Recently, there has been a renewed interest in the quantification of coherence or other coherencelike concepts within the framework of quantum resource theory. However, rigorously defined or not, the notion of coherence or decoherence has already been used by the community for decades since the advent of quantum theory. Intuitively, the definitions of coherence and decoherence should be two sides of the same coin. Therefore, a natural question is raised: How can the conventional decoherence processes, such as the von Neumann--L\"uders (projective) measurement postulation or partially dephasing channels, fit into the bigger picture of the recently established theoretical framework? Here we show that the state collapse rules of the von Neumann or L\"uders-type measurements, as special cases of genuinely incoherent operations (GIOs), are consistent with the resource theories of quantum coherence. New hierarchical measures of coherence are proposed for the L\"uders-type measurement and their relationship with measurement-dependent discord is addressed. Moreover, utilizing the fixed-point theory for ${C}^{*}$ algebra, we prove that GIOs indeed represent a particular type of partially dephasing (phase-damping) channels which have a matrix representation based on the Schur product. By virtue of the Stinespring dilation theorem, the physical realizations of incoherent operations are investigated in detail and we find that GIOs in fact constitute the core of strictly incoherent operations and generally incoherent operations and the unspeakable notion of coherence induced by GIOs can be transferred to the theories of speakable coherence by the corresponding permutation or relabeling operators.
Recently, there has been a renewed interest in the quantification of coherence or other coherencelike concepts within the framework of quantum resource theory. However, rigorously defined or not, the notion of coherence or decoherence has already been used by the community for decades since the advent of quantum theory. Intuitively, the definitions of coherence and decoherence should be two sides of the same coin. Therefore, a natural question is raised: How can the conventional decoherence processes, such as the von Neumann--L"uders (projective) measurement postulation or partially dephasing channels, fit into the bigger picture of the recently established theoretical framework? Here we show that the state collapse rules of the von Neumann or L"uders-type measurements, as special cases of genuinely incoherent operations (GIOs), are consistent with the resource theories of quantum coherence. New hierarchical measures of coherence are proposed for the L"uders-type measurement and their relationship with measurement-dependent discord is addressed. Moreover, utilizing the fixed-point theory for C∗ algebra, we prove that GIOs indeed represent a particular type of partially dephasing (phase-damping) channels which have a matrix representation based on the Schur product. By virtue of the Stinespring dilation theorem, the physical realizations of incoherent operations are investigated in detail and we find that GIOs in fact constitute the core of strictly incoherent operations and generally incoherent operations and the unspeakable notion of coherence induced by GIOs can be transferred to the theories of speakable coherence by the corresponding permutation or relabeling operators.
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.
Quantum coherence is a fundamental feature of quantum mechanics and an underlying requirement for most quantum information tasks. In the resource theory of coherence, incoherent states are diagonal with respect to a fixed orthonormal basis; i.e., they can be seen as arising from a von Neumann measurement. Here, we introduce and study a generalization to a resource theory of coherence defined with respect to the most general quantum measurements, i.e., to arbitrary positive-operator-valued measures (POVMs). We establish POVM-based coherence measures and POVM-incoherent operations that coincide for the case of von Neumann measurements with their counterparts in standard coherence theory. We provide a semidefinite program that allows us to characterize interconversion properties of resource states and exemplify our framework by means of the qubit trine POVM, for which we also show analytical results.
Quantum coherence, like entanglement, is a fundamental resource in quantum information. In recent years, remarkable progress has been made in formulating resource theory of coherence from a broader perspective. The notions of block-coherence and POVM-based coherence have been established. Certain challenges, however, remain to be addressed. It is difficult to define incoherent operations directly, without requiring incoherent states, which proves a major obstacle in establishing the resource theory of dynamical coherence. In this paper, we overcome this limitation by introducing an alternate definition of incoherent operations induced via coherence measures and quantify dynamical coherence based on this definition. Finally, we apply our proposed definition to quantify POVM-based dynamical coherence.
Abstract Any quantum resource theory is based on free states and free operations, i.e. states and operations which can be created and performed at no cost. In the resource theory of coherence free states are diagonal in some fixed basis, and free operations are those which cannot create coherence for some particular experimental realization. Recently, some problems of this approach have been discussed, and new sets of operations have been proposed to resolve these problems. We propose here the framework of genuine quantum coherence. This approach is based on a simple principle: we demand that a genuinely incoherent operation preserves all incoherent states. This framework captures coherence under additional constrains such as energy preservation and all genuinely incoherent operations are incoherent regardless of their particular experimental realization. We also introduce the full class of operations with this property, which we call fully incoherent. We analyze in detail the mathematical structure of these classes and also study possible state transformations. We show that deterministic manipulation is severely limited, even in the asymptotic settings. In particular, this framework does not have a unique golden unit, i.e. there is no single state from which all other states can be created deterministically with the free operations. This suggests that any reasonably powerful resource theory of coherence must contain free operations which can potentially create coherence in some experimental realization.
Quantum coherence is one of the most important resources in quantum information theory. Indeed, preventing the loss of coherence is one of the most important technical challenges obstructing the development of large-scale quantum computers. Recently, there has been substantial progress in developing mathematical resource theories of coherence, paving the way towards its quantification and control. To date however, these resource theories have only been mathematically formalised within the realms of convex-geometry, information theory, and linear algebra. This approach is limited in scope, and makes it difficult to generalise beyond resource theories of coherence for single system quantum states. In this paper we take a complementary perspective, showing that resource theories of coherence can instead be defined purely compositionally, that is, working with the mathematics of process theories, string diagrams and category theory. This new perspective offers several advantages: i) it unifies various existing approaches to the study of coherence, for example, subsuming both speakable and unspeakable coherence; ii) it provides a general treatment of the compositional multi-system setting; iii) it generalises immediately to the case of quantum channels, measurements, instruments, and beyond rather than just states; iv) it can easily be generalised to the setting where there are multiple distinct sources of decoherence; and, iv) it directly extends to arbitrary process theories, for example, generalised probabilistic theories and Spekkens toy model---providing the ability to operationally characterise coherence rather than relying on specific mathematical features of quantum theory for its description. More importantly, by providing a new, complementary, perspective on the resource of coherence, this work opens the door to the development of novel tools which would not be accessible from the linear algebraic mind set.
Quantum coherence is one of the most important resources in quantum information theory. Indeed, preventing the loss of coherence is one of the most important technical challenges obstructing the development of large-scale quantum computers. Recently, there has been substantial progress in developing mathematical resource theories of coherence, paving the way towards its quantification and control. To date however, these resource theories have only been mathematically formalised within the realms of convex-geometry, information theory, and linear algebra. This approach is limited in scope, and makes it difficult to generalise beyond resource theories of coherence for single system quantum states. In this paper we take a complementary perspective, showing that resource theories of coherence can instead be defined purely compositionally, that is, working with the mathematics of process theories, string diagrams and category theory. This new perspective offers several advantages: i) it unifies various existing approaches to the study of coherence, for example, subsuming both speakable and unspeakable coherence; ii) it provides a general treatment of the compositional multi-system setting; iii) it generalises immediately to the case of quantum channels, measurements, instruments, and beyond rather than just states; iv) it can easily be generalised to the setting where there are multiple distinct sources of decoherence; and, iv) it directly extends to arbitrary process theories, for example, generalised probabilistic theories and Spekkens toy model---providing the ability to operationally characterise coherence rather than relying on specific mathematical features of quantum theory for its description. More importantly, by providing a new, complementary, perspective on the resource of coherence, this work opens the door to the development of novel tools which would not be accessible from the linear algebraic mind set.