Coherence Quantifies Precision Gains in Quantum Phase Estimation Protocols

Research demonstrates a direct relationship between quantum coherence and the precision of phase estimation, a crucial process in technologies like metrology and quantum algorithms. Every quantifiable element of a quantum state’s coherence contributes to improved estimation accuracy, establishing coherence as a fundamental resource.

Precise determination of phase – the relative timing of wave functions – underpins a diverse range of technologies, from atomic clocks and magnetic resonance imaging to quantum computing algorithms. Researchers have now established a quantifiable link between a quantum state’s inherent coherence – its ability to exist in multiple states simultaneously – and its efficacy in phase estimation protocols. Felix Ahnefeld, Thomas Theurer, and Martin B. Plenio, in their work entitled ‘Coherence as a resource for phase estimation’, demonstrate that coherence directly enhances the precision with which unknown phases can be determined. By constructing and analysing resource theories of incoherent networks – systems incapable of generating coherence – the team, based at the University of Ulm and the University of Copenhagen, derive optimal protocols and a family of coherence measures that explicitly connect a quantum state’s coherence with its performance in phase estimation tasks.

Coherence Directly Quantifies the Limits of Phase Estimation

Precise phase estimation is fundamental to the operation of many quantum technologies, including quantum computation and sensing, and establishes a quantifiable relationship between performance and the maintenance of quantum coherence – a property describing the superposition of quantum states. Recent research has employed a resource-theoretic framework to rigorously define the limits of phase estimation accuracy, linking it directly to the coherence present in the initial quantum state.

Researchers formulated the problem by considering quantum networks incapable of generating coherence. This restriction allows them to isolate the contribution of pre-existing coherence to the estimation process. By limiting the permissible quantum operations to those that cannot create additional coherence, they characterised optimal strategies for phase estimation and derived a minimal average cost – a measure of the resources required – for achieving a given level of accuracy.

The key finding demonstrates that each quantifiable unit of coherence directly improves estimation accuracy. This establishes a clear link between a quantum state’s coherence and its effectiveness in phase estimation, solidifying coherence as a crucial resource. In essence, the more coherence a state possesses, the more accurately a phase can be estimated, given a fixed expenditure of resources.

To formulate and solve the complex optimisation problem, the researchers employed semidefinite programming (SDP) – a mathematical technique used to optimise linear objective functions subject to linear matrix inequalities. SDP provides a robust and rigorous framework for analysing the fundamental limits of estimation.

This work highlights the critical role of coherence in any technology reliant on accurate phase estimation. Precision sensors, for example, suffer performance degradation from even slight phase errors. Similarly, quantum computers utilise phase estimation as a key subroutine in numerous algorithms, including Shor’s algorithm for factoring large numbers and quantum simulation. The ability to quantify the value of coherence provides a pathway for optimising quantum states and protocols, potentially leading to more powerful and efficient technologies.

This research provides a foundational understanding of the resources required for precise quantum measurements, and will likely inform future advancements in both fundamental science and practical applications.