Optimal quantum estimation of nonobservable quantities

We study the optimal way to estimate the quantum expectation value of a physical observable when a finite number of copies of a quantum pure state are presented. The optimal estimation is determined by minimizing the squared error averaged over all pure states distributed in a unitary invariant way. We find that the optimal estimation is ``biased'' though the optimal measurement is given by successive projective measurements of the observable. The optimal estimate is not the sample average of observed data, but the arithmetic average of observed and ``default nonobserved'' data, with the latter consisting of all eigenvalues of the observable.We study the optimal way to estimate the quantum expectation value of a physical observable when a finite number of copies of a quantum pure state are presented. The optimal estimation is determined by minimizing the squared error averaged over all pure states distributed in a unitary invariant way. We find that the optimal estimation is ``biased'' though the optimal measurement is given by successive projective measurements of the observable. The optimal estimate is not the sample average of observed data, but the arithmetic average of observed and ``default nonobserved'' data, with the latter consisting of all eigenvalues of the observable.

arXiv:quant-ph/0512037Physical Review AJun 14, 20064 citations

Quantum Estimation Theory

Sanjoy K. Mitter, Stephen K. Young

Springer eBooksJan 1, 19792 citations

Optimal distinguishing observables for quantum parameter estimation

Zhong Wei, Xiao-Ming Lu, Xiao-Xing Jing +1 more

Nov 26, 20130 citations

Nonobservable nature of the nuclear shell structure: Meaning, illustrations, and consequences

T. Duguet, H. Hergert, J. D. Holt +1 more

Background: The concept of single-nucleon shells constitutes a basic pillar of our understanding of nuclear structure. Effective single-particle energies (ESPEs) introduced by French [Proceedings of the International School of Physics ``Enrico Fermi,'' Course XXXVI, Varenna 1965, edited by C. Bloch (Academic Press, New York, 1966)] and Baranger [Nucl. Phys. A 149, 225 (1970)] represent the most appropriate tool to relate many-body observables to a single-nucleon shell structure. As briefly discussed in Duguet and Hagen [Phys. Rev. C 85, 034330 (2012)], the dependence of ESPEs on one-nucleon transfer probability matrices makes them purely theoretical quantities that ``run'' with the nonobservable resolution scale $\ensuremath{\lambda}$ employed in the calculation.Purpose: Given that ESPEs provide a way to interpret the many-body problem in terms of simpler theoretical ingredients, the goal is to specify the terms, i.e., the exact sense and conditions, in which this interpretation can be conducted meaningfully.Methods: While the nuclear shell structure is both scale and scheme dependent, the present study focuses on the former. A detailed discussion is provided to illustrate the scale (in)dependence of observables and nonobservables and the reasons why ESPEs, i.e., the shell structure, belong to the latter category. State-of-the-art multireference in-medium similarity renormalization group and self-consistent Gorkov Green's function many-body calculations are employed to corroborate the formal analysis. This is done by comparing the behavior of several observables and of nonobservable ESPEs (and spectroscopic factors) under (quasi) unitary similarity renormalization group transformations of the Hamiltonian parametrized by the resolution scale $\ensuremath{\lambda}$.Results: The formal proofs are confirmed by the results of ab initio many-body calculations in their current stage of implementation. In practice, the unitarity of the similarity transformations is broken owing to the omission of induced many-body interactions beyond three-body operators and to the nonexact treatment of the many-body Schr\"odinger equation. The impact of this breaking is first characterized by quantifying the artificial running of observables over a (necessarily) finite interval of $\ensuremath{\lambda}$ values. Then the genuine running of ESPEs is characterized and shown to be convincingly larger than the one of observables (which would be zero in an exact calculation).Conclusions: The nonobservable nature of the nuclear shell structure, i.e., the fact that it constitutes an intrinsically theoretical object with no counterpart in the empirical world, must be recognized and assimilated. Indeed, the shell structure cannot be determined uniquely from experimental data and cannot be talked about in an absolute sense as it depends on the nonobservable resolution scale employed in the theoretical calculation. It is only at the price of fixing arbitrarily (but conveniently) such a scale that one can establish correlations between observables and the shell structure. To some extent, fixing the resolution scale provides ESPEs (and spectroscopic factors) with a quasi-observable character. Eventually, practitioners can refer to nuclear shells and spectroscopic factors in their analyses of nuclear phenomena if, and only if, they use consistent structure and reaction theoretical schemes based on a fixed resolution scale they have agreed on prior to performing their analysis and comparisons.Background: The concept of single-nucleon shells constitutes a basic pillar of our understanding of nuclear structure. Effective single-particle energies (ESPEs) introduced by French [Proceedings of the International School of Physics ``Enrico Fermi,'' Course XXXVI, Varenna 1965, edited by C. Bloch (Academic Press, New York, 1966)] and Baranger [Nucl. Phys. A 149, 225 (1970)] represent the most appropriate tool to relate many-body observables to a single-nucleon shell structure. As briefly discussed in Duguet and Hagen [Phys. Rev. C 85, 034330 (2012)], the dependence of ESPEs on one-nucleon transfer probability matrices makes them purely theoretical quantities that ``run'' with the nonobservable resolution scale $\ensuremath{\lambda}$ employed in the calculation.Purpose: Given that ESPEs provide a way to interpret the many-body problem in terms of simpler theoretical ingredients, the goal is to specify the terms, i.e., the exact sense and conditions, in which this interpretation can be conducted meaningfully.Methods: While the nuclear shell structure is both scale and scheme dependent, the present study focuses on the former. A detailed discussion is provided to illustrate the scale (in)dependence of observables and nonobservables and the reasons why ESPEs, i.e., the shell structure, belong to the latter category. State-of-the-art multireference in-medium similarity renormalization group and self-consistent Gorkov Green's function many-body calculations are employed to corroborate the formal analysis. This is done by comparing the behavior of several observables and of nonobservable ESPEs (and spectroscopic factors) under (quasi) unitary similarity renormalization group transformations of the Hamiltonian parametrized by the resolution scale $\ensuremath{\lambda}$.Results: The formal proofs are confirmed by the results of ab initio many-body calculations in their current stage of implementation. In practice, the unitarity of the similarity transformations is broken owing to the omission of induced many-body interactions beyond three-body operators and to the nonexact treatment of the many-body Schr\"odinger equation. The impact of this breaking is first characterized by quantifying the artificial running of observables over a (necessarily) finite interval of $\ensuremath{\lambda}$ values. Then the genuine running of ESPEs is characterized and shown to be convincingly larger than the one of observables (which would be zero in an exact calculation).Conclusions: The nonobservable nature of the nuclear shell structure, i.e., the fact that it constitutes an intrinsically theoretical object with no counterpart in the empirical world, must be recognized and assimilated. Indeed, the shell structure cannot be determined uniquely from experimental data and cannot be talked about in an absolute sense as it depends on the nonobservable resolution scale employed in the theoretical calculation. It is only at the price of fixing arbitrarily (but conveniently) such a scale that one can establish correlations between observables and the shell structure. To some extent, fixing the resolution scale provides ESPEs (and spectroscopic factors) with a quasi-observable character. Eventually, practitioners can refer to nuclear shells and spectroscopic factors in their analyses of nuclear phenomena if, and only if, they use consistent structure and reaction theoretical schemes based on a fixed resolution scale they have agreed on prior to performing their analysis and comparisons.

arXiv:1411.1237Physical Review CSep 14, 201576 citations

Quantum Estimation Theory

Yu Watanabe

Springer thesesDec 17, 20138 citations

Nonuniversality of tricritical behavior of Fermi-Bose mixtures

K. K. Singh

Tricritical amplitude ratios have been calculated for a mixture of weakly interacting fermions and bosons. The ratios are characterized by a quantum parameter \ensuremath{\alpha}\ensuremath{\propto}( ${b}^{3}$ /${\ensuremath{\lambda}}_{B}^{4}$ ${K}_{F}$ ${)}^{1/2}$ where ${K}_{F}$ , ${\ensuremath{\lambda}}_{B}$, and b denote, respectively, the Fermi momentum of the fermion component, the boson thermal wavelength, and the scattering length associated with the fermion-boson interaction. To first order in \ensuremath{\alpha}, the results are formally the same as those obtained by Fisher and Sarbach for a classical n-component spin model in the limit n\ensuremath{\rightarrow}\ensuremath{\infty}; to order ${\ensuremath{\alpha}}^{2}$ they differ. It is argued that nonobservance of nonuniversal behavior in $^{3}\mathrm{\ensuremath{-}}^{4}$He mixtures is not inconsistent with the theoretical estimate following from \ensuremath{\alpha}. Better experimental accuracy in the determination of the amplitude ratios may exhibit nonuniversal behavior.Tricritical amplitude ratios have been calculated for a mixture of weakly interacting fermions and bosons. The ratios are characterized by a quantum parameter \ensuremath{\alpha}\ensuremath{\propto}(${b}^{3}$/${\ensuremath{\lambda}}_{B}^{4}$${K}_{F}$${)}^{1/2}$ where ${K}_{F}$, ${\ensuremath{\lambda}}_{B}$, and b denote, respectively, the Fermi momentum of the fermion component, the boson thermal wavelength, and the scattering length associated with the fermion-boson interaction. To first order in \ensuremath{\alpha}, the results are formally the same as those obtained by Fisher and Sarbach for a classical n-component spin model in the limit n\ensuremath{\rightarrow}\ensuremath{\infty}; to order ${\ensuremath{\alpha}}^{2}$ they differ. It is argued that nonobservance of nonuniversal behavior in $^{3}\mathrm{\ensuremath{-}}^{4}$He mixtures is not inconsistent with the theoretical estimate following from \ensuremath{\alpha}. Better experimental accuracy in the determination of the amplitude ratios may exhibit nonuniversal behavior.

Physical review. B, Condensed matterJan 1, 19873 citations

Tricritical amplitude ratios have been calculated for a mixture of weakly interacting fermions and bosons. The ratios are characterized by a quantum parameter \ensuremath{\alpha}\ensuremath{\propto}(

{b}^{3}

/${\ensuremath{\lambda}}_{B}^{4}$

{K}_{F}

{)}^{1/2}

where

{K}_{F}

, ${\ensuremath{\lambda}}_{B}$, and b denote, respectively, the Fermi momentum of the fermion component, the boson thermal wavelength, and the scattering length associated with the fermion-boson interaction. To first order in \ensuremath{\alpha}, the results are formally the same as those obtained by Fisher and Sarbach for a classical n-component spin model in the limit n\ensuremath{\rightarrow}\ensuremath{\infty}; to order ${\ensuremath{\alpha}}^{2}$ they differ. It is argued that nonobservance of nonuniversal behavior in $^{3}\mathrm{\ensuremath{-}}^{4}$He mixtures is not inconsistent with the theoretical estimate following from \ensuremath{\alpha}. Better experimental accuracy in the determination of the amplitude ratios may exhibit nonuniversal behavior.