Abstract The pivotal role of complex numbers in quantum mechanics underpins the resource theory of imaginarity. We investigate imaginarity dynamics in a single-qubit open system coupled to a non-Markovian environment. Crucially, cavity field detuning emerges as the dominant regulator, driving continuous conversion between the real and imaginary components of coherence. Nonzero detuning induces characteristic non-periodic oscillations of imaginarity between zero and maximal values, preventing complete decoherence at specific times. We establish that imaginarity resources stem from both intrinsic system evolution and environmental feedback. Significantly, detuning-driven imaginarity generation persists even in Markovian regimes, demonstrating its origin beyond environmental memory effects. These insights offer new perspectives for understanding and harnessing quantum coherence.
Abstract The pivotal role of complex numbers in quantum mechanics underpins the resource theory of imaginarity. We investigate imaginarity dynamics in a single-qubit open system coupled to a non-Markovian environment. Crucially, cavity field detuning emerges as the dominant regulator, driving continuous conversion between the real and imaginary components of coherence. Nonzero detuning induces characteristic non-periodic oscillations of imaginarity between zero and maximal values, preventing complete decoherence at specific times. We establish that imaginarity resources stem from both intrinsic system evolution and environmental feedback. Significantly, detuning-driven imaginarity generation persists even in Markovian regimes, demonstrating its origin beyond environmental memory effects. These insights offer new perspectives for understanding and harnessing quantum coherence.
Abstract The use of imaginary numbers in modelling quantum mechanical systems encompasses the wave-like nature of quantum states. Here we introduce a resource theoretic framework for imaginarity, where the free states are taken to be those with density matrices that are real with respect to a fixed basis. This theory is closely related to the resource theory of coherence, as it is basis dependent, and the imaginary numbers appear in the off-diagonal elements of the density matrix. Unlike coherence however, the set of physically realizable free operations is identical to both completely resource non-generating (RNG) operations, and stochastically RNG operations. Moreover, the resource theory of imaginarity does not have a self-adjoint resource destroying map. After introducing and characterizing the free operations, we provide several measures of imaginarity, and give necessary and sufficient conditions for pure state transformations via physically consistent free operations in the single shot regime.
We theoretically investigate the non-Markovian dynamical decoherence of a quantum system coupled to nonequilibrium environments with nonstationary statistical properties. We show the time evolution of the decoherence factor in real-imaginary space to study the environment-induced energy renormalization and backaction of coherence which are associated with the unitary and nonunitary parts of the quantum master equation, respectively. It is also shown that the nonequilibrium decoherence dynamics displays a transition between Markovian and non-Markovian and the transition boundary depends on the environmental parameters. The results are helpful for further understanding non-Markovian dynamics and coherence backaction on an open quantum system from environments.
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.
The precise characterization of dynamics in open quantum systems often presents significant challenges, leading to the introduction of various approximations to simplify a model. One commonly used strategy involves Markovian approximations, assuming a memoryless environment. In this study, such approximations are not used and an analytical dynamical depiction of an open quantum system is provided. The system under consideration is an oscillator that is surrounded by a bath of oscillators. The resulting dynamics are characterized by a second-order complex coefficient linear differential equation, which may be either homogeneous or inhomogeneous. Moreover, distinct dynamical regions emerge, depending on certain parameter values. Notably, the steady-state average excitation number (AEN) of the system shows rapid escalation with increasing non-Markovianity, reflecting the intricacies of real-world dynamics. In cases where there is detuning between the system frequency and the environmental central frequency within a non-Markovian regime, the AEN maintains its initial value for an extended period. Furthermore, the application of pulse control can effectively protect the quantum system from decoherence effects without using approximations. The pulse control can not only prolong the relaxation time of the oscillator, but can also be used to speed up the relaxation process, depending on the specifications of the pulse. By employing a kick pulse, the Mpemba effect can be observed in the non-Markovian regime in a surprisingly super-cooling-like effect.
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.
We establish a rigorous framework that identifies imaginarity as a fundamental resource inherent in quantum coherence. By means of a geometric decomposition, we partition coherence into distinct imaginarity and residual components, thereby revealing a universal hierarchical relationship among these resources. For bipartite systems, this decomposition provides explicit bounds on the extent to which nonlocal correlations and imaginarity limit local coherence generation. Furthermore, we devise an explicit operational protocol that converts imaginarity into usable coherence, demonstrating the direct interconvertibility of these resources under physically admissible operations. The dynamical evolution under diagonal Hamiltonians is fully characterized, showing that while total coherence is conserved, imaginarity and residual coherence exhibit complementary oscillations. Our results provide a rigorous geometric and operational characterization of imaginarity as a fundamental constituent of quantum coherence, offering concrete insights for resource management in distributed quantum technologies. The geometric framework and theoretical bounds established herein are fully general, while the explicit conversion protocol and dynamical analysis serve as a compelling proof-of-principle demonstration in qubit systems.