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Colloquium: quantum coherence as a resource

30 Oct 2017-Reviews of Modern Physics (American Physical Society)-Vol. 89, Iss: 4, pp 041003
TL;DR: In this article, the authors 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.
Abstract: 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.

Summary (14 min read)

Jump to: [Introduction][A. Constraints, operations, and resources][1. Incoherent states][2. Classes of incoherent operations][1. Maximally coherent states and state transformations via incoherent operations][2. States and maps][C. Quantum coherence in distributed scenarios][D. Connection between coherence and entanglement theory][A. Postulates for coherence monotones and measures][B. Distillable coherence and coherence cost][C. Distance-based quantifiers of coherence][1. Relative entropy of coherence][2. Coherence quantifiers based on matrix norms][3. Geometric coherence][D. Convex roof quantifiers of coherence][E. Coherence monotones from entanglement][F. Robustness of coherence][G. Coherence quantifiers from interferometric visibility][H. Coherence of assistance][I. Coherence and quantum correlations beyond entanglement][J. Coherence in continuous variable systems][1. Asymmetry monotones][2. Quantifying superpositions][3. Coherence rank and general quantifiers of nonclassicality][4. Optical coherence and nonclassicality][1. General distance-based coherence quantifiers][2. Quantum-incoherent relative entropy][3. Distillable coherence of collaboration][4. Recoverable coherence][5. Uncertainty relations and monogamy of coherence][IV. DYNAMICS OF QUANTUM COHERENCE][A. Freezing of coherence][B. Coherence in non-Markovian evolutions][C. Cohering power of quantum channels and evolutions][D. Average coherence of random states and typicality][1. Average relative entropy of coherence][2. Average l1 norm of coherence][3. Average recoverable coherence][E. Quantum speed limits][V. APPLICATIONS OF QUANTUM COHERENCE][A. Quantum thermodynamics][1. Thermal operations][2. State transformations via thermal operations][3. Work extraction and quantum thermal machines][B. Quantum algorithms][C. Quantum metrology][D. Quantum channel discrimination][E. Witnessing quantum correlations][F. Quantum biology and transport phenomena][G. Quantum phase transitions] and [VI. CONCLUSIONS]

Introduction

  • The coherent superposition of states, in combination with the quantization of observables, represents one of the most fundamental features that mark the departure of quantummechanics from the classical realm.
  • The development of a rigorous theory of quantum coherence as a physical resource has been initiated only recently.
  • In a nutshell, any such theory first considers constraints that are imposed on us in a specific physical situation (e.g., the inability to perform joint quantum operations between distant laboratories due to the impossibility to transfer quantum systems from one location to the other while preserving their quantum coherence, and thus restricting us to local operations and classical communication).
  • These applications include so-called “QuantumTechnologies 2.0,” such as quantum-enhancedmetrology and communication protocols, and extend farther into other fields, such as thermodynamics and even certain branches of biology.

A. Constraints, operations, and resources

  • These constraints may be due to either fundamental conservation laws, such as superselection rules and energy conservation, or constraints due to the practical difficulty of executing certain operations, e.g., the restriction to local operations and classical communication (LOCC) which gives rise to the resource theory of entanglement (Plenio and Virmani, 2007; Horodecki et al., 2009).
  • The states that can be generated from the maximally mixed state1 by the application of free operations in F alone, are considered to be available free of charge, forming the set I of free states.
  • All the other states attain the status of a resource, whose provision carries a cost.
  • For the purposes of the present exposition of the resource theory of coherence, the authors begin by adopting the latter point of view and then proceed to require additional desirable properties of their classes of free operations.

1. Incoherent states

  • Coherence is naturally a basis dependent concept, which is why the authors first need to fix the preferred or reference basis in which to formulate their resource theory.
  • If the reference basis for a single qubit is taken to be the computational basis fj0i; j1ig, i.e., the eigenbasis of the Pauli σz operator, then any density matrix with a nonzero off-diagonal element jϱ01j ¼ jh0jϱj1ij ≠ 0 is outside the set I of incoherent states, and hence has a resource content.
  • If the Hamiltonian is nondegenerate, the corresponding set of free states is exactly the set of incoherent states described, where the incoherent basis is defined by the eigenbasis of the Hamiltonian.
  • In the following, whenever the authors refer to incoherent states, they explicitly mean states of the form (1).

2. Classes of incoherent operations

  • The definition of free operations for the resource theory of coherence is not unique and different choices, often motivated by suitable practical considerations, are being examined in the literature.
  • The authors also mention genuinely incoherent operations (GIO) and fully incoherent operations (FIO) (de Vicente and Streltsov, 2017).
  • A classification of the different frameworks of coherence, motivated by the notion of speakable and unspeakable information (Peres and Scudo, 2002; Bartlett, Rudolph, and Spekkens, 2007), was proposed by Marvian and Spekkens (2016).

1. Maximally coherent states and state transformations via incoherent operations

  • The authors start by identifying a d-dimensional maximally coherent state as a state that allows for the deterministic generation of all other d-dimensional quantum states by means of the free operations.
  • The authors note that not all frameworks of coherence that were discussed in Sec. II.
  • One may also determine maximally coherent states under certain additional constraints, such as the degree of mixedness, which gives rise to the class of maximally coherent mixed states (Singh, Bera, Dhar, and Pati, 2015).
  • A notable result in this context was provided by Chitambar and Gour 2016a, 2016b, 2017), who gave a full characterization of single-qubit state conversion via SIO, DIO, IO, or MIO.
  • The optimal rate for this process can be evaluated analytically; see Sec. III.

2. States and maps

  • A more complex task beyond state preparation is that of the generation of a general quantum operation from a supply of coherent states and incoherent operations.
  • Just as the maximally entangled state allows for the generation of all quantum operations (Eisert et al., 2000) via LOCC, so does the maximally coherent state allow for the generation of all quantum operations via IO.
  • The explicit construction for an arbitrary single-qubit unitary can be found in Baumgratz, Cramer, and Plenio (2014), and the extension to general quantum operations of arbitrary dimension was studied by Chitambar and Hsieh (2016) and Ben Dana et al. (2017).
  • The corresponding construction makes use of maximally coherent states even if the targeted quantum operation may only be very slightly coherent.
  • By virtue of the monotonicity of coherence quantifiers under incoherent operations, lower bounds to the amount of coherence required to implement a quantum operation can be provided (Mani and Karimipour, 2015; Ben Dana et al., 2017; Bu et al., 2017).

C. Quantum coherence in distributed scenarios

  • Based on the framework of LOCC known from entanglement theory (Plenio and Virmani, 2007; Horodecki et al., 2009), one can introduce the framework of local incoherent operations and classical communication (LICC) (Chitambar and Hsieh, 2016; Streltsov, Rana, Bera, and Lewenstein, 2017).
  • While LICC operations in general have a difficult mathematical structure, it is possible to introduce the more general class of separable incoherent (SI) operations which has a simple mathematical form (Streltsov, Rana, Bera, and Lewenstein, 2017): Λ½ϱAB ¼ X i ðAi ⊗ BiÞϱABðA†i ⊗ B†i Þ; ð14Þ where Ai and Bi are local incoherent operators.
  • Asymmetric scenarios where only one of the parties is restricted to IO locally have also been considered (Chitambar et al., 2016; Streltsov, Rana, Bera, and Lewenstein, 2017).
  • The latter task was also performed experimentally (Wu et al., 2017) and will be discussed in more detail in Sec. III.L.3.
  • Here σAi are arbitrary states on the subsystem A, while jiiB are incoherent states on the subsystem B. Moreover, LQICC is a strict subset of SQI (Streltsov, Rana, Bera, and Lewenstein, 2017) and GOIB (Matera et al., 2016).

D. Connection between coherence and entanglement theory

  • The resource theory of coherence exhibits several connections to the resource theory of entanglement.
  • These results were generalized to quantum discord (Ma et al., 2016) and general types of nonclassicality (Killoran, Steinhoff, and Plenio, 2016); see Secs. III.
  • Another relation between entanglement and coherence was provided by Streltsov, Chitambar et al. (2016), where they introduced and studied the task of incoherent quantum state merging.
  • An important open question in this context is whether any of the aforementioned classes of incoherent operations Rev. Mod.

A. Postulates for coherence monotones and measures

  • The first axiomatic approach to quantify coherence was presented by Åberg (2006), and an alternative framework was more recently developed by Baumgratz, Cramer, and Plenio (2014).
  • At this point it is instrumental to compare conditions C2 and C3 to the corresponding conditions in entanglement theory (Vedral et al., 1997; Vedral and Plenio, 1998; Plenio and Virmani, 2007; Horodecki et al., 2009).
  • The authors remark that the terminology adopted here differs from the one used in some recent literature, which is mainly based on Baumgratz, Cramer, and Plenio (2014).
  • With the more stringent approach presented here, inspired from entanglement theory, the authors aim to distinguish two important coherence quantifiers: distillable coherence (which is equal to the relative entropy of coherence) and coherence cost (which is equal to the coherence of formation).

B. Distillable coherence and coherence cost

  • The distillable coherence is the optimal number of maximally coherent single-qubit states jΨ2iwhich can be obtained per copy of a given state ϱ via incoherent operations in the asymptotic limit.
  • For pure states this result was independently found by Yuan et al. (2015).
  • Condition C5 also follows from Eq. (27) and the definition of coherence of formation; see Sec. 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 the coherence cost CdðϱÞ ≤ CcðϱÞ: ð28Þ.
  • For pure states this inequality becomes an equality, which implies that the resource theory of coherence is reversible for pure states.

C. Distance-based quantifiers of coherence

  • A general distance-based coherence quantifier is defined as (Baumgratz, Cramer, and Plenio, 2014) 10A similar situation occurs in the case of entanglement, for which a fully reversible resource theory can also be constructed if one considers the largest set of operations preserving separability (Brandão and Plenio, 2008, 2010), which is a strict superset of separable operations.
  • Monotonicity C2 is also fulfilled for any set of operations discussed in Sec. II.
  • A.2 if the distance is contractive (Baumgratz, Cramer, and Plenio, 2014), DðΛ½ϱ ;Λ½σ Þ ≤ Dðϱ; σÞ ð30Þ for any quantum operation Λ. Moreover, any distance-based coherence quantifier fulfills convexity C4 whenever the corresponding distance is jointly convex (Baumgratz, Cramer, and Plenio, 2014), D X i piϱi; X j pjσj ≤ X i piDðϱi; σiÞ: ð31Þ.
  • In the following, the authors explicitly discussed three important distance-based coherence quantifiers.

1. Relative entropy of coherence

  • If the distance is chosen to be the quantum relative entropy Sðϱ∥σÞ ¼ Tr½ϱlog2ϱ − Tr½ϱlog2σ ; ð32Þ the corresponding quantifier is known as the relative entropy of coherence11 (Baumgratz, Cramer, and Plenio, 2014): CrðϱÞ ¼ min σ∈I Sðϱ∥σÞ: ð33Þ.
  • The relative entropy of coherence fulfills conditions C1, C2, and C4 for any set of operations discussed in Sec. II.
  • Moreover, it is equal to the distillable coherence Cd, and therefore both quantities admit the same closed expression (Gour, Marvian, and Spekkens, 2009; Baumgratz, Cramer, and Plenio, 2014; Winter and Yang, 2016) CrðϱÞ ¼ CdðϱÞ ¼ SðΔ½ϱ Þ − SðϱÞ; ð34Þ where Δ½ϱ is the dephasing operation defined in Eq. (5).
  • As further shown by Singh, Bera, Misra, and Pati (2015), the relative entropy of coherence can also be interpreted as the minimal amount of noise required for fully decohering the state ϱ.

2. Coherence quantifiers based on matrix norms

  • The authors now consider coherence quantifiers based on matrix norms, i.e., such that the corresponding distance has the form Dðϱ; σÞ ¼ kϱ − σk with some matrix norm k·k.
  • Note first that any such distance is jointly convex, i.e., fulfills Eq. (31), as long as the corresponding norm k·k fulfills the triangle inequality and absolute homogeneity12 (Baumgratz, Cramer, and Plenio, 2014).
  • Furthermore, the quantity on the right-hand side of Eq. (34) was independently proposed as a coherence quantifier by Herbut (2005) under the name coherence information.
  • Since the trace norm is contractive under quantum operations, the corresponding coherence quantifier C1 satisfies the conditions C1, C2, and C4 for any set of operations discussed in Sec. II.
  • Similar results were obtained for general X states: also in this case C1 is equivalent to Cl1 , and condition C3 is fulfilled for the set IO (Rana, Parashar, and Lewenstein, 2016).

3. Geometric coherence

  • The geometric coherence fulfills conditions C1, C2, and C4 for any set of operations discussed in Sec. II.
  • A.2. For pure states, the geometric coherence takes the form CgðjψiÞ ¼ 1 −maxijhijψij2, which means that Cg does not meet conditions C5 and C6.
  • Another related quantity was introduced by Baumgratz, Cramer, and Plenio (2014) and studied by Shao et al. (2015) where it was called fidelity of coherence: CFðϱÞ ¼ 1 −maxσ∈I ffiffiffiffiffiffiffiffiffiffiffiffiffiffi Fðϱ; σÞp .
  • While CF fulfills conditions C1, C2, and C4 for any set of operations discussed in Sec. II.

D. Convex roof quantifiers of coherence

  • Provided that a quantifier of coherence has been defined for all pure states, it can be extended to mixed states via the standard convex roof construction (Yuan et al., 2015): CðϱÞ ¼ inf fpi;jψ iig X i piCðjψ iiÞ; ð41Þ where the infimum is taken over all pure state decompositions of ϱ ¼.
  • Thus, 13Yuan et al. (2015) called this quantity intrinsic randomness.

E. Coherence monotones from entanglement

  • The supremum is performed over all bipartite incoherent operations Λi ∈ IO, i.e., such that the corresponding Kraus operators map the product basis jkijli onto itself.
  • In particular, CE fulfills conditions C1–C4 for the set IO whenever E fulfills the corresponding conditions in entanglement theory (Streltsov et al., 2015).
  • For distillable entanglement, relative entropy of entanglement, and geometric entanglement, the supremum in Eq. (47) is achieved when Λi is the generalized CNOT operation, i.e., the optimal incoherent operation is the unitary UCNOTjiij0i ¼ jiijii: ð49Þ.
  • In both cases, the bound is saturated by the generalized CNOT operation; see Eq. (49).
  • These results also imply that a state ϱS can be used to create entanglement via incoherent operations if and only if ϱS has nonzero coherence (Streltsov et al., 2015).

F. Robustness of coherence

  • Another coherence monotone was introduced by Napoli et al. (2016) and Piani et al. (2016) and termed robustness of coherence.
  • A similar quantity was studied earlier in entanglement theory under the name robustness of entanglement (Vidal and Tarrach, 1999; Steiner, 2003).
  • The latter result also implies that RC does not comply with C5 and C6.
  • Interestingly, for any state ϱ there exists a witnessW saturating this inequality.
  • The robustness of coherence is moreover a figure of merit in the task of quantum phase discrimination; see Sec. V.D for its definition and detailed discussion.

G. Coherence quantifiers from interferometric visibility

  • Clearly, quantum coherence is required for the observation of interference patterns, e.g., in the double slit experiment.
  • Recently, this idea was formalized by von Prillwitz, Rudnicki, and Mintert (2015), where they studied the problem to determine coherence properties from interference patterns.
  • Bagan et al. (2016) derived two exact complementarity relations between quantifiers of coherence and path information in a multipath interferometer, using, respectively, the l1 norm and the relative entropy of coherence.
  • Then by placing an output detector which implements a measurement described by a positiveoperator valued measure M ¼ ðMωÞ, one observes outcomes ω sampled from the Born probability pMjϱðωjφ⃗Þ ¼ Tr½ϱðφ⃗ÞMω , which constitute the interference pattern.
  • Various examples of visibility functionals and the corresponding coherence monotones were further discussed by Biswas, García-Díaz, and Winter (2017), showing, in particular, that the robustness of coherence presented in Sec. III.

H. Coherence of assistance

  • This quantity is dual to the coherence of formation defined in Eq. (42) as the minimum over all decompositions.
  • For this reason the authors get Cað1=2Þ ¼ 1, i.e., the coherence of assistance is maximal for the maximally mixed state.
  • On the one hand, this means that the coherence of assistance is not a coherence monotone, as it indeed violates condition C1.
  • Moreover, for all single-qubit states ϱ the coherence of assistance is n-copy additive and thus can be written as CaðϱÞ ¼ SðΔ½ϱ Þ. Further, there is a close relation between the coherence of assistance and the entanglement of assistance Rev. Mod.

I. Coherence and quantum correlations beyond entanglement

  • The states of Eq. (58) are also known as classical quantum (Piani, Horodecki, and Horodecki, 2008), and the corresponding set will be denoted by CQ.
  • This notion can be straightforwardly extended to more than two parties (Piani et al., 2011).
  • They showed that the creation of general quantum correlations from a state ϱ is bounded above by its coherence.

J. Coherence in continuous variable systems

  • The resource theory framework for quantum coherence adopted in this Colloquium assumes a finite-dimensional Hilbert space.
  • Some of the previously listed quantifiers of coherence have also been studied in continuous variable systems and specifically in bosonic modes of the radiation field.
  • These systems are characterized by an infinitedimensional Hilbert space, spanned by the Fock basis fjnig∞n¼0 of eigenstates of the particle number operator a†a (Braunstein and van Loock, 2005).
  • Similarly to what was done by Eisert, Simon, and Plenio (2002) in entanglement theory, Y.-R. Zhang et al. (2016) imposed a finite mean energy constraint ha†ai≡ n̄ < ∞ to address the quantification of coherence in such systems with respect to the Fock reference basis.

1. Asymmetry monotones

  • This framework is based on the notion of TIO, which were already introduced in Sec. II.
  • Thus, in this case the relative entropy of asymmetry.

2. Quantifying superpositions

  • A very general approach to quantify coherence was presented by Åberg (2006) within the framework of quantum superposition.
  • Pk all have rank 1, the total operation Π corresponds to the total dephasing Δ. However, in general the projectors Pk can have rank larger than 1.
  • Åberg (2006) also proposed a set of conditions a faithful quantifier of superposition ought to satisfy and showed that the relative entropy of superposition fulfills these conditions.
  • The relative entropy of superposition is a special case of the relative entropy of asymmetry presented in Sec. III.K.1 and admits the following expression (Åberg, 2006): SrðϱÞ ¼ min Π½σ ¼σ Sðϱ∥σÞ ¼ Sðϱ∥Π½ϱ Þ: ð73Þ.

3. Coherence rank and general quantifiers of nonclassicality

  • An alternative approach was taken by Killoran, Steinhoff, and Plenio (2016), Mukhopadhyay et al. (2017), Regula et al. (2017), and Theurer et al. (2017), who investigated a very general form of nonclassicality, also going beyond the framework of coherence.
  • Killoran, Steinhoff, and Plenio (2016) introduced the coherence rank of a general pure state.
  • Note, however, that the incoherent channels of Levi and Mintert (2014) are in general different from the IO defined by Baumgratz, Cramer, and Plenio (2014) and can be rather identified with the SIO (Winter and Yang, 2016; Yadin et al., 2016).
  • The authors also note that a related framework was presented recently by Yadin and Vedral (2016) to quantify macroscopic coherence.
  • The possibility to establish superpositions of unknown quantum states via universal quantum protocols was investigated by Oszmaniec et al. (2016).

4. Optical coherence and nonclassicality

  • The framework of Killoran, Steinhoff, and Plenio (2016) and Theurer et al. (2017) is partly motivated by the seminal theory of optical coherence in continuous variable systems 15The coherence rank can be generalized to mixed states via a procedure similar to the convex roof described in Sec. III.D.
  • The resulting quantity rCðϱÞ ¼ inffpi;jψ iigmaxirCðjψ iiÞ is called the coherence number of ϱ (Regula et al., 2017).
  • If the set of classical states is identified with the convex hull of Glauber-Sudarshan coherent states, as in the theory of optical coherence, then the corresponding CO includes so-called passive operations, i.e., operations preserving the mean energy ha†ai, which can be implemented by linear optical elements such as beam splitters and phase shifters.
  • In particular, Asbóth, Calsamiglia, and Ritsch (2005) proposed to quantify optical nonclassicality for a single-mode state ϱ in terms of the maximum two-mode entanglement that can be generated from ϱ using linear optics, auxiliary classical states, and ideal photodetectors.
  • Furthermore, Vogel and Sperling (2014) independently defined a notion analogous to the coherence rank rC of Eq. (74) for optical nonclassicality, i.e., with fjciig being a subset of (linearly independent) Glauber-Sudarshan coherent states.

1. General distance-based coherence quantifiers

  • S, the aforementioned quantities are related via the following inequality (Yao et al., 2015): CDðϱÞ ≥ QDðϱÞ ≥ EDðϱÞ: ð75Þ.
  • These results can be straightforwardly generalized to more than two parties (Yao et al., 2015).

2. Quantum-incoherent relative entropy

  • Br ðϱABÞ ¼ min σAB∈QI SðϱAB∥σABÞ; ð76Þ where the minimum is taken over the set of quantumincoherent statesQI defined in Eq. (15).
  • As further discussed by Chitambar et al. (2016), the quantum-incoherent relative entropy admits the following closed expression: CAjBr ðϱABÞ ¼ SðΔB½ϱAB Þ − SðϱABÞ; ð77Þ whereΔB denotes a dephasing operation on subsystem B only.

3. Distillable coherence of collaboration

  • The distillable coherence of collaboration was introduced and studied by Chitambar et al. (2016) as the figure of merit for the task of assisted coherence distillation.
  • The distillable coherence of collaboration is the highest achievable rate for this procedure (Chitambar et al., 2016): CAjBLQICCðϱÞ¼ sup n.
  • Here the infimum is taken over all LQICC operations Λ, and τ ¼ jΨ2ihΨ2jB is the maximally coherent single-qubit state on Bob’s subsystem.
  • Streltsov, Rana, Bera, and Lewenstein (2017) extended this framework to other sets of operations, such as LICC, SI, and SQI; see Sec. II.C for their definitions.
  • The experiment used polarization-entangled photon pairs for creating pure entangled states and also mixed Werner states.

4. Recoverable coherence

  • It is defined in the same way as the distillable coherence of collaboration in Eq. (78), but with the set of LQICC operations replaced by GOIB; see Sec. II.C for their definition.
  • Following the analogy to distillable coherence of collaboration, the authors denote the recoverable coherence by CAjBGOIB.
  • As shown by Matera et al. (2016), the recoverable coherence is additive, convex, monotonic on average under GOIB operations, and upper bounded by the quantum-incoherent relative entropy.
  • Notably the recoverable coherence has an operational interpretation, as it is directly related to the precision of estimating the trace of a unitary via the deterministic quantum computation with one qubit (DQC1) quantum algorithm (Matera et al., 2016); see also Sec. V.B for a more general discussion on the role of coherence in quantum algorithms.
  • Additionally, minimizing the recoverable coherence over all local bases leads to an alternative quantifier of discord (Matera et al., 2016).

5. Uncertainty relations and monogamy of coherence

  • Uncertainty relations for quantum coherence, both for a single party and for multipartite settings, have been studied by Peng et al. (2016) and Singh, Pati, and Bera (2016).
  • If coherence is defined with respect to two different bases fjiig and fjaig, the corresponding relative entropies of coherence Cir and Car fulfill the following uncertainty relation (Singh, Pati, and Bera, 2016): CirðϱÞ þ Car ðϱÞ ≥ −2 log2 max i;a jhijaij −.
  • The discussion on monogamy of quantum coherence is also inspired by results from entanglement theory (Coffman, Kundu, and Wootters, 2000; Horodecki et al., 2009).
  • Further results on monogamy of coherence were also presented by Radhakrishnan et al. (2016).

IV. DYNAMICS OF QUANTUM COHERENCE

  • Quantum coherence is typically recognized as a fragile feature: the vanishing of coherence in open quantum systems exposed to environmental noise, commonly referred to as decoherence (Breuer and Petruccione, 2002; Zurek, 2003; Schlosshauer, 2005), is perhaps the most distinctive manifestation of the quantum-to-classical transition observed at their macroscopic scales.
  • Numerous efforts have been invested into devising feasible control schemes to preserve coherence in open quantum systems, with notable examples including dynamical decoupling (Viola, Knill, and Lloyd, 1999), quantum feedback control (Rabitz et al., 2000), and error correcting codes (Shor, 1995).
  • In this section the authors review more recent work concerning the dynamical evolution of coherence quantifiers (defined in Sec. III) subject to relevant Markovian or non-Markovian evolutions.
  • Coherence effects in biological systems and their potential functional role are discussed in Sec. V.
  • Here the authors also discuss generic properties of coherence in mixed quantum states, the cohering (and decohering) power of quantum channels, and the role played by coherence quantifiers in defining speed limits for closed and open quantum evolutions.

A. Freezing of coherence

  • One of the most interesting phenomena observed in the dynamics of coherence is the possibility for its freezing, that is, complete time invariance without any external control, in the presence of particular initial states and noisy evolutions.
  • Cr is strictly decreasing (Bromley, Cianciaruso, and Adesso, 2015).
  • Specifically, all coherence monotones (respecting, in particular, property C2 for the set SIO) are frozen for an initial state subject to a strictly incoherent channel if and only if the relative entropy of coherence is frozen for such initial state (Yu, Zhang, Liu, and Tong, 2016).
  • Liu et al. (2016) theoretically explored freezing of coherence for a system of two-level atoms interacting with the vacuum fluctuations of an electromagnetic field bath.

B. Coherence in non-Markovian evolutions

  • Some attention has been devoted to the study of coherence in non-Markovian dynamics.
  • Addis et al. (2014) studied the phenomenon of coherence trapping in the presence of nonMarkovian dephasing.
  • Addis et al. (2014) showed that the specifics of coherence trapping depend on the environmental spectrum: its low-frequency band determines the presence or absence of information backflow, while its high-frequency band determines the maximum coherence trapped in the stationary state.
  • The dynamics of the l1 norm of coherence for two qubits globally interacting with a harmonic oscillator bath was investigated by Bhattacharya, Banerjee, and Pati (2016), finding that non-Markovianity slows down the coherence decay.
  • A proposal to witness non-Markovianity of incoherent evolutions via a temporary increase of coherence quantifiers was discussed by Chanda and Bhattacharya (2016), inspired by more general approaches to witness and measure nonMarkovianity based on revivals of distinguishability, entanglement, or other informational quantifiers (Rivas, Huelga, and Plenio, 2014; Breuer et al., 2016).

C. Cohering power of quantum channels and evolutions

  • When considering unitary channels U and adopting the l1 norm, García-Díaz, Egloff, and Plenio (2016) and Bu et al. (2017) proved that PCðUÞ ¼ ~PCðUÞ in the case of a single qubit, but PCðUÞ < ~PCðUÞ strictly in any dimension larger than 2.
  • Finally, the authors mention that similar studies have been done in entanglement theory (Zanardi, Zalka, and Faoro, 2000; Linden, Smolin, and Winter, 2009).

D. Average coherence of random states and typicality

  • While some dynamical properties of coherence may be dependent on specific channels and initial states, it is also interesting to study typical traits of coherence quantifiers on randomly sampled pure or mixed states.
  • Note that generic random states, exhibiting the typical features of coherence summarized in the following, can be in fact generated by a dynamical model of a quantized deterministic chaotic system, such as a quantum kicked top (Puchała, Pawela, and Życzkowski, 2016).
  • This was proven rigorously resorting to Lévy’s lemma, hence showing that states with coherence bounded away from its average value occur with exponentially small probability.

1. Average relative entropy of coherence

  • The exact average of the relative entropy of coherence for pure d-dimensional states jψi ∈.
  • This shows that random pure states have 17More precisely, this would quantify the asymmetry or symmetry power rather than the cohering or decohering power.
  • Typicality of the relative entropy of coherence for random mixed states was investigated by Puchała, Pawela, and Życzkowski (2016), Zhang (2017), and Zhang, Singh, and Pati (2017).
  • Considering the probability measure induced by partial tracing, that is, corresponding to random mixed states ϱ ¼ TrCd0 jψihψ j, with jψi ∈.
  • The concentration of measure phenomenon for Cr was then proven by Puchała, Pawela, and Życzkowski (2016) and Zhang, Singh, and Pati (2017).

2. Average l1 norm of coherence

  • Concerning the l1 norm of coherence, Singh, Zhang, and Pati (2016) derived a bound to the average ECl1ðjψiÞ for pure Haar-distributed d-dimensional states jψi, exploiting a relation between the l1 norm of coherence and the so-called classical purity (Cheng and Hall, 2015).
  • 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, while the l1 norm of coherence of random mixed states scales only with the square root of d.

3. Average recoverable coherence

  • Miatto et al. (2015) considered a qubit interacting with a ddimensional environment, of which an a-dimensional subset is considered accessible, while the remaining k-dimensional subset (with d ¼ ak) is unaccessible.
  • While for d ≫ 1 such an interaction leads to decoherence of the principal qubit, its coherence can be partially recovered by quantum erasure, which entails measuring (part of) the environment in an appropriate basis to erase the information stored in it about the system, hence restoring coherence of the latter.
  • Cd of the system plus environment composite and studied the average recoverable l1 norm of coherence ECl1ðϱÞ of the marginal state ϱ of the principal qubit, following an optimal measurement on the accessible a-dimensional subset of the environment.
  • By virtue of typicality, this means that, regardless of how a high-dimensional environment is partitioned, suitably measuring half of it generically suffices to project a qubit immersed in such environment onto a nearmaximally coherent state, a fact reminiscent of the quantum Darwinism approach to decoherence (Zurek, 2009; Brandão, Piani, and Horodecki, 2015).

E. Quantum speed limits

  • In the dynamics of a closed or open quantum system, quantum speed limits dictate the ultimate bounds imposed by quantum mechanics on the minimal evolution time between two distinguishable states of the system.
  • Interestingly, for pure states, even the inverses of the Mandelstam-Tamm and Margolus-Levitin quantities appearing on the right-hand side of Eq. (99) are themselves asymmetry monotones, bounding the asymmetry monotone given by 1=τ⊥ðjψiÞ.
  • Marvian, Spekkens, and Zanardi (2016) then derived new Mandelstam-Tamm–type quantum speed limits for unitary dynamics based on various measures of distinguishability, including a bound featuring the Wigner-Yanase skew information with respect to H (obtained when D is set to the relative Rényi entropy of order 1=2), which was also independently obtained by Mondal, Datta, and Sazim (2016).
  • Λ½ϱ under a (generally open) dynamics Λ. Quantum speed limits then ensue from a simple geometric observation, namely, that the geodesic connecting ϱ to ϱτ, whose length can be indicated byLgðϱ; ϱτÞ, is the path of shortest length among all physical evolutions between the given initial and final states Lgðϱ; ϱτÞ ≤ lgΛðϱ; ϱτÞ∀Λ.
  • The authors finally mention that looser speed limits involving the skew information were also recently presented by Pires, Céleri, and Soares-Pinto (2015) and Mondal, Datta, and Sazim (2016).

V. APPLICATIONS OF QUANTUM COHERENCE

  • Ranging from quantum information processing to quantum sensing and metrology, thermodynamics, and biology.the authors.
  • Particular emphasis is given to those settings in which a specific coherence monotone introduced in Sec. III acquires an operational interpretation, hence resulting in novel insights stemming from the characterization of quantum coherence as a resource.

A. Quantum thermodynamics

  • Recently, the role of coherence in quantum thermodynamics was discussed by several authors.
  • The authors review the main concepts in the following, being in large part based on the resource theory of quantum thermodynamics (Brandão et al., 2013; Gour et al., 2015; Goold et al., 2016) defined by the framework of thermal operations (Janzing et al., 2000).

1. Thermal operations

  • The importance of thermal operations arises from the fact that they are consistent with the first and second laws of thermodynamics (Lostaglio et al., 2015).
  • First, they are TIO with respect to the Hamiltonian HS, i.e., Λth½e−iHStϱSeiHSt ¼ e−iHStΛth½ϱS eiHSt: ð104Þ.
  • Here preservation of the thermal state is the only requirement on the quantum operation.
  • Interestingly, Faist, Oppenheim, and Renner (2015) showed that these maps are strictly more powerful than thermal operations.

2. State transformations via thermal operations

  • Several recent works studied necessary and sufficient conditions for two states ϱ and σ to be interconvertible via thermal operations.
  • More general conditions which allow for the addition of ancillas and catalytic conversions (Jonathan and Plenio, 1999), known as the second laws of quantum thermodynamics, were presented by Brandão et al. (2015).
  • As discussed by Lostaglio, Jennings, and Rudolph (2017), it is important to distinguish between two cases: namely, whether the state γE is incoherent, or has nonzero coherence, with respect to the eigenbasis of HE.
  • And apart from that the process can also create coherence in the system.
  • In particular, they investigated the creation of coherence for a quantum system (with respect to the eigenbasis of its Hamiltonian H) via unitary operations from a thermal state and also explored the energy cost for such coherence creation.

3. Work extraction and quantum thermal machines

  • Here hWiðϱSÞ denotes the amount of work that can be extracted from the state ϱS, and Π½ϱS ¼ Pi Tr½ΠiϱS Πi, where Πi are projectors onto the eigenspaces of HS.
  • Further results on the role of coherence for work extraction have also been presented by 20Note that thermomajorization is also related to the mixing distance studied by Ruch, Schranner, and Seligman (1978); see Egloff et al. (2015), where a relation between majorization and optimal guaranteed work extraction up to a risk of failure was investigated.
  • Moreover, it was shown by Vacanti, Elouard, and Auffeves (2015) that work is typically required for keeping coherent states out of thermal equilibrium.
  • Quantum coherence was also shown to be useful for transient cooling in absorption refrigerators (Mitchison et al., 2015).

B. Quantum algorithms

  • The role of coherence in quantum algorithms was discussed by Hillery (2016), with particular focus on the Deutsch-Jozsa algorithm (Deutsch and Jozsa, 1992).
  • Another important quantum algorithm is known as the DQC1 (Knill and Laflamme, 1998).
  • The role of quantum discord for DQC1 was later questioned by Dakić, Vedral, and Brukner (2010), who showed that certain nontrivial instances do not involve any quantum correlations.
  • This issue was further discussed by Datta and Shaji (2011).
  • The latter work indicates that coherence is indeed a suitable figure of merit for this protocol.

C. Quantum metrology

  • As the bound in Eq. (108) is asymptotically achievable for n ≫ 1 by means of a suitable optimal measurement, the quantum Fisher information directly quantifies the optimal precision of the estimation procedure and is thus regarded as the main figure of merit in quantum metrology (Paris, 2009; Giovannetti, Lloyd, and Maccone, 2011).
  • Quantum coherence in the form of asymmetry is the primary resource behind the power of quantum metrology.
  • In the absence of noise, the Heisenberg scaling can be equivalently achieved using n entangled probes in parallel, each subject to one instance of Uφ (Huelga et al., 1997; Giovannetti, Lloyd, and Maccone, 2006).

D. Quantum channel discrimination

  • Consider a d-dimensional probe and a set of unitary channels fUφg generated by H ¼ P d−1 i¼0 ijiihij, where fjiig sets the reference incoherent basis for the probe system and φ can take any of the d values f2πk=dgd−1k¼0 with uniform probability 1=d.
  • This provides a direct operational interpretation for the robustness of coherence RC in quantum discrimination tasks.
  • Such an interpretation can be extended to more general channel discrimination scenarios (i.e., with nonuniform prior probabilities and including nonunitary incoherent channels) and carries over to the robustness of asymmetry with respect to arbitrary groups (Napoli et al., 2016; Piani et al., 2016).

E. Witnessing quantum correlations

  • Recently, several authors tried to find Bell-type inequalities for various coherence quantifiers.
  • They found a Bell-type bound for this quantity for all product states ϱA ⊗ ϱB and showed that the bound is violated for maximally entangled states and a certain choice of observables X and Y.
  • In a similar spirit, the interplay between coherence and quantum steering was investigated by Mondal and Mukhopadhyay (2015) and Mondal, Pramanik, and Pati (2017), where steering inequalities for various coherence quantifiers were found, and by X. Hu and Fan (2016) and Hu et al. (2016), where the maximal coherence of steered states was investigated.
  • As further shown by Girolami and Yadin (2017), detection of coherence can also be used to witness multipartite entanglement.
  • In particular, an experimentally accessible lower bound on the quantum Fisher information (which does not require full state tomography) can serve as a witness for multipartite entanglement, as explicitly demonstrated for mixtures of Greenberger-Horne-Zeilinger states (Girolami and Yadin, 2017).

F. Quantum biology and transport phenomena

  • Transport is fundamental to a wide range of phenomena in the natural sciences and it has long been appreciated that coherence can play an important role for transport, e.g., in the solid state (Deveaud-Plédran, Quattropani, and Schwendimann, 2009; Li et al., 2012).
  • It is now recognized that typically both coherent and noise dynamics are required to achieve optimal performance.
  • A range of mechanisms to support this claim and understand its origin qualitatively [see Huelga and Plenio (2013) for an overview] has been identified.
  • In the studies of the impact of coherence on transport dynamics, formal approaches using coherence and asymmetry quantifiers based on the Wigner-Yanase skew information were used (Vatasescu, 2015, 2016), but the connection to function has remained tenuous so far.
  • Another question of interest in this context concerns that of the distinction between classical and quantum coherence (O’Reilly and Olaya-Castro, 2014) and dynamics (Wilde, McCracken, and Mizel, 2009; Li et al., 2012) in biological systems, most notably photosynthetic units.

G. Quantum phase transitions

  • Coherence and asymmetry quantifiers have been employed to detect and characterize quantum phase transitions, i.e., changes in the ground state of many-body systems occurring at or near zero temperature and driven purely by quantum fluctuations.
  • Karpat, Çakmak, and Fanchini (2014) and Çakmak, Karpat, and Fanchini (2015) showed that single-spin coherence reliably identifies the second-order quantum phase transition in the thermal ground state of the anisotropic spin-1=2 XY chain in a transverse magnetic field.
  • In particular, the singlespin skew information with respect to the Pauli spin-x operator σx, as well as its experimentally friendly lower bound which can be measured without state tomography (Girolami, 2014), exhibits a divergence in its derivative at the critical point, even at relatively high temperatures.
  • Malvezzi et al. (2016) extended the previous analysis to ground states of spin-1 Heisenberg chains.
  • Focusing on the one-dimensional XXZ model, they found that no coherence and asymmetry quantifier (encompassing skew information, relative entropy, and l1 norm) is able to detect the triple point Rev. Mod.

VI. CONCLUSIONS

  • Quantum thermodynamics, and quantum biology, as well as physics more widely.the authors.
  • The set F allows one to create any incoherent state from any other state.
  • A.2. In place of the first condition, the authors give the corresponding literature reference, where suitable motivations can be found for each set.
  • The authors further reviewed in Sec. III the current progress on quantifying coherence and related manifestations of nonclassicality in compliance with the underlying framework of resource theories, in particular, highlighting interconnections between different measures and, where possible, their relations to entanglement measures.
  • Most of these advances are still at a very early stage, and the operational value of coherence still needs to be pinpointed clearly in many contexts.

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Colloquium: Quantum coherence as a resource
Alexander Streltsov
*
Faculty of Applied Physics and Mathematics, Gdańsk University of Technology,
80-233 Gdańsk, Poland,
National Quantum Information Centre in Gdańsk, 81-824 Sopot, Poland,
Dahlem Center for Complex Quantum Systems, Freie Universita
t Berlin,
D-14195 Berlin, Germany,
and ICFOInstitut de Cie` ncies Foto` niques, The Barcelona Institute of Science and
Technology, ES-08860 Castelldefels, Spain
Gerardo Adesso
Centre 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
Martin B. Plenio
Institute of Theoretical Physics and IQST, Ulm University,
Albert-Einstein-Allee 11, D-89069 Ulm, Germany
(published 30 October 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 considerable interest.
Despite the fundamental importance of quantum coherence, the development of a rigorous theory
of quantum coherence as a physical resource has been initiated only recently. This Colloquium
discusses and reviews the development of this rapidly growing research field that encompasses the
characterization, quantification, manipulation, dynamical evolution, and operational application of
quantum coherence.
DOI: 10.1103/RevModPhys.89.041003
CONTENTS
I. Introduction 2
II. Resource Theories of Quantum Coherence 3
A. Constraints, operations, and resources 3
1. Incoherent states 4
2. Classes of incoherent operations 4
B. Coherence as a resource 6
1. Maximally coherent states and state
transformations via incoherent operations 6
2. States and maps 7
C. Quantum coherence in distributed scenarios 7
D. Connection between coherence and
entanglement theory 8
III. Quantifying Quantum Coherence 9
A. Postulates for coherence monotones and measures 9
B. Distillable coherence and coherence cost 10
C. Distance-based quantifiers of coherence 10
1. Relative entropy of coherence 11
2. Coherence quantifiers based on matrix norms 11
3. Geometric coherence 12
D. Convex roof quantifiers of coherence 12
E. Coherence monotones from entanglement 13
F. Robustness of coherence 13
G. Coherence quantifiers from interferometric visibility 14
H. Coherence of assistance 14
I. Coherence and quantum correlations beyond
entanglement 15
J. Coherence in continuous variable systems 15
K. Coherence, asymmetry, and nonclassicality 16
1. Asymmetry monotones 16
2. Quantifying superpositions 17
3. Coherence rank and general quantifiers of
nonclassicality 17
4. Optical coherence and nonclassicality 17
L. Multipartite settings 18
1. General distance-based coherence quantifiers 18
2. Quantum-incoherent relative entropy 18
3. Distillable coherence of collaboration 19
4. Recoverable coherence 19
5. Uncertainty relations and monogamy of coherence 19
IV. Dynamics of Quantum Coherence 20
A. Freezing of coherence 20
*
streltsov.physics@gmail.com
gerardo.adesso@nottingham.ac.uk
martin.plenio@uniulm.de
REVIEWS OF MODERN PHYSICS, VOLUME 89, OCTOBERDECEMBER 2017
0034-6861=2017=89(4)=041003(34) 041003-1 © 2017 American Physical Society

B. Coherence in non-Markovian evolutions 21
C. Cohering power of quantum channels and evolutions 21
D. Average coherence of random states and typicality 22
1. Average relative entropy of coherence 22
2. Average l
1
norm of coherence 23
3. Average recoverable coherence 23
E. Quantum speed limits 23
V. Applications of Quantum Coherence 24
A. Quantum thermodynamics 24
1. Thermal operations 24
2. State transformations via thermal operations 25
3. Work extraction and quantum thermal machines 25
B. Quantum algorithms 26
C. Quantum metrology 26
D. Quantum channel discrimination 27
E. Witnessing quantum correlations 27
F. Quantum biology and transport phenomena 28
G. Quantum phase transitions 28
VI. Conclusions 29
Acknowledgments 29
References 30
I. INTRODUCTION
Coherence marks the departure of todays theories of the
physical world from the principles of classical physics. The
theory of electromagnetic waves, which may exhibit optical
coherence and interference, represents a radical departure
from classical ray optics. Energy quantization and the rise of
quantum mechanics as a unified picture of waves and particles
in the early part of the 20th century have further amplified the
prominent role of coherence in physics. Indeed, by combi-
nation of energy quantization and the tensor product structure
of the state space, coherence underlies phenomena such as
quantum interference and multipartite entanglement that play
a central role in the applications of quantum physics and
quantum information science.
The investigation and exploitation of coherence of quantum
optical fields has a long-standing history. It has enabled the
realization of now mature technologies, such as the laser
and its applications, that are often classified as Quantum
Technologies 1.0 as they rely mainly on single particle
coherence. At the mathematical level the coherence of
quantum optical fields is described in terms of phase space
distributions and multipoint correlation functions, approaches
that find their roots in classical electromagnetic theory
(Glauber, 1963; Sudarshan, 1963; Mandel and Wolf, 1965 ).
However, quantum coherence is not restricted to optical
fields. More importantly, as the key ingredient that drives
quantum technologies, it is highly desirable to be able to
precisely quantify the usefulness of coherence as a resource
for such applications. These pressing questions are calling for
further development of the theory of quantum coherence.
The emergence of quantum information science over the
last three decades has, among other insights, led to a reassess-
ment of quantum physical phenomena as resources that may
be exploited to achieve tasks that are otherwise not possible
within the realm of classical physics. This resource-driven
viewpoint has motivated the development of a quantitative
theory that captures the resource character of physical traits in
a mathematically rigorous fashion.
In a nutshell, any such theory first considers constraints that
are imposed on us in a specific physical situation (e.g., the
inability to perform joint quantum operations between distant
laboratories due to the impossibility to transfer quantum
systems from one location to the other while preserving their
quantum coherence, and thus restricting us to local operations
and classical communication). Executing general quantum
operations under such a constraint then requires quantum
states that contain a relevant resource (e.g., entangled states
that are provided to us at a certain cost) and can be consumed
in the process. The formulation of such resource theories was
in fact initially pursued with the quantitative theory of
entanglement (Plenio and Virmani, 2007; Horodecki et al.,
2009) but has since spread to encompass a wider range of
operational settings (Horodecki and Oppenheim, 2013b;
del Rio, Kraemer, and Renner, 2015; Coecke, Fritz, and
Spekkens, 2016).
The theory of quantum coherence as a resource is a case in
point. Following an early approach to quantifying superposi-
tions of orthogonal quantum states by Åberg (2006) and
progressing alongside the independent yet related resource
theory of asymmetry (Gour and Spekkens, 2008; Vaccaro et al.,
2008; Gour, Marvian, and Spekkens, 2009; Marvian and
Spekkens, 2014a, 2014b), a resource theory of coherence
was primarily proposed by Baumgratz, Cramer, and Plenio
(2014) and Levi and Mintert (2014) and further developed by
Chitambar and Gour (2016a, 2016b, 2017), Winter and Yang
(2016),andYadin
et al. (2016). Such a theory asks the question
what can be achieved and at what resource cost when the
devices that are available to us are essentially classical, that is,
they cannot create coherence in a preferred basis. This analysis,
currently still under development, endeavors to provide a
rigorous framework to describe quantum coherence in analogy
with what has been done for quantum entanglement and other
nonclassical resources (Plenio and Virmani, 2007; Horodecki
et al.,2009; Modi et al.,2012; Horodecki and Oppenheim,
2013b; Sperling and Vogel, 2015; Streltsov, 2015; Adesso,
Bromley, and Cianciaruso, 2016). Within such a framework,
recent progress has shown that a growing number of applica-
tions can be certified to rely on various incarnations of quantum
coherence as a primary ingredient, and appropriate figures of
merit for such applications can be precisely linked back to
specific coherence monotones, providing operational interpre-
tations for the latter.
Theseapplications include so-calledQuantumTechnologies
2.0, such as quantum-enhanced metrology and communication
protocols, and extend farther into other fields, such as thermo-
dynamics and even certain branches of biology. Beyond such
application-drivenviewpoints, which may provide new insights
intoall theseareas,one canalso considerthetheoryof coherence
as a resource as a novel approach toward the demarcation of the
fundamental difference between classical and quantum physics
in a quantitative manner: a goal that may eventually lead to a
better understanding of the classical-quantum boundary.
This Colloquium collected the most up-to-date knowledge
on coherence in single and composite quantum systems from
a modern information theory perspective. We reviewed this
fascinating and fundamental subject in an accessible manner,
yet without compromising any rigor.
Streltsov, Adesso, and Plenio: Colloquium: Quantum coherence as a resource
Rev. Mod. Phys., Vol. 89, No. 4, OctoberDecember 2017 041003-2

The Colloquium is organized as follows (see Fig. 1).
Section II gives a comprehensive overview of recent develop-
ments to construct a resource theory of quantum coherence,
including a hierarchy of possible cl asses of incoherent
operations, and the conditions to which any valid coherence
quantifier should abide. It also discusses established links with
other resource theories, most prominently those of asymmetry
and quantum entanglement. Section III presents a compen-
dium of recently proposed monotones and measures of
quantum coherence, based on different physical approaches
endowed with different mathematical properties, in single and
multipartite systems; interplays with other measures of non-
classicality are highlighted as well. Section IV reviews the
phenomenology of quantum coherence in the dynamical
evolution of open quantum systems, further reporting results
on the average coherence of random states, and on the
cohering power of quantum channels. Section V focuses on
the plethora of applications and operational interpretations
highlighted so far for quantum coherence in thermodynamics,
interference phenomena, quantum algorithms, metrology and
discrimination, quantum biology, many-body physics, and the
detection of quantum correlations. Section VI concludes with
a summary and discussion of some currently open issues in the
theoretical description of coherence and its role in quantum
physics and beyond.
We emphasize that, due to limitations in space and
focus, this Colloquium cannot cover all ramifications of
the concept of quantum coherence. It is nevertheless our
expectation that this Colloquium, while being self-
contained, can stimulate the interested reader to undertake
further research toward 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 an
optimization of existing ones) in a variety of physical and
biological contexts of high technological interest.
II. RESOURCE THEORIES OF QUANTUM COHERENCE
Coherence is a property of the physical world that is used
to drive a wide variety of phenomena and devices. Hence
coherence adopts the quality of a resource, as it may be
provided at a certain cost, manipulated by otherwise incoher-
ent means and consumed to achieve useful tasks. The
quantitative study of these processes and their attainable
efficiencies requires careful definitions of the accessible
operations and gives rise to a framework that has become
known as a resource theory. In line with earlier developments
in quantum information science, for example, in the context of
entanglement (Vedral and Plenio, 1998; Plenio and Virmani,
2007; Brandão and Plenio, 2008, 2010; Horodecki et al.,
2009), quantum thermodynamics (Ruch, 1975; Janzing et al.,
2000; Brandão et al., 2013; Gour et al., 2015; Goold et al.,
2016), purity (Horodecki, Horodecki, and Oppenheim, 2003),
and referenc e frames (Gour and Spekkens, 2008; Gour,
Marvian, and Spekkens, 2009; Marvian and Spekkens,
2014a), the formulation of the resource theory of coherence
(Åberg, 2006 ; Baumgratz, Cramer, and Plenio, 2014; Levi and
Mintert, 2014; Winter and Yang, 2016) extends the family of
resources theories of knowledge (del Rio, Kraemer, and
Renner, 2015).
A. Constraints, operations, and resources
A resource theory is fundamentally determined by con-
straints that are imposed on us and which determine the set of
the freely accessible quantum operations F . These constraints
may be due to either fundamental conservation laws, such
as superselection rules and energy conservation, or constraints
due to the practical difficulty of executing certain operations,
e.g., the restriction to local operations and classical commu-
nication (LOCC) which gives rise to the resource theory
of entanglement (Plenio and Virmani, 2007; Horodecki et al.,
2009).
The states that can be generated from the maximally mixed
state
1
by the application of free operations in F alone, are
considered to be available free of charge, forming the set I of
free states. All the other states attain the status of a resource,
whose provision carries a cost. These resource states may be
FIG. 1. Plan of the Colloquium. (Top) Sec. II: Resource theories
of quantum coherence; the inset depicts a comparison between
some classes of incoherent operations. Adapted from Chitambar
and Gour, 2016b. (Right) Sec. III: Quantifying quantum coher-
ence; the inset depicts the construction of coherence monotones
from entanglement. Adapted from Streltsov et al., 2015. (Bottom)
Sec. IV: Dynamics of quantum coherence; the inset depicts an
illustration of coherence freezing under local incoherent chan-
nels. Adapted from Bromley, Cianciaruso, and Adesso, 2015.
(Left) Sec. V: Applications of quantum coherence; the insets
depicts a schematic of a quantum phase discrimination protocol.
Adapted from Napoli et al., 2016. The Introduction (Sec. I) and
Conclusions (Sec. VI) complete the Colloquium.
1
The maximally mixed state can always be obtained by erasing all
information about the system. Hence it is fair to assume that it is
devoid of any useful resource and freely available.
Streltsov, Adesso, and Plenio: Colloquium: Quantum coherence as a resource
Rev. Mod. Phys., Vol. 89, No. 4, OctoberDecember 2017 041003-3

used to achieve operations that cannot be realized by using
only members of F . Alternatively, one may also begin by
defining the set of free states I , and then consider classes of
operations that map this set into itself and use this to define F .
For the purposes of the present exposition of the resource
theory of coherence, we begin by adopting the latter point of
view and then proceed to require additional desirable proper-
ties of our classes of free operations.
1. Incoherent states
Coherence is naturally a basis dependent concept, which is
why we first need to fix the preferred or reference basis in
which to formulate our resource theory. The reference basis
may be dictated by the physics of the problem under
investigation (e.g., one may focus on the energy eigenbasis
when addressing coherence in transport phenomena and
thermodynamics) or by a task for which coherence is required
(e.g., the estimation of a magnetic field in a certain direction
within a quantum metrology setting). Given a d-dimensional
Hilbert space H (with d assumed finite, even though some
extensions to infinite d can be considered), we denote its
reference orthonormal basis by fjiig
i¼0;;d1
. The density
matrices that are diagonal in this specific basis are called
incoherent, i.e., they are accessible free of charge and form the
set I BðHÞ, where BðHÞ denotes the set of all bounded
trace class operators on H. Hence, all incoherent density
operators ϱ I are of the form
ϱ ¼
X
d1
i¼0
p
i
jiihi1Þ
with probabilities p
i
.
In the case of more th an one party, the preferred basis with
respect to which coherence is studied will be constructed
as the tensor product of the corresponding local reference
basis states for each subsystem. General multipartite incoher-
ent states are then defined as convex combinations of such
incoherent pure product states (Bromley, Cianciaruso, and
Adesso, 2015; Streltsov et al., 2015; Winter and Yang, 2016).
For example, if the reference basis for a single qubit is taken
to be the computational basis fj0i; j1ig, i.e., the eigenbasis of
the Pauli σ
z
operator, then any density matrix with a nonzero
off-diagonal element jϱ
01
j¼jh0jϱj1ij 0 is outside the set I
of incoherent states, and hence has a resource content.
Similarly, for an N-qubit system, the set of incoherent states
I is formed by all and only the density matrices ϱ diagona l in
the composite computational basis fj0i; j1ig
N
, with any
other state being coherent, that is, resourceful.
Note that some frameworks of coherence may allow for a
larger set of free states. This is, in particular, the case for the
resource theory of asymmetry ( Gour and Spekkens, 2008;
Gour, Marvian, and Spekkens, 2009; Marvian and Spekkens,
2014a, 2014b), where the set of free states is defined by all
states which commute with a given Hamiltonian H. If the
Hamiltonian is nondegenerate, the corresponding set of
free states is exactly the set of incoherent states described,
where the incoherent basis is defined by the eigenbasis
of the Hamiltonian. However, the situation changes if
the Hamiltonian has degeneracies, in which case any
superposition of the eigenstates corresponding to the degen-
erate subspaces is also considered as free. This has important
implications in quantum thermodynamics as described in
more detail in Sec. V. A. In the following, whenever we refer
to incoherent states, we explicitly mean states of the form (1).
2. Classes of incoherent operations
The definition of free operations for the resource theory of
coherence is not unique and different choices, often motivated
by suitable practical considerations, are being examined in the
literature. Here we present the most important classes and
briefly discuss their properties and relations among each other.
We start with the largest class, the maximally incoherent
operations (MIO) (Åberg, 2006) (also known as incoherence
preserving operations), which are defined as any trace
preserving completely positive and nonselective quantum
operations Λ BðHÞ BðHÞ such that
Λ½I I : ð2Þ
As with every quantum operation, this mathematically natural
set of operations can also always be obtained by a Stinespring
dilation, i.e., the provision of an ancillary environment in
some state σ, a subsequent unitary operation U between
system and environment, followed by the tracing out of the
environment,
Λ½ϱ¼Tr
E
½Uðϱ σÞU
: ð3Þ
If an operation can be implemented a s in Eq. (3) by using an
incoherent state σ of the environment and a global incoher-
ent unitary U (a unitary that is diagonal in the preferred
basis), we say that the operation has a free dilation. Note
that despite the fact that MIO cannot create coherence,
these operations in general do not have a free dilation
2
(Chitambar and Gour, 2016a , 2016b, 2017; Marvian and
Spekkens, 2016).
A smaller and more relevant class of free operations for the
theory of coherence is that of incoherent operations (IO)
(Baumgratz, Cramer, and Plenio, 2014) which are character-
ized as the set of trace preserving completely positive maps
Λ BðHÞ BðHÞ admitting a set of Kraus operators
3
fK
n
g
such that
P
n
K
n
K
n
¼ 1 (trace preservation) and, for all n and
ϱ I ,
K
n
ϱK
n
Tr½K
n
ϱK
n
I: ð4Þ
This definition of IO ensures that, in any of the possible
outcomes of such an operation, coherence can never be
generated from an incoherent input state, not even
2
This mirrors the situation in entanglement theory where separable
operations cannot create entanglement but in general cannot be
implemented via LOCC (Bennett et al., 1999).
3
According to the Kraus decomposition, the maps act as Λ½ϱ¼
P
n
K
n
ϱK
n
.
Streltsov, Adesso, and Plenio: Colloquium: Quantum coherence as a resource
Rev. Mod. Phys., Vol. 89, No. 4, OctoberDecember 2017 041003-4

probabilistically.
4
Also this class of operations does not admit
a free dilation in general (Chitambar and Gour, 2016a, 2016b,
2017; Marvian and Spekkens, 2016).
In the previous two definitions, the focus was placed on the
inability of incoherent operations to generate coherence. One
may, however, be more stringent by adding further desirable
properties to the set of free operations. One such approa ch
requires that admissible operations are not capable of making
use of coherence in the input state. Defining the dephasing
operation
Δ½ϱ¼
X
d1
i¼0
jiihijϱjiihij; ð5Þ
an operation Λ is called strictly incoherent (SIO) (Winter and
Yang, 2016; Yadin et al., 2016) if it can be written in terms of a
set of incoherent Kraus operators fK
n
g, such that the out-
comes of a measurement in the reference basis applied to the
output are independent of the coherence of the input state, i.e.,
hijK
n
ϱK
n
jii¼hijK
n
Δ½ϱK
n
ji6Þ
for all n and i. Equivalently, SIO can be characterized as those
operations that have an incoherent Kraus decomposition fK
n
g
such that the operators K
n
are also incoherent (Winter and
Yang, 2016; Yadin et al., 2016). As shown by Chitambar and
Gour (2016a, 2016b, 2017), SIO in general do not admit a free
dilation either. Nevertheless, a special type of dilation for SIO
of the form (3) was provided by Yadin et al. (2016), which
consists of (i) unitary operations on the environment con-
trolled by the incoherent basis of the system
P
j
jjihjj U
j
,
(ii) measurements on the environment in any basis, and
(iii) incoherent unitary operations on the system conditioned
on the measurement outcome
P
j
e
iθ
j
jπðjÞihjj, with fjπðjÞig
denoting a permutation of the reference basis of the system.
The classes of operations defined so far include permuta-
tions of the reference basis states for free. This is natural when
any such operation is considere d from a passive point of view,
amounting to a relabeling of the states. Viewed from an active
point of view instead, i.e., asking for a unitary operation that
realizes this permutation, it may be argued that such an
operation does in fact cost coherence resources in order to be
performed in the laboratory. This suggests that, from an
operational point of view, permutations should be excluded
from the free operations, as is done in the more stringent set
of translationally invariant operations (TIO). The latter are
defined as those commuting with phase randomization (Gour
and Spekkens, 2008; Gour, Marvian, and Spekkens, 2009;
Marvian and Spekkens, 2013 , 2014a, 2014b, 2016; Marvian,
Spekkens, and Zanardi, 2016). More precisely, given a
Hamiltonian H, an operation Λ is translationally invariant
with respect to H if it fulfills the condition (Gour, Marvian,
and Spekkens, 2009; Marvian and Spekkens, 2014a; Marvian,
Spekkens, and Zanardi, 2016)
e
iHt
Λ½ϱe
iHt
¼ Λ½e
iHt
ϱe
iHt
: ð7Þ
TIO play an important role in the resource theory of
asymmetry (see Sec. III.K.1) and quantum thermodynamics
(see Sec. V. A). Interestingly, Marvian and Spekkens (2016)
showed that TIO have a free dilation if one additionally
allows postselection with an incoherent measurement on the
environment.
As mentioned, the sets MIO, IO, and SIO in general do not
have a free dilation, i.e., they cannot be implemented by
coupling the system to an environment in an incoherent state
followed by a global incoherent unitary. Motivated by this
observation, Chitambar and Gour (2016a, 2016b, 2017)
introduced the set of physical incoherent operations (PIO).
These are all operations which can instead be implemented in
the aforementioned way, additionally allowing for incoherent
measurements on the environment and classical postprocess-
ing of the measurement outcomes.
Clearly, MIO is the largest set of free operations for a
resource theory of coherence, and all other sets listed are strict
subsets of it. Inclusion relations between each of these sets are
nontrivial in general. Here we mention that (see also Fig. 1,
top panel)
PIO SIO IO MIO; ð8Þ
and refer the interested reader to Chitambar and Gour (2016a,
2016b, 2017) and de Vicente and Streltsov (2017) for more
detailed discussions.
Another interesting set is given by dephasing-covariant
incoherent operations (DIO), which were introduced inde-
pendently by Chitambar and Gour (2016a, 2016b, 2017) and
Marvian and Spekkens (2016). These are all operations Λ
which commute with the dephasing map Eq. (5), i.e.,
Λ½Δ½ϱ ¼ Δ½Λ½ϱ. It is an interesting open question whether
DIO have a free dilation.
We also mention genuinely incoherent operations (GIO)
and fully incoherent operations (FIO) (de Vicente and
Streltsov, 2017). GIO are operations which preserve all
incoherent states, i.e., Λ½ϱ¼ϱ for any incoherent state ϱ.
In particular, every GIO is incoherent regardless of the
particular Kraus decomposition, i.e., for every experimental
realization of the operation. Since GIO do not allow for
transformations between different incoherent states, notably,
for example, between the energy eigenstates (when coherence
is measured with respect to the eigenbasis of the Hamiltonian
of the system), they capture the framework of coherence in the
presence of additional constraints, such as energy conserva-
tion. FIO are in turn the most general set of operations which
are incoherent for every Kraus decomposition (
de Vicente and
Streltsov, 2017). The GIO framework is closely related to the
concept of resource destroying maps introduced by Liu, Hu,
and Lloyd (2017). The latter studies quantum operations
which transform any quantum state onto a free state and
moreover preserve all free states. If the set of free states is
4
Note that relaxing the condition of trace preservation here may
have nontrivial consequences, as one should then ensure that the
missing Kraus operators share the property that they map I into
itself. That this is possible for IO was recently proven by Theurer
et al. (2017).
Streltsov, Adesso, and Plenio: Colloquium: Quantum coherence as a resource
Rev. Mod. Phys., Vol. 89, No. 4, OctoberDecember 2017 041003-5

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TL;DR: In this article, the quantum Fourier transform and its application in quantum information theory is discussed, and distance measures for quantum information are defined. And quantum error-correction and entropy and information are discussed.
Abstract: Part I Fundamental Concepts: 1 Introduction and overview 2 Introduction to quantum mechanics 3 Introduction to computer science Part II Quantum Computation: 4 Quantum circuits 5 The quantum Fourier transform and its application 6 Quantum search algorithms 7 Quantum computers: physical realization Part III Quantum Information: 8 Quantum noise and quantum operations 9 Distance measures for quantum information 10 Quantum error-correction 11 Entropy and information 12 Quantum information theory Appendices References Index

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TL;DR: Probability in classical and quantum physics has been studied in this article, where classical probability theory and stochastic processes have been applied to quantum optical systems and non-Markovian dynamics in physical systems.
Abstract: PREFACE ACKNOWLEDGEMENTS PART 1: PROBABILITY IN CLASSICAL AND QUANTUM MECHANICS 1. Classical probability theory and stochastic processes 2. Quantum Probability PART 2: DENSITY MATRIX THEORY 3. Quantum Master Equations 4. Decoherence PART 3: STOCHASTIC PROCESSES IN HILBERT SPACE 5. Probability distributions on Hilbert space 6. Stochastic dynamics in Hilbert space 7. The stochastic simulation method 8. Applications to quantum optical systems PART 4: NON-MARKOVIAN QUANTUM PROCESSES 9. Projection operator techniques 10. Non-Markovian dynamics in physical systems PART 5: RELATIVISTIC QUANTUM PROCESSES 11. Measurements in relativistic quantum mechanics 12. Open quantum electrodynamics

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TL;DR: In this paper, Dirac Formulation of Quantum Mechanics Elementary Quantum Systems Operator Algebra Quantization of the Electromagnetic Field Interaction of Radiation with Matter Quantum Theory of Damping--Density Operator Methods Quantum Theory-Langevin Approach Lamb's Semiclassical Theory of a Laser [1] Statistical properties of a laser Appendices Index
Abstract: Dirac Formulation of Quantum Mechanics Elementary Quantum Systems Operator Algebra Quantization of the Electromagnetic Field Interaction of Radiation with Matter Quantum Theory of Damping--Density Operator Methods Quantum Theory of Damping--Langevin Approach Lamb's Semiclassical Theory of a Laser [1] Statistical Properties of a Laser Appendices Index

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Journal ArticleDOI
TL;DR: In this article, a review of quantum-enhanced measurements can be found, including the use of more general quantum correlations such as quantum discord, identical particles, or non-trivial hamiltonians, and the estimation of thermodynamical parameters or parameters characterizing non-equilibrium states.
Abstract: Quantum-enhanced measurements exploit quantum mechanical effects for increasing the sensitivity of measurements of certain physical parameters and have great potential for both fundamental science and concrete applications. Most of the research has so far focused on using highly entangled states, which are, however, difficult to produce and to stabilize for a large number of constituents. In the following we review alternative mechanisms, notably the use of more general quantum correlations such as quantum discord, identical particles, or non-trivial hamiltonians; the estimation of thermodynamical parameters or parameters characterizing non-equilibrium states; and the use of quantum phase transitions. We describe both theoretically achievable enhancements and enhanced sensitivities, not primarily based on entanglement, that have already been demonstrated experimentally, and indicate some possible future research directions.

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Frequently Asked Questions (8)
Q1. What is the pressing open question in the theory of optical coherence?

In particular, determining the suitable set of free operations for the theory of optical coherence stands as one of the most pressing open questions. 

In the case of more than one party, the preferred basis with respect to which coherence is studied will be constructed as the tensor product of the corresponding local reference basis states for each subsystem. 

Transport is fundamental to a wide range of phenomena in the natural sciences and it has long been appreciated that coherence can play an important role for transport, e.g., in the solid state (Deveaud-Plédran, Quattropani, and Schwendimann, 2009; Li et al., 2012). 

The importance of thermal operations arises from the fact that they are consistent with the first and second laws of thermodynamics (Lostaglio et al., 2015). 

The explicit construction for an arbitrary single-qubit unitary can be found in Baumgratz, Cramer, and Plenio (2014), and the extension to general quantum operations of arbitrary dimension was studied by Chitambar and Hsieh (2016) and Ben Dana et al. (2017). 

This quantum algorithm provides an exponential speedup over the best known classical procedure for estimating the trace of a unitary matrix (given as a sequence of two-qubit gates). 

This demonstrates that continuous variable states exhibiting optical nonclassicality can be seen essentially as the limiting case of the same resource states identified by Baumgratz, Cramer, and Plenio (2014), when the incoherent basis is chosen as the set of Glauber-Sudarshan coherent states. 

General quantum operations of the form in Eq. (14), but without the incoherence restriction, have been studied extensively in entanglement theory, where they are called separable operations (Vedral and Plenio, 1998).