[Paper Review] Coherent control of quantum systems as a resource theory

J. M. Matera, Dario Egloff+2 more|arXiv (Cornell University)|Dec 23, 2015

Quantum Information and Cryptography32 references3 citations

TL;DR

This paper establishes a resource theory for coherent control of quantum systems, showing that coherence and entanglement are interconvertible resources—termed 'recoverable coherence'—which quantify the precision of quantum protocols like DQC1. The key result is that the estimation accuracy in DQC1 scales as half the logarithm of the recoverable coherence, linking quantum advantage to a unified measure of quantumness via discord.

ABSTRACT

Control at the interface between the classical and the quantum world is fundamental in quantum physics. In particular, how classical control is enhanced by coherence effects is an important question both from a theoretical as well as from a technological point of view. In this work, we establish a resource theory describing this setting and explore relations to the theory of coherence, entanglement and information processing. Specifically, for the coherent control of quantum systems the relevant resources of entanglement and coherence are found to be equivalent and closely related to a measure of discord. The results are then applied to the DQC1 protocol and the precision of the final measurement is expressed in terms of the available resources.

Motivation & Objective

• To formalize a resource theory for coherent control in quantum systems, where classical control interfaces with quantum targets.
• To identify and quantify the quantum resources enabling enhanced control beyond classical limits.
• To clarify the role of coherence, entanglement, and quantum discord in quantum advantage protocols such as DQC1.
• To demonstrate how recoverable coherence quantifies the precision of quantum measurements in DQC1.
• To unify measures of quantumness—coherence, entanglement, and discord—under a single operational framework.

Proposed method

• Proposes a resource theory where incoherent operations are free, and coherent superpositions are the resource, defined via a fixed basis Z.
• Introduces 'recoverable coherence' as a quantifier of usable quantum resources, equivalent to entanglement and related to quantum discord.
• Uses the resource theory to analyze the DQC1 protocol, modeling the estimation of trace(U)/dim via repeated measurements on maximally coherent probes.
• Derives the asymptotic precision of the trace estimation as a function of the number of probes, linked to the recoverable coherence via standard error of the mean.
• Shows that the precision scales as 1/2 log₂(C_Z^REC), where C_Z^REC is the recoverable coherence of the input state.
• Demonstrates that the optimal precision is achieved when the input state is purified to maximally coherent probes, with the error bounded by the standard error of the mean.

Experimental results

Research questions

• RQ1 How can coherent control of quantum systems be formalized as a resource theory?
• RQ2 What is the operational equivalence between coherence, entanglement, and quantum discord in the context of control?
• RQ3 How does the recoverable coherence quantify the precision of quantum algorithms like DQC1?
• RQ4 Can the quantum advantage in DQC1 be fully attributed to recoverable coherence rather than entanglement?
• RQ5 What is the fundamental limit on measurement precision in DQC1, and how is it determined by quantum resource measures?

Key findings

• Recoverable coherence is the fundamental resource for coherent control, equivalent to both entanglement and quantum discord in this framework.
• The precision of the DQC1 protocol's trace estimation scales as 1/2 log₂(C_Z^REC), where C_Z^REC is the recoverable coherence of the input state.
• In the asymptotic limit, the standard error of the mean determines the precision, and this error is bounded by the inverse square root of the number of probes.
• The optimal precision is achieved when the input state is purified into maximally coherent probes, maximizing the recoverable coherence.
• The correction term in the precision formula is bounded and negligible in the asymptotic regime, confirming the leading-order scaling with recoverable coherence.
• The framework shows that coherence and entanglement are interconvertible resources in this operational setting, with discord providing a natural measure of quantumness.