Quantum Theory of Coherence

John H. Selby, Ciarán M. Lee

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.

Miguel Orszag

Selby, JH, Ciarán M. Lee

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.

Miguel Orszag

Jianbin Liu, Yu Zhou, Hui Chen +4 more

Compared to classical optical coherence theory based on Maxwell's electromagnetic theory and Glauber's quantum optical coherence theory based on matrix mechanics formulation of quantum mechanics, quantum optical coherence theory based on Feynman's path integral formulation of quantum mechanics provides a novel tool to study optical coherence. It has the advantage of understanding the connection between mathematical calculations and physical interpretations better. Quantum optical coherence theory based on Feynman's path integral is introduced and reviewed in this paper. Based on the results of transient first-order interference of two independent light beams, it is predicted that the classical model for electric field of thermal light introduced by classical optical textbooks may not be accurate. The physics of two-photon bunching of thermal light and Hong-Ou-Mandel dip of entangled photon pairs is the same, which can be interpreted by constructive and destructive two-photon interference, respectively. Quantum optical coherence theory based on Feynman's path integral is helpful to understand the coherence properties of light, which may eventually lead us to the answer of the question: what is a photon?