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arXiv:2412.07449

Basis-independent Coherence and its Applications

Zhi‐Xiang Jin, Yuan-Hong Tao, Bing Yu, Shao-Ming Fei

Published Dec 10, 2024• 0 citations
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Abstract

In the quantitative theory of quantum coherence, the amount of coherence for given states can be meaningfully discussed only when referring to a preferred basis. One of the objections to this quantification is that the amount of coherence is an intrinsically basis-dependent quantity. This limitation can, however, be lifted when considering a set of quantum states invariant under arbitrary unitary transformations. Thus, we analyze a basis-independent definition of quantum coherence, and the incoherent state is taken as the maximally mixed state. We describe the relationship between the basis-independent and the basis-dependent approaches and give several applications to show the advantages of the former method. The relations among basis-independent coherence, quantum entanglement, and quantum discord are discussed by using the relative entropy within a multipartite system.

DOI
10.48550/arxiv.2412.07449

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In the quantitative theory of quantum coherence, the amount of coherence for given states can be meaningfully discussed only when referring to a preferred basis. One of the objections to this quantification is that the amount of coherence is an intrinsically basis-dependent quantity. This limitation can, however, be lifted when considering a set of quantum states invariant under arbitrary unitary transformations. Thus, we analyze a basis-independent definition of quantum coherence, and the incoherent state is taken as the maximally mixed state. We describe the relationship between the basis-independent and the basis-dependent approaches and give several applications to show the advantages of the former method. The relations among basis-independent coherence, quantum entanglement, and quantum discord are discussed by using the relative entropy within a multipartite system.
In the quantitative theory of quantum coherence, the amount of coherence is defined as the distance between the given state to the closest incoherent state. The set of incoherent states is conventionally defined as any state with a diagonal density matrix. One of the objections to this formulation is that the incoherent states are intrinsically basis-dependent, which makes the amount of coherence also a basis-dependent quantity. Basis-independent measures have recently been proposed where the incoherent state is taken as the maximally mixed state. We show that this is the only possible choice of reference incoherent state, without modifying the original definition of coherence. We find a relation between the two formulations by defining a contribution to the coherence due to the basis choice. The hierarchical relationship between quantum coherence and the various quantum correlations is explored in detail. Finally, we illustrate some operational uses of the basis-independent quantum coherence in quantum information theory tasks.
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Quantum coherence, incompatibility, and quantum correlations are fundamental features of quantum physics. A unified view of those features is crucial for revealing quantitatively their intrinsic connections. We define the relative quantum coherence of two states as the coherence of one state in the reference basis spanned by the eigenvectors of another one and establish its quantitative connections with the extent of mutual incompatibility of two states. We also show that the proposed relative quantum coherence, which can take any form of measures such as l1{l}_{1}l1​ norm and relative entropy, can be interpreted as or connected to various quantum correlations such as quantum discord, symmetric discord, entanglement of formation, and quantum deficits. Our results reveal conceptual implications and basic connections of quantum coherence, mutual incompatibility, and quantum correlations.
Since quantum coherence is an undoubted characteristic trait of quantum physics, the quantification and application of quantum coherence has been one of the long-standing central topics in quantum information science. Within the framework of a resource theory of quantum coherence proposed recently, a fiducial basis should be preselected for characterizing the quantum coherence in specific circumstances, namely, the quantum coherence is a basis-dependent quantity. Therefore, a natural question is raised: what are the maximum and minimum coherences contained in a certain quantum state with respect to a generic basis? While the minimum case is trivial, it is not so intuitive to verify in which basis the quantum coherence is maximal. Based on the coherence measure of relative entropy, we indicate the particular basis in which the quantum coherence is maximal for a given state, where the Fourier matrix (or more generally, complex Hadamard matrices) plays a critical role in determining the basis. Intriguingly, though we can prove that the basis associated with the Fourier matrix is a stationary point for optimizing the l1{l}_{1}l1​ norm of coherence, numerical simulation shows that it is not a global optimal choice.
Quantum coherence is known as an important resource in many quantum information tasks, which is a basis-dependent property of quantum states. In this paper, we discuss quantum incoherence based simultaneously on k bases using Matrix Theory Method. First, by defining a correlation function m(e,f) of two orthonormal bases e and f, we investigate the relationships between sets I(e) and I(f) of incoherent states with respect to e and f. We prove that I(e)=I(f) if and only if the rank-one projective measurements generated by e and f are identical. We give a necessary and sufficient condition for the intersection I(e)⋂I(f) to include a state except the maximally mixed state. Especially, if two bases e and f are mutually unbiased, then the intersection has only the maximally mixed state. Secondly, we introduce the concepts of strong incoherence and weak coherence of a quantum state with respect to a set B of k bases and propose a measure for the weak coherence. In the two-qubit system, we prove that there exists a maximally coherent state with respect to B when k=2 and it is not the case for k=3.
Recent results in quantum information theory characterize quantum coherence in the context of resource theories. Here, we study the relation between quantum coherence and quantum discord, a kind of quantum correlation which appears even in nonentangled states. We prove that the creation of quantum discord with multipartite incoherent operations is bounded by the amount of quantum coherence consumed in its subsystems during the process. We show how the interplay between quantum coherence consumption and creation of quantum discord works in the preparation of multipartite quantum correlated states and in the model of deterministic quantum computation with one qubit.