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arXiv:2605.14299

Imaginarity Resource Theory of Gaussian Quantum Channels

Ting Zhang, Jinchuan Hou, Xiaofei Qi

quant-phSubmitted May 14, 2026• Announced May 15, 2026• 0 citations
arXivHTMLPDF

Abstract

Complex numbers play an indispensable role in quantum mechanics and quantum information, as validated by both theoretical analysis and experimental verification. Since quantum information processing inherently relies on quantum channels, the resource theory for quantum channels is equally fundamental to that for quantum states. In this paper, we propose two frameworks for quantifying the imaginarity of Gaussian channels. The first framework regards all real superchannels as free superchannels. Within this setting, we introduce two concrete imaginarity measures for Gaussian channels: I_s^GC based on existing imaginarity measures of Gaussian states, and I_d^GC derived directly from the intrinsic parameters of Gaussian channels, which enjoys high computational simplicity. The second framework adopts only a proper subset of real superchannels as free superchannels. Under this framework, we put forward another imaginarity measure I_c^GC , which is fully determined by the inherent parameters of Gaussian channels and features continuity as well as tractable computation. As a practical application, we employ I_c^GC to investigate the dynamical behavior of Quantum Brownian Motion Gaussian channels throughout the entire evolutionary process.

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Complex numbers play an indispensable role in quantum mechanics and quantum information, as validated by both theoretical analysis and experimental verification. Since quantum information processing inherently relies on quantum channels, the resource theory for quantum channels is equally fundamental to that for quantum states. In this paper, we propose two frameworks for quantifying the imaginarity of Gaussian channels. The first framework regards all real superchannels as free superchannels. Within this setting, we introduce two concrete imaginarity measures for Gaussian channels: I_s^GC based on existing imaginarity measures of Gaussian states, and I_d^GC derived directly from the intrinsic parameters of Gaussian channels, which enjoys high computational simplicity. The second framework adopts only a proper subset of real superchannels as free superchannels. Under this framework, we put forward another imaginarity measure I_c^GC , which is fully determined by the inherent parameters of Gaussian channels and features continuity as well as tractable computation. As a practical application, we employ I_c^GC to investigate the dynamical behavior of Quantum Brownian Motion Gaussian channels throughout the entire evolutionary process.
It is a long-standing debate that why quantum mechanics uses complex numbers but not real numbers only. To address this topic, in recent years, the imaginarity theory has been developed in the way of quantum resource theory. However, the existing imaginarity theory mainly focuses on the quantum systems with finite dimensions. Gaussian states are widely used in many fields of quantum physics, but they are in the quantum systems with infinite dimensions. In this paper we establish a resource theory of imaginarity for bosonic Gaussian states. To do so, under the Fock basis, we determine the real Gaussian states and real Gaussian channels in terms of the means and covariance matrices of Gaussian states. Also, we provide two imaginary measures for Gaussian states based on the fidelity.
Imaginarity, stemming from the complex structure of quantum mechanics, has recently emerged as a fundamental resource, yet its dynamical generation remains largely unexplored. In this work, we introduce the notion of imaginarity-generating power (IGP) of unitary dynamics, which quantifies the ability of unitary operations to produce imaginarity from initially real quantum states. To quantify imaginarity, we employ a measure based on the Hilbert--Schmidt norm, which we show to be monotone under real unital operations. Within the framework of dynamical resource theories, we derive an exact expression for the purity-constrained IGP in arbitrary dimensions and show that, for pure real input states, it depends solely on intrinsic and experimentally accessible properties of the unitary. We further analyze its average behavior over ensembles of states with varying purity under both uniform and Hilbert--Schmidt distributions. We prove that it satisfies the essential properties of a valid resource monotone within the dynamical resource theory of imaginarity. We also characterize the unitaries that maximize the IGP and determine the corresponding bounds. Moreover, for Haar-random unitaries, we show that the IGP concentrates near its maximal value in high dimensions with small fluctuations, indicating that typical high-dimensional quantum dynamics are highly effective at generating imaginarity.
Complex numbers are widely used in quantum physics and are indispensable components for describing quantum systems and their dynamical behavior. The resource theory of imaginarity has been built recently, enabling a systematic research of complex numbers in quantum information theory. In this work, we develop two theoretical methods for quantifying imaginarity, motivated by recent progress within resource theories of entanglement and coherence. We provide quantifiers of imaginarity by the convex roof construction and quantifiers of the imaginarity by the least imaginarity of the input pure states under real operations. We also apply these tools to study the state conversion problem in the resource theory of imaginarity.
Complex numbers are widely used in both classical and quantum physics and are indispensable components for describing quantum systems and their dynamical behavior. Recently, the resource theory of imaginarity has been introduced, allowing for a systematic study of complex numbers in quantum mechanics and quantum information theory. In this work we develop theoretical methods for the resource theory of imaginarity, motivated by recent progress within theories of entanglement and coherence. We investigate imaginarity quantification, focusing on the geometric imaginarity and the robustness of imaginarity, and apply these tools to the state conversion problem in imaginarity theory. Moreover, we analyze the complexity of real and general operations in optical experiments, focusing on the number of unfixed wave plates for their implementation. We also discuss the role of imaginarity for local state discrimination, proving that any pair of real orthogonal pure states can be discriminated via local real operations and classical communication. Our study reveals the significance of complex numbers in quantum physics and proves that imaginarity is a resource in optical experiments.
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 It is a fundamental question why quantum mechanics employs complex numbers rather than solely real numbers. In this work, we conduct the first analysis of imaginarity quantification in neutrino flavor and spin-flavor oscillations. As quantum systems in coherent superposition, neutrinos are ideal candidates for quantifying imaginarity within the resource theoretic framework, using measures such as the ℓ 1 -norm and the relative entropy of imaginarity. We show that in the case of two-flavor mixing, these measures of imaginarity are nonzero. The measures of imaginarity reach their extreme values when the probabilistic features of quantum theory are fully maximized, i.e., both the transitional and survival probabilities are approximately equal. Our study reveals that the imaginarity, as a resource, can be harnessed not solely from the presence of a complex phase in the mixing matrix but also from the intrinsic quantum dynamics of time evolution itself. We further extend our analysis to explore the dynamics of three-flavor neutrino mixing, incorporating the effects of a nonzero CP phase.