Time-energy uncertainty relation for noisy quantum metrology

TL;DR

This paper establishes a fundamental trade-off in noisy quantum metrology linking the loss of clock accuracy to environmental information leakage, introduces metrological codes, and explores their potential for noise-resilient quantum sensing.

Contribution

It introduces a quantum trade-off relation for noisy metrology, characterizes metrological codes, and proposes new schemes for noise-resistant quantum measurements.

Findings

Bob's Fisher information loss equals Eve's gain about energy.

Metrological codes can preserve sensitivity despite noise.

Conditions for noise-robust quantum clocks are derived.

Abstract

Detection of weak forces and precise measurement of time are two of the many applications of quantum metrology to science and technology. We consider a quantum system initialized in a pure state and whose evolution is governed by a Hamiltonian $H$; a measurement can later estimate the time $t$ for which the system has evolved. In this work, we introduce and study a fundamental trade-off which relates the amount by which noise reduces the accuracy of a quantum clock to the amount of information about the energy of the clock that leaks to the environment. Specifically, we consider an idealized scenario in which Alice prepares an initial pure state of the clock, allows the clock to evolve for a time $t$ that is not precisely known, and then transmits the clock through a noisy channel to Bob. The environment (Eve) receives any information that is lost. We prove that Bob's loss of quantum Fisher information (QFI) about $t$ is equal to Eve's gain of QFI about a complementary energy parameter. We also prove a more general trade-off that applies when Bob and Eve wish to estimate the values of parameters associated with two noncommuting observables. We derive the necessary and sufficient conditions for the accuracy of the clock to be unaffected by the noise. These are a subset of the Knill-Laflamme error-correction conditions; states satisfying these conditions are said to form a metrological code. We provide a scheme to construct metrological codes in the stabilizer formalism. We show that there are metrological codes that cannot be written as a quantum error-correcting code with similar distance in which the Hamiltonian acts as a logical operator, potentially offering new schemes for constructing states that do not lose any sensitivity upon application of a noisy channel. We discuss applications of our results to sensing using a many-body state subject to erasure or amplitude-damping noise.