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Physical Review A

Analysis of the relationship between imaginarity and entanglement

Yanna Sun, Ruonan Ren, Yiran Wang, Yongming Li

Published Mar 21, 2025• 10 citations
Publisher

Abstract

Imaginary numbers play an important role in physics, especially in quantum physics. They are not only a part of describing the dynamic behavior of quantum systems, but also play a key role in the phase and interference phenomena of quantum states. Similar to coherence and entanglement, the imaginarity has been regarded as a quantum resource, further revealing the nonclassical properties of quantum systems and providing new research directions and applications for fields such as quantum computing and quantum communication. In our work, we mainly explore the quantitative relationship between imaginarity and entanglement. For an arbitrary quantum state, there exists a general quantitative relationship between imaginarity and entanglement. For a given class of quantum states, where the closest real states to them are diagonal states, there exists an imaginarity-entanglement inequality. We demonstrate that for these quantum states, any degree of imaginarity with respect to some reference basis can be transformed into entanglement through incoherent real operations. This discovery allows us to define an entanglement-based imaginarity quantifier for these states in quantum systems of arbitrary dimensions and to study their properties. It is shown that for these quantum states, imaginarity and entanglement are operationally equivalent. Our work provides a clear quantitative and operational link between imaginarity and entanglement.

DOI
10.1103/physreva.111.032425

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Imaginary numbers play an important role in physics, especially in quantum physics. They are not only a part of describing the dynamic behavior of quantum systems, but also play a key role in the phase and interference phenomena of quantum states. Similar to coherence and entanglement, the imaginarity has been regarded as a quantum resource, further revealing the nonclassical properties of quantum systems and providing new research directions and applications for fields such as quantum computing and quantum communication. In our work, we mainly explore the quantitative relationship between imaginarity and entanglement. For an arbitrary quantum state, there exists a general quantitative relationship between imaginarity and entanglement. For a given class of quantum states, where the closest real states to them are diagonal states, there exists an imaginarity-entanglement inequality. We demonstrate that for these quantum states, any degree of imaginarity with respect to some reference basis can be transformed into entanglement through incoherent real operations. This discovery allows us to define an entanglement-based imaginarity quantifier for these states in quantum systems of arbitrary dimensions and to study their properties. It is shown that for these quantum states, imaginarity and entanglement are operationally equivalent. Our work provides a clear quantitative and operational link between imaginarity and entanglement.
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 Baumgratz, Cramer and Plenio established a rigorous framework (BCP framework) for quantifying the coherence of quantum states (2014, Phys. Rev. Lett. 113 , 140401). In BCP framework, a quantum state is called incoherent if it is diagonal in the fixed orthonormal basis, and a coherence measure should satisfy some conditions. For a fixed orthonormal basis, if a quantum state ρ has nonzero imaginary part, then ρ must be coherent. How to quantitatively characterize this fact? In this work, we show that any coherence measure C in BCP framework has the property C ( ρ ) − C (Re ρ )≥0 if C is invariant under state complex conjugation, i.e. C ( ρ ) = C ( ρ *), here ρ *is the conjugate of ρ , Re ρ is the real part of ρ . If C does not satisfy C ( ρ ) = C ( ρ *), we can define a new coherence measure C ′ ( ρ ) = 1 2 [ C ( ρ ) + C ( ρ * ) ] such that C ′ ( ρ ) = C ′ ( ρ * ) . We also establish some similar results for bosonic Gaussian states.
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.
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 Complex numbers play a key role in classical and quantum physics. Recently, the comprehensive formulation of the resource theory of imaginarity was proposed and various computable and meaningful measures of imaginarity were identified. In this work, we investigate the bounds for imaginarity of quantum superpositions in high dimension using the geometric imaginarity. We establish the relationship between the imaginarity of the superposition of quantum states and the imaginarity of the states being superposed.