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Advanced Quantum Technologies

Experimentally Accessible Imaginarity Measure for Composite Quantum Systems

Bo Wu, Zhanyuan Fu, Lei Li, Ping Li, Yongming Li

Published Sep 1, 2026• 0 citations
Publisher

Abstract

ABSTRACT In this work, we develop an experimentally accessible framework for quantifying imaginarity resources in composite quantum systems. We first formulate an observable‐based imaginarity measure using expectation values of antisymmetric Hermitian observables, thereby identifies the minimal observable sector required for evaluating imaginarity. By extending the trace‐norm imaginarity measure to multipartite systems, we introduce local imaginarity and imaginarity correlation to distinguish local and correlation contributions of imaginarity resources. We further establish observable representations for these quantities and investigate their evaluation using incomplete measurements through truncated measures based on the accessible antisymmetric sector. The performance of the truncated scheme is quantified by analyzing the retained observable ratio, lower‐bound recovery, and finite‐shot statistical convergence. Numerical investigations on qubit and qutrit systems demonstrate the behavior of local and correlation imaginarity in nonextremal states, higher‐dimensional systems, and under phase‐damping noise. Finally, we reveal a quantitative connection between imaginarity correlation and entanglement, derive an upper bound, and show that imaginarity correlation can serve as an experimentally accessible witness of entanglement. Our results provide an operational framework for characterizing and measuring imaginarity resources in realistic quantum systems.

DOI
10.1002/qute.70447

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Abstract Exploiting partial knowledge of a qubit in terms of its basis-dependent measure of imaginarity, we formulate a steering criterion for bipartite qubit systems, which requires two dichotomic measurements at the untrusted side and two mutually unbiased bases at the trusted side. We show that quantum correlations embodied through our proposed imaginarity steering inequality (ISI) can be witnessed by suitably constructed Hermitian operators that depend on fewer state parameters and require measurement of fewer observables than for witnessing other forms of quantum non-local correlations. The monogamy of such correlations is also demonstrated. We further underscore the steering of imaginarity as an efficient non-local resource in the presence of white noise and under unsharp measurements, showing the robustness of imaginarity steering compared with certain other steering criteria.

arXiv:2508.17140•Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences•Aug 26, 2026•1 citation
ABSTRACT In this work, we develop an experimentally accessible framework for quantifying imaginarity resources in composite quantum systems. We first formulate an observable‐based imaginarity measure using expectation values of antisymmetric Hermitian observables, thereby identifies the minimal observable sector required for evaluating imaginarity. By extending the trace‐norm imaginarity measure to multipartite systems, we introduce local imaginarity and imaginarity correlation to distinguish local and correlation contributions of imaginarity resources. We further establish observable representations for these quantities and investigate their evaluation using incomplete measurements through truncated measures based on the accessible antisymmetric sector. The performance of the truncated scheme is quantified by analyzing the retained observable ratio, lower‐bound recovery, and finite‐shot statistical convergence. Numerical investigations on qubit and qutrit systems demonstrate the behavior of local and correlation imaginarity in nonextremal states, higher‐dimensional systems, and under phase‐damping noise. Finally, we reveal a quantitative connection between imaginarity correlation and entanglement, derive an upper bound, and show that imaginarity correlation can serve as an experimentally accessible witness of entanglement. Our results provide an operational framework for characterizing and measuring imaginarity resources in realistic quantum systems.
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 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.
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.
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.
This work studies the decay of three imaginarity measures, namely the l1l_1l1​-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 l1l_1l1​-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++=∣+⟩⟨+∣⊗∣+⟩⟨+∣Ω_{++}=|+\rangle\langle+|\otimes|+\rangle\langle+|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.
Abstract Exploiting partial knowledge of a qubit in terms of its basis-dependent measure of imaginarity, we formulate a steering criterion for bipartite qubit systems, which requires two dichotomic measurements at the untrusted side and two mutually unbiased bases at the trusted side. We show that quantum correlations embodied through our proposed imaginarity steering inequality (ISI) can be witnessed by suitably constructed Hermitian operators that depend on fewer state parameters and require measurement of fewer observables than for witnessing other forms of quantum non-local correlations. The monogamy of such correlations is also demonstrated. We further underscore the steering of imaginarity as an efficient non-local resource in the presence of white noise and under unsharp measurements, showing the robustness of imaginarity steering compared with certain other steering criteria.