Abstract
Quantum coherence, present whenever a quantum system exists in a superposition of multiple classically distinct states, marks one of the fundamental departures from classical physics. Quantum coherence has recently been investigated rigorously within a resource-theoretic formalism. However, the finer-grained notion of multilevel coherence, which explicitly takes into account the number of superposed classical states, has remained relatively unexplored. A comprehensive analysis of multilevel coherence, which acts as the single-party analogue to multipartite entanglement, is essential for understanding natural quantum processes as well as for gauging the performance of quantum technologies. Here, we develop the theoretical and experimental groundwork for characterizing and quantifying multilevel coherence. We prove that nontrivial levels of purity are required for multilevel coherence, as there is a ball of states around the maximally mixed state that do not exhibit multilevel coherence in any basis. We provide a simple, necessary, and sufficient analytical criterion to verify the presence of multilevel coherence, which leads to a complete classification of multilevel coherence for three-level systems. We present the robustness of multilevel coherence, a bona fide quantifier, which we show to be numerically computable via semidefinite programming and experimentally accessible via multilevel coherence witnesses, which we introduce and characterize. We further verify and lower bound the robustness of multilevel coherence by performing a semi-device-independent phase discrimination task, which is implemented experimentally with four-level quantum probes in a photonic setup. Our results contribute to understanding the operational relevance of genuine multilevel coherence, also by demonstrating the key role it plays in enhanced phase discrimination—a primitive for quantum communication and metrology—and suggest new ways to reliably and effectively test the quantum behavior of physical systems.
- Received 2 May 2018
- Revised 15 August 2018
DOI:https://doi.org/10.1103/PhysRevX.8.041007
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
Published by the American Physical Society
Physics Subject Headings (PhySH)
Popular Summary
Unlike everyday objects, quantum particles can, counterintuitively, be in a superposition of two locations at once. The feature behind this is quantum coherence, which underpins virtually every quantum phenomenon. Things quickly get complicated when the particle may be in three or more possible locations. Is the particle really in a superposition of all three locations—A, B, and C—at once, or is there some probability of a superposition A and B with some other probability of A and C? Here, we provide a comprehensive toolset to characterize, detect, quantify, and experimentally measure such multilevel coherence.
We start by developing a theoretical framework in which to characterize multilevel coherence, along with analytical criteria that together lead to a complete characterization of such systems. We demonstrate this toolset experimentally in a photonic four-level quantum system. We show that multilevel coherence obeys a strict hierarchy, where higher levels of coherence are more challenging to establish and maintain but also offer advantages in practical tasks.
Given the fundamental role played by quantum coherence, our results have far-reaching applicability for the foundations of physics, quantum technologies, and exciting fields such as quantum biology.