Of Local Operations and Physical Wires

In this work (multipartite) entanglement, discord and coherence are unified as different aspects of a single underlying resource theory defined through simple and operationally meaningful elemental operations. This is achieved by revisiting the resource theory defining entanglement, Local Operations and Classical Communication (LOCC), placing the focus on the underlying quantum nature of the communication channels. Taking the natural elemental operations in the resulting generalization of LOCC yields a resource theory that singles out coherence in the wire connecting the spatially separated systems as an operationally useful resource. The approach naturally allows to consider a reduced setting as well, namely the one with only the wire connected to a single quantum system, which leads to discord-like resources. The general form of free operations in this latter setting is derived and presented as a closed form. We discuss in what sense the present approach defines a resource theory of quantum discord and in which situations such an interpretation is sound -- and why in general discord is not a resource. This unified and operationally meaningful approach makes transparent many features of entanglement that in LOCC might seem surprising, such as the possibility to use a particle to entangle two parties, without it ever being entangled with either of them, or that there exist different forms of multipartite entanglement.

In this work (multipartite) entanglement, discord and coherence are unified as different aspects of a single underlying resource theory defined through simple and operationally meaningful elemental operations. This is achieved by revisiting the resource theory defining entanglement, Local Operations and Classical Communication (LOCC), placing the focus on the underlying quantum nature of the communication channels. Taking the natural elemental operations in the resulting generalization of LOCC yields a resource theory that singles out coherence in the wire connecting the spatially separated systems as an operationally useful resource. The approach naturally allows to consider a reduced setting as well, namely the one with only the wire connected to a single quantum system, which leads to discord-like resources. The general form of free operations in this latter setting is derived and presented as a closed form. We discuss in what sense the present approach defines a resource theory of quantum discord and in which situations such an interpretation is sound -and why in general discord is not a resource. This unified and operationally meaningful approach makes transparent many features of entanglement that in LOCC might seem surprising, such as the possibility to use a particle to entangle two parties, without it ever being entangled with either of them, or that there exist different forms of multipartite entanglement.
One of the oldest questions in the field of quantum mechanics asks in what consists the difference between classical and quantum states. While certainly it is hard to compare frameworks that are so fundamentally different in nature,the main qualitative difference on a formal level is that quantum mechanics deals with probability amplitudes instead of probabilities. As a consequence, one of the main predictions of quantum mechanics is that physical systems can exhibit coherent superpositions of those states associated to sharply defined values of the observables [1] which can for instance be observed in interference experiments. It is mainly this difference that prevents quantum theory from being explicable by a deterministic hidden variable model with variables that are local [2][3][4][5][6] or non-contextual [7][8][9]. The question then becomes how one can understand this difference in detail and furthermore how one can quantify it. This is relevant in its own right and also to give solid foundations to the debates about how quantum mechanical observed coherences in biological systems [10] are, or to objectively compare different platforms for quantum computers by measuring how much more resources they provide than their classical counterparts.
A particularly transparent approach is given by the theory of local operations and classical communication (LOCC) (see [11,12], for reviews), which incorporates the idea of the impossibility of creating non-local superpositions (entanglement) if one has two distant parties that can only communicate classically, akin to Bell's argument for the non-classicality of quantum physics [3] (see section I).
However, entanglement is not the only non-classical feature of quantum mechanics. There are other forms of superposition that can also give advantages over classical states. An instance in which this became apparent is the protocol of deterministic quantum computing with one qubit (DQC1) that can outperform any known classical algorithms even if no bipartite entanglement is present [13][14][15][16][17]. It was argued that a property denoted as discord [18][19][20] would be the resource responsible for this operational advantage. While discord is Q A Q B cc? an interesting measure, it is hard to argue that it is a resource of non-classicality, since it can be created already by mixing discord-free states (in fact product states) and mixing states amounts to forgetting information -a task that is easily accomplished classically [18]. Some advances in understanding the resources in DQC1 have recently been made using a different form of non-classicality, called coherence, which is exhibited by superpositions of states in a fixed orthonormal basis, whose elements and their statistical mixtures are the incoherent states [21][22][23][24]. Based on this theory, complementing recently studied connections between entanglement and coherence [25][26][27][28][29][30][31][32][33][34][35][36][37][38] it was shown that the precision of the DQC1 protocol is a function of the coherence of the qubit one uses as a control and that any state with some property called basis depended discord is a resource in this setting [39,40].
The standard (basis independent) discord is recovered when a minimization over this quantity is considered [40]. Ironically, in the theories used in these papers coherent states and entangled states are interconvertible resources and optimal instances of basis depended discord, making this more general variant of superposition in some sense equivalent to entanglement again. Even more generally, there is an operation that maps any superposition of a given set of linearly independent states (not necessarily orthogonal) to entangled states and non-superposition states to separable states [41][42][43]. This means that while entanglement does not seem to be the most generic non-classical resource, as it does not easily encompass all forms of superposition that seem useful for quantum tasks, the resources that enable to produce entanglement might well be. Seemingly, the ability to do something truly quantum entails the possibility to produce entanglement. It is this "universal character" [44] of entanglement, that motivates the present paper.
Starting from LOCC, we treat the wire needed to do classical communication as a full physical (quantum) system, which can in principle realize states with some degree of coherence. We will show how the flow of this coherence can be used to create entanglement shared among the ends of the wire (see Fig. 1). We find a set of natural elemental operations on system and wires -designated free in this setting -that are equivalent to LOCC when wires are in incoherent states but, if not, allow to convert coherence into entanglement between the systems connected by the wire. In this setting, coherence in the wires is exactly as useful as entanglement between the two quantum systems, similarly as one might expect from a similar relationship between different forms of quantum cryptography (see e.g. [45,46]). This setting thus explains the recently studied relation between these different resources [25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40] as the interplay of different facets of the same resource theory. Additionally, the present framework reduces to an effective theory of basis dependent quantum discord in the case of a wire connected with just one quantum system. We will argue in what sense one can see it as a theoretical framework of quantum discord, and also discuss in some detail whyeven so -discord cannot be called a resource. Indeed it even makes sense to focus on the effective theory on the wire alone, yielding a theory of coherence which we show to be very similar -though not identical -to the theory of coherence defined in [22], but being motivated operationally instead of abstractly. Finally we generalize the setting to the multipartite case and show how in the tripartite case the non-equivalence of the W and the GHZ state (see [47]) can be understood operationally. All proofs are in the appendix.

I. LOCAL OPERATIONS AND PHYSICAL WIRES
As explained in the introduction, we are aiming at better understanding and quantifying non-classicality. The tool of choice to develop this understanding are resource theories, since they provide a systematic guide for analysing situations where one wants to find and/or quantify the properties that can be useful for some tasks.
Abstractly, and glossing over details, to get a resource theory one puts a meaningful, but artificial restriction to what operations are allowed (free), within a given framework. The restriction should be chosen such that the connection to the property one wants to focus on is as clear as possible (which does not need to coincide with the distinction between "easy" and "hard"). There may be some preparations that are free op-erations and accordingly some states that are free. Since the states that are not free cannot be prepared by free operations, in some cases they might help to do an operation that otherwise would not be free, these are the resource states. There are more useful states and less useful states (if from one state one can reach another one with free operations the first is more useful, since it can be used for anything the second can be), imposing a partial order on the states. A measure for the resource can therefore only be meaningful if it is monotonous under the free operations, restricting strongly possible candidates.
One of the most successful resource theories in quantum information, and the starting point of our considerations, is the theory of local operations and classical communication (LOCC). This theory aims to capture the idea of EPR and Bell that in classical physics it is not possible to reproduce the effect of having non-local superpositions of states [2,3]. LOCC can be described by its elemental operations consisting of arbitrary local quantum operations on one system, postselection and classical broadcasting of the result. Any LOCC operation is a concatenation of such operations (potentially depending on the results of the previous ones). Here we want to treat the broadcasting as an operation using a physical wire, instead of how it is usually done as an implicit exchange of classical information. The broadcasting is then simply given by forwarding the state of the wire as an ancilla to the party in question.
In this way, it makes sense to talk about the state of the communication channel as a quantum state. The standard LOCC theory is recovered by assuming that the state of the wire is forced to be incoherent, i.e, ρ wires = i p(i)|i i|, and in a product state with both parties, meaning that the probability distribution is encoded in the diagonals. Note that while the basis one uses to encode a probability distribution in the wire is in principle arbitrary, one needs to fix it in advance; to be precise, while it may change in time, this change must not depend on the measurement outcomes of the protocol, but must be defined at the beginning of the protocol. Henceforth we will call this basis incoherent and denote it by Z. With this definition it also makes sense to allow some classical processing in the wire as a free operation on this extended theory, that is, permutations of these basis states. As an example of why it is important to fix the basis, and therefore which states are incoherent, in advance to get a fair description of the role coherence plays, let us consider the BB84 protocol [48].
The goal of BB84 is to distribute random keys in a safe way. This is achieved by sending qubits through a quantum channel where each of them is encoded by Alice in one of two possible non-commuting bases depending on the outcome of a random measurement. On the other side, Bob measures the qubit in a random basis. After repeating this many times, Alice and Bob publicly compare the bases chosen and keep only the data of the measurements that were done in the same bases. They can then compare a fraction of the remaining data to be certain (in the asymptotic limit) that nobody interfered. A naive approach to describe this protocol is that, as on each round the state of the qubit sent is diagonal in some basis, then the full protocol is classical. Why does this algorithm beat classical key distribution? The crucial departure from classical physics of this protocol is the random choice of bases and the fact that the security of the algorithm relies on the fact that the choice of basis in which the qubits are diagonal in is unknown to a possible eavesdropper Eve. Therefore, to analyse the security of this protocol, we need to take the perspective of Eve. For her, whatever basis she assume to be incoherent, there will be some states sent by Alice that will be coherent, for a long enough sequence. That is because she does not know the outcomes of Alice measurements that defined the encoding bases. For this reason a fair description of what Eve can do needs to assume that she chooses the basis before the protocol starts.
As noted in the introduction, we only need one quantum system Q together with a wire W to make sense of this theory. We start by describing this case in some detail, before coming back to the case with multiple parties Q = ⊗ i Q i and wires W = ⊗ j W j . For simplicity, to change the phases of the basis states is also assumed to be a free operation -but as shown below this does not significantly alter the theory. We assume that both Q and W are finite dimensional systems, but don't keep their dimensions fixed (see footnote [49] for more details). The free operations in this case consist of iteratively applying the following elemental Kraus operations (which should form a CPTP map on W ⊗ Q, that is α K α † K α = 1) and postselecting (for a neater notation we only write out the spaces on which the Kraus operations act non-trivially, on all others the identity operation is assumed): 3. Observed quantum operations: Any generalized measurements on Q, encoding its outcomes as an incoherent state of an ancillary subsystem W a of W, |α W a ⊗ F α Q , for Kraus operators F α Q acting on Q. 4. Classical to quantum forwarding: Transfer a subsystem W s of W to a subsystem Q t of Q: j j| W s ⊗ | j Q t .
For simplicity, here we assume the incoherent basis Z to be the same for all times, but one can easily get the more general setting by replacing the free operations K α by (U W (t, t 0 ) ⊗ 1 Q )K α , where U W (t, t 0 ) defines the (necessarily predefined, see above) change of basis for the time-step t 0 → t. Note that this includes replacing the identity by (U W (t, t 0 )⊗1 Q ) and that the identity operation itself is not a free operation any more. Sticking to this rule the results stay unchanged, because concatenating free operations on t 0 , t 1 and t 1 , t 2 yields free operations on t 0 , t 2 . This is relevant if one wants to change the basis one calls classical in time, as is usually the case in settings where discord is thought to be a meaningful measure. We will come back to this in subsection I A.
Note also that we treat encoding and decoding asymmetrically. The rational we employ here is that it is hard to encode quantum information in a wire, but if coherence is provided and it is possible to sustain, transport and control it, you may very well also be able to use it. That is why we only allow to encode classical information in the wire, while any state can be retrieved. We call the set of free operations in this theory of local operations and physical wires LOP(W ← Q) = LOP, were here we use the symbols ← and → for classical encoding, while and are used for transferring a quantum system. Items 3. and 4. are depicted in Fig. 2. As promised, we a) b) Figure 2. The picture depicts items 3. and 4. of the elemental operations defining LOP. Item 3., depicted in sub-figure a), means that to do operations on the quantum state, measure what happens and use the incoherent basis of the wire to encode the observation is considered to be a classical operation on the wire and therefore free. Item 4, depicted in sub-figure b), means that the wire is effectively treated as a quantum system. This means that it can interact with the quantum system in a quantum way. However if the wire is an extended object it might be hard to change its state in a controlled way by this interaction. Therefore it is considered a free operation to forward the state of a subsystem of the wire to the quantum side, but not to alter the state of the wire in a quantum way.
first show that the phases are not really relevant (remember that all proofs can be found in the appendix): Proposition 1. Any operation in LOP can be done with arbitrarily high probability of success by a combination of permutations, observed quantum operations and classical to quantum forwarding.
It is easy to see that the set of free states that can be prepared (probabilistically or not) by these operations is given by the so-called incoherent-quantum states, CQ (n) Z = {ρ | ρ = m p m |m m| W ⊗ σ mQ , |m ∈ Z} [25,39]. Therefore LOP is a subset of the incoherent-quantum operations (IQO), that is given by any operations that map incoherent-quantum states to incoherent-quantum states, even after post-selection [39]. It is not so easy to see how strict the inclusion is. We will discuss this question later in Prop. 6, once we have gathered more insight into the theory.
To get a clearer idea of the operations at hand, we will need to find a more compact form of the operations belonging to LOP. To this end and of independent interest it is helpful to know bijections that are elements of LOP. Knowing bijections in a resource theory is useful because they define equivalence classes on states, in that all states that can be reached by bijections can be freely interconverted, making them equivalent resources (as for any task one state can be used for, any state that can be mapped into it is at least as useful). A trivial bijection in LOP is given by any unitary on Q, which means that any measure of the theory necessarily has to be invariant under basis changes of Q. The following lemma establishes such a bijection between the states on the wire W and maximally correlated states on W ⊗ Q (also see [25,30]). We are now ready to give an equivalent characterization of the operations in LOP, which has the advantage of being an explicitly finite concatenation of maps of one fixed form. On one hand this might help to find a minimal number of necessary Kraus operators [50], on the other, a priori it is not obvious that such a simplification exists, since for LOCC, which is a very similar theory, the number of rounds needed for a transformation is unbounded [51]. The proposition states that one can write any element of LOP as a concatenation of N LOP maps (and N is bounded by the Hilbert space dimension of the wire) that are composed of Kraus operators K α j j having a specific functional form. Due to the possibility of post-selection, one has to consider different paths given by the outcomes of the generalized measurements, that is, which Kraus operators K α j j have been measured; after t maps (labelled from 1 to t), these paths are denoted by α t , and α t+1 denotes the possible outcomes of the map labelled by t + 1. Both the maps and the length of the protocol may vary depending on the outcomes of previous measurements, but for any path α, the total length N = N( α) can be restricted to be less or equal to the dimension of the Hilbert space of the wire. Fig. 3 depicts a generic example for a protocol which acts on an initially three-dimensional wire and a quantum system. Figure 3. Depicted is a diagram of a possible protocol defining an LOP map as described in Prop. 3 for the special case of an input state that is three-dimensional on the wire. All the arrows starting from the same node represent a CPTP map. Each arrow represents a sub-map with given outcome (labelled from 1 on the left to n on the right). The dimension of the maximal support of the input or output states on the wire is stated on the left of the diagram. The branches can be grouped into four different types, where the types differ by how the dimension of the maximal support of the state on the wire (given wlog by r( α t−1 ), after applying t maps with the outcomes α t ), changes during the protocol. Any possible branch type is depicted at least once. In this picture the respective groups of outcomes for the four branch types are (we only name the outcome vectors up to the point that the group is defined): {(1), (2)}, {(3), (4), (5)}, {(6, 1)} and {(6, 2), (7)}. See the Appendix for more details on the branch types. The highlighted branch is given by Λ 2 (2) • Λ 2 1 , that is, the CPTP map Λ 2 (2) = Λ 1 2 (2) + Λ 2 2 (2) (composed by two sub-maps), after getting the outcome 2 in the first map. The length of the protocol N is 2 for these outcomes, while it is 3 for e.g. the outcomes (7,2). In principle the dimension of the support of the outcome state on the wire after applying an LOP map is arbitrary (though for the depicted protocol it is at most 3 × 3 + 1 × 2 + 7 × 1 = 18). Proposition 3. Let Λ be a CPTP map acting on W ⊗ Q. Λ is an LOP operation exactly if it can be written as a finite sequence of maps is the length of the protocol, possibly depending on the previous outcomes α t = (α 1 , . . . , α t ), with σ α t being an injective map to the positive labels of a new incoherent basis (see footnote [49]) and E α t t α t−1 , i being an arbitrary operator acting on Q, potentially depending on previous outcomes and controlled by the populations of the wire and 1 ≤ r( α t ) < r( α t−1 ) for any t ∈ {1, . . . , N − 1} and r(0) = dim(W).
The above proposition is useful to connect other theories that have been discussed in the literature with the one presented here. From Prop. 3 it follows that destructive measurements on any subsystem W s of W are free (destructive measurements being a set of Kraus operators mapping to a one-dimensional Hilbert space, which consists of only one, trivially incoherent, state). This can also be more directly seen, as one can forward the subsystem W s from the wire to the quantum side and perform the measurement there. Note that one can understand any Positive Operator Valued Measure (POVM) as a destructive measurement, since one is not interested in the outcome [52], which can be stated as: any POVM can be implemented inside LOP. Also note that any special incoherent operation SIO can be performed on W. Special incoherent operations have Kraus operators which commute with dephasing, forcing them to have the form F α = i c α (i)|σ α (i) i|, for permutations σ α and complex c α (i) [23,33]. Even more restrictive, physical incoherent operations PIO are special incoherent operations where the permutations are fixed for all the Kraus operators (σ α = σ). Both are obviously special cases of the form given in Prop. 3. Indeed, Proposition 4 (SIO, PIO and LOP). Let Λ be a CPTP map acting on W ⊗ Q. Λ is an LOP operation exactly if it can be written as a sequence of maps One can equivalently replace item (a) by "a special incoherent operation on I".
Noting that items (a), (c) and (d) together form an effective theory of the special incoherent operations by restricting the theory to the effect on the wire [33], we find that the present approach (apart from giving a completely different motivation) only differs by allowing measurements. This difference however, is crucial; that special incoherent operations commute with dephasing means that apart from being unable to create coherences, they affect populations only depending on populations, making them incoherent in a very strict sense. So strict in fact that coherences don't have any effect that can be measured by free operations, meaning that coherences are useless for any observable task in that resource theory and there are thus no resource states in the theory. In contrast, ideally (but maybe not always necessary) one would find that having enough non-free states at hand removes completely any restrictions of the theory, showing that this is really the resource that helps overcome the restriction and giving sense to questions of how much resources one needs for a given task. Exactly this we find in the present approach: if supplemented by enough coherent ancillary states, LOP can be used to achieve any desired quantum operation.
We conclude that coherence is a meaningful resource in LOP. Furthermore, from Lem. 2 it follows that the resource of coherence d i=1 1 √ d |i W can be reversibly converted into entanglement between the wire and the system Q, d i=1 1 √ d |ii WQ which hence is an equivalent resource of the theory (also see [25,40]).
The above properties resemble much those found for incoherent-quantum operations (IQO) [39] and as stated above, the free operations of this theory contain LOP. But how strict is this inclusion? In the following proposition, we show that in general incoherent-quantum operations cannot be performed by LOP operations. However, Proposition 6 (LOP and IQO). Be Λ an incoherent-quantum operation on W 1 ⊗ Q, which is exactly the case if it is CPTP and has a Kraus decomposition with Kraus operators of the for some functions f α acting on the labels of the incoherent basis and some operators E α (i) acting on Q. Let d := dim(W 1 ). If d = 2, Λ ∈ LOP. For d ≥ 3 LOP IQO, but there is a stochastic implementation of the map in LOP with a success rate of at least 1/d; i.e. there is an Meaning that even though the theories LOP and IQO differ, they are very similar. Equally close are the theories they induce on the wire (by tracing out Q at the end). We conclude that the effective theory we obtain by restricting LOP to the wire -even though being operationally motivatedis in many aspects similar to the abstractly motivated theory of coherence introduced in [22], were the free operations are any that can be decomposed into Kraus operators that cannot create coherence. Specifically, the free states are the same, the resource states are the same and the same amount of resources is needed to remove all restrictions of the theories, in both theories any destructive measurements can be performed, for qubits the theories are equivalent and for higher dimensions the theories are stochastically equivalent. Furthermore, for a given , having enough copies of any state which is not incoherent-quantum, LOP allows to prepare a maximally coherent state with fidelity f > 1 − and probability p > 1 − (see Appendix D). Therefore, since maximally coherent states can be used as a resource to implement any quantum operation (Thm. 5), all not incoherent-quantum states are resources for non-classicality of the wire in the present setting. This is true independently of how one defines nonclassicality, as long as it does not include full quantum theory. More specifically, since for any free operation the state remains incoherent-quantum if it was at the beginning, one can make a stochastic model for any free operation, which correctly describes the change of the reduce state of the wire, as long as at the beginning the full state is incoherent quantum. Explicitly, one just needs to replace the elemental operations, that is, permutations, tracing out and encoding a classical outcome of a measurement by their obvious non-contextual (and local) classical counterparts. For the measurement this means encoding the classical probabilities as defined by the Born rule and the current state of the external quantum system to measure, but this can only depend on some previous populations of the wire: its previous classical states. Even so, having enough copies of any state which is not incoherent-quantum will allow to do any operation on the full system, including those that have no non-contextual (or non-local, see the next section about entanglement) classical counterparts.

A. Quantum Discord
Before continuing and applying the tools we have just developed to multipartite entanglement, let us stop a moment and draw the connection to quantum discord. In a sense to be made precise now, LOP can be seen as a theory of quantum discord. As the free states of LOP are the incoherent-quantum states defined above, the theory can be seen as quantifying how much a state differs from an incoherent-quantum state, that is, by definition, how much basis-dependent quantum discord it has (called measurement dependent in [20]). Take any measure of basis-dependent discord which can at the same time be normalized such that it yields 1 for a singlet state. Assume also that the measure is upper bounded by the entropy of the local state of the wire. Then, by minimizing this quantity over all possible bases we obtain a measure of discord satisfying the properties stated in section II.A.1. of the review [20]. While starting from coherence theory it seems strange to do this optimization, since one starts by choosing a natural basis as incoherent and only then the notion of coherence makes sense, the present approach is much nearer to one original setting in which quantum discord was introduced [18]. Indeed the theory of coherence of the wire in the present approach is an effective theory and the incoherent basis is arbitrary, it just depends on what one assumes to be the default basis to encode the information in. This allows to interpret quantum discord as a natural lower bound that defines the non-classical resource one has to assume at least, independently of the basis chosen. However, this is not the same as to say that quantum discord is a resource of non-classicality. What happens is, that if one has two quantum states that have zero discord, there still might be no basis for which both have zero basis dependent discord simultaneously and it is thus not surprising that their mixture might not be discord-free (also see the discussion in [18] and Fig. 2 in [40]). This means that in the present framework we reproduce the interpretation of quantum discord as how non-classical one has to assume the state of the wire at least, when seen as a part of the full system W ⊗ Q (similarly as was the motivation for its introduction in [18])if one considers only one state (even at different times, since, as noted above, one can change the basis in time). But having more than one possible state (for instance if one does not consider the evolution of one given state, but a protocol defined for any input state, such as in quantum cryptography), it is necessary to fix one basis (possibly a different one at different times) for all the states one considers, to have a meaningful quantity. It is in this sense that the present approach defines a resource theory of quantum discord, even though discord is not (and should not be!) a monotone and thus a measure of the resource theory, as discord is not a resource, it is an indicator of non-classicality: if a state has non-zero discord, this means that for whatever basis one chooses as incoherent, the state is a resource for non-classicality with the present framework.

II. COHERENCE COST OF ENTANGLEMENT
The aim of this section is to better understand multipartite entanglement by looking at the coherence needed to generate it. This is a natural approach as entanglement is always a result of coherent interactions happened in the past. In the case of bipartite entanglement the theory in our setting is the one depicted in Fig. 1, which we denote by LOP(Q 1 → W ← Q 2 ) and consists of two quantum systems Q 1 and Q 2 connected by a wire W. The elemental free operations are the free operations of LOP(W ← Q 1 ) together with the ones in LOP(W ← Q 2 ), meaning that any operation is given by the composition of these operations, possibly with post-selection. In general, we will call LOP(Q 1 → W 1 ← Q 2 → W 2 ← . . .) the set of operations consisting of concatenating operations in the corresponding sets LOP(W j ← Q j (or j+1) ). The notation is explained in Fig. 4 The minimal amount of coherence needed to create pure bipartite entanglement directly follows from Lem. 2 (also see [25]). One can produce a pure state on Q 1 ⊗ Q 2 which in its Schmidt-decomposition [52] is given by |ψ Q 1 ,Q 2 = i c i |ii Q 1 ,Q 2 from i c i |i W on W by applying Lem. 2 to get |ψ Q 1 ,W and then using classical to quantum for-warding on W ⊗ Q 2 to get the wanted state. On the other hand one also sees that the production is optimal since one needs to have at least that amount of entanglement on the bipartite cut Q 1 | W ⊗ Q 2 (remember that entanglement and coherence are equivalent resources for LOP(Q 1 → W)). As shown below, the connection between coherence and entanglement in the bipartite case is even stronger; a maximally correlated state can be used as a resource for a LOCC transformation exactly if the equivalent coherent state can be used as a resource to do the transformation under LOP. This puts the connection between coherence and entanglement that has recently attracted a lot of attention on the level of resources and operations instead of measures [25][26][27][28][29][30][31][32][33][34][35][36][37][38][39]. It will be useful to introduce the notation W j = W j 1 ⊗W j 2 . . . and Q j = Q j 1 ⊗Q j 2 . . ., with the convention that the upper index labels the local systems and the lower their respective subsystems, whose number may vary. The full systems are referred to by Q = ⊗ j Q j , W = ⊗ j W j . The first wire connects the systems Q 1 and Q 2 the second connects Q 2 , Q 3 and Q 4 , resulting in the theory Q 4 )). Each party (and also the wires) might consist of more than one quantum system and changing the number of subsystems of each party is a free operation in LOP.
Theorem 7. Let η LOCC = i j r i, j |i j| Q 1 1 ⊗ |i j| Q 2 1 be a maximally correlated state (in arbitrary orthonormal local bases of Q 1 1 ⊗ Q 2 1 ) and η LOP = i j r i, j |i j| W be a corresponding state in the incoherent basis Z of W. If Λ is a CPTP map on Q 1 ⊗ Q 2 , then the following statements are equivalent: We now use the common definition that ρ O → σ means that there is a map Λ ∈ O, with Λ[ρ] = σ (for some space of superoperators O). The following corollary then follows directly by taking Λ in the theorem, to be the preparation of the state σ Q 1 ,Q 2 : Corollary 8. Let η LOCC = i j r i, j |i j| Q 1 1 ⊗ |i j| Q 2 1 be a maximally correlated state (in arbitrary orthonormal local bases) and η LOP = i j r i, j |i j| W be a corresponding state in the inco-  Figure 5. Two wiring schemes that are equivalent in the LOCC paradigm, but inequivalent according to LOP. For LOCC, i.e. for incoherent states of the wiring, every operation can be performed by acting over the extremes of each wire, without a direct interaction among wires. On the other hand, if wires are quantum, direct interaction among wires enlarge the set of possible inequivalent operations. As a result, the initial coherence necessary to prepare a multipartite entangled state depends on the wiring topology.
herent basis Z of W. Then: Having shown this very strong connection between coherence and bipartite entanglement, we move to the multipartite LOCC case. While in the bipartite case it is quite clear how one needs to explicitly implement the wires (as there is one clearly simplest way to connect two parties), in the multipartite case the situation is more complex. To be able to do any operation by using enough coherence, each party needs to be connected to all others (possibly indirectly over third parties). On the other hand one also does not need to have two parties connected in two different ways and avoiding to have double connections simplifies the theories. We use the short-hand notation LOP(W 1 ← (Q 1 , Q 2 , . . .)) for the case of one wire connecting different quantum systems (i.e. concatenating operations in the sets LOP(W 1 ← Q i )). For the example of three parties Q = A ⊗ B ⊗ C, we are left with the 3+1 theories of the two types depicted in figure 5; namely the three ways (A, B or C) and LOP(W ← (A, B, C)). In general we get that following the rules explained above, the possible generalization of Npartite LOCC are exactly given by the theories where σ is a permutation of the indexes denoting the local quantum systems. Of course one could also look at the union of all these theories, which has the advantage of providing a completely unified view, while having the disadvantage of being excessively complicated. What all these theories have in common is that they are generalizations of multipartite LOCC on the quantum side, that reduce to it if one starts with an incoherent state on W, which is made precise in the following remark.
In the bipartite case, this means that if the initial state of the wire is incoherent and the wire is in a product states with both parties, tracing out the wire at the end reduces the theory to LOCC. The reason is that the state of the wire can be copied and stored in a local register on both sides. Every operation above can then just be obtained by local operations and broadcasting of classical information, by updating the two local registers (just do the same permutations and phases on both registers, copy the information of measurements to both registers and trace out the corresponding part of a register on Bob's side if Alice's has been used in a quantum operation). We make this seemingly obvious statement precise, because intuition sometimes might be misleading; for instance it is possible to produce entanglement by sending a quantum particle back and forth that is never entangled with either of the two parties [53]. While in entanglement theory this is surprising, since one can entangle two parties without using entanglement to do so, with the present approach the same statement is very intuitive, as entanglement is simply not the only resource present, coherence and basis dependent discord are resources as well. In the present approach without having resources one cannot entangle two parties, while any amount of basis-dependent discord allows to do that (that is because, as noted above, one can distillate coherence from any resource and the statement then just follows from Thm. 7). The above remark makes it easy to prove that the coherence needed to create entanglement gives a bound on the possible entanglement conversions, namely: Theorem 10. Let ρ, σ be states on W. If ∃τ W , a state on W, Note that for any of these theories (independently of the wiring) the free states are given by: with Z ⊗ the product basis of the incoherent basis of each wire, and S EP(n) the set of n−partite separable states This implies that for any of these theories, with S (ρ σ) := Tr(ρ(log(ρ)−log(σ))) the relative entropy, defines an additive geometric monotone (see e.g. [40]). In general, the evaluation of this quantity is an NP-Complete problem [54]. However, due to the monotonicity of the relative entropy, we notice that being R Z ⊗ (Tr Q ρ) the relative entropy of coherence on the wires, and R E n (Tr W ρ) the relative entropy of entanglement [55][56][57]. Due to the additivity of the relative entropy, if either R Z ⊗ (Tr Q ρ) = 0 or R E n (Tr W ρ) = 0, Eq. (2) turns into an equality, providing a lower bound to the amount of initial coherence required to prepare an entangled state among the parties of Q [25,39].
As an example of how this perspective can be applied in the multipartite case let's revisit the case of the |W and of the |GHZ state [47] (here only the results are discussed, the protocols for the conversions can be found in the Appendix).
The relative entropy of entanglement of |GHZ n = |0 ⊗n −|1 ⊗n √ 2 is 1 [56] and indeed one can prepare the |GHZ n state by LOP(W ← (Q 1 , Q 2 , . . . , Q n )) from |0 +|1 √ 2 . For |W n = n−1 k=0 |0 ⊗k |1 |0 ⊗n−k−1 √ n the relative entropy of entanglement is (n − 1) log 2 (n/(n − 1)) > 1∀n > 2 [57]. It is then a simple corollary of Thm. 10 that |GHZ n LOCC(Q 2 ,...,Q n ) |W n . The second thing to note is that on any bipartition the |GHZ n state is LOCC equivalent to |GHZ 2 , while the |W 3 state on any bipartition is LOCC equivalent to 1/ . Indeed one can in all three possible two-wire settings while the bipartite entanglement one can produce on the bipartition Q 1 , (Q 2 ⊗Q 3 ) is not enough to prepare |GHZ 3 : as on any bipartition one needs to prepare a fully entangled qubit and this is equivalent to a maximally coherent qubit, the state with minimal coherence to prepare |GHZ 3 is given by 1/2(|0 + |1 ) W 1 ⊗ (|0 + |1 ) W 2 , which is strictly more coherent on W 1 than 1/ Again as a corollary of Thm. 10, we have that |GHZ 3 . A strong indication that the resources in the different types of wirings correspond to differ-ent types of entanglement.

III. CONCLUSION
Recently there has been considerable interest in the connection between coherence, discord and entanglement [25][26][27][28][29][30][31][32][33][34][35][36][37][38][39]. But the connection was made on the level of quantifiers and measures. The current paper shows that by generalizing the fundamental theory of entanglement -LOCC, one obtains a connection between coherence, discord and entanglement that is even deeper, namely on the level of the operations themselves. In this sense the current approach defines a theory that lies at the root of entanglement and it seems natural to assume that it will be useful to assess the interplay between different resources of quantumness in complex settings, as is needed for instance if one wants to quantify the resources needed for quantum algorithms (see e.g. [58]). Moreover, while we exemplified here that the present approach can give a clear explanation of the difference between some very basic forms of multipartite entanglement by exemplifying that different forms are optimal in different settings, it remains an interesting open question whether it yields such an explicative power also in more general cases and how it connects to known structures in entanglement theory, such as explained in e.g. [59][60][61][62][63].
Furthermore, there is an ongoing discussion over what is the "right" theory of (speakable) coherence [33,36,[64][65][66][67]. While we do not claim to close this discussion (nor actually that it can be conclusively closed, as it always will depend on the setting one is interested in), we note that the effective resource theory of coherence emerging from our approach is to our knowledge the first which is built on operationally meaningful elemental operations while still having coherence as a resource in the sense of the name, meaning that enough ancillary coherence can completely lift the restrictions imposed by the resource theory. Proof. The operator B = i |i i| W ⊗ |i Q can be implemented by a sequence of maps that will be described in the following in terms of their Kraus operators. To this end we start with W = W 1 and Q = 1 and apply where the first map is a single outcome measurement of Q storing the outcome in an ancilla W 2 (observed quantum operation), the second is a permutation on W, and the third is the forwarding of the system W 2 to Q. Identifying W = W 1 , we get the desired operation.
For the converse, we first apply a measurement in the Fourier basis, followed by a correction of the phase on W.
|i i| W ⊗ j| Q . Obviously the map defined by these Kraus operators is an element of LOP and a left-inverse of B, as required.
Note that the basis on the quantum side can be chosen arbitrary in the above lemma. We continue with the proof of Prop 1.
Proposition (1). Any operation in LOP can be done with arbitrarily high probability of success by a combination of permutations, observed quantum operations and classical to quantum forwarding.
Proof. Let W have dimension d. The only thing to show is that indeed one can change the phases on W in the above framework. Let U = j e i φ( j) | j j| W be the wanted phase shift. We start by identifying Q = Q 1 . The protocol is: Else: redefine U = U • j e −2π i k j/d | j j| and restart with item 1.
The probability of success in each round is given by 1/d, independently of the initial state. After M iterations we therefore have a probability of success given by The next two pages are devoted to the proof of the closed form of LOP operations, which simplifies its use for both theoretical as well as practical purposes, for instance for the comparison of LOP to other resource theories.
Proposition (3). Let Λ be a CPTP map acting on W ⊗ Q. Λ is an LOP operation exactly if it can be written as a finite sequence of maps is the length of the protocol, possibly depending on the previous outcomes α t = (α 1 , . . . , α t ), with σ α t being an injective map to the positive labels of a new incoherent basis (see footnote [49]) and E α t t α t−1 , i being an arbitrary operator acting on Q, potentially depending on previous outcomes and controlled by the populations of the wire and 1 ≤ r( α t ) < r( α t−1 ) for any t ∈ {1, . . . , N − 1} and r(0) = dim(W).
One of the non-trivial results that we need to show in the proof of prop 3 is that classical to quantum forward can indeed be decomposed as described in the proposition. In the following lemma we slightly generalize this statement, as it does not significantly complicate the proof and the statement might be of independent interest. Lemma 11. Let Λ = α F α · F α † be a CPTP map acting on both W and Q with where f maps indices to indices. Then Λ admits a decomposition as in Prop. 3.
Proof. For simplicity, we will only prove that there is a finite protocol of the given form. That the length of the protocol is bounded by the dimension of W will be proven independently later in the proof of Prop 3. The function f in Eq. A2 can map different members of the incoherent basis to the same output state. The main idea of the proof is to first reorder the incoherent basis of the wire (using a bijection) in such a way, that we can then use a sequence of maps with Kraus operators of the form Eq. 1 to implement the same map and the same subselection possibilities as with the given operators A2. The main trick is to iteratively collapse the subspaces belonging to the pre-image of | f (i) .
Let us begin with the case that the image of f (i) is {1, . . . s} for a s ∈ N ≤ dim(W). Define W k = {i | f (i) = k} and a permutation σ 1 that maps the elements of W k to {1 + k−1 j=1 |W j |, . . . , k j=1 |W j |}. This implements the announced reordering of the incoherent basis of the wire and corresponds to a unitary Λ 1 given by K 1 = |σ 1 (i) i| W . Next we define r t = t + s−t j=1 |W j |, e.g. r 0 = dim(W), r 1 = 1 + s−1 j=1 |W j |, r s−1 = s − 1 + |W 1 | and r s = s. With this, we then define (for t ∈ {2, . . . s + 1}) where the permutation σ l ⊕ is defined by the mapping i → i+1, for i < l and l → 1. These Kraus operators are of the form given in Eq. 1.
From the CPTP condition for the Kraus operators defined in Eq. A2, we get that i, It is straightforward to see by induction that Now we redefine the map Λ s+1 → Tr Q 2 Λ s+1 , from which follows the statement in the case that the image of f (i) is {1, . . . s} for a s ∈ N ≤ dim(W). Now assume that the image of f is not {1, . . . s}. In this case, we can proceed in the same way and change the permutation of Λ s+1 (before we trace out Q 2 ) such that we implement the correct f .
We are now ready to give the proof of Prop. 3.
Proof of Prop. 3. We need to prove four statements: 1. Any elemental LOP map can be decomposed into an arbitrarily long sequence of CPTP maps represented by Kraus operators of the form given in Eq. 1.
2. Any CPTP map given by Kraus operators as in Eq. 1 can be decomposed as an LOP map.
3. Induction +1: The composition of two CPTP maps that can be decomposed into an arbitrary long sequence of CPTP maps represented by Kraus operators of the form given in Eq. 1 can again be decomposed into this form. This statement is trivial.
4. Any CPTP map that can be decomposed into an arbitrary long sequence of CPTP maps represented by Kraus operators of the form given in Eq. 1 can also be decomposed into such a sequence with r( α t+1 ) < r( α t ). From this follows that the choice N ≤ dim(W) is always possible, since r(0) is w.l.o.g equal to dim W.
The first statement is easy to see; permutations of the basis Z of W, diagonal unitaries on W, and observed quantum operations on Q all have the form of K α 1 1 . That classical to quantum forwarding has the form 1 is a direct corollary of Lem. 11.
To implement a map given by Kraus operators of the form 1 i ) by elemental operations, the first step is to implement the trivial observed quantum operation followed by a permutation on W given by Then we do a classical to quantum forwarding of system W 2 to an ancillary system Q 2 . Up to here we can summarize the concatenation of these operations by The next step is a quantum operation on Q defined by the Kraus operators which is a quantum operation exactly if the Kraus operators K α t t ( α t−1 ) form one. In total, we then have After a permutation on W 1 that implements σ α t , we obtain in total The last step is to use the operation B −1 W 1 ,Q 2 from Lem. 2 to get rid of Q 2 and we end up with the wanted operation.
As already mentioned, the third statement is trivial. The hard part is the fourth statement. Assume we have two arbitrary sets of Kraus operators K α t t , K α t−1 t−1 of the given form corresponding to Λ t and Λ t−1 . Then we distinguish two cases. In the first case, we show that we can find two new sets of Kraus operators L α t t , L α t−1 t−1 of the required form such that K α t t • K α t−1 t−1 = L α t t • L α t−1 t−1 , r( α t−2 ) remains the same and r( α t−1 ) < r( α t−2 ) for the two new sets (the place where one cuts the Hilbert space dimension of the wire in the step t (r( α t−1 )) depends on the previous outcomes, but not on the current one. That is why its index is t − 1 and not t). In the second case, one can replace the two CPTP maps by one CPTP map of the required form such that r(α t−2 ) remains unchanged.
Assume r( α t−1 ) ≥ r( α t−2 ) (otherwise there is nothing to show). First we split up the injection σ α t−1 (i) into a permutation and an order-preserving injection. Formally, we define the permutation η α t−1 on {1, . . . r( α t−2 )} and the injection a : and a(i) < a( j) for i < j. Then there is some l ≤ r( α t−2 ) such that with f (x, ·) denoting the function f (x, y) for fixed x, as a function of y and we use f (A) for a function f and a set A to denote the image of the set A under f .
We first consider the case l < r( α t−2 ). We first define with i ∈ {1, . . . r( α t−2 )}. By further defining the injection η α t (k) = σ α t (a(k)), we can finally define: and First we note that the map defined by L α t−1 t−1 ( α t−2 ) is CPTP exactly if the map defined by K α t−1 t−1 ( α t−2 ) is (as they only differ by a permutation at the end). Now we show that L α t t ( α t−1 ) forms a CPTP map as well. Remember that min[r( α t−1 ), a(·)] = a(min[l, ·]) and since a is a bijection on its image, min[r( α t ), a(i)]| min[r( α t ), a( j)] = min[l, i]| min[l, j] . Then where we used in the last line that η α t is a bijection as well.
Finally we need to check that we get indeed the right map- ). This follows from the equalities The case l = r( α t−1 ) can be handled similarly, just by noting that the action of cut l gets trivial and one can therefore express the concatenation of the two maps as a single map of the same form. This proves that one can chose r such, that: dim W ≥ r(0) > r( α 1 ) > . . . > r( α N−1 ) ≥ 1. It follows that it is always possible to have N ≤ dim W.
We can now use the above proposition to prove one direction of the connection between LOP and S IO operations given in proposition 4. One can equivalently replace item (a) by "a special incoherent operation on I".
Proof. We start the proof by noting that any destructive measurements on a subsystem of W can be done by LOP operations by classical to quantum forwarding of the subsystem in question to an ancilla in Q, Q 2 and then doing the measurement there. Special incoherent operations are those with a Kraus decomposition with Kraus operators of the form [23], which obviously is a special case of the form K 1 in Prop. 3 (i.e. Eq. 1), the same is true for control unitaries of the form (c). Next we note that PIO is a subset of SIO [64], so that any operation having a decomposition as in the proposition is an element of LOP.
For the converse we only need to show that we can do classical to quantum forwarding using only the operations (a)-(d). This goes by virtually the same protocol we already applied for the inverse part of Lem. 2. We first do a controlled unitary from the system W s in question to an ancillary system Q s prepared in the state |0 , to which we want to teleport, apply a measurement in the Fourier basis of W s (note that measurements in different fully coherent bases only differ by a diagonal unitary, which is an element of PIO, so we can assume w.l.o.g that the basis we can measure in is given by the Fourier basis), followed by a correction of the phase on Q s : |l l| Q s e −2π i kl/ dim(W s ) , resulting in the action i i| W s ⊗ |i Q s .
The most general set of meaningful operations, if one just has the restriction that one wants to keep the wire incoherent and classically correlated with the quantum system, are the operations that do not allow to generate states which are not incoherent-quantum from those that are. If one allows postselection this means that the same should be true for each Kraus operator defining the operation. This set, introduced in [39], is called IQO, for incoherent-quantum operations. Our approach by contrast, is operational, in the sense that we only allow some specific elemental operations which are meaningful. This has the advantage that it is more transparent and does not allow operations that could be not classical in a way that might not be obvious, but the disadvantage, that there might be some operations that we do not allow, that are still meaningful. Fortunately we can prove that the gap between the theory we propose and the completely abstractly defined maximal possible theory IQO is not that big. The details of this statement is the content of proposition 6.
Proposition (6). Be Λ an incoherent-quantum operation on W 1 ⊗ Q, which is exactly the case if it is CPTP and has a Kraus decomposition with Kraus operators of the form for some functions f α acting on the labels of the incoherent basis and some operators E α (i) acting on Q. Let d := dim(W 1 ). If d = 2, Λ ∈ LOP. For d ≥ 3 LOP IQO, but there is a stochastic implementation of the map in LOP with a success rate of at least 1/d; i.e. there is an operation Λ ∈ LOP( Proof. The form of the incoherent-quantum operations directly follows from applying each Kraus operators to a product state incoherent on W and requiring that it is still (up to normalization) a product state incoherent on W. The converse, namely that any Kraus operator of that form is incoherentquantum (that is preserves the set of incoherent-quantum states) is a trivial consequence of the convexity of the set of incoherent-quantum states.
The protocol for doing the operation given by the Kraus operators K α = i | f α (i) i| W ⊗E α (i) Q , by LOP operations with a probability of 1/d is given by (identifying W = W 1 and Q = Q 1 at the beginning and the end): In total the operation is given by the Kraus operators If the outcome is k = 0, k |i = 1/ √ d ∀i and the protocol is successful. Note that the probability for this is 1/d independently of the initial state the operation is applied to. If k 0 there is an i-dependent phase and in general the protocol fails (the information about i is lost, so that at this point there is no way to correct the phases).
Let's consider the case d = 2. We define the set R = {α | f α (1) = f α (2)} and R c its complement, separating the injective from the non-injective functions on W. The idea in the following is to first check (on Q) whether one has an injective or a non-injective case on W and then change the form on W accordingly, while inverting the check and applying the final operation on Q. Let wlog. α ∈ {1, . . . N}. We note that since the K α form a CPTP map we have in particular that α∈R E α (1) † B • E α (2) B = 0. For this reason it makes sense to define the operations E 0 (i) = α∈R E α (i) † B • E α (i) B and K 0 = i |i i| ⊗ E 0 (i) and it is easy to check that K α with α ∈ {0} ∪ R c , again forms a CPTP map. This map has the form of Eq. 1 and is therefore an element of LOP. For α ∈ R we then need a second step and we define the oper- is a diagonal matrix and D(i) −1 is its diagonal right-inverse on its support and vice versa). We also need the operation E 0 Here it is useful to note that E 0 (i) • E 0 (i) −1 = E 0 (i) −1 • E 0 (i) is the projection on the image of E 0 (i). We can then define in the notation of Prop.
Noticing that the probability to measure α = 0 in the second step, provided that the outcome in the first was α = 0, is 0 and that E α (i) • E 0 (i) −1 • E 0 (i) = E α (i) (since the support of E 0 (i) contains the support of E α (i)), we find that indeed applying the protocol as in Prop. 3 with the above defined operations yields the right map. The proof of the statement LOP IQO is done in section E, where we provide an explicit counterexample for a wire with Hilbert-space dimension 3.
Whether one can meaningfully call states that are not free in a given resource theory "resources", depends on whether they can be used to do tasks that are impossible under the application of free operations alone. Theorem 5 shows that maximally coherent states are resources in the maximal sense of the word: they enable to do anything within quantum mechanics.  repeatedly a state with l1-coherence 0.01 is used to increase the l1-coherence p/2 of a given state (also initialized with coherence 0.01). If the coherence gets negative the state is dropped. This is repeated 10000 times to get the average behaviour. One sees that at some point the protocol saturates with coherence p/2 ≈ 0.5 Notice that repeating the sequence many times (adding many times a state with coherence q), if one gets symmetric outcomes (same number of + and -) the resulting state is equal to the initial one, while the probability to get outcomes with more + is higher than to get outcomes with more −. There is therefore a bias to get more coherent states over time. Also the protocol only saturates at p f inal = 1. This means that for a given , having enough copies of any state which is not incoherent-quantum, LOP allows to prepare a maximally coherent state with fidelity f > 1 − and probability p > 1 − . A simulation of this protocol is given in Fig. 6. Of course the protocol is sub-optimal in many ways: it only considers two-dimensional coherence and destroys even a part of that by making the state symmetric at the beginning. Furthermore one could maybe improve the algorithm by grouping states. In any case it shows that one can distillate coherence in this setting, similar as it is done in [23] using incoherent operations, even though the specific protocol given there does not seem to be easily adapted to LOP. It remains an interesting open question what is the best possible distillation rate of coherence in LOP.