week ending PRL 113, 140401 (2014) PHYSICAL REVIEW LETTERS 3 OCTOBER 2014 Quantifying Coherence T. Baumgratz, M. Cramer, and M. B. Plenio Institut für Theoretische Physik, Albert-Einstein-Allee 11, Universität Ulm, 89069 Ulm, Germany (Received 24 February 2014; revised manuscript received 18 July 2014; published 29 September 2014) We introduce a rigorous framework for the quantification of coherence and identify intuitive and easily computable measures of coherence. We achieve this by adopting the viewpoint of coherence as a physical resource. By determining defining conditions for measures of coherence we identify classes of functionals that satisfy these conditions and other, at first glance natural quantities, that do not qualify as coherence measures. We conclude with an outline of the questions that remain to be answered to complete the theory of coherence as a resource. DOI: 10.1103/PhysRevLett.113.140401 PACS numbers: 03.65.Aa, 03.67.Mn Introduction.—Coherence, being at the heart of interfer- physical operations to be realized define resources (e.g., ence phenomena, plays a central role in physics as it entangled states) that help to overcome the imposed con- enables applications that are impossible within classical straints [7,8]. This viewpoint has proven fruitful not only mechanics or ray optics. The rise of quantum mechanics as for the development of applications, but also in providing a unified picture of waves and particles further strengthened the impetus for theory to establish a unified and rigorously the prominent role of coherence in physics. Indeed, in defined framework for a quantitative theory of physical conjunction with energy quantization and the tensor prod- resources by addressing the three principal issues: (i) the uct structure of state space, coherence underlies phenomena characterization, (ii) the quantification, and (iii) the manipu- such as multiparticle interference and entanglement that lation of quantum states under the imposed constraints play a central role in applications of quantum physics and [9,10]. This framework is being explored for entanglement quantum information science. [5,6], thermodynamics [11,12], and reference frames [13] Quantum optical methods provide an important set of and has led to the recognition of deep interrelations between tools for the manipulation of coherence, and indeed, at its the theories of entanglement and the second law [7,8]. basis lies the formulation of the quantum theory of coherence In contrast, a wide variety of measures of coherence [1,2]. Here, coherence is studied in terms of phase space is in use (often functions of a density matrix’ off-diagonal distributions and multipoint correlation functions to provide entries) whose use tends to be justified principally on the a framework that relates closely to classical electromagnetic grounds of physical intuition. Here, we put such measures phenomena. While this is helpful in drawing intuition from on a sound footing by establishing a quantitative theory of classical wave mechanics and identifies those aspects for coherence as a resource following the approach that has which quantum coherence deviates from classical coherence been established for entanglement in Refs. [6–8] and for phenomena, it does not provide a rigorous and unambiguous reference frames in Ref. [13]. We present the basic framework. The development of such a quantitative frame- assumptions of our approach and use these to identify work for coherence gains further urgency in the light of recent various quantitative and easy-to-compute valid measures of discussions concerning the role of coherence in biological coherence while rejecting others. Results.— systems [3] which can benefit from a more rigorous approach At the heart of our discussion lies the charac- to the quantification of coherence properties. terization of incoherent states together with the notion of The development of quantum information science over incoherent operations, i.e., quantum operations that map the the last two decades has led to a reassessment of quantum set of incoherent states onto itself. We distinguish between physical phenomena such as entanglement, elevating them quantum operations with and without subselection. These from mere tools to “subtly humiliate the opponents of technical definitions lead to an operationally well-defined quantum mechanics” [4] to resources that may be exploited maximally coherent state which may serve as a unit for to achieve tasks that are not possible within the realm of coherence. We collect a set of conditions any proper measure classical physics. This viewpoint, then, motivates the devel- of coherence should satisfy. Prime among them is the opment of a quantitative theory that captures this resource requirement of monotonicity under incoherent operations. character in a mathematically rigorous fashion. The formu- We, then, discuss several examples—some of which take the lation of such resource theories was initially pursued with the form of easy to evaluate analytical expressions. For instance, quantitative theory of entanglement [5,6] which led to the we find that the relative entropy of coherence view that constraints [e.g., the restriction to local operations C ϱˆ S ϱˆ − S ϱˆ ; and classical communication (LOCC)] that prevent certain rel:ent:ð Þ¼ ð diagÞ ð Þ ð1Þ 0031-9007=14=113(14)=140401(5) 140401-1 © 2014 American Physical Society week ending PRL 113, 140401 (2014) PHYSICAL REVIEW LETTERS 3 OCTOBER 2014 S ϱˆ where is the von Neumann entropy and diag denotes the subselection according to these measurement outcomes. ϱˆ Kˆ state obtained from by deleting all off-diagonal elements, PThese are also defined by Kraus operators n with l ˆ † ˆ and the intuitive 1 norm of coherence nKnKn ¼ 1, which now, however, may each have a X Kˆ d d different output dimension ( n is a n × in matrix) and are Cl ðϱˆÞ¼ jϱi;jj; ð2Þ ˆ ˆ † 1 again required to fulfil KnIKn ⊂ I for each n. Retaining i;j i≠j the knowledge of outcomes of the measurement, the state ˆ ˆ † corresponding to outcome n is given by ϱˆ n ¼ KnϱˆKn=pn are both proper measures of coherence. In contrast, we ˆ ˆ † and occurs with probability pn ¼ tr½KnϱˆKn. find that the sum of the squared absolute values of all Incoherent Kraus operators that are of particular impor- off-diagonal elements violates monotonicity. tance for decoherence mechanisms of single qubits are, Incoherent states.—The first step to defining a coherence e.g., the ones that define the depolarizing, the phase- measure is to agree which states are incoherent. A natural damping, and the amplitude-damping channels [17,18]. definition is to fix a particular basis, fjiigi 1;…;d, of the ¼ Moreover, permutations of modes of dual-rail qubits in d-dimensional Hilbert space H in which we consider our linear optics experiments are examples of incoherent quantum states [14]. We call all density matrices that are operations. With this, we set the framework for a resource diagonal in this basis incoherent and, henceforth, label this theory for quantum coherence. All that follows is deduced set of quantum states by I ⊂ H [15]. Hence, all density from these physically well motivated definitions. δˆ ∈ I operators are of the form Maximally coherent state.—We start by identifying a d Xd -dimensional maximally coherent state as a state that ˆ allows for the deterministic generation of all other δ ¼ δijiihij: ð3Þ d i¼1 -dimensional quantum states by means of incoherent operations. Note that this definition (i) is independent of Incoherent operations.—The definition of coherence a specific measure for the coherence and (ii) allows us to monotones (and, thus, coherence quantifiers) requires the identify a unit for coherence to which all measures may be definition of operations that are incoherent—just as in normalized. A maximally coherent state is given by entanglement theory the definition of entanglement monot- ones requires a definition of nonentangling operations. 1 Xd jΨdi≔ pffiffiffi jii; ð4Þ There, this definition is determined by practical consid- d erations, namely locality constraints, which leads to the i¼1 definition of LOCC operations. Here, we characterize the because by means of incoherent operations [of type (A) or set of incoherent physical operations as follows. Quantum (B)] alone, any d × d state ϱˆ may be prepared from Ψd Kˆ j i operationsP are specified by a set of Kraus operators f ng with certainty. We show this by explicitly constructing an Kˆ †Kˆ 1 satisfying n n n ¼ . We require the incoherent oper- incoherent operation that achieves the transformation in ˆ ˆ † ators to fulfil KnIKn ⊂ I for all n [16]. This definition the Supplemental Material [19]. guaranteesP that in an overall quantum operation Two natural questions arise immediately. First, is this ˆ ˆ † ϱˆ↦ nKnϱˆKn, even if one does not have access to maximally coherent state a resource which, when con- individual outcomes n, no observer (e.g., one who does sumed, allows for the generation of all other coherent have access to these outcomes) would conclude that operations by means of incoherent operations? We dem- coherence has been generated from an incoherent state. onstrate in the Supplemental Material [19] that this is, Hence, we do not allow, not even probabilistically, that in indeed, the case: Provided with jΨ2i, every unitary oper- any of the arms of the quantum operation coherence is ation on a qubit may be implemented by incoherent generated from incoherent input states. operations. Second, one may ask whether incoherent We distinguish two classes of quantum operations. operations introduce an order on the set of quantum states, (A) The incoherent completely positive and trace preserv- i.e., whether, given two states ϱˆ and σˆ, either ϱˆ can be Φ Φ ϱˆ σˆ Ping quantum operations ICPTP, which act as ICPTPð Þ¼ transformed into or vice versa. We have to leave this ˆ ˆ † ˆ nKnϱˆKn, where the Kraus operators Kn are all of the as an open question, but report small progress in the d d Kˆ IKˆ † ⊂ I same dimension out × in and satisfy n n .