Genuine quantum coherence
Any quantum resource theory is based on free states and free operations, i.e., states and operations which can be created and performed at no cost. In the resource theory of coherence free states are diagonal in some fixed basis, and free operations are those which cannot create coherence for some particular experimental realization. Recently, some problems of this approach have been discussed, and new sets of operations have been proposed to resolve these problems. We propose here the framework of genuine quantum coherence. This approach is based on a simple principle: we demand that a genuinely incoherent operation preserves all incoherent states. This framework captures coherence under additional constrains such as energy preservation and all genuinely incoherent operations are incoherent regardless of their particular experimental realization. We also introduce the full class of operations with this property, which we call fully incoherent. We analyze in detail the mathematical structure of these classes and also study possible state transformations. We show that deterministic manipulation is severely limited, even in the asymptotic settings. In particular, this framework does not have a unique golden unit, i.e., there is no single state from which all other states can be created deterministically with the free operations. This suggests that any reasonably powerful resource theory of coherence must contain free operations which can potentially create coherence in some experimental realization.
Quantum mechanics offers a radically different description of reality that collides with the intuition behind that of classical physics. At first this was only regarded from the foundational point of view. However, in recent decades it has been realized that the fundamentally different features of quantum theory can be exploited to realize revolutionary applications Nielsen and Chuang (2010). Quantum information theory has taught us that quantum technologies can outperform classical ones in a large variety of tasks such as communication, computation or metrology. This has led to identify and study nonclassical salient properties of quantum theory, like entanglement Plenio and Virmani (2007); Horodecki et al. (2009) or nonlocality Brunner et al. (2014) which stem from the tensor product structure. This is done in order to understand better from a theoretical perspective the full potential of quantum resources and, also, to seek for new paths for applications. Undoubtedly, the superposition principle, which leads to coherence, is another characteristic trait of quantum mechanics; however, a rigorous theoretical study of this phenomenon on the analogy of the aforementioned resources has only been initiated very recently Aberg (2006); Gour and Spekkens (2008); Gour et al. (2009); Levi and Mintert (2014); Marvian and Spekkens (2013, 2014); Baumgratz et al. (2014); Winter and Yang (2016). Nevertheless, quantum coherence is the basis of single-particle interferometry Oi (2003); Åberg (2004); Oi and Åberg (2006) and it is believed to play a nontrivial role in the outstanding efficiency of several biological processes Lloyd (2011); Li et al. (2012); Huelga and Plenio (2013); Singh Poonia et al. (2014). This grants coherence the status of a resource and makes necessary to develop a solid framework allowing to asses and quantify this phenomenon together with the rules for its manipulation.
Resource theories have proven to be a very successful framework to build a rigorous and systematic study of the possibilities and limitations of distinct features of quantum information theory. Originally developed in the case of entanglement theory Plenio and Virmani (2007); Horodecki et al. (2009), the conceptual elegance and applicability of this approach has led to consider in the last years, resource theories for several quantum features such as frame alignment Gour and Spekkens (2008), stabilizer computation Veitch et al. (2014), nonlocality de Vicente (2014) or steering Gallego and Aolita (2015). In such theories one considers a set of free states and of free operations. The latter constitutes the set of transformations that the physical setting allows to implement. Free states must be mapped to free states under all free operations and they are useless in this physical setting (it is usually assumed that they can be prepared at no cost). With this, non-free states can be regarded as resource states: they allow to overcome the limitations imposed by state manipulation under the set of free operations. Furthermore, free operations then provide all possible protocols to manipulate the resource and induce the most natural ordering among states since the resource cannot increase under this set of transformations. This allows to rigorously construct resource measures: these quantifiers must not increase under free operations. For instance, in entanglement theory the set of free operations is local (quantum) operations and classical communication (LOCC) while free states are separable states (entangled states are then the resource states) and the basic principle behind entanglement measures is that they must not increase under LOCC Plenio and Virmani (2007); Horodecki et al. (2009). Recent literature has set the first steps to build a resource theory of coherence Aberg (2006); Gour and Spekkens (2008); Gour et al. (2009); Levi and Mintert (2014); Marvian and Spekkens (2013, 2014); Baumgratz et al. (2014); Winter and Yang (2016), which has allowed to study the role of coherence in quantum theory Chen et al. (2016a); Cheng and Hall (2015); Hu and Fan (2016); Mondal et al. (2016); Deb (2016); Du et al. (2015); Bai and Du (2015); Bera et al. (2015); Liu et al. (2016); Xi et al. (2015); Li et al. (2014); von Prillwitz et al. (2015); Karpat et al. (2014); Çakmak et al. (2015); Du et al. (2015); Hillery (2016); Yadin et al. (2016); Bu et al. (2016); Matera et al. (2016); Chitambar et al. (2016); Chitambar and Hsieh (2016); Ma et al. (2016); Streltsov et al. (2016), its dynamics under noisy evolution Bromley et al. (2015); Singh et al. (2015); Mani and Karimipour (2015); Singh et al. (2015); Chanda and Bhattacharya (2016); Bu et al. (2015); Singh et al. (2016); Zhang et al. (2015); Allegra et al. (2016); García-Díaz et al. (2016); Silva et al. (2016); Peng et al. (2016), and to obtain new coherence measures Girolami (2014); Du and Bai (2015); Streltsov et al. (2015a); Killoran et al. (2016); Girolami and Yadin (2015); Yuan et al. (2015); Yao et al. (2015); Pires et al. (2015); Qi et al. (2015); Xu (2016); Yadin and Vedral (2016); Rana et al. (2016); Zhang et al. (2016); Radhakrishnan et al. (2016); Chen et al. (2016b); Chitambar and Gour (2016); Napoli et al. (2016); Piani et al. (2016).
However, there is an ongoing intense debate on how this theory should be exactly formulated with several alternatives being considered Aberg (2006); Gour et al. (2009); Baumgratz et al. (2014); Marvian and Spekkens (2014); Marvian et al. (2016); Winter and Yang (2016); Yadin et al. (2016); Chitambar and Gour (2016); Marvian and Spekkens (2016). Notice that in the case of entanglement the physical setting clearly identifies the set of allowed operations: the parties, who might be spatially separated, can only act locally. However, while in the case of coherence it is clear that free states should correspond to incoherent states (see below for definitions), the physical setting does not impose any clear restriction on what free operations should be. Thus, any set of operations that map incoherent states to incoherent states might qualify in principle as a good candidate. It is therefore fundamental to identify reasonable sets of free operations from the physical and/or mathematical perspective and to study the different features of such resource theories. In this paper, we analyze in detail two such sets and we thoroughly study the possibilities and limitations of the emergent resource theories for state manipulation.
In the framework of coherence, the physical setting identifies a particular set of basis states as classical. A state on is called incoherent if it is diagonal in the fixed aforementioned basis () and otherwise coherent. In the standard resource theory of coherence of Baumgratz et al. Baumgratz et al. (2014), free operations are given by the so-called incoherent operations. These correspond to those maps that admit an incoherent Kraus decomposition:
where, besides the normalization condition , the Kraus operators fulfill that the unnormalized states remain incoherent for every if is. On an experimental level, this definition means that a quantum operation can be implemented in an incoherent way. By performing the measurements given by the aforementioned Kraus operators, no coherence can be created from an incoherent state even if we allow for postselection. Thus, this set of operations appears to be a very sensible choice and it leads to a reasonably rich resource theory Winter and Yang (2016). However, as mentioned above, no physical reason is known why these operations should be regarded as free in this setting. This naturally leads to consider other possibilities either because they seem physically justified or because they have a convenient structure or properties in a given setting Aberg (2006); Gour et al. (2009); Marvian and Spekkens (2014); Marvian et al. (2016); Winter and Yang (2016); Yadin et al. (2016); Chitambar and Gour (2016); Marvian and Spekkens (2016). In particular, one can think of scenarios where the particular forms of implementing quantum operations are restricted. Moreover, one can consider resource theories of speakable or unspeakable information Marvian and Spekkens (2016). The former are theories where the quality of the resource is independent of the physical encoding while in the latter it depends on the underlying degrees of freedom. It turns out that in the resource theory of incoherent operations, coherence is a speakable resource and it might be desirable to consider theories where coherence is unspeakable Marvian and Spekkens (2016). In this paper we introduce and study the sets of genuinely incoherent operations (GIO) and fully incoherent operations (FIO). GIO leads to a resource theory of unspeakable coherence and is derived from one simple condition, namely that the operation preserves all incoherent states. We will show in this work that these operations are incoherent irrespectively of the implementation (i.e. of the Kraus decomposition), and exhibit several interesting properties which are not present for the set introduced in Baumgratz et al. (2014). On the other hand the set FIO is the most general set of operations which are incoherent for all Kraus decompositions and leads to a theory of speakable coherence.
This paper is structured as follows. In Sec. II we consider GIO. We define and motivate this set of operations and we analyze in detail its mathematical structure. This allows us to derive genuine coherence measures and to study extensively the possibilities and limitations of GIO for state manipulation. We show that deterministic transformations are very constrained in this framework. However, we show that stochastic transformations have a much richer structure. We also consider the asymptotic setting, which plays a key role in resource theories, and show that any general form of distillation and dilution is impossible. Motivated by these limitations in Sec. III we study the set of FIO. We characterize mathematically this kind of operations, which allows us to prove that this set is a strict superset of GIO. Although this is reflected in a strictly more powerful capability for state manipulation, we provide several results suggesting that FIO are still rather limited. Finally, in Sec. IV we provide a discussion on the relation of the operations presented in this work to other alternative incoherent operations presented in the literature.
Ii Genuinely incoherent operations
ii.1 Definition and motivation
The underlying principle in every resource theory of coherence is that incoherent states should not be a resource. Thus, lacking a clear intuition on the operations the physical setting allows to implement, any set of operations that maps incoherent states to incoherent states is a good candidate to be regarded as the free operations from the mathematical point of view. The largest class of maps with this property has been studied in previous literature and is referred to as maximally incoherent operations Aberg (2006). It is natural to consider the opposite extreme case, i.e those operations for which all incoherent states are fixed points. We define genuinely incoherent (GI) operations to be quantum operations which have this property of preserving all incoherent states:
for any incoherent state . Thus, in the GI formalism, incoherent states are not resourceful in an extreme way: if we are provided with such a state, we are bound to it and no protocol is possible. In particular, incoherent states cannot even be transformed to other incoherent states. From this point of view, in this setting incoherent states are not free states, i.e. they cannot be prepared at no cost with the allowed set of operations. What grants coherent states the status of a resource in this case is the fact that these are the only states for which non-trivial protocols are in principle possible.
One reason to study the resource theory of genuine coherence is that this is arguably the most contrived theory one can think of at the level of allowed operations. In this sense, GI manipulation can be regarded as a building block for protocols in other resource theories in which the free operations have a richer structure. More importantly, genuine coherence is an extreme form of a resource theory of unspeakable coherence. Another example of such a theory is the resource theory of asymmetry Gour et al. (2009); Marvian and Spekkens (2014); Marvian et al. (2016), while the framework of Baumgratz et al. Baumgratz et al. (2014) is a representative of speakable coherence. As discussed in Marvian and Spekkens (2016), resource theories of speakable coherence are those where the means of encoding the information is irrelevant. On the other hand, unspeakable information can only be encoded in certain degrees of freedom. In this case, coherent states given by a particular superposition of classical states need not be equally useful as the same superposition of a different set of classical states. The resource theory of genuine coherence is an extreme form of unspeakable coherence in the sense that no coherent state is interchangeable with any other. This might be particularly meaningful from the physical point of view in scenarios where the classical states are constrained by e.g. energy preservation rules. Indeed, if the states are the eigenstates of some nondegenerate Hamiltonian, GI operations correspond to energy-preserving operations as defined in Chiribella and Yiang (2015). Another example would be noiseless excitation transport. If the classical states (i.e. the position of the excitation) can be freely mapped to each other, in the absence of dissipation the process of transport is trivial.
Another interesting feature of GI operations, as we will see in the next subsection, is that GI maps are incoherent independently of the Kraus decomposition (cf. Eq. (1)). This is not the case in the standard framework of Baumgratz et al. Baumgratz et al. (2014). To see this, notice that a quantum operation admits different Kraus representations as characterized in the following well-known theorem (see e. g. Watrous (2016); Holevo (2012)).
Two sets of Kraus operators and correspond to Kraus representations of the same map if and only if there exists a partial isometry matrix such that
Thus, as an example for the above claim take the single-qubit operation given by the following Kraus operators:
with . These operators are incoherent as can be seen by noting that holds true for any pure state . If we now apply Theorem 1 to the channel defined above with with the Hadamard matrix , we get the Kraus operators
Note that the Kraus operators are not incoherent, which can be directly checked by applying them to the state : . Thus, these Kraus operators convert the incoherent state into one of the maximally coherent states or .
The above example shows that a quantum operation which is incoherent in one Kraus decomposition is not necessarily incoherent in another Kraus decomposition. This observation is not surprising, a similar effect appears also in entanglement theory. There, the set of LOCC operations is also defined via a certain structure of Kraus operators which is lost when other Kraus decompositions are considered. However, this feature is clearly justified by the physical setting: as long as a map has a Kraus representation of the LOCC form, there are physical means for the spatially separated parties to implement the corresponding protocol. On the other hand, the definition of incoherent operations is rather abstract and we do not have a physical reason that guarantees that a certain map is implementable. It is certainly admissible to take the analogy of LOCC and consider maps which have one Kraus representation that does not create coherence. However, it also seems reasonable a priori to explore the alternative case in which maps must not create coherence independently of the Kraus decomposition. This would be relevant in resource theories of coherence where one is not granted with the power to choose different particular experimental implementations of a map.
ii.2 Mathematical characterization
In order to understand the potential of a resource theory based on GI operations, it will be useful to have a more detailed mathematical description of these operations beyond its mere definition given in Eq. (2). The framework of Schur operations plays a key role here. A quantum operation acting on a Hilbert space of dimension is called a Schur operation if there exists a matrix such that
Here, denotes the Schur or Hadamard product, i. e. entry-wise product for two matrices of the same dimension:
The fact that is a quantum channel – and thus a trace preserving completely positive map – adds additional constraints on the matrix Watrous (2016):
must be positive semidefinite (PSD),
the diagonal elements of must be .
The following theorem provides a simple characterization of GIO that will be used throughout this paper.
The following statements are equivalent:
is a genuinely incoherent quantum operation, i. e. for every incoherent state .
Any Kraus representation of as in Eq. (1) has all Kraus operators diagonal.
can be written as
with a PSD matrix such that .
Let us start by showing that 1 implies 2. For any GI operation and any pure incoherent state it must hold that
where are Kraus operators of . However, this equality can only hold true if is an eigenstate of every Kraus operator:
This shows that every Kraus operator is diagonal. The fact that is straightforward. Any operation defined as in Eq. (8) is indeed genuinely incoherent, i.e., it preserves all incoherent states. It remains to prove that 2 implies 3. This is a direct consequence of Theorem 4.19 in Watrous (2016). ∎
Part 2 of this theorem clearly shows that the set of GI operations is a strict subset of incoherent operations introduced by Baumgratz et al. Baumgratz et al. (2014). Furthermore, this also immediately proves our claim of the previous subsection that every Kraus decomposition of a GI map does not create coherence. By definition, every GI operation is unital, i.e. it preserves the maximally mixed state: . An important example for a GIO is a convex combination of unitaries diagonal in the incoherent basis:
with probabilities and unitaries defined as . It is now natural to ask if any GIO can be written in this form. The characterization provided in Theorem 2 allows to study in great detail the mathematical structure of the set of GIOs and, in particular, to answer the above question.
For qubits and qutrits any GIO is of the form (11). This is no longer true for dimension 4 and above.
For the proof of the theorem we refer to Appendix A.
In the context of resource theories, one can also be interested in tasks that, although impossible deterministically, might be implemented with a certain non-zero probability of success. Interistingly, Theorem 2 can also be generalized to the stochastic scenario. In this case we need to consider trace non-increasing genuinely incoherent operations, i. e. transformations of the form
where the Kraus operators are all diagonal in the incoherent basis but do not need to form a complete set, i.e., . This means that the process is not deterministic, but occurs with probability given by . We will call such a map stochastic genuinely incoherent (SGI) operation. It is important to note that any SGI operation can be completed to a genuinely incoherent operation by another SGI operation. Thus, a transformation among two states can be implemented with some non-zero probability of success if and only if there exists an SGI operation connecting them (up to normalization). The following theorem generalizes Theorem 2 to the stochastic scenario.
A quantum operation is SGI if and only if it can be written as
with a PSD matrix such that .
Proposition 4.17 and theorem 4.19 in Watrous (2016) establish the equivalence of Schur maps with a PSD matrix and maps with diagonal Kraus operators independently of whether the maps are trace-preserving or not. SGI maps correspond to the case of trace non-increasing maps. Thus, it only remains to check that this condition is fulfilled if and only if . First of all it must hold that in order for the matrix to be PSD. Then, on the one hand, is clearly sufficient for the Schur map to be trace non-increasing. On the other hand, looking at the action of the map on the states the bound is also found to be necessary. ∎
The power of the above theorem lies in the fact that it gives a simple characterization of all SGI operations, which will be very useful when we study stochastic state transformations in Sec. II.5.
ii.3 Coherence rank and coherence set
In entanglement theory Horodecki et al. (2009), local unitaries are invertible local operations, and states related by local unitaries have the same amount of entanglement. Thus, for most problems concerning bipartite pure-state entanglement, it is sufficient to consider the Schmidt coefficients of the corresponding states.
In direct analogy, we notice that diagonal unitaries are invertible GI operations. Hence, for any measure of genuine coherence, states related by diagonal unitaries are equally coherent. Thus, without loss of generality, we can restrict our considerations to pure states
such that . Obviously, a pure state is incoherent if and only if for some .
In analogy to the Schmidt rank in entanglement theory Horodecki et al. (2009), one can define the coherence rank of a pure state as the number of basis elements for which Killoran et al. (2016). The coherence rank, like its analogous in entanglement theory, provides useful information about the coherence content of a state and constrains the possible transformations among resource states. For instance, the coherence rank cannot increase under incoherent operations Winter and Yang (2016). As we will see later, one particularity of genuine coherence is the following. It is not only relevant the coherence rank but also for which basis elements a state has zero components. We encode this information in the coherence set.
The coherence set of denotes the subset of for which .
The coherence set captures one of the crucial differences between the formalism of GI operations and that of incoherent operations of Baumgratz et al. (2014). In the latter, incoherent states are exchangeable. However, diagonal unitaries do not allow to permute basis elements and by its very definition an incoherent state cannot be transformed by GI operations into a different incoherent state. Thus, the relevance of the coherence set arises from the fact that we are dealing with a resource theory of unspeakable coherence. Unless otherwise stated, in the following when we start with a state it should be assumed that all sums go over the elements of . Finally, we will always use the notation for the density matrix corresponding to the pure state (i. e. ).
ii.4 Quantifying genuine coherence
Having introduced the framework of genuine coherence, we will provide methods to quantify the amount of genuine coherence in a given state. For this, we will follow established notions for entanglement and coherence quantifiers Vedral et al. (1997); Vedral and Plenio (1998); Horodecki et al. (2009); Baumgratz et al. (2014).
A measure of genuine coherence should have at least the following two properties.
Nonnegativity: is nonnegative, and zero if and only if the state is incoherent.
Monotonicity: does not increase under GI operations, .
It is instrumental to compare the above conditions to the corresponding conditions in entanglement theory Vedral et al. (1997); Vedral and Plenio (1998); Horodecki et al. (2009). There, the condition corresponding to G2 implies that an entanglement measure does not increase under LOCC. This condition and nonnegativity are regarded as the most fundamental conditions for an entanglement measure Horodecki et al. (2009).
The following two conditions will be regarded as desirable but less fundamental.
Strong monotonicity: does not increase on average under the action of GI operations for any set of Kraus operators , i.e., with probabilities and states .
Convexity: is a convex function of the state, .
The entanglement equivalent of G2’ states that the entanglement measure does not increase on average under selective LOCC. This condition as well as convexity are not mandatory for a good entanglement measure Horodecki et al. (2009). Following the notion from entanglement theory, we will consider conditions G1 and G2 to be more fundamental than G2’ and G3. A measure which fulfills conditions G1, G2, and G2’ will be called genuine coherence monotone. Additionally, the corresponding measure (monotone) will be called convex if condition G3 is fulfilled as well. Note that G2’ and G3 in combination imply G2.
Since the set of GI operations is a subset of general incoherent operations, any coherence monotone in the sense of Baumgratz et al. is also a genuine coherence monotone. Examples for such monotones are the relative entropy of coherence, the -norm of coherence, and the geometric coherence Baumgratz et al. (2014); Streltsov et al. (2015a). However, it is possible that some quantities which do not give rise to a good coherence measure in the framework of Baumgratz et al. are still good measures of genuine coherence. This is indeed the case for the Wigner-Yanase skew information Wigner and Yanase (1963): with the commutator , and is some nondegenerate Hermitian operator diagonal in the incoherent basis. As is shown in Appendix B, is a convex measure of genuine coherence, it fulfills conditions G1, G2, and G3. It remains open if it also fulfills the condition G2’.
Alternatively, a very general measure of genuine coherence can be defined as follows:
where is the set of incoherent states and is an arbitrary distance which does not increase under unital operations , i.e., for any operation which preserves the maximally mixed state . The following theorem shows that fulfills the corresponding conditions.
is a measure of genuine coherence, it satisfies conditions G1 and G2. If is jointly convex, also satisfies G3.
The proof that satisfies condition G1 follows from the fact that any distance is nonnegative and zero if and only if . For proving G2, let be the closest incoherent state to , i.e., . The fact that any genuinely incoherent operation is unital together with the requirement that does not increase under unital maps implies:
where in the last inequality we used the fact that is incoherent.
We will now show that is also convex if the distance is jointly convex, i.e., if it satisfies
For this, let be the closest incoherent state to : . Then we have
which is the desired statement. ∎
An example for such a distance is the quantum relative entropy , and the corresponding measure is known as the relative entropy of coherence Baumgratz et al. (2014). As mentioned above, this measure also satisfies strong monotonicity G2’ Baumgratz et al. (2014).
Remarkably, Theorem 6 also holds for all distances based on Schatten -norms
with the Schatten -norm and . This follows from the fact that Schatten -norms do not increase under unital operations Pérez-García et al. (2006). This result is surprising since Schatten -norms are generally problematic in quantum information theory. In particular, the attempt to quantify entanglement via these norms leads to quantities which can increase under local operations for Ozawa (2000); Streltsov et al. (2015b). Similar problems arise for other types of quantum correlations such as quantum discord Piani (2012); Paula et al. (2013).
For and the corresponding distances are also known as trace distance and Hilbert-Schmidt distance. In the case of the Hilbert-Schmidt distance the coherence measure can be evaluated explicitly Baumgratz et al. (2014):
where denotes complete dephasing in the incoherent basis. However, for general Schatten norms is not the closest incoherent state to Baumgratz et al. (2014).
While the distance-based measure of coherence defined in Eq. (14) does not admit a closed expression for a general distance , we prove now that the following simple quantity is also a valid measure of coherence:
In particular, satisfies conditions G1 and G2 if the distance is contractive under unital operations. To see this, notice that the distance is nonnegative and zero if and only if . Hence, G1 is fulfilled. For condition G2, recall that all Kraus operators of a GI operation are diagonal in the incoherent basis, and thus any GI operation commutes with the dephasing operation :
It follows that
where the inequality follows from the fact that is unital. Additionally, is convex if the distance is jointly convex, which is true for all distances based on Schatten -norms Baumgratz et al. (2014). The above claim can be proven directly via the following calculation:
It is worth noticing that it remains unclear if the measures are also genuine coherence monotones, i.e., if they satisfy G2’ 111Note that quantifiers based on Schatten -norms as defined in Eq. (14) do not give rise to coherence monotones in the sense of Baumgratz et al. for Rana et al. (2016). However, this does not exclude the possibility of genuine coherence monotones based on Schatten norms..
ii.5 State manipulation under GI operations
In any resource theory the free operations play a key role regarding the ordering of states relative to their usefulness. If can be transformed to , then cannot be less useful than . This is because any task that can be achieved by can also be implemented by since the latter can be transformed at no cost to the former but not necessarily the other way around. Thus, as we have seen in the previous section, any measure of genuine coherence should be non-increasing under GI operations (property G2). Therefore, in order to understand the power of the resource theory of genuine coherence it is important to clarify the possibilities and limitations of GI operations for state transformation. In this section we carry out a thorough analysis considering both pure and mixed-state conversions, deterministic and stochastic manipulation and single and many-copy scenarios. As we will see, it turns out that state manipulation under GI operations is rather contrived.
ii.5.1 Single-state transformations
In the standard resource theory of coherence the state
represents the golden unit: it can be transformed into any other state on via incoherent operations Baumgratz et al. (2014) and, therefore, it can be considered as the maximally coherent state. We will see now that this is no longer the case for genuine coherence. As we will show in the following theorem, the situation for state manipulation under GI operations is much more drastic.
A pure state can be deterministically transformed into another pure state via GI operations if and only if with a genuinely incoherent unitary .
From Theorem 2 it follows that for any GI operation the states and have the same diagonal elements, i.e., . For pure states and this means that , and thus the states are the same up to a unitary which is diagonal in the incoherent basis. ∎
This theorem shows that deterministic GI transformations among pure states are trivial. We can only use invertible operations to transform states within the classes of equally coherent states. This resembles the case of multipartite entangled states where almost no pure state can be transformed to any other state outside its respective equivalence class de Vicente et al. (2013).
We have therefore seen that the framework of genuine coherence does not have a golden unit, i.e., there is no unique state from which all other states can be prepared via GI operations. However, it is still possible that for every mixed state there exists some pure state from which can potentially be created via GI operations. In the following theorem we will show that this is indeed the case.
For every mixed state there exists a pure state and a GI operation such that
Let be an arbitrary mixed state. Since is PSD, we can assume that for all . We will now provide a pure state and a GI operation such that . As we will prove in the following, the desired state and GI operation are given as
with the matrix 222Note that is not normalized, i.e., in general.
This theorem shows that in the framework of genuine quantum coherence the set of all pure states can be regarded as a resource: all mixed states can be obtained from some pure states via GI operations. Thus, although there is no maximally genuinely coherent state, there is a maximal genuinely coherent set in the terminology of de Vicente et al. (2013). Moreover, noticing that transformations under GI operations require that the diagonal entries of the density matrices are preserved, the theorem further implies the following corollary, which characterizes all conversions from pure states to mixed states.
A pure state can be deterministically transformed by GI operations into the mixed state if and only if .
At this point we also note that mixed states cannot be deterministically transformed to a pure state. Indeed, let a mixed state have spectral decomposition with . Then, if there existed a GI map such that for some pure state , we would need that , which is forbidden by Theorem 7 (unless for all for some genuinely incoherent unitary, which would imply that is pure).
The impossibility of deterministic GI conversions among pure states calls for the analysis of probabilistic transformations. As explained above this amounts to the use of SGI operations. In the following theorem we evaluate the optimal probability for pure state conversion via SGI operations. In this theorem we will also explicitly use Definition 5 for the coherence set .
A probabilistic transformation by GI operations from to is possible if and only if . The optimal probability of conversion is
Without loss of generality we can write
We will first show that the condition is necessary for a probabilistic transformation. Let be an SGI map such that . Then, it must be that . Hence, since the Kraus operators are diagonal, if for some index we have then must be true as well. This proves that is a necessary condition for probabilistic transformation.
We will now show that for there exists a protocol implementing the transformation with the aforementioned probability. For this, we additionally define the matrix
and the number . Using again the Schur product theorem, we have that the matrix is PSD with . Hence, there exists an SGI map such that . Moreover, and . Thus, can be transformed to with probability (notice that ).
In the final step, we show that there cannot exist a protocol with larger probability of success. We do this by contradiction. Suppose that there exists an SGI map with Schur representation given by the PSD matrix such that with . Let be the index for which . Since , we would have that , which is in contradiction with the fact that is a trace non-increasing map. ∎
Notice that the state is not maximally genuinely coherent even under the stochastic point of view. Indeed, let be the state for which give rise to . Then, for , . Given the impossibility to relate pure states by deterministic GI operations, it would be tempting to order the set of pure coherent states by if . Unfortunately, it turns out that such an order would be not well defined. To see that, consider the state with squared components . For , we have that and but . On the other hand, by arguing as in Vidal (2000), we have that, fixing a target state, the function gives a computable genuine coherence monotone for all pure states. Furthermore, iff and are related via diagonal unitaries. Hence, by changing the target state we obtain different monotones, each of them being maximal for a different state in the maximal genuinely coherent set.
Finally, one may wonder whether mixed states can be transformed by GI operations with some non-zero probability into a pure coherent state. A simple example is given by where the two pure states and are respectively supported on the orthogonal subspaces and . If we denote by the projector onto the subspace , then the GI map with Kraus operators and transforms into with probability and into with probability . The next theorem shows that this is essentially the only possibility.
Let denote a subspace spanned by a subset of two elements of the incoherent basis. Then, a non-pure coherent state can be transformed by GI operations with non-zero probability to some pure coherent state if and only if is a pure coherent state for some choice of .
The “if” part of the theorem is immediate since a map with a unique Kraus operator given by is clearly an SGI operation. To prove the “only if” part we will show that if is not pure for any possible choice of , then cannot be transformed by GI operations with non-zero probability into a pure coherent state. We will proceed by assuming the opposite and arriving at a contradiction. Suppose that there exists an SGI map with Schur representation given by the PSD matrix such that with and . By our premise, the projection of on every 2-dimensional subspace spanned by two elements of the incoherent basis must be positive definite (not PSD). This, together with the fact that is coherent implies that there must exist such that and . The existence of the SGI map imposes the following three equations
which altogether yield
However, this implies that cannot be PSD and, hence, a contradiction. ∎
Hence, most mixed states cannot be stochastically transformed to any pure state. Thus, if, as discussed above, we regard pure states as the most resourceful states over mixed states, it turns out that most less resourceful states cannot be transformed, even with small probability, to a resource state in the one-copy regime.
ii.5.2 Multiple-state and multiple-copy transformations
So far we have just discussed possible transformations acting on a single copy of a state. However, in quantum information theory it is standard to find that multiple-state transformations broaden the possibilities for resource manipulation. An important example of this are activation phenomena. This means that the transformation (or, more generally, with ) is possible even though it is impossible to implement the conversion . In this case the state is called an activator. In the particular case when the activator can be returned, i.e. , the process is known as catalysis.
Another example, and probably the most paradigmatic one, is distillation. In these protocols one aims at transforming many copies of a less useful state into less copies of a maximally useful state in the asymptotic limit of infinitely many available copies. For instance, in entanglement theory this target state that acts as a golden standard to measure the usefulness of the resource is the maximally entangled two-qubit state .
In general, we say that a state can be distilled from the state at rate if and is -close to and as . As a measure of closeness we will use the trace norm; that is, it must hold that where . The optimal rate at which distillation is possible, i. e. the supremum of over all protocols fulfilling the aforementioned conditions, is a very relevant figure of merit known as distillable resource and plays a key role for the quantification of usefulness in resource theories. The reversed protocol, which is known as dilution, is also an interesting object of study. In this case one seeks for the optimal rate at which less copies of maximally useful state can be converted into more copies of a less useful state. This leads to another figure of merit: the resource-cost. In more detail, the cost of is the infimum of the rate over all protocols with such that (where is a golden unit maximally resourceful state) transforms -close to and as . The distillable entanglement and entanglement-cost have been widely studied in entanglement theory Horodecki et al. (2009) and allow to establish the phenomenon of irreversibility. More recently, the distillable coherence and coherence-cost have been characterized and irreversibility has also been identified in this setting Winter and Yang (2016).
In order to discuss multiple-state and multiple-copy manipulation under GI operations, it should be made clear what the set of allowed maps is in this setting. If we are allowed to act jointly on different states each of them acting on the Hilbert space , we define the incoherent basis in the total Hilbert space as ( ), where is the incoherent basis in each Hilbert space Bromley et al. (2015); Streltsov et al. (2015a). This can be further justified by the no superactivation postulate (cf. Ref. Chitambar and Gour (2016)). Thus, joint GI operations should preserve incoherent states in this basis and they will be characterized by having Kraus operators diagonal in the joint incoherent basis. By the same reasons as in Section II.2, these GI maps will also admit a Schur representation in the joint incoherent basis.
We are now in the position to state our results on multiple-state and multiple-copy manipulation under joint GI operations. It turns out that these protocols are out of reach: activation and any non-trivial form of distillation and dilution are impossible. This claim is a consequence of the following lemma.
For every two states and every GI map acting on such that , there exists another GI map acting on such that .
By assumption together with Theorem 2, there exists a PSD matrix A with diagonal entries equal to 1,
which induces the GI operation such that
The state is obtained by taking the partial trace over the second subsystem:
with the matrix with entries . The proof is complete if we can show that the operator is PSD and that holds for all , since in this case by Theorem 2 there must exist a GI operation such that . It is straightforward to verify that holds true for all . To see that is PSD, notice that , where the operator can be written as
with the (unnormalized) vector . Thus, is PSD (and so is by assumption). Hence, by the Schur product theorem, is the partial trace of a PSD matrix and it must me PSD too. ∎
The above lemma shows that there cannot be any activation phenomena in the resource theory of genuine coherence and it will be very useful in our study of coherence distillation and dilution with GI operations. In order to analyze the possibility of distillation one first needs to discuss what the target state is going to be. However, this is not at all clear under GI operations since, as we pointed out in the previous subsection, there is no unique state which would allow to create all other states via GI operations. In the following, we will show that in general it is not possible to distill the state from via GI operations if has more coherence than . Here, we measure the coherence by the relative entropy of coherence Baumgratz et al. (2014)
with the quantum relative entropy and denotes the set of all incoherent states. The relative entropy of coherence is known to be equal to the distillable coherence Winter and Yang (2016), and is also a faithful genuine coherence monotone (cf. Sec. II.4). We are now in the position to prove the following theorem.
Given two states and with
it is not possible to distill from at any rate via GI operations.
We will prove the statement by contradiction, assuming that distillation is possible for some state with . In particular, this would imply that for large enough it is possible to approximate one copy of the state . To be more precise, for any there exists an integer and a GI operation such that
where the partial trace is taken over some subset of copies.
In the next step we use Lemma 12 to note that the map can always be written as a GI operation acting on just one copy of :
Combining the aforementioned arguments, we conclude that for any there exists a GI operation such that
In the final step we will use the asymptotic continuity of the relative entropy of coherence (see Lemma 12 in Winter and Yang (2016)). It implies that
with the binary entropy and the fixed dimension of the Hilbert space. Using the fact that the bound on the right-hand side of this inequality is continuous for and that it goes to zero as , we can say that for any there exists some GI operation such that
On the other hand, the assumption implies that there exists some such that
Recalling that the relative entropy of coherence is a genuine coherence monotone, i.e., , we arrive at the following result:
for some and any GI operation . This is a contradiction to Eq. (42), and the proof of the theorem is complete. ∎
From the above theorem it follows that it is not possible to distill the state from any non-equivalent single-qubit state , since .
In the final part of this section we address the impossibility of dilution. Interestingly, it turns out that diluting less copies of a state into more copies of another is generically impossible independently of which state is picked as a golden unit.
Given any two coherent states and of the same dimensionality it is not possible to dilute to at any rate via GI operations.
If dilution at rate (i. e. leaving aside one-copy deterministic transformations when possible) was possible, this would require that there existed integers such that
Notice that the presence of some junk incoherent part is indispensable as GI operations cannot increase the dimensionality. Since the trace distance cannot increase by quantum operations, by tracing out the particles in the system the above equation requires in particular that
Now, Lemma 12 implies that for some GI map . Hence, it must hold that
However, the junk part is incoherent and, therefore, so must be . Any coherent state is bounded away from the set of incoherent states and, thus, the above inequality cannot hold for any coherent state . ∎
Thus, dilution with rate from a more useful qudit state into a less useful qudit state is impossible by GI operations independently of the measure of coherence used.
It is known that quantum resource theories where the free operations are maximal (resource-non-generating maps in the asymptotic limit) are asymptotically reversible and the optimal rate is given by the regularized relative entropy Brandão and Gour (2015). GI operations are more contrived and represent the opposite extreme: non-trivial forms of distillation and dilution are impossible.
ii.6 Relation to entanglement theory of maximally correlated states
Recently, several authors conjectured Winter and Yang (2016); Streltsov et al. (2015a); Chitambar et al. (2016) that the resource theory of coherence as introduced by Baumgratz et al. is equivalent to the resource theory of entanglement, if the latter is restricted to the set of maximally correlated states. Given a state , we can always associate with it a bipartite maximally correlated state . This connection has led to several important results, e.g., the distillable coherence, the coherence cost, and the coherence of assistance of are all equal to the corresponding entanglement equivalent of Winter and Yang (2016); Chitambar et al. (2016). Moreover, it has been conjectured Chitambar et al. (2016) that for any two states and related via some incoherent operation such that , the maximally correlated states and are related via some LOCC operation: . As we will see in the following theorem, this conjecture is true if genuinely incoherent operations are considered.
Given two states and related via a genuinely incoherent operation such that , there always exists an LOCC operation relating the corresponding maximally correlated states: .
We will prove this statement by showing that can be chosen as , i.e., the genuinely incoherent operation acting on Alice’s subsystem. For this, we first write explicitly:
with and .
We will now apply the same operation on Alice’s subsystem of the maximally correlated state :
This completes the proof of the theorem. ∎
This theorem lifts the relation between coherence and entanglement to a new level. For an arbitrary coherence-based task involving genuinely incoherent operations, we can immediately make statements about the corresponding entanglement-based task involving LOCC operations.
Iii Fully incoherent operations
iii.1 General concept
As discussed in Sec. II.1, one of the reasons to consider GI operations was to rule out any form of hidden coherence in the free operations of a resource theory of coherence. However, we have seen in Sec. II.5 that state manipulation under GI operations might be too limited. This leads to think whether one could consider a larger set of allowed operations still having the property that every Kraus representation is incoherent but that could allow for a richer structure for state manipulation.
We recall that the incoherent operations considered in the resource theory of coherence introduced by Baumgratz et al. in Baumgratz et al. (2014) are given by Kraus operators such that for every incoherent state , is (up to normalization) an incoherent state as well . It will be relevant in the following to notice that such are characterized by having at most one non-zero entry in every column Du et al. (2015). As we used already in Sec. II.1, the fact that the Kraus operators of a different Kraus representation of some incoherent operation might not be incoherent can be easily seen using Theorem 1. On the contrary, with this theorem it is straightforward to check that GI maps are incoherent (in fact, even diagonal) in every Kraus representation. As discussed above, it comes as a natural question whether GI maps constitute the most general class of operations having this property. Interestingly, as we will show in the following, the answer is no: there exist operations which are not GI but still incoherent in every Kraus representation. A simple example for such an operation is the erasing map, which puts every input state onto the state . This operation is incoherent in every Kraus representation because it must hold that for every Kraus operator. On the other hand, the operation is clearly not GI because any incoherent state that is not does not remain invariant under this map. We will call this class of operations which are incoherent in every Kraus representation fully incoherent (FI).
One might wonder then what are the properties of the class of FI maps and which differences it has with the class of GI maps. In particular, we want to compare these sets of operations in the task of state transformation to see whether FI induces a richer structure. For this, we will first provide a full characterization of FI operations in the following theorem.
A quantum operation is FI if and only if all Kraus operators are incoherent and have the same form.
Before we prove the theorem some remarks are in place. The requirement that all Kraus operators have the same form means that their nonzero entries are all at the same position (i. e. whenever there is a non-zero entry in a given column, it must occur at the same row for every Kraus operator). As an example, according to the theorem any single-qubit quantum operation defined by the Kraus operators
is fully incoherent, since both Kraus operators are incoherent and have the same form. The completeness condition puts constraints on the complex parameters: and . Note that – according to the theorem – this map is FI, but it is not GI since the Kraus operators are not diagonal. Indeed, this is exactly the erasing map which maps every state onto . We will now provide the proof of Theorem 16.
That maps with this property are FI is immediate. If the Kraus operators in one representation are all incoherent and have a particular given form, then, by Theorem 1, so does every Kraus representation since Eq. (3) preserves this structure. Hence, every Kraus representation is incoherent.
Thus, to complete the proof we only need to see that any map which has one (incoherent) Kraus representation in which not all operators are of the same form cannot be FI, i. e. it must then admit another Kraus representation which is not incoherent. For such, we can take without loss of generality that
where denotes the -th column of the matrix and an arbitrary non-zero number. Moreover, we define two unitary matrices and :
Since is unitary the set of Kraus operators constructed using Eq. (3) is another Kraus representation of the map given by . However, we find that
and the representation given by is therefore not incoherent. ∎
From Theorem 16 it is clear that GI maps are contained in the class of FI maps since every Kraus operator in every Kraus representation is diagonal, hence fulfilling the condition of Theorem 16. As also noted above Theorem 16, there exist operations which are FI but not GI, and the erasing map is one example. Another instance of FI maps, taking for example maps on , are those for which the Kraus operators are given by
A property fulfilled by FI maps is that these operations allow to prepare any pure incoherent state from an arbitrary input state. For that one would use the corresponding erasure map with the property for every state . It can be easily seen that any such map is always FI for any choice of incoherent state . This means that, contrary to the case of GI operations, in this setting pure incoherent states are indeed free states, i.e. they can be prepared with the free operations.
Another feature of FI maps is that any incoherent unitary transformation can be implemented. Thus, elements of the incoherent basis can be permuted and the coherence set is no longer meaningful. Hence, the coherence rank takes the relevant role instead, similarly to state manipulation under incoherent operations. Thus, in this case coherence is regarded as a speakable resource.
On the other hand, a striking feature of FI maps is that they constitute a non-convex set. More precisely, given two FI operations and , their convex combination
is not always fully incoherent. This can be demonstrated with the following single-qubit operations:
with the Pauli matrices
While both operations and are fully incoherent, their convex combination in Eq. (55) is not fully incoherent for . This can be seen by noting that the Kraus operators of are given by and . Since these Kraus operators do not have the same form for , by Theorem 16 the operation cannot be fully incoherent.
This non-convexity is, thus, a consequence of the fact that FI maps are characterized by a property of the set of implemented Kraus operators and not of each individual operator as it is the case for incoherent or GI operations. In practice, this means that if one considers state manipulation under FI maps, it turns out that two particular operations might be free but not to implement each of them with a given probability. In particular, this implies that, although pure incoherent states can be prepared with the free operations as pointed out above, this result does not need to extend to mixed incoherent states despite the fact that they constitute free states as well. This is because it is not allowed to mix different FI fixed-output maps. Actually, it can be checked that it is not the case that every state can be transformed to any mixed incoherent state by some FI map. The interested reader is referred to Appendix C for a particular example. This construction relies on an observation on the structure of FI maps used in Theorem 17 below.
iii.2 Permutations as basic FI operations
As mentioned above, any genuinely incoherent operation is also fully incoherent. Here, we will consider another important class of FI operations, which we call permutations. A permutation with is a unitary which interchanges the states and , and preserves all states for :
The above definition involves the permutation of only two states and . In the following, we will also consider more general permutations with more than two elements. We will denote an arbitrary general permutation by . Any such permutation can be decomposed as a product of permutations of only two states. Notice that any such corresponds to an FI unitary transformation.
iii.3 State manipulation under FI operations
iii.3.1 Pure state deterministic transformations
Given the limitations of GI operations, in this section we study the potential for deterministic state manipulation if the allowed set of operations is given by FI maps. Interestingly, we will see in the following that – contrary to the case of GI maps – transformations among pure states are possible. However, these are rather limited as shown in the following theorem.
A deterministic FI transformation from to is possible only if . Moreover, for