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Journal of Physics A Mathematical and Theoretical

Freezing imaginarity measures with real operations

Bingke Zheng, Zhihua Guo, Chengyang Zhang, Huaixin Cao

Published Jan 10, 2025• 5 citations
Publisher

Abstract

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.

DOI
10.1088/1751-8121/ada8e8

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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.
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 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, well‐defined measures of imaginarity are proposed using various distance metrics, drawing inspiration from recent advancements in quantum entanglement and coherence. Specifically, quantitatively evaluating imaginarity is focused through measures such as Tsallis relative ‐entropy, Sandwiched Rényi relative entropy, and Tsallis relative operator entropy. Additionally, the decay rates of these measures are analyzed. These findings reveal that the Tsallis relative ‐entropy of imaginarity exhibits higher decay rate under quantum channels compared to other measures. Finally, the ordering of single‐qubit states are examined under these imaginarity measures, demonstrating that the order remains invariant under the bit‐flip channel for specific parameter ranges. This study enhances the understanding of imaginarity as a quantum resource and its potential applications in quantum information theory.
Recent studies have shown that Bargmann invariants provide effective detectors of set imaginarity. In this paper, we investigate set imaginarity as a quantum resource in qubit systems. By exploiting the structure of Bargmann invariants, we show that the free operations for qubit set imaginarity consist precisely of common unital operations and common planarized operations. Based on this characterization, we introduce an axiomatic framework for set-imaginarity measures (SIMs). In particular, we propose two refined notions, namely unified SIMs and complete SIMs, which allow a more fine-grained quantification of set imaginarity. To make these notions concrete, we construct two qubit SIMs from the Bargmann invariants of three-state subsets. We prove that one of them is a unified SIM, while the other satisfies the stronger requirements of a complete SIM. Furthermore, we revisit the robustness of set imaginarity previously introduced in the literature. We show that, although this robustness is a valid SIM for qubit systems, it is neither a unified SIM nor a complete SIM. To overcome this limitation, we propose an improved robustness-type measure and rigorously prove that it defines a complete qubit SIM.
Abstract Imaginarity has proven to be a valuable resource in various quantum information processing tasks. A natural question arises: can the imaginarity of quantum states be broadcast via real operations? In this work, we present explicit structures for nonreal states whose imaginarity can be broadcast and cloned. That is, for a nonreal state, its imaginarity can be cloned if and only if it is a direct sum of several maximally imaginary states under orthogonal transformation, and its imaginarity can be broadcast if and only if it is a direct sum of a real state and some nonreal qubit states which are mixtures of two orthogonal maximally imaginary states under orthogonal transformation. In particular, we show that for a nonreal pure state, its imaginarity cannot be broadcast unless it is a maximally imaginary state. Furthermore, we derive a trade-off relation on the imaginarity broadcasting of pure states in terms of the measure of irreversibility of quantum states concerning real operations and the geometric measure of imaginarity. In addition, we demonstrate that any faithful measure of imaginarity is not superadditive.
This work studies the decay of three imaginarity measures, namely the l1l_1l1​-norm-based imaginarity, the robustness of imaginarity, and the relative-entropy-based imaginarity, under typical noise channels. For arbitrary single-qubit pure states subject to real quantum channels, we prove an exact attenuation theorem: the l1l_1l1​-norm-based imaginarity and the robustness exhibit identical attenuation governed by a common channel-dependent factor, whereas the relative-entropy-based imaginarity additionally depends on the input-state orientation relative to the reference basis. We extend the investigation to two-qubit entangled and dual-rail states, establishing a quantitative link between photon-loss probability and residual imaginarity for photonic dual-rail encodings. For separable two-qubit states, we define a real-operation resource preorder and show that Ω++=∣+⟩⟨+∣⊗∣+⟩⟨+∣Ω_{++}=|+\rangle\langle+|\otimes|+\rangle\langle+|Ω++​=∣+⟩⟨+∣⊗∣+⟩⟨+∣ is operationally maximal. We further distinguish operational de-imaginary power from reference-state imaginarity loss. For several real two-qubit channels we derive exact operational de-imaginary powers, while for other channels we provide analytic reference-state losses serving as benchmarks and rigorous lower bounds.