Compositional resource theories of coherence

John H. Selby1,2 and Ciarán M. Lee3

1International Centre for Theory of Quantum Technologies, University of Gdańsk, Wita Stwosza 63, 80-308 Gdańsk, Poland
2Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario, N2L 2Y5, Canada
3Department of Physics and Astronomy, University College London, Gower Street, WC1E 6BT, UK

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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.

One of the biggest barriers to building usable, large-scale quantum computers is the loss of quantum coherence, a key resource underlying quantum superpositions. Given the crucial role of coherence, researchers have begun to develop theories to quantify it as a resource—that is, how it can be manipulated and controlled. Much like our mathematical understanding of how petrol is converted to the energy that powers cars, we want to understand how a quantum computer converts coherence to computation. If we understand how to use it as a resource we will be able to protect quantum computers from losing it too quickly.

To date however, the mathematical understanding of coherence as a resource has been largely focused on studying single qubits. This is a major obstacle for understanding how coherence powers quantum computers, as large-scale quantum computers consist of a huge number of qubits interacting with each other and evolving through time. We need to understand coherence as a resource in this challenging multiple interacting state context if we are to deliver useful quantum computers.

Using category theory—an area of mathematics studying the abstract structures that appear in different branches of mathematics—our paper solves this challenging problem. This provides a roadmap for how coherence should be manipulated and controlled to ensure the correct functioning of large-scale quantum computers.

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