Superposition Measures in Quantum Resource Theories

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.