Optimizing qubit phase estimation

The theory of quantum state estimation is exploited here to investigate the most efficient strategies for this task, especially targeting a complete picture identifying optimal conditions in terms of Fisher information, quantum measurement, and associated estimator. The approach is specified to estimation of the phase of a qubit in a rotation around an arbitrary given axis, equivalent to estimating the phase of an arbitrary single-qubit quantum gate, both in noise-free and then in noisy conditions. In noise-free conditions, we establish the possibility of defining an optimal quantum probe, optimal quantum measurement, and optimal estimator together capable of achieving the ultimate best performance uniformly for any unknown phase. With arbitrary quantum noise, we show that in general the optimal solutions are phase dependent and require adaptive techniques for practical implementation. However, for the important case of the depolarizing noise, we again establish the possibility of a quantum probe, quantum measurement, and estimator uniformly optimal for any unknown phase. In this way, for qubit phase estimation, without and then with quantum noise, we characterize the phase-independent optimal solutions when they generally exist, and also identify the complementary conditions where the optimal solutions are phase dependent and only adaptively implementable

Entanglement-assisted quantum parameter estimation from a noisy qubit pair: A Fisher information analysis

Benefit from entanglement in quantum parameter estimation in the presence of noise or decoherence is investigated, with the quantum Fisher information to asses the performance. When an input probe experiences any (noisy) transformation introducing the parameter dependence, the performance is always maximized by a pure probe. As a generic estimation task, for estimating the phase of a unitary transformation on a qubit affected by depolarizing noise, the optimal separable probe and its performance are characterized as a function of the level of noise. By entangling qubits in pairs, enhancements of performance over that of the optimal separable probe are quantified, in various settings of the entangled pair. In particular, in the presence of the noise, enhancement over the performance of the one-qubit optimal probe can always be obtained with a second entangled qubit although never interacting with the process to be estimated. Also, enhancement over the performance of the two-qubit optimal separable probe can always be achieved by a two-qubit entangled probe, either partially or maximally entangled depending on the level of the depolarizing noise

Quantum state discrimination and enhancement by noise

Quantum state discrimination and enhancement by noise

La physique quantique pour le traitement de l'information et du signal

La physique quantique pour le traitement de l\u27information et du signal

Information quantique et calcul quantique – une introduction

Information quantique et calcul quantique – une introduction

Optimization of quantum states for signaling across an arbitrary qubit noise channel with minimum-error detection

For discrimination between two signaling states of a qubit, the optimal detector minimizing the probability of error is applied to the situation where detection has to be performed from a noisy qubit affected by an arbitrary quantum noise separately characterized. With no noise, any pair of orthogonal pure quantum states is optimal for signaling as it enables error-free detection. In the presence of noise, detection errors are in general inevitable, and the pairs of signaling states best resistant to such noise are investigated. With an arbitrary quantum noise, modeled as a channel affecting the qubit, and when minimum-error detection is performed from the output, a characterization of the optimal input signaling pairs and of their best detection performance is obtained through a simple maximization of a quadratic scalar criterion in three constrained real variables. This general characterization enables to establish that such optimal signaling pairs are always made of two orthogonal pure quantum states, but that they must be specifically selected to match the noise properties and prior probabilities. The maximization is explicitly solved for several generic quantum noise processes relevant to the qubit, such as the squeezed generalized amplitude damping noise which describes interaction with a thermal bath representing a decohering environment and which includes as special cases both the generalized and the regular amplitude damping noise processes, and such as general Pauli noise processes which include for instance the bit-flip noise and the depolarizing noise. Also, examined is the situation of one imposed (pure or mixed) signaling state, for which the other associated signaling state optimal for noisy detection is determined as a pure state, yet not necessarily orthogonal

Analyse 3D de microstructures de graines en microtomographie par rayons X

Analyse 3D de microstructures de graines en microtomographie par rayons X

Cramér-Rao bounds and condition number in SPECT: Comparison between conventional thin holes collimator and emission tomography project with large and long holes collimators

Objectives: The project of emission tomography with large and long holes collimators (CACAO-TROLL), was proposed some time ago. The use of collimators with larger holes is intended to increase the number of photons detected and therefore the information available to reconstruct the images. This project is however exploratory and most research works in SPECT stick today to the conventional thin hole collimator (CTHC). It may be objected that if the number of photons increases, the information conveyed by each photon is lower. This thought is however inconsistent with our previously published demonstration using information theory. We develop here another approach. Methods: We first derived a formula to express the response function of the CACAO-TROLL acquisition, taking a complete account of the depth dependence and the attenuation of the gamma ray in the collimator. The conventional SPECT response function was modelled by using the formula of Youngho Seo (JNM 2005 vol 46 n 5 pp 868) standing for a VPC-45 LEHR collimator. For both projects, various parameters were tested in a 2D reduction of the problem in the transverse plane. Results: The results show a slight shift between the behaviour of the condition numbers and the Cramér-Rao bounds. For small image size (less than 30x30) the CACAO-TROLL project exhibits a lower condition number than CTHC, and higher Cramér-Rao bounds. For larger sizes, both factors increase steeply for CTHC. Finally, for a proper choice of the holes geometry, the Cramér-Rao bound is more than an order of magnitude better for the CACAO-TROLL project than for CTHC. Conclusions: This calculation confirms, at least in theory, that increasing the number of collected photons and the accuracy of the collimation can lead to better estimates in emission tomography. A good algorithm to fully benefit from this improved acquisition may remain a challenging point. It is to be expected that this calculation may stimulate such research in a near future

The minimum description length principle for probability density estimation by regular histograms

The minimum description length principle is a general methodology for statistical modeling and inference that selects the best explanation for observed data as the one allowing the shortest description of them. Application of this principle to the important task of probability density estimation by histograms was previously proposed. We review this approach and provide additional illustrative examples and an application to real-world data, with a presentation emphasizing intuition and concrete arguments. We also consider alternative ways of measuring the description lengths, that can be found to be more suited in this context. We explicitly exhibit, analyze and compare, the complete forms of the description lengths with formulas involving the information entropy and redundancy of the data, and not given elsewhere. Histogram estimation as performed here naturally extends to multidimensional data, and offers for them flexible and optimal subquantization schemes. The framework can be very useful for modeling and reduction of complexity of observed data, based on a general principle from statistical information theory, and placed within a unifying informational perspective

Ressources quantiques et traitement numérique des images

Ressources quantiques et traitement numérique des images