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arXiv:2209.03532 · Results in Physics

Methods of constructing superposition measures

Jia-Lin Teng, Fengli Yan, Ting Gao

Published Sep 18, 2023• 1 citation
arXivHTMLPDF

Abstract

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.

DOI
10.1016/j.rinp.2023.106984

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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 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.
In this Letter, we detail an orthogonalization procedure that allows for the quantification of the amount of coherence present in an arbitrary superposition of coherent states. The present construction is based on the quantum coherence resource theory introduced by Baumgratz, Cramer, and Plenio and the coherence resource monotone that we identify is found to characterize the nonclassicality traditionally analyzed via the Glauber-Sudarshan P distribution. This suggests that identical quantum resources underlie both quantum coherence in the discrete finite dimensional case and the nonclassicality of quantum light. We show that our construction belongs to a family of resource monotones within the framework of a resource theory of linear optics, thus establishing deeper connections between the class of incoherent operations in the finite dimensional regime and linear optical operations in the continuous variable regime.
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.
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Quantum coherence, as a direct manifestation of the quantum superposition principle, is a crucial resource in quantum information processing. Block coherence resource theory generalizes the traditional coherence framework by defining coherence via a set of orthogonal projectors. Within this framework, we investigates the construction and comparison of block coherence measures. First, we propose two universal methods for constructing coherence measures and introduce a two-parameter family of measures based on the ααα-zzz Rényi relative entropy and a family of measures based on the Tsallis relative operator entropy. Second, through theoretical proofs and numerical counterexamples, we compares the ordering relations and numerical magnitudes among different block coherence measures and establishes a series of universal numerical inequalities to constrain their values. Besides, we also use Cα,1C_{α,1}Cα,1​ to show the role of coherence in complex dynamic evolution of the Kominis master equation that includes recombination reactions.