{'fraction_non_alphanumeric': 0.030186410102224896, 'fraction_numerical': 0.004810583283223091, 'mean_word_length': 4.525249169435216, 'pattern_counts': {'":': 0, '<': 2, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 1, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Are the laws of entanglement theory thermodynamical?

We argue that on its face, entanglement theory satisfies laws equivalent to thermodynamics if the theory can be made reversible by adding certain bound entangled states as a free resource during entanglement manipulation. Subject to plausible assumptions, we prove that this is not the case in general, and discuss the implications of this for the thermodynamics of entanglement.

The pioneering papers in quantum information theory [1,2,3] revealed a potential irreversibility in entanglement processing. They suggested that by local operations and classical communication (LOCC), one needs more pure entangled states to produce a state than can be drawn from it. Some researchers expressed the intuition that this is due to some "second law" in entanglement processing. The first paper making this analogy rigorous was [4]. However the authors did not discuss irreversibility, instead building the analogy in the region of full reversibility -in the domain of pure states (see also [5]). It was sometimes argued that irreversibility is where entanglement and thermodynamics differ, as e.g. the Carnot cycle is reversible. In [6] a different point of view was presented which attempted to account for this irreversibility. The leading idea was that entanglement is analogous to energy and that distillation of pure entanglement is like drawing work. (the amount of pure entanglement drawn from a state is E d while the amount needed to create a state we denote by E c ). Meanwhile the extreme case of irreversibility was discovered: the bound entangled (BE) state. One cannot draw any pure entanglement from them, but entanglement is needed to create them.
In this paper we will investigate aspects of the latter analogy and argue that it is extremely useful for understanding the basic laws of entanglement processing. We study the consequences of our proposal, and make the theory precise so that testing is possible. We first show that the very fact of irreversibility does not cancel the analogy, rather it is a constitutive element. This enables us to state the three laws of entanglement for such a theory. Next we however show that the analogy with reversible thermodynamics does not appear hold in general. We then speculate on possible ways the basic laws of thermodynamics may still yield insights into quantum information theory.
The main observation we will need is that what constitutes thermodynamics, and distinguishes it e.g. from mechanics, is that there are two forms of energy: disordered (heat) and ordered (work). The first law does not "feel" any difference between these. It says that to create a single heat bath of energy E we need precisely this amount of work. The second law says that this creation is irreversible: from a single heat bath one cannot draw work. Thus the second law accounts for the basic irreversibility of the heat bath formation: it is due to loss of information. In its ideal version, thermodynamics offers also a reversible change of work into heat and vice versa (the Carnot cycle). However this is only possible if one has heat reservoirs from the very beginning. This is depicted on Fig. 1. To summarize: 1) there is a form of energy, that cannot be used to draw work (single heat baths); 2) it can however be used to store work, but: 3) work can be stored in heat only if there is some heat at the beginning 4) in the latter case work can be stored reversibly.
Let us now argue that thus far, entanglement processing not only does not exclude a perfect analogy with the features of thermodynamics outlined above, but suggests strongly that there is one. (We will then provide a new result, which, up to a plausible conjecture, precludes such a prima-facia possibility.) We will follow an idea related to the entanglement-energy analogy of [6].
First it is reasonable to assume that pure state entanglement is analogous to mechanical energy, while mixed state entanglement corresponds to energy that may be partly accumulated in the form of heat. Now, distillation [1] i.e. drawing pure entanglement from mixed states is like running an engine that produces work out of heat. There is a question: in thermodynamics we have systems that have energy but no work can be drawn from them -single heat baths. If we didn't have a counterpart of it in entanglement processing, then a constitutive element would be missing. However there exist so-called bound entangled states [7] that are entangled (have "energy") but this "energy" cannot be used i.e. changed into an ordered form. Thus BE states are the counterpart of a "single heat bath" -a completely disordered form of entanglement. Other mixed states would represent partially ordered entanglement: work stored in thermodynamical systems (e.g. pairs of reservoirs of different temperatures). The process of forming a BE state out of pure states could be thought of as analogues to Joule's experiment establishing the equivalence between work and heat where all work is (irreversibly) changed into heat. The energy of the created heat bath is equal to the amount of work added to the system. This leads us to the conclusion: The counterpart of total energy would be entanglement cost (amount of pure entanglement needed to produce the state). At this moment one can ask a pointed question: does a counterpart to the first law hold in entanglement theory? Is the entanglement in a BE state still present? Or perhaps to form the state one needs entanglement but during the formation process it gets dissipated [8]. This is very much connected with the main question which we will now address: Can we have an analogue of reversible work extraction? One might say: 'no, because in entanglement processing there is irreversibility'. This is however not an adequate answer. Let us argue that we can have a perfect analogy of reversible work extraction, despite the fact that generically we need more singlets to create a state than can be draw from it.
First note, that any thermodynamical system has more energy than we can draw from it (excluding the case when one of the reservoirs has zero temperature). This is due to the second law. The same is true for entanglement: for a generic mixed state E D < E c . Thus the irreversibility of entanglement processing could be due to a second law of entanglement that says "the disorder of entanglement can only increase". Now, according to Figure 1, for a given state ̺, two conditions should be matched • one should be able to distill E d of pure entanglement, but in such a way that the garbage left over from the distillation procedure (denoted by • one should be able to form the state ̺ out of E d (̺) singlets plus a BE state of entanglement cost equal This is a consequence of conservation of E c which would be analogous to the first law. Even though we will show that in general this will not hold, we think it is important to realize how this theory would look. Indeed perhaps reversible thermodynamics of entanglement is an important element of the total, more complicated picture. Besides, the independent development [9] suggests that for some classes of mixed states this theory may be valid.
Thus, we would have the following reversible thermodynamics of entanglement: • There is a unique measure of entanglement (E c ) • E d is not an autonomous measure of entanglement: it is the "work" that can be drawn in units of E c .
• Processes of formation and distillation are reversible in the sense described above.
• The process of changing pure entanglement into mixed entanglement is irreversible [10].
Exactly as in statistical physics we would have two kinds of irreversible processes: dissipation (where E c decreases) and pure decoherence (where E c is constant, but E d decreases). The first irreversibility does not exist in optimal processes (which we consider). The second could be removed if the processes start with some initial supply of BE states. Then to form a state, one would not need to create disordered entanglement out of the ordered form but only dilute the ordered one into the noisy entanglement of BE states. Usually hypothetical reversibility in entanglement processing is associated with a unique measure of entanglement, in the sense that there is only one function monotonic under operations. Indeed we can state the above in such a way that there will only be one measure of entanglement. Since bound entanglement is used only as a source of fully disordered entanglement, it is not actually a resource. Thus we can treat it as free of charge (say heat is cheap, only work costs). Our class of operations would then be as postulated in [11]: LOCC plus BE ancillas. This does not make the theory trivial, since we still cannot obtain singlets for free. Then however, the only monotone would be E D . Actually, it would be equal to the (regularised) relative entropy distance to the set of nondistillable states [12,13,14]. At this point we can even formulate a precise theorem Theorem 1 If under a given class of operations C that includes mixing states, one can reversibly transform ̺ into pure states, then any asymptotically continuous function E that is monotonic under the class C satisfies where E C R is the relative entropy distance from the set of states that is closed under C. Proof. Uniqueness comes from reversibility and uniqueness of the entanglement measure in the pure state case [4,15] (that again comes from reversibility between pure states [3]). E C R is a monotone under C [16], and is also asymptotically continuous if the considered set is convex [15] (which is the case since C includes mixing).
Thus we can formulate the second law. Surprisingly it has the same form as in statistical physics: it is Uhlmann monotonicity see e.g. [17] saying that under completely positive maps the relative entropy does not increase. Indeed from this, one gets that the relative entropy distance is monotonic, from which our theorem follows. In this way the law gives the optimal entanglement that can be distilled exactly as the second law gives the Carnot efficiency. This also suggests a feedback to thermodynamics: it seems possible to express the extractable work by means of the relative entropy distance from the single heat bath states (for a fixed Hamiltonian). Indeed, theorem 1 appears to be satisfied in thermodynamics. We hope to come back to this interesting point elsewhere. It is also rather amusing that in this theory we would have an analogue to the third law of thermodynamics: one cannot distill singlets with perfect fidelity Let us return to the question of whether we can have reversibility for all states. Recall that for this proposal: Reversible thermodynamics of entanglement = possibility of reversible separation of bound and pure entanglement.
Below we show a counterexample . To this end we need to be more rigorous. The first problem is that if there are NPT [18] bound entangled states, then BE cannot be for free, as such states together with some PPT states are distillable, so that singlets would be for free too. Thus we shouldn't speak about BE states, but rather about the smaller class of states that is closed under LOCC and under tensor product. Call this set the Hyper-Set (HS ). Thus to produce a counter-example, we should show that there is a state for which E LOCC+HS where LOCC+HS is LOCC operations plus states from HS as free ancillas. One can also consider the class of Hyper-Maps that includes the class LOCC+HS -namely the maps that do not move states outside of HS or the subclass of these consisting of the maps that are closed under tensor product.
We do not know the set HS or the class of Hyper-Maps. However, the following result [19] adds an additional restriction: any NPT state can be distilled with the help of some PPT state. The HS would therefore not be able to include PPT states and any NPT state. The conditions on other sets are also very restrictive, and we therefore conjecture that the HS is PPT and the class is PPT maps [20] introduced in [21].
Let us therefore define PPT entanglement of formation E ppt f . Recall that the traditional entanglement of formation was defined as [2] E f (̺) = inf i p i S(̺ i A ) where the infimum runs over all ensembles ̺ = i p i |ψ i ψ i | and ̺ i A is obtained by partial trace of ψ i . One knows that the entanglement cost is given by the regularized entan-glement of formation [22] To define the PPT version of E f we need the notion of phase-separated states. A state is called phase-separated if it is either pure or PPT or a tensor product of both. Then we define where the infimum runs over decompositions of ̺ into phase-separated states, and f (̺ ppt we count only the pure entanglement. This quantity intuitively measures the entanglement cost under LOCC+PPT, like entanglement of formation E f was associated with entanglement cost under LOCC. Indeed following [22] Proposition 1 If E ppt f is monotonic under PPT maps, and asymptotically continuous [23], then its regularization is equal to the entanglement cost under PPT maps. If the assumptions hold, then we have counterpart of (3) for PPT maps. Consider now the state [2,13].
We will now state the main result of the paper Theorem 2 For the state (5) p = 1/2, 0, 1 the PPT distillable entanglement is strictly smaller than the regularized PPT entanglement of formation Proof. We will prove that E ppt We will actually show that it is true for all maximally correlated (MC) states [13] defined as ̺ = ij a ij |ii jj|.
We need the following facts about MC states:  (2) can be found e.g. in [13]. Proof of (3) is straightforward. Now we can prove the theorem. From (1) it follows that if ̺ is MC then so is ̺ ⊗n . Now take any ensemble ̺ = i p i ̺ i , with phase-separated ̺ i . Due to (2) we have that all ̺ i are MC. Then (3) implies that if ̺ i is PPT, it must be separable. Otherwise it is either pure or product |ψ ψ| ⊗ σ ppt of a pure state and a PPT state. However due to (1), σ ppt must be MC, hence it is separable. Thus we do not have PPT states in decomposition, so that is monotonic under PPT and asymptotically continuous and that HS is PPT, we obtain that there is no reversible thermodynamics of entanglement. Instead we in general have some dissipation of entanglement even in optimal processes. Since E ∞ R is the entanglement that could be distilled in the reversible case one can be tempted to argue that the difference between E c and E ∞ R gives the amount of entanglement dissipated during formation, while E ∞ R − E d is that dissipated during distillation. Then the process of formation of BE states could be non-dissipative, while for maximally correlated states, the whole bound entanglement would be dissipated during formation of the state. I.e. bound entanglement is needed to create the state, but is lost during creation (this is supported by the fact that one can localize the information corresponding to the bound entanglement for these state [26]).
Does it mean that thermodynamical analogies should be abandoned? First of all, let us emphasize that even if we do not have reversible thermodynamics of entanglement it is definitely instructive to know how far we are from this regime. Thus one might be able to understand the theory looking at deviations from the desired behavior. Moreover, there is still place for the analogy. It may be that one can account for the E c that is dissipated. Perhaps we sometimes operate in the non-equilibrium regime, or perhaps when distilling entanglement we have phase transitions (cf. [27]). Then we may have irreversiblity due to the release of heat. A result disproving the conjecture that E ppt f is asymptotically continuous would also reopen the possibility that these three laws may still hold as is.
Finally, in a recent development [9] a nontrivial state was exhibited, for which there is reversibility under PPT maps, even though E d < E c . If one can show that the PPT map can be realized by means of LOCC+PPT, we would obtain a nontrivial regime where reversible thermodynamics of entanglement holds.
To summarize, the features that distinguish entanglement theory from reversible thermodynamics are not due to the existence of bound entanglement (or distillationformation irreversibility). Rather the problem is that two "phases", bound entanglement and free (pure) entanglement cannot in general be separated without loss of entanglement cost. Finally, we believe that our paper provides a new, clearer picture of entanglement theory, and that further investigation into the thermodynamics of entanglement is warranted.