Finally, we show that the relative entropy of POVM coherence is equal to the cryptographic randomness gain, providing an important operational meaning to the concept of coherence with respect to a general measurement. This leads to the introduction of strongly monotonic resource measures that neatly generalize well-known standard coherence measures. This is related to the similar concepts in the resource theory of quantum coherence for orthonormal bases 5759. As in any resource theory, we define the free states and free operations in this scenario. Moreover, we introduce a rigorous, probabilisitic framework for POVM-based coherence measures and free operations. First, we attempt to construct a resource theory of quantum coherence for arbitrary bases by relaxing the restriction of orthogonality. In this work, we explore features of this framework which arise due to the rich structure of POVMs compared to projective measurements. In particular, POVM-incoherent (free) states and operations were established. In recent work, the notion of coherence with respect to a general quantum measurement, i.e., positive operator-valued measure (POVM), was introduced and embedded into a resource-theoretic framework that generalizes the standard resource theory of coherence. Copyright remains with the original copyright holders such as the authors or their institutions.Coherence is a cornerstone of quantum theory and a prerequisite for the advantage of quantum technologies. This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license (). Type:Ĭompositional resource theories of coherenceĪn open access version is available from UCL Discovery The underlying physical principle we follow is that the free dynamical objects are those that cannot preserve or distribute coherence. (3.39), the latter just means that whatever such a quantum computer does can be equally performed. In the present paper, we formulate the quantum resource theory of dynamical coherence. coherence theory with a set of USI operations given in Eq. 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. Hence, much work has been devoted in recent years to quantify the coherence present in a system. 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. ![]() 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. Such tasks can in theory be solved on a large-scale quantum computer whereas classical computers would not finish computations in any. This approach is limited in scope, and makes it difficult to generalise beyond resource theories of coherence for single system quantum states. To date however, these resource theories have only been mathematically formalised within the realms of convex-geometry, information theory, and linear algebra. ![]() ![]() Recently, there has been substantial progress in developing mathematical resource theories of coherence, paving the way towards its quantification and control. Indeed, preventing the loss of coherence is one of the most important technical challenges obstructing the development of large-scale quantum computers. Quantum coherence is one of the most important resources in quantum information theory.
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