End-Point Measurement (EPM) Scheme

• End-Point Measurement (EPM) Scheme is a collection of methods that quantify and validate process outcomes using statistical, physical, and quantum measurements.
• It employs techniques such as hypothesis testing, reference benchmarking, and simulation-based validation to detect bias, precision loss, and drift.
• EPM methods are vital across disciplines—from battery diagnostics to network performance—supporting robust experimental design and error mitigation.

End-Point Measurement (EPM) Scheme describes a class of methodologies used to extract, diagnose, or validate process outcomes by quantifying system states—often statistical or physical quantities—at the end of the process. EPM schemes are fundamental to experimental design, statistical process control, quantum diagnostics, network performance measurement, nuclear and chemical analysis, and battery degradation studies. They connect observable measurement endpoints (e.g., terminal state energies, voltages, test sample means) to underlying process fidelity, error sources, or physical mechanisms. Modern EPM approaches leverage statistical hypothesis testing, reference-based benchmarking, error projections, and quantum coherence-sensitive protocols to ensure robust and sensitive assessment of experimental quality, process drift, or device stability.

1. Statistical Frameworks in EPM

A rigorous EPM scheme for evaluating the quality of measurement processes is established in the statistical testing framework (Nyssen et al., 2013). This approach critiques conventional methods that compare the observed mean to an error interval constructed from the test sample alone, showing that larger test sample standard deviation (sₘ) can paradoxically increase acceptance probability even as process quality degrades.

Instead, the framework employs a reference ("gold standard") set of measurements performed under controlled conditions by skilled operators. The core statistical tool is a Student t-test comparing the test mean (m_T) to the gold standard mean (m_R) with t-statistic:

$t = \frac{m_T - m_R}{s_R \sqrt{1/N + 1/n}}$

where $s_R$ is the standard deviation from $N$ reference measurements, and $n$ is the number of test samples. The acceptance interval is defined by:

$$m_R \pm t_{\alpha, N-1} \cdot \sqrt{1/N + 1/n} \cdot s_R$$

with $t_{\alpha, N-1}$ the critical value for significance level $\alpha$ and $N-1$ degrees of freedom.

Numerical simulation confirms that this approach tightly controls Type-I error at $\alpha$ and is highly sensitive to both bias (mean shift) and precision loss (variance increase) in test measurements. In EPM contexts (e.g., chemico-physical endpoint tracking), a gold standard increase specificity in detecting deviations due to instrumental drift or operator error, subject to the representativeness and quality of reference data and normality assumptions on distribution shape.

2. Physical Measurement and Simulation in EPM

In nuclear and radiation physics, EPM schemes are used to determine endpoint energies (e.g., β-decay endpoint) and reaction cross sections. Measurement of β-decay endpoints using segmented planar HPGe detectors combines singles and β–γ coincidence techniques, exploiting timing coincidence with auxiliary detectors to discriminate individual decay branches and improve selectivity (Bhattacharjee et al., 2014). Experimental spectra are validated against detailed GEANT3 Monte Carlo detector simulations, with endpoint energies extracted via Fermi-Kurie linear fits:

$E_p = \frac{b}{a} + d$

where $b$ and $a$ are parameters from the Fermi-Kurie plot fit, and $d$ corrects for entrance window degradation, and also via χ² minimization between simulated and experimental spectra:

$$\chi^2 = \frac{1}{N-1} \sum_i \frac{(d_i - t_i)^2}{\sigma_i}$$

This dual approach ensures accuracy in extracting Q-values, which critically inform atomic mass, binding energy, and nuclear model development.

For photonuclear reactions, EPM schemes are used to derive average cross sections and isomeric ratios at defined bremsstrahlung endpoint energies (e.g., 30 MeV, 40 MeV) on natural rhenium, using activation and high-purity HPGe gamma spectrometry, with corrections for flux-weighted photon distribution (Avetisyan et al., 2021):

$$\bar{\sigma} = \frac{\int \sigma(E) \phi(E) dE}{\int \phi(E) dE}$$

$ICR = \frac{O_m}{O_{tot}}$

These methods enable differentiation of ground vs. metastable state populations—a significant capability for nuclear model refinement and applications requiring isomeric quantification.

3. EPM Schemes in Quantum Systems

Quantum EPM protocols are distinguished by their treatment of measurement-induced decoherence and their ability to extract resource-theoretic quantities from process endpoints. A leading protocol diagnoses quantum-gate coherence by analyzing the statistics of local energy measurements on output quantum states (Gianani et al., 2022). Rather than full state tomography, the scheme tracks quantum coherence via differences in characteristic functions of energy-change PDFs, decomposing the initial state’s density matrix into its diagonal and off-diagonal (coherent) components:

$$\eta_\chi(\mathcal{E} \circ \mathcal{U}, \mathcal{U}) = \mathbb{E}_{|\psi\rangle} \left| \mathcal{G}_\chi(i ; \mathcal{E} \circ \mathcal{U}) - \mathcal{G}_\chi(i ; \mathcal{U}) \right|$$

Experimental quantum-optics data confirm that coherence loss (e.g., from unitary gate errors) is well captured by end-point energy statistics, enabling rapid diagnostics without entangled two-copy measurements. This approach generalizes in quantum stochastic thermodynamics, where EPM schemes permit fluctuation theorems accurately reflecting quantum coherence in non-equilibrium energy exchange, unlike conventional Two-Point Measurement (TPM) protocols, which erase off-diagonal initial state elements (Artini et al., 5 Aug 2025).

The entropy production in EPM-based heat-exchange fluctuation relations incorporates both classical (energy difference) and coherence-dependent terms:

$$\Sigma_{if} = (\beta_S - \beta) \Delta E_{if} + \Gamma_{if}$$

where $\Gamma_{if}$ quantifies coherence-induced deviations, revealing quantum signatures in prethermal phases and non-classical entropy production.

4. Error Mitigation and Drift-Resilient EPM Protocols

EPM-inspired protocols for quantum error mitigation address mid-circuit measurement errors in dynamic circuits—a significant bottleneck for quantum error correction (QEC) and gate benchmarking (Santos et al., 12 Jun 2025). These methods exploit repeated measurement with parity-based postprocessing or reset-and-feedforward strategies to computationally amplify and then cancel temporal drift in readout error without frequent calibration.

Mitigation order $m$ is controlled via summation over Taylor-series coefficients, yielding assignment matrices that eliminate error up to order $\epsilon^{m+1}$:

$$M_{mit}^{(m)} = \sum_{k=0}^m a_k^{(m)} M^{2k+1} = I + O(\epsilon^{m+1})$$

Experiments on superconducting (IBMQ) and trapped-ion (Quantinuum) devices demonstrate improved readout fidelity and integration with Layered-KIK gate error mitigation, enabling "end-to-end" EPM strategies for QEC and fast gate-set tomography. This drift-resilient paradigm eliminates calibration overhead and is robust against uncharacterized noise.

5. Endpoint Analysis in Battery Diagnostics

In electrochemical systems, EPM schemes track shifts in endpoints (end-of-charge, end-of-discharge) to resolve parasitic side-reactions and degradation mechanisms (Rodrigues, 4 Aug 2025). The "endpoint slippage" metric quantitatively links observed endpoint displacements to oxidation/reduction side reactions, impedance rise, and loss of active material (LAM). Mathematical relationships (e.g.,

$$D_{\text{slip},k} = 2q_{\text{red},k}(1-\alpha) + N_0 L_{\text{AM, liNE},k}(x_{EOC,k-1}-x_{EOD,k-1})$$

$$C_{\text{slip},k} \approx q_{\text{ox},k} + q_{\text{ox},k-1}$$

$$Q_{\text{loss},k} = D_{\text{slip},k} - C_{\text{slip},k}$$

)

account for additive contributions from side reactions and aging modes. Correction of endpoint data—for LAM and impedance rise—requires additional information (e.g., voltage profile slope, Li content of disconnected domains), but enables precise quantification of parasitic capacity and improved interpretation of battery degradation mechanisms.

6. EPM in Network Measurement and Performance Monitoring

In network systems, EPM methodologies are integrated into testbeds, mobile apps, and measurement libraries for assessment of end-to-end throughput, latency, and discriminations such as traffic shaping (Goel et al., 2014). Common endpoint metrics include throughput ($T = S / t$), latency, jitter, and packet loss, with data aggregated and normalized according to measurement conditions and testbed configurations.

Challenges in mobile EPM include sensitivity to device churn, detection of traffic shaping, resource management, and cross-platform standardization. Recent trends favor open-source, cross-platform measurement libraries and standard API development for improved reproducibility and comparability. Incorporating EPM statistics enables robust diagnostics and benchmarking across heterogeneous devices and network conditions.

7. Comparative EPM Strategies and Limitations

EPM schemes, whether statistical, physical, or computational, universally enhance sensitivity to both bias and precision loss in process outcomes by leveraging external benchmarks, multidimensional characterization (e.g., energy, capacity, network statistics), and error-mitigated data aggregation. Nonetheless, their efficacy depends critically on the accuracy, quality, and representativeness of reference datasets; the validity of additive models in decomposing endpoint shifts; and the feasibility of extracting auxiliary parameters (normalization coefficients, electrode properties, assignment matrices, etc.) in practical contexts.

Key limitations include: requirement for well-characterized gold standards in statistical EPM; model accuracy in physical EPM; nonlinearity and measurement bias in quantum EPM; and practical challenges in endpoint correction due to unknowable quantities in battery diagnostics. Future progress will depend on methodological advances that enhance reference dataset fidelity, integrate drift-robust designs, and facilitate the extraction of underlying physical parameters from endpoint observables.

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End-Point Measurement (EPM) schemes are architected to maximize detection sensitivity, diagnostic accuracy, and robustness against common error sources across disciplines. Their evolution is tightly coupled to advances in statistical hypothesis testing, simulation-based validation, quantum protocol development, and network engineering, with ongoing research addressing the challenges in correction, standardization, drift mitigation, and reference quality assessment.