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.
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.
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.
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.
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.
Wave-particle duality is one of the basic features of quantum mechanics, giving rise to the use of complex numbers in describing states of quantum systems and their dynamics and interaction. Since the inception of quantum theory, it has been debated whether complex numbers are essential or whether an alternative consistent formulation is possible using real numbers only. Here, we attack this long-standing problem theoretically and experimentally, using the powerful tools of quantum resource theories. We show that, under reasonable assumptions, quantum states are easier to create and manipulate if they only have real elements. This gives an operational meaning to the resource theory of imaginarity. We identify and answer several important questions, which include the state-conversion problem for all qubit states and all pure states of any dimension and the approximate imaginarity distillation for all quantum states. As an application, we show that imaginarity plays a crucial role in state discrimination, that is, there exist real quantum states that can be perfectly distinguished via local operations and classical communication but that cannot be distinguished with any nonzero probability if one of the parties has no access to imaginarity. We confirm this phenomenon experimentally with linear optics, discriminating different two-photon quantum states by local projective measurements. Our results prove that complex numbers are an indispensable part of quantum mechanics.
Abstract The resource‐theoretic frameworks for quantum imaginarity have been developed in recent years. Within these frameworks, many imaginarity measures for finite‐dimensional systems have been proposed. However, for imaginarity of Gaussian states in continuous‐variable (CV) systems, there are only two known Gaussian imaginarity measures, which exhibit prohibitive computational complexity when applied to multi‐mode Gaussian states. In this paper, a computable Gaussian imaginarity measure is proposed for ‐mode Gaussian systems. The value of is simply formulated by the displacement vectors and covariance matrices of Gaussian states. A comparative analysis of with existing two Gaussian imaginarity measures indicates that can be used to detect imaginarity in any ‐mode Gaussian states more efficiently. As an application, the dynamics behavior of ‐mode Gaussian states is studied in Gaussian Markovian noise environments for two‐mode CV system by utilizing . Moreover, it is proved that, can induce a quantification of any ‐multipartite multi‐mode CV systems which satisfies all requirements for measures of multipartite multi‐mode Gaussian correlations, which unveils that, ‐mode Gaussian imaginarity can also be regarded as a kind of multipatite multi‐mode Gaussian correlation and is a multipartite Gaussian quantum resource.
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.