Certification and Quantification of Multilevel Quantum Coherence

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 multi-level coherence, which acts as the single-party analogue to multi-partite 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 non-trivial 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 multilevel coherence, which leads to a complete classification 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. We further verify and lower-bound the robustness of multilevel coherence by performing a semi-device-independent phase discrimination task, implemented experimentally with four-level 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 test the quantum behaviour of physical systems.

20 pages, 6 figures. (v2) Title changed, presentation improved. (v3) Expanded and updated with new results, including a necessary and sufficient analytical criterion to detect multilevel coherence; one co-author added

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• Experimental progress on quantum coherence: detection, quantification, and manipulation
• Benchmarking one-shot distillation in general quantum resource theories
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• More entanglement implies higher performance in tailored channel discrimination tasks
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• Tomographic Witnessing and Holographic Quantifying of Coherence
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• Witnessing quantum coherence with prior knowledge of observables
• The Zero-error Entanglement Cost is Highly Non-Additive
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• Certifying high-dimensional quantum channels
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• Characterizing high-dimensional quantum contextuality
• Coordinately Assisted Distillation of Quantum Coherence in Multipartite System
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• The coherent measurement cost of coherence distillation
• A resource theory of Maxwell's demons
• Scalable protocol to coherence estimation from scarce data: Theory and experiment