Coherence in quantum estimation

TL;DR

This paper explores the role of quantum coherence in estimation precision, linking geometric quantum state properties to measurement bounds, and demonstrates how coherence can be harnessed and analyzed in various quantum estimation protocols, including noisy and critical systems.

Contribution

It introduces a framework connecting quantum state geometry and coherence to estimation bounds, providing new insights into resource management and subsystem factorization in quantum estimation.

Findings

Coherence is identified as a key resource for optimizing quantum estimation.

The framework applies to both noiseless and noisy multi-qubit estimation protocols.

Coherence exhibits non-analyticities and scaling behavior near quantum phase transitions.

Abstract

The geometry of quantum states provides a unifying framework for estimation processes based on quantum probes, and it allows to derive the ultimate bounds of the achievable precision. We show a relation between the statistical distance between infinitesimally close quantum states and the second order variation of the coherence of the optimal measurement basis with respect to the state of the probe. In Quantum Phase Estimation protocols, this leads to identify coherence as the relevant resource that one has to engineer and control to optimize the estimation precision. Furthermore, the main object of the theory i.e., the Symmetric Logarithmic Derivative, in many cases allows to identify a proper factorization of the whole Hilbert space in two subsystems. The factorization allows: to discuss the role of coherence vs correlations in estimation protocols; to show how certain estimation processes can be completely or effectively described within a single-qubit subsystem; and to derive lower bounds for the scaling of the estimation precision with the number of probes used. We illustrate how the framework works for both noiseless and noisy estimation procedures, in particular those based on multi-qubit GHZ-states. Finally we succinctly analyze estimation protocols based on zero-temperature critical behaviour. We identify the coherence that is at the heart of their efficiency, and we show how it exhibits the non-analyticities and scaling behaviour proper of a large class of quantum phase transitions.