Noisy frequency estimation with noisy probes

TL;DR

This paper investigates frequency estimation in noisy environments using mixed initial states and noisy encoding, deriving formulas for Fisher information and demonstrating that quantum enhancements are achievable regardless of initial mixedness.

Contribution

The study introduces a frequency estimation protocol with mixed initial states and noisy encoding, showing quantum enhancements are independent of initial mixedness and achievable under realistic noise conditions.

Findings

Quantum enhancements do not depend on initial mixedness.

Zeno scaling is attainable with time inhomogeneous noise.

Sensitivity is invariant under qubit permutations and related to purity.

Abstract

We consider frequency estimation in a noisy environment with noisy probes. This builds on previous studies, most of which assume that the initial probe state is pure, while the encoding process is noisy, or that the initial probe state is mixed, while the encoding process is noiseless. Our work is more representative of reality, where noise is unavoidable in both the initial state of the probe and the estimation process itself. We prepare the probe in a GHZ diagonal state, starting from $n+1$ qubits in an arbitrary uncorrelated mixed state, and subject it to parameter encoding under dephasing noise. For this scheme, we derive a simple formula for the (quantum and classical) Fisher information, and show that quantum enhancements do not depend on the initial mixedness of the qubits. That is, we show that the so-called `Zeno' scaling is attainable when the noise present in the encoding process is time inhomogeneous. This scaling does not depend on the mixedness of the initial probe state, and it is retained even for highly mixed states that can never be entangled. We then show that the sensitivity of the probe in our protocol is invariant under permutations of qubits, and monotonic in purity of the initial state of the probe. Finally, we discuss two limiting cases, where purity is either distributed evenly among the probes or concentrated in a single probe.