Incompatibility in Quantum Parameter Estimation

In this paper we introduce a measure of genuine quantum incompatibility in the estimation task of multiple parameters, that has a geometric character and is backed by a clear operational interpretation. This measure is then applied to some simple systems in order to track the effect of a local depolarizing noise on the incompatibility of the estimation task. A semidefinite program is described and used to numerically compute the figure of merit when the analytical tools are not sufficient, among these we include an upper bound computable from the symmetric logarithmic derivatives only. Finally we discuss how to obtain compatible models for a general unitary encoding on a finite dimensional probe.Comment: We clarified the relation between LU and LAC measurements. 35 pages, 3 figure

Optimizing quantum-enhanced Bayesian multiparameter estimation in noisy apparata

Achieving quantum-enhanced performances when measuring unknown quantities requires developing suitable methodologies for practical scenarios, that include noise and the availability of a limited amount of resources. Here, we report on the optimization of quantum-enhanced Bayesian multiparameter estimation in a scenario where a subset of the parameters describes unavoidable noise processes in an experimental photonic sensor. We explore how the optimization of the estimation changes depending on which parameters are either of interest or are treated as nuisance ones. Our results show that optimizing the multiparameter approach in noisy apparata represents a significant tool to fully exploit the potential of practical sensors operating beyond the standard quantum limit for broad resources range

Non-asymptotic Heisenberg scaling: experimental metrology for a wide resources range

Adopting quantum resources for parameter estimation discloses the possibility to realize quantum sensors operating at a sensitivity beyond the standard quantum limit. Such approach promises to reach the fundamental Heisenberg scaling as a function of the employed resources $N$ in the estimation process. Although previous experiments demonstrated precision scaling approaching Heisenberg-limited performances, reaching such regime for a wide range of $N$ remains hard to accomplish. Here, we show a method which suitably allocates the available resources reaching Heisenberg scaling without any prior information on the parameter. We demonstrate experimentally such an advantage in measuring a rotation angle. We quantitatively verify Heisenberg scaling for a considerable range of $N$ by using single-photon states with high-order orbital angular momentum, achieving an error reduction greater than $10$ dB below the standard quantum limit. Such results can be applied to different scenarios, opening the way to the optimization of resources in quantum sensing

Incompatibility in Quantum Metrology

One of the characterizing feature of quantum mechanics is the incompatibility between observables, understood since the beginning as the Heisenberg uncertainty principle. This is a trade-off relation between the measurements precision of two conjugated physical quantities. In the modern approach of quantum information theory some vagueness about the uncertainty principle has been cleared out, and a whole new theory of incompatibility has been developed. Among today cornerstones of quantum information there is quantum estimation theory, that deals with the precision limits in extraction of parameters characterizing a certain physical state. These parameters may not have an associated observable (temperature, phase shift, noise, ...) and to recover information about them one needs to perform complex data processing of the measured data. When going to the full realm of quantum metrology, entanglement can be exploited to boost the estimation strategy. These developments help in clarifying what can and can't be done within the domain of quantum mechanics. However, far from being just theoretical curiosities they have had and will have great relevance in the fields of applied physics concerning gravitational waves, biology and enhanced imaging, just to cite some applications. As a matter of fact it's hard to overestimate the importance of quantum metrology for the development of quantum technologies: it naturally pops out of fundamental tasks like the calibration of gates for quantum computation, the quantification of the amount of entanglement and/or noise in a given state, the characterization of thermal environments and many others. The objective of this thesis is to explore quantum incompatibility within the domain of quantum metrology. We give a precise quantification of the amount of incompatibility in the estimation of multiple parameters by defining a suitable figure of merit. Its mathematical properties are extensively characterized, and it is computed for some relevant estimations on qubit and qutrit systems; deeply relying on the theory of Quantum Local Asymptotic Normality (QLAN). Then we analyze the metrological process in presence of noise by taking as example the simple scenario of a qubit subject to various disturbances. The computation of the figure of merit points toward the presence of an information-compatibility trade-off, i.e. we can have an improvement in compatibility by dropping some of the information content and vice versa. Further work is dedicated to enforcing full compatibility at the expense of a finite amount of information sacrificed. Then we perform calculations of the noisy figure of merit for the more elaborate scenarios of sequential estimation and multiqubit entangled input, and we expose the interplay between the enhancement in compatibility given by noise and that given by entanglement. Since now we have been talking about incompatibility at the measurement level, but in multiparameter quantum metrology there is also incompatibility at the level of probes, meaning that each parameter has its own optimal input, and in the simultaneous estimation they compete for the emergence of a joint optimal state. We explore this issue by defining a figure of merit for probes incompatibility. Being this work completely theoretical the hope is that it will contribute to advance the edge of quantum technologies and in particular that of today NISQ (Noisy Intermediate Scale Quantum) technologies. It is worth mentioning that in the final appendix we formulate and solve a very harmful loophole in entanglement enhanced metrology (Heisenberg scaling) which could potentially destroy its advantage. We feel that the loophole and its resolution have been overlooked in the literature and were never given a proper treatment

Untwining multiple parameters at the exclusive zero-coincidence points with quantum control

In this paper we address a special case of ‘sloppy’ quantum estimation procedures which happens in the presence of intertwined parameters. A collection of parameters are said to be intertwined when their imprinting on the quantum probe that mediates the estimation procedure, is performed by a set of linearly dependent generators. Under this circumstance the individual values of the parameters can not be recovered unless one tampers with the encoding process itself. An example is presented by studying the estimation of the relative time-delays that accumulate along two parallel optical transmission lines. In this case we show that the parameters can be effectively untwined by inserting a sequence of balanced beam splitters (and eventually adding an extra phase shift on one of the lines) that couples the two lines at regular intervals in a setup that remind us a generalized Hong-Ou-Mandel interferometer. For the case of two time delays we prove that, when the employed probe is the frequency-correlated biphoton state, the untwining occurs in correspondence of exclusive zero-coincidence (EZC) point. Furthermore we show the statistical independence of two time delays and the optimality of the quantum Fisher information at the EZC point. Finally we prove the compatibility of this scheme by checking the weak commutativity condition associated with the symmetric logarithmic derivative operators

Experimental metrology beyond the standard quantum limit for a wide resources range

Adopting quantum resources for parameter estimation discloses the possibility to realize quantum sensors operating at a sensitivity beyond the standard quantum limit. Such an approach promises to reach the fundamental Heisenberg scaling as a function of the employed resources N in the estimation process. Although previous experiments demonstrated precision scaling approaching Heisenberg-limited performances, reaching such a regime for a wide range of N remains hard to accomplish. Here, we show a method that suitably allocates the available resources permitting them to reach the same power law of Heisenberg scaling without any prior information on the parameter. We demonstrate experimentally such an advantage in measuring a rotation angle. We quantitatively verify sub-standard quantum limit performances for a considerable range of N (O(30,000)) by using single-photon states with high-order orbital angular momentum, achieving an error reduction, in terms of the obtained variance, >10 dB below the standard quantum limit. Such results can be applied to different scenarios, opening the way to the optimization of resources in quantum sensing