Quantum Coherence As a Resource

Colloquium: Quantum Coherence as a Resource

Alexander Streltsov,1, 2, 3, 4, ∗ Gerardo Adesso,5, † and Martin B. Plenio6, ‡ 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] † [email protected] ‡ [email protected] 2

I. INTRODUCTION point. Following an early approach to quantifying superpositions 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 combination 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 states fier should abide; it also discusses established links with other that contain a relevant resource (e.g. entangled states that are resource theories, most prominently those of asymmetry and provided to us at a certain cost) and can be consumed in the of quantum entanglement. Section III presents a compendium process. The formulation of such resource theories was in fact of recently proposed monotones and measures of quantum co- initially pursued with the quantitative theory of entanglement herence, based on different physical approaches endowed with (Horodecki et al., 2009; Plenio and Virmani, 2007) but has different mathematical properties, in single and multipartite since spread to encompass a wider range of operational set- systems; interplays with other measures of nonclassicality are tings (Coecke et al., 2016; del Rio et al., 2015; Horodecki and highlighted as well. SectionIV reviews the phenomenology of Oppenheim, 2013b). quantum coherence in the dynamical evolution of open quan- The theory of quantum coherence as a resource is a case in tum systems, further reporting results on the average coher- 3

Section II II. RESOURCE THEORIES OF QUANTUM COHERENCE

PIO Coherence is a property of the physical world that is used to drive a wide variety of phenomena and devices. Hence co- Resource herence adopts the quality of a resource, as it may be pro- Theories vided at a certain cost, manipulated by otherwise incoherent means, and consumed to achieve useful tasks. The quantita- Section V Section III tive study of these processes and their attainable eﬃciencies coherent incoherent S A requires careful deﬁnitions of the accessible operations and

Quantum incoherent operation gives rise to a framework that has become known as a resource

Coherence entangled S:A theory. In line with earlier developments in quantum infor- ? ✓ mation science, for example in the context of entanglement Applications Quantification (Brandão and Plenio, 2008, 2010; Horodecki et al., 2009; Ple- Section IV nio and Virmani, 2007; Vedral and Plenio, 1998), quantum thermodynamics (Brandão et al., 2013; Goold et al., 2016; Gour et al., 2015; Janzing et al., 2000; Ruch, 1975), purity (Horodecki et al., 2003), and reference frames (Gour et al., 2009; Gour and Spekkens, 2008; Marvian and Spekkens, Dynamics 2014a), the formulation of the resource theory of coherence (Åberg, 2006; Baumgratz et al., 2014; Levi and Mintert, 2014; Winter and Yang, 2016) extends the family of resources theo- Figure 1 (Color online) Plan of the Colloquium. (Top) SectionII: ries of knowledge (del Rio et al., 2015). Resource theories of quantum coherence; the inset depicts a com- parison between some classes of incoherent operations, adapted and reproduced with permission from (Chitambar and Gour, 2016b). (Right) Section III: Quantifying quantum coherence; the inset depicts the construction of coherence monotones from entanglement, A. Constraints, operations and resources adapted and reproduced with permission from (Streltsov et al., 2015). (Bottom) SectionIV: Dynamics of quantum coherence; the inset A resource theory is fundamentally determined by con- depicts an illustration of coherence freezing under local incoherent straints that are imposed on us and which determine the set channels, adapted and reproduced with permission from (Bromley et al., 2015). (Left) SectionV: Applications of quantum coherence; of the freely accessible quantum operations F . These con- the insets depicts a schematic of a quantum phase discrimination pro- straints may be due to either fundamental conservation laws, tocol, adapted and reproduced with permission from (Napoli et al., such as superselection rules and energy conservation, or con- 2016). Introduction (SectionI) and Conclusions (SectionVI) com- straints due to the practical difficulty of executing certain op- plete the Colloquium. erations, e.g. the restriction to local operations and classical communication (LOCC) which gives rise to the resource theory of entanglement (Horodecki et al., 2009; Plenio and Vir- ence of random states, and on the cohering power of quantum mani, 2007). channels. SectionV focuses on the plethora of applications The states that can be generated from the maximally mixed and operational interpretations highlighted so far for quan- state1 by the application of free operations in F alone, are tum coherence in thermodynamics, interference phenomena, considered to be available free of charge, forming the set I quantum algorithms, metrology and discrimination, quantum of free states. All the other states attain the status of a re- biology, many-body physics and the detection of quantum cor- source, whose provision carries a cost. These resource states relations. SectionVI concludes the Colloquium with a sum- may be used to achieve operations that cannot be realized by mary and a discussion of some currently open issues in the using only members of F . Alternatively, one may also begin theoretical description of coherence and its role in quantum by defining the set of free states I, and then consider classes physics and beyond. of operations that map this set into itself and use this to define We emphasize that, due to limitations in space and focus, F . For the purposes of the present exposition of the resource this Colloquium cannot cover all ramifications of the concept theory of coherence, we begin by adopting the latter point of of quantum coherence. It is nevertheless our expectation that view and then proceed to require additional desirable proper- this Colloquium, while being self-contained, can stimulate the ties of our classes of free operations. reader to undertake further research towards achieving a fully satisfactory and physically consistent characterization of the wide-interest topic of coherence as a resource in quantum systems of arbitrary dimensions. This, we hope, may also lead to the formulation of novel direct applications of coherence (or 1 The maximally mixed state can always be obtained by erasing all informa- an optimization of existing ones) in a variety of physical and tion about the system. Hence it is fair to assume that it is devoid of any biological contexts of high technological interest. useful resource and freely available. 4

1. Incoherent states 2. Classes of incoherent operations

Coherence is naturally a basis dependent concept, which is The definition of free operations for the resource theory of why we first need to fix the preferred, or reference basis in coherence is not unique and different choices, often motivated which to formulate our resource theory. The reference basis by suitable practical considerations, are being examined in the may be dictated by the physics of the problem under inves- literature. Here, we present the most important classes and tigation (e.g. one may focus on the energy eigenbasis when briefly discuss their properties and relations among each other. addressing coherence in transport phenomena and thermody- We start with the largest class, the maximally incoherent namics) or by a task for which coherence is required (e.g. the operations (MIO) (Åberg, 2006) (also known as incoherence estimation of a magnetic field in a certain direction within a preserving operations), which are defined as any trace preserv- quantum metrology setting). Given a d-dimensional Hilbert ing completely positive and non-selective quantum operations space H (with d assumed finite, even though some exten- Λ : B(H) 7→ B(H) such that sions to infinite d can be considered), we will denote its ref- Λ[I] ⊆ I. (2) erence orthonormal basis by {|ii}i=0,...,d−1. The density matri- incoher- ces that are diagonal in this specific basis are called As with every quantum operation, also this mathematically ent , i.e., they are accessible free of charge, and form the set natural set of operations can always be obtained by a Stine- I ⊂ B H B H ( ), where ( ) denotes the set of all bounded trace spring dilation, i.e. the provision of an ancillary environment H class operators on . Hence, all incoherent density operators in some state σ, a subsequent unitary operation U between ∈ I % are of the form system and environment, followed by the tracing out of the environment, Xd−1 % = pi|iihi| (1) † Λ[%] = TrE[U(% ⊗ σ)U ]. (3) i=0 If an operation can be implemented as in Eq. (3) by using an with probabilities pi. incoherent state σ of the environment and a global incoherent In the case of more than one party, the preferred basis with unitary U (a unitary that is diagonal in the preferred basis), we respect to which coherence is studied will be constructed as say that the operation has a free dilation. It should be noted the tensor product of the corresponding local reference ba- that despite the fact that MIO cannot create coherence, these sis states for each subsystem. General multipartite incoher- operations in general do not have a free dilation2 (Chitambar ent states are then defined as convex combinations of such in- and Gour, 2016a,b, 2017; Marvian and Spekkens, 2016). coherent pure product states (Bromley et al., 2015; Streltsov A smaller and more relevant class of free operations for et al., 2015; Winter and Yang, 2016). the theory of coherence is that of incoherent operations (IO) For example, if the reference basis for a single qubit is taken (Baumgratz et al., 2014) which are characterized as the set of to be the computational basis {|0i , |1i}, i.e., the eigenbasis of trace preserving completely positive maps Λ : B(H) 7→ B(H) the Pauli σ operator, then any density matrix with a nonzero 3 P † z admitting a set of Kraus operators {Kn} such that n Kn Kn = offdiagonal element |%01| = |h0|%|1i| , 0 is outside the set I 1 (trace preservation) and, for all n and % ∈ I, of incoherent states, hence has a resource content. Similarly, K %K† for an N-qubit system, the set of incoherent states I is formed n n ∈ I † . (4) by all and only the density matrices % diagonal in the compos- Tr[Kn%Kn ] ite computational basis {|0i , |1i}⊗N , with any other state being This definition of IO ensures that, in any of the possible out- coherent, that is, resourceful. comes of such an operation, coherence can never be generated We also note that some frameworks of coherence may al- from an incoherent input state, not even probabilistically4. low for a larger set of free states. This is in particular the case Also this class of operations does not admit a free dilation for the resource theory of asymmetry (Gour et al., 2009; Gour in general (Chitambar and Gour, 2016a,b, 2017; Marvian and and Spekkens, 2008; Marvian and Spekkens, 2014a,b), where Spekkens, 2016). the set of free states is defined by all states which commute In the previous two definitions, the focus was placed on the with a given Hamiltonian H. If the Hamiltonian is nondegen- inability of incoherent operations to generate coherence. One erate, the corresponding set of free states is exactly the set of incoherent states described above, where the incoherent basis is defined by the eigenbasis of the Hamiltonian. However, the situation changes if the Hamiltonian has degeneracies, in 2 This mirrors the situation in entanglement theory where separable opera- which case also any superposition of the eigenstates corre- tions cannot create entanglement but in general cannot be implemented via sponding to the degenerate subspaces is also considered as LOCC (Bennett et al., 1999). 3 P † free. This has important implications in quantum thermody- According to the Kraus decomposition, the maps act as Λ[%] = n Kn%Kn . 4 Note that relaxing the condition of trace preservation here may have non- namics, as is described in more detail in Section V.A. In the trivial consequences, as one should then ensure that the “missing” Kraus following, whenever we refer to incoherent states, we explic- operators share the property that they map I into itself. That this is possible itly mean states of the form (1). for IO was recently proven in (Theurer et al., 2017). 5 may however be more stringent by adding further desirable As already mentioned above, the sets MIO, IO, and SIO in properties to the set of free operations. One such approach general do not have a free dilation, i.e., they cannot be im- requires that admissible operations are not capable of making plemented by coupling the system to an environment in an use of coherence in the input state. Defining the dephasing incoherent state followed by a global incoherent unitary. Mo- operation tivated by this observation, (Chitambar and Gour, 2016a,b, 2017) introduced the set of physical incoherent operations Xd−1 (PIO). These are all operations which can instead be imple- ∆[%] = |iihi|%|iihi|, (5) mented in the aforementioned way, additionally allowing for i=0 incoherent measurements on the environment and classical an operation Λ is called strictly incoherent (SIO) (Winter and postprocessing of the measurement outcomes. Yang, 2016; Yadin et al., 2016) if it can be written in terms of a Clearly, MIO is the largest set of free operations for a resource theory of coherence, and all other sets listed above are set of incoherent Kraus operators {Kn}, such that the outcomes of a measurement in the reference basis applied to the output strict subsets of it. Inclusion relations between each of these are independent of the coherence of the input state, i.e., sets are nontrivial in general. Here, we mention that (see also Fig.1, top panel) † † hi|Kn%Kn |ii = hi|Kn∆[%]Kn |ii (6) PIO ⊂ SIO ⊂ IO ⊂ MIO, (8) for all n and i. Equivalently, SIO can be characterized as those and refer to (Chitambar and Gour, 2016a,b, 2017; de Vicente operations which have an incoherent Kraus decomposition and Streltsov, 2017) for a more detailed discussion. † {Kn} such that the operators Kn are also incoherent (Winter Another interesting set is given by dephasing-covariant in- and Yang, 2016; Yadin et al., 2016). As was shown by (Chita- coherent operations (DIO), which were introduced indepen- mbar and Gour, 2016a,b, 2017), SIO in general do not admit dently by (Chitambar and Gour, 2016a,b, 2017) and (Marvian a free dilation either. Nevertheless, a special type of dilation and Spekkens, 2016). These are all operations Λ which com- for SIO of the form (3) was provided in (Yadin et al., 2016), mute with the dephasing map Eq. (5), i.e., Λ ∆[%] = ∆ Λ[%] . which consists of: (i) unitary operations on the environment It is an interesting open question whether DIO have a free di- P controlled by the incoherent basis of the system: j | jih j|⊗U j; lation. (ii) measurements on the environment in any basis; (iii) inco- We also mention genuinely incoherent operations (GIO) herent unitary operations on the system conditioned on the and fully incoherent operations (FIO) (de Vicente and P iθ j measurement outcome: j e |π( j)ih j|, with {|π( j)i} denoting Streltsov, 2017). GIO are operations which preserve all in- a permutation of the reference basis of the system. coherent states, i.e., Λ[%] = % for any incoherent state %. In The classes of operations defined so far include permuta- particular, every GIO is incoherent regardless of the particu- tions of the reference basis states for free. This is natural lar Kraus decomposition, i.e., for every experimental realiza- when any such operation is considered from a passive point tion of the operation. Since GIO do not allow for transforma- of view, amounting to a relabeling of the states. Viewed from tions between different incoherent states, notably for example an active point of view instead, i.e. asking for a unitary op- between the energy eigenstates (when coherence is measured eration that realizes this permutation, it may be argued that with respect to the eigenbasis of the Hamiltonian of the sys- such an operation does in fact cost coherence resources in or- tem), they capture the framework of coherence in the presence der to be performed in laboratory. This suggests that, from an of additional constraints, such as energy conservation. FIO is operational point of view, permutations should be excluded in turn the most general set of operations which is incoher- from the free operations, as is done in the more stringent ent for every Kraus decomposition (de Vicente and Streltsov, set of translationally-invariant operations (TIO). The latter 2017). The GIO framework is closely related to the concept are defined as those commuting with phase randomization of resource destroying maps introduced by (Liu et al., 2017). (Gour et al., 2009; Gour and Spekkens, 2008; Marvian and The latter studies quantum operations which transform any Spekkens, 2013, 2014a,b, 2016; Marvian et al., 2016). More quantum state onto a free state, and moreover preserve all free precisely, given a Hamiltonian H, an operation Λ is transla- states. If the set of free states is taken to be incoherent, any tionally invariant with respect to H if it fulfills the condition such coherence destroying map is also GIO. (Gour et al., 2009; Marvian and Spekkens, 2014a; Marvian The role of energy in the context of coherence has also et al., 2016) been investigated by (Chiribella and Yang, 2015). In particular, (Chiribella and Yang, 2015) defined the class of energy e−iHtΛ[%]eiHt = Λ[e−iHt%eiHt]. (7) preserving operations (EPO) as all operations which have a free dilation as in Eq. (3), with the additional requirement that TIO play an important role in the resource theory of asym- the unitary commutes with the Hamiltonian of the system and metry (see Sec. III.K.1) and quantum thermodynamics (see environment individually. Note that the set of EPO is a strict Sec. V.A). Interestingly, (Marvian and Spekkens, 2016) subset of TIO (Chiribella and Yang, 2015). showed that TIO have a free dilation if one additionally allows That there is such a wide variety of possible definitions of postselection with an incoherent measurement on the environ- incoherent operations should not come as a surprise, as it mir- ment. rors the situation in entanglement theory where the set LOCC 6

Sets of free operations idealization which, arguably, cannot be achieved in the real Speakable coherence FIO PIO SIO IO DIO MIO world. Hence, already from this practical consideration, one Unspeakable coherence GIO EPO TIO should also permit approximate state transformations such Table I Classification of different frameworks of coherence into that, instead of the exact map Λ[%] = σ, one would be sat- speakable and unspeakable type (Marvian and Spekkens, 2016). isfied with achieving√ ||Λ[%] − σ||1 ≤ for some small , where † ||M||1 = Tr M M denotes the trace norm. Beyond the mere practical motivation, allowing for approximate transforma- is operationally well-defined, though mathematically cumber- tions has some relevance because it can considerably change some, while larger sets such as separable operations (Vedral the set of available state transformations. In particular, when and Plenio, 1998) and positive partial transpose preserving op- permitting small errors to occur in catalysts their power may erations (Rains, 2001) have mathematically very convenient increase further, in fact so much that any state transformation properties, but in general do require a finite amount of free may potentially become possible through embezzling of quan- entanglement for their implementation (Bennett et al., 1999). tum states (van Dam and Hayden, 2003). Stronger constraints Nevertheless, they are very useful as they provide bounds on such as substituting by / log D where D = dim[η] and η is achievable efficiencies for transformations under the difficult the state of the catalyst can prevent embezzling (see (Brandão to handle LOCC constraints. Equally, the wealth of definitions et al., 2015a) for examples in quantum thermodynamics). On for alternative incoherent operations can be expected to pro- the other hand, the class of asymptotically exact incoherent vide both conceptual and quantitative insights into the proper- operations becomes particularly relevant when one wishes to ties of coherence as a resource. consider the mapping % 7→ σ not for a single copy, but start- A classification of the different frameworks of coherence, ing from N copies, %⊗N . In this case, one typically asks for the ⊗brNc ⊗N motivated by the notion of speakable and unspeakable infor- minimal rate r such that limN→∞{infΛ ||Λ[% ]−σ ||1} = 0, mation (Bartlett et al., 2007; Peres and Scudo, 2002), has been where r is the asymptotic cost of σ in terms of %, and bxc de- proposed by (Marvian and Spekkens, 2016). Speakable infor- notes the largest integer lower than or equal to the real num- mation includes any type of information for which the means ber x. Indeed, (Winter and Yang, 2016) have carried out such of encoding is not relevant, while unspeakable information de- a program defining the appropriate quantities and obtaining pends on the way of encoding. In general, a framework of results that will be presented in more detail in Sec. III.B. Ap- coherence is speakable, if it allows√ for a free transformation√ proximate transformations for the class TIO have also been between the states (|0i + |1i)/ 2 and (|0i + |2i)/ 2. Oth- investigated recently by (Marvian and Lloyd, 2016). erwise, the framework of coherence is unspeakable (Marvian and Spekkens, 2016). In TableI we summarize the aforementioned frameworks of coherence according to this classifica- B. Coherence as a resource tion. Before we move on, we would like to discuss two possi- We now proceed to explore the use of coherence as a re- ble extensions of incoherent operations which also apply to source for enabling operations that would not otherwise be other classes that have been presented above. First, motivated possible if only incoherent operations were accessible to the by observations in entanglement theory (Jonathan and Plenio, experimenter. 1999) one might define the set of catalytic incoherent operations (Baumgratz et al., 2014) by allowing for the use of an 1. Maximally coherent states and state transformations via additional physical system of arbitrary dimension, the cata- incoherent operations lyst, in a state η which has to be returned unchanged after the transformation. That is, for arbitrary states % and σ of the We start by identifying a d-dimensional maximally coher- system and (bipartite) incoherent operations Λ we require ent state as a state that allows for the deterministic generation of all other d-dimensional quantum states by means of the free ⊗ ⊗ Λ[% η] = σ η. (9) operations. Note that this definition is independent of a specific coherence quantifier and allows to identify a unit for co- Indeed, in the theory of entanglement the addition of catalysts herence (coherence bit, or cobit)5 to which all measures may is known to confer additional power in that it makes possible be normalized; see also Sec. III for more details on quantify- state transformations that would otherwise be impossible un- ing coherence. The canonical example of a maximally coher- der LOCC alone (Jonathan and Plenio, 1999). As was shown ent state is by (Bu et al., 2016), these results also hold in the resource − theory of coherence: catalytic incoherent operations allow for 1 Xd 1 pure state transformations which cannot be achieved with in- |Ψdi = √ |ii (10) d coherent operations alone. The role of catalysts for the class i=0 TIO has also been discussed in (Marvian and Lloyd, 2016; Marvian and Spekkens, 2013). Concerning the second possible extension, up until this 5 The name “cobit” for a unit of coherence has been first used in (Chitambar point we have considered only exact state transformations, an and Hsieh, 2016) 7

T as it is easy to see that by IO, any d-dimensional state % may state with Bloch vector s = (sx, sy, sz) via SIO, DIO, IO be prepared from |Ψdi with certainty (Baumgratz et al., 2014). or MIO if and only if the following inequalities are fulfilled We note that not all frameworks of coherence that were dis- (Streltsov et al., 2017): cussed in Sec. II.A.2 have a maximally coherent state, i.e., a 2 2 2 2 state from which all other states can be created via the corre- sx + sy ≤ rx + ry , (12) sponding set of operations. In particular, such a state does not 1 − s2 1 − r2 exist for PIO (Chitambar and Gour, 2016a,b, 2017), GIO, and z ≥ z . 2 2 2 2 (13) FIO (de Vicente and Streltsov, 2017). sx + sy rx + ry The full set of maximally coherent states is obtained as the In the asymptotic setting, any state which is not incoher- orbit of |Ψdi under all incoherent unitaries (Bai and Du, 2015). ent allows for the distillation of maximally coherent states via For coherence theory the maximally coherent state |Ψdi plays a role analogous to the maximally entangled state in entan- IO (Winter and Yang, 2016). The optimal rate for this pro- glement theory. As we shall see later, both concepts are very cess can be evaluated analytically. We refer to Sec. III.B for closely related. One may also determine maximally coherent more details, where also the asymptotic state formation from states under certain additional constraints, such as the degree maximally coherent states is discussed. of mixedness, which gives rise to the class of maximally coherent mixed states (Singh et al., 2015a). For a d-dimensional d system and any fixed spectrum {pn}n=1,(Streltsov et al., 2. States and maps Pd 2016b) have proven that the state %max = n=1 pn |n+ihn+|, where {|n+i} denotes a mutually unbiased basis with respect A more complex task beyond state preparation is that of the 2 1 to the incoherent basis {|ii} (that is, |hi|n+i| = d ), is a univer- generation of a general quantum operation from a supply of sal maximally coherent mixed state with respect to any coher- coherent states and incoherent operations. Just as the maxi- ence monotone under the set MIO. Complementarity relations mally entangled state allows for the generation of all quantum between measures of purity and coherence have been further operations (Eisert et al., 2000) via LOCC, so does the maxi- investigated in (Cheng and Hall, 2015; Giorda and Allegra, mally coherent state allow for the generation of all quantum 2016a; Kumar, 2017; Streltsov et al., 2016b; Xi et al., 2015b). operations via IO. The explicit construction for an arbitrary The interconversion of pure states via incoherent operations single-qubit unitary can be found in (Baumgratz et al., 2014), has been studied by (Chitambar and Gour, 2016a,b, 2017; Du and the extension to general quantum operations of arbitrary et al., 2015, 2017; Winter and Yang, 2016; Zhu et al., 2017). dimension was studied by (Ben Dana et al., 2017; Chitam- In particular, a pure state |φi can be converted into another bar and Hsieh, 2016). In particular, it was shown by (Ben pure state |ψi via IO or SIO if and only if ∆[|ψihψ|] majorizes Dana et al., 2017) that any quantum operation acting on a ∆[|φihφ|](Winter and Yang, 2016; Zhu et al., 2017), i.e., Hilbert space of dimension d can be implemented via IO by consuming a maximally coherent state of dimension d. The ∆[|ψihψ|] ∆[|φihφ|] (11) corresponding construction makes use of maximally coherent states even if the targeted quantum operation may only is a necessary and sufficient condition for the transformation be very slightly coherent. It is an open question how much |φi 7→ |ψi via IO or SIO6. Here, the majorization relation for coherence is required for creating an arbitrary unitary, or an density matrices % σ means that their spectra spec(%) = arbitrary quantum operation in general. However, by virtue (p ≥ ... ≥ p ) and spec(σ) = (q ≥ ... ≥ q ) fulfill the 1 d 1 d of the monotonicity of coherence quantifiers under incoherent relation Pt p ≥ Pt q for all t < d, and Pd p = Pd q . i=1 i j=1 j i=1 i j=1 j operations, lower bounds to the amount of coherence required Note that a similar majorization condition is also known in to implement a quantum operation can be provided (Ben Dana entanglement theory (Nielsen, 1999). et al., 2017; Bu et al., 2017; Mani and Karimipour, 2015). Less progress has been made for the single copy transformations of mixed coherent states and only isolated results are known. A notable result in this context was provided by (Chi- tambar and Gour, 2016a,b, 2017), who gave a full charac- C. Quantum coherence in distributed scenarios terization of single-qubit state conversion via SIO, DIO, IO, or MIO. In particular, a single-qubit state with Bloch vector Based on the framework of LOCC known from entangle- T r = (rx, ry, rz) can be converted into another single-qubit ment theory (Horodecki et al., 2009; Plenio and Virmani, 2007), one can introduce the framework of local incoherent operations and classical communication (LICC) (Chitambar and Hsieh, 2016; Streltsov et al., 2017). In both concepts 6 For IO, this result was first claimed by (Du et al., 2015), however the proof one has two separated parties, Alice and Bob, who are con- turned out to be incomplete (Chitambar and Gour, 2016a,b, 2017; Winter nected via a classical channel (such as a telephone). While and Yang, 2016). In a recent erratum, (Du et al., 2017) addressed the criti- cism of (Chitambar and Gour, 2016a,b), and presented an alternative proof in the LOCC framework Alice and Bob are allowed to locally for systems of dimension 3. A complete proof for IO in any dimension was perform any quantum operation which is compatible with the presented recently by (Zhu et al., 2017). laws of quantum mechanics, in the framework of LICC the 8 parties are restricted to local incoherent operations only7. states onto itself, defined in (Ma et al., 2016). We also note While LICC operations in general have a difficult mathe- that GOIB can create entanglement whilst using up coherence matical structure, it is possible to introduce the more general (Matera et al., 2016), which is not possible for LQICC and class of separable incoherent (SI) operations which has a sim- SQI operations. Finally, we note that the free states under ple mathematical form (Streltsov et al., 2017): LICC and SI operations are bipartite incoherent states, i.e., convex combinations of incoherent product states (Streltsov AB X AB † † Λ[% ] = (Ai ⊗ Bi)% (Ai ⊗ Bi ), (14) et al., 2017). i where Ai and Bi are local incoherent operators. General quantum operations of the form (14), but without the incoherence D. Connection between coherence and entanglement theory restriction, have been studied extensively in entanglement theory, where they are called separable operations (Vedral and The resource theory of coherence exhibits several connec- Plenio, 1998). It is an interesting open question whether the tions to the resource theory of entanglement. One of the first intersection of LOCC and SI operations is equal to LICC op- approaches in this direction was presented in (Streltsov et al., erations (Streltsov et al., 2017). 2015), where it was shown that any state with nonzero co- Asymmetric scenarios where only one of the parties is re- herence can be used for entanglement creation via bipartite stricted to IO locally have also been considered (Chitambar incoherent operations; we refer to Sec. III.E for a detailed dis- et al., 2016; Streltsov et al., 2017). In the case where the cussion. These results were generalized to quantum discord second party (Bob) is restricted to incoherent operations, the (Ma et al., 2016) and general types of nonclassicality (Killo- corresponding sets of operations are called local quantum- ran et al., 2016); see Secs. III.I and III.K.3 for more details. incoherent operations and classical communication (LQICC) The interplay between coherence and entanglement in dis- and separable quantum-incoherent operations (SQI) respec- tributed scenarios has been first studied by (Chitambar and tively. LQICC operations are in particular important for the Hsieh, 2016), who investigated state formation and distillation tasks of incoherent quantum state merging (Streltsov et al., of entanglement and local coherence via LICC operations. In- 2016a) (see also next Section) and assisted coherence distil- terestingly, it was shown that a bipartite state has distillable lation (Chitambar et al., 2016; Streltsov et al., 2017). The entanglement if and only if entanglement can be distilled via latter task has also been performed experimentally (Wu et al., LICC (Chitambar and Hsieh, 2016). 2017), and will be discussed in more detail in Sec. III.L.3. Another relation between entanglement and coherence was A related set of operations has been presented by (Matera provided in (Streltsov et al., 2016a), where the authors intro- et al., 2016) in the context of a resource theory of control of duced and studied the task of incoherent quantum state merg- quantum systems, and called global operations incoherent on ing. The latter is an incoherent version of standard quan- B (GOIB). Those operations allow for incoherent operations tum state merging (Horodecki et al., 2005, 2007), where two on the subsystem B, and moreover they also allow for con- parties aim to merge their parts of a tripartite quantum state trolled operations in the incoherent reference basis from B to while preserving correlations with a third party. While stan- A. The latter amount to arbitrary control unitaries of the form dard quantum state merging can lead to a gain of entangle- P A B ment (Horodecki et al., 2005, 2007), no merging protocol can UC = i Ui ⊗ |iihi| (Matera et al., 2016). Moreover, the framework also allows to attach or discard subsystems of A, lead to a gain of entanglement and coherence simultaneously and to perform measurements with postselection on this sub- (Streltsov et al., 2016a). system (Matera et al., 2016). Finally, we mention that the resource theory of coherence The sets LQICC, SQI, and GOIB lead to the same set QI of shares many similarities with the resource theory of entangle- free states, which are called quantum-incoherent states (Chi- ment, if the latter is restricted to maximally correlated states (Chitambar et al., 2016; Winter and Yang, 2016). In particu- tambar et al., 2016; Matera et al., 2016; Streltsov et al., 2017) P lar, given a quantum state % = %i j |iih j|, we can assign to it and take the form i j P a bipartite maximally correlated state as %mc = i j %i j |iiih j j|. X A B % = piσi ⊗ |iihi| . (15) An important open question in this context is whether any i of the aforementioned classes of incoherent operations Λi is equivalent to LOCC operations ΛLOCC on maximally corre- σA A |iiB Here, i are arbitrary states on the subsystem , while are lated states, i.e., incoherent states on the subsystem B. Moreover, LQICC is a strict subset of SQI (Streltsov et al., 2017) and GOIB (Matera ? σ = Λ [%] ⇔ σ = Λ [% ]. (16) et al., 2016). In turn, the latter is a strict subset of the maxi- i mc LOCC mc mal set of operations mapping the set of quantum-incoherent Partial answers to this questions were presented for IO in (Chitambar et al., 2016; Winter and Yang, 2016) and for GIO in (de Vicente and Streltsov, 2017). As we will also discuss in 7 In the frameworks studied in (Chitambar and Hsieh, 2016; Streltsov et al., the following Section, several quantifiers of coherence known 2017) local operations were restricted to the IO class, but other classes of today coincide with their equivalent entanglement quantifiers local free operations can also be considered. for the corresponding maximally correlated state. 9

III. QUANTIFYING QUANTUM COHERENCE Conditions C1 and C2 can be seen as minimal requirements: any quantity C should at least fulfill these two condi- A. Postulates for coherence monotones and measures tions in order to be a meaningful resource quantifier for some coherence-based task. Condition C3 quantifies the intuition The first axiomatic approach to quantify coherence has that coherence should not increase under incoherent measure- been presented by (Åberg, 2006), and an alternative frame- ments even if one has access to the individual measurement work has been developed more recently by (Baumgratz et al., outcomes. This condition C3, when combined with convex- 2014). The latter approach can be seen as parallel to the ax- ity C4, implies monotonicity C2 (Baumgratz et al., 2014). iomatic quantification of entanglement, first introduced two The reason for this overcompleteness comes from entangle- decades ago (Vedral and Plenio, 1998; Vedral et al., 1997). ment theory: there are meaningful quantifiers of entanglement The basis of the axiomatic approach consists of the follow- which satisfy only some of the above conditions (Horodecki ing postulates that any quantifier of coherence C should ful- et al., 2009; Plenio and Virmani, 2007). fill (Åberg, 2006; Baumgratz et al., 2014; Levi and Mintert, Following standard notions from entanglement theory (Ple- 2014): nio and Virmani, 2007), we call a quantity C which fulfills condition C1 and either condition C2 or C3 (or both) a coher- (C1) Nonnegativity: ence monotone. A quantity C will be further called a coher- C(%) ≥ 0 (17) ence measure if it fulfills C1–C4 together with the following two additional conditions: in general, with equality if and only if % is inco- (C5) Uniqueness for pure states: For any pure state |ψi herent. C takes the form (C2) Monotonicity: C does not increase under the ac- | ih | | ih | tion of incoherent operations8, i.e., C( ψ ψ ) = S (∆[ ψ ψ ]), (21) − C(Λ[%]) ≤ C(%) (18) where S (%) = Tr[% log2 %] is the von Neumann entropy9. for any incoherent operation Λ. (C6) Additivity: C is additive under tensor products, (C3) Strong monotonicity: C does not increase on av- i.e., erage under selective incoherent operations, i.e., C(% ⊗ σ) = C(%) + C(σ). (22) X qiC(σi) ≤ C(%) (19) i We wish to remark that the terminology adopted here differs from the one used in some recent literature, which is mainly † with probabilities qi = Tr[Ki%Ki ], post- based on (Baumgratz et al., 2014). In particular, several au- † thors require that a coherence measure only fulfills conditions measurement states σi = Ki%Ki /qi, and incoherent Kraus operators Ki. C1–C4. With the more stringent approach presented here, inspired from entanglement theory, we aim to distinguish two (C4) Convexity: C is a convex function of the state, i.e., important coherence quantifiers: distillable coherence (which   is equal to the relative entropy of coherence) and coherence X X  ≥   cost (which is equal to the coherence of formation). As we piC(%i) C  pi%i . (20) i i will see in the following, these two quantities fulfill all conditions C1–C6 and are thus elevated to the status of measures. At this point it is instrumental to compare conditions C2 and Most of the other coherence quantifiers presented in this Col- C3 to the corresponding conditions in entanglement theory loquium remain coherence monotones, in the sense that they (Horodecki et al., 2009; Plenio and Virmani, 2007; Vedral and fulfill conditions C1 and C2; several of them also turn out to Plenio, 1998; Vedral et al., 1997). There, C2 is equivalent to fulfill C3 and C4, however most violate C5 and C6. the requirement that an entanglement quantifier E should not Finally, we note that alternative or additional desirable re- increase under LOCC. Similarly, C3 is equivalent to the requirements for coherence monotones may be proposed. In quirement that E should not increase on average under selective LOCC operations, i.e., when the communicating parties are able to store their measurement outcomes. 9 Note that this condition may be considered too strong in view of the fact that the quantity in the right-hand side of Eq. (21), i.e. the relative entropy of coherence (discussed in Sec. III.C.1), adopts its operational meaning in terms of coherence distillation and dilution only in the asymptotic 8 While this condition was originally proposed for the set IO (Baumgratz limit. In a weaker form, postulate C5 might read instead: ∀ > 0 there ex- ⊗n et al., 2014), it can be generalized in a straightforward way to any set of ists a family of states Φ(n) ∈ B (|ψihψ| ) such that limn→∞ C(Φ(n))/n = operations discussed in Sec. II.A.2 S (∆[|ψihψ|]). 10 particular, as was shown by (Yu et al., 2016b), for the set IO Interestingly, restricting again Λi to the set IO, the coherence the conditions C3 and C4 can be replaced by the following cost admits a single-letter expression. In particular, it is equal single requirement of additivity on block-diagonal states: to the coherence of formation (Winter and Yang, 2016):

Cp% ⊕ (1 − p)σ = pC(%) + (1 − p)C(σ). (23) Cc(%) = C f (%). (27) When combining Eq. (23) with C1 and C2, these three conditions are equivalent to C1–C4 (Yu et al., 2016b). Furthermore, The coherence of formation will be defined and discussed in (Peng et al., 2016a) postulated that any valid coherence quan- detail in Sec. III.D. The result in Eq. (27) can be seen as par- tifier should be maximal only on the set of pure maximally allel to the well known fact that the entanglement cost is equal et al. coherent states |Ψdi, a property satisfied in particular by the to the regularized entanglement of formation (Hayden , two coherence measures mentioned just above; however, such 2001). However, in the case of coherence no regularization is states do not represent a golden unit in all versions of the re- required, which significantly simplifies the evaluation of Cc. source theory of coherence, as remarked in Sec. II.B.1. The coherence cost is a coherence measure, i.e., it ful- In the following Sections, we review the most relevant co- fills all conditions C1–C6. Conditions C1–C4 follow from herence monotones and measures defined in current literature, Eq. (27) and the fact that the coherence of formation fulfills as well as other recent quantitative studies of coherence — C1–C4 (Yuan et al., 2015). Condition C5 also follows from and nonclassicality more broadly — in single and multipartite Eq. (27) and the definition of coherence of formation, see Sec- quantum systems. tion III.D. Moreover, the coherence cost is additive, i.e., condition C6 is also satisfied (Winter and Yang, 2016). In general, the distillable coherence cannot be larger than B. Distillable coherence and coherence cost the coherence cost: The distillable coherence is the optimal number of maxi- Cd(%) ≤ Cc(%). (28) mally coherent single-qubit states |Ψ2i which can be obtained per copy of a given state % via incoherent operations in the For pure states this inequality becomes an equality, which im- asymptotic limit. The formal definition of distillable coher- plies that the resource theory of coherence is reversible for ence can be given as follows (Winter and Yang, 2016; Yuan pure states. In particular, any state |ψ i with a distillable co- et al., 2015): 1 herence of c cobits can be asymptotically converted into any ( ! ) 1 other state |ψ i with a distillable coherence of c cobits at a C % R %⊗n − | ih |⊗bnRc . 2 2 d( ) = sup : lim inf Λi[ ] Ψ2 Ψ2 1 = 0 n→∞ Λi rate c1/c2. However, there exist mixed states with Cd strictly (24) smaller than Cc (Winter and Yang, 2016). Nevertheless, this From this definition it is tempting to believe that exact evalua- phenomenon does not appear in its extremal form, i.e., there tion of distillable coherence is out of reach. Surprisingly, this are no states with zero distillable coherence but nonzero co- is not the case, and a simple expression for distillable coher- herence cost (Winter and Yang, 2016). Therefore, in contrast ence of an arbitrary mixed state, with Λi belonging to the set to entanglement theory (Horodecki et al., 1998), there is no IO, was given in (Winter and Yang, 2016): ‘bound coherence’ within the resource theory of coherence based on the set IO (Winter and Yang, 2016). − Cd(%) = S (∆[%]) S (%). (25) Notice instead that, if one considers the maximal set MIO For pure states this result was independently found by (Yuan of incoherent operations, the corresponding resource theory et al., 2015). Interestingly, the distillable coherence also co- of coherence becomes reversible also for mixed states, as the incides with the relative entropy of coherence which was in- general framework of (Brandão and Gour, 2015) applies in troduced in (Baumgratz et al., 2014) and will be discussed in this case (Winter and Yang, 2016). This means that the distil- Sec. III.C.1. Since the relative entropy of coherence is a co- lable coherence under MIO remains equal to the relative en- herence measure (Baumgratz et al., 2014), the same is true for tropy of coherence, i.e. Eq. (25) still holds under the largest the distillable coherence, i.e., it fulfills all requirements C1– set of free operations, while the coherence cost under MIO C6. also reduces to the same quantity, Cc(%) = Cd(%), thus closing 10 Another important quantifier is the coherence cost (Winter the irreversibility gap . We further note that not all sets of and Yang, 2016; Yuan et al., 2015): it quantifies the minimal operations presented in Sec. II.A.2 allow for free coherence distillation; this is further discussed in Sec.VI. rate of maximally coherent single-qubit states |Ψ2i required to produce a given state % via incoherent operations in the asymptotic limit. Its formal definition is similar to that of entanglement cost, and can be given as follows: 10 ( ! ) A similar situation occurs in the case of entanglement, for which a fully re- ⊗n h ⊗bnRci versible resource theory can also be constructed if one considers the largest Cc(%) = inf R : lim inf % − Λi |Ψ2ihΨ2| = 0 . n→∞ Λi 1 set of operations preserving separability (Brandão and Plenio, 2008, 2010), (26) which is a strict superset of separable operations. 11

C. Distance-based quantifiers of coherence note that the relative entropy between an arbitrary state % and another incoherent state σ ∈ I can be written as A general distance-based coherence quantifier is defined as (Baumgratz et al., 2014) S (%||σ) = S (∆[%]) − S (%) + S (∆[%]||σ), (35)

CD(%) = inf D(%, σ), (29) which is straightforward to prove using the relation σ∈I Tr[% log2 σ] = Tr[∆[%] log2 σ]. Minimizing the right-hand where D is a distance and the infimum is taken over the set of side of Eq. (35) over all incoherent states σ ∈ I we imme- incoherent states I. Clearly, any quantity defined in this way diately see that the minimum is achieved for σ = ∆[%], which fulfills C1 for any distance D(%, σ) which is nonnegative and completes the proof of Eq. (34). From Eq. (34) it also fol- zero if and only if % = σ. Monotonicity C2 is also fulfilled for lows that the relative entropy of coherence fulfills conditions any set of operations discussed in Sec. II.A.2 if the distance is C5 and C6. contractive (Baumgratz et al., 2014), As was further shown by (Singh et al., 2015b), the relative entropy of coherence can also be interpreted as the mini- D(Λ[%], Λ[σ]) ≤ D(%, σ) (30) mal amount of noise required for fully decohering the state %. Moreover, it was shown in (Rodríguez-Rosario et al., 2013) for any quantum operation Λ. that the relative entropy of coherence is related to the devia- Moreover, any distance-based coherence quantifier fulfills tion of the state from thermal equilibrium. convexity C4 whenever the corresponding distance is jointly Possible extensions of the relative entropy of coherence to convex (Baumgratz et al., 2014), quantifiers based on the relative Rényi and Tsallis entropies X X X have also been discussed (Chitambar and Gour, 2016a,b, D pi%i, p jσ j ≤ piD (%i, σi) . (31) 2017; Rastegin, 2016). i j i

In the following, we will discuss explicitly three important 2. Coherence quantifiers based on matrix norms distance-based coherence quantifiers. We will now consider coherence quantifiers based on ma- 1. Relative entropy of coherence trix norms, i.e., such that the corresponding distance has the form D(%, σ) = ||% − σ|| with some matrix norm || · ||. We If the distance is chosen to be the quantum relative entropy first note that any such distance is jointly convex, i.e., fulfills Eq. (31), as long as the corresponding norm || · || fulfills the 12 S (%||σ) = Tr[% log2 %] − Tr[% log2 σ], (32) triangle inequality and absolute homogeneity (Baumgratz et al., 2014). Thus, any matrix norm with the aforementioned the corresponding quantifier is known as the relative entropy properties gives rise to a convex coherence quantifier. Rele- 11 of coherence (Baumgratz et al., 2014): vant norms with these properties are the lp-norm || · ||lp and the Schatten p-norm || · ||p: Cr(%) = min S (%||σ). (33) σ∈I  1/p X   p The relative entropy of coherence fulfills conditions C1, C4, kMkl =  Mi j  , (36) p   and C2 for any set of operations discussed in Sec. II.A.2. For i, j 1/p the set IO it also fulfills C3 (Baumgratz et al., 2014). More- † p/2 kMkp = Tr M M (37) over, it is equal to the distillable coherence Cd, therefore both quantities admit the same closed expression (Baumgratz et al., 2014; Gour et al., 2009; Winter and Yang, 2016) with p ≥ 1. The corresponding coherence quantifiers will be

denoted by Clp and Cp respectively.

Cr(%) = Cd(%) = S (∆[%]) − S (%), (34) The l1-norm of coherence Cl1 was ﬁrst introduced and stud-

ied in (Baumgratz et al., 2014). In particular, Cl1 fulfills the where ∆[%] is the dephasing operation defined in Eq. (5). The conditions C1–C4 for the set IO and has the following simple proof of this equality is remarkably simple: it is enough to expression (Baumgratz et al., 2014): X Cl1 (%) = min k% − σkl = |%i j|. (38) σ∈I 1 i, j 11 The relative entropy of coherence defined in Eq. (33) coincides with a special instance of the relative entropy of superposition (Åberg, 2006) and of the relative entropy of asymmetry (Gour et al., 2009; Vaccaro et al., 2008), which will be discussed in Sec. III.K. Furthermore, the quantity in the right- hand side of Eq. (34) was independently proposed as a coherence quantifier 12 Absolute homogeneity of a matrix norm || · || means that ||aM|| = |a| × ||M|| by (Herbut, 2005) under the name coherence information. holds for any complex number a and any matrix M. 12

For the maximally coherent state Cl1 takes the value quantities in the single-qubit case. As we will see in the fol-

Cl1 (|Ψdi) = d − 1, which also means that Cl1 does not ful- lowing subsections, Cg can also be considered as a convex ﬁll conditions C5 and C6. It has also been shown in (Bu and roof quantiﬁer of coherence, and is also closely related to the

Xiong, 2016) that Cl1 is a DIO and MIO monotone for d = 2, geometric entanglement. Upper and lower bounds on the ge- while it violates the monotonicity condition C2 for DIO and ometric coherence have been investigated in (Zhang et al., MIO for d > 2. 2017a). For p = 1 the corresponding Schatten norm reduces to the Another related quantity was introduced by (Baumgratz trace norm. Since the trace norm is contractive under quantum et al., 2014) and studied in (Shao et al., 2015) where it was p operations, the corresponding coherence quantifier C1 satis- called fidelity of coherence: CF (%) = 1 − maxσ∈I F(%, σ). fies the conditions C1, C4, and C2 for any set of operations While CF fulfills conditions C1, C4, and C2 for any set of op- discussed in Sec. II.A.2. However, C1 violates the condition erations discussed in Sec. II.A.2, it violates the strong mono- C3 for the set IO, as was shown in (Yu et al., 2016b) based on tonicity condition C3 for IO (Shao et al., 2015). the violation of property in Eq. (23). Nevertheless, for single- qubit states C1 is equivalent to Cl1 , and thus the condition C3 is fulfilled in this case (Shao et al., 2015). Similar results were D. Convex roof quantifiers of coherence obtained for general X-states: also in this case C1 is equiva- Provided that a quantifier of coherence has been defined lent to Cl1 , and condition C3 is fulfilled for the set IO (Rana et al., 2016). The characterization of the closest incoherent for all pure states, it can be extended to mixed states via the state with respect to the trace norm is nontrivial in general, standard convex roof construction (Yuan et al., 2015): and partial results for pure states and X-states have been ob- X C(%) = inf piC(|ψii), (41) tained in (Chen et al., 2016a; Rana et al., 2016). {pi,|ψii} i For p = 2 the Schatten norm is equivalent to the Hilbert- where the infimum is taken over all pure state decompositions Schmidt norm, and also equal to the l2-norm. In this con- P text, (Baumgratz et al., 2014) studied the case where coher- of % = i pi |ψiihψi|. Constructions of this type have been pre- ence is quantified via the squared Hilbert-Schmidt norm. They viously introduced and widely studied in entanglement theory found that the coherence quantifier obtained in this way vio- (Bennett et al., 1996; Uhlmann, 1998). lates strong monotonicity C3 for the set IO. For any p ≥ 1, If for pure states the amount of coherence is quantified by the distillable coherence C (|ψi) = S (∆[|ψihψ|]), as in Cp is a convex coherence monotone for all single-qubit states, d i.e., it fulfills C1, C4, and C2 for the set IO (Rana et al., 2016). Eq. (21), the corresponding convex roof quantifier is known as the coherence of formation (Åberg, 2006; Winter and Yang, However, in general Clp and Cp violate conditions C2 and C3 for the set IO for all p > 1 in higher-dimensional systems 2016; Yuan et al., 2015): (Rana et al., 2016). Interestingly, Cp is a coherence monotone X C f (%) = inf piS (∆[|ψiihψi|]). (42) for the set GIO for all p ≥ 1 (de Vicente and Streltsov, 2017). {p ,|ψ i} i i i

As already noted in Sec. III.B, C f is equal to the coherence 3. Geometric coherence cost under IO. As was shown by (Yuan et al., 2015), the coherence of formation13 fulfills conditions C1–C4 for the set Here we will consider the geometric coherence defined in IO. By definition, C f also fulfills C5, and additivity C6 was (Streltsov et al., 2015) as follows: proven in (Winter and Yang, 2016). Remarkably, C f violates monotonicity C2 for the class MIO (Hu, 2016). P Cg(%) = 1 − max F(%, σ) (39) For a general state % = i j %i j |iih j|, the coherence of for- σ∈I mation is equal to the entanglement of formation (Bennett √ √ with the fidelity F(%, σ) = || % σ||2. The geometric coher- et al., 1996) of the corresponding maximally correlated state 1 P ence fulfills conditions C1, C4, and C2 for any set of opera- %mc = i j %i j |iiih j j| (Winter and Yang, 2016): tions discussed in Sec. II.A.2. Additionally, it also fulfills C3 C (%) = E (% ). (43) for the set IO (Streltsov et al., 2015). For pure states, the ge- f f mc 2 ometric coherence takes the form Cg(|ψi) = 1 − maxi |hi|ψi| , Using the formula for the entanglement of formation of two- which means that Cg does not meet conditions C5 and C6. qubit states (Wootters, 1998), Eq. (43) implies that the coher- If % is a single-qubit state, Cg admits the following closed ence of formation can be evaluated exactly for all single-qubit expression (Streltsov et al., 2015): states (Yuan et al., 2015):  p  q ! − | |2 1 2 1 + 1 4 %01  Cg(%) = 1 − 1 − 4|%01| , (40) C (%) = h   , (44) 2 f  2  where %01 = h0|%|1i is the offdiagonal element of % in the incoherent basis. Note that for all single-qubit states we have 13 (Yuan et al., 2015) called this quantity intrinsic randomness. C1 = Cl1 = 2|%01|, and thus Cg is a simple function of these 13

S where h(x) = −x log2 x − (1 − x) log2(1 − x) is the binary % has nonzero coherence, some incoherent operation Λi can entropy. If we compare this expression with the expression in fact create entanglement between the system S and the an- for the geometric coherence in Eq. (40), it follows that C f = cilla A (see also Fig.1, right panel). This finding is quantified h(1 − Cg) holds for any single-qubit state. Thus, C f is also a in Eq. (47): CE is a coherence monotone whenever E is an simple function of C1 and Cl1 in this case. entanglement monotone. In particular, CE fulfills conditions The coherence concurrence was defined in (Qi et al., 2016) C1–C4 for the set IO whenever E fulfills the corresponding as the convex roof of the l1-norm of coherence, conditions in entanglement theory (Streltsov et al., 2015). X In various situations CE admits an explicit formula. In par- C (%) = inf p C (|ψ ihψ |). (45) c1 i l1 i i ticular, if E is the distillable entanglement, CE amounts to the {pi,|ψii} i distillable coherence (Streltsov et al., 2015):

It follows by deﬁnition that Cc1 (|ψihψ|) = Cl1 (|ψihψ|) for any CEd (%) = Cd(%). (48) pure state |ψi, while Cc1 (%) ≥ Cl1 (%) for an arbitrary mixed state %. The coherence concurrence satisﬁes properties C1– If E is the relative entropy of entanglement, CE is the relative C4 for the set IO as proven in (Qi et al., 2016), but it violates entropy of coherence, which again is equal to Cd. A simi- C5 and C6. The relation between coherence concurrence and lar relation can be found for the geometric entanglement and entanglement concurrence (Wootters, 1998) was also investi- the geometric coherence: CEg (%) = Cg(%)(Streltsov et al., gated in (Qi et al., 2016). In particular, for a single-qubit state 2015). For distillable entanglement, relative entropy of en-

%, Cc1 (%) = Cl1 (%) = 2|%01|, which means that the coherence tanglement, and geometric entanglement, the supremum in concurrence of % is equal to the entanglement concurrence of Eq. (47) is achieved when Λi is the generalized CNOT op- the corresponding maximally correlated two-qubit state %mc, eration, i.e., the optimal incoherent operation is the unitary and that the functional relation between C f and Cc1 for a qubit, obtained from Eq. (44), is the same as that between the entan- UCNOT |ii |0i = |ii |ii . (49) glement of formation and the entanglement concurrence for It is not known if this unitary is the optimal incoherent opera- two qubits, established in (Wootters, 1998). tion for all entanglement monotones E. The geometric coherence C introduced in Eq. (39) can also g If entanglement is quantified via the distance-based ap- be regarded as a convex roof quantifier: proach (Vedral et al., 1997), X Cg(%) = inf piCg(|ψii). (46) {p ,|ψ i} ED(%) = inf D(%, σ) (50) i i i σ∈S This can be proven using a general theorem in (Streltsov et al., with a contractive distance D, it was further shown in 2010) (see Theorem 2 in the appendix there). Moreover, (Streltsov et al., 2015) that the creation of entanglement from Eq. (43) also holds for the geometric coherence and entan- a state %S via incoherent operations is bounded above by its glement: Cg(%) = Eg(%mc), where the geometric entanglement distance-based coherence: is defined as E (%) = 1 − max F(%, σ), and S is the set h i g σ∈S S :A S ⊗ | i h |A ≤ S of separable states (Streltsov et al., 2010; Wei and Goldbart, ED Λi % 0 0 CD % . (51) 2003). While this result was originally proven for Λi ∈ IO in (Streltsov et al., 2015), it generalizes to any set of incoherent operations presented in Sec. II.A.2. In case the distance D E. Coherence monotones from entanglement is chosen to be the quantum relative entropy, there exists an S An alternative approach to quantify coherence has been in- incoherent operation saturating the bound for any state % as ≥ troduced in (Streltsov et al., 2015). In particular, the authors long as dA dS . The same is true if the distance is chosen as − showed that any entanglement monotone E gives rise to a co- D(%, σ) = 1 F(%, σ). In both cases, the bound is saturated by herence monotone C via the following relation: the generalized CNOT operation, see Eq. (49). These results E also imply that a state %S can be used to create entanglement ( ) S S S :A h S Ai via incoherent operations if and only if % has nonzero coher- CE(% ) = lim sup E Λi % ⊗ |0i h0| . (47) d →∞ ence (Streltsov et al., 2015). A Λi

Here, %S is a state of the system S , and A is an additional F. Robustness of coherence ancilla of dimension dA. The supremum is performed over all bipartite incoherent operations Λi ∈ IO, i.e., such that the corresponding Kraus operators map the product basis |ki |li Another coherence monotone was introduced by (Napoli onto itself. et al., 2016; Piani et al., 2016), and termed robustness of co- The intuition behind the entanglement-based coherence herence. For a given state %, it quantifies the minimal mixing S required to make the state incoherent: quantifiers CE is the following: if the state % is incoherent, then the total state Λ [%S ⊗ |0i h0|A] will remain separable for i % + sτ RC(%) = min s ≥ 0 ∈ I , (52) any bipartite incoherent operation Λi. However, if the state τ 1 + s 14 where the minimum is taken over all quantum states τ. A sim- applied in each arm, the output state of the particle can be ilar quantity was studied earlier in entanglement theory un- written as %(ϕ~) = U(ϕ~)%U†(ϕ~), where % is the input state and P iϕ j der the name robustness of entanglement (Steiner, 2003; Vidal U(ϕ~) = j e | jih j|. Then, by placing an output detector and Tarrach, 1999). The robustness of coherence fulfills C1, which implements a measurement described by a positive- C4, and C2 for all sets of operations discussed in Sec. II.A.2, operator valued measure M = (Mω), one observes outcomes ω and additionally it also fulfills C3 for IO (Napoli et al., 2016; sampled from the Born probability pM|%(ω|ϕ~) = Tr[%(ϕ~)Mω], Piani et al., 2016). Moreover, RC coincides with the l1-norm which constitute the interference pattern. One can then de- of coherence for all single-qubit states, all X-states, and all fine suitable visibility functionals V[pM|%], which intuitively pure states. The latter result also implies that RC does not capture the degree of variability of the interference pattern as comply with C5 and C6. a function of the phases {ϕ j}. It was then shown in (Biswas The robustness of coherence has an operational interpreta- et al., 2017) that, for any visibility functional V satisfying cer- tion which is related to the notion of coherence witnesses.A tain physical requirements, the corresponding optimal visibil- coherence witness is defined in a similar way as an entangle- ity (maximized over all output measurements M), defines a ment witness in entanglement theory: it is a Hermitian oper- valid interference-based coherence quantifier, ator W such that Tr[Wσ] ≥ 0 is true for all incoherent states σ ∈ I. Under the additional constraint W ≤ 11 it was shown CV (%) = sup V[pM|%], (54) in (Napoli et al., 2016; Piani et al., 2016) that the following M inequality holds true: which satisfies properties C1–C3 for the set SIO, and also C4 if V is convex in p. Various examples of visibility function- RC(%) ≥ max {0, −Tr[%W]} . (53) als and the corresponding coherence monotones were further discussed in (Biswas et al., 2017), showing in particular that Interestingly, for any state % there exists a witness W saturat- the robustness of coherence presented in Sec. III.F can be ing this inequality. On the one hand, this result means that the alternatively interpreted (up to a normalization factor) as an robustness of coherence is accessible in laboratory by measur- interference-based quantifier of the form (54) for a suitable V, ing the expectation value of a suitable witness W, as recently and that one can also obtain variants of the asymmetry mono- demonstrated in a photonic experiment (Wang et al., 2017). tones such as the Wigner-Yanase skew information discussed On the other hand, it also means that RC can be evaluated in Sec. III.K.1, which satisfy the necessary monotonicity re- via a semidefinite program. The robustness of coherence is quirements for coherence (with respect to the set SIO). These moreover a figure of merit in the task of quantum phase dis- results establish important links between the somehow more crimination; we refer to Sec. V.D for its definition and detailed abstract aspects in the resource theory of coherence and oper- discussion. Moreover, the results presented in (Napoli et al., ational notions in the physics of interferometers. 2016; Piani et al., 2016) also carry over to the resource theory of asymmetry, which is discussed in Sec. III.K.1.

H. Coherence of assistance G. Coherence quantifiers from interferometric visibility The coherence of assistance was introduced by (Chitambar Clearly, quantum coherence is required for the observation et al., 2016) as follows: of interference patterns, e.g. in the double slit experiment. Re- cently, this idea was formalized in (von Prillwitz et al., 2015), X C (%) = sup p S (∆[|ψ ihψ |]), (55) where the authors studied the problem to determine coherence a i i i {pi,|ψii} i properties from interference patterns. Similar ideas were also put forth in (Bera et al., 2015), where the authors studied the where the maximum is taken over all pure state decomposi- P role of the l1-norm of coherence in general multislit experi- tions of % = i pi |ψiihψi|. This quantity is dual to the coher- ments. The authors of (Bagan et al., 2016) derived two ex- ence of formation defined in Eq. (42) as the minimum over all act complementarity relations between quantifiers of coher- decompositions. If the state % is the maximally mixed single- ence and path information in a multipath interferometer, us- qubit state, it can be written as a√ mixture of the maximally ing respectively the l1-norm and the relative entropy of co- coherent states |±i = (|0i ± |1i)/ 2 with equal probabilities. herence. Studies on the quantification of interference and its For this reason we get Ca(11/2) = 1, i.e., the coherence of as- relationship with coherence have also been performed earlier, sistance is maximal for the maximally mixed state. On the one see e.g. (Braun and Georgeot, 2006). hand, this means that the coherence of assistance is not a co- The authors of (Biswas et al., 2017) recently presented a herence monotone, as it indeed violates condition C1. On the general framework to quantify coherence from visibility in in- other hand, the coherence of assistance plays an important role terference phenomena. Consider a multi-path interferometer, for the task of assisted coherence distillation, see Sec. III.L.3 in which a single particle can be in one of d paths, denot- for a detailed discussion. d−1 ing the path variable by a set of orthogonal vectors {| ji} j=0 While finding a closed expression for the coherence of as- which define the reference basis. If a local phase shift ϕ j is sistance seems difficult in general, its regularization admits a 15 simple expression (Chitambar et al., 2016): Recently, (Ma et al., 2016) studied the role of coherence for creating general quantum correlations. The authors showed ∞ 1 ⊗n Ca (%) = lim Ca % = S (∆[%]). (56) that the creation of general quantum correlations from a state n→∞ n % is bounded above by its coherence. In particular, if D is a Moreover, for all single-qubit states % the coherence of assis- contractive distance, the following relation holds for any pair tance is n-copy additive, and thus can be written as Ca(%) = of quantifiers QD and CD (Ma et al., 2016): S (∆[%]). There is further a close relation between the coher- S |A h S Ai S ence of assistance and the entanglement of assistance intro- QD Λi % ⊗ σ ≤ CD % . (62) duced in (DiVincenzo et al., 1999). In particular, the coher- P S ence of assistance of a state % = i j %i j |iih j| is equal to the Here, S is a system in an arbitrary state % and A is an an- entanglement of assistance of the corresponding maximally cilla in an incoherent state σA. While this result was orig- P ∈ correlated state %mc = i j %i j |iiih j j| (Chitambar et al., 2016): inally proven for Λi IO in (Ma et al., 2016), it generalizes to any set of operations discussed in Sec. II.A.2. These Ca(%) = Ea(%mc). (57) results are parallel to the discussion on creation of entanglement from coherence presented in (Streltsov et al., 2015), see also Sec. III.E. In particular, any state %S with nonzero co- I. Coherence and quantum correlations beyond herence can be used for creating discord via incoherent op- entanglement erations, since any such state can be used for the creation of entanglement (Streltsov et al., 2015). Quantum discord (Henderson and Vedral, 2001; Ollivier A general framework to define quantifiers of discord and and Zurek, 2001; Zurek, 2000) is a measure of quantum corre- quantumness in a bipartite system from corresponding quan- lations going beyond entanglement (Adesso et al., 2016; Modi tifiers of quantum coherence, by minimizing the latter over all et al., 2012; Oppenheim et al., 2002; Streltsov, 2015). A bi- local bases for one or both subsystems respectively, has been partite quantum state %AB is said to have zero discord (with formalized recently in (Adesso et al., 2016). respect to the subsystem A) if and only if it can be written as

AB X A B % = pi |eii hei| ⊗ % , (58) cq i J. Coherence in continuous variable systems i where the states {|eii} form an orthonormal basis, but are not The resource theory framework for quantum coherence necessarily incoherent. The states of Eq. (58) are also known adopted in this Colloquium assumes a finite-dimensional as classical-quantum (Piani et al., 2008), and the correspond- Hilbert space. However, some of the previously listed quanti- ing set will be denoted by CQ. Correspondingly, a state %AB fiers of coherence have also been studied in continuous vari- is called (fully) classically correlated, or classical-classical, if able systems, and specifically in bosonic modes of the radi- and only if it can be written as (Piani et al., 2008) ation field. These systems are characterized by an infinite- ∞ X dimensional Hilbert space, spanned by the Fock basis {|ni}n=0 AB A B † %cc = pi j |eii hei| ⊗ |e ji he j| , (59) of eigenstates of the particle number operator a a (Braunstein i, j and van Loock, 2005). and the corresponding set will be denoted by CC. If a state Similarly to what was done by (Eisert et al., 2002) in entanglement theory, (Zhang et al., 2016b) imposed a finite mean is not fully classically correlated, we say that it possesses † nonzero general quantum correlations. This notion can be energy constraint, ha ai ≡ n¯ < ∞, to address the quan- straightforwardly extended to more than two parties (Piani tification of coherence in such systems with respect to the et al., 2011). Fock reference basis. The relative entropy of coherence (see The amount of discord in a given state can be quantified Sec. III.C.1) was found to maintain its status as a valid mea- via a distance-based approach similar to the distance-based sure of coherence, in particular reaching a finite maximum max − ∞ approach for coherence presented in Sec. III.C. In particular, Cr = (¯n + 1) log(¯n + 1) n¯ logn ¯ < for any state with one can define general distance-based quantifiers of discord finite mean energyn ¯ < ∞. On the contrary, it was shown that A|B A|B the l1-norm of coherence Cl1 (see Sec. III.C.2) admits no fi- δD and quantumness QD as follows (Adesso et al., 2016; Modi et al., 2012; Roga et al., 2016): nite maximum and can diverge even on states with finite mean energy (Zhang et al., 2016b). This suggests that the l1-norm A|B AB AB AB does not provide a suitable quantifier of coherence in contin- δD (% ) = inf D(% , σ ), (60) σAB∈CQ uous variable systems. A|B AB AB AB QD (% ) = inf D(% , σ ) (61) (Xu, 2016) focused on the quantification of coherence σAB∈CC in bosonic Gaussian states of infinite-dimensional systems, with some distance D. If D is chosen to be the quantum which form an important subset of states entirely specified by relative entropy, the corresponding quantities are respectively their first and second moments and useful for theoretical and known as the relative entropy of discord and of quantumness experimental investigations of continuous variable quantum (Modi et al., 2010; Piani et al., 2011). information processing (Adesso and Illuminati, 2007; Weed- 16 brook et al., 2012). In particular, (Xu, 2016) defined a Gaus- An example for an asymmetry monotone which is not a co- sian relative entropy of coherence for a Gaussian state % (with herence monotone is the Wigner-Yanase skew information respect to the Fock basis) as the relative entropy difference 1 h √ i between % and the closest incoherent Gaussian state, which W(%, H) = − Tr H, %2 , (65) is a thermal state expressible in terms of the first and sec- 2 ond moments of %. However, this is only an upper bound on where [X, Y] = XY −YX is the commutator. This quantity was the true relative entropy of coherence of a Gaussian state %, first introduced by (Wigner and Yanase, 1963) as a measure of which is still given by Eq. (34) for any % with finite mean en- information, and was proven to be an asymmetry monotone ergy (Zhang et al., 2016b), since the closest incoherent state in (Girolami, 2014; Marvian, 2012; Marvian and Spekkens, to % given by its diagonal part ∆[%] in the Fock basis is not in 2014a). While (Girolami, 2014) originally proposed the skew general a Gaussian state. More recently, (Buono et al., 2016) information as a coherence monotone, it was later shown that studied geometric quantifiers of coherence for Gaussian states such a quantity can instead increase under IO, in particular in terms of Bures and Hellinger distance from the set of inco- under permutations of the reference basis states, hence violat- herent Gaussian states (thermal states); once more, these are ing C2 for this set of operations (Du and Bai, 2015; Marvian upper bounds to the corresponding distance-based coherence et al., 2016). monotones as defined in Sec. III.C. The quantity in Eq. (65) can be generalized to the class Relations between coherence, optical nonclassicality, and of Wigner-Yanase-Dyson skew informations (Wehrl, 1978; entanglement in continuous variable systems have been inves- Wigner and Yanase, 1963) tigated recently in (Killoran et al., 2016; Sperling and Vogel, h 2i h a 1−a i 2015; Vogel and Sperling, 2014), and will be discussed in the Wa(%, H) = Tr %H − Tr % H% H , (66) next Section. with 0 < a < 1, which are also asymmetry monotones (Marvian, 2012; Marvian and Spekkens, 2014a), reducing to 1 K. Coherence, asymmetry and nonclassicality Eq. (65) for a = 2 . In particular, the average monotone R 1 W %, H a W %, H quantum variance ( ) = 0 d a( ), branded as the , 1. Asymmetry monotones has recently found applications in quantum many-body systems (Frérot and Roscilde, 2016). The relation between coherence and the framework of Another instance of an asymmetry monotone is given by asymmetry has been discussed most recently in (Marvian and the quantum Fisher information (Braunstein and Caves, 1994; Spekkens, 2014a,b, 2016; Marvian et al., 2016; Piani et al., Girolami and Yadin, 2017; Marvian and Spekkens, 2014a; 2016). This framework is based on the notion of transla- Zhang et al., 2016a), which can be defined (under a smooth- tionally invariant operations (TIO), which were already intro- ness hypothesis) as duced in Sec. II.A.2.

In (Marvian et al., 2016), the authors proposed postulates ∂2F(% ,% ) I(%, H) = −2 t t+ε , (67) for quantifying the asymmetry of a state with respect to time 2 ∂ε ε→0 translations e−iHt induced by a given Hamiltonian H. Any 14 asymmetry monotone A should vanish for all states which where F denotes the fidelity defined after Eq. (39), and %t = are invariant under time translations, i.e., which are incoherent e−iHt%eiHt. The quantum Fisher information quantifies the sen- in the eigenbasis of H. If we denote the latter set by IH, we sitivity of the state % to a variation in the parameter t charac- have terizing a unitary dynamics generated by H, hence playing a central role in quantum metrology (Braunstein and Caves, A(%) = 0 ⇐⇒ % ∈ IH. (63) 1994; Giovannetti et al., 2011). The Wigner-Yanase-Dyson skew information Eq. (66) and Moreover, A should not increase under translationally invari- the quantum Fisher information Eq. (67) can be seen as spe- ant operations Λ ∈ TIO (see Eq. (7)): cial instances of the entire family of quantum generalizations of the classical Fisher information (Petz and Ghinea, 2011), A(Λ[%]) ≤ A(%). (64) which are defined in terms of the Riemannian contractive metrics on the quantum state space as classified by (Morozova For nondegenerate Hamiltonians, the set IO is strictly larger and Cencovˇ , 1991; Petz, 1996). All these quantities have been than the set TIO (Marvian et al., 2016). It follows that the proven to be asymmetry monotones, i.e. worthwhile quanti- set of coherence monotones (intended as IO monotones) is a fiers of unspeakable coherence, in (Zhang et al., 2016a). Fur- strict subset of the set of asymmetry monotones (intended as ther details on the applications of these asymmetry monotones TIO monotones). in the contexts of quantum speed limits, quantum estimation and discrimination (Marvian and Spekkens, 2016; Mar- vian et al., 2016; Mondal et al., 2016; Napoli et al., 2016; Piani et al., 2016; Pires et al., 2015, 2016), are reported in 14 Note that (Marvian et al., 2016) call these quantities asymmetry measures. Secs. IV.E and V.C. 17

The concept of asymmetry is closely related to the concept the relative entropy of superposition fulfills these conditions. of quantum reference frames (Bartlett et al., 2007; Gour et al., The latter is defined as follows: 2009; Gour and Spekkens, 2008; Vaccaro et al., 2008). Ini- tially, these two concepts were defined for an arbitrary Lie S r(%) = S (Π[%]) − S (%). (72) group G. If U(g) is a unitary representation of the group with The relative entropy of superposition is a special case of the g ∈ G, the set of invariant states G consists of all states which relative entropy of asymmetry presented in Sec. III.K.1, and are invariant under the action of all U(g). The relative entropy admits the following expression (Åberg, 2006): of asymmetry (also known as relative entropy of frameness) is then defined as (Gour et al., 2009) S r(%) = min S (%||σ) = S (%||Π[%]). (73) Π[σ]=σ

Ar(%) = min S (%||σ). (68) σ∈G 3. Coherence rank and general quantifiers of nonclassicality Remarkably, it was shown by (Gour et al., 2009) that the relative entropy of asymmetry admits the following expression, An alternative approach was taken by (Killoran et al., 2016; first independently introduced as G-asymmetry by (Vaccaro Mukhopadhyay et al., 2017; Regula et al., 2017; Theurer et al., 2008): et al., 2017), who investigated a very general form of nonclassicality, also going beyond the framework of coherence. Ar(%) = S Γ % − S (%) = S %||Γ % , (69) In particular, depending on the task under consideration, it R can be useful to identify a set of pure states {|cii} as classical. where Γ[%] = dgU(g)%U†(g) is the average with respect These states do not have to be mutually orthogonal in general. dg to the Haar measuren . If theo unitary representation of the As an example, in entanglement theory those are all states of group is given by e−iHt : t ∈ R with a Hermitian nondegen- the product form |ci = |αi ⊗ |βi. P erate matrix H = i hi |iihi|, the set G of invariant states is pre- (Killoran et al., 2016) introduced the coherence rank of cisely the set I of states which are incoherent in the eigenbasis a general pure state15. Analogously to the Schmidt rank in {|ii} of H. Thus, in this case the relative entropy of asymmetry entanglement theory, the coherence rank counts the minimal Ar is equal to the relative entropy of coherence Cr with respect number of classical states needed in the expansion of the gen- to the eigenbasis of H taken as a reference basis. eral state |ψi: The robustness of asymmetry with respect to an arbitrary  R−1  Lie group G is an asymmetry monotone defined in (Napoli  X  r (|ψi) = min R : |ψi = a |c i . (74) et al., 2016; Piani et al., 2016) as C  i i   i=0  % + sτ The authors of (Killoran et al., 2016) then proved that non- RA(%) = min s ≥ 0 ∈ G . (70) τ 1 + s classicality can always be converted into entanglement in the following sense: if the set of classical states {|cii} is linearly −iHt If the unitary representation of the group is given by e as independent, there always exists a unitary operation which above, then the robustness of asymmetry R reduces to the A converts each state |ψi with coherence rank rC into a bipartite robustness of coherence RC with respect to the eigenbasis of state |ψ˜i with the same Schmidt rank. The authors of (Regula H, defined in Eq. (52). et al., 2017) extended this result to the multipartite setting, by constructing a unitary protocol for converting nonclassicality of a d-level system, prepared in any input state with coherence 2. Quantifying superpositions rank 2 ≤ rC ≤ d, into genuine (rC + 1)-partite entanglement between the system and up to d ancillary qubits. A very general approach to quantify coherence was pre- A concept related to the coherence rank was discussed in sented by (Åberg, 2006) within the framework of quantum (Levi and Mintert, 2014) in the specific context of coherent superposition. In this approach, the Hilbert space H is di- delocalization. In this framework, a state |ψi is called k- K vided into K subspaces L1,..., LK such that ⊕ Lk = H. If Pk−1 k=1 coherent if it can be written as |ψi = i=0 ai |ii with all coeffi- P is the projector corresponding to the subspace L , the total k k cients ai being nonzero. Here the integer k corresponds to the operation Π is defined as coherence rank rC(|ψi), where the set of classical states {|cii} in the definition of Eq. (74) is identified with the reference ba- K X sis of incoherent states {|ii}.(Levi and Mintert, 2014) also pro- Π[%] = Pk%Pk. (71) k k=1 posed quantifiers for this concept of -coherence, and showed

If the projectors Pk all have rank one, the total operation Π corresponds to the total dephasing ∆. However, in general the 15 The coherence rank can be generalized to mixed states via a procedure projectors Pk can have rank larger than one. similar to the convex roof described in Sec. III.D. The resulting quantity

(Åberg, 2006) also proposed a set of conditions a faithful rC (%) = inf{pi,|ψii} maxi rC (|ψii) is called the coherence number of % (Reg- quantifier of superposition ought to satisfy and showed that ula et al., 2017). 18 that these quantifiers do not grow under incoherent channels ing as an inspiration for the more recent studies of (Streltsov in their framework. Note however that the incoherent chan- et al., 2015) and (Killoran et al., 2016; Theurer et al., 2017) nels of (Levi and Mintert, 2014) are in general different from reviewed in the previous Sections. the IO defined by (Baumgratz et al., 2014), and can be rather In particular, (Asbóth et al., 2005) proposed to quantify op- identified with the SIO (Winter and Yang, 2016; Yadin et al., tical nonclassicality for a single-mode state % in terms of the 2016). We also note that a related framework was presented maximum two-mode entanglement that can be generated from recently by (Yadin and Vedral, 2016) to quantify macroscopic % using linear optics, auxiliary classical states, and ideal pho- coherence. The possibility to establish superpositions of un- todetectors. Any output entanglement monotone E defines known quantum states via universal quantum protocols has a corresponding nonclassicality quantifier PE for the input been investigated by (Oszmaniec et al., 2016). state %, referred to as entanglement potential. The authors of (Asbóth et al., 2005) considered in particular the quantities PE derived by choosing E to be the logarithmic negativ- 4. Optical coherence and nonclassicality ity or the relative entropy of entanglement. The definition of entanglement-based coherence monotones by (Streltsov et al., The framework of (Killoran et al., 2016; Theurer et al., 2015) as presented in Sec. III.E can be seen as the finite- 2017) is partly motivated by the seminal theory of optical co- dimensional counterpart to the study of (Asbóth et al., 2005). herence in continuous variable systems (Glauber, 1963; Man- Furthermore, (Vogel and Sperling, 2014) independently de- del and Wolf, 1965; Sudarshan, 1963). In this theory, the pure fined a notion analogous to the coherence rank rC of Eq. (74) classical states are identified with the Glauber-Sudarshan co- for optical nonclassicality, i.e. with {|cii} being a subset of (lin- herent states |αi of the radiation field, defined (for a single early independent) Glauber-Sudarshan coherent states. They bosonic mode) as the right eigenstates of the annihilation op- then showed that a single-mode state |ψi with nonclassical- erator, a |αi = α |αi, with α ∈ C2. These states form an over- ity rank rC can always be mapped into a two-mode entangled complete, non-orthogonal basis for the infinite-dimensional state with the same Schmidt rank, by means of a balanced Hilbert space. Any quantum state % which cannot be written beam splitter acting on the input mode and a vacuum ancillary as a mixture of Glauber-Sudarshan coherent states is hence mode. This can be seen as a special instance of the general regarded as nonclassical. We wish to highlight a semantic theorem of (Killoran et al., 2016) presented in Sec. III.K.3. subtlety here: a Glauber-Sudarshan coherent state |αi (with Finally, a connection between optical nonclassicality and α , 0) is in fact coherent if one is interested in charac- the theory of coherence as reviewed in Sec.II with respect to terizing coherence with respect to the Fock basis {|ni} as in the set IO has been recently established by (Tan et al., 2017). Sec. III.J, since it can be written as a superposition thereof: −| |2 P∞ n They introduced an orthogonalization procedure, according |αi = e α /2 √α |ni. However, in the theory of optical n=0 n! to which one can define quantifiers of optical nonclassical- coherence the Glauber-Sudarshan coherent states play rather ity PC, i.e. coherence with respect to the non-orthogonal basis the role of classical, or free, states (i.e., the analogous of inco- of Glauber-Sudarshan coherent states, in terms of any coher- herent states in the resource theory of coherence discussed so ence monotone C applied to N-dimensional subspaces of the far), as they can be generated by classical currents acting on a Hilbert space, in a suitable limit N → ∞; details of the map- quantum field (Louisell, 1973). Hence it is well motivated that ping can be found in (Tan et al., 2017). They then proved they form the reference set with respect to which the resource that any such PC is a monotone under linear optical passive of optical nonclassicality is defined and quantified. operations if the corresponding C is an IO monotone. This A general resource theory of nonclassicality, as suggested demonstrates that continuous variable states exhibiting optical in (Brandão and Plenio, 2008), has not been completely for- nonclassicality can be seen essentially as the limiting case of malized yet. In particular, determining the suitable set of free the same resource states identified in (Baumgratz et al., 2014), operations for the theory of optical coherence stands as one of when the incoherent basis is chosen as the set of Glauber- the most pressing open questions. The set CO of classical op- Sudarshan coherent states. erations, defined as the maximal set of operations preserving a reference set of (not mutually orthogonal) classical states, has been studied in (Sperling and Vogel, 2015), where it was L. Multipartite settings shown that CO is convex and obeys the semigroup property. If the set of classical states is identified with the convex hull of 1. General distance-based coherence quantifiers Glauber-Sudarshan coherent states, as in the theory of optical coherence, then the corresponding CO includes so-called pas- In the bipartite setting it is possible to obtain coherence sive operations, i.e. operations preserving the mean energy monotones CD by using the distance-based approach as in ha†ai, which can be implemented by linear optical elements Eq. (29), where I is now the set of bipartite incoherent states, such as beam splitters and phase shifters. i.e., convex combinations of states of the form |ki |li, with {|ki} A series of works investigated the conversion from optical and {|li} the incoherent reference bases for each subsystem re- nonclassicality into entanglement by means of passive opera- spectively (Bromley et al., 2015; Streltsov et al., 2015). It is tions (Asbóth et al., 2005; Kim et al., 2002; Sperling and Vo- instrumental to compare the quantities obtained in this way gel, 2015; Vogel and Sperling, 2014; Wolf et al., 2003), serv- to other corresponding distance-based quantifiers of bipartite 19

B nonclassicality such as quantumness QD and entanglement S (% ). Remarkably, this improvement does not depend on the ED. Due to the inclusion relation I ⊂ CC ⊂ S, the afore- particular choice of the incoherent reference basis. mentioned quantities are related via the following inequality In (Streltsov et al., 2017) this framework was extended (Yao et al., 2015): to other sets of operations, such as LICC, SI, and SQI, see Sec. II.C for their definitions. In general, if X is one of the CD(%) ≥ QD(%) ≥ ED(%). (75) sets described above, then the distillable coherence of collaboration can be generalized as follows (Streltsov et al., 2017): These results can be straightforwardly generalized to more than two parties (Yao et al., 2015). A|B h h ⊗nii ⊗bRnc C (%) = sup R : lim inf TrA Λ % − τ = 0 . X n→∞ Λ∈X 1 (80) AB A|B 2. Quantum-incoherent relative entropy Interestingly, for any mixed state % = % the quantities CSI A|B A|B A|B and CSQI are equal, and all quantities CX are between CLICC The quantum-incoherent relative entropy was defined by A|B and Cr (Streltsov et al., 2017): (Chitambar et al., 2016) as follows: CA|B ≤ CA|B ≤ CA|B = CA|B ≤ CA|B. (81) A|B AB AB AB LICC LQICC SI SQI r Cr % = min S % ||σ , (76) σAB∈QI Moreover, for bipartite pure states |ψiAB, Eq. (79) generalizes where the minimum is taken over the set of quantum- as follows (Streltsov et al., 2017): incoherent states QI defined in Eq. (15). As further discussed A|B AB A|B AB A|B AB A|B AB in (Chitambar et al., 2016), the quantum-incoherent relative CLICC(|ψi ) = CLQICC(|ψi ) = CSI (|ψi ) = CSQI(|ψi ) entropy admits the following closed expression: A|B AB ∞ B B = Cr (|ψi ) = Ca (% ) = S (∆[% ]). (82) A|B AB B h ABi AB Cr % = S ∆ % − S % , (77) Very recently (Wu et al., 2017) provided the first experimental demonstration of assisted coherence distillation. The B where ∆ denotes a dephasing operation on subsystem B only. experiment used polarization-entangled photon pairs for cre- As we will see in the following, the quantum-incoherent rel- ating pure entangled states, and also mixed Werner states. Af- ative entropy is a powerful upper bound on the distillable co- ter performing a suitable measurement on one of the photons, herence of collaboration. the second photon was found in a state with a larger amount of coherence. Finally, we note that the distillable coherence of collabora- 3. Distillable coherence of collaboration tion can be regarded as the coherence analogue of the entanglement of collaboration presented in (Gour and Spekkens, The distillable coherence of collaboration was introduced 2006). and studied by (Chitambar et al., 2016) as the figure of merit for the task of assisted coherence distillation. In this task AB Alice and Bob share a bipartite state % and aim to extract 4. Recoverable coherence maximally coherent single-qubit states |Ψ2i on Bob’s side via LQICC operations. The distillable coherence of collaboration The recoverable coherence was introduced by (Matera is the highest achievable rate for this procedure (Chitambar et al., 2016) in the context of a resource theory of control of et al., 2016): quantum systems. It is defined in the same way as the distillable coherence of collaboration in Eq. (78), but with the set of A|B h h ⊗nii ⊗bRnc C (%) = sup R : lim inf TrA Λ % − τ = 0 . LQICC n→∞ Λ 1 LQICC operations replaced by GOIB, see Sec. II.C for their (78) definition. Following the analogy to distillable coherence of collaboration, we will denote the recoverable coherence by Here, the infimum is taken over all LQICC operations Λ, and CA|B . As was shown by (Matera et al., 2016), the recov- B GOIB τ = |Ψ2ihΨ2| is the maximally coherent single-qubit state on erable coherence is additive, convex, monotonic on average Bob’s subsystem. As was shown in (Chitambar et al., 2016), under GOIB operations, and upper bounded by the quantum- for a pure state |ψiAB the distillable coherence of collaboration incoherent relative entropy. Since LQICC is a subset of GOIB is equal to the regularized coherence of assistance of Bob’s operations, we get the following inequality: reduced state: A|B A|B A|B CLQICC ≤ CGOIB ≤ Cr . (83) A|B AB ∞ B B CLQICC(ψ ) = Ca (% ) = S (∆[% ]). (79) Notably, the recoverable coherence has an operational inter- It is interesting to compare this result to the distillable coher- pretation, as it is directly related to the precision of estimat- B B B B ence of Bob’s local state: Cd(% ) = S (∆ [% ]) − S (% ). This ing the trace of a unitary via the DQC1 quantum algorithm means that assistance provides an improvement on the distil- (Matera et al., 2016); see also Sec. V.B for a more general lation rate given exactly by the local von Neumann entropy discussion on the role of coherence in quantum algorithms. 20

Additionally, minimizing the recoverable coherence over all In this Section we review more recent work concerning local bases leads to an alternative quantifier of discord (Mat- the dynamical evolution of coherence quantifiers (defined in era et al., 2016). Sec. III) subject to relevant Markovian or non-Markovian evolutions. Coherence effects in biological systems and their potential functional role will be discussed in Sec.V. Here we 5. Uncertainty relations and monogamy of coherence also discuss generic properties of coherence in mixed quantum states, the cohering (and decohering) power of quan- Uncertainty relations for quantum coherence, both for a sin- tum channels, and the role played by coherence quantifiers in gle party and for multipartite settings, have been studied in defining speed limits for closed and open quantum evolutions. (Peng et al., 2016b; Singh et al., 2016a). If coherence is defined with respect to two different bases, {|ii} and {|ai}, the i a corresponding relative entropies of coherence Cr and Cr ful- A. Freezing of coherence fill the following uncertainty relation (Singh et al., 2016a): ! One of the most interesting phenomena observed in the dy- i a namics of coherence is the possibility for its freezing, that Cr(%) + Cr (%) ≥ −2 log2 max |hi|ai| − S (%). (84) i,a is, complete time invariance without any external control, in the presence of particular initial states and noisy evolutions. For bipartite states %XY ,(Singh et al., 2016a) derived the fol- (Bromley et al., 2015) identified a set of dynamical condi- lowing uncertainty relation for the bipartite relative entropies i j ab tions under which all distance-based coherence monotones of coherence Cr and Cr : CD obeying postulates C1, C2 (for the set IO), and C4 stay i j XY ab XY XY simultaneously frozen for indefinite time (see also Fig.1, bot- Cr (% ) + Cr (% ) ≤ 2 log2 dXY − 2K(% ). (85) tom panel). We can summarize these conditions as follows. Here, dXY is the dimension of the composite Hilbert space, and Let us consider an open quantum system of N qubits, each K(%XY ) arises from the Lewenstein-Sanpera decomposition16: subject to a nondissipative Markovian decoherence channel, XY XY XY K(% ) = λS (%s ) + (1 − λ)S (%e ). representing dephasing the eigenbasis of the kth Pauli matrix The discussion on monogamy of quantum coherence is also σk, where k = 1 corresponds to bit flip noise, k = 2 to bit- inspired by results from entanglement theory (Coffman et al., phase flip noise, and k = 3 to phase flip noise (the latter equiv- 2000; Horodecki et al., 2009). In particular, a coherence quan- alent to conventional phase damping in the computational ba- tifier C is called monogamous with respect to the subsystem sis). Such ‘k-flip’ channels on each qubit are described by a X for a tripartite state %XYZ if (Kumar, 2017; Yao et al., 2015) set of Kraus operators (Nielsen and Chuang, 2010) K (t) = p p 0 1 − q(t)/2 11, K (t) = 0, K (t) = q(t)/2 σ , where C(%XYZ) ≥ C(%XY ) + C(%XZ). (86) i, j,k k k {i, j, k} is a permutation of {1, 2, 3} and q(t) = 1 − e−γt is As was shown in (Kumar, 2017; Yao et al., 2015), the relative the strength of the noise, with γ the decoherence rate. Any entropy of coherence is not monogamous in general, although such dynamics, mapping a N-qubit state %(0) into %(t) = P3 K (t) ⊗ ... ⊗ K (t)%(0)K (t) ⊗ ... ⊗ K (t)†, it can be monogamous for certain families of states. Further m1,...,mN =0 m1 mN m1 mN results on monogamy of coherence have also been presented is incoherent (in particular, strictly incoherent) with respect to ⊗N by (Radhakrishnan et al., 2016). any product basis {|mi} , with {|mi} being the eigenbasis of any of the three canonical Pauli operators σm on each qubit. Let us now consider a family of N-qubit mixed states with IV. DYNAMICS OF QUANTUM COHERENCE all maximally mixed marginals, defined by    X3  Quantum coherence is typically recognized as a fragile fea- 1  ⊗N ⊗N  %(t) = 11 + c j(t)σ  , (87) 2N  j  ture: the vanishing of coherence in open quantum systems ex- j=1 posed to environmental noise, commonly referred to as deco- ⊗N herence (Breuer and Petruccione, 2002; Schlosshauer, 2005; with c j = Tr[%σ j ]. For any even N, these states span Zurek, 2003), is perhaps the most distinctive manifestation a tetrahedron with vertices {1, (−1)N/2, 1}, {−1, −(−1)N/2, 1}, of the quantum-to-classical transition observed at our macro- {1, −(−1)N/2, −1}, and {−1, (−1)N/2, −1}, in the three- scopic scales. Numerous efforts have been invested into de- dimensional space of the correlation functions {c1, c2, c3}. In vising feasible control schemes to preserve coherence in open the special case N = 2, they reduce to Bell diagonal states quantum systems, with notable examples including dynami- of two qubits, that is, arbitrary convex mixtures of the four cal decoupling (Viola et al., 1999), quantum feedback control maximally entangled Bell states. Freezing of coherence un- (Rabitz et al., 2000) and error correcting codes (Shor, 1995). der k-flip channels then manifests for the subclass of states of N/2 Eq. (87) with initial condition ci(0) = (−1) c j(0)ck(0), for any even N ≥ 2. For all such states, and for any distance functional D(%, σ) being zero for σ = %, contractive under 16 A Lewenstein-Sanpera decomposition (Lewenstein and Sanpera, 1998) of quantum channels, and jointly convex, one has a state % is a decomposition of the form % = λ%s +(1−λ)%e with a separable state % , an entangled state % , and probability 0 ≤ λ ≤ 1. s e CD %(t) = CD %(0) ∀ t ≥ 0 , (88) 21 where CD is a corresponding distance-based quantifier of co- B. Coherence in non-Markovian evolutions herence (see Sec. III.C), and coherence is measured with respect to the product eigenbasis {| ji}⊗N of the Pauli operator Some attention has been devoted to the study of coher- ⊗N σ j (Bromley et al., 2015; Silva et al., 2016). Notice that ence in non-Markovian dynamics. In (Addis et al., 2014), the the freezing extends as well to the l1-norm of coherence, as it phenomenon of coherence trapping in the presence of non- amounts to the trace distance of coherence in the considered Markovian dephasing was studied. Namely, for a single qubit states (up to a normalization factor). For odd N, including the subject to non-Markovian pure dephasing evolutions (i.e., a k- general case of a single qubit, no measure-independent freez- flip channel with γt replaced by a non-monotonic function of ing of coherence can occur instead for the states of Eq. (87), t), the stationary state at t → ∞ may retain a nonzero coher- apart from trivial instances; this means that, for all nontrivial ence in the eigenbasis of σk, as quantified e.g. by the l1-norm evolutions preserving e.g. the l1-norm of coherence Cl1 , some of coherence. This can only occur in the presence of non- other quantifier such as the relative entropy of coherence Cr is Markovian dynamics, and is different from the previously dis- strictly decreasing (Bromley et al., 2015). cussed case of coherence freezing, in which coherence is mea- In (Yu et al., 2016a), the relative entropy of coherence sured instead with respect to a reference basis transversal to was found to play in fact a special role in identifying condi- the dephasing direction. It was shown in (Addis et al., 2014) tions such that any coherence monotone is frozen at all times. that the specifics of coherence trapping depend on the envi- Specifically, all coherence monotones (respecting in particu- ronmental spectrum: its low-frequency band determines the lar property C2 for the set SIO) are frozen for an initial state presence or absence of information backflow, while its high- subject to a strictly incoherent channel if and only if the rel- frequency band determines the maximum coherence trapped ative entropy of coherence is frozen for such initial state (Yu in the stationary state. In (Zhang et al., 2015) the coherence et al., 2016a). In formulae, in the stationary state of a qubit initially correlated with a zero- temperature Ohmic-like bath, realizing a non-Markovian pure C %(t) = C %(0) ∀ C ⇔ Cr %(t) = Cr %(0) , (89) dephasing channel, was further studied. The best dynamical conditions were identified (in terms of initial qubit-bath corre- where %(t) = Λt[%(0)] with Λt ∈ SIO (see Sec. II.A.2). Using lations and bath spectral density) to optimize coherence trap- this criterion, one can identify other classes of initial states ping, that is, to maximize coherence in the stationary qubit exhibiting measure-independent frozen coherence under local state and to minimize the evolution time towards such a state. k-flip channels (Yu et al., 2016a). The dynamics of the l1-norm of coherence for two qubits We remark that the described freezing effect differs from an globally interacting with a harmonic oscillator bath was in- instance of decoherence-free subspace (Lidar, 2014), where vestigated in (Bhattacharya et al., 2016), finding that non- an open system dynamics acts effectively as a unitary evo- Markovianity slows down the coherence decay. A proposal to lution on a subset of quantum states, preserving their infor- witness non-Markovianity of incoherent evolutions via a tem- mational properties. Here, instead, the purity of the involved porary increase of coherence quantifiers was finally discussed states is degraded with time, but their coherence in the cho- in (Chanda and Bhattacharya, 2016), inspired by more general sen reference basis remains unaffected. Other signatures of approaches to witness and measure non-Markovianity based quantumness like measures of discord-type correlations can on revivals of distinguishability, entanglement, or other infor- also freeze under the same dynamical conditions (Ciancia- mational quantifiers (Breuer et al., 2016; Rivas et al., 2014). ruso et al., 2015; Mazzola et al., 2010; Modi et al., 2012), albeit only for a finite time in the case of Markovian dynamics. A unified geometric analysis of these phenomena is given C. Cohering power of quantum channels and evolutions in (Bromley et al., 2015; Cianciaruso et al., 2015; Silva et al., 2016). The cohering power of a quantum channel (Baumgratz In (Liu et al., 2016), freezing of coherence was explored et al., 2014; Mani and Karimipour, 2015) can be defined as theoretically for a system of two-level atoms interacting with the maximum amount of coherence that the channel can cre- the vacuum fluctuations of an electromagnetic field bath. A ate when acting on an incoherent state, more comprehensive analysis of the dynamics of the l1-norm of coherence for one qubit subject to various types of com- PC(Λ) = max C(Λ[%]) , (90) mon noisy channels was reported in (Pozzobom and Maziero, %∈I 2017). The dynamics of the l1-norm of coherence for general d-dimensional systems was further investigated in (Hu where I denotes the set of incoherent states (with respect to a and Fan, 2016), where a factorization relation for the evolu- chosen reference basis) and C is any quantifier of coherence. In a similar way, (Mani and Karimipour, 2015) defined the tion equation of Cl1 was derived, leading in particular to a condition for its freezing. decohering power of a quantum channel Λ as the maximum The phenomenon of frozen quantum coherence was demon- amount by which the channel reduces the coherence of a max- strated experimentally in a room temperature nuclear mag- imally coherent state, netic resonance setup (Silva et al., 2016), with two-qubit and DC(Λ) = max {C(%) − C(Λ[%])} , (91) four-qubit ensembles prepared in states of the form (87). %∈M 22 where M denotes the set of maximally coherent states (with coherence resulting from the action of the channel on an arbi- respect to a chosen reference basis) and C is any quantifier trary state, of coherence. If C is chosen to be convex (i.e., respecting PeC(Λ) = max {C(Λ[%]) − C(%)} , (93) property C4), then the optimizations in Eqs. (90) and (91) are % achieved by pure (respectively incoherent and maximally coherent) states, simplifying the evaluation of the cohering and where C is any quantifier of coherence and, unlike Eq. (90), decohering power of a quantum channel. % is not restricted to be an incoherent state. By definition, Refs. (Bu et al., 2017; Bu and Wu, 2016; García-Díaz et al., PC(Λ) ≤ PeC(Λ). When considering unitary channels U and 2016; Mani and Karimipour, 2015; Situ and Hu, 2016; Xi adopting the l1-norm, it was proven in (Bu et al., 2017; García- et al., 2015a) calculated the cohering and decohering power Díaz et al., 2016) that PC(U) = PeC(U) in the case of a single of various unitary and non-unitary quantum channels adopt- qubit, but PC(U) < PeC(U) strictly in any dimension larger ing different quantifiers of coherence. In particular, (Mani than 2. These results were then shown to hold for arbitrary non-unitary channels Λ in (Bu and Wu, 2016), meaning that and Karimipour, 2015) showed that PC(Λ) = DC(Λ) for all single-qubit unitary channels when adopting the Wigner- in general the maximum coherence gain due to a qudit channel Yanase skew information as a quantifier of coherence C.17 In is obtained when acting on an input state with already nonzero (Bu et al., 2017), the authors derived a closed expression for coherence. the cohering power of a unitary channel U when adopting the In addition to channels one may also consider evolutions l -norm as a quantifier of coherence, that are generated by a Hamiltonian H or a Lindbladian L 1 Lt (García-Díaz et al., 2016). For a time evolution Φt = e we P (U) = kUk2 − 1 , (92) determine the coherence rate Cl1 1→1 1 L∆t k k k k Υ(L) = lim max[C(e %) − C(%)] (94) where U 1→1 = maxkxk1=1 Ux 1 denotes the maximum col- ∆t→0 ∆t % umn sum matrix norm. This implies that, for the cohering N and in case of unitary evolutions U(t) = e−iHt we write power of a tensor product of unitaries j U j with respect to a N Q h i product basis, one has PC U j + 1 = j PC (U j) + 1 , 1 l1 j l1 Υ(H) = lim max[C(e−iH∆t%eiH∆t) − C(%)]. (95) which generalizes an expression already obtained in (Mani ∆t→0 ∆t % and Karimipour, 2015) in the case of all equal U j. In (Bu et al., 2017) it was also proven that the N-qubit unitary op- Further alternative approaches to define and quantify the cohering power of quantum channels have been pursued re- eration with the maximal l1-norm cohering power (even including arbitrary global unitaries) is the tensor product H⊗N cently in (Zanardi et al., 2017a,b). Finally, we mention that of N single-qubit Hadamard gates, with P (H⊗N ) = 2N − 1. similar studies have been done in entanglement theory (Lin- Cl1 Further examples of cohering and decohering power of quan- den et al., 2009; Zanardi et al., 2000). In particular, (Linden et al., 2009) showed that entangling and disentangling power tum channels with respect to l1-norm, relative entropy, and other coherence quantifiers were presented in (Bu et al., 2017; of unitaries are not equivalent in general. García-Díaz et al., 2016; Situ and Hu, 2016; Xi et al., 2015a). The authors of (Bu et al., 2017) provided an operational in- D. Average coherence of random states and typicality terpretation for the cohering power. Given a quantum channel Λ : B(H) → B(H) acting on a principal system, it is said While some dynamical properties of coherence may be very to be implementable by incoherent operations supplemented dependent on specific channels and initial states, it is also in- by an ancillary quantum system if there exists an incoherent 0 0 teresting to study typical traits of coherence quantifiers on ran- operation IO 3 Λi : B(H ⊗ H ) → B(H ⊗ H ) and states domly sampled pure or mixed states. Note that generic ran- σ, σ0 ∈ B(H 0) of the ancilla such that, for any state % ∈ B(H) 0 dom states, exhibiting the typical features of coherence sum- of the principal system, one has Λi[% ⊗ σ] = Λ[%] ⊗ σ . marized in the following, can be in fact generated by a dynam- In this setting, the cohering power of Λ quantifies the min- ical model of a quantized deterministic chaotic system, such imum amount of coherence to be supplied in the ancillary as a quantum kicked top (Puchała et al., 2016). state σ to make Λ implementable by incoherent operations: In (Singh et al., 2016b) the authors showed that the rela- C(σ) ≥ PC(Λ), where C is any coherence monotone fulfilling tive entropy of coherence (equal to the distillable coherence), C1–C4. the coherence of formation, and the l1-norm of coherence, On the other hand, (Bu et al., 2017; García-Díaz et al., all exhibit the concentration of measure phenomenon, mean- 2016) considered a more general definition of cohering power ing that, with increasing dimension of the Hilbert space, the of a quantum channel Λ, given by the maximum increase of overwhelming majority of randomly sampled pure states have coherence (according to those quantifiers) taking values very close to the average coherence over the whole Hilbert space. This was proven rigorously resorting to Lévy’s lemma, hence 17 More precisely, this would quantify the asymmetry/symmetry power rather showing that states with coherence bounded away from its av- than the cohering/decohering power. erage value occur with exponentially small probability. 23

1. Average relative entropy of coherence 3. Average recoverable coherence

The exact average of the relative entropy of coherence for In (Miatto et al., 2015), the authors considered a qubit pure d-dimensional states |ψi ∈ Cd sampled according to the interacting with a d-dimensional environment, of which an Haar measure was computed in (Singh et al., 2016b), ﬁnding a-dimensional subset is considered accessible, while the re- maining k-dimensional subset (with d = ak) is unaccessi-

ECr(|ψi) = Hd − 1 , (96) ble. For illustration, one can think of the environment being constituted by N additional qubits, of which NA are ac- Pd cessible and N = N − N are unaccessible; in this case where Hd = k=1(1/k) is the dth harmonic number. As shown K A in (Puchała et al., 2016), for large dimension d 1 the av- d = 2N , a = 2NA , k = 2Nk . While for d 1 such an inter- erage in Eq. (96) tends to ECr(|ψi) ' log d − (1 − γ), with action leads to decoherence of the principal qubit, its coher- γ ≈ 0.5772 denoting the Euler constant. This shows that ran- ence can be partially recovered by quantum erasure, which dom pure states have relative entropy of coherence close to entails measuring (part of) the environment in an appropriate (but strictly smaller than by a constant value) the maximum basis to erase the information stored in it about the system, log d (Puchała et al., 2016; Singh et al., 2016b). hence restoring coherence of the latter. The authors consid- 2 d Typicality of the relative entropy of coherence for random ered random pure states |ψi ∈ C ⊗ C of the system plus mixed states was investigated in (Puchała et al., 2016; Zhang, environment composite, and studied the average recoverable 2017; Zhang et al., 2017b). Considering the probability mea- l1-norm of coherence ECl1 (%) of the marginal state % of the sure induced by partial tracing, that is, corresponding to ran- principal qubit, following an optimal measurement on the ac- 0 0 d d 0 cessible a-dimensional subset of the environment. They found dom mixed states % = TrCd |ψi hψ|, with |ψi ∈ C ⊗C (d ≤ d ) 18 sampled according to the Haar measure, (Zhang, 2017; Zhang that the average recoverable coherence√ stays close to zero if ∝ et al., 2017b) derived a compact analytical formula for the av- a < k, scaling as ECl1 (%) 1/ k, but it transitions to a value 0 erage ECr(%), given by ECr(%) = (d − 1)/(2d ). In particular, close to unity as soon as at least half of the environment be- − when considering the ﬂat Hilbert-Schmidt measure (obtained comes accessible, scaling linearly as ECl1 (%) = 1 k/(4a) for by setting d0 = d in the previous expression), one gets a ≥ k. With increasing dimension d, the transition at a = k

becomes sharper and the distribution of Cl1 (%) becomes more 1 1 concentrated near its average value, the latter converging to 1 EC (%) = − , (97) r 2 2d in the limit d → ∞. By virtue of typicality, this means that, regardless of how a high-dimensional environment is parti- which tends asymptotically to a constant 1/2 for d → ∞. The tioned, suitably measuring half of it generically suﬃces to expression in Eq. (97) was also independently calculated in project a qubit immersed in such environment onto a near- (Puchała et al., 2016), in the limit of large dimension d 1. maximally coherent state, a fact reminiscent of the quantum The concentration of measure phenomenon for Cr was then Darwinism approach to decoherence (Brandão et al., 2015b; proven in (Puchała et al., 2016; Zhang et al., 2017b). These Zurek, 2009). results show that random mixed states have signiﬁcantly less (relative entropy of) coherence than random pure states. E. Quantum speed limits

In the dynamics of a closed or open quantum system, quan- 2. Average l -norm of coherence 1 tum speed limits dictate the ultimate bounds imposed by quantum mechanics on the minimal evolution time between two Concerning the l -norm of coherence, (Singh et al., 2016b) 1 distinguishable states of the system. In particular, consider derived a bound to the average EC (|ψi) for pure Haar- l1 a quantum system which evolves, according to a unitary dy- distributed d-dimensional states |ψi, exploiting a relation be- namics generated by a Hamiltonian H, from a pure state |ψi tween the l1-norm of coherence and the so-called classical to a ﬁnal orthogonal state |ψ i = e−iHτ⊥/~ |ψi with hψ |ψi = 0. purity (Cheng and Hall, 2015). Then (Puchała et al., 2016) ⊥ ⊥ Then, seminal investigations showed that the evolution time obtained exact results for the average C of pure and mixed l1 τ is bounded from below as follows: states in large dimension d 1, respectively distributed ac- ⊥ cording to the Haar and Hilbert-Schmidt measures, ﬁnding τ⊥(|ψi) ≥ max{π~/(2∆E), π~/(2E)} , (99) √ √ 2 2 2 π π where (∆E) = hH iψ − hHi and E = hHiψ − E0 (with E0 ECl (|ψi) ' (d − 1) , ECl (%) ' d . (98) ψ 1 4 1 2 the ground state energy), and the two bounds in the right-hand

This shows that, for asymptotically large d, the l1-norm of coherence of random pure states scales linearly with d and stays smaller than the maximal value (d − 1) by a factor π/4, 18 Note that the term “recoverable coherence” is here used in a different con- while the l1-norm of coherence of random mixed states scales text and refers to a different concept than the one introduced in (Matera only with the square root of d. et al., 2016) and discussed in Sec. III.L.4. 24 side of Eq. (99) are due to (Mandelstam and Tamm, 1945) and geodesic length. In formula, the tightest speed limit is ob- (Margolus and Levitin, 1998), respectively. In the last seven tained by minimizing the ratio decades, a great deal of work has been devoted to identifying g g more general speed limits, for pure as well as mixed states, g `Λ(%, %τ) − L (%, %τ) δΛ = g , (101) and unitary as well as non-unitary evolutions; see e.g. (del L (%, %τ) Campo et al., 2013; Pires et al., 2016; Taddei et al., 2013) and references therein. Recent studies have shown in particular over the metric g. Several examples are presented in (Pires that quantifiers of coherence, or more precisely of asymmetry, et al., 2016), demonstrating the importance of choosing dif- play a prominent role in the determination of quantum speed ferent information metrics for open system dynamics, as well limits. as clarifying the roles of classical populations versus quantum The authors of (Marvian et al., 2016) studied, for any > 0, coherences in the determination and saturation of the speed D limits. In particular, in the case of a single qubit, while for any the minimum time τ necessary for a state % to evolve under the Hamiltonian H into a partially distinguishable state unitary dynamics the speed limit based on the quantum Fisher −iHt iHt %t = e %e , such that D(%, %t) ≥ according to a distance information (Taddei et al., 2013) is always tighter than the functional D. In formula, one based on the Wigner-Yanase skew information, this is no longer true when considering non-unitary dynamics. Specif-  ically, for parallel and transversal dephasing, as well as am- ∞, if D(%, % ) < ∀t > 0; D  t plitude damping dynamics, (Pires et al., 2016) derived new τ (%) =  min {t : D(%, %t) ≥ }, otherwise. t>0 tighter speed limits based on the Wigner-Yanase skew infor- (100) mation. We finally mention that looser speed limits involv- Then, for any > 0 and for any distance D which is con- ing the skew information have also been recently presented in tractive and jointly convex, (Marvian et al., 2016) proved (Mondal et al., 2016; Pires et al., 2015). D that 1/τ (%), which represents the average speed of evolu- The speed of a two-qubit photonic system — quantified by tion, is an asymmetry monotone of % with respect to time the family of asymmetry monotones associated with the quan- translations generated by the Hamiltonian H, being in par- tum Riemannian contractive metrics — undergoing a con- ticular monotonically nonincreasing under the corresponding trolled unitary evolution, has been measured experimentally TIO. Interestingly, for pure states, even the inverses of the in (Zhang et al., 2016a), by means of an all-optical direct de- Mandelstam-Tamm and Margolus-Levitin quantities appear- tection scheme requiring less measurements than full state to- ing in the right-hand side of Eq. (99) are themselves asym- mography. metry monotones, bounding the asymmetry monotone given by 1/τ⊥(|ψi). The authors of (Marvian et al., 2016) then derived new Mandelstam-Tamm–type quantum speed limits for V. APPLICATIONS OF QUANTUM COHERENCE unitary dynamics based on various measures of distinguishability, including a bound featuring the Wigner-Yanase skew In this Section we discuss applications of quantum coher- information with respect to H (obtained when D is set to the ence to a variety of fields, ranging from quantum information relative Rényi entropy of order 1/2), which was also indepen- processing to quantum sensing and metrology, thermodynam- dently obtained in (Mondal et al., 2016). ics and biology. Particular emphasis will be given to those The authors of (Pires et al., 2016) developed a general ap- settings in which a specific coherence monotone introduced proach to Mandelstam-Tamm–type quantum speed limits for in Sec. III acquires an operational interpretation, hence result- all physical processes. Given a metric g on the quantum state ing in novel insights stemming from the characterization of g space, let `Λ(%, %τ) denote the length of the path connecting quantum coherence as a resource. an initial state % to a final state %τ = Λ[%] under a (generally open) dynamics Λ. Quantum speed limits then ensue from a simple geometric observation, namely that the geodesic con- A. Quantum thermodynamics g necting % to %τ, whose length can be indicated by L (%, %τ), is the path of shortest length among all physical evolutions be- Recently, the role of coherence in quantum thermodynam- g g tween the given initial and final states: L (%, %τ) ≤ `Λ(%, %τ) ics has been discussed by several authors. We will review ∀Λ. Then (Pires et al., 2016) considered the infinite family of the main concepts in the following, being in large part based quantum speed limits derived from this geometric principle, on the resource theory of quantum thermodynamics (Brandão with g denoting any possible quantum Riemannian contrac- et al., 2013; Goold et al., 2016; Gour et al., 2015) defined by tive metric (Morozova and Cencovˇ , 1991; Petz, 1996; Petz and the framework of thermal operations (Janzing et al., 2000). Ghinea, 2011), including two prominent asymmetry monotones: quantum Fisher information and Wigner-Yanase skew information (see Sec. III.K.1). For any given dynamics Λ and 1. Thermal operations pair of states %, %τ, one can identify the tightest speed limit as the one corresponding to the metric g such that the length In the following, we consider a system S and an environ- g of the dynamical path `Λ is the closest to the corresponding ment E with a total Hamiltonian HSE = HS + HE. Given 25 a state of the system %S , a thermal operation on this state is Plenio, 1999), known as second laws of quantum thermody- defined as (Janzing et al., 2000) namics, were presented in (Brandão et al., 2015a). They can also be applied to the situation where the state %S has coher- h S i h S E †i Λth % = TrE U% ⊗ γ U , (102) ence. Interestingly, for interconversion of single-qubit states via thermal operations, (Cwikli´ nski´ et al., 2015) found a sim- where γE = e−βHE /Tr[e−βHE ] is a thermal state of the envi- ple set of necessary and sufficient conditions, in terms of so- ronment with the inverse temperature β = 1/kBT and we de- called damping matrix positivity. Similar considerations were mand that the unitary U commutes with the total Hamiltonian: also presented for other classes of operations such as enhanced [U, HSE] = 0. thermal operations (Cwikli´ nski´ et al., 2015) and cooling maps The importance of thermal operations arises from the fact (Narasimhachar and Gour, 2015). that they are consistent with the first and the second law of thermodynamics (Lostaglio et al., 2015b). In particular, since E the unitary U commutes with the total Hamiltonian, these op- In the above discussion the state of the environment γ was erations preserve the total energy of system and environment. assumed to be a thermal state. In recent literature on quantum Moreover, they do not increase the Helmholtz free energy19 thermodynamics, this constraint has been relaxed, allowing γE to be a general state of the environment. As discussed in S S S F(% ) = Tr[% HS ] − kBTS (% ), (103) (Lostaglio et al., 2017), it is important to distinguish between two cases: namely, whether the state γE is incoherent, or has S S S i.e., for any two states % and σ = Λth[% ] it holds that nonzero coherence, with respect to the eigenbasis of HE. F(σS ) ≤ F(%S ). Thermal operations also have two other important properties (Lostaglio et al., 2015a,b). First, they are TIO with respect In both cases it is also assumed that an arbitrary number of copies of γE are available, which is a usual assumption from to the Hamiltonian HS , i.e., the point of view of resource theories. An important result in h i h i −iHS t S iHS t −iHS t S iHS t Λth e % e = e Λth % e . (104) this respect was obtained in (Lostaglio et al., 2015b). There, it was shown that by allowing for an arbitrary number of copies S S Second, they preserve the Gibbs state: Λth[γ ] = γ . As dis- of the Gibbs state together with some other incoherent state cussed in (Lostaglio et al., 2015a,b), these two properties are γE, the operation in Eq. (102) can be used to approximate any related to the first and the second law of thermodynamics re- incoherent state of the system, i.e., any state which is diag- spectively. In particular, preservation of the Gibbs state im- onal in the eigenbasis of the Hamiltonian HS . Moreover, in plies that no work can be extracted from a thermal state. this case it is possible to implement any TIO (Lostaglio et al., A closely related concept is known as Gibbs-preserving op- 2015b). Although these processes can be used to perform an erations (Faist et al., 2015; Ruch and Mead, 1976). Here, arbitrary amount of work, they are still limited in the sense preservation of the thermal state is the only requirement on that they cannot create coherence (Lostaglio et al., 2017). The the quantum operation. Interestingly, it was shown by (Faist situation changes if the state γE has coherence. In this case the et al., 2015) that these maps are strictly more powerful than process in Eq. (102) can also be used to perform an arbitrary thermal operations. In particular, Gibbs-preserving operations amount of work, and apart from that the process can also cre- can create coherence from incoherent states, while this cannot ate coherence in the system. However, even in this case the be done via thermal operations. theory is nontrivial, i.e., not all transformations are possible (Lostaglio et al., 2017, 2015b).

2. State transformations via thermal operations The role of coherence in quantum thermodynamics was fur- Several recent works studied necessary and sufficient con- ther studied in (Misra et al., 2016), where the authors ana- ditions for two states % and σ to be interconvertible via thermal lyzed the physical situation in which the resource theories of operations. In the absence of coherence, i.e., if %S is diago- coherence and thermodynamics play competing roles. In par- nal in the eigenbasis of HS , such conditions were presented ticular, they investigated creation of coherence for a quantum in (Horodecki and Oppenheim, 2013a) and termed thermo- system (with respect to the eigenbasis of its Hamiltonian H) majorization20. More general conditions which allow for the via unitary operations from a thermal state, and also explored addition of ancillas and catalytic conversions (Jonathan and the energy cost for such coherence creation. Given an initial −βH −βH thermal state %T = e /Tr[e ] at temperature T = 1/β (setting kB = 1), (Misra et al., 2016) showed that there always exists a unitary transformation (in fact, a real orthog- 19 As customary in thermodynamics, in this section we define the von Neu- 0 onal one) which maps %T into a state % such that its diag- mann entropy S (%) = −Tr[% log %] with logarithm to base e. 0 20 onal ∆[% ] = %T 0 amounts to a thermal state at temperature Note that thermo-majorization is also related to the mixing distance studied 0 in (Ruch et al., 1978). We also refer to (Egloff et al., 2015), where a relation T > T. This creates the maximal relative entropy of co- ∆E between majorization and optimal guaranteed work extraction up to a risk herence Cr,max = S (%T 0 ) − S (%T ), at the cost of spending an of failure was investigated. amount of energy ∆E = Tr[H(%T 0 − %T )]. 26

3. Work extraction and quantum thermal machines on the backaction on the clock, and therefore its resource con- sumption, in terms of energy and coherence. Interestingly, coherence cannot be converted to work in a direct way. This phenomenon is known as work locking (Horodecki and Oppenheim, 2013a; Lostaglio et al., 2015a; B. Quantum algorithms Skrzypczyk et al., 2014), and can be formalized as follows (Korzekwa et al., 2016): The role of coherence in quantum algorithms was discussed by (Hillery, 2016), with particular focus on the Deutsch-Jozsa hWi %S ≤ hWi Π%S . (105) algorithm (Deutsch and Jozsa, 1992). This quantum algorithm can decide whether a boolean function is constant or Here, hWi (%S ) denotes the amount of work that can be ex- balanced by just one evaluation of the function, while in the S S P S tracted from the state % , and Π[% ] = i Tr Πi% Πi, where classical case the number of evaluations grows exponentially Πi are projectors onto the eigenspaces of HS . Note that the in the number of input bits. As was shown by (Hillery, 2016), operation Π is in general different from the full dephasing ∆ coherence is a resource in this protocol in the sense that a defined in Eq. (5), since the latter removes all offdiagonal el- smaller amount of coherence in the protocol increases the er- ements, while Π preserves some offdiagonal elements if the ror of guessing whether the underlying function was constant Hamiltonian HS has degeneracies. A detailed study of this or balanced. problem was also presented in (Korzekwa et al., 2016), where A similar investigation with respect to the Grover algorithm it was shown that work extraction from coherence is still pos- (Grover, 1997) was performed in (Anand and Pati, 2016; Shi sible in certain scenarios. This relies on the repeated use of et al., 2017). In (Anand and Pati, 2016), the authors stud- a coherent ancilla in a catalytic way as shown by (Åberg, ied the relation between coherence and success probability in 2014). Further results on the role of coherence for work ex- the analog Grover algorithm, which is a version of the orig- traction have also been presented in (Kammerlander and An- inal Grover algorithm based on adiabatic Hamiltonian evolu- ders, 2016). Moreover, it was shown in (Vacanti et al., 2015) tion. It was found that the success probability psucc of the that work is typically required for keeping coherent states out algorithm is related to the amount of coherence in the corre- of thermal equilibrium. The role of coherence in determining sponding quantum state as follows (Anand and Pati, 2016): the distribution of work done on a quantum system has been p − also studied in (Solinas and Gasparinetti, 2015, 2016). Cl1 (psucc) = 2 psucc(1 psucc), (106) The role played by coherence in the operation of quantum Cr(psucc) = −psucc log2 psucc − (1 − psucc) log2(1 − psucc). thermal machines, such as heat engines and refrigerators, has (107) been investigated recently (Rahav et al., 2012; Scully et al., 2011). Various authors have explored the use of optical co- Another important quantum algorithm is known as deter- herence, in the form of squeezing in a thermal bath, to push ministic quantum computation with one qubit (DQC1) (Knill the performance of nanoscale heat engines and quantum ab- and Laflamme, 1998). This quantum algorithm provides an sorption refrigerators beyond their classical limitations (Abah exponential speedup over the best known classical procedure and Lutz, 2014; Correa et al., 2014; Manzano et al., 2016; for estimating the trace of a unitary matrix (given as a se- Niedenzu et al., 2016; Roßnagel et al., 2014). However, the quence of two-qubit gates). Interestingly, this algorithm re- 21 advantages found in these studies are not directly related to quires vanishingly little entanglement (Datta et al., 2005), a processing of coherence, but originate at least in part from but a typical instance of DQC1 has nonzero quantum discord the fact that, in energetic terms, a squeezed bath has an en- (Datta et al., 2008). The role of quantum discord for DQC1 ergy content which is equivalent to that of a thermal bath at was later questioned by (Dakic´ et al., 2010), who showed that a higher effective temperature. Quantum coherence was also certain nontrivial instances do not involve any quantum corre- shown to be useful for transient cooling in absorption refriger- lations. This issue was further discussed in (Datta and Shaji, ators (Mitchison et al., 2015). More generally, the authors of 2011). Thus, the question of which type of quantumness cor- (Uzdin et al., 2015) established the thermodynamical equiva- rectly captures the performance of this algorithm remained lence of all engine types in the quantum regime of small ac- open. The role of coherence for DQC1 was first studied by tion (compared to ~). They then identified generic coherent (Ma et al., 2016), and later by (Matera et al., 2016). The lat- and incoherent work extraction mechanisms, and showed that ter work indicates that coherence is indeed a suitable figure coherence enables power outputs that can reach significantly of merit for this protocol. In particular, (Matera et al., 2016) beyond the power of incoherent (i.e., stochastic) engines. showed that the precision of the algorithm is directly related It is noteworthy that the control of any engine, especially an to the recoverable coherence, defined in Sec. III.L.4. autonomous device, requires a clock in order to switch on and off an interaction at specified moments in time, and thereby control the device. At the quantum level, such a control leads 21 In this context we mention that only bipartite entanglement was considered to correlations and thus a possible loss of coherence in the in (Datta et al., 2005), and the role of multipartite entanglement in DQC1 clock. (Woods et al., 2016) addresses the question of the co- remains unclear. We refer to (Parker and Plenio, 2000, 2002) for similar herence cost of such control via clocks and establishes limits considerations with respect to Shor’s algorithm. 27

C. Quantum metrology 2016a), as discussed in Sec. III.K.1, acquire an interpretation as the speed of evolution of the probe state % under the dynam- The main goal of quantum metrology (Braunstein and ics Ut generated by H. This highlights the role of coherence Caves, 1994; Braunstein et al., 1996; Giovannetti et al., 2004, quantifiers in the determination of quantum speed limits, as 2006) is to overcome classical limitations in the precise esti- reviewed in Sec. IV.E. mation of an unknown parameter ϕ encoded e.g. in a unitary In absence of noise, the Heisenberg scaling can be equiva- −iϕH evolution Uϕ = e . Applications of quantum metrology in- lently achieved using n entangled probes in parallel, each sub- clude phase estimation for accelerometry, optical and gravita- ject to one instance of Uϕ (Giovannetti et al., 2006; Huelga tional wave interferometry, high precision clocks, navigation et al., 1997). A great deal of work has been devoted to char- devices, magnetometry, thermometry, remote sensing, and su- acterizing possibilities and limitations for quantum metrology perresolution imaging (Giovannetti et al., 2011; Paris, 2009). in the presence of various sources of noise, which result in As one can appreciate by the following simple example, loss of coherence or entanglement of the probes (Braun et al., quantum coherence plays a fundamental role in this task. For 2017; Chaves et al., 2013; Chin et al., 2012; Demkowicz- simplicity, let H be a nondegenerate single-qubit Hamilto- Dobrzanski et al., 2012; Escher et al., 2011; Giovannetti et al., nian H = E0 |0i h0| + E1 |1i h1|. A very simple possibility 2011; Huelga et al., 1997; Demkowicz-Dobrzanski´ and Mac- to estimate ϕ is to apply the unitary to a single-qubit state cone, 2014; Nichols et al., 2016; Smirne et al., 2016). Typ- |ψi = a |0i + b |1i, and to perform a measurement on the final ically the Heisenberg scaling is not retained, except under −iϕE −iϕE state Uϕ |ψi = ae 0 |0i + be 1 |1i. If the probe state |ψi some error models which allow for the successful implemen- has no coherence in the eigenbasis of H (i.e., a = 0 or b = 0), tation of suitable quantum error correcting procedures (Arrad the final state Uϕ |ψi will be the same as |ψi up to an irrelevant et al., 2014; Dür et al., 2014; Kessler et al., 2014; Macchi- global phase, i.e., from the final measurement we cannot gain avello et al., 2002; Preskill, 2000; Sekatski et al., 2016; Unden any information about the parameter ϕ. On the other hand, et al., 2016). if a and b are both nonzero, it is always possible to extract An alternative investigation on the role of coherence in information about ϕ via a suitable measurement. quantum metrology has been carried out in (Giorda and Al- In general, given an initial probe state % and assuming the legra, 2016b), where a relation has been derived between the probing procedure is repeated n times, the mean square error quantum Fisher information and the second derivative of the (∆ϕ)2 in the estimation of ϕ is bounded below by the quantum relative entropy of coherence, the latter evaluated with respect Cramér-Rao bound (Braunstein and Caves, 1994) to the optimal measurement basis in a (unitary or noisy) pa- 1 rameter estimation process. (∆ϕ)2 ≥ , (108) nI(%, H) where I(%, H) is the quantum Fisher information, a quanti- D. Quantum channel discrimination fier of asymmetry (i.e., of unspeakable coherence) (Marvian and Spekkens, 2016) defined in Eq. (67). As the bound in Quantum coherence also plays a direct role in quantum Eq. (108) is asymptotically achievable for n 1 by means of channel discrimination, a variant of quantum metrology where a suitable optimal measurement, the quantum Fisher informa- the task is not to identify the value of an unknown parameter tion directly quantifies the optimal precision of the estimation ϕ, but to distinguish between a set of possible values ϕ can procedure, and is thus regarded as the main figure of merit in take. In the case of a binary channel discrimination, in particu- −iϕH quantum metrology (Giovannetti et al., 2011; Paris, 2009). lar deciding whether a unitary Uϕ = e is applied or not to a Using only probe states without any coherence or entangle- local probe (i.e., distinguishing between Uϕ and the identity), ment, the quantum Fisher information can scale at most lin- the Wigner-Yanase skew information defined in Eq. (65) has early with n, I(%, H) ∼ n. However, starting from a probe state been linked to the minimum error probability of the discrim- % with coherence√ (e.g., the maximally coherent state given by ination (Farace et al., 2014; Girolami, 2014; Girolami et al., a = b = 1/ 2 in the previous example) and applying Uϕ se- 2013). quentially n times before the final measurement, allows one More recently, the task of quantum phase discrimination to reach the so-called Heisenberg scaling, I(%, H) ∼ n2, which has been studied in (Napoli et al., 2016; Piani et al., 2016) (see yields a genuine quantum enhancement in precision (Giovan- also Fig.1, left panel). Consider a d-dimensional probe and Pd−1 netti et al., 2006). In this clear sense, quantum coherence a set of unitary channels {Uϕ} generated by H = i=0 i |iihi|, in the form of asymmetry is the primary resource behind the where {|ii} sets the reference incoherent basis for the probe 2πk d−1 power of quantum metrology. system and ϕ can take any of the d values d k=0 with uni- More generally, (Marvian and Spekkens, 2016) proved that form probability 1/d. One such channel acts on the probe, any function which quantifies the performance of probe states initialized in a state %, and the goal is to guess which channel % in the metrological task of estimating a unitarily encoded pa- instance has occurred (i.e., to identify the correct value of ϕ) rameter ϕ should be a quantifier of asymmetry with respect to with the highest probability of success. Using any probe state translations Uϕ induced by the generator H. Notice that if the σ ∈ I which is incoherent with respect to the eigenbasis of H, parameter ϕ is identified with time t, the quantum Fisher in- no information about ϕ is imprinted on the state and the prob- formation and related quantifiers of asymmetry (Zhang et al., ability psucc(σ) of guessing its correct value is simply given by 28

1/d, corresponding to a random guess. On the other hand, a solid state (Deveaud-Plédran et al., 2009; Li et al., 2012). Re- probe state % with coherence in the eigenbasis of H, accompa- cently, however, some research efforts have started to inves- nied by an optimal measurement at the output, allows one to tigate the role of coherence in the perhaps surprising arena achieve a better discrimination, leading to a higher probability of “warm, wet and noisy” biological systems. Motivated in of success psucc(%) ≥ psucc(σ). part by experimental observations using ultrafast electronic The enhancement in the probability of success for this task spectroscopy of light-harvesting complexes in photosynthesis when exploiting a coherent state %, compared to the use of any (Collini et al., 2010; Engel et al., 2007) the beneficial inter- incoherent state σ ∈ I, is given exactly by the robustness of play of coherent and incoherent dynamics has been identified coherence of % defined in Eq. (52)(Napoli et al., 2016; Piani as a key theme (Caruso et al., 2009; Mohseni et al., 2008; et al., 2016): Plenio and Huelga, 2008; Rebentrost et al., 2009a,b) in biological transport and more generally in the context of biolog- psucc(%) = 1 + RC(%). (109) ical function (Huelga and Plenio, 2013). It is now recognized psucc(σ) that typically both coherent and noise dynamics are required to achieve optimal performance. This provides a direct operational interpretation for the robustness of coherence RC in quantum discrimination tasks. Such A range of mechanisms to support this claim and under- an interpretation can be extended to more general channel dis- stand its origin qualitatively (see (Huelga and Plenio, 2013) crimination scenarios (i.e., with non-uniform prior probabili- for an overview) have been identified. These include con- ties, and including non-unitary incoherent channels) and car- structive and destructive interference due to coherence and its ries over to the robustness of asymmetry with respect to arbi- suppression by decoherence (Caruso et al., 2009) as well as trary groups (Napoli et al., 2016; Piani et al., 2016). the interaction between electronic and long-lived vibrational degrees of freedom (coherent) (Chin et al., 2010, 2013; Chris- tensson et al., 2012; Kolli et al., 2012; O’Reilly and Olaya- E. Witnessing quantum correlations Castro, 2014; Prior et al., 2010; Roden et al., 2016; Womick et al., 2012) in the environment and with a broadband vibra- Recently, several authors tried to find Bell-type inequali- tional background (incoherent) (Caruso et al., 2009; Mohseni ties for various coherence quantifiers. In particular, (Bu and et al., 2008; Plenio and Huelga, 2008; Rebentrost et al., Wu, 2016) considered quantifiers of coherence of the form 2009a,b). Nevertheless, the detailed role played by coherence C(X, Y,%AB), where X and Y are local observables on the sub- and coherent dynamics in these settings remains to be unrav- system A and B respectively, and the coherence is considered eled and quantified and it is here where the detailed quanti- with respect to the eigenbasis of X ⊗ Y. They found a Bell- tative understanding of coherence emerging from its resource type bound for this quantity for all product states %A ⊗ %B, and theory development may be beneficial. showed that the bound is violated for maximally entangled Initially, researchers studied the entanglement properties of states and a certain choice of observables X and Y. In a similar states (Caruso et al., 2009; Fassioli and Olaya-Castro, 2010; spirit, the interplay between coherence and quantum steering Sarovar et al., 2010) and evolutions (Caruso et al., 2010) that was investigated in (Mondal and Mukhopadhyay, 2015; Mon- emerge in biological transport dynamics. It should be noted dal et al., 2017), where steering inequalities for various co- though that in the regime of application the quantities consid- herence quantifiers were found, and in (Hu and Fan, 2016; Hu ered by (Sarovar et al., 2010) amount to coherence quantifiers et al., 2016), where the maximal coherence of steered states rather than purely entanglement quantifiers. In the studies of was investigated. the impact of coherence on transport dynamics, formal ap- As was further shown by (Girolami and Yadin, 2017), de- proaches using coherence and asymmetry quantifiers based on tection of coherence can also be used to witness multipar- the Wigner-Yanase skew information were used (Vatasescu, tite entanglement. In particular, an experimentally accessi- 2015, 2016), but the connection to function has remained ten- ble lower bound on the quantum Fisher information (which uous so far. It was indeed noted that it may become necessary does not require full state tomography) can serve as a witness to separately quantify real and imaginary part of coherence as for multipartite entanglement, as was explicitly demonstrated these can have significantly different effects on transport (Ro- for mixtures of GHZ states (Girolami and Yadin, 2017). This den and Whaley, 2016). Another question of interest in this builds on previous results on detecting different classes of context concerns that of the distinction between classical and multipartite entanglement using the quantum Fisher informa- quantum coherence (O’Reilly and Olaya-Castro, 2014) and tion (Hyllus et al., 2012; Pezzé and Smerzi, 2009, 2014; Tóth, dynamics (Li et al., 2012; Wilde et al., 2010) in biological 2012; Tóth and Apellaniz, 2014). systems, most notably photosynthetic units. Finally, it should be noted that there are other biological F. Quantum biology and transport phenomena phenomena that are suspected to benefit from coherent and incoherent dynamics, notably magnetoreception in birds (Cai Transport is fundamental to a wide range of phenomena in et al., 2010; Gauger et al., 2011; Ritz et al., 2000) where the the natural sciences and it has long been appreciated that co- role of coherence in the proposed radical pair mechanism was herence can play an important role for transport e.g. in the studied on the basis of coherence quantifiers (Kominis, 2015) 29 and the molecular mechanisms underlying olfaction (Turin, latter has a natural approach which is defined by the set of 1996). Unlike photosynthesis, however, experimental evi- LOCC operations. This set of operations has a clear physi- dence is still limited and not yet at a stage where conclusions cal motivation and several nice properties which make exact drawn from the quantitative theory of coherence as a resource evaluation tractable in many relevant situations. In particular, can be verified. this approach has a golden unit, since any bipartite quantum state can be created via LOCC operations from a maximally entangled state. G. Quantum phase transitions In the following, we provide six simple conditions which we believe any physical theory of quantum coherence as a re- Coherence and asymmetry quantifiers have been employed source should be tested on. These conditions are motivated to detect and characterize quantum phase transitions, i.e., by recent developments on the resource theories of coherence changes in the ground state of many-body systems occurring and entanglement. In particular, we propose that any resource at or near zero temperature and driven purely by quantum fluc- theory of coherence should have a set of free operations F tuations. The critical points can be identified by witnessing a with the following properties: particular feature in a chosen coherence quantifier, such as a F divergence, a cusp, an inflexion, or a vanishing point. 1. Physical motivation: the set of operations has a well defined physical justification. The authors of (Çakmak et al., 2015; Karpat et al., 2014) showed that single-spin coherence reliably identifies the 2. Post-selection: The set F allows for post-selection, second-order quantum phase transition in the thermal ground i.e., there is a well-defined prescription for perform- 1 state of the anisotropic spin- 2 XY chain in a transverse mag- ing multi-outcome measurements, and obtaining corre- netic field. In particular, the single-spin skew information sponding probabilities and post-measurement states. with respect to the Pauli spin-x operator σx, as well as its experimentally friendly lower bound which can be measured 3. No coherence creation: The set F (including post- without state tomography (Girolami, 2014), exhibit a diver- selection) cannot create coherence from incoherent gence in their derivative at the critical point, even at relatively states. high temperatures. 4. Free incoherent states: The set F allows to create any The authors of (Malvezzi et al., 2016) extended the previ- incoherent state from any other state. ous analysis to ground states of spin-1 Heisenberg chains. Fo- cusing on the one-dimensional XXZ model, they found that 5. Golden unit: The set F allows to convert the maximally no coherence and asymmetry quantifier (encompassing skew coherent state |Ψdi to any other state of the same dimen- information, relative entropy, and l1-norm) is able to detect sion. the triple point of the infinite-order Kosterlitz-Thouless tran- 6. No bound coherence: Given many copies of some (co- sition, while the single-spin skew information with respect to herent) state %, the set F allows to extract maximally a pair of complementary observables σ and σ can instead be x z coherent single-qubit states |Ψ i at nonzero rate. employed to successfully identify both the Ising-like second- 2 order phase transition and the SU(2) symmetry point. While conditions 2–6 can be tested directly, the first condition Further applications of coherence and asymmetry quanti- seems to be the most demanding. In TableII we list the sta- fiers to detecting quantum phase transitions in fermionic and tus of conditions 2–6 for the existing sets MIO, IO, SIO, DIO, spin models have been reported in (Chen et al., 2016b; Li and TIO, PIO, GIO, and FIO, introduced in Sec. II.A.2. In place Lin, 2016). of the first condition, we give the corresponding literature reference, where suitable motivations can be found for each set. As the Table shows, several frameworks of coherence do not VI. CONCLUSIONS fulfill all of our criteria, and several entries still remain open. We further reviewed in Section III the current progress on In this Colloquium we have seen that quantum coherence quantifying coherence and related manifestations of nonclas- plays an important role in quantum information theory, quan- sicality in compliance with the underlying framework of re- tum thermodynamics, and quantum biology, as well as physics source theories, in particular highlighting interconnections be- more widely. Similar to entanglement, but even more funda- tween different measures and, where possible, their relations mental, coherence can be regarded as a resource, if the exper- to entanglement measures. Several open questions remain to imenter is limited to quantum operations which cannot create be addressed in these topics, as we pointed out throughout the coherence. The latter set of operations is not uniquely spec- text. The rest of the Colloquium (SectionsIV andV) was ified: in SectionII we have reviewed the main approaches dedicated to investigate more physical aspects of coherence in this direction, their motivation, and their main differences. in quantum systems and its applications to quantum technolo- Most of these approaches have some desirable properties gies, many-body physics, biological transport, and thermody- which distinguish them from the other frameworks. namics. Most of these advances are still at a very early stage, It is instrumental to compare once again the resource the- and the operational value of coherence still needs to be pin- ory of coherence to the resource theory of entanglement. The pointed clearly in many contexts. 30

1 2 3 4 5 6 of the authors and do not necessarily reflect the views of the MIO (Åberg, 2006) yes yes yes yes yes John Templeton Foundation. IO (Baumgratz et al., 2014; Winter yes yes yes yes yes and Yang, 2016) SIO (Winter and Yang, 2016; Yadin yes yes yes yes ? et al., 2016) DIO (Chitambar and Gour, 2016b; yes yes yes yes ? REFERENCES Marvian and Spekkens, 2016) TIO (Marvian and Spekkens, 2016; yes yes yes no ? Åberg, J. (2006), arXiv:quant-ph/0612146. Marvian et al., 2016) Åberg, J. (2014), Phys. Rev. Lett. 113, 150402. PIO (Chitambar and Gour, 2016b) yes yes yes no ? Abah, O., and E. Lutz (2014), Europhys. Lett. 106 (2), 20001. GIO yes yes no no no Addis, C., G. Brebner, P. Haikka, and S. Maniscalco (2014), Phys. (de Vicente and Streltsov, 2017) FIO yes yes no no ? Rev. A 89, 024101. Adesso, G., T. R. Bromley, and M. Cianciaruso (2016), J. Phys. A Table II List of alternative frameworks of coherence with respect to 49 (47), 473001. our criteria 1–6 provided in the text. 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