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Measuring quantum coherence in coherence-vector representation

DING Lei

Published Jan 1, 2020• 0 citations
OpenAlex

Abstract

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.

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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.
Quantum coherence is an essential ingredient in quantum information processing and plays an important role in quantum computation. Therefore, it is a hot issue about how to quantify the coherence of quantum states in theoretical framework. The coherence effect of a state is usually described by the off-diagonal elements of its density matrix with respect to a particular reference basis. Recently, based on the established notions from quantitative theory of entanglement, a resource theory of coherence quantification has been proposed[1,2]. In the theory framework, a proper measure of coherence should satisfy three criteria: the coherence should be zero for all incoherent state; the coherence should not increase under mixing quantum states; the coherence should not increase under incoherent operations. Then, a number of coherence measures have been suggested, such as l1 norm of coherence and the relative entropy of coherence[2]. Wigner function is known as an important tool to study the non-classical property of quantum states for continuous-variable quantum systems. It has been generalized to finite-dimensional Hilbert spaces, and named as discrete Wigner function[9-16]. The magic property of quantum states, which promotes stabilizer computation to universal quantum computation, can be generally measured by the absolute sum of the negative items (negativity sum) in the discrete Wigner function of the observed quantum states. In this paper we investigate quantum coherence from the view of discrete Wigner function. From the definition of the discrete Wigner function of the quantum systems with odd prime dimensions, for a given density matrix we analyze in phase space the performance of its diagonal and off-diagonal items. We find that, the discrete Wigner function of a quantum state contains two aspects: the true quantum coherence and the classical mixture, where the part of classical mixture can be excluded by only considering the discrete Wigner function of the diagonal items of the density matrix. Thus, we propose a possible measure method for quantum coherence from the discrete Wigner function of the off-diagonal items of the density matrix. We show that the proposed measure method satisfies the criteria (C1) and (C2) of coherence measure perfectly. For the criteria (C3), we give a numerical proof in three-dimensional quantum system. Meanwhile, we compare the proposed coherence measure with l1 norm coherence, and get an inequality relationship between them. Finally, an inequality is obtained to discuss the relation between quantum coherence and the negativity sum of discrete Wigner function, which shows that the quantum coherence is only necessary but not sufficient for quantum computation speed-up.
Given a preferred orthonormal basis BBB in the Hilbert space of a quantum system we define a measure of the coherence generating power of a unitary map with respect to BBB. We give its explicit analytical form and provide an operational protocol to directly detect it. This measure vanishes if and only if the unitary map transforms all the incoherent states into themselves and is non-increasing under pre- or post-processing by incoherent operations. We characterize the set of unitaries with maximal coherence generating power and study the properties of our measure when the unitary is drawn at random from the Haar distribution. For large state-space dimension a random unitary has, with overwhelming probability, nearly maximal coherence generating power with respect to any basis. Finally, extensions to general unital completely positive maps and the relation to the concept of asymmetry are discussed.
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.
Given a preferred orthonormal basis BBB in the Hilbert space of a quantum system, we define a measure of the coherence-generating power of a unitary operation with respect to BBB. This measure is the average coherence generated by the operation acting on a uniform ensemble of incoherent states. We give its explicit analytical form in any dimension and provide an operational protocol to directly detect it. We characterize the set of unitaries with maximal coherence-generating power and study the properties of our measure when the unitary is drawn at random from the Haar distribution. For a large state-space dimension a random unitary has, with overwhelming probability, nearly maximal coherence-generating power with respect to any basis. Finally, extensions to general unital quantum operations and the relation to the concept of asymmetry are discussed.
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.
One of the main problems in any quantum resource theory is the characterization of the conversions between resources by means of the free operations of the theory. In this work we advance on this characterization within the quantum coherence resource theory by introducing the generalized coherence vector of an arbitrary quantum state. The generalized coherence vector is a probability vector that can be interpreted as a concave roof extension of the pure states coherence vector. We show that it completely characterizes the notions of being incoherent, as well as being maximally coherent. Moreover, using this notion and the majorization relation, we obtain a necessary condition for the conversion of general quantum states by means of incoherent operations. These results generalize the necessary conditions of conversions for pure states given in the literature, and show that the tools of the majorization lattice are useful also in the general case. Finally, we introduce a family of coherence quantifiers by considering concave and symmetric functions applied to the generalized coherence vector. We compare this proposal with the convex roof measure of coherence and others quantifiers given in the literature.