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arXiv:1803.02221 · Physical Review A

Quantum reciprocity relations for fluctuations of position and momentum

Mahasweta Pandit, Anindita Bera, Aditi Sen, Ujjwal Sen

Published Jul 29, 2019• 3 citations
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Abstract

We propose a trade-off between the Lipschitz constants of the position and momentum probability distributions for arbitrary quantum states. We refer to the trade-off as a quantum reciprocity relation. The Lipschitz constant of a function may be considered to quantify the extent of fluctuations of that function and is, in general, independent of its spread. The spreads of the position and momentum distributions are used to obtain the celebrated Heisenberg quantum uncertainty relations. We find that the product of the Lipschitz constants of position and momentum probability distributions is bounded below by a number that is of the order of the inverse square of the Planck's constant.

DOI
10.1103/physreva.100.012131

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We propose a trade-off between the Lipschitz constants of the position and momentum probability distributions for arbitrary quantum states. We refer to the trade-off as a quantum reciprocity relation. The Lipschitz constant of a function may be considered to quantify the extent of fluctuations of that function and is, in general, independent of its spread. The spreads of the position and momentum distributions are used to obtain the celebrated Heisenberg quantum uncertainty relations. We find that the product of the Lipschitz constants of position and momentum probability distributions is bounded below by a number that is of the order of the inverse square of the Planck's constant.
The authors of this paper [M. Pandit et al., Phys. Rev. A 100, 012131 (2019)] proposed an alternative formulation of the uncertainty relation in quantum mechanics in terms of the Lipschitz constants. They evaluated these constants in various cases and decided that the product of the square roots of Lipschitz constants for position and momentum probability distributions is bounded below by $0.3/\ensuremath{\hbar}$. In this comment, I invalidate this claim by giving an example of wave functions for which the value of the product of the Lipschitz constants for position and momentum does not obey this bound and tends to zero.
We study reciprocity relations between fluctuations of the probability distributions corresponding to position and momentum, and other observables, in quantum theory. These kinds of relations have been previously studied in terms of quantifiers based on the Lipschitz constants of the concomitant distributions. However, it turned out that they were not valid for all states. Here, we ask the following question: can those relations be described using other quantifiers? By appealing to the Fisher information, we study reciprocity relations for different families of states. In particular, we look for a connection of this problem with previous works.
The position-momentum uncertainty-like inequality based on moments of arbitrary order for ddd-dimensional quantum systems, which is a generalization of the celebrated Heisenberg formulation of the uncertainty principle, is improved here by use of the R\'enyi-entropy-based uncertainty relation. The accuracy of the resulting lower bound is physico-computationally analyzed for the two main prototypes in ddd-dimensional physics: the hydrogenic and oscillator-like systems.
The usual Heisenberg uncertainty relations are generalized to the case of nonsimultaneous measurements of position and momentum. The product of the corresponding uncertainties is bounded from below by a quantity depending on the dynamics of the system (the potential), and which may be lower than ħ/2 and even vanish. The case of two successive measurements of momentum and position is also treated, and one recovers the standard phenomenon of spreading of the wave packet. Whereas the case of an arbitrary potential can only be treated to lowest orders in the lime delay, a complete solution is given for the harmonic oscillator.
We consider successive measurements of position and momentum of a single particle. Let P be the conditional probability to measure the momentum k with precision dk, given a previously successful position measurement q with precision dq. Several upper bounds for the probability P are determined. For arbitrary, but given precision dq, dk, these bounds refer to the variation of q, k and the state vector of the particle. A weak bound is given by the inequality P <= dkdq/h, where h is Planck's quantum of action. It is non-trivial for all measurements with dkdq < h. A sharper bound is obtained by applying the Hilbert-Schmidt-norm. As our main result the least upper bound of P is determined. All bounds are independent of the order with which the measuring of position and momentum is made.
Heisenberg's uncertainty principle is usually taken to express a limitation of operational possibilities imposed by quantum mechanics.Here we demonstrate that the full content of this principle also includes its positive role as a condition ensuring that mutually exclusive experimental options can be reconciled if an appropriate trade-off is accepted.The uncertainty principle is shown to appear in three manifestations, in the form of uncertainty relations: for the widths of the position and momentum distributions in any quantum state; for the inaccuracies of any joint measurement of these quantities; and for the inaccuracy of a measurement of one of the quantities and the ensuing disturbance in the distribution of the other quantity.Whilst conceptually distinct, these three kinds of uncertainty relations are shown to be closely related formally.Finally, we survey models and experimental implementations of joint measurements of position and momentum and comment briefly on the status of experimental tests of the uncertainty principle.