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arXiv:2002.01507 · Physical Review A

Mean value of the quantum potential and uncertainty relations

F. Nicacio, F. T. Falciano

Published May 7, 2020• 3 citations
arXivHTMLPDF

Abstract

In this paper we determine a lower bound to the mean value of the quantum potential for an arbitrary state. Furthermore, we derive a generalized uncertainty relation that is stronger than the Robertson-Schr\"odinger inequality and hence also stronger than the Heisenberg uncertainty principle. The mean value is then associated to the nonclassical part of the covariances of the momenta operator. This imposes a minimum bound for the nonclassical correlations of momenta and gives a physical characterization of the classical and semiclassical limits of quantum systems. The results obtained primarily for pure states are then generalized for density matrices describing mixed states.

DOI
10.1103/physreva.101.052105

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In this paper we determine a lower bound to the mean value of the quantum potential for an arbitrary state. Furthermore, we derive a generalized uncertainty relation that is stronger than the Robertson-Schr"odinger inequality and hence also stronger than the Heisenberg uncertainty principle. The mean value is then associated to the nonclassical part of the covariances of the momenta operator. This imposes a minimum bound for the nonclassical correlations of momenta and gives a physical characterization of the classical and semiclassical limits of quantum systems. The results obtained primarily for pure states are then generalized for density matrices describing mixed states.
The Heisenberg uncertainty relation sets a fundamental limit for quantum measurement of incompatible observables. Its standard form derived by Weyl and Robertson is of purely quantum nature when the state is pure. However, for mixed states, because the variance involving a mixed state is a hybrid of classical and quantum uncertainty, the conventional uncertainty relation is of a ``mixed'' flavor. It is desirable to seek some decomposition of variance into classical and quantum parts and to cast the Heisenberg uncertainty relation for mixed states in a more quantum form. By use of the skew information introduced by Wigner and Yanase in 1963, we make such an attempt and establish a different uncertainty relation which is stronger than the conventional one.
For a quantum state and a set of observables, we can construct an associated covariance matrix and a natural quantum Fisher information matrix. These two matrices characterize the uncertainty and information content of the observables in the relevant state. An inequality between these two matrices is established. This inequality may be interpreted as a general quantification of the Heisenberg uncertainty principle from a statistical estimation perspective. In particular, it implies a new uncertainty relation which refines the celebrated Schrödinger uncertainty relation.
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 study how correlations in a state of an open quantum system affect the intrinsic uncertainties of the expectation values of an arbitrary pair of noncommuting observables. We show that for such observables, there exist Heisenberg-type uncertainty relations that take fully into account correlations in the state of the system. If the quantum system is in a pure state, such uncertainty relations reduce to the conventional one. We obtain an equation for the density operator of a general state that minimizes the new uncertainty relations, and demonstrate that in the important case of coordinate and momentum operators, the minimum-uncertainty states are displaced squeezed thermal states.
We analyze the Schwarz inequality and its generalizations, as well as inequalities resulting from the Jensen inequality. They are used in quantum theory to derive the Heisenberg-Robertson (HR) and Schroedinger-Robertson (SR) uncertainty relation for two non-commuting observables and their generalizations to three or more non-commuting observables. Jensen's inequality, in turn, is helpful in deriving various the "sum uncertainty relations" for two or more observables. Using these inequalities, we derive various types of generalized uncertainty relations for more than two non--commuting observables and analyze their properties and critical points. We also study the connections between the generalizations of the HR and SR uncertainty relations for two and more observables and the correlations of these observables in the state of the quantum system under study. In this analysis, we pay special attention to the consequences of the generalized SR uncertainty relation for three non--commuting observables on their correlations in a given state of a quantum system and to the connections of this relation with the appropriate correlation matrix, whose matrix elements are the quantum versions of the Pearson coefficient. We show also that the SR uncertainty relation (including the generalized ones) can be written in an equivalent way using these Pearson coefficients.
Abstract We report a universal improvement to the standard Robertson–Schrödinger uncertainty relation. Our result shows that the Robertson–Schrödinger lower bound can be supplemented by a new noncommutativity-induced term. This term represents a previously overlooked quantum contribution and becomes more pronounced as the state becomes more mixed. Moreover, it is expressed as the expectation value of a positive observable, namely the squared modulus of the commutator, and therefore preserves the direct, experimentally accessible character of the Robertson–Schrödinger relation. For two-level quantum systems, our relation becomes an exact equality for any state and any pair of observables, thereby ensuring the tightness of the bound in the strongest possible sense. The relation also yields, as a corollary, a complete proof of a general uncertainty bound that had previously been supported only by numerical evidence.