Entropic Energy-Time Uncertainty Relation

TL;DR

This paper establishes a new entropic energy-time uncertainty relation applicable to quantum systems, incorporating quantum memory and linking it to the relative entropy of asymmetry, with potential implications for quantum information processing.

Contribution

It introduces a strong, entropic energy-time uncertainty relation for general Hamiltonians, including quantum memory effects, and interprets it as a bound on the relative entropy of asymmetry.

Findings

The relation applies to both discrete and continuous time.

Quantum memory can reduce the uncertainty bound.

The relation has operational significance in quantum information.

Abstract

Energy-time uncertainty plays an important role in quantum foundations and technologies, and it was even discussed by the founders of quantum mechanics. However, standard approaches (e.g., Robertson's uncertainty relation) do not apply to energy-time uncertainty because, in general, there is no Hermitian operator associated with time. Following previous approaches, we quantify time uncertainty by how well one can read off the time from a quantum clock. We then use entropy to quantify the information-theoretic distinguishability of the various time states of the clock. Our main result is an entropic energy-time uncertainty relation for general time-independent Hamiltonians, stated for both the discrete-time and continuous-time cases. Our uncertainty relation is strong, in the sense that it allows for a quantum memory to help reduce the uncertainty, and this formulation leads us to reinterpret it as a bound on the relative entropy of asymmetry. Due to the operational relevance of entropy, we anticipate that our uncertainty relation will have information-processing applications.