How to quantify coherence: Distinguishing speakable and unspeakable notions
Abstract
Quantum coherence is a critical resource for many operational tasks. Understanding how to quantify and manipulate it also promises to have applications for a diverse set of problems in theoretical physics. For certain applications, however, one requires coherence between the eigenspaces of specific physical observables, such as energy, angular momentum, or photon number, and it makes a difference which eigenspaces appear in the superposition. For others, there is a preferred set of subspaces relative to which coherence is deemed a resource, but it is irrelevant which of the subspaces appear in the superposition. We term these two types of coherence unspeakable and speakable respectively. We argue that a useful approach to quantifying and characterizing unspeakable coherence is provided by the resource theory of asymmetry when the symmetry group is a group of translations, and we translate a number of prior results on asymmetry into the language of coherence. We also highlight some of the applications of this approach, for instance, in the context of quantum metrology, quantum speed limits, quantum thermodynamics, and NMR. The question of how best to treat speakable coherence as a resource is also considered. We review a popular approach in terms of operations that preserve the set of incoherent states, propose an alternative approach in terms of operations that are covariant under dephasing, and we outline the challenge of providing a physical justification for either approach. Finally, we note some mathematical connections that hold among the different approaches to quantifying coherence.
Contents

I Introduction and summary
 I.1 ResourceTheoretic approach to unspeakable coherence
 I.2 ResourceTheoretic approaches to speakable coherence
 I.3 Criticism of resourcetheoretic approaches to speakable coherence
 I.4 A proposal for the resource theory of speakable coherence
 I.5 Relation between different approaches
 I.6 The choice of preferred subspaces
 I.7 Composite systems
 I.8 Outline
 II Preliminaries

III Coherence via translationallycovariant operations

III.1 Free operations as translationallycovariant operations
 III.1.1 Definition of translationallycovariant operations
 III.1.2 Translationallycovariant measurements
 III.1.3 Translationallycovariant unitary operations
 III.1.4 Characterization via Stinespring dilation
 III.1.5 Characterization via modes of translational asymmetry
 III.1.6 Characterization via Kraus decomposition
 III.2 Physical justifications for the restriction to translationallycovariant operations
 III.3 Covariance with respect to independent translations

III.1 Free operations as translationallycovariant operations
 IV Coherence via dephasingcovariant operations
 V Coherence via incoherencepreserving and incoherent operations
 VI Methods for deriving measures of coherence
 VII Concluding Remarks
 VIII Acknowledgements
I Introduction and summary
Many properties of quantum states can be better understood by considering them as constituting a resource Coecke et al. (2014). The properties of entanglement Horodecki et al. (2009); V. Vedral and M. Plenio (1998), asymmetry Gour and Spekkens (2008); Gour et al. (2009); Marvian and Spekkens (2013, 2014a, 2014b); Marvian (2012); Marvian and Spekkens (2014c); Gour (2005); Skotiniotis and Gour (2012); Narasimhachar and Gour (2014), and athermality Brandão et al. (2015); Brandao et al. (2013); Gour et al. (2015); Skrzypczyk et al. (2014); Horodecki and Oppenheim (2013); Åberg (2013) are good examples. We are here concerned with the property of having coherence relative to some decomposition of the Hilbert space. This property appears to be necessary for certain types of tasks, and as such it is natural to attempt to understand coherence from the resourcetheoretic perspective. This has led to some proposals for how to define a resource theory of coherence and in particular how to quantify coherence Baumgratz et al. (2014); Marvian and Spekkens (2014c); Marvian et al. (2015); Yadin et al. (2015); Yadin and Vedral (2015).
The following is a list of operational tasks for which quantum coherence seems to be a resource.

Quantum metrology. An example is the task of estimating the phase shift on a field mode, used in quantum accelerometers and gravitometers. Here one requires the ability to prepare and measure coherent superpositions of different occupation numbers of the mode. Another example is estimating the rotation of a quantum gyroscope about some axis, where one must be able to prepare and measure coherent superpositions of eigenstates of the angular momentum operator along that axis, a problem that is relevant for developing high precision measurements of magnetic field strength Ajoy and Cappellaro (2012). A third example is building high precision clocks, where one must be able to prepare and measure coherent superpositions of energy eigenstates V. Giovannetti, S. Lloyd, and L. Maccone (2006); Giovannetti et al. (2001, 2011); Schnabel et al. (2010).

Reference frame alignment. Examples include aligning distant gyroscopes, synchronizing distant clocks, and phaselocking distant phase references. Each example requires communicating quantum states that carry the appropriate sort of information (orientation, time, and phase, for instance) and therefore, like the metrology examples, requires coherence relative to the appropriate eigenspaces Bartlett et al. (2007).

Thermodynamic tasks. An example is the task of extracting as much work as possible from a given quantum state given a bath at some fixed background temperature. This require states that are not in thermal equilibrium at the background temperature. Resources here include not only those states having a nonthermal distribution of energy eigenstates, but also those that have coherence between the energy eigenspaces Lostaglio et al. (2015a, b); Janzing et al. (2000); Ćwikliński et al. (2015).

Computational, cryptographic and communication tasks. For these sorts of tasks, it is well known that having access only to preparations and measurements that are all diagonal in some basis, hence incoherent, is not sufficient for achieving any quantum advantage. So it is natural to seek to study such coherence as a resource.
For many resource theories, there are also applications to problems in theoretical physics. For example, while entanglement theory was originally developed through its role as a resource in operational tasks such as quantum teleportation Bennett et al. (1993) and dense coding Bennett and Wiesner (1992), the possibility of quantifying entanglement has since found applications in diverse problems, including the study of phase transitions, characterizing the ground states of manybody systems Latorre et al. (2003); Kitaev and Preskill (2006); Vidal et al. (2003); Marvian (2013), holography in quantum field theories Ryu and Takayanagi (2006), and the black hole informationloss paradox Almheiri et al. (2013); Verlinde and Verlinde (2012); Hayden and Preskill (2007). Similarly, the possibility of quantifying coherence is expected to shed light on various problems in theoretical physics. The following are a few examples.

Quantum speed limits. The MandelstamTamm bound Mandelstam and Tamm (1945) and the MargolusLevitin bound N. Margolus and L.B. Levitin (1998) are upper bounds on the minimum time it takes for a system in some state to evolve to a (partially) distinguishable state. This time is clearly related to the amount of coherence between energy eigenstates and therefore quantifying this coherence can shed light on quantum speed limits Marvian et al. (2015); Mondal et al. (2016).

Magnetic resonance techniques. For such techniques, in particular in NMR, if the system one is probing consists of many spins, then the large dimension of the Hilbert space together with constraints on the measurements are such that full tomography is not possible. Still, one can obtain much useful information about the state by measuring the degree of coherence relative to the quantization axis Cappellaro (2014). If the system is quantized along the axis, then coherence of order of the state is defined as the norm of the sum of the offdiagonal terms with , where is the eigenstate with eigenvalue of , the magnetic moment in the direction Cappellaro (2014)^{1}^{1}1This concept is closely related to the idea of decomposition into modes of asymmetry, introduced in Refs. Marvian (2012); Marvian and Spekkens (2014b) (See Sec.III.1.5 for a short review). In particular, the thquantum coherence component in the language of Cappellaro (2014) is the same as the mode component in the language of Marvian (2012); Marvian and Spekkens (2014b). Furthermore, the Frobenius norm of the thquantum coherence component, which can be measured in NMR experiments, provides lower and upper bounds on a measure of coherence studied in Marvian (2012); Marvian and Spekkens (2014b), which quantifies the asymmetry of a state in mode (See Eq.(81)).. Measuring the quantum coherence of different orders is relatively straightforward and has been useful in many NMR experiments, in particular, in the context of quantum information processing, as well as in simulations of manybody dynamics (See e.g. Cappellaro et al. (2007); Cho et al. (2006); Cappellaro (2014)).

Coherence lengths. The spatial extent over which a quantum state is coherent is an important concept in manybody physics Frérot and Roscilde (2015), for instance, in the onset of BoseEinstein condensation Ketterle and Miesner (1997), and in quantum biology, for instance, in excitation transport in photosynthetic complexes Levi et al. (2015); Mohseni et al. (2008); Rebentrost et al. (2009).

Order parameters. Quantum phase transitions in the ground states of quantum manybody systems, such as a spin chain, can be studied in terms of the degree of coherence contained in local reductions of the state, such as singlespin or twospin, density operators Karpat et al. (2014); Malvezzi et al. (2016).

Decoherence theory. It is well known that interaction of a system with its environment can lead to the loss of coherence relative to preferred subspaces that depend on the nature of the interaction Joos et al. (2013). For instance, if an environment couples to the spatial degree of freedom of a system, then it will reduce the spatial extent over which the system exhibits coherence Joos et al. (2013). Such decoherence plays a significant role in many accounts of the emergence of classicality. Measures of coherence, therefore, can be used as a tool for studying such emergence.
It is critical, however, to distinguish two types of coherence that arise in these various applications. The distinction can be explained as follows. Consider the states
(1) 
If we are interested in quantum computation using qutrits and the elements of the set are the computational basis states, then we would expect the states and to be equivalent resources because the particular identities of the computational basis states appearing in the superposition are not relevant for any computational task. In this case, is simply an arbitrary label or flag for different distinguishable pure states. If, on the other hand, we are considering a phase estimation task and the elements of the set are eigenstates of the number operator, then there is a significant difference between the states and . For instance, can detect a phase shift of while cannot. Conversely, if one’s task is to estimate a very small phase shift, then is a better resource than , because the former becomes more orthogonal to itself than the latter under a small phase shift. Similarly, starting from the incoherent state , to prepare a state close to one needs to have access to a phase reference with higher precision than the phase reference required to prepare a state close to Marvian and Spekkens (2014b). Here, is not an arbitrary label, but an eigenvalue of the number operator, and its value is relevant for the task of phase estimation.
These two types of coherence pertain to two types of information which have been termed speakable and unspeakable Bartlett et al. (2007); Peres and Scudo (2002). Speakable information is information for which the means of encoding is irrelevant. This is exemplified by the fact that if one seeks to transmit a bitstring, it is irrelevant what degree of freedom one uses to encode the bits. Unspeakable information is information which can only be encoded in certain degrees of freedom. Information about orientation, for instance, is unspeakable because it can only be transmitted using a system that transforms nontrivially under rotations. Information about time is also unspeakable because it can only be transmitted by a system that transforms nontrivially under timetranslations. We shall therefore refer to the two types of coherence we have outlined above as speakable and unspeakable respectively.
For the list of operational tasks we have provided above, the relevant notion of coherence is the unspeakable one in all cases except for the last item. This is also the case for most of the physical applications listed above. This is because from the point of view of speakable coherence, the eigenvalues of the observable that defines the preferred subspaces are not relevant: the set of preferred subspaces is a set without any order. However, for the examples of quantum speed limits, coherence lengths and magnetic resonance, for instance, it is clear that the eigenvalues of the relevant observable, the Hamiltonian, position, and magnetic moment observables respectively, has important physical meaning, and there is a natural order defined on the preferred subspaces. Therefore, the notion of unspeakable coherence seems to be the more appropriate one in these cases.
Most recent work on coherence as a resource, however, considers only speakable coherence. One might think, therefore, that there is work to be done in defining a resource theory of unspeakable coherence. In fact, however, such a resource theory already exists. It simply goes by another name: the resource theory of asymmetry.^{2}^{2}2In early work on the topic, which emphasized the role of asymmetric quantum states in defining quantum reference frames, the resource theory of asymmetry was termed the resource theory of frameness Gour and Spekkens (2008). To be precise, the resource of unspeakable coherence is nothing more than the resource of asymmetry relative to a group of translations.
i.1 ResourceTheoretic approach to unspeakable coherence
Consider the task of phase estimation as an example. A state of some field mode has coherence relative to the eigenspaces of the number operator if and only if it is asymmetric (i.e., symmetrybreaking) relative to the group of phaseshifts generated by , where such translations of the phase are represented by the group of unitaries . This follows from the fact that if a state is symmetric under phaseshifts, that is, , then it must be blockdiagonal with respect to the eigenspaces of , while if it is not symmetric under phaseshifts, then it cannot have this form.
Other examples are treated in a similar fashion^{3}^{3}3Throughout this article, we use units where .: coherence relative to the eigenspaces of a Hamiltonian is simply asymmetry relative to the group of timetranslations generated by this Hamiltonian, ; coherence relative to the eigenspaces of the momentum operator is simply asymmetry relative to the group of spatial translations, ; coherence relative to the eigenspaces of the angular momentum operator is simply asymmetry relative to the group of rotations around , .
Any resource theory must not only partition the states into those that are resources and those that can be freely prepared at no cost, it must also partition the operations into those that are resources and those that can be freely implemented at no cost. The free set of operations is required to be closed under composition and convex combination Coecke et al. (2014). In entanglement theory, for instance, not only are the states partitioned into those that are unentangled, hence free, and those that are entangled, hence resources, but the operations are also partitioned into those that can be achieved by Local Operations and Classical Communications (LOCC), which are deemed to be free, and those which cannot, which are deemed to be resources.
If one considers each of the tasks for which unspeakable coherence is a resource, one sees that the freelyimplementable operations are those that are covariant under translations, that is, for which first translating and then implementing the operation is equivalent to first implementing the operation and then translating (See Def. 2). For instance, in the task of reference frame alignment, the setup of the problem is that there are two parties, each of which has a local reference frame (e.g. a gyroscope, a clock, a phase reference), but the group element that relates these two frames (e.g. the rotation, the time translation, the phase shift) is unknown. It is not difficult to show that the operations that one party can implement relative to the other party’s reference frame are precisely those that are covariant under the group action Bartlett et al. (2007); Marvian and Spekkens (2013, 2014a); Marvian (2012). In fact, there are many ways of providing a physical justification of the translationallycovariant operations, and we shall review these at length further on.
These considerations imply that the problem of quantifying and classifying unspeakable coherence can be considered a special case of the resource theory of asymmetry where the group under consideration describes a translational symmetry. (The resource theory of asymmetry is more general than this, however, because it is also capable of dealing with nonAbelian groups where asymmetry does not simply correspond to the existence of coherence between some preferred set of subspaces.)
The notion that the resource of coherence should be understood as asymmetry relative to the action of a translational symmetry and that the free operations defining the resource theory are the translationallycovariant ones was first proposed in Ref. Marvian and Spekkens (2014c) and developed in Appendix A of Ref. Marvian and Spekkens (2014b) and in Ref. Marvian et al. (2015) (See also Yadin and Vedral (2015)).
This connection implies that most questions about unspeakable coherence as a resource find their answers in prior work on the resource theory of asymmetry. It suffices to specialize known results to the particular translational symmetry of interest. One of the goals of this article is to explicitly translate some of these known results from the language of asymmetry to the language of coherence, to describe the measures of coherence that result, and to review some of the applications.
This approach to coherence has already been applied to shed light on the various applications of unspeakable coherence outlined above: quantum metrology Marvian and Spekkens (2014b, c), aligning reference frames Bartlett et al. (2007); Marvian and Spekkens (2013), quantum thermodynamics Lostaglio et al. (2015a); Yang and Chiribella (2015); Å berg (2014); Janzing et al. (2000); Ćwikliński et al. (2015), quantum speed limits Marvian et al. (2015); Mondal et al. (2016). Furthermore, it was shown in Ref. Marvian and Spekkens (2014c) that for symmetic opensystem dynamics, measures of asymmetry are monotonically nonincreasing, thereby yielding a significant generalization of Noether’s theorem. Translated into the language of coherence, this result states that for open system dynamics that is translationallycovariant, every measure of coherence which is derived within the translationalcovariance approach to coherence provides a monotone of the dynamics. Such measures, therefore, provide a powerful new tool for studying decoherence.
i.2 ResourceTheoretic approaches to speakable coherence
The second topic we address in this work is whether and how one can develop a resource theory of speakable coherence.
When one considers recent work on quantifying coherence from the perspective of the speakable/unspeakable distinction, it is clear that it concerns itself only with the speakable notion (See e.g. Baumgratz et al. (2014); Winter and Yang (2015); Streltsov (2015); Girolami and Yadin (2015); Ma et al. (2015); Yadin et al. (2015); Streltsov et al. (2015); Napoli et al. (2016); Piani et al. (2016)). Most of this work builds on a proposal by Baumgratz, Cramer and Plenio (BCP) Baumgratz et al. (2014). The set of free operations in the BCP approach, called incoherent operations, is defined based on the Kraus decomposition of quantum operations, and is closely related to another set which is called incoherenceperserving. These are operations which take every incoherent state to an incoherent state.
Because it concerns speakable coherence, this approach is only appropriate for tasks concerning speakable information. Nonetheless, if one is content to accept that a resource theory of speakable coherence has a more limited scope of applications than one might have naïvely expected, the question arises of whether the BCP approach is the right way to define the resource theory of speakable coherence.
i.3 Criticism of resourcetheoretic approaches to speakable coherence
As we noted earlier, to take a resourcetheoretic approach to any given property one must first of all make a proposal for which set of operations can be freely implemented. But a given proposal for how to do so is only expected to have physical relevance if it can be provided with a physical justification, that is, if one can provide a restriction on experimental capabilities that yields all and only the operations in the free set that is proposed. For instance, in entanglement theory, the restriction on experimental capabilities that yields all and only the LOCC operations between two parties is the absence of any quantum channel between the two parties.
Despite the amount of attention that the BCP approach has received, no one has yet described a physical justification for the set of incoherent operations or the set of incoherencepreserving operations. The property of taking incoherent states to incoherent states is certainly a mathematically welldefined constraint; whether there is an experimental constraint that corresponds to this property is the question of interest here. Of course one can imagine physical scenarios in which preparing coherent states is hard, for instance, because of the challenge of isolating one’s systems from environmental decoherence. But this does not justify the claim that the set of incoherent operations or the set of incoherencepreserving operations is the natural one to study; to do so one needs to argue that all of the operations in a given set can be easily implemented in that physical scenario. However, it is not clear whether such a justification can be found.
For one, because the free states in the resource theory of coherence should be restricted to those that have no coherence between the preferred subspaces, one would expect that the free measurements in a resource theory of coherence should be similarly restricted. However, we show that the BCP proposal places no constraint on the sorts of measurements that can be implemented, in the following sense: for any POVM, it is possible to find a measurement that realizes it which is considered free in the BCP approach. Therefore, to justify the BCP approach to the resource theory of speakable coherence, one needs to argue that there are physical or experimental constraints which lead to a significant restriction on statepreparations and transformations, but no restriction on the possibilities for discriminating states.
For another, it turns out that even if one finds physical scenarios in which the set of free unitaries is the set of incoherent unitaries (as defined by BCP), this still does not justify the set of incoherent or incoherencepreserving operations as the set of free operations for a resource theory of speakable coherence. As we will show, a general incoherencepreserving (incoherent) operation cannot be implemented using only incoherent states, incoherent unitaries and incoherent measurements. Using a more technical language, this means that incoherencepreserving (incoherent) operations do not admit dilation using only incoherent resources (at least, not in a straightforward way, when we treat all systems evenhandedly).
This lack of dilation for the set of free operations is not necessarily a problem in its own right ^{4}^{4}4In entanglement theory, the free operations are the LOCC operations, which do not all admit of a dilation in terms of the free unitaries (because the only unitaries in the LOCC set are tensor products of local unitaries, which do not support any sort of communication, classical or quantum, between the two parties), and yet this is not considered a problem with entanglement theory because there is a natural physical restriction that yields LOCC as the set of free operations, namely, the fact that classical channels between parties are technologically easy to implement, while quantum channels are not. . One can imagine physical scenarios where the set of operations that we can implement in a controlled fashion using the free unitaries and free states, are smaller than the set of all free operations we can implement on the systems of interest, because the latter operations might result from uncontrolled interaction of these systems with an environment over which the experimenter has limited control, such as a thermal bath. The lack of a dilation that is evenhanded in its treatment of systems implies that this is the only sort of avenue open for providing a physical justification of the incoherencepreserving or incoherent operations. Furthermore, we show that there are other natural proposals for the set of free operations for the resource theory of speakable coherence that share precisely the same set of free unitaries, namely, the incoherent unitaries. Therefore, even if one accepts that the set of incoherent unitaries is the appropriate set of free unitaries for a resource theory of speakable coherence, this does not resolve the question of which of the many sets of free operations consistent with this choice one should use to define the resource theory.
Our concerns about the suitability of the incoherencepreserving operations in a resource theory of coherence are bolstered by comparing them to the nonentangling operations in entanglement theory^{5}^{5}5One can also define a set of operations that plays a role in entanglement theory which is parallel to the role played by incoherent operations in the resource theory of coherence, namely, those bipartite operations for which there is a Kraus decomposition where each term is nonentangling.. The nonentangling operations are those that map unentangled states to unentangled states Brandao and Plenio (2008). Like the incoherencepreserving operations, therefore, they are the largest set that maps the free states to the free states. The nonentangling operations are a strictly larger set than the LOCC operations because they include nonlocal operations such as swapping systems between the two parties, and because they allow the implementation of arbitrary POVM measurements on the bipartite system. It is difficult to imagine any restriction on experimental capabilities that yields all and only the nonentangling operations. Indeed, it is widely acknowledged that the LOCC operations—which do arise from a natural restriction, having classical channels but not quantum channels—is the physically interesting set, while the nonentangling operations are studied primarily as a mathematical technique for making inferences about LOCC. Incoherencepreserving and incoherent operations may ultimately have a similarly subservient role to play in the resource theory of coherence.
i.4 A proposal for the resource theory of speakable coherence
In the absence of a physical justification for the BCP approach, the question arises of whether an alternative choice of the set of free operations might be more suited to a resourcetheoretic treatment of speakable coherence. Once the question is raised, a natural alternative for the set of free operations immediately suggests itself, namely, those that are covariant under dephasing, that is, those that commute with the operation that achieves complete dephasing relative to the preferred subspaces. We call this the dephasingcovariance approach to coherence. A variant of this proposal has recently been considered in Ref. Yadin and Vedral (2014)^{6}^{6}6Ref. Yadin and Vedral (2014) came to our attention as we were preparing this manuscript..
This proposal does not have one of the counterintuitive features of the BCP approach that was outlined above: the set of free measurements includes only the POVMs whose elements are incoherent, as one would expect. Nonetheless, it is still not clear whether the dephasingcovariance approach to coherence has much physical relevance because it is still unclear whether there is any restriction on experimental capabilities that picks out all and only the dephasingcovariant operations. (In particular, it is unclear whether every dephasingcovariant operation admits of a dilation in terms of incoherent states, incoherent measurements and dephasingcovariant unitaries.) We do not settle the issue here.
i.5 Relation between different approaches
In addition to providing a characterization and assessment of both the dephasingcovariance approach and the BCP approach for treating speakable coherence as a resource, we explore the mathematical relation between the free set of operations that each adopt. In particular, we show that the dephasingcovariant operations relative to a choice of preferred subspaces are a strict subset of the incoherent (incoherencepreserving) operations relative to the same choice. This implies that any measure of coherence in the incoherent (incoherencepreserving) approach is also a measure of coherence in the dephasingcovariance approach.
We also compare the translationalcovariance approach to coherence with the dephasingcovariance approach (and, via the connection noted above, with the incoherent and incoherenceperserving approaches).
Any given translational symmetry defines a decomposition of the Hilbert space via the joint eigenspaces of the generators of this symmetry. Thus, for any given translational symmetry, one can consider the sets of dephasingcovariant, incoherent, and incoherencepreserving operations defined relative to this decomposition of the Hilbert space. For instance, the incoherencepreserving operations defined by a given translational symmetry is the set of operations under which any state which is incoherent with respect to the eigenspaces of its generators, is mapped to a state which is still incoherent relative to these eigenspaces.
We show that the set of translationallycovariant operations is a strict subset of the dephasingcovariant operations and thus also a strict subset of the incoherent (incoherencepreserving) operations. This implies that any measure of coherence in the dephasingcovariance proposal is also a measure of coherence in the translationalcovariance proposal. We also show that this inclusion relation is strict.
Given these inclusion relations, the question arises of whether the measures of coherence that have been identified recently as valid in the BCP approach were already identified in prior work on the resource theory of asymmetry. We show that this is indeed the case for most such measures of coherence.^{7}^{7}7Note that the present work expands on some of the comparisons between the BCP approach to coherence and the one based on translational asymmetry made in Ref. Marvian et al. (2015). In addition to noting these relations, we discuss two general techniques for deriving measures of coherence, one that infers them from measures of information and the other that appeals to a certain kind of decomposition of operators into socalled modes of asymmetry.
i.6 The choice of preferred subspaces
The notion of the state of a system being coherent is only meaningful relative to a choice of decomposition of the Hilbert space of the system into subspaces. The latter must be dictated by physical considerations, which is to say, operational criteria. This is because, from a purely mathematical point of view, any state is coherent in some basis and incoherent in another basis. If , then the state is coherent if one judges relative to the basis, but by the same token, is deemed coherent if one judges relative to the basis. One consequently has no alternative but to appeal to physical considerations in defining the notion of coherence^{8}^{8}8The fact that the directsum decomposition of Hilbert space that defines the resource of coherence is determined by the physical problem of interest is completely analogous to how the factorization of the Hilbert space of a composite system that defines the resource of entanglement is so determined..
Furthermore, physical considerations often dictate that the relevant notion of coherence is relative to a decomposition of the Hilbert space into subspaces that are not 1dimensional. A few examples serve to illustrate this.
In the context of decoherence theory, environmental decoherence does not always pick out 1dimensional subspaces of the system Hilbert space. The dimensions of the decohering subspaces depend on which degree of freedom of the system couples to the environment and generally this is a degenerate observable. Indeed, this fact, i.e., the existence of Decoherence Free Subspaces with dimension larger than one, has been exploited to protect quantum information against decoherence Lidar and Whaley (2003); Zanardi and Rasetti (1997).
Another common example is where the notion of coherence that is of interest is coherence relative to the eigenspaces of some particular physical observable, such as the system’s Hamiltonian (as happens when the coherence is a resource for building a quantum clock) or the photon number operator in a particular mode (as happens when the coherence is a resource for phase estimation). Even if the notion of coherence of interest is the speakable one, some physical degree of freedom must be used to encode the coherence, and practical considerations might dictate the use of a particular physical observable. And in all such cases, there is a priori no reason that the physical observable should be nondegenerate.
A final example is if there are degrees of freedom over which the experimenter has no control. In this case, coherence in that degree of freedom is neither observable nor usable.
Thus, if the physically relevant observable is , and labels its eigenspaces while is a degeneracy index, the state is an incoherent state in the resource theory insofar as it has no coherence between the eigenspaces of ^{9}^{9}9This is analogous to how entanglement between laboratories can be a resource while entanglement within a given laboratory is not.. Coherence within an eigenspace of might be made to be a resource as well, but this requires the degeneracy to be broken, for instance, by introducing another physical observable which picks out a basis of that eigenspace.
We conclude that any resource theory of coherence should be able to quantify and characterize coherence, not only with respect to 1dimensional subspaces, but also with respect to subspaces of arbitrary dimension. As we will see in the following, the resource theory of unspeakable coherence based on translationallycovariant operations has this capability. In the case of speakable coherence, we define dephasingcovariant and incoherencepreserving operations to incorporate this possibility, and we generalize the definition of incoherent operations in BCP Baumgratz et al. (2014), which assumed 1dimensional subspaces, to do so as well.
i.7 Composite systems
How should the resource theory of coherence be defined on composite systems? In particular, how should we define the set of free states and free operations in this case? For instance, suppose we are interested in quantifying coherence with respect to the energy eigenbasis, which is relevant, for instance, in the context of thermodynamics and clock synchronization. Consider two noninteracting systems having identical Hamiltonians, and , with energy eigenbases and respectively. One can then imagine two different ways of defining coherence on the composite system . Definition (1): coherence is defined relative to the products of eigenspaces of the single system Hamiltonians. In this case, a joint state of systems and is coherent if it contains coherence with respect to either of these two Hamiltonians. In particular, in this approach, the state is considered a resource. Definition (2): coherence is defined relative to the eigenspaces of the total Hamiltonian, that is, of , where and are, respectively, the identity operators on systems and . In this case, states which do not contain coherence relative to the total Hamiltonian, such as , are deemed to be incoherent.
As the set of preferred subspaces and incoherent states on a single system should be chosen based on physical considerations, the set of incoherent states on composite systems should also be defined in a similar fashion. It turns out that each of the above definitions can be relevant in some physical scenarios. For instance, in the scenarios where the two subsystems cannot exchange energy (for instance, because they are held by two distant parties) then approach (1) is relevant, and entangled states such as are resources. On the other hand, in the scenarios where we can easily apply operations that allow energy exchange between the two subsystems, then the relevant observable is the total energy, and not the energy of the individual subsystems. Therefore, in this situation approach (2) is the relevant one, and entangled states such as are not resources. The fact that the resource of coherence is only defined relative to a choice of basis which depends on the physical scenario is precisely analogous to how the resource of entanglement is only defined relative to a choice of factorization of the Hilbert space which depends on the physical scenario. (For instance, in the distant laboratories paradigm, entanglement between laboratories is a resource, while entanglement between systems in the same laboratory is not.)
i.8 Outline
The article is organized as follows. Sec. II covers preliminary material, including a discussion of certain features that are common to the various different proposals for a resource theory of coherence, what counts as a physical justification of a proposal for the set of free operations, and the definition of a measure of coherence. Sec. III presents the resource theory of unspeakable coherence that one obtains by taking the free operations to be those that are translationally covariant. In particular, various different characterizations and physical justifications of the free operations are provided. Sec. IV presents the proposal for speakable coherence based on dephasingcovariant operations, together with a discussion of the relation to the translationallycovariant operations and physical justifications. In Sec. V, we review the BCP proposal for speakable coherence, which is defined by the incoherent operations, as well as a related proposal, defined by the set of incoherencepreserving operations. The relation to the dephasingcovariance appraoch is considered, as well as possibilities for a physical justification. Finally, in Sec. VI, we consider measures of coherence within the various approaches, and in Sec. VII we provide some concluding remarks.
Ii Preliminaries
Any resource theory is specified by a set of free states and a set of free operations. These are states and operations which are easy or allowed to prepare and implement under a practical or fundamental constraint.
ii.1 Free states
The notion of coherence is only defined relative to a preferred decomposition of the Hilbert space into subspaces. This preferred decomposition is determined based on practical restrictions or physical considerations, although in some cases a preferred decomposition may be considered as a purely mathematical exercise. For a system with Hilbert space , we denote the preferred subspaces by , so that . Here, the index may be discrete or continuous. We denote the projectors onto these subspaces by .
The free states, which are termed incoherent states, are those states which are blockdiagonal relative to the preferred subspaces,
(2) 
An alternative way of characterizing the set of free states is via map that dephases between the preferred subspaces. This dephasing map has the form
(3) 
As a superoperator acting on the vector space of operators, is a projector, and hence idempotent, . In fact, it projects onto the subspace of operators that are blockdiagonal relative to the decomposition , so that the set of incoherent states can be characterized as those that are invariant under ,
(4) 
Note that for any choice of preferred subspaces, the set of incoherent states is closed under convex combinations. We will denote this set by .
ii.2 Free measurements
If a system survives a quantum measurement, then the outcome of the measurement provides the ability to predict the outcomes of future measurements on the system. To do so, one must specify the state update map associated to the measurement. The von Neumann projection postulate is an example. This is a specification of the measurement’s predictive aspect. Whether the system survives or not, every quantum measurement also allows one to make retrodictions about earlier interventions of the system. We here focus only on the retrodictive aspect of a measurement, which in quantum theory is represented by a postiveoperatorvalued measure (POVM). An element of a POVM, denoted , satisfies , and wlll be termed an effect.
We will call an effect incoherent if it is blockdiagonal relative to the preferred subspaces. Although one might have expected all proposals for the resource theory of coherence to allow as free only those POVMs made up of incoherent effects, we will see that this is not the case for the proposal based on incoherencepreserving or incoherent operations.
ii.3 Free operations
We turn now to the set of free operations. We consider not only those operations wherein the input and output spaces are the same (i.e., transformations of a system) but also those where they may be different (in which case the operation involves adding or taking away some or all of the system). In particular, the free operations from a trivial input space to a nontrivial output space specify which preparations of the output system can be freely implemented, so that a specification of the free operations implies a specification of the free states.
The minimal property that the set of free operations should have is to be incoherencepreserving, which is to say that each free operation takes every incoherent state on the input space to an incoherent state on the output space. Note that the incoherencepreserving property implies that for the special case where the operation is a state preparation, the free set corresponds to the incoherent states. ^{10}^{10}10This is because we can think of the input in a state preparation as a 1dimensional system, which is necessarily in an incoherent state.
All proposals we consider here are such that every free operation is incoherencepreserving. Nonetheless, different proposals for how to treat coherence as a resource differ in their choice of the set of free operations, subject to this constraint.
Note that we here use the term quantum operation to refer to a tracenonincreasing completely positive linear map. If the operation is tracepreserving, we will refer to it as a quantum channel.
ii.4 Physical justification of the free operations through dilation
It is widely believed that physical systems undergoing closedsystem dynamics evolve according to a unitary map. In this view, the only circumstance in which a nonunitary map is used to describe the evolution of a system’s state is when the system is known to undergo opensystem dynamics, that is, when it interacts with some auxiliary system (perhaps its environment) via a unitary map, but one chooses to not describe the auxiliary system, by marginalizing or postselecting on it. It is straigthforward to show that in any situation wherein a system interacts unitarily with the auxiliary system and one subsequently marginalizes or postselects on the latter (through a partial trace or a partial trace with some measurement effect respectively), the effective evolution of the system’s state is always described by a completely positive tracenonincreasing map (what we are here calling a quantum operation). The Stinespring dilation theorem Nielsen and Chuang (2000) guarantees that the viceversa is also true: every quantum operation on the system can be achieved in this fashion.
For a given triple of state of the auxiliary system, effect measured thereon, and unitary coupling of the system to its auxiliary, we will term the effective quantum operation on the system that it defines the marginal operation on the system. For a given quantum operation on the system, we will term any triple of state of the auxiliary, effect on the auxiliary and unitary coupling of system to auxiliary that yields the operation as a marginal, a dilation of that operation.
In the context of resource theories, one way to define the free set of operations on a system is by specifying the free states and free effects on the auxiliary system, as well as the free unitaries that couple the system to the auxiliary, and then defining the free set of operations on the system as all and only those that can be obtained as a marginal of these. If a proposal for the free set of operations is not defined in this way, then one can and should ask whether it admits of such a definition or not. In other words, one should seek to determine whether the free set of operations in a given proposal can be understood as those that admit of a dilation in terms of the free states and effects on, and free unitary couplings with, the auxilary system. We refer to such dilations as free dilations.
We shall here ask this question of various proposals for how to choose the free set of operations in a resource theory of coherence. If, for any given proposal, one finds that the free set of operations on the system of interest includes an operation that does not admit of a free dilation, then this may imply that some nontrivial resource on the composite of system and auxiliary must be consumed in order to realize the operation.
We will show that in cases where one considers a translation group that acts collectively on all physical systems, the translationallycovariant operations have free dilation. We will also show that the set of incoherencepreserving operations and the set of incoherent operations does not have this property, at least not if we treat all systems evenhandedly. For the set of dephasing covariant operations, the question remains open.
ii.5 Measures of coherence
In any resource theory, a measure of the resource is a function from states to real numbers which defines a partial order on the set of states. The essential property that any such function must have is to be a monotone (i.e, to be monotonically nonincreasing) under the free operations^{11}^{11}11See Ref. Gour and Spekkens (2008) for an operational justification of monotonicity, that is, an account of why monotonicity is required if a measure of a resource is to characterize the degree of success achievable in some operational task.. We are therefore going to use the following definition of a measure of coherence:
Definition 1
A function from states to real numbers is a measure of coherence according to a given proposal if
(i) For any tracepreserving quantum operation which is free according to the proposal,
it holds that .
(ii) For any incoherent state , it holds that .
Because any incoherent state can be mapped to any other incoherent state via a free tracepreserving operation, condition (i) implies that the value of the function must be the same for all incoherent states. Condition (ii) merely expresses a choice of convention for this value: that all incoherent states should be assigned measure zero. Of course, given any function satisfying condition (i), one can define a shift of this function which satisfies condition (ii).
It is worth noting that any measure of a resource is constant on states that are connected by a free unitary operation. That is, if is a free unitary operation, then any resource measure must satisfy
The proof is simply that if and are connected by a free unitary, then state conversion in both directions are possible under the free operations, and , which in turn implies that and .
We distinguish the three resourcetheoretic approaches to coherence that we consider in this article by the set of free operations that define them: translationallycovariant, dephasingcovariant and incoherencepreserving operations. A measure of coherence within a given approach is also defined relative to the set of free operations within that approach. Therefore, we refer to measures of coherence within the different approaches as measures of TCcoherence, DCcoherence, and IPcoherence respectively. In Sec. VI, we provide a list of examples for each type.
Iii Coherence via translationallycovariant operations
We begin by demonstrating that if one is interested in an unspeakable notion of coherence, then coherence can be understood as asymmetry relative to a symmetry group of translations. In this approach, the coherence is defined based on a given observable , such as the Hamiltonian, the linear momentum, or the angular momentum. Then, to characterize coherence relative to the eigenbasis of , we consider the asymmetry relative to the group of translations generated by , i.e., the group of unitaries
(5) 
The superoperator representation of the translation is then
(6)  
Note that this group has often a natural physical interpretation. For instance, if is the Hamiltonian, then it generates the group of time translations, and if is the component of angular momentum in some direction, then it generates the group of rotations about this direction.^{12}^{12}12Note that, depending on the spectrum of , this group could be isomorphic to U(1) or to (under addition).
The free states are taken to be those that are translationallyinvariant or translationallysymmetric,
(7) 
One can easily see that the set of translationallyinvariant states coincides with the set of states that are incoherent with respect to the eigenspaces of , i.e.,
(8) 
Therefore, in the translationalcovariance approach to coherence the preferred subspaces relative to which coherence is a resource are the eigenspaces of the generator .
iii.1 Free operations as translationallycovariant operations
For a given choice of symmetry transformations, the resource theory of asymmetry is defined by taking the set of free operations to be those that are covariant relative to the symmetry transformations. We here particularize this definition to the case of a translational symmetry, and provide several ways of characterizing this set.
iii.1.1 Definition of translationallycovariant operations
Definition 2
We say that a quantum operation is translationallycovariant relative to the translational symmetry generated by if
(9) 
Note that condition (9) is equivalent to
(10) 
If the input and output spaces of the map are distinct, then the generator may be different on the input and output spaces. For instance, in the case where corresponds to the angular momentum in a certain direction, then this observable may have different representations on the input and output spaces. For simplicity, we do not indicate such differences in our notation. A preparation of the state is an operation with a trivial input space and translationalcovariance in this case implies translationalinvariance of , confirming that translationallyinvariant states are the free states in this approach.
Sec. II.3 articulated a minimal constraint on the free operations, that they should be incoherencepreserving. Translationallycovariant operations have this property because from Eq. (7) one can deduce that for a translationallycovariant operation , and an incoherent state for input,
(11) 
which implies that is translationallyinvariant, hence incoherent. Therefore incoherent states are mapped to incoherent states.
If one thinks of incoherence as translational symmetry, then the incoherencepreserving property formalizes the simple intuition known as Curie’s principle: If the initial state does not break the translational symmetry and the evolution does not break the translational symmetry either, then the final state cannot break the translational symmetry.
iii.1.2 Translationallycovariant measurements
If an operation has a trivial output space, so that it corresponds to tracing with a measurement effect on the input space, that is, , then Eq. (9) reduces to
(12) 
which in turn implies
(13) 
i.e., the effect is translationallyinvariant. This condition is equivalent to , so that the effect is incoherent with respect to the eigenspaces of .
Proposition 3
A POVM is translationallycovariant if and only if all of its effects are incoherent relative to the eigenspaces of .
iii.1.3 Translationallycovariant unitary operations
Finally, if is a unitary translationallycovariant operation, that is, for some unitary operator (in which case the input and output spaces are necessarily the same), then Eq. (9) reduces to
(14) 
which implies
(15) 
for some phase . Taking the traces of both sides, we find that in finitedimensional Hilbert spaces, this condition can hold if and only if , that is, if and only if
(16) 
which is equivalent to , so that the unitary operator is also blockdiagonal with respect to the eigenspaces of . If denotes the set of eigenspaces of , the projectors onto these, and an arbitrary set of unitaries within each such subspace, then any such unitary can be written as
(17) 
If the Hilbert space is infinitedimensional, on the other hand, then the characterization above need not apply. Indeed, in this case, there are translationallycovariant unitaries that need not map every eigenspace of to itself. For instance, if the generator is a charge operator with integer eigenvalues, , then the unitary that applies a rigid shift of the charge by an integer , that is,
defines a unitary operation that is covariant relative to the group of shifts of the phase conjugate to charge, where . As another example, if the system is a particle in one dimension, then the unitary operation that boosts the momentum by is translationallycovariant relative to the group of spatial translations. This is because the unitary operation associated with a boost by , where is the position operator, and the unitary operation associated with a translation by , where is the momentum operator, commute with one another for all .
iii.1.4 Characterization via Stinespring dilation
We show that every translationallycovariant operation on a system can arise by coupling the system to an ancilla in an incoherent (translationallyinvariant) state, subjecting the composite to a translationallycovariant unitary, and postselecting on the outcome of a measurement on the ancilla which is assoicated to an incoherent (translationallyinvariant) effect. Such an implementation is termed a translationallycovariant dilation of the operation.
To make sense of the notion of a translationallycovariant dilation, however, one needs to specify not only the representation of the translation group on the system and ancilla individually, but on the composite of system and ancilla as well. Recall that we allow the operation to have different input and output spaces, so that to make sense of a translationallycovariant dilation, we must also specify the representation of the translation group on the output versions of the system and ancilla.
Some notation is helful here. We denote the Hilbert spaces corresponding to the input and output of the map by and respectively. Denoting the Hilbert space of the ancilla by , the composite Hilbert space of system and ancilla is . We denote the subsystem that is complementary to by (this is the subsystem over which one traces), so that .
In the physical situations to which TCcoherence applies—which we will discuss at length in Sec. III.2—one can always choose an ancilla system such that translation is represented collectively on the composite of system and ancilla. Specifically, if is the generator of translations on and is the generator of translations on , then the generator of translations on the composite is . Similarly, we have . It follows that the translation operation on the composite is collective on the factorization , that is, and on the factorization , that is, . In the discussion below, we use to denote the generator of translations, regardless of the system it is acting upon.
Proposition 4
A quantum operation is translationallycovariant if and only if it can be implemented by coupling the system to an ancilla prepared in an incoherent state via a translationallycovariant unitary quantum operation , and then postselecting on the outcome of a measurement on the ancilla which is associated with an incoherent effect . Formally, the condition is that for all quantum states ,
(18) 
where and and where for all .
Proof. The proof that any operation of the form of Eq. (18) is translationallycovariant is as follows:
(19a)  
(19b)  
(19c)  
(19d)  
(19e)  
(19f)  
(19g) 
where in the second equality, we have used that fact that is an incoherent state; in the third equality, we have used the fact that is a translationallycovariant operation; in the fifth equality, we have used the fact that is an incoherent effect; and in the last equality, we have used Eq. (18).
For the converse implication, we refer the reader to the result on the form of the Stinespring dilation for groupcovariant quantum operations by Keyl and Werner Keyl and Werner (1999).
iii.1.5 Characterization via modes of translational asymmetry
We begin by introducing some technical machinery.
Denote the preferred subspaces of relative to which coherence is evaluated, that is, the eigenspaces of , by , where is the set of eigenvalues of (these may be discrete or continuous). Let the set of modes be the set of the gaps between all eigenvalues, i.e. . In the case where is the system Hamiltonian, each element of can be interpreted as a frequency of the system.
Elements of the set label different modes in the system. For any , define the superoperator
(20) 
where . This superoperator is the projector that erases all the terms in the input operator except those which connect eigenstates of whose eigenvalues are different by , i.e., all except those which are of the form , where and are eigenstates of with eigenvalues and respectively. One can easily show that
(21a)  
(21b)  
(21c)  
(21d) 
where is the identity superoperator, and is the Kronecher delta.
The set of superoperators are a complete set of projectors to different subspaces of the operator space . It can be easily shown that these subspaces are orthogonal according to the HilbertSchmidt inner product, defined by for arbitrary pair of operators . Therefore, the operator space can be decomposed into a direct sum of operator subspaces, , where each is the image of .
Note that any operator in the operator subspace transforms distinctively under translations,
(22) 
We refer to as the “mode ” operator subspace. For any operator , the component of that operator in the operator subspace , denoted
is termed the “mode component of ”.
Clearly, every incoherent (i.e. translationally symmetric) state lies entirely within the mode 0 operator subspace, while a coherent (i.e. translationally asymmetric) state has a component in at least one mode operator subspace with .
Operator subspaces associated with distinct values have been called “modes of asymmetry” in Ref. Marvian and Spekkens (2014b), where the decomposition of states, operations and measurements into their different modes was shown to constitute a powerful tool in the resource theory of asymmetry.
Example 5
Consider the special case where is the angular operator in the direction. For simplicity, assume that is nondegenerate and let be its orthonormal eigenbasis, where is the eigenstate of with eigenvalue . Since the eigenvalues of the angular momentum operator are all separated by integers, it follows the set of modes is a subset of the integers, . Then, for each integer we have
(23) 
Furthermore, the mode component of any operator is given by
(24) 
The mode of the density operator corresponds to coherence of order in the context of magnetic resonance techniques Cappellaro (2014).
With these notions in hand, we can provide the modebased characterization of the translationallycovariant operations.
Proposition 6
A quantum operation is translationally covariant relative to the generator if and only if it preserves the modes of asymmetry associated to , that is, if and only if the mode component of the input state is mapped to the mode component of the output state. Formally, the condition is that whenever , we have where and . Note that is only a normalized state if is a channel (i.e. tracepreserving) and is otherwise subnormalized.
iii.1.6 Characterization via Kraus decomposition
Proposition 7
A quantum operation is translationally covariant if and only if it admits of a Kraus decomposition of the form
(25) 
where the elements of the set are all mode operators.
To see that any quantum operation with such a Kraus decomposition is translationally covariant, we note that
(26) 
and then use the fact that and Eq. (22) to infer that , which in turn implies
(27) 
which, from Eq. (10), simply asserts the translational covariance of .
The proof that every translationallycovariant operation has a Kraus decomposition of the form specified can be inferred from a result in Ref. Gour and Spekkens (2008) which characterizes the Kraus decomposition of any groupcovariant operation, by specializing the result to the case of a translation group. Alternatively, it can be inferred from the Stinespring dilation, Proposition 4, using a slight generalization (from channels to all operations) of the argument provided in Appendix A.1 of Ref. Marvian et al. (2015).
Proposition 7 also implies:
Corollary 8
A quantum operation is translationallycovariant if and only if it admits of a Kraus decomposition every term of which is translationallycovariant.
It suffices to note that each term of the Kraus decomposition specified in proposition 7 is translationallycovariant. It follows that if the different terms in this Kraus decomposition correspond to the different outcomes of a measurement, then even one who postselected on a particular outcome would describe the resulting operation as translationallycovariant.
iii.2 Physical justifications for the restriction to translationallycovariant operations
As noted in the introduction, it is critical that any definition of the restricted set of operations in a resource theory must be justifiable operationally. In this section, we discuss different physical scenarios in which the set of translationallycovariant operations are naturally distinguished as the set of easy or freelyavailable operations.
iii.2.1 Fundamental or effective symmetries of Hamiltonians
If, for a set of systems, the Hamiltonians one can access are symmetric and the states and measurements that one can implement are also symmetric, then for any given system only symmetric operations are possible^{13}^{13}13This is the easy half of the dilation theorem in Prop. 18.. Such symmetry constraints can sometimes be understood to be consequences of fundamental or effective symmetries in the problem.
A constraint of translational symmetry on the Hamiltonian is fundamental if it arises from a fundamental symmetry of nature, such as a symmetry of spacetime. It is effective if it arises from practical constraints, for instance, if one is interested in time scales or energy scales for which a symmetrybreaking term in the Hamiltonian becomes negligible. A translational symmetry constraint on the states and measurements can sometimes arise as a consequence of this symmetry of the Hamiltonian. For instance, if the only states that one can freely prepare are those that are thermal, then given that thermal states depend only on the Hamiltonian, any fundamental or effective symmetry of the Hamiltonian is inherited by the thermal states.
iii.2.2 Lack of shared reference frames
The most natural experimental restriction that leads to translationallycovariant operations is when one lacks access to any reference frame relative to which the translations can be defined. Such a lack of access can arise in a few ways.
For a pair of separated parties, each party may have a local reference frame, but no information about the relation between the two reference frames. For instance, a pair of parties may each have access to a Caretesian reference frame (or clock, or phase reference), but not know what rotation (or timetranslation or phaseshift) relates one to the other.
Under this kind of restriction, each party essentially lacks access to the reference frame of the other. It has been shown that this lack of a shared reference frame implies that the only operations that one party can implement, relative to the reference frame of the other, are those that are groupcovariant (see Refs. Bartlett et al. (2007); Marvian and Spekkens (2013)). For instance, if two parties lack of a shared phase reference, then the only operations whose descriptions they can agree on are phasecovariant operations.
It is also possible that the reference frame that one requires cannot even be prepared locally, due to technological limitations. For instance, only after the experimental realization of BoseEinstein condensation in atomic systems Anderson et al. (1995); Davis et al. (1995), was it possible to prepare a system that could serve as a reference frame for the phase conjugate to atom number.
iii.2.3 Metrology and phase estimation
Unspeakable coherence is the main resource for quantum metrology, and in particular phase estimation. In this context, a state is a resource to the extent that it allows one to estimate an unknown translation applied to the state (such as a phaseshift, a rotation, or evolution for some time interval). Suppose one prepares a system in the state prior to it being subjected to a unitary translation , where is unknown. In this case, one knows that the state after the translation is an element of the ensemble and the task is to estimate . Clearly, if is invariant under translations (i.e., incoherent), then it is useless for the estimation task. In this sense, translationally asymmetry, and hence coherence relative to the eigenspaces of the generator of translations, is a necessary resource for metrology.
Furthermore, as we show in the following, in this context, the set of translationallycovariant operations has also a simple and natural interpretation. Suppose one is interested in determining which of two states, and , is the better resource for the task of estimating an unknown translation. To do so, one must determine which of the two encodings, and , carries more information about . But suppose there exists a quantum operation such that for all , it transforms to , i.e.,
(28) 
Here, the quantum operation can be thought as an information processing which we perform on the state before performing the measurement which yields the value of . If such a quantum operation exists, then we can be sure that the state is more useful than for this metrological task. Because any information that we can obtain using the state , we can also obtain if we use the state .
It turns out that any such information processing can be chosen to be translationally covariant with respect to translation , i.e.
Proposition 9
For any given pair of states and the following statements are equivalent:
(i) There exists a translationallycovariant quantum operation
such that .
(ii) There exists a quantum operation such that , for all .
This is the specialization to the case of a translational symmetry group of a similar proposition for an arbitrary symmetry group the proof of which is presented in Ref. Marvian and Spekkens (2013), where we have also presented a version of this duality for pure states and unitaries, its interpretation in terms of reference frames, and some of its applications.
Statement (ii) in proposition 9 concerns the relative quality of and as resources for metrology, while statement (i) concerns the relative quality of and within the resource theory defined by the restriction to translationallycovariant operations. The partial order of quantum states under translationallycovariant operations, therefore, determines their relative worth as resources for metrology. Note that in this context, translationallycovariant operations are all and only the operations that are relevant.
It follows from proposition 9 that any function which quantifies the performance of states in this metrological task should be a measure of unspeakable coherence.
iii.2.4 Thermodynamics
The resource theory of athermality seeks to understand states deviating from thermal equilibrium as a resource Brandão et al. (2015); Brandao et al. (2013); Gour et al. (2015); Skrzypczyk et al. (2014); Horodecki and Oppenheim (2013); Åberg (2013); Janzing et al. (2000); Ćwikliński et al. (2015) . The free operations defining the theory, termed thermal operations are all and only those that can be achieved using thermal states, unitaries that commute with the free Hamiltonian, and the partial trace operation. (The restriction on unitaries is motivated by the fact that were one to allow more general unitaries, one could increase the energy of a system, thereby allowing thermodynamic work to be done for free.)
Noting that: (i) if a unitary commutes with the free Hamiltonian, then it is covariant under timetranslations, and (ii) because thermal states are defined in terms of the free Hamiltonian, they are symmetric under timetranslations, it follows from the dilation theorem for translationallycovariant operations (Prop. 4) that the restriction to thermal operations implies a restriction to timetranslationcovariant operations.
iii.2.5 Control theory
Suppose we are trying to prepare a quantum system in a desired state by applying a sequence of control pulses to the system. Then, there is an important distinction between the pulses which commute with the system Hamiltonian , and hence are invariant under time translations, and those which are not. Namely, to apply the pulses which do not commute with the system Hamiltonian, we need be careful about the timing of the pulses, and also the duration that the pulse is acting on the system.
To see this, first assume that the pulses are applied instantaneously, i.e., the width of the pulse is sufficiently small that the intrinsic evolution of the system generated by the Hamiltonian during the pulse is negligible. Then, if instead of applying the control unitary at the exact time , we apply it at time , the effect of applying this pulse would be equivalent to applying the pulse at time , instead of the desired pulse . If does not commute with the Hamiltonian , then in general and are different unitaries, and so the final state is different from the desired state.
Furthermore, if the control pulse commutes with the system Hamiltonian , then dealing with the nonzero width of the pulse is much easier and we do not need to be worried about the intrinsic evolution of the system during the pulse, as we now demonstrate. In general, to apply a control unitary we need to apply a control field to the system. The effect of this control field can be described by a term which is added to the system Hamiltonian . Then, to implement a control unitary which does not commute with the Hamiltonian , we need to apply a control field which does not commute with the Hamiltonian . In this case, the width of the control pulse, i.e., the duration over which we apply the control field , becomes an important parameter. In practice, in many situations we need to choose the control field to be strong enough so that the evolution of the system during the pulse width is negligible. On the other hand, if the control field commutes with the system Hamiltonian, then the effect of finite width can be easily taken into account, and so we do not need to apply strong fields to the system.
It follows that in this context, operations which are covariant under time translations are easy to implement, because for this type of operation, there is no sensistivity to the exact timing and the width of the control pulses. So it is natural to consider the operations that are covariant under timetranslations as the set of freelyimplementable operations, and this again leads us to treat coherence as translational asymmetry.
iii.3 Covariance with respect to independent translations
It can happen that the set of all systems is partitioned into subsets and that the action of the translation group is only collective for those systems within a given subset, while it is independent for different subsets. Suppose that the subsets are labelled by and that for the set of systems of type , the generator of collective translations on these systems is denoted . Consider the group element consisting of a translation by for all the systems of type . We label this group element by the independent translation parameters where denotes the number of different types of system. The unitary representation of this group element is
(29) 
The superoperator representation of this group element is then
(30) 
In this case, the set of free states are translationallyinvariant relative to translations generated by the set of generators . These are the states that are blockdiagonal relative to the distinct joint eigenspaces of .
The free operations are those that are translationallycovariant relative to the set of generators , that is,
(31) 
Indeed, all of the results expressed in this section can be generalized by substituting the translations with , the superoperator with , and the eigenspaces of with the joint eigenspaces of .
Iv Coherence via dephasingcovariant operations
Much recent work seeking to quantify coherence as a resource has considered speakable coherence. The article of Baumgratz, Cramer and Plenio Baumgratz et al. (2014) (BCP) provides one such proposal, which has been taken up by most other authors who have sought to characterize coherence as a resource. Nonetheless, we postpone our discussion of the BCP proposal to Sec. V and instead begin our discussion of speakable coherence with a very different proposal, based on operations that are dephasingcovariant. We here assess the dephasingcovariance approach and compare it to the translationalcovariance approach discussed in the last section.
iv.1 Free operations as dephasingcovariant operations
iv.1.1 Definition of dephasingcovariant operations
As before, suppose that the preferred subspaces relative to which coherence is to be quantified are and are associated with the projectors .
Definition 10
We say that a quantum operation is dephasingcovariant relative to the preferred subspaces if it commutes with the associated dephasing operation, of Eq. (3), i.e., if
(32) 
Note that if the input and output spaces of the map are distinct, then the dephasing map is different on the input and output spaces, but we do not indicate this difference in our notation.
Dephasingcovariant quantum operations are easily seen to be incoherencepreserving. It suffices to note that if is dephasingcovariant, then for any incoherent state ,
(33) 
and therefore is invariant under dephasing and hence incoherent.
iv.1.2 Dephasingcovariant measurements
If the output space is trivial, so that the map corresponds to tracing with a measurement effect on the input space, that is, , then Eq. (32) reduces to
(34) 
where we have used the fact that is selfadjoint relative to the HilbertSchmidt inner product, and this in turn implies
(35) 
where we have used the fact that the set of all quantum states form a basis of the operator space. Thus is an incoherent effect, i.e., it is blockdiagonal with respect to the preferred subspaces.
Proposition 11
A POVM is dephasingcovariant if and only if all of its effects are incoherent.
Comparing to proposition 3, we see that a POVM is dephasingcovariant if and only if it is translationallycovariant.
iv.1.3 Dephasingcovariant unitary operations
Because the dephasingcovariant operations are all incoherencepreserving, the set of unitary dephasingcovariant operations are included within the set of unitary incoherencepreserving operations. As it turns out, the two sets are in fact equivalent. We postpone the proof until Sec. V.2, Proposition 18, where we also present the general form of such unitaries.
iv.1.4 Considerations regarding the existence of a free dilation
By analogy with the considerations of Sec. III.1.4, in order to discuss the possibility of dilating a dephasingcovariant operation with the use of an ancilla in an incoherent state and a dephasingcovariant unitary on the composite of system and ancilla, one needs to specify not only the preferred subspaces (relative to which coherence is defined) on the system and ancilla individually, but on the composite of system and ancilla as well. When the input and output spaces differ this needs to be specified on the outputs as well, as discussed in Sec. III.1.4.
Recall that the system input and output spaces are denoted and , the ancilla input and output spaces are denoted and , and the composite of system and ancilla is . We also denote the associated sets of incoherent states and dephasing maps with the subscripts , and (or ).
We here assume that the preferred subspaces for the composite are just the tensor products of those for the system and for the ancilla, so that
(36) 
Proposition 12
A quantum operation is dephasingcovariant if it can be implemented by coupling the system to an ancilla in a state that is incoherent, via a unitary quantum operation that is dephasingcovariant, and then postselecting on a measurement outcome associated to an incoherent effect . Suppose can be implemented as a dilation of the form
(37) 
where is a state on , is a state on , for some unitary operator on , and is an effect on . Then formally, is dephasingcovariant if there is such a dilation where , and .
Proof. The proof is as follows:
(38)  
(39)  
(40)  
(41)  
(42)  
(43) 
where in the second line, we have used the fact that together with Eq. (36); in the third line, we have used the fact that is a dephasingcovariant operation; in the fourth line, we have used the fact that and Eq. (36); in the fifth line, we have used Eq. (36) again; and in the sixth line, we have used Eq. (47).
It is an open question whether every dephasingcovariant operation on a system can be implemented in this fashion for a suitable choice of ancilla.
iv.1.5 Characterization via diagonal and offdiagonal modes
A useful way of distinguishing incoherent states from coherent states is by considering their representation as vectors in , the space of linear operators on . The dephasing operations is a projector on the operator space, i.e., it satisfies , and it induces a direct sum decomposition on the operator space as , where and are respectively the image and the kernel of . For an arbitrary operator , we define the diagonal component of to be
(44) 
and the offdiagonal component of to be
(45) 
Clearly, all incoherent states lie entirely within , while every coherent state has some nontrivial component in .
Then, the fact that dephasingcovariant operations by definition commute with the dephasing operation , immediately implies that these operations are blockdiagonal with respect to this decomposition of the operator space .
It is useful to consider how a dephasingcovariant operation is represented as a matrix on the operator space . If is an orthonormal basis (with respect to the Hilbert Schmidt inner product) for the space of operators , then can be represented by the matrix elements . is dephasingcovariant iff its matrix representation has the following form relative to the decomposition ,
(46) 
where and are matrices.
Alternatively, the modebased characterization of dephasingcovariant operations can be given as follows.
Proposition 13
A quantum operation is dephasingcovariant relative to a preferred set of subspaces if and only if it preserves the diagonal and offdiagonal modes. Formally, the condition is that whenever , we have and .
iv.2 Physical justification for the restriction to dephasingcovariant operations?
We noted that whether every dephasingcovariant operation admits of a dilation in terms of an incoherent ancilla state, an incoherent effect on the ancilla, and a dephasingcovariant unitary on the systemancilla composite is currently an open question.
If its answer is positive, then the problem of finding a physical justification for the dephasingcovariant operations reduces to finding a physical scenario wherein the only free states and effects on the ancilla are incoherent and the only free unitaries on the systemancilla composite are those that are dephasingcovariant. It is not obvious how to justify the latter constraint in particular. However, even if the answer is negative, there remains the possibility that one can find a physical justification for the free set of operations being the set of dephasingcovariant operations. This is the same possibility that remained for justifying the incoherencepreserving or incoherent operations as the free set, namely, by allowing that different systems may not be treated evenhandedly.
Overall, therefore, it is at present unclear whether a restriction to dephasingcovariant operations arises from a natural experimental restriction.
iv.3 Relation of dephasingcovariant operations to translationallycovariant operations
iv.3.1 Relation between the sets of free operations
Here we study the relation between the dephasingcovariant operations and the translationallycovariant operations for the same choice of the preferred subspaces. In the dephasingcovariance approach to coherence, one must begin with a choice of preferred subspaces relative to which dephasing occurs. If one is given a translational symmetry, then one can choose these subspaces to be the eigenspaces of the generator of that symmetry group (or, if the symmetry group incorporates independent commuting translations and therefore multiple commuting generators, then as the joint eigenspaces of the generators). Conversely, if a set of preferred subspaces is given, one can always construct a Hermitian operator that has these subspaces as eigenspaces with distinct eigenvalues and consider this to be a generator of translations. Because a given choice of preferred subspaces might only be physically justified in one of the two approaches, the comparison we are making here is best understood as probing the mathematical relation between the two approaches to quantifying coherence as a resource.
To understand this connection it is useful to note that the dephasing operation relative to the eigenspaces of can be realized by applying a random translation to the system, that is, a translation where is chosen uniformly at random,
(47) 
It is also useful to note the connection between the two approaches from the perspective of mode decompositions. The diagonal mode relative to the eigenspaces of corresponds to the mode of translational asymmetry relative to the generator ,
(48) 
while the offdiagonal mode relative to the eigenspaces of corresponds to the direct sum of the modes of translational asymmetry relative to the generator ,
(49) 
Intuitively then, to choose the dephasingcovariant operations as the free set of operations is to disregard the distinction between the different nonzero modes.
Denote the the set of quantum operations that are translationallycovariant with respect to a generator by TC and the set of quantum operations that are dephasingcovariant with respect to the eigenspaces of (which we will denote ) by DC.
Proposition 14
The operations that are translationallycovariant relative to translations generated by are a proper subset of those that are dephasingcovariant relative to the eigenspaces of ,
(50) 
Proof. The subset relation can be understood easily within any of the characterizations of translationallycovariant and dephasingcovariant operations that we have provided. For instance, starting with the expression for the translationalcovariance of an operation , Eq. (9), if one integrates over , one obtains the expression for the dephasing covariance of , Eq. (32), where we have made use of Eq. (47).
To show that the inclusion is strict, it suffices to show that there are dephasingcovariant operations which are not translationallycovariant. Any example of a dephasingcovariant operation wherein one nonzero mode, , is mapped to another, distinct, nonzero mode is sufficient.
For example, consider the unitary that swaps a pair of states living in different eigenspaces of , and leaves the rest of the states unchanged. This operation in general will not be translationallycovariant while it is dephasingcovariant.
Despite the strict inclusion of translationallycovariant operations in the set of dephasingcovariant operations, if we focus on the POVMs associated to measurements (i.e. the retrodictive aspect of the measurement) the two approaches pick out the same set, as we noted earlier.
iv.3.2 Relation between measures of coherence
Prop. 14 implies that if a transformation from initial state to final state is allowed under operations, then it is also allowed under operations. This means that any measure of coherence is also a measure of coherence. In fact, one can show that
Proposition 15
Any measure of coherence is also a measure of coherence, but not viceversa.
V Coherence via incoherencepreserving and incoherent operations
In this section, we consider approaches to coherence wherein the free operations are incoherencepreserving or incoherent operations.
v.1 Free operations as incoherencepreserving operations
Definition 16
A quantum operation is said to be incoherencepreserving if it maps incoherent states on the input space to incoherent states on the output space,
(51) 
Just as was the case with the dephasingcovariant operations, the incoherencepreserving operations can be characterized in terms of their interaction with the dephasing map:
Proposition 17
A quantum operation is incoherencepreserving if and only if
(52) 
We can also characterize incoherencepreserving operations in terms of their representations as matrices on the operator space , just as we did for dephasingcovariant operations.
We deduce from Eq. (52) that an operation is incoherencepreserving if and only if its matrix representation has the following form relative to the decomposition ,
(53) 
where , and are matrices. A comparison with the analogous characterization of dephasingcovariant operations, Eq. (46), shows how incoherencepreserving operations do not preserve the diagonal and offdiagonal modes.
We postpone our characterization of the incoherencepreserving measurements until Sec. V.4 because it forms the basis of one of our criticisms of this approach.