Reconstruction-Aware Losses in Quantum Calibration

• Reconstruction-Aware Losses are objective functions that jointly estimate transmission losses and photon noise in quantum channels.
• The method employs multi-efficiency measurements with photon-number-resolving detectors and regularized least-squares minimization to reconstruct channel parameters.
• This approach advances quantum communication and metrology by enabling reliable channel calibration, error mitigation, and adaptive protocol design.

Reconstruction-aware losses are objective functions specifically designed to promote the accurate recovery of both noise and loss characteristics within quantum channels. In the context of quantum communication and quantum metrology, such losses are essential to ensure that the calibration of a quantum channel—particularly the determination of transmission losses and the identification of channel noise—is representative of the physical phenomena affecting the transferred quantum states. The framework centers on the joint estimation of transmission losses and photon number statistics of noise through photon-number-resolving detectors (PNRDs). This class of losses, operationalized via regularized least-squares functionals, provides a principled approach for quantum channel analysis and for the calibration of quantum-enabled technologies.

1. Joint Reconstruction Protocol for Losses and Noise

The protocol is built upon the simultaneous estimation of channel transmittance and the photon statistics of background noise using a single measurement series encompassing multiple detector efficiencies. An experimental setup combines a single-photon source (SPS) with an unknown noise source at a beam splitter. The photon number distribution at the output, $p(m)$, accounts for both the signal and the noise source: $p(m) = b(m-1)\varsigma + b(m)(1-\varsigma)$ where $\varsigma$ is the probability of a single photon at the channel input, and $b(m)$ is the probability of $m$ noise photons.

Upon transmission through the quantum channel, a photon-number-resolving detector with adjustable efficiency $\eta$ is employed. The likelihood of measuring $k$ photo-counts is given by the model: $$\Pi(k, \eta, T_{\mathrm{ch}}) = \sum_{m>k} \frac{k!(m-k)!}{m!} (T_{\mathrm{ch}}\eta)^k (1-T_{\mathrm{ch}}\eta)^{m-k} p(m)$$ where $T_{\mathrm{ch}}$ is the transmittance parameter to be estimated.

By performing measurements across a series of detection efficiencies $\{\eta_i\}$ and observing the corresponding empirical photo-count probabilities $P_i(k)$, the protocol reconstructs $T_{\mathrm{ch}}$ and $b(m)$ via minimization: $$\sum_{i, k} [P_i(k) - \Pi(k, \eta_i, T_{\mathrm{ch}})]^2$$ This minimization includes a regularizing "smoothness" constraint to ensure the physicality of the reconstructed distribution, preventing nonphysical oscillations in the estimated noise statistics.

The reconstruction is performed over a truncated Fock space, with cut-off $M$ chosen so that $p(m > M)$ is negligible.

2. Photon-Number-Resolving Detector and Multi-Efficiency Strategy

A superconducting Transition Edge Sensor (TES) serves as the photon-number-resolving detector, allowing discrimination of multiple photon number states with high efficiency (experimentally around 67%). The quantum efficiency $\eta$ is varied using a calibrated variable attenuator, enabling the acquisition of photocount histograms $\{P_i(k)\}$ at different effective detection efficiencies.

This strategy is critical because varying the efficiency reveals the interplay of loss and noise in the measured statistics: each value of $\eta$ provides a different mixture of the signal and noise due to the probabilistic reduction in detected photons. Collectively, these measurements form an over-constrained system that supports the reliable joint estimation of both the transmission loss and the noise statistics.

The recursive least-squares minimization, with an appropriate smoothness constraint, is carried out for each efficiency, using the recorded detector response. The protocol thus leverages the entire multi-efficiency dataset for global reconstruction.

3. Transmission Losses and Noise Estimation: Algorithmic Considerations

The protocol estimates the transmission efficiency and noise model by fitting the theoretical distribution $\Pi(k, \eta, T_{\mathrm{ch}})$ against the observed multi-efficiency photocount distributions. Calibration of the actual detector efficiency is conducted independently using a heralded single-photon source and InGaAs single-photon avalanche diodes (SPADs). The system requires that all involved efficiencies are independently measured, including the total TES detection efficiency and the insertion losses from the optical system (beam splitter, coupling, etc).

Key conditions for estimation accuracy include:

• Reliable calibration of detector efficiencies;
• Minimization of higher-order photon contributions from the SPS (reported $\alpha = 0.005 \pm 0.007$);
• Proper control and measurement of all optical losses.

Experimental demonstrations yielded reconstructed channel transmittance values ($T_{\mathrm{ch}}=0.84 \pm 0.04$) in strong agreement with independently measured results ($T_{\mathrm{ch}}=0.85 \pm 0.02$), underscoring the protocol’s accuracy.

4. Noise Statistics Reconstruction and Metrics

The joint reconstruction also yields the full photon number statistics of the channel noise, $b(m)$, for noise modeled as either Poissonian (laser source) or pseudo-thermal (ground glass disk). The mean photon number per pulse is varied (typically $\langle n \rangle$ from $\sim 0.45$ to $\sim 2$), and the fidelity between the reconstructed and theoretically expected distributions is evaluated as: $F = \sum_{m=0}^M \sqrt{b^{(r)}(m) b^{(e)}(m)}$ Experimental fidelities range from $90.4\%$ to $99.1\%$, confirming the precision of statistical noise characterization even in the presence of varying noise types and levels.

Limitations emerge in the regime of low overall efficiencies or when the Fock space dimension must be expanded (high mean photon numbers), where the dominance of zero-count events and lower detector sensitivity diminish statistical power.

5. Applications to Quantum Communication and Tomography

The protocol supports more realistic quantum communication and quantum metrology systems by enabling accurate, joint calibration of channel loss and stochastic noise—both major sources of decoherence and error. In quantum key distribution and quantum state/process tomography, reconstruction-aware losses derived from this protocol permit improved resource allocation and error-correction strategy design.

The approach facilitates:

• Direct, experiment-driven identification of decoherence/reconstruction losses for plugging into tomography or calibration workflows;
• Quantitative determination of operational efficiency thresholds below which reliable reconstruction fails;
• Use as a self-consistent protocol given its cross-validation with independent loss measurements.

In addition, it supports adaptive quantum protocol design by allowing the real-time adjustment of error-mitigation schemes or resource allocation on the basis of up-to-date, data-driven noise and loss reconstructions.

6. Impact and Future Directions

Reconstruction-aware losses, implemented through joint minimization over optical loss and noise statistics and constrained for physicality, represent a robust methodology for the calibration and operation of quantum channels. This experimental approach quantifies and mitigates a common bottleneck in quantum information processing—the indistinguishability of loss and noise effects.

By formalizing the estimation of both error sources, the protocol not only advances self-consistency and cross-validation within device calibration routines but also enables the construction of data-driven, loss-aware objective functions for quantum state reconstruction and quantum-enhanced sensing. The approach serves as a foundational tool in the development of error-correcting and adaptive protocols across the quantum technology landscape.