The superposition principle lies at the heart of many non-classical properties of quantum mechanics. Motivated by this, we introduce a rigorous resource theory framework for the quantification of superposition of a finite number of linear independent states. This theory is a generalization of resource theories of coherence. We determine the general structure of operations which do not create superposition, find a fundamental connection to unambiguous state discrimination, and propose several quantitative superposition measures. Using this theory, we show that trace decreasing operations can be completed for free which, when specialised to the theory of coherence, resolves an outstanding open question and is used to address the free probabilistic transformation between pure states. Finally, we prove that linearly independent superposition is a necessary and sufficient condition for the faithful creation of entanglement in discrete settings, establishing a strong structural connection between our theory of superposition and entanglement theory.Introduction. -During the last decades, there has been an increasing interest in quantum technologies. The main reason for this is the operational advantages of protocols or devices working in the quantum regime over those relying on classical physics. Early examples include entanglement-based quantum cryptography [1], quantum dense coding [2] and quantum teleportation [3], where entanglement is a resource which is consumed and manipulated. Therefore the detection, manipulation and quantification of entanglement was investigated, leading to the resource theory of entanglement [4]. Typical quantum resource theories (QRTs) are built by imposing an additional restriction to the laws of quantum mechanics [5][6][7]. In the case of entanglement theory, this is the restriction to local operations and classical communication (LOCC). From such a restriction, the two main ingredients of QRTs emerge: The free operations and the free states (which are LOCC and separable states in the case of entanglement theory). All states which are not free contain the resource under investigation and are considered costly. Therefore free operations must transform free states to free states, allowing for the resource to be manipulated but not freely created. Once these main ingredients are defined, a resource theory investigates the manipulation, detection, quantification and usage of the resource.In principle, not only entanglement but every property of quantum mechanics not present in classical physics could lead to an operational advantage [8,9]. This motivates the considerable interest in the rigorous quantification of nonclassicality [10][11][12][13][14][15]. The superposition principle underlies many non-classical properties of quantum mechanics including entanglement or coherence. Recently resource theories of coherence [11,16,17] and their role in fields as diverse as quantum computation [8,18,19], quantum phase discrimination [20] and quantum thermodynamics [21] attracted considerable attenti...

To describe certain facets of non-classicality, it is necessary to quantify properties of operations instead of states. This is the case if one wants to quantify how well an operation detects non-classicality, which is a necessary prerequisite for its use in quantum technologies. To do so rigorously, we build resource theories on the level of operations, exploiting the concept of resource destroying maps. We discuss the two basic ingredients of these resource theories, the free operations and the free super-operations, which are sequential and parallel concatenations with free operations. This leads to defining properties of functionals that are well suited to quantify the resources of operations. We introduce these concepts at the example of coherence. In particular, we present two measures quantifying the ability of an operation to detect, i.e. to use, coherence, one of them with an operational interpretation, and provide methods to evaluate them. arXiv:1806.07332v2 [quant-ph]

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The resource theory of coherence [1-6] studies the operational value of superpositions in quantum technologies. A key question in this theory concerns the efficiency of manipulation and inter-conversion [3, 7-10] of the resource. Here we solve this question completely for qubit states by determining the optimal probabilities for mixed state conversions via stochastic incoherent operations. Extending the discussion to distributed scenarios, we introduce and address the task of assisted incoherent state conversion where the process is enhanced by making use of correlations with a second party. Building on these results, we demonstrate experimentally that the optimal state conversion probabilities can be achieved in a linear optics set-up. This paves the way towards real world applications of coherence transformations in current quantum technologies.Practical constraints on our ability to manipulate physical systems restrict the control we can exert on them. It is, for example, exceedingly difficult to exchange quantum systems undisturbed over long distances [11]. When manipulating spatially separated subsystems, effectively, this limits us to Local Operations and Classical Communication (LOCC). Under these operations, it is only possible to prepare certain states, i.e. separable ones. The states which cannot be produced under LOCC are called entangled and elevated to resources: Consuming them allows to implement operations such as quantum state teleportation [12] to achieve perfect quantum state transfer which would not be possible with LOCC alone. This has important consequences, e.g. in quantum communication and other quantum technologies, but also for our understanding of the view of the fundamental laws of nature [11,[13][14][15].Entanglement is explored within the framework of quantum resource theories, which can also be used to investigate other non-classical features of quantum mechanics in a systematic way. A concept underlying many * These authors contributed equally to this work. †

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