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Uncertainty principle and quantum Fisher information. II

Gibilisco, P., Daniele Imparato, Isola, T.

Published Jan 1, 2007• 88 citations
OpenAlex

Abstract

Heisenberg and Schr{\"o}dinger uncertainty principles give lower bounds for the product of variances $Var_{\rho}(A)\cdot Var_{\rho}(B)$, in a state $\rho$, if the observables $A,B$ are not compatible, namely if the commutator $[A,B]$ is not zero. In this paper we prove an uncertainty principle in Schr{\"o}dinger form where the bound for the product of variances $Var_{\rho}(A)\cdot Var_{\rho}(B)$ depends on the area spanned by the commutators $[\rho,A]$ and $[\rho,B]$ with respect to an arbitrary quantum version of the Fisher information.

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Uncertainty principle and quantum Fisher information. II.

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Heisenberg and Schr{"o}dinger uncertainty principles give lower bounds for the product of variances Varρ(A)⋅Varρ(B)Var_{\rho}(A)\cdot Var_{\rho}(B)Varρ​(A)⋅Varρ​(B), in a state ρ\rhoρ, if the observables A,BA,BA,B are not compatible, namely if the commutator [A,B][A,B][A,B] is not zero. In this paper we prove an uncertainty principle in Schr{"o}dinger form where the bound for the product of variances Varρ(A)⋅Varρ(B)Var_{\rho}(A)\cdot Var_{\rho}(B)Varρ​(A)⋅Varρ​(B) depends on the area spanned by the commutators [ρ,A][\rho,A][ρ,A] and [ρ,B][\rho,B][ρ,B] with respect to an arbitrary quantum version of the Fisher information.
The Heisenberg–Robertson (HR) and Robertson–Schrödinger (RS) uncertainty relations are studied. Analyzing the RS uncertainty relation we found, that there can exist a large set of states of the quantum system under considerations, for which the lower bound of the product of the standard deviations of a pair of noncommuting observables, A and B, is zero, and which differ from those described in the literature for the HR uncertainty relation. These states are not eigenstates of either the observable A or B. The correlation function for these observables in such states is equal to zero. We have also shown that the so-called “sum uncertainty relations” also do not provide any information about lower bounds on the standard deviations calculated for these states. We additionally show that the uncertainty principle in its most general form has two faces: One is that it is a lower bound on the product of standard deviations, and the other is that the product of standard deviations is an upper bound on the modulus of the correlation function of a pair of the noncommuting observables in the state under consideration.
We explore quantum uncertainty relations involving the Fisher information functionals I x and I p evaluated, respectively, on a wavefunction Ψ( x ) defined on a D -dimensional configuration space and the concomitant wavefunction on the conjugate momentum space. We prove that the associated Fisher functionals obey the uncertainty relation I x I p ⩾ 4 D 2 when either Ψ( x ) or is real. On the other hand, there is no lower bound to the above product for arbitrary complex wavefunctions. We give explicit examples of complex wavefunctions not obeying the above bound. In particular, we provide a parametrized wavefunction for which the product I x I p can be made arbitrarily small.
The Heisenberg-Robertson uncertainty relation expresses a limitation in the possible preparations of the system by giving a lower bound to the product of the variances of two observables in terms of their commutator. Notably, it does not capture the concept of incompatible observables because it can be trivial; i.e., the lower bound can be null even for two noncompatible observables. Here we give two stronger uncertainty relations, relating to the sum of variances, whose lower bound is guaranteed to be nontrivial whenever the two observables are incompatible on the state of the system.
We prove a few state-dependent uncertainty relations for the product as well as the sum of variances of two incompatible observables. These uncertainty relations are shown to be tighter than the Robertson-Schr\"odinger uncertainty relation and other ones existing in the current literature. Also, we derive a state-dependent upper bound to the sum and the product of variances using the reverse Cauchy-Schwarz inequality and the Dunkl-Williams inequality. Our results suggest that not only can we not prepare quantum states for which two incompatible observables can have sharp values, but also we have both the lower and the upper limits on the variances of quantum mechanical observables at a fundamental level.