Precision Bounds for Characterising Quantum Measurements

TL;DR

This work establishes a comprehensive framework for characterising quantum detectors by introducing the detector quantum Fisher information (DQFI) and deriving fundamental, probe-optimal precision bounds for both single- and multi-parameter detector estimation. It provides two concrete DQFI constructions—the spectral bound ${\\mathcal{J}_{\\|,\\theta}}$ and the trace bound ${\\mathcal{J}_{\\mathrm{Tr},\\theta}}$—and proves attainability criteria, including a tight bound via detector extension bounds that in many cases are achievable with separable probes. The authors validate the framework with a platform demonstration of dephasing-noise qubit detectors on IBM hardware and extend the method to photodetector tomography and multi-parameter detector tomography, revealing the nuanced role of probe compatibility and ancilla resources. Overall, the work completes the triad of optimal state, detector, and process tomography, offering practical benchmarks and guiding principles for precise, efficiently calibrated quantum measurements with broad impact on quantum sensing and quantum information processing.

Abstract

Quantum measurements, alongside quantum states and processes, form a cornerstone of quantum information processing. However, unlike states and processes, their efficient characterisation remains relatively unexplored. We resolve this asymmetry by introducing a comprehensive framework for efficient detector estimation that reveals the fundamental limits to extractable parameter information and errors arising in detector analysis - the \emph{detector quantum Fisher information}. Our development eliminates the need to optimise for the best probe state, while highlighting aspects of detector analysis that fundamentally differ from quantum state estimation. Through proofs, examples and experimental validation, we demonstrate the relevance and robustness of our proposal for current quantum detector technologies. By formalising a dual perspective to state estimation, our framework completes and connects the triad of efficient state, process, and detector tomography, advancing quantum information theory with broader implications for emerging technologies reliant on precisely calibrated measurements.

Figures (15)

• Figure 1: Triad of quantum state, detector and process tomography.a, Characterising an unknown quantum state invokes the SQFI, which specifies the optimal measurements. b, Characterising an unknown measurement invokes the DQFI, which specifies the optimal probe states. c, Characterising an unknown process invokes the process QFI, or maximum of output-state SQFI over all process-input states, simultaneously requiring the optimal probe states and measurements.
• Figure 2: Estimating qubit measurements subject to Pauli errors.a-c, Noisy $Z$-measurement with bit-flip error, noisy ${(X+Z)}/{\sqrt{2}}$-measurement with phase-flip error, and noisy ${(X+Z)}/{\sqrt{2}}$-measurement with bit-phase-flip error, respectively. The Fisher information for probe states notated as $\ket{\theta,\phi}\coloneqq \cos(\theta/2) \ket{0} + \exp{i\phi} \sin(\theta/2) \ket{1}$ is depicted as circular markers (orange, purple and blue) on dashed lines for non-optimal probes, and as star-shaped markers (golden) for the optimal probes. The DQFI ${\mathcal{J}_{\vert \vert, \theta}}$ (dashed black curve) is tight, as it equals the maximum CFI ${\mathcal{F}_\theta}_\mathrm{max}$. d, Two-parameter estimation of a noisy ${(X+Z)}/{\sqrt{2}}$-measurement subject to independent bit-flip and phase-flip errors. Here, the two detector QCRBs, i.e., ${\mathcal{J}_{\mathrm{Tr},\theta}}$ (blue) and ${\mathcal{J}_{\vert \vert, \theta}}$ (green) both overestimate the tight bound (golden) for total MSE.
• Figure 3: DQFI applied to dephased projection measurements of qubits.a, The optimal probe state for estimating dephasing strength $p$ lies in a vertical section (orange plane) of the Bloch sphere containing the noiseless measurement $\Pi_\mathrm{ideal}$ corresponding to $p=0$. b, The intersection that contains the measurement projectors $\ket{\theta,\phi}_{\pm}$ (arrowheads) depicts the dynamics of probes under dephasing: the dephasing action $\mathcal{N}_\mathrm{dep}$ contracts the blue disk horizontally (towards the vertical dashed line) by a factor of $(1-2p)$ to form the grey elliptic region. The optimal probe states $\rho_\mathrm{opt}^{\pm}$ (black dots) are those for which the dephased states $\mathcal{N}_\mathrm{dep}(\rho_\mathrm{opt}^{\pm})$ (red dots) align with the measurement direction $\ket{\theta, \phi}_\pm$. c, Circuit for estimating the noisy measurement $\Pi_p$ comprised of environment-assisted dephasing $\mathcal{N}_{\mathrm{dep}}$ followed by the projection $\Pi_\mathrm{ideal}$. The varying probe states, $\rho_\mathrm{in}$, are controlled by polar angles $\theta_\mathrm{in}$. (For details of the circuit implementation see Methods.) d, The experimental MSE (blue dots), superimposed by the noiseless simulation results (green dots), agrees well with the theoretical black curve. The MSEs are plotted against the DQFI bound (dashed blue line). The minimum-MSE probe angle from simulation ($90\%$ confidence interval shown in green shading) and from experiment (in blue shading) match the theory, shown as dashed black line. The inset shows a zoomed-in range of probe angles near this optimal point. The simulation error bars are statistical arising due to finite samples whereas the experimental error bars also include the effect of platform noise (see Supp. Mat. \ref{['app:IBMExp']}).
• Figure 4: Optimal process estimation through the dual approaches of detector and state estimation. a,b, Current approaches to achieve optimal process estimation that attains the process QFI (see Supp. Mat. \ref{['supp:QPT']} for definition). The approach involves maximising the SQFI of the process output state over input probes $\rho_\mathrm{in}$. d,e, Alternatively, we show that optimality can be achieved by maximising the DQFI ${\mathcal{J}_{\vert \vert, \theta}}$ of the effective measurement $\Pi_{\mathrm{eff}, \theta}$ over final measurements $\Pi$. c, The two approaches treat the optimisation over probes and measurements in reverse order and are equivalent. b,e,f-h, An example demonstrating the equivalence for estimating qubit $Z$-rotations. In (b), we consider generic mixed qubit states $\rho_\mathrm{in}$ with $\langle X\rangle, \langle Y\rangle$ & $\langle Z\rangle$ Pauli expectation, whereas in (e), we consider projection measurements $\Pi$ with arbitrary polar angle and zero azimuthal angle (from rotational symmetry). The state approach (b) says, for any given probe state (f), the optimal measurement is in the equatorial plane (EP) and is unbiased in direction to the state. The detector approach (e) says, for any given measurement (g), the optimal probe lies in the EP in a direction unbiased to the measurement. The optimal process estimation strategy (h) requires both to hold true, fixing both probe and measurement in the EP, while still unbiased to each other (star markers in (b), (e)).
• Figure 5: Comparison of detector quantum Fisher information (DQFI) measures for single-parameter estimation from randomly-generated qubit measurement models. (a) Across 10,000 random models, the extended DQFI SDP (red) provides a tighter bound for single-parameter estimation than the trace DQFI (light orange) and the spectral DQFI (black). (b) A zoomed-in version of (a) shows that the trace DQFI can be far from the extended DQFI but the spectral DQFI is typically close. In (a) and (b), the scatter points are sorted in the increasing order of the extended DQFI.

Theorems & Definitions (41)

• Definition 1: Spectral DQFI
• Definition 2: Trace DQFI
• Theorem 1: DQFI upper-bounds maximum CFI
• Theorem 1: DQFI upper-bounds maximum CFI
• proof
• Example 1
• Theorem 2: Attainability for diagonal measurements
• proof
• Example 2
• Example 3