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arXiv:2404.06210 · Physica Scripta

Coherence and imaginarity of quantum states

Jianwei Xu

Published Dec 2, 2024• 10 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1402-4896/ad99a1

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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.
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.
Recently, how to measure quantum coherence is a hot topic in the quantum resource theory. In 2014, Baumgratz et al. proposed a rigorous framework to quantify coherence. In this framework, a particular basis of the Hilbert space is required to define the coherence measure, and hence, the proposed coherence measure of the same quantum state does not remain the same under the basis transformations. In this paper, a measure of coherence based on the normalized modulus of the coherent vector is proposed in coherent vector representation, where the property of the coherent vector under basis transformation is considered. The measure of coherence proposed here is independent of the choice of the basis. The incoherent states and incoherent operators based on this measure can also be well defined as the classical maximal mixed states and the unitary operators respectively. And three important properties can be obtained, namely, non-negativity, convexity and invariance on the incoherent operators.
Abstract Since both coherence and imaginarity have proven to be fundamental resources in quantum information processing and their definitions are both relative to a given basis, it is natural to investigate the conversion between them. In this work, we prove that a quantum state can be transformed into a state with nonvanishing imaginarity resource by incoherent operations if and only if it has nonvanishing coherence resource. On the other hand, we demonstrate that a quantum state can be transformed into a state with vanishing coherence resource by orthogonal operations if and only if it has vanishing imaginarity resource. Based on these results, we show that for any imaginarity measure, a corresponding coherence measure can be defined as the maximal amount of imaginarity generated by incoherent operations, and that for any coherence quantifier, a corresponding imaginarity quantifier can be defined as the minimal coherence generated by orthogonal operations.
The notion of measurement is of both foundational and instrumental significance in quantum mechanics, and coherence destroyed by measurements (decoherence) lies at the very heart of quantum to classical transition. Qualitative aspects of this spirit have been widely recognized and analyzed ever since the inception of quantum theory. However, axiomatic and quantitative investigations of coherence are attracting great interest only recently with several figures of merit for coherence introduced [Baumgratz, Cramer, and Plenio, Phys. Rev. Lett. 113, 140401 (2014)]. While these resource theoretic approaches have many appealing and intuitive features, they rely crucially on various notions of incoherent operations which are sophisticated, subtle, and not uniquely defined, as have been critically assessed [Chitambar and Gour, Phys. Rev. Lett. 117, 030401 (2016)]. In this paper, we elaborate on the idea that coherence and quantum uncertainty are dual viewpoints of the same quantum substrate, and address coherence quantification by identifying coherence of a state (with respect to a measurement) with quantum uncertainty of a measurement (with respect to a state). Consequently, coherence measures may be set into correspondence with measures of quantum uncertainty. In particular, we take average quantum Fisher information as a measure of quantum uncertainty, and introduce the corresponding measure of coherence, which is demonstrated to exhibit desirable properties. Implications for interpreting quantum purity as maximal coherence, and quantum discord as minimal coherence, are illustrated.
Abstract Quantum coherence is a fundamental manifestation of the quantum superposition principle. Recently, Baumgratz et al (2014 Phys. Rev. Lett . 113 140401) presented a rigorous framework to quantify coherence from the view of theory of physical resource. Here we propose a new valid quantum coherence measure which is a convex roof measure, for a quantum system of arbitrary dimension, essentially using the generalized Gell-Mann matrices. Rigorous proof shows that the proposed coherence measure, coherence concurrence, fulfills all the requirements dictated by the resource theory of quantum coherence measures. Moreover, strong links between the resource frameworks of coherence concurrence and entanglement concurrence is derived, which shows that any degree of coherence with respect to some reference basis can be converted to entanglement via incoherent operations. Our work provides a clear quantitative and operational connection between coherence and entanglement based on two kinds of concurrence. This new coherence measure, coherence concurrence, may also be beneficial to the study of quantum coherence.