Recent results in quantum information theory characterize quantum coherence in the context of resource theories. Here, we study the relation between quantum coherence and quantum discord, a kind of quantum correlation which appears even in nonentangled states. We prove that the creation of quantum discord with multipartite incoherent operations is bounded by the amount of quantum coherence consumed in its subsystems during the process. We show how the interplay between quantum coherence consumption and creation of quantum discord works in the preparation of multipartite quantum correlated states and in the model of deterministic quantum computation with one qubit.

A major signature of quantum mechanics beyond classical physics is coherence, the existence of superposition states. The recently developed resource theory of quantum coherence allows the formalisation of incoherent operations -those operations which cannot create coherence. We identify the set of operations which additionally do not use coherence. These are such that coherence cannot be exploited by a classical observer, who measures incoherent properties of the system, to go beyond classical dynamics. We give a physical interpretation in terms of interferometry and prove a dilation theorem, showing how these operations can always be constructed by interacting the system in an incoherent way with an ancilla. Such a physical justification is not known for the incoherent operations, thus our results lead to a physically well-motivated resource theory of coherence. Next, we investigate the implications for coherence in multipartite systems. We show that quantum correlations can be defined naturally with respect to a fixed basis, providing a link between coherence and quantum discord. We demonstrate the interplay between these two quantities under our studied operations, and suggest implications for the theory of quantum discord by relating the studied operations to those which cannot create discord. * benjamin.yadin@physics.ox.ac.uk †

We propose a universal language to assess macroscopic quantumness in terms of coherence, with a set of conditions that should be satisfied by any measure of macroscopic coherence. We link the framework to the resource theory of asymmetry. We show that the quantum Fisher information gives a good measure of macroscopic coherence, enabling a rigorous justification of a previously proposed measure of macroscopicity. This picture lets us draw connections between different measures of macroscopicity and evaluate them; we show that another widely studied measure fails one of our criteria. PACS numbers: 03.65.Ta, 03.67.MnIntroduction.-One of the most difficult challenges in quantum theory is to explain how it can be compatible with the classical world observed at the macroscopic scale. We are accustomed to the fact that microscopic systems can exist in superposition states, but quantum theory also allows this behaviour in principle at the macroscopic scale. Various attempts have been made to address this; one common approach may be broadly termed 'decoherence', in which the quantumness of a large system is highly susceptible to being destroyed by interactions with a noisy environment [1]; similarly, it has been argued that classical behaviour emerges from limitations on the precision of measurements [2][3][4]. Others have suggested fundamental modifications to quantum theory that are negligible at microscopic scales but cause macroscopic superpositions to quickly appear classical [5].Ultimately, it is up to experiments to probe the boundary between the quantum and classical worlds, and much progress has been made in this direction in recent decades [6][7][8]. A wide variety of systems have been explored, including interference of large molecules [9, 10], superpositions of coherent states in photonic systems [11], superconducting circuits behaving as large-scale qubits [12], and the control of low-lying vibrational states of micromechanical oscillators [13].Given this diversity, we need a general way to quantify how well each experiment has achieved its aim of creating large-scale quantum coherence. Beginning with ideas by Leggett [6,14], a variety of measures of 'quantum macroscopicity' have been proposed, each motivated differently and aiming to capture a different potentially macroscopic quantum property. Some assume that the given state is a superposition of the form ψ 0 ⟩ + ψ 1 ⟩ and quantify to what extent the two branches are macroscopically distinct [15][16][17] or useful for interferometry [18]. Others apply to more general states of systems of many qubits [19], continuous-variable modes [20] and mechanical objects [21].These various approaches do not share any precise unifying principles. Inspired by recent work on quantifying

Genuinely quantum states of a harmonic oscillator may be distinguished from their classical counterparts by the Glauber-Sudarshan P-representation -a state lacking a positive P-function is said to be nonclassical. In this paper, we propose a general operational framework for studying nonclassicality as a resource in networks of passive linear elements and measurements with feed-forward. Within this setting, we define new measures of nonclassicality based on the quantum fluctuations of quadratures, as well as the quantum Fisher information of quadrature displacements. These lead to fundamental constraints on the manipulation of nonclassicality, especially its concentration into subsystems, that apply to generic multi-mode non-Gaussian states. Special cases of our framework include no-go results in the concentration of squeezing and a complete hierarchy of nonclassicality for single mode Gaussian states.