Abstract The resource theory of superposition extends resource theories of coherence, offering a rigorous framework to quantify the superposition within a finite set of linearly independent states. Based on this theory, we propose and analyze several legitimate superposition measures, including the superposition of formation (SOF) and the logarithmic measure, demonstrating that both satisfy the required conditions for a valid superposition measure. We then investigate the intriguing properties of various superposition measures. We find that the relative entropy of superposition cannot be expressed as a closed form, and it is superadditive under tensor products and monotone under partial trace. Furthermore, we show that the SOF also exhibits superadditive properties. It is shown that the upper bound of the relative entropy of superposition measure is determined by the SOF. In contrast, we find that the logarithmic superposition measure exhibits subadditive properties, and its regularized version also exist. Additionally, we discuss the rank-measure of superposition and show that it satisfies additivity. Finally, we investigate the general quantum resource measure via the standard convex roof construction in quantum resource theory, and some properties of a general resource measure are also discussed.
Abstract The resource theory of superposition extends resource theories of coherence, offering a rigorous framework to quantify the superposition within a finite set of linearly independent states. Based on this theory, we propose and analyze several legitimate superposition measures, including the superposition of formation (SOF) and the logarithmic measure, demonstrating that both satisfy the required conditions for a valid superposition measure. We then investigate the intriguing properties of various superposition measures. We find that the relative entropy of superposition cannot be expressed as a closed form, and it is superadditive under tensor products and monotone under partial trace. Furthermore, we show that the SOF also exhibits superadditive properties. It is shown that the upper bound of the relative entropy of superposition measure is determined by the SOF. In contrast, we find that the logarithmic superposition measure exhibits subadditive properties, and its regularized version also exist. Additionally, we discuss the rank-measure of superposition and show that it satisfies additivity. Finally, we investigate the general quantum resource measure via the standard convex roof construction in quantum resource theory, and some properties of a general resource measure are also discussed.
The resource theory of quantum superposition is an extension of the quantum coherent theory, in which linear independence relaxes the requirement of orthogonality. It can be used to quantify the nonclassical in superposition of finite number of optical coherent states. Based on convex roof extended, state transformation and weight, we give three methods of constructing superposition measures of quantum states, respectively. We also generalize the superposition resource theory from two perspectives.
Abstract We introduce a framework unifying the mathematical characterisation of different measures of general quantum resources and allowing for a systematic way to define a variety of faithful quantifiers for any given convex quantum resource theory. The approach allows us to describe many commonly used measures such as matrix norm-based quantifiers, robustness measures, convex roof-based measures, and witness-based quantifiers together in a common formalism based on the convex geometry of the underlying sets of resource-free states. We establish easily verifiable criteria for a measure to possess desirable properties such as faithfulness and strong monotonicity under relevant free operations, and show that many quantifiers obtained in this framework indeed satisfy them for any considered quantum resource. We derive various bounds and relations between the measures, generalising and providing significantly simplified proofs of results found in the resource theories of quantum entanglement and coherence. We also prove that the quantification of resources in this framework simplifies for pure states, allowing us to obtain more easily computable forms of the considered measures, and show that many of them are in fact equal on pure states. Further, we investigate the dual formulation of resource quantifiers, which provide a characterisation of the sets of resource witnesses. We present an explicit application of the results to the resource theories of multi-level coherence, entanglement of Schmidt number k , multipartite entanglement, as well as magic states, providing insight into the quantification of the four resources by establishing novel quantitative relations and introducing new quantifiers, such as a measure of entanglement of Schmidt number k which generalises the convex roof–extended negativity, a measure of k -coherence which generalises the ℓ 1 norm of coherence, and a hierarchy of norm-based quantifiers of k -partite entanglement generalising the greatest cross norm.
The superposition principle lies at the heart of many nonclassical 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 specialized 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.
The quantification of state-based quantum ``resources'' such as entanglement, coherence, and nonstabilizer states lies at the heart of quantum science and technology, providing potential advantages over classical methods. In a realistic scenario, due to the imperfections and uncertainties in physical devices, we are unable to perfectly prepare or detect the true quantum states. Consequently, it is necessary to study the quantification of quantum resources under such circumstances. In this work, by focusing on the state-based quantum resource theory, we introduce a family of resource measures called $\ensuremath{\epsilon}$ measure that relies on a precision parameter to address this issue. This family of resource measures inherits the fundamental properties of the original resource measure, such as weak monotonicity, convexity, monogamy, and so forth. Furthermore, the $\ensuremath{\epsilon}$ measure remains continuous irrespective of whether the original measure is continuous or not. We also investigate the $\ensuremath{\epsilon}$ measure of distance-based resource quantifiers, and some interesting properties are presented. As part of the applications, we derived several formulas for the $\ensuremath{\epsilon}$ measure of coherence, nonstabilizerness, asymmetry, nonuniformity, and imaginarity. Additionally, we offered an upper bound for the $\ensuremath{\epsilon}$ measure of the resource rank. Finally, we outline how this work can be extended to channel-based resources among others.
Chapter 10 delves into the quantification of quantum resources, an essential aspect of quantum resource theories that determines the value of quantum states for specific applications. It begins by defining resource measures and investigating their fundamental properties such as monotonicity under free operations and convexity. The chapter discusses distance-based resource measures, which quantify how far a given quantum state is from the set of free states. Such measures often utilize divergences and metrics explored in earlier chapters. Techniques to compute the relative entropy of a resource are also covered. To refine resource measures, the chapter introduces the concept of smoothing, which considers small deviations from the ideal state to make the measures more robust against perturbations. This approach is crucial in single-shot scenarios where finite resources are available. Furthermore, the chapter examines resource monotones and support functions, offering a comprehensive framework for the theoretical and practical assessment of quantum resources.
To describe the nonclassicality of quantum channels, it is necessary to quantify their properties. In the framework of resource theory of quantum channel, we present three new entanglement measures of channels for a quantum system of arbitrary finite dimension, based on the Choi relative entropy of channels, concurrence, and k-ME concurrence, respectively. Rigorous proofs show that these proposed measures fulfill all the requirements dictated by the resource theory of quantum channels, including non-negativity, weak monotonicity, strong monotone, convexity, additivity, and subadditivity. By using the measure, we describe the probability that the initial hypothesis is true in the hypothesis testing of channels. Compared with other known measures of the quantum channels, these measures have an obvious advantage in computability. According to the power of generating entanglement, we give a classification of quantum channels. Moreover, we provide several examples that characterize the properties of quantum channels and the conversions between quantum coherence states and entangled states and between fully separable states and multipartite entangled states.