Quantifying the imaginarity via different distance measures (2501.07775v1)

Abstract: The recently introduced resource theory of imaginarity facilitates a systematic investigation into the role of complex numbers in quantum mechanics and quantum information theory. In this work, we propose well-defined measures of imaginarity using various distance metrics, drawing inspiration from recent advancements in quantum entanglement and coherence. Specifically, we focus on quantitatively evaluating imaginarity through measures such as Tsallis relative $\alpha$-entropy, Sandwiched R\'{e}nyi relative entropy, and Tsallis relative operator entropy. Additionally, we analyze the decay rates of these measures. Our findings reveal that the Tsallis relative $\alpha$-entropy of imaginarity exhibits higher decay rate under quantum channels compared to other measures. Finally, we examine the ordering of single-qubit states under these imaginarity measures, demonstrating that the order remains invariant under the bit-flip channel for specific parameter ranges. This study enhances our understanding of imaginarity as a quantum resource and its potential applications in quantum information theory.

• The paper presents novel imaginarity measures derived from entropic distances such as Tsallis and Rényi entropies to capture the imaginary part of quantum states.
• It evaluates the decay of these measures under noise channels like bit flip, phase damping, and amplitude damping, with the Tsallis measure showing notable resilience.
• A detailed state ordering analysis confirms invariant quantum state ranking under specific transformations, reinforcing the measures' reliability for quantum information processing.

Quantifying the Imaginarity via Different Distance Measures

The paper by Meng-Li Guo et al. presents a rigorous exploration of imaginarity as a quantum resource, focusing on the development of well-defined measures using diverse entropic distance metrics within the framework of quantum resource theory. This investigation is informed by recent advances in the paper of quantum entanglement and coherence, emphasizing the essential role of complex numbers in quantum mechanics. By proposing measures such as Tsallis relative $\alpha$-entropy, Sandwiched Rényi relative entropy, and Tsallis relative operator entropy, the authors contribute significant insights into the quantification of imaginarity, a relatively novel aspect of quantum resource theory.

Core Contributions

1. Imaginarity Measures: The authors introduce several innovative imaginarity measures based on different entropy types, specifically Tsallis relative $\alpha$-entropy, Sandwiched Rényi relative entropy, and Tsallis relative operator entropy. Each measure adheres to the foundational properties of quantum resource measures: nonnegativity, monotonicity under real operations, strong imaginarity monotonicity, and convexity. The authors present these measures analytically, ensuring they effectively characterize the imaginary component of quantum states.
2. Quantum Channel Effects: The paper meticulously examines how these imaginatory measures decay when subjected to various quantum channels, such as bit flip (BF), phase damping (PD), and amplitude damping (AD) channels. The findings suggest that the Tsallis relative $\alpha$-entropy of imaginarity is more resilient to quantum noise compared to the other measures, indicating its potential applicability in scenarios where quantum coherence and stability are paramount.
3. State Ordering Analysis: A detailed investigation into the ordering of single-qubit states under the proposed imaginatory measures is conducted. The paper demonstrates that, under certain conditions, the order of qubit states remains invariant when undergoing transformations through specific quantum channels, notably the bit-flip channel. This invariance reinforces the utility of these measures for reliable quantum information processing tasks.

Theoretical and Practical Implications

The theoretical implications of this work extend to enhancing our comprehension of complex numbers as indispensable elements in quantum systems. By establishing imaginatory as a quantifiable resource, the paper offers new perspectives on the functional role of complex numbers in quantum mechanics, furthering the foundational understanding of quantum systems' imaginary components.

From a practical standpoint, the resilience of the Tsallis relative $\alpha$-entropy of imaginatory under noisy conditions signifies its potential in maintaining information integrity in quantum communication systems. This robustness against decoherence suggests a viable path towards more stable quantum computational architectures, where information loss is minimized.

Future Directions

The paper paves the way for future research to explore more complex multi-qubit systems and a broader range of quantum channels. It would be worthwhile to investigate the applicability of these imaginatory measures in quantum protocols such as teleportation and cryptography, where quantum resources are critically leveraged. Additionally, a comparative analysis with other quantum resource theories could provide further insights into the universality and adaptability of these measures.

In summary, Meng-Li Guo and collaborators have substantially contributed to quantum mechanics by formalizing and advancing the concept of imaginatory as a quantum resource through innovative entropic measures. The potential applications of this research are vast, poised to influence both theoretical advancements and practical implementations in quantum information science.