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arXiv:2404.00637 · Reports on Mathematical Physics

Imaginarity Measures Induced by Relative Entropy

Xiangyu Chen, Qiang Lei

Published Feb 1, 2025• 6 citations
arXivHTMLPDF
DOI
10.1016/s0034-4877(25)00003-5

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Abstract Complex numbers provide a remarkably powerful formalism in quantum mechanics. In this paper, we propose a method to construct imaginarity measures by real part states. Specifically, we propose an imaginarity measure in terms of fidelity and explore its properties. The analytical expression of the imaginarity measure is presented in qubit systems. The relations between the proposed imaginarity measure and some other imaginarity measures (such as geometric imaginarity, Tsallis relative entropy imaginarity and trace norm imaginarity) are derived. The complementarity relations of the imaginarity measure under a complete set of mutually unbiased bases (MUBs) are provided in low-dimensional systems. This work not only highlights the prominent role of the real part state in the imaginarity resource theory, but also reveals the constraint of imaginarity on a complete set of MUBs physically.

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Freezing imaginarity measures with real operations

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Abstract Imaginarity plays an essential role in many quantum information processing tasks. Usually, real operations do not increase imaginarity measures. In present work, we aim to discuss freezing of imaginarity measures with real operations. It is proved that a real operation Φ freezes all imaginarity measures of a state ρ if and only if it freezes the relative entropy measure of imaginarity of the state ρ . In the qubit case, real operations that freeze imaginarity measures based on trace distance and relative entropy are characterized, respectively. Moreover, the freezing of imaginarity measure based on trace distance is not necessarily dilated into multipartite systems, while that based on the Tsallis relative entropy can be dilated into multipartite systems. Finally, we characterize real local operations that freeze the imaginarity measure based on trace distance of a special type of X-states in a multi-qubit system.

Journal of Physics A Mathematical and Theoretical•Jan 10, 2025•5 citations
Abstract Complex numbers provide a remarkably powerful formalism in quantum mechanics. In this paper, we propose a method to construct imaginarity measures by real part states. Specifically, we propose an imaginarity measure in terms of fidelity and explore its properties. The analytical expression of the imaginarity measure is presented in qubit systems. The relations between the proposed imaginarity measure and some other imaginarity measures (such as geometric imaginarity, Tsallis relative entropy imaginarity and trace norm imaginarity) are derived. The complementarity relations of the imaginarity measure under a complete set of mutually unbiased bases (MUBs) are provided in low-dimensional systems. This work not only highlights the prominent role of the real part state in the imaginarity resource theory, but also reveals the constraint of imaginarity on a complete set of MUBs physically.
Abstract Imaginarity plays an essential role in many quantum information processing tasks. Usually, real operations do not increase imaginarity measures. In present work, we aim to discuss freezing of imaginarity measures with real operations. It is proved that a real operation Φ freezes all imaginarity measures of a state ρ if and only if it freezes the relative entropy measure of imaginarity of the state ρ . In the qubit case, real operations that freeze imaginarity measures based on trace distance and relative entropy are characterized, respectively. Moreover, the freezing of imaginarity measure based on trace distance is not necessarily dilated into multipartite systems, while that based on the Tsallis relative entropy can be dilated into multipartite systems. Finally, we characterize real local operations that freeze the imaginarity measure based on trace distance of a special type of X-states in a multi-qubit system.