Joint Phase and Phase-Diffusion Estimation

• Joint phase and phase-diffusion estimation is a quantum metrology topic that addresses extracting both a deterministic phase shift and stochastic dephasing effects via the quantum Fisher information framework.
• The analysis outlines measurement trade-offs using separable and collective strategies, highlighting optimal probe state designs and precision limits under varying noise regimes.
• Experimental implementations in photonic and atomic interferometry demonstrate that tailored probes and adaptive measurement techniques can significantly improve joint estimation accuracy.

Joint phase and phase-diffusion estimation addresses the simultaneous extraction of a deterministic phase shift, typically denoted $\varphi$ or $\phi$, and the diffusion strength $\Delta$ associated with stochastic phase kicks or decoherence in quantum systems. This problem is central to quantum metrology and precision interferometry, especially in environments where dephasing is non-negligible and itself informative. The study of its quantum limits, probe and measurement optimization, classical and quantum Fisher information matrices, collective measurement advantage, and practical strategies forms a mature and rapidly evolving subfield.

1. Theoretical Framework and Quantum Fisher Information

The estimation scenario considers quantum probes—frequently two-mode states such as split single photons, coherent, N00N, or Holland–Burnett states—subjected to a deterministic phase shift $\phi$ (unitary transformation) and additional phase diffusion, typically modeled as a Gaussian random variable of standard deviation $\Delta$ applied to the phase. The dynamical map for the probe state thus comprises both a coherent unitary and a dephasing (non-unitary) noise channel.

A paradigmatic case, reducing to a two-dimensional Hilbert space, considers a general qubit state (expressed e.g. as $$|\psi_0\rangle = \cos\frac{\theta}{2} |0\rangle + \sin\frac{\theta}{2} |1\rangle$$) evolving via

$$\rho(\phi,\Delta) = \begin{pmatrix} \cos^2(\theta/2) & \cos(\theta/2)\sin(\theta/2) e^{-\Delta^2 - i\phi} \ \cos(\theta/2)\sin(\theta/2) e^{-\Delta^2 + i\phi} & \sin^2(\theta/2) \end{pmatrix}.$$

The joint estimation of $\phi$ and $\Delta$ is quantified by the quantum Fisher information matrix (QFIM), whose entries are

$$H_{\phi\phi} = \sin^2\theta\, e^{-2\Delta^2}, \quad H_{\Delta\Delta} = \sin^2\theta\, \frac{4\Delta^2}{e^{2\Delta^2}-1}, \quad H_{\phi\Delta} = 0,$$

for the two-parameter set $(\phi, \Delta)$ and for equatorial probes ($\theta = \pi/2$ the QFIM is diagonal) (Vidrighin et al., 2014). This structure, with vanishing off-diagonals, is generic for symmetric probes and large spin ensembles in the continuous-variable or high-dimensional limit, and remains valid for more general states in the leading-order asymptotics (Knysh et al., 2013, Szczykulska et al., 2017, Jayakumar et al., 2024).

2. Fundamental Precision Bounds and Trade-off Relations

The quantum Cramér–Rao bound (QCRB) determines the lower achievable variances for unbiased joint estimators: $$\mathrm{Cov}(\hat{\phi},\hat{\Delta}) \geq \frac{1}{M} H^{-1},$$ with $M$ the number of independent repetitions. For diagonal QFIM, the marginal bounds are simply

$$\mathrm{Var}(\phi) \geq \frac{1}{M H_{\phi\phi}}, \qquad \mathrm{Var}(\Delta) \geq \frac{1}{M H_{\Delta\Delta}}.$$

For single-copy, separable measurement strategies, it has been shown that the normalized sum of attainable Fisher informations is bounded by unity: $$\frac{F_{\phi\phi}}{H_{\phi\phi}} + \frac{F_{\Delta\Delta}}{H_{\Delta\Delta}} \leq 1,$$ enforcing a linear trade-off between phase and phase-diffusion precision (Vidrighin et al., 2014, Altorio et al., 2015). This boundary is tight for measurements in the equatorial Bloch plane or double homodyne detection. In high dimensions, for states such as Holland–Burnett or the non-Gaussian "Cosine" state, higher scaling (Heisenberg or quadratic) of $H_{\phi\phi}$ and $H_{\Delta\Delta}$ is possible, but the trade-off remains: for small $\Delta$, maximal precision for one parameter leads asymptotically to vanishing information about the other (Szczykulska et al., 2017).

In the limit of large spin-$j$ (or particle number $N$), the QFIM entries for optimal probe states approach (Knysh et al., 2013): $$J_{\varphi\varphi} = \frac{1}{\Delta+\pi^2/N^2},\qquad J_{\Delta\Delta} = \frac{1}{2\Delta^2 + 4\pi^2\Delta/N^2}.$$

3. Optimal Probes and Measurement Strategies

Optimal probe states depend on the regime (level of phase diffusion and dimensionality). For low $\Delta$ (weak dephasing), non-Gaussian Cosine-profile states or generalized Holland–Burnett (gHB) states maximize joint information (Jayakumar et al., 2024, Knysh et al., 2013). For fixed-particle number probes in high diffusion, the surviving coherence arises from specific off-diagonals, and symmetry constraints affect the measurement design (Szczykulska et al., 2017).

Measurement optimization is constrained by the incompatibility of the SLDs for $\phi$ and $\Delta$; there is no single projective measurement that extracts both at the quantum limit for single copies. However, specific POVMs—such as equatorial plane measurements

$$\Pi_j = \frac{n_j}{2}\left( \begin{matrix} 1/2 & 1/2 e^{-i\chi_j} \ 1/2 e^{i\chi_j} & 1/2 \end{matrix} \right)$$

with a symmetric distribution of phases $\chi_j$—saturate the separable bound (Vidrighin et al., 2014, Altorio et al., 2015). Double homodyne detection in optical architectures (measurement of orthogonal quadratures after a 50:50 beamsplitter) is both theoretically optimal and experimentally accessible for many probe classes, achieving the equality in the summed trade-off bound (Vidrighin et al., 2014, Jayakumar et al., 2024).

Weak measurements, composed of a weak $\sigma_z$ stage followed by strong $\sigma_x$, tune the balance between $\mathrm{Var}(\phi)$ and $\mathrm{Var}(\Delta)$ and explore the full Pareto boundary of the trade-off (Altorio et al., 2015). The optimal weighting parameter can be determined analytically or numerically as a function of the noise regime.

4. Collective Measurements and Multiparameter Enhancement

The strict trade-off for single-copy, separable measurement strategies is surpassed when collective (entangled) measurements are allowed. For two copies of the probe, deterministic Bell-basis measurements yield

$$\frac{F_{\phi\phi}}{2 H_{\phi\phi}} + \frac{F_{\Delta\Delta}}{2 H_{\Delta\Delta}} \leq 1.5,$$

demonstrating an approximately $50\%$ improvement over the separable bound (Wang et al., 27 Dec 2025, Vidrighin et al., 2014). The classical Fisher information for Bell measurements on equatorial qubit probes attains the two-copy bound for sufficiently small $\Delta$. Experimental results corroborate that Bell-basis projections with optimal probe encoding realize this enhancement at accessible noise levels (Wang et al., 27 Dec 2025).

For higher-dimensional states and collective measurement on $k$ copies, further enhancement is in principle possible, limited by the commutativity structure of the SLDs (weak commutativity condition). In large ensembles with high $N$, the joint optimality of canonical phase measurements for both parameters is restored in the asymptotic limit (Knysh et al., 2013). However, complexity and loss-robustness become practical limitations.

5. Experimental Implementations and Applications

Joint phase and phase-diffusion estimation protocols have been implemented in both photonic and atomic interferometric contexts. In quantum optical platforms, double homodyne detection (equatorial POVMs on polarization-encoded qubits) and deterministic Bell measurements in linear-optical circuits have been realized (Vidrighin et al., 2014, Wang et al., 27 Dec 2025). Detector tomography, calibration with known probe polarization, and maximum-likelihood estimation based on measured experimental outcome distributions enable extraction of both $\phi$ and $\Delta$.

In atomic settings, such as quantum or atom interferometers with differential readout, explicit maximum-likelihood procedures for extracting both a differential phase shift and the width of uncorrelated phase noise have been demonstrated (Pezzè et al., 23 Mar 2025). The analytic form of the likelihood and Fisher information enable assignment of optimal error bars, outperforming traditional curve-fitting or ellipse-fitting techniques.

Practical architectures include Mach–Zehnder and Sagnac interferometers (photonic), and coupled atomic sensors with correlated and independent noise sources. The availability of photon-number-resolving detection and advances in collective measurement implementation are extending feasible probe classes and dimensionality (Jayakumar et al., 2024).

6. Regimes of Precision, Noise, and Scaling

The structure of the attainable estimation precisions varies with the phase-diffusion regime:

• Low $\Delta$ regime: Heisenberg scaling ($\sim 1/N$) in both phase and diffusion estimation is attainable for optimized non-Gaussian probe states (Knysh et al., 2013, Jayakumar et al., 2024). Trade-off boundaries approach minimal joint estimation performance (sum bound $\to 1$), reflecting measurement incompatibility (Szczykulska et al., 2017).
• Intermediate $\Delta$: There exists a sweet-spot in phase diffusion which maximizes the attainable joint information; for Holland–Burnett states and multi-mode, multi-particle probes, trade-off curves approach their theoretical maxima (Szczykulska et al., 2017).
• High $\Delta$ regime: Quantum enhancement is lost; QFIM entries decay exponentially with $\Delta$, and optimal strategies revert to robust, shot-noise-limited schemes (Knysh et al., 2013, Szczykulska et al., 2017).
• Critical threshold: Above $\Delta_c \approx 0.2512$, entanglement confers no net advantage in phase or diffusion estimation compared to optimal separable strategies (Knysh et al., 2013).

Probe resilience to losses has been investigated, with "all-photons-in-one-port" generalized Holland–Burnett states retaining quantum scaling for moderate transmission ($\eta \simeq 0.5$) and outperforming N00N and balanced HB states (Jayakumar et al., 2024).

7. Outlook and Extensions

Open directions include scaling collective measurement protocols to more than two copies, generalizing the framework to include additional noise types (amplitude damping, losses), and exploring adaptive and error-corrected protocols for robust operation at large $\Delta$ (Wang et al., 27 Dec 2025). Additionally, the integration of optimal estimation routines into standard data analysis workflows (maximum-likelihood over differential clouds, numerical Fisher calculation) stands to improve performance for existing interferometric and gradiometric architectures (Pezzè et al., 23 Mar 2025).

These results collectively define the quantum-metrological landscape for joint estimation of a unitary parameter and non-unitary phase noise, with rigorous bounds on achievable precision, optimal probe design, feasibility of collective measurements, and realistic strategies for precision-limited applications in quantum technologies.