Colloquium: Quantum Coherence as a Resource Alexander Streltsov,1, 2, 3, 4, ∗ Gerardo Adesso,5, y and Martin B. Plenio6, z 1Faculty of Applied Physics and Mathematics, Gdansk´ University of Technology, 80-233 Gdansk,´ Poland 2National Quantum Information Centre in Gdansk,´ 81-824 Sopot, Poland 3Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, D-14195 Berlin, Germany 4ICFO – Institut de Ciències Fotòniques, The Barcelona Institute of Science and Technology, ES-08860 Castelldefels, Spain 5Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems (CQNE), School of Mathematical Sciences, The University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom 6Institute of Theoretical Physics & IQST, University of Ulm, Albert-Einstein-Allee 11, D-89069 Ulm, Germany (Dated: July 10, 2017) The coherent superposition of states, in combination with the quantization of observables, represents one of the most fundamental features that mark the departure of quantum mechanics from the classical realm. Quantum coherence in many-body systems embodies the essence of entanglement and is an essential ingredient for a plethora of physical phenomena in quantum optics, quantum information, solid state physics, and nanoscale thermodynamics. In recent years, research on the presence and functional role of quantum coherence in biological systems has also attracted a considerable interest. Despite the fundamental importance of quantum coherence, the development of a rigorous theory of quantum coherence as a physical resource has only been initiated recently. In this Colloquium we discuss and review the development of this rapidly growing research field that encompasses the characterization, quantification, manipulation, dynamical evolution, and operational application of quantum coherence. CONTENTS 2. Quantum-incoherent relative entropy 19 3. Distillable coherence of collaboration 19 I. Introduction2 4. Recoverable coherence 19 5. Uncertainty relations and monogamy of coherence 20 II. Resource theories of quantum coherence3 A. Constraints, operations and resources3 IV. Dynamics of quantum coherence 20 1. Incoherent states4 A. Freezing of coherence 20 2. Classes of incoherent operations4 B. Coherence in non-Markovian evolutions 21 B. Coherence as a resource6 C. Cohering power of quantum channels and evolutions 21 1. Maximally coherent states and state transformations via D. Average coherence of random states and typicality 22 incoherent operations6 1. Average relative entropy of coherence 23 2. States and maps7 2. Average l1-norm of coherence 23 C. Quantum coherence in distributed scenarios7 3. Average recoverable coherence 23 D. Connection between coherence and entanglement theory8 E. Quantum speed limits 23 III. Quantifying quantum coherence9 V. Applications of quantum coherence 24 A. Quantum thermodynamics 24 A. Postulates for coherence monotones and measures9 1. Thermal operations 24 B. Distillable coherence and coherence cost 10 2. State transformations via thermal operations 25 C. Distance-based quantifiers of coherence 11 3. Work extraction and quantum thermal machines 26 1. Relative entropy of coherence 11 B. Quantum algorithms 26 2. Coherence quantifiers based on matrix norms 11 C. Quantum metrology 27 3. Geometric coherence 12 D. Quantum channel discrimination 27 D. Convex roof quantifiers of coherence 12 E. Witnessing quantum correlations 28 E. Coherence monotones from entanglement 13 F. Quantum biology and transport phenomena 28 F. Robustness of coherence 13 G. Quantum phase transitions 29 G. Coherence quantifiers from interferometric visibility 14 arXiv:1609.02439v3 [quant-ph] 10 Jul 2017 H. Coherence of assistance 14 VI. Conclusions 29 I. Coherence and quantum correlations beyond entanglement 15 J. Coherence in continuous variable systems 15 Acknowledgments 30 K. Coherence, asymmetry and nonclassicality 16 1. Asymmetry monotones 16 References 30 2. Quantifying superpositions 17 3. Coherence rank and general quantifiers of nonclassicality 17 4. Optical coherence and nonclassicality 18 L. Multipartite settings 18 1. General distance-based coherence quantifiers 18 ∗ [email protected] y [email protected] z [email protected] 2 I. INTRODUCTION point. Following an early approach to quantifying superposi- tions of orthogonal quantum states by (Åberg, 2006), and pro- Coherence marks the departure of today’s theories of the gressing alongside the independent yet related resource theory physical world from the principles of classical physics. The of asymmetry (Gour et al., 2009; Gour and Spekkens, 2008; theory of electro-magnetic waves, which may exhibit opti- Marvian and Spekkens, 2014a,b; Vaccaro et al., 2008), a re- cal coherence and interference, represents a radical departure source theory of coherence has been primarily proposed in from classical ray optics. Energy quantisation and the rise of (Baumgratz et al., 2014; Levi and Mintert, 2014) and further quantum mechanics as a unified picture of waves and particles developed in (Chitambar and Gour, 2016a,b, 2017; Winter and in the early part of the 20th century has further amplified the Yang, 2016; Yadin et al., 2016). Such a theory asks the ques- prominent role of coherence in physics. Indeed, by combina- tion what can be achieved and at what resource cost when the tion of energy quantization and the tensor product structure devices that are available to us are essentially classical, that is, of the state space, coherence underlies phenomena such as they cannot create coherence in a preferred basis. This analy- quantum interference and multipartite entanglement, that play sis, currently still under development, endeavors to provide a a central role in applications of quantum physics and quantum rigorous framework to describe quantum coherence, in anal- information science. ogy with what has been done for quantum entanglement and The investigation and exploitation of coherence of quantum other nonclassical resources (Adesso et al., 2016; Horodecki optical fields has a longstanding history. It has enabled the re- and Oppenheim, 2013b; Horodecki et al., 2009; Modi et al., alization of now mature technologies, such as the laser and its 2012; Plenio and Virmani, 2007; Sperling and Vogel, 2015; applications, that are often classified as ‘Quantum Technolo- Streltsov, 2015). Within such a framework, recent progress gies 1.0’ as they rely mainly on single particle coherence. At has shown that a growing number of applications can be cer- the mathematical level the coherence of quantum optical fields tified to rely on various incarnations of quantum coherence is described in terms of phase space distributions and multi- as a primary ingredient, and appropriate figures of merit for point correlation functions, approaches that find their roots in such applications can be precisely linked back to specific co- classical electromagnetic theory (Glauber, 1963; Mandel and herence monotones, providing operational interpretations for Wolf, 1965; Sudarshan, 1963). the latter. However, quantum coherence is not restricted to optical These applications include so-called ‘Quantum Technolo- fields. More importantly, as the key ingredient that drives gies 2.0’, such as quantum-enhanced metrology and commu- quantum technologies, it would be highly desirable to be able nication protocols, and extend further into other fields, like to precisely quantify the usefulness of coherence as a resource thermodynamics and even certain branches of biology. Be- for such applications. These pressing questions are calling for yond such application-driven viewpoint, which may provide a further development of the theory of quantum coherence. new insights into all these areas, one can also consider the The emergence of quantum information science over the theory of coherence as a resource as a novel approach towards last three decades has, amongst other insights, led to a re- the demarcation of the fundamental difference between classi- assessment of quantum physical phenomena as resources that cal and quantum physics in a quantitative manner: a goal that may be exploited to achieve tasks that are otherwise not possi- may eventually lead to a better understanding of the classical- ble within the realm of classical physics. This resource-driven quantum boundary. viewpoint has motivated the development of a quantitative The present Colloquium collects the most up-to-date theory that captures the resource character of physical traits knowledge on coherence in single and composite quantum in a mathematically rigorous fashion. systems, from a modern information theory perspective. We In a nutshell, any such theory first considers constraints that set to review this fascinating and fundamental subject in an are imposed on us in a specific physical situation (e.g. the in- accessible manner, yet without compromising any rigor. ability to perform joint quantum operations between distant The Colloquium is organized as follows (see Figure1). laboratories due the impossibility to transfer quantum systems SectionII gives a comprehensive overview of recent develop- from one location to the other while preserving their quan- ments to construct a resource theory of quantum coherence, tum coherence, and thus restricting us to local operations and including a hierarchy of possible classes of incoherent opera- classical communication). Executing general quantum oper- tions, and the conditions to which any valid coherence quanti- ations under such a constraint then requires quantum