in principle (Yu et al., 16 Sep 2025, Yu et al., 2022).

The realization of this scaling—previously accessible only in idealized Hermitian models—is now experimentally validated for both $\mathcal{PT}$-symmetric and “generic” non-Hermitian Hamiltonians.

3. Criteria for Optimal Measurement in Non-Hermitian Quantum Metrology

To actually achieve the ultimate precision theoretically allowed by the QFI, the measurement observable must be chosen such that the error-propagation formula for an unbiased estimator attains the quantum Cramér–Rao bound: $$(\Delta \theta)^2 = \frac{(\Delta A)^2}{n |\partial_\theta \langle A \rangle|^2} = \frac{1}{n \mathcal{F}_\theta}$$ where $A$ is a Hermitian measurement operator and $n$ the number of trials. In the non-Hermitian context, the derived optimality condition is (Yu et al., 16 Sep 2025, Yu et al., 2022): $|f\rangle = i c |g\rangle$ where $$|f\rangle = (\hat{h} - \langle \hat{h} \rangle_\alpha)|\varphi_\alpha\rangle$$, $$|g\rangle = (A - \langle A\rangle_\alpha)|\varphi_\alpha\rangle$$, and $c \in \mathbb{R}$. This means that the fluctuation of the generator (of parameter encoding) and of the measurement must be aligned (up to a phase factor), saturating the induced generalized uncertainty relation: $$(\Delta A)^2 (\Delta h)^2 \geq |\langle \hat{h}^\dagger A\rangle_\alpha - \langle \hat{h}^\dagger \rangle_\alpha \langle A \rangle_\alpha |^2$$ Measurements constructed according to this criterion for the specific evolution and initial state ensure that the QCRB is saturated even in the presence of nonunitarity.

4. Experimental Demonstration and Practical Protocols

The theoretical advances have been experimentally validated using single-photon interferometry employing post-selection to simulate nonunitary evolution (Yu et al., 16 Sep 2025). The probe qubit is encoded in photon polarization, with ancillary degrees of freedom represented by spatial modes. Two distinct non-Hermitian evolutions were realized:

• (i) a $\mathcal{PT}$-symmetric Hamiltonian with entirely real spectrum in the unbroken regime;
• (ii) a non-symmetric (neither $\mathcal{PT}$ nor anti-$\mathcal{PT}$) Hamiltonian.

The nonunitary channels are generated by embedding in an enlarged Hilbert space (unitary dilation) followed by post-selection, ensuring correct normalization of the output state. Projective polarization measurements corresponding to optimal observables, as prescribed above (e.g., $A = |0\rangle\langle 0|$ for horizontal polarization), were implemented.

Experimental data confirmed that the scaling of precision with time $t$ tracked the theoretical $1/t$ prediction, with the standard deviation of the estimator decreasing accordingly and the QFI showing the expected $t^2$ dependence. These results constitute the first full experimental demonstration that Heisenberg scaling is achievable in general non-Hermitian quantum metrological protocols.

5. Extension to General Non-Hermitian and Open-System Metrology

The universal QFI expression applies broadly, including cases where parameter encoding is nonlinear in the generator or where the Hamiltonian is explicitly time-dependent or involves gain/loss channels. The only necessary adaptation is in the calculation of the normalized output state $|\varphi_\alpha\rangle$ and the appropriate generator $\hat{h}_\alpha$.

Moreover, the general approach connects naturally to open-system and decoherence scenarios. Effective non-Hermitian evolutions can arise via post-selection in monitored dissipative channels, continuous measurement, or reservoir engineering, making this framework applicable to a wide range of realistic quantum sensors (Alipour et al., 2014, Zeng et al., 8 May 2025).

6. Broader Context and Implications

The demonstration that Heisenberg scaling can be achieved in non-Hermitian quantum metrology carries several implications:

• Enhanced sensing precision is attainable under nonunitary evolution, provided optimal measurement strategies are employed.
• The formalism unifies Hermitian and non-Hermitian estimation theory, and is usable irrespective of system symmetries such as $\mathcal{PT}$ or anti-$\mathcal{PT}$ symmetry.
• Practical metrological platforms, including photonics, trapped ions, and materials with engineered gain/loss, can achieve quantum-limited performance using these prescriptions.

The analysis also opens avenues for designing sensors that exploit features such as exceptional points, non-Markovian dynamics, or criticality to further enhance sensitivity, provided the interplay between information flow, normalization, and measurement is well controlled.

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Table: Core Components in Non-Hermitian Quantum Metrology (as per (Yu et al., 16 Sep 2025, Yu et al., 2022))

Quantity	General Expression	Role
Evolution operator	$U_\alpha = \exp(-i H_\alpha t)$	Parameter encoding (nonunitary)
Generator	$$\hat{h}_\alpha = i (\partial_\alpha U_\alpha) U_\alpha^{-1}$$	Sensitivity generator
QFI	$$4[\langle \hat{h}_\alpha^\dagger \hat{h}_\alpha \rangle_\alpha - \langle \hat{h}_\alpha^\dagger \rangle_\alpha \langle \hat{h}_\alpha \rangle_\alpha]$$	Ultimate precision limit
Optimal measurement	$|f\rangle = i c |g\rangle$	Saturates Quantum Cramér–Rao Bound

This precise, formalized approach provides a rigorous basis for the analysis, optimization, and realization of quantum parameter estimation under general non-Hermitian evolutions, laying the foundation for both theoretical investigations and practical quantum-enhanced measurement technology.