Noise-resolution uncertainty principle in classical and quantum systems

Timur E Gureyev,Alexander Kozlov,Yakov I Nesterets,Harry M Quiney,David M Paganin

doi:10.1038/s41598-020-64539-7

Abstract

We show that the width of an arbitrary function and the width of the distribution of its values cannot be made arbitrarily small simultaneously. In the case of ergodic stochastic processes, an ensuing uncertainty relationship is then demonstrated for the product of correlation length and variance. A closely related uncertainty principle is also established for the average degree of fourth-order coherence and the spatial width of modes of bosonic quantum fields. However, it is shown that, in the case of stochastic and quantum observables, certain non-classical states with sub-Poissonian statistics, such as for example photon number squeezed states in quantum optics, can overcome the “classical” noise-resolution uncertainty limit. This uncertainty relationship, which is fundamentally different from the Heisenberg and related uncertainty principles, can define an upper limit for the information capacity of communication and imaging systems. It is expected to be useful in a variety of problems in classical and quantum optics and imaging.

Highlights

We show that the width of an arbitrary function and the width of the distribution of its values cannot be made arbitrarily small simultaneously
This provides a strong indication that, unlike number-phase and many other uncertainty relationships in physics, the NRU cannot be derived from the Heisenberg uncertainty principle (HUP) or similar relationships
Following a general mathematical description, we demonstrate how the proposed approach can be applied to stochastic processes, where it provides a relationship between correlation length and variance

Summary

Introduction

We show that the width of an arbitrary function and the width of the distribution of its values cannot be made arbitrarily small simultaneously. When applied to stochastic distributions, the NRU implies that a distribution, such as a detected spatial distribution of identical particles, cannot be made arbitrarily spatially narrow and have arbitrary high signal-to-noise ratio (SNR) at the same time, if the total mean number of particles in the system is kept constant This implies a fundamental trade-off between the (spatial) resolution and the SNR of a distributed classical or quantum measurement. The approach developed in the present paper allows us to overcome this difficulty and obtain NRU-type relations for certain correlation functions in quantum optics In this context, we give an example of a quantum system that can defeat the classical limit for the product of the variance and the width of a single-mode field. It is likely that the NRU may impose effective bounds on the precision of certain quantum measurements and quantum information capacity[4,20,21]

Results

Discussion

Conclusion