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Laser Physics Letters

Quantum imaginarity measure sensitivity to deimaginarizing maps

Peng-Fei Wang, Xu-Zheng Lu, Zhihua Guo, Chengyang Zhang

Published Apr 30, 2026• 0 citations
Publisher

Abstract

Abstract In this paper, we introduce a measure of quantum imaginarity based on the deimaginarization strength exerted by a class of quantum operations on quantum states, and derive an upper bound for this measure. For the qubit scenario, we establish a tight upper bound on the sum of imaginarity measures with respect to three mutually unbiased bases. Finally, we investigate the decay behavior of quantum imaginarity for maximally imaginary states under the action of the bit-flip, phase-damping, and amplitude-damping channels.

DOI
10.1088/1612-202x/ae631e

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Abstract In this paper, we introduce a measure of quantum imaginarity based on the deimaginarization strength exerted by a class of quantum operations on quantum states, and derive an upper bound for this measure. For the qubit scenario, we establish a tight upper bound on the sum of imaginarity measures with respect to three mutually unbiased bases. Finally, we investigate the decay behavior of quantum imaginarity for maximally imaginary states under the action of the bit-flip, phase-damping, and amplitude-damping channels.
We investigate the trade-off relations between imaginarity and mixedness in arbitrary ddd-dimensional quantum systems. For given mixedness, a quantum state with maximum imaginarity is defined to be a "maximally imaginary mixed state" (MIMS). By using the l1l_{1}l1​ norm of imaginarity and the normalized linear entropy, we conclusively identify the MIMSs for both qubit and qutrit systems. For high-dimensional quantum systems, we present a comprehensive class of MIMSs, which also gives rise to complementarity relations between the 111-norm of imaginarity and the 111-norm of mixedness, as well as between the relative entropy of imaginarity and the von Neumann entropy. Furthermore, we examine the evolution of the trade-off relation for single-qubit states under four specific Markovian channels: bit flip channel, phase damping channel, depolarizing channel and amplitude damping channel.
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 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.
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.
Quantum imaginarity is an essential physical resource that underpins key functionalities of modern quantum technologies. We improve imaginarity witnessing via the prior knowledge of imaginarity-witness operators. To this end, we derive a rigorous upper bound on the maximum expectation value of an imaginarity witness operator over the set of all free (real) quantum states, which is given by the spectral norm of the real component of the corresponding witness operator. We demonstrate via detailed examples that this bound substantially improves imaginarity detection. We further classify all imaginarity-witness operators into four distinct families based on this bound. For these four witness classes, we perform a comprehensive analysis of their completeness and finite completeness, the joint detection of shared imaginary quantum states by different witnesses, and the conditions for distinct witnesses to identify identical imaginary states. Our results advance the fundamental understanding of imaginarity detection and offer useful insights for both theoretical studies and experimental implementations of quantum imaginarity.