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 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.

arXiv:1703.10943Physical Review LettersDec 5, 2017141 citations

Superposition measures in quantum resource theories

Hai‐Bo Xing

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.

Physica ScriptaAug 1, 20250 citations

A theoretical justification for the superposition principle

Mohammad Barzegar-Jalali

International Journal of PharmaceuticsJan 1, 19832 citations

Optimal Superposition Enhancement of a Mixed State by Stochastic Diagonal Superposition-Free Operations

Hai‐Bo Xing

The resource theory of superposition is a generalization of coherence theory. A rigorous framework for quantifying superposition of linear independent states has been given by previous studies. In this paper, the optimal superposition enhancement of a mixed state is studied. Based on the $l_{1}$ measure of superposition the optimal superposition enhancement value of a mixed state can be obtained under the stochastic diagonal superposition-free operations(or incomplete diagonal superposition-free operations). In addition, the optimal probability corresponding to the optimal superposition enhancement value of a mixed state is also discussed. The results show that although the superposition-free operations can decrease the superposition of a state, the superposition enhancement of a state is possible, a single mixed state can be probabilistically transformed into another state with higher superposition than before by superposition-free operations. These studies will give us a better understanding of the superposition manipulation and the resource theory of superposition.The resource theory of superposition is a generalization of coherence theory. A rigorous framework for quantifying superposition of linear independent states has been given by previous studies. In this paper, the optimal superposition enhancement of a mixed state is studied. Based on the $l_{1}$ measure of superposition the optimal superposition enhancement value of a mixed state can be obtained under the stochastic diagonal superposition-free operations(or incomplete diagonal superposition-free operations). In addition, the optimal probability corresponding to the optimal superposition enhancement value of a mixed state is also discussed. The results show that although the superposition-free operations can decrease the superposition of a state, the superposition enhancement of a state is possible, a single mixed state can be probabilistically transformed into another state with higher superposition than before by superposition-free operations. These studies will give us a better understanding of the superposition manipulation and the resource theory of superposition.

Results in PhysicsJan 1, 20221 citation

l_{1}

measure of superposition the optimal superposition enhancement value of a mixed state can be obtained under the stochastic diagonal superposition-free operations(or incomplete diagonal superposition-free operations). In addition, the optimal probability corresponding to the optimal superposition enhancement value of a mixed state is also discussed. The results show that although the superposition-free operations can decrease the superposition of a state, the superposition enhancement of a state is possible, a single mixed state can be probabilistically transformed into another state with higher superposition than before by superposition-free operations. These studies will give us a better understanding of the superposition manipulation and the resource theory of superposition.