Quantum error correction assisted quantum metrology without entanglement

TL;DR

This paper demonstrates that quantum error correction assisted quantum metrology can achieve noise-free evolution and Heisenberg scaling without entanglement, highlighting the role of quantum resources beyond entanglement.

Contribution

It reveals that entanglement is not necessary for noise-free quantum metrology in certain QECQM schemes, especially over long and short time scales.

Findings

Entanglement is not required for noise-free evolution in some QECQM problems.

Noise-free quantum metrology is achievable without quantum correlations over short time scales.

Long-term noise-free evolution can occur without entanglement in QECQM schemes.

Abstract

In this article we study the role that quantum resources play in quantum error correction assisted quantum metrology (QECQM) schemes. We show that there exist classes of such problems where entanglement is not necessary to retrieve noise free evolution and Heisenberg scaling in the long time limit. Over short time scales, noise free evolution is also possible even without any form of quantum correlations. In particular, for qubit probes, we show that whenever noise free quantum metrology is possible via QECQM, entanglement free schemes over long time scales and correlation free schemes over short time scales are always possible.

Full text

I Introduction

Quantum information science has gained prominence as an area of research in the recent decades. One of the key promises of the field is that the the quantum regime contains intrinsic advantages over classical theories that can be exploited for a variety of informational tasks. A promising area of study which has gained considerable attention recently is the application of quantum error correction techniques to enhance the precision of quantum metrology Kessler2014 ; Arrad2014 ; Dur2014 . Quantum metrology concerns itself with the precise estimation of some unknown physical parameter, but the precision of such tools often require the preparation of nonclassical quantum states that are sensitive to decoherence effects Huelga1997 ; Dobrzanski2009 ; Escher2011 . Quantum error correction thus offers the promise of enhancing precision by reducing the amount of noise acting on the system.

Another key concern in quantum information is the study of the differences between quantum and classical theories, leading to the development of a theory of quantum resources. Examples of quantum resources include entanglement Horodecki2001 and quantum coherence Streltsov2017 . Quantum entanglement is at present a well established quantum resource with many applications such as cryptography Ekert1991 , teleportation Bennett1991 and superdense coding Bennett1992 . In comparison, the resource theory of quantum coherence is a recent theoretical development, with applications in topics as diverse as quantum macroscopicity Yadin2015 ; Kwon2016 , quantum optics Bagan2016 ; Tan2017 and quantum metrology Tan2018 . It is worth nothing that entanglement and coherence are not entirely separate quantum resources, since entangled states generally contains coherence, though the converse is not necessarily trueStreltsov2015 ; Tan2016 ; Tan2018-2 .

In this article, we will examine the problem of the quantum resources that are necessary for quantum error correction protocols to succeed while simultaneously allowing for quantum enhanced metrology Sekatski2017 ; Zhou2018 . Interestingly, we find that there exist regimes where this can occur without the presence of quantum entanglement, thus requiring us to invoke more general notions of nonclassicality such as quantum discord Ollivier2001 ; Henderson2001 in order to account for the success such protocols. This joins a list of known applications for quantum discord in quantum information  Cavalcanti2011 ; Datta2008 ; Chuan2012 ; Davic2012 ; Bobby2014 . Quantum discord was also considered previously in various other specialized metrological scenarios  Modi2011 ; Cable2016 ; Girolami2014 ; Braun2018 .

We also show that in the extremal case of short interaction times, product states containing zero quantum correlation, but nonzero quantum coherence is sometimes sufficient to generate nontrivial Fisher information in a noise free manner. For qubit probes in particular, we prove that whenever quantum error correction assisted protocols are possible, then an entanglement free protocol over long time scales, or a quantum correlation free protocol over short time scales is also possible.

II Preliminaries

Here, we review some basic notions concerning nonclassical quantum states that will be used in the paper. A more detailed description of quantum metrology, and the role of quantum error correction in metrology, will be provided in the next section.

First, we define the notion of coherence. Let $\rho$ be the density matrix of a quantum state. Then for a fixed basis $\{\left|i\right>\}$, if $\rho$ is not diagonal with respect to this basis, then we say that the state is coherent, or that the state contains coherence.

Second, a pure, bipartite quantum state of the form $\left|\psi\right>_{1}\langle\psi|\otimes\left|\phi\right>_{2}\langle\phi|$ is referred to as a product state. A quantum density matrix $\rho$ that is expressible as a convex sum of product states $\rho=\sum_{i}p_{i}\left|\psi_{i}\right>_{1}\langle\psi_{i}|\otimes\left|\phi_{i}\right>_{2}\langle\phi_{i}|$ is called a separable state. Furthermore, if a state $\rho$ is not separable, then we say that the state is entangled.

We now introduce some notations. We will denote the canonical Pauli matrices on the $m$th qubit as $X_{m}$, $Y_{m}$ and $Z_{m}$ respectively. The computational basis refers to the basis $\{\left|0\right>,\left|1\right>\}$, from which we can define the states $\left|+\right>\coloneqq\frac{1}{\sqrt{2}}(\left|0\right>+\left|1\right>)$ and $\left|-\right>\coloneqq\frac{1}{\sqrt{2}}(\left|0\right>-\left|1\right>)$. The unitary performing a CNOT operation between the $m$th and $n$th qubits is denoted $U^{\mathrm{CNOT}}_{mn}$ where the first subindex $m$ is the control qubit, i.e. $U^{\mathrm{CNOT}}_{mn}\left|0\right>_{m}\left|\psi\right>_{n}=\left|0\right>_{m}\left|\psi\right>_{n}$ and $U^{\mathrm{CNOT}}_{mn}\left|1\right>_{m}\left|\psi\right>_{n}=\left|1\right>_{m}X_{n}\left|\psi\right>_{n}$.

III Error Correction in the Sequential Scheme for quantum metrology