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.
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.
Imaginarity has shown to be an important resource in quantum information. The witness theory of quantum resource, such as entanglement witness, coherence witness, and imaginarity witness, has been established, in particular entanglement witness and coherence witness have been extensively explored. We present here another class of imaginarity witnesses beyond the one in [{Phys. Lett. A} \textbf{530}, 130135 (2025)]. Within our framework, any nonreal Hermitian operator under the reference basis is an imaginarity witness and only finite of them can detect all the imaginarity states. As the common approaches that were explored in the witness theories of entanglement and coherence, we then explore the relations between these imaginarity witnesses in different cases: (i) when they can detect common imaginary states, (ii) when they can detect the same sets of imaginary states, and (ii) when they obey the finer relation. Finally, we define an imaginarity measure in terms of witnesses, termed witnessed imaginarity, and prove that it coincides with both the trace norm of imaginarity and the robustness of imaginarity.
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 theory is traditionally formulated using complex numbers. This imaginarity of quantum theory has been quantified as a resource with applications in discrimination tasks, pseudorandomness generation, and quantum metrology. In the standard formulation, a quantum state is said to have "imaginarity" if the associated density matrix is not real-valued in a given, fixed basis. If instead we consider a set of states, it is possible to devise tests that guarantee imaginarity of some state in the set, independently of the basis chosen. Here we propose such basis-independent witnesses for imaginarity that rely on measurements of unitary-invariant properties of sets of states. For three pure states, we completely characterize the invariant values attainable by quantum theory, and give a partial characterization for four pure states. We show that simple pairwise overlap measurements suffice to witness imaginarity of sets of four states, but not for sets of three. Our witnesses are experimentally friendly, opening up a new path for measuring and using imaginarity as a 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.
Recently, the resource theory of imaginarity was proposed, viewing the imaginary part of a quantum state as a basic resource leading to quantum advantages. As with other quantum resources such as correlations, coherence, and asymmetry, a natural question arises: Can imaginarity resource be broadcast by a real operation which is free in the resource theory of imaginarity? In this work, we introduce two concepts of imaginarity broadcasting. The first pertains to nonreal states, while the second is related to the imaginarity resource. For nonreal states, we demonstrate that a specific type of nonreal states can be broadcast. However, when it comes to broadcasting the imaginarity resource, we get a surprising result that any imaginarity resource can be perfectly broadcast via a real operation. By comparing different (no-)broadcasting theorems, we distinguish the resource theory of imaginarity from the resource theories of correlations, coherence, and asymmetry, and reveal certain distinctive features of the resource theory of imaginarity.
The imaginary in quantum theory plays a crucial role in describing quantum coherence and is widely applied in quantum information tasks such as state discrimination, pseudorandomness generation, and quantum metrology. A recent paper by Fernandes et al. [Phys. Rev. Lett. 133, 190201 (2024)] showed how to use the Bargmann invariant to witness the imaginarity of a set of quantum states. In this work, we delve into the structure of Bargmann invariants and their quantum realization in qubit systems. First, we present a characterization of special sets of Bargmann invariants (also studied by Fernandes et al. for a set of four states) for a general set of n quantum states. Then, we study the properties of the relevant Bargmann invariant set Bn and its quantum realization in qubit systems. Our results provide new insights into the structure of Bargmann invariants, contributing to the advancement of quantum information techniques, particularly within qubit systems.