Geometrical structures of quantum mechanics provide us with new insightful results about the nature of quantum theory. In this work we consider mixed quantum states represented by finite rank density operators. We review our geometrical framework that provide the space of density operators with Riemannian and symplectic structures, and we derive a geometric uncertainty relation for observables acting on mixed quantum states. We also give an example that visualizes the geometric uncertainty relation for spin- $\frac{1}{2}$ particles.Geometrical structures of quantum mechanics provide us with new insightful results about the nature of quantum theory. In this work we consider mixed quantum states represented by finite rank density operators. We review our geometrical framework that provide the space of density operators with Riemannian and symplectic structures, and we derive a geometric uncertainty relation for observables acting on mixed quantum states. We also give an example that visualizes the geometric uncertainty relation for spin-$\frac{1}{2}$ particles.

arXiv:1412.3927Physica ScriptaJan 26, 20153 citations

Quantizing the quantum uncertainty

Etera R. Livine

Annals of PhysicsOct 4, 20231 citation

Geometric formulation of the uncertainty principle

G. M. Bosyk, Tristán M. Osán, Pedro W. Lamberti +1 more

A geometric approach to formulate the uncertainty principle between quantum observables acting on an $N$ -dimensional Hilbert space is proposed. We consider the fidelity between a density operator associated with a quantum system and a projector associated with an observable, and interpret it as the probability of obtaining the outcome corresponding to that projector. We make use of fidelity-based metrics such as angle, Bures, and root infidelity to propose a measure of uncertainty. The triangle inequality allows us to derive a family of uncertainty relations. In the case of the angle metric, we recover the Landau-Pollak inequality for pure states and show, in a natural way, how to extend it to the case of mixed states in arbitrary dimension. In addition, we derive and compare alternative uncertainty relations when using other known fidelity-based metrics.A geometric approach to formulate the uncertainty principle between quantum observables acting on an $N$-dimensional Hilbert space is proposed. We consider the fidelity between a density operator associated with a quantum system and a projector associated with an observable, and interpret it as the probability of obtaining the outcome corresponding to that projector. We make use of fidelity-based metrics such as angle, Bures, and root infidelity to propose a measure of uncertainty. The triangle inequality allows us to derive a family of uncertainty relations. In the case of the angle metric, we recover the Landau-Pollak inequality for pure states and show, in a natural way, how to extend it to the case of mixed states in arbitrary dimension. In addition, we derive and compare alternative uncertainty relations when using other known fidelity-based metrics.

arXiv:1308.4029Physical Review AMar 10, 201415 citations

Universal geometric approach to uncertainty, entropy, and information

Michael J. W. Hall

It is shown that a unique measure of volume is associated with any statistical ensemble, which directly quantifies the inherent spread or localization of the ensemble. It is applicable whether the ensemble is classical or quantum, continuous or discrete, and may be derived from a small number of theory-independent geometric postulates. Remarkably, this unique ensemble volume is proportional to the exponential of the ensemble entropy, and hence provides an interesting geometric characterization of the latter quantity. Applications include unified volume-based derivations of results in quantum and classical information theory, a precise geometric interpretation of thermodynamic entropy for equilibrium ensembles, a geometric derivation of semiclassical uncertainty relations, a means for defining classical and quantum localization for arbitrary evolution processes, and a proposed definition for the spot size of an optical beam. Advantages of ensemble volume over other measures of localization (root-mean-square deviation, Renyi entropies, and inverse participation ratio) are discussed.It is shown that a unique measure of volume is associated with any statistical ensemble, which directly quantifies the inherent spread or localization of the ensemble. It is applicable whether the ensemble is classical or quantum, continuous or discrete, and may be derived from a small number of theory-independent geometric postulates. Remarkably, this unique ensemble volume is proportional to the exponential of the ensemble entropy, and hence provides an interesting geometric characterization of the latter quantity. Applications include unified volume-based derivations of results in quantum and classical information theory, a precise geometric interpretation of thermodynamic entropy for equilibrium ensembles, a geometric derivation of semiclassical uncertainty relations, a means for defining classical and quantum localization for arbitrary evolution processes, and a proposed definition for the spot size of an optical beam. Advantages of ensemble volume over other measures of localization (root-mean-square deviation, Renyi entropies, and inverse participation ratio) are discussed.

arXiv:physics/9903045Physical Review AApr 1, 1999152 citations

Quantizing the Quantum Uncertainty

Etera R. Livine

Jan 1, 20230 citations

\frac{1}{2}

particles.

A geometric approach to formulate the uncertainty principle between quantum observables acting on an

N

-dimensional Hilbert space is proposed. We consider the fidelity between a density operator associated with a quantum system and a projector associated with an observable, and interpret it as the probability of obtaining the outcome corresponding to that projector. We make use of fidelity-based metrics such as angle, Bures, and root infidelity to propose a measure of uncertainty. The triangle inequality allows us to derive a family of uncertainty relations. In the case of the angle metric, we recover the Landau-Pollak inequality for pure states and show, in a natural way, how to extend it to the case of mixed states in arbitrary dimension. In addition, we derive and compare alternative uncertainty relations when using other known fidelity-based metrics.