In single-shot Bayesian multiparameter quantum estimation, measurement incompatibility can increase the minimum joint quadratic loss by at most a factor of two relative to the bound obtained by optimising each parameter separately. Using the generalised Barnum--Knill inequality, we establish the same bound for a wider class of additive losses generated by positive score functions, showing that this limitation extends beyond the standard quadratic setting. To illustrate this result, we consider a broad family of periodic losses, including the sine loss sometimes employed in interferometry, and derive the corresponding incompatibility bound for Bayesian multiphase estimation with arbitrary prior knowledge. We further extend the framework to unbounded losses, such as absolute losses; this also recovers the case of quadratic losses.
How much detected thermal light is needed to identify several emitting sites? This working paper studies binary sources on a known grid of addressable source-image modes, using one fixed passive optical mask, photon counting, and a predetermined observation length. It distinguishes the original one-reference apparatus, complete Fourier reference collection, and a separate apparatus with one added vacuum input. All recorded photons count toward the measurement cost. The central result establishes the optimal quadratic detected-photon order at unit brightness for every fixed support limit on every grid, without geometric restrictions or primality assumptions. A reciprocal transmission mask attains this order using target counts alone, while recorded reference photons remain included in the cost. The proof reconstructs short cyclic arcs from two rational-probe values, then uses contour integration to isolate entire clusters of poles. This gives uniform quantitative separation without requiring a fixed minimum distance between individual poles. A bounded candidate list removes the grid-size logarithm from the final comparisons. On prime grids, a shorter finite-field argument recovers the support from two integer edge profiles. The quadratic law also holds uniformly over every nonempty reference subset when known brightness remains in a fixed interval bounded away from zero. The contour argument extends to a general stability theorem for rational interaction observables: under explicit locality and conditioning assumptions, quantitative distinguishability within clusters yields quantitative separation of the global polynomial data. For irregular registered dictionaries, this gives a separation margin proportional to the minimum site spacing, with an optimal spacing exponent. This extension assumes exactly resolved distance channels; their optical acquisition and robustness to unknown position errors require further analysis. Explicit dependence on the support limit also yields a growing-support regime: the photon cost remains
n2+o(1)log(1/δ) whenever
s2log(2s)=o(logn). The accompanying computational bound is stated in real arithmetic, and the dependence on sparsity is deliberately conservative. The manuscript retains explicit ramp-mask constructions for every configuration of at most four emitters, for short-span sources, and for sources with distinct cyclic pair distances. The last class has no span restriction and contains an asymptotically overwhelming fraction of fixed-size supports. For four emitters, ordinary-correlation rigidity and cyclic separation together distinguish reflected configurations. An explicit passive mask also attains the exact optimal Fourier-intensity separation under a prescribed transmission floor. In both the original and complete-reference experiments, sufficiently faint light produces a different law: at fixed sparsity, the optimal photon order grows as the grid size raised to twice the support limit. For at most two emitters, the paper determines the optimal detected-photon order throughout the known-brightness range, from arbitrarily faint light through unit brightness, on odd grids in the original apparatus and on every grid with complete reference collection. The result includes the dependence on grid size, brightness, and error probability, including an additional logarithmic information cost between the faint and bright limits. At fixed confidence, the quadratic regime begins at brightness of order
(logn)/n2. A mask combining an index ramp with a quadratic-residue code attains the sharp order. On even physical grids, one known vacuum input and odd-dimensional Fourier optics give the same law; omitting one target output preserves the detector count. This is a distinct apparatus. For the specified two-scale transmissions on an infinite even-grid family, an exact lower bound using the complete count record proves that changing phases or decoders alone cannot remove their intermediate-brightness limitation. A collected-light model makes the physical scope explicit. For a known collection operator, binary-source identifiability is equivalent to injectivity of the induced thermal covariance map. Covariance collisions give identical quantum states and therefore prevent discrimination by any subsequent receiver. A passive multiport establishes population identifiability when the map is injective. The paper also derives the exact throughput for uniform passive mode separation from the smallest Gram eigenvalue and accounts separately for detected and incident photons. These results connect the assumed mode access to a specified collection model; they do not establish a general quantum-optimal receiver or a complete astronomical implementation. Reference excess variance supplies a normalization that identifies the spatial source despite an unknown finite common-profile thermal-mode spectrum and independent unknown Poisson backgrounds. The finite-sampling guarantee includes the empty source under explicit mean and nonzero-signal bounds. Population identifiability of the mode spectrum is distinguished from stable estimation. When background means are known, the arm means recover source mass and improve a guaranteed scale-free precision radius. Further results include exact thermal count variances, confidence exponents, calibration bounds, a photon penalty for discarding detector labels, and an exact target-only localization obstruction when singleton brightness is unrestricted and unknown. The results have analytical proofs, supported by 34 reproducible finite and symbolic verification suites. These checks provide corroboration, not experimental validation or external peer review. The unrestricted fixed-sparsity unit-brightness problem is resolved on all grids. Remaining questions include the optimized even-grid intermediate-brightness transition in the original apparatus, transitions for three or more emitters, sharp growing-support rates, sharp reference-coverage dependence at vanishing brightness, and optimal covariance-only scale-free precision. Continuous off-grid recovery requires a tolerance-based criterion and additional stability analysis. The work makes no literature-priority claim.
Recovery from a coherent rotation with an unobserved angle depends on both the encoding and what calibration reveals about the noise. We determine their combined limit for a known finite-dimensional generator and one final recovery. Two polynomial projections organize the answer. A physical Krylov chain separates correctable error amplitudes from their first state-dependent distortion; projection onto lower-angle powers identifies the part that a coherent decoder can cancel. Under explicit rank and moment conditions, their combination gives the exact leading optimal entanglement infidelity at weak noise and a recovery attaining it for any finite logical dimension. The optimal worst-input error lies between one and
d times that infidelity at leading order, for logical dimension
d. When the angle distribution is unknown, we optimize one decoder for every distribution compatible with the calibration. When spectral and calibration frequencies are integer multiples of a common frequency, moment positivity turns this into an exact finite optimization. Small measured means can still allow rare finite rotations. If the sole calibration datum is
βr=E[sin2r(θ/2)], with
θ the wrapped rotation phase at that common frequency, the optimal worst-prior infidelity is
O(βr) exactly when generator powers through degree
r−1 are correctable. When the first uncorrectable power has degree
m<r, the optimal loss scales as
βrm/r, with a sharp coefficient fixed by a universal moment problem. In a loss-free four-photon memory, population measurements at two calibration durations retain the fully informed quartic error coefficient for a fixed symmetric narrow-noise family with positive variance and finite fourth moment; a single duration permits loss quadratic in the noise width.
Results concerning the construction of quantum Bayesian error regions as a means to certify the quality of parameter point estimators have been reported in recent years. This task remains numerically formidable in practice for large dimensions and so far, no analytical expressions of the region size and credibility (probability of any given true parameter residing in the region) are known, which form the two principal region properties to be reported alongside a point estimator obtained from collected data. We first establish analytical formulas for the size and credibility that are valid for a uniform prior distribution over parameters, sufficiently large data samples and general constrained convex parameter estimation settings. These formulas provide a means to an efficient asymptotic error certification for parameters of arbitrary dimensions. Next, we demonstrate the accuracies of these analytical formulas as compared to numerically computed region quantities with simulated examples in qubit and qutrit quantum-state tomography where computations of the latter are feasible.
What must a quantum error-correction decoder know about temporally correlated noise, and what does it cost to learn? We study this question for fixed stabilizer memory experiments with known system-environment interactions. We formulate calibration in terms of the fidelity differences that select a Pauli recovery and relate missing information to the resulting fidelity loss. Our main example uses weak coherent
X rotations in a class of CSS codes including Shor, Steane, and odd-distance rotated surface codes. All physical qubits share a sign either fixed throughout a run or independently redrawn each interval. The two models have identical syndrome statistics and the same spatiotemporal Pauli process (SPP), obtained by Pauli-twirling every noise interval. Yet at suitable storage times, they require different logical corrections. In the weak-noise limit, each model is almost perfectly correctable when known. Without model information, even the best final recovery has a worst-case fidelity loss approaching
1/2 relative to the noise-aware optimum, despite access to the full syndrome record. For this pair of models, at fixed code and fixed positive target loss below
1/2, we prove matching calibration bounds. With logical preparation and final readout, complete QEC cycles require a number of noise intervals scaling as the inverse rotation angle raised to the code's
X distance. Allowing two uninterrupted intervals before one extraction reduces the optimal cost to inverse-square scaling. Physical Pauli twirling of every noise interval changes both recovery performance and its information requirements. Discarding the record of applied random Pauli operations makes the SPP sufficient for optimal recovery. Keeping this record can improve the attainable fidelity, but reaching that optimum may require additional noise information.
Abstract The quest for precision in parameter estimation is a fundamental task in different scientific areas. The relevance of this problem thus provided the motivation to develop methods for the application of quantum resources to estimation protocols. Within this context, Bayesian estimation offers a complete framework for optimal quantum metrology techniques, such as adaptive protocols. However, the use of the Bayesian approach requires extensive computational resources, especially in the multiparameter estimations that represent the typical operational scenario for quantum sensors. Hence, the requirement to characterize protocols implementing Bayesian estimations can become a significant challenge. This work focuses on the crucial task of robustly benchmarking the performances of these protocols in both single and multiple-parameter scenarios. By comparing different figures of merits, evidence is provided in favor of using the median of the quadratic error in the estimations in order to mitigate spurious effects due to the numerical discretization of the parameter space, the presence of limited data, and numerical instabilities. These results, providing a robust and reliable characterization of Bayesian protocols, find natural applications to practical problems within the quantum estimation framework.