Uncertainty principle of quantum processes

Yunlong Xiao,Kuntal Sengupta,Siren Yang,Gilad Gour

doi:10.1103/physrevresearch.3.023077

Yunlong Xiao, Kuntal Sengupta + Show 2 more

Open Access

https://doi.org/10.1103/physrevresearch.3.023077

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Abstract

Heisenberg's uncertainty principle, which imposes intrinsic restrictions on our ability to predict the outcomes of incompatible quantum measurements to arbitrary precision, demonstrates one of the key differences between classical and quantum mechanics. The physical systems considered in the uncertainty principle are static in nature and described mathematically with a quantum state in a Hilbert space. However, many physical systems are dynamic in nature and described with the formalism of a quantum channel. In this paper, we show that the uncertainty principle can be reformulated to include process-measurements that are performed on quantum channels. Since both quantum states and quantum measurements are themselves special cases of quantum channels, our formalism encapsulates the uncertainty principle in its utmost generality. More specifically, we obtain expressions that generalize the Maassen-Uffink uncertainty relation and the universal uncertainty relations from quantum states to quantum channels.

Highlights

Counterintuitive as it may seem, the uncertainty principle has been firmly rooted as a fundamental restriction that lies at the heart of quantum mechanics [1]
We have addressed the question of whether quantum mechanics will obstruct us from predicting the outcomes of incompatible process-channel measurements to arbitrary precision
We studied uncertainty relations in three distinct forms: Maassen–Uffink form; direct-sum form; and direct-product form, which reduces to the well-known Maassen–Uffink entropic uncertainty relations [21] and universal uncertainty relations (UURs) [23,25,26] as our special cases by choosing the process to be a state-preparation channel ρ, i.e., = ρ

Summary

INTRODUCTION

Counterintuitive as it may seem, the uncertainty principle has been firmly rooted as a fundamental restriction that lies at the heart of quantum mechanics [1]. The conventional description of the uncertainty principle captures the statistical properties of the system to be studied at a fixed moment, leading to the mathematical representation of uncertainty relations in terms of the initial state. This approach has serious insufficiencies when it comes to describing many realistic scenarios and has impeded progress in understanding the physical nature of the quantum world.

PRELIMINARIES

MAASSEN–UFFINK UNCERTAINTY RELATIONS

CONCLUSIONS AND DISCUSSIONS