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 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.
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 Quantum imaginarity is one of the core quantum resources in the field of quantum information science. In this work, we define the quantum α – z Jensen-Tsallis divergence, and construct three kinds of imaginarity quantifiers based on it, by considering the distance from a quantum state to its conjugate state, real part, and the set of free states, respectively. We systematically investigate the fundamental properties of the above imaginarity quantifiers, and prove that the first and second kinds of divergences are well-defined quantum imaginarity measures, while the third one is an imaginarity monotone. Meanwhile, we investigate the intrinsic connections between them and the existing relative entropy-induced imaginarity measures. We discuss the evolutionary behavior of our imaginarity quantifiers with different parameters by typical examples.
Abstract Resource theory of imaginarity provides a useful framework to understand the role of complex numbers, which are essential in the formulation of quantum mechanics, in a mathematically rigorous way. In the first part of this article, we study the properties of ‘real’ (quantum) operations both in single-party and bipartite settings. As a consequence, we provide necessary and sufficient conditions for state transformations under real operations and show the existence of ‘real entanglement’ monotones. In the second part of this article, we focus on the problem of single copy state transformation via real quantum operations. When starting from pure initial states, we completely solve this problem by finding an analytical expression for the optimal fidelity of transformation, for a given probability of transformation and vice versa. Moreover, for state transformations involving arbitrary initial states and pure final states, we provide a semidefinite program to compute the optimal achievable fidelity, for a given probability of transformation.
This work studies the decay of three imaginarity measures, namely the l1-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 l1-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++=∣+⟩⟨+∣⊗∣+⟩⟨+∣ 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.
Quantum imaginarity and quantum nonlocality capture the essence of quantumness of a physical system from different aspects. We establish a connection between the imaginarity and quantum nonlocality in any two-qubit states. Based on the imaginarity l1-norm and relative entropy, we derive complementary relations among the quantum imaginarities with respect to any set of mutually unbiased bases for arbitrary qubit states. Based on these complementary relations, we introduce the nonlocal advantage of quantum imaginarity (NAQI). We show that any two-qubit state with NAQI is quantum steerable, but not vice versa. In addition, we investigate the exclusion property of NAQI in three-qubit pure states.