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On quantum non-signalling boxes

A classical non-signalling (or causal) box is an operation on classical bipartite input with classical bipartite output such that no signal can be sent from a party to the other through the use of the box. The quantum counterpart of such boxes, i.e. completely positive trace-preserving maps on bipartite states, though studied in literature, have been investigated less intensively than classical boxes. We present here some results and remarks about such maps. In particular, we analyze: the relations among properties as causality, non-locality and entanglement; the connection between causal and entanglement breaking maps; the characterization of causal maps in terms of the classification of states with fixed reductions. We also provide new proofs of the fact that every non-product unitary transformation is not causal, as well as for the equivalence of the so-called semicausality and semilocalizability properties.

that causal boxes are rare in the set of all maps. Then we investigate the connection between causal maps that are not localizable and the set of entanglement breaking maps. We link the classification of causal maps to the classification of states with same reductions. Subsequently, we study the trade-off between nonlocality and entangling power for a family of non-signalling maps. We also relate the issue of irreversibility for causal boxes to a similar one in entanglement theory. We interpret the result of [3], that using non-signalling boxes distant parties can compute any one-bit valued function of distributed data by use of one bit of communication, in terms of thermodynamical-like reversibility, that was sought to present entanglement theory [7].
As far as new proofs are concerned, we provide a compact proof of the theorem of [2] characterizing semicausal maps, of the fact that nonlocal unitaries are signalling, and that semicausal and semilocalizable maps are equivalent.
The paper is organized as follows. In Section II we recall the mathematical definition of causality and localizability, and provide some basic results. In Section III we focus on properties of unitary transformations, giving a new proof that non-factorized unitary transformations are non-causal, i.e. they allow communication. Section IV is devoted to the equivalence of semicausality and semilocalizability. In Section V we study some classes of causal maps; we define for causal maps their reduced maps, depicting a classification in classes of equivalence parallel to the ones for states with same reductions.In Section VI the relation between quantum causal maps and classical non-signalling boxes is investigated. In Section VII we analyze the relationship between the standard quantum non-locality related to entanglement and the property of non-localizability of some causal maps. The communication irreversibility exhibited by boxes is discussed in analogy to bound entanglement in Section VIII.

A. Definitions
In this section we provide basic definitions. Then we prove some properties of causal maps. In particular, we show that they are rare in the set of all physical maps, and provide a new short proof of characterization of causal maps of [2].
Completely positive trace-non increasing maps (superoperators) from density operators to density operators, on Hilbert space H, are the standard tool to describe the change (e.g. due to temporal evolution or to measurement) a quantum system undergoes. We will now consider such maps on a bipartite system A+B with associated Hilbert space H AB = H A ⊗H B . We denote by S A , S B and S AB the sets of states of system A, B and A+B, respectively. The causality properties of a map regard whether the map allows communication or not.
Definition 1. A map Λ : S AB → S AB is A B semicausal if for every state ρ AB and every map Γ A : S A → S A we have that is no action of Alice before the global operation Λ has any detectable consequence on Bob's side. A similar definition holds for a map which is A B semicausal. A map is causal if it is both A B and A B semicausal.
If a map is not A B semicausal, we say it is A → B signalling. If a map is not causal, we say it is signalling.
The localizability properties of a map regard whether the map needs communication (either classical or quantum) to be realized. Definition 2. A map Λ : S AB → S AB is A → B semilocalizable if the transformation it describes can be performed by one-way (either quantum or classical) communication from A to B, i.e.
where R is an ancilla system and • denotes composition. An operation is localizable if it can be performed locally by sharing an ancilla system and no communication at all, i.e.
It is immediate to check that A → B semilocalizability implies A B semicausality, and therefore localizability implies causality. It is also evident [2] that semicausal (semilocalizable) maps form a convex set, therefore the same holds for causal maps, whose set is given by the intersection of the two convex sets of A B and A B semicausal maps. Also the set of localizable maps is convex [2], but to show this is in principle less obvious, exactly because there are maps which, being causal, are both A → B and A ← B semilocalizable, but are not localizable.

B. Causal maps are rare
The following proposition asserts that semicausal maps are "rare" in the set of all CPT maps, because the smallest combination of A → B signalling map with a A B semicausal map ruins the semicausality of the latter.
Proposition 3. The set of semicausal maps has an empty algebraic interior [36].
Proof. Given a A B semicausal map Λ and a A → B signalling map Σ the map Λ(p) = pΣ + (1 − p)Λ is A → B signalling for any 0 < p ≤ 1. Indeed, consider a state ρ AB and a map Γ A such that Then, for 0 < p ≤ 1, i.e. Λ(p) is signalling.
As a consequence, the set of semicausal maps has zero measure with respect to any reasonable measure on the set of maps.
One could wonder whether the previous theorem says that the set of non-signalling maps is a face of all CPT maps, i.e. whether the fact that a causal map Λ C can be written as a non-trivial convex combination of two maps Λ 1 and Λ 2 forces also both the latter to be causal. It is not so. For example let us consider the totally depolarizing channel D AB [X] = Tr[X]½/(d AB ), which is clearly causal. Fixing a basis {|0 , ...|d AB − 1 } in H AB , let us define the unitary operators L|j = |(j + 1)mod(d AB ) and F |j = e 2πij/dAB |j .
Then D AB [X] can be written as Since the unitary operators L k F l are in general entangling and thus signalling (see Section III), a causal superoperator as D AB may be written as the convex combination of signalling maps.

C. Characterization of semicausal maps
We present here a compact proof of Theorem 8 of [2] characterizing the set of semicausal maps.

Theorem 4. A map Λ is A B semicausal if and only if
Proof. (if) Obvious, since the totally depolarizing channel D has the property D • Γ = D for every trace-preserving map Γ. (only if) By the definition of semicausality, we have that for every ρ AB and Γ A : Therefore we must have for every Γ A , in particular for Γ A = D A . This ends the proof.
Condition (9) can be rewritten as Equality (12) can be checked through the Choi-Jamio lkowski isomorphism [8,9]: given a map Γ AB , we associate to it the unique operator with |i A ⊗ |i A ′ , and similarly for P + BB ′ . Remarkably, it follows that if a map is signalling, then a possible procedure for signalling is the following. One takes the state violating the equality (9) (it is easy to see that one can always chose it to be pure product one). Then Alice encodes "0" by doing nothing, and she encodes "1" by depolarizing her system (replaces her system with one in the maximally mixed state). Then Alice and Bob apply the map, and by theorem, the two situations lead to different density matrix of Bob's system, which creates a channel of nonzero capacity from Alice to Bob.
Condition (12) can be used to check the semigroup property proved in [2], i.e. that if Λ 1,2 are two A B semicausal maps, then also Λ 1 • Λ 2 is A B semicausal. In fact, D. Reduced maps and states with fixed marginals Theorem 4 says immediately that if Λ AB is A B semicausal, then the reduced density matrix ρ ′ B = Tr(ρ ′ AB ) = Tr(Λ[ρ ′ AB ]) depends only on ρ B = Tr(ρ AB ). This means that an A B semicausal map Λ AB maps two density matrices ρ AB and σ AB such that ρ B = σ B into density matrices ρ ′ AB and σ ′ AB such that ρ ′ B = σ ′ B . Let us divide the set of states S AB into the following convex classes of equivalence: and, given a set of operators S, we denote by the image of S through a map Λ. The classes of equivalence [[ρ A ⊗ ρ B ]] AB have been studied in [10,11]; in particular their extremal point have been partly characterized. We notice that the property of causality of maps is strictly related to such a classification. Indeed, we have while it is causal if and only if (see Figure 1) Given an A B semicausal map Λ AB we can therefore consistently define the reduced map with the choice ρ B = ½ B /d B arbitrary and irrelevant. For semicausal maps (i.e. for maps for which the reduced map is well defined) we can then introduce classes parallel to the ones for states: with Λ AB A B semicausal and Ω A and Λ A reduced maps. For causal maps we can define Like the set S AB is partitioned in convex subsets It has already been proved [2,12] that unitary maps X → U AB XU AB such that U AB = U A ⊗ U B are signalling. We provide here an alternative proof, which we consider to be simpler and more direct.
We notice that every unitary operator U AB : H AB → H AB such that for all |ψ A ⊗ ψ B ∈ H AB we have has one of the following forms [13]: It is evident that all unitaries corresponding to case (b) are signalling, while in case (a) signalling is not possible. Therefore we focus on unitary transformations with entangling power, i.e. there is a separable state with the same Schmidt number r and the same set {(λ i , |i A )} for any arbitrary state |ψ B ∈ H B , so that for any possible choice of |ψ B on side B, the reduction ρ A after the action of the unitary transformation is the same. Indeed, according to [14] if we consider two purifications |ξ 1 AB and |ξ 2 AB of the same state ρ A , they are linked by a local unitary transformation acting on B; in our case, we must therefore have if we want to exclude A ← B signalling. A similar request holds to forbid A → B signalling. The previous requirement is not satisfied by any entanglement creating unitary operation U AB . Let us fix a state |φ in H A and an orthonormal basis {|ψ a } in H B . As just explained, to satisfy the non-signalling condition it must be with r the Schmidt number and the dependence of the Schmidt decomposition on |ψ a appearing only on side B. The reduced density matrix obtained tracing over the B degrees of freedom after the action of U AB is Let us now consider the action of U AB on the separable state |φ ⊗ (αψ a + βψ b ) , a = b, with i.e.
In this case we have For the transformation to be non-signalling ρ ′ A = ρ A ; because of the freedom in the choice of α, β it must be r i,j=1 i.e.
with i, j = 1, ..., r and a, b = 1, ..., d B . Therefore there should be rd B orthonormal states in H B , which is possible only if r = 1, i.e. if the unitary transformation does not create entanglement. This ends the proof.
It is worth noting that as regards the partitioning of causal maps in convex sets

IV. SEMICAUSALITY IS EQUIVALENT TO SEMILOCALIZABILITY: NEW PROOF
That semicausality implies semilocalizability (the converse implication being quite trivial) was proved in [4], using the uniqueness of the Stinespring representation for a completely positive map. Here we provide a slightly different reasoning leading to the same result, using the Choi-Jamio lkowski isomorphism and Theorem 4.
Given a A B semicausal map Λ AB we consider its isomorphic state According to condition (9), ρ AA ′ BB ′ has reduction with where Λ B is the reduced map of Λ AB as defined in (21). Since Λ B is a CPT map, with E an auxiliary system of dimension at most d 2 B and |0 E a reference state of its. Therefore ρ A ′ BB ′ admits a purification of the form with E embedded in C, so that |φ AA ′ BB ′ E → |φ AA ′ BB ′ C , U BC → U BE and |0 E → |0 C , and U AC a unitary operation on A and C only. So we have found Because of the Choi-Jamio lkowski isomorphism, the right-hand side of (45) defines a unique map S AB → S AB , which is A ← B semilocalizable and coincides with the starting map Λ AB .

V. CLASSES OF CAUSAL MAPS
In this section we discuss the relation between causal maps and entanglement breaking maps, as well as some positive but not completely positive maps. We provide also a general scheme to "construct" causal maps.

A. Causal maps and entanglement-breaking maps
Note that all known examples of non-trivial causal maps, in particular the ones that are not localizable, belong to the class of entanglement breaking trace-preserving (EBT) maps [15]: FIG. 2: Relations between the different sets of maps: entanglement breaking trace-preserving (EBT), localizable, causal (light grey), convex hull of localizable and causal EBT maps (dark grey). No example of a map which is not in the latter set but it is causal is known at present. Notice that because of Proposition 3 any non-trivial convex combination of a localizable map and a non-causal EBT map is non-causal.
We can apply Theorem 4 to this class of maps. We find that an EBT map Λ is A B semicausal if and only if Such condition is clearly satisfied if for example Notice that even if a map Λ as in (46) is an EBT map, i.e.
is always separable with respect to the cut AB|C, for any state ρ ABC , it may create entanglement between A and B, since the states σ i AB may be entangled. An interesting question is the following: is the set of all causal maps contained in the convex hull of localizable and causal EBT maps? If so, it would mean that non-trivial causal maps (e.g. non-localizable) have some "classical" feature: they destroy (completely, if EBT) quantum information contained in the initial state. The current knowledge about the set of causal maps is presented on Figure 2.

B. Non-signalling maps from positive maps
It is possible to find examples of non-signalling maps starting from a non-signalling map Λ P which is positive but not completely positive and considering the convex combination pΛ P + (1 − p)D AB with p such that the resulting map is completely positive [16]. Here are some examples: • Λ P = T , the transposition with respect to a fixed basis: 0 ≤ p ≤ 1 dAB+1 ; α,β=0 ξ αβ ρ α,β σ α ⊗ σ β , with ξ αβ = −1 if α = 0 and β = 0 and ξ αβ = 1 otherwise: 0 ≤ p ≤ 1 3 . Unfortunately, such examples do not provide any more insight as regards the structure of causal maps, since they are all localizable; in particular they can been realized by shared randomness. In fact, any positive map Λ admits a diagonal representation , the set of bounded operators on H, and λ i ∈ R. Note that given any orthonormal operator basis {F i } we have with λ ij ∈ R; for example, the global transposition is given by the composition of local transpositions, Since this is a basis, any trace-preserving linear map Λ may be thought as completely defined by the assignment In general, with arbitrary choice of ρ i , the resulting map is not only non-completely positive, but not even positive [37]. In any case, given an arbitrary assignment (55), i.e. an arbitrary trace-preserving linear map, we can always construct a convex combination pΛ + (1 − p)D with p such that the resulting map is completely positive. In our case we can therefore consider an operator basis in B(H AB ) given by some separable product states P A 1 ⊗ P B 1 , . . . , P A . A trace-preserving linear map Λ is defined by the assignment Λ[P A i ⊗ P B j ] = ρ ij AB , i = 1, . . . , d 2 A , j = 1, . . . , d 2 B , with ρ ij AB ∈ S AB . A sufficient and necessary requirement for the resulting map to be causal is given by Proposition 5 and in this case may be stated as Table I

VI. RELATION WITH CLASSICAL NON-SIGNALLING BOXES
Classical non-signalling boxes take the input from two parties and furnish an output such that the parties can not communicate, i.e. Alice can not understand from her output what was Bob's input. Still, when they meet afterwards they may find that the recorded outputs are more correlated than in the case in which the output on one side was completely independent from the input on the other side. In this case we speak of non-signalling non-local classical boxes, since the implementation of such boxes requires communication.
Classical boxes are described by conditional probabilities p ab|xy of obtaining outputs a and b with inputs x and y, on Alice's and Bob's side respectively [17]. Such conditional probabilities form a convex set. In the case of two possible distinct inputs, x, y ∈ {0, 1}, and d A , d B possible distinct outputs a ∈ {0, 1, ..., d A − 1}, b ∈ {0, 1, ..., d B − 1} on Alice's and Bob's side respectively, we have a characterization of the (non-local) extreme points of such a convex set (apart local reversible transformation, e.g. relabellings): }. This means that for fixed k, on each side the output is an apparently random number among {0, ..., k − 1} even if the outputs are strictly correlated. It is not difficult to check that a corresponding quantum non-local box may be given by the map: and where m ⊕ k n = (m + n) mod k. Indeed, "feeding" such map with a separable state |ψ in = |x ⊗ y , x, y ∈ {0, 1}, and measuring in the factorized basis |a ⊗ b , a ∈ {0, 1, ..., d A − 1}, b ∈ {0, 1, ..., d B − 1}, we obtain exactly the conditional probabilities (57). In general, with input ρ in , the action of the quantum box corresponds to with p = 11|ρ in |11 and Σ a permutation of order k realizing the transformation Σ|b = |b ⊕ k 1 . Such map is therefore of the form (46) (with the slight difference that input and output dimensions are different) and condition (48) is valid both for A and B, so that the map is causal. Another possible choice for a quantum version of the non-local box is given by The two possible quantum versions differ in coherence: in the second case local measurements are performed, and then the classical box is operated. Note that the map Λ NL can be realized either with classical communication having at disposal a shared maximally entangled state or with quantum communication. The map Λ NL requires only classical communication, as the classical version.
In the next section we will focus on the case k = d A = d B = 2, that is on the map with, as before, p = 11|ρ in |11 and where σ 1 = 0 1 1 0 . Its "incoherent" version was considered in [2]: It was there proved that with such a transformation at disposal, it is possible to violate Bell inequalities up to their algebraic maximum, so that the map can not be localizable (if the map is localizable Cirel'son's bound holds [18]).
In the next section we analyze the interplay between such a non-localizability feature and the non-locality related to entanglement. Notice that considering boxes as primitives, i.e. as if they existed in nature so that we do not care about their realization, the coherent non-local quantum box Λ NL can be used to produce a singlet. Hence it can reproduce all phenomena exhibited by the latter state. This is in contrast with the classical PR-machine (or the incoherent quantum box Λ ′ NL ), which can reproduce results of local measurement performed on the singlet, but cannot for example reproduce teleportation. Indeed, the classical box can be implemented by use of classical communication, which cannot convey quantum information as teleportation does.

VII. ENTANGLEMENT, ENTANGLING POWER, NON-LOCALITY AND COMMUNICATION COMPLEXITY
We will consider a family of coherent maps, which exhibit a trade-off between the degree of nonlocality measured by violation of the CHSH inequality and the average entanglement created out of product states [19]. We also compare the degree of maximal violation obtainable with coherent and incoherent versions of non-local boxes.
Let us consider the standard CHSH inequality, involving the quantity where the label a, a ′ distinguish two possible setups for an experiment on A side, with each run of the experiment conveying one result r a(a ′ ) = ±1; b, b ′ label similarly two possible setups on B side, and E(c, d) stays for the expectation value of the product r c s d of the result r c of experiment c on side A and s d of experiment d on side B. Local hidden variables models obey while when the expectation values are calculated according to quantum mechanics with respect to a fixed state |ψ , i.e. E(a, b) = ψ|A a ⊗ B b |ψ , with A a and B b dichotomic observables for A and B respectively, (67) can be violated, that is one can have I(a, a ′ ; b, b ′ ) > 2. Anyway, there is a limit for the violation of (67) in this framework, the so-called Cirel'son bound [18], I(a, a ′ ; b, b ′ ) ≤ 2 √ 2. We consider two different experimental setups on side A which comprise the preparation of one between two possible orthogonal initial state |0 A , |1 A ; each state corresponds to an observable a x · σ, x = 0, 1. Similarly for B, with observables b y · σ, y = 0, 1. The experimental procedure is the following: 1. Alice prepares a state |x A , with either x = 0 or x = 1; similarly for Bob, with |y B , y = 0, 1; 2. the state |ψ in = |x A ⊗ |y B is fed to the quantum non-local box (63); 3. Alice and Bob measure observables a x · σ and b y · σ with respect to the output Λ k [|ψ in ψ in |].
It is easy to check that Choosing we obtain for the experimental procedure depicted above I( a 0 , a 1 ; b 0 , b 1 ) = 4.
Contrary to its dephased version, the map (63) has an entangling power, i.e. it can map separable states into entangled states. Therefore, it is not only non-localizable, but also gives raise to "standard" quantum non-locality. We will study the interplay between these two features considering the one-parameter map for α = 1 we recover the map (63), while for α = 0 we get an operation that replaces the input state with |ψ 0 ψ 0 |, i.e. a constant map with maximally entangled output. Such an operation can be realized by Alice and Bob through the use, apart of the resources needed for Λ NL , of local shared randomness. They check their shared random number λ ∈ [0, 1]: if 0 ≤ λ ≤ α they proceed with the protocol realizing Λ NL , otherwise their output is |ψ 0 ψ 0 |. The communication complexity of implementing Λ α is therefore (at most) α times the communication complexity of realizing Λ NL .

A. Entanglement and communication complexity for the maximal violation of the CHSH inequality
We generalize the previous reasoning about the violation of the CHSH inequality, which we want to maximize for every 0 ≤ α ≤ 1. We follow an experimental procedure like the one depicted above, taking into account that, in principle, the input orthonormal states |x A , x = 0, 1, and |y B , y = 0, 1, on Alice's and Bob's sides are arbitrary and independent of the POVM entering in the definition of Λ NL . This means that we have we may write the quantity we want to find as For each fixed α, I M (α) is attained by and choosing ϕ to maximize c 00 + c 01 + c 10 − c 11 = 3 cos ϕ − cos 3ϕ + α(1 + cos 3ϕ).
The optimal angle is (see Figure 3(a)) so that We consider also the family of maps for α = 1 we recover the map (65), while for α = 0 we get a constant map with separable output. Looking for the maximal violation of CHSH inequality with our experimental procedure, we have in this case We find p 0 A = p 0 B = 0, p 1 A = p 1 B = 1 and a i = b j = (0, 0, 1), so that I ′ M (α) = 2(1 + α).
In Figure 4(a) it is shown how, for fixed α, the map Λ α allows in general a violation of the CHSH inequality greater than the one obtained through the map Λ ′ α . The maximal violation becomes equal for the two maps in the range 2/3 ≤ α ≤ 1. Conversely, as can be more appreciated in Figure 4(b), the map Λ α allows the same amount of violation of the CHSH inequality than the map Λ ′ α with lower α, i.e., considering the scheme depicted above to realize Λ α , with a lower amount of communication. Entanglement can therefore be interpreted as a resource to reduce the communication complexity of violating the CHSH inequality. It is interesting it seems to be of no help for 2/3 ≤ α ≤ 1; in such range of values, the non-locality, as measured by I M (α), appears to be essentially due to the communication involved in the realization of the operation.

B. Trade-off between nonlocality and entangling power for non-signalling boxes
One may define the entangling power of a map Λ with respect to a measure of entanglement E as [19] where the average is taken with respect to the initial separable pure state |ψ 1 ψ 1 | ⊗ |ψ 2 ψ 2 |, i.e., in general, with p(ψ 1 , ψ 2 ) a probability distribution over separable pure states. Since we are considering a map acting on qubits, we adopt the Bloch sphere parametrization with 0 ≤ θ i < π, 0 ≤ φ i < 2π, and the corresponding uniform probability distribution dψ 1 dψ 2 p(ψ 1 , ψ 2 ) = 1 (4π) 2 dφ 1 dφ 2 dθ 1 dθ 2 sin θ 1 sin θ 2 . (89) As entanglement measure we will adopt the relative entropy of entanglement [20,21], E = E R ; it takes an easy expression in the case of a Bell diagonal state ρ Bell of two qubits [20,21]: where λ is the highest eigenvalue of ρ Bell and H(x) is the binary Shannon entropy In our case the output of Λ α is always a rank-two Bell diagonal state, so that with p = 11|ρ|11 . Given the parametrization (88), we have so that  In Figure 5 the entangling power E pow (Λ α ) is plotted against the degree of violation of the CHSH inequality I M (α). The two quantities are inversely correlated: the more the degree of non-locality, the lesser the average amount of entanglement created.
Thus we have obtained an example of trade-off capability of creating nonlocal correlations and creating entanglement. The box that violates Bell's inequalities in the strongest way, is not so good in producing entanglement out of product states on average. This can be explained by the fact that if an operation is to be strongly nonlocal, and yet, is not to allow signalling, it must produce some masking noise.
Note that we can find an analogue of such behaviour also for classical boxes. Obviously, we see that in PR box, the outputs, apart from satisfying equation a ⊕ b = xy are otherwise completely random, to ensure the property of non-signalling. Now, let us discuss how such randomness (i.e. noise) implies a trade-off analogous to that between nonlocality and entangling power. Namely, instead of entangling power we can think of the "correlating power" defined as follows. We first choose some measure C of correlations for classical states. Then the quantity is defined in a similar way as entangling power: it is the average output C produced out of initial product input states by the box. Thus C plays here a role analogous to that of an entanglement measure in the definition of the entangling power. It is not hard to agree that for classical states, the uniformly, perfectly correlated state must have maximal C, if only we assume that C does not increase under local operations, which is quite reasonable for correlation measure [22]. Thus, if we take a box that replaces any state with maximally correlated one, it will have maximal possible correlating power. On the other hand, PR box cannot all the time produce maximally correlated states. Thus again, the price for being nonlocal is smaller capability of creating correlations.
It is interesting though, that in quantum case there is some effect that does not have analogue in classical case. Namely, we can consider "power of producing classical correlations". There can be many measures of classical correlations contents of quantum state (cf. [22,23,24,25,26,27]). Nevertheless it is plausible to assume, that a quantum state has maximal possible classical correlations when there exists local measurements that produce classical, maximally correlated state. This is exactly the situation with quantum nonlocal box, since it always produces a mixture of two singlets. Thus the power of producing classical correlations is the maximal possible one and it does not go down with increasing nonlocality of the box.

VIII. BOXES, BOUND ENTANGLEMENT AND CARNOT REVERSIBILITY
In this section we will discuss irreversibility exhibiting by boxes in analogy to bound entanglement. The latter was connected with thermodynamics [7], and a sort of Carnot reversibility was found to be impossible in general. Below we analyze this issue for (mainly classical) boxes, basing on the result of van Dam [3] on communication complexity and nonlocal boxes.

A. Carnot reversibility and entanglement
Bipartite bound entangled states are states from which one cannot distill singlets by local operations and classical communication. However they are entangled, so that to create them, one needs singlets [28,29]. In general, for many states we have that the amount of singlets one can draw is smaller that the amount needed to create them. In other words, entanglement cost E C is usually greater than distillable entanglement E D , and for bound entangled states E C > 0 and E D = 0 [30]. The difference between E C and E D can be called content of bound entanglement in the state, E B = E C − E D . For bound entangled states the whole entanglement is bound E C = E B . Thus there are two types of entanglement: "free" entanglement, that can be used to perform quantum communication, and bound entanglement. This was compared with thermodynamics, where we have two types of energy: free energy, capable to perform work, and bound energy. In a sense bound entanglement is like a single heat bath, which though contains energy, cannot be used for drawing work cyclically. Irreversibility between formation and distillation is analogous to irreversibility exhibited in the process of dissipation, where e.g. mechanical energy is changed into heat.
We know that in thermodynamics, apart from irreversibility, we have also reversibility in Carnot cycle. Carnot cycle shows that it is possible to dilute work into heat and get it back. In [31,32] we have shown that similar reversibility holds for purity and noise. That is changing pure state into mixed one is irreversible, but one can regain reversibility, if one creates a given state with initial maximally mixed states and pure states in proper proportions. Thus purity can be reversibly diluted into noise, like work into heat. In [7] we have analyzed whether this is possible in entanglement theory, i.e. whether singlets can be diluted into bound entanglement reversibly. It would mean that it is possible to reversibly obtain any state from singlets (analogue of work) and bound entangled states (analogue of heat). The amount of needed singlets to create ρ would be equal to E D (ρ). Apart from singlets, one would use some bound entangled state σ b with E C (σ b ) = E B (ρ). Thus the singlets would contribute to free entanglement only, while bound entangled state σ would contribute to bound entanglement. We have found that up to unproven, but plausible assumptions, it is not the case in general (cf. [33]).
Before we pass to discuss boxes, let us mention that we have a hierarchy of sets in entanglement theory. There is the set of separable states, which can be created solely by local operations and classical communication (LOCC), without singlets (equivalently without quantum communication). These states thus do not represent a resource in a LOCC framework, we can have them for free. Then we have the set of bound entangled states, which cannot be created for free, but can not be distilled (through LOCC) to give singlets. Finally there is "the most valuable" set of states, distillable states.

B. Quantum boxes and Carnot reversibility
Let us discuss the issue of boxes in this context. First of all let us say what is the resource, and what is for free. The resource is in this case communication, whether classical or quantum. The states, whether product, separable or entangled are for free. Also local operations are public resource. Therefore we may consider the communication resource to be only classical, the quantum one being obtainable via teleportation.
The analogue of worthless separable states are localizable boxes. They can be produced for free, i.e. without communication (though perhaps using entangled states as ancillae). Then there is set of causal boxes, analogue of nondistillable states. Such boxes cannot be used to communicate. The boxes that are causal, but are not localizable, are analogous to bound entangled states: one needs communication to implement them, however they do not "return" the communication, it is lost. Finally, for a generic box, one expects that the amount of communication needed to implement the box, C C , will usually exceed the amount of communication that can be obtained by use of the box, C D . Thus we can speak about bound, C B = C C − C D , and free communication. In the following we will consider classical boxes, and in the end of the subsection we will go back to quantum boxes.
Let us discuss the possibility of Carnot reversibility. For boxes it would mean that any box could be built out of causal boxes and communication, in such a way that the amount of used communication is equal to the communication that we can get back from the box. For example consider inner product where x and y are n-bit strings, respectively in Alice's and Bob's hands, and addition is modulo 2. The function has one bit output (to make it a two-bit output map, we could for example return f (x, y) to Bob and a fixed bit to Alice). It is known that to perform such box n bits of communication are needed [34], i.e. there is no better way than the following: Alice sends all bits to Bob, and Bob computes the value f (x, y). The amount of communication that can be obtained from this box cannot be of course more than one, being the output (on Bob's side) one bit. Thus we have large irreversibility, the "bound communication" amounting to n − 1 bits. Now, we ask if we can gain reversibility when we implement this box out of causal boxes and communication. We can use a result by van Dam [3], who shows that by use of n PR-boxes (each has at most 1 bit of bound communication) inner product can be computed at additional cost of only 1 bit of communication. Thus in this case there is no loss of free communication. It is interesting then to ask the following question: Given that causal boxes are a public resource, what boxes can be implemented by use of the same amount of communication that can then be recovered from them?
Actually, it turns out, that any function with (n + n)-bit input and 1 bit output, can be realized by n PR boxes plus a single bit of communication [3]. If such a function can be used for one bit of communication (like inner product), then we do not lose free communication. This happens as soon as the function to be computed is not independent of the input of the party which we consider the sender of the message (in our case Alice). Indeed, if it is independent then the total communication needed to compute the function is zero (neither causal boxes nor free communication are involved in the process); if it is not, then for sure there is a fixed input configuration of the receiver, Bob, such that the Alice can send a 1-bit-message changing her input.
Note that our discussion serves also as interpretation of van Dam result. The fact that causal boxes can be so helpful, can be interpreted as follows: the functions that have 1 bit output, have a lot of bound communication. Thus they are themselves almost like causal boxes. Thus it is less surprising, that one can build such function out of many causal boxes and small communication. What remain surprising, is the very existence of causal boxes characterized by C C > 0 and C D = 0.
For quantum boxes, the question of Carnot reversibility appears to be more involved. We can ask if there is analogous result to that of van Dam: can quantum communication complexity be reduced to a single qubit for boxes with 2n-qubit input and 1-qubit output, if causal boxes are public resource?

IX. CONCLUSIONS
We have provided new results on non-signalling boxes, and alternative proofs for existing results. We have shown that causal boxes are rare in the set of all maps. Then we have investigated connection of causal maps that are not localizable with the set of entanglement breaking maps. An open question is whether there exist maps that are causal, but do not belong to convex hull of causal entanglement breaking maps and localizable maps. We have related the "construction"/classification of causal maps to the study of classes of states with fixed reductions. Subsequently, we have found a trade-off between nonlocality and entangling power for a family of non-signalling maps. It would be interesting to prove such a trade-off for the whole family of causal maps: indeed, it must be that the very condition of causality imposes such a trade-off. Finally, we have related the issue of irreversibility for causal boxes (they need communication, but do not return it) to a similar one in entanglement theory and the result of [3], that with use of non-signalling boxes distant parties can compute any 1 bit valued function by use of one bit of communication, with thermodynamical reversibility, that was sought to present entanglement theory [7].
As far as new proofs are concerned, we have provided a compact proof of theorem of [2] characterizing semicausal maps, of the fact that nonlocal unitaries are signalling, and that semicausalilty and semilocalizability are equivalent.
There are many interesting questions that one can pose in the context of non-signalling maps. One of them is the following: can causal boxes play a role in design of fault tolerant schemes? The reason why they could be useful, is that because of causality, they do not propagate errors. So one could try to design circuits in such a way, that if only possible, a causal box is put instead of usual gate. Of course, the fact that the causal boxes to not propagate errors, means that they also do not propagate information, hence it is not clear, if they can be useful elements of circuits.
Another important question concerns complexity of distributed quantum computing. Can we have quantum analogue of van Dam result, i.e. can any map with one qubit output be implemented by causal boxes and only one qubit of communication?
Still the basic problem of characterizing the set of causal boxes in terms of extremal points remains open. We hope we have provided here an interesting direction by relating it to the issue of entanglement breaking maps.
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