Instytut Podstawowych Problemów Techniki

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Quantum entanglement and coherence are two fundamental features of nature, arising from the superposition principle of quantum mechanics. While considered as puzzling phenomena in the early days of quantum theory, it is only very recently that entanglement and coherence have been recognized as resources for the emerging
quantum technologies, including quantum Metrology, quantum communication, and quantum computing. In this work we study the limitations for the interconversion between coherence and entanglement. We prove a fundamental no-go theorem, stating that a general resource theory of superposition does not allow for
entanglement activation. By constructing a quantum controlled- NOT gate as a free operation, we experimentally show that such activation is possible within the more constrained framework of quantum coherence. By using recent results from coherence theory, we further show that the trace norm entanglement is not a strong entanglement monotone.

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The search for a simple description of fundamental physical processes is an important part of quantum
theory. One example for such an abstraction can be found in the distance lab paradigm: if two separated
parties are connected via a classical channel, it is notoriously difficult to characterize all possible operations these parties can perform. This class of operations is widely known as local operations and classical communication. Surprisingly, the situation becomes comparably simple if the more general class of
separable operations is considered, a finding that has been extensively used in quantum information theory
for many years. Here, we propose a related approach for the resource theory of quantum coherence, where
two distant parties can perform only measurements that do not create coherence and can communicate their outcomes via a classical channel. We call this class local incoherent operations and classical communication. While the characterization of this class is also difficult in general, we show that the larger
class of separable incoherent operations has a simple mathematical form, yet still preserves the main
features of local incoherent operations and classical communication. We demonstrate the relevance of our
approach by applying it to three different tasks: assisted coherence distillation, quantum teleportation, and single-shot quantum state merging. We expect that the results we obtain in this work also transfer to other concepts of coherence that are discussed in recent literature. The approach we present here opens new ways to study the resource theory of coherence in distributed scenarios.

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Understanding the resource consumption in distributed scenarios is one of the main goals of quantum information theory. A prominent example for such a scenario is the task of quantum state merging, where two parties aim to merge their tripartite quantum state parts. In standard quantum state merging, entanglement is considered to be an expensive resource, while local quantum operations can be performed at no additional cost. However, recent developments show that some local operations could be more expensive than others: it is reasonable to distinguish between local incoherent operations and local operations which can create coherence. This idea leads us to the task of incoherent quantum state merging, where one of the parties has free access to local incoherent operations only. In this case the resources of the process are quantified by pairs of entanglement and coherence. Here, we develop tools for studying this process and apply them to several relevant scenarios. While quantum state merging can lead to a gain of entanglement, our results imply that no merging procedure can gain entanglement and coherence at the same time. We also provide a general lower bound on the entanglement-coherence sum and show that the bound is tight for all pure states. Our results also lead to an incoherent version of Schumacher compression: in this case the compression rate is equal to the von Neumann entropy of the diagonal elements of the corresponding quantum state.

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We introduce and study the task of assisted coherence distillation. This task arises naturally in bipartite systems where both parties work together to generate the maximal possible coherence on one of the subsystems. Only incoherent operations are allowed on the target system, while general local quantum operations are permitted on the other; this is an operational paradigm that we call local quantum-incoherent operations and classical communication. We show that the asymptotic rate of assisted coherence distillation for pure states is equal to the coherence of assistance, an analog of the entanglement of assistance, whose properties we characterize. Our findings imply a novel interpretation of the von Neumann entropy: it quantifies the maximum amount of extra quantum coherence a system can gain when receiving assistance from a collaborative party. Our results are generalized to coherence localization in a multipartite setting and possible applications are discussed

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Entanglement distribution is key to the success of secure communication schemes based on quantum mechanics, and there is a strong need for an ultimate architecture able to overcome the limitations of recent proposals such as those based on entanglement percolation or quantum repeaters. In this work we provide a broad theoretical background for the development of such technologies. In particular, we investigate the question of whether entanglement distribution is more efficient if some amount of entanglement—or some amount of correlations in general—is available prior to the transmission stage of the protocol. We show that in the presence of noise the answer to this question strongly depends on the type of noise and on the way the entanglement is quantified. On the one hand, subadditive entanglement measures do not show an advantage of preshared correlations if entanglement is established via combinations of single-qubit Pauli channels. On the other hand, based on the
superadditivity conjecture of distillable entanglement, we provide evidence that this phenomenon occurs for this measure. These results strongly suggest that sending one half of some pure entangled state down a noisy channel is the best strategy for any subadditive entanglement quantifier, thus paving the way to a unified approach for entanglement distribution which does not depend on the nature of noise. We also provide general bounds for entanglement distribution involving quantum discord and present a counterintuitive phenomenon of the advantage of arbitrarily little entangled states over maximally entangled ones, which may also occur for quantum channels relevant in experiments.

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