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Two or more quantum systems can be correlated in a way that defies any explanation in terms of classical shared randomness - correlations that can be created solely by the communication of classical bits. This type of quantum correlations, entanglement, is responsible for the impossibility of giving a local realistic interpretation to quantum theory as exemplified by Bell inequalities and plays a distinguished role in quantum information processing [1]. It turns out that entangled quantum systems can be harnessed to transmit, store, and manipulate information in a more efficient and secure way than possible in the realm of classical physics. Given this resource character of entanglement, it is an important problem to characterize ways to manipulate it and meaningful approaches to its quantification. This is the objective of entanglement theory [1].

Given an entangled state shared by two observers, Alice and Bob, that can operate locally on their quantum particles and communicate classical bits to each other, can they transform their entangled state into another by such local operations and classical communication (LOCC) procedure? This question is particularly relevant e.g. when the entangled state at their hand is very noisy and hence not directly useful and they would like to distil it into a more pure form. An interesting regime to address is the interconvertibility of entangled states by LOCC is the limit of an asymptotic large number of copies of the state. Here, in analogy to the situation encountered in information theory, one expects that the problem can be solved in simpler terms. The first result in this direction was encouraging. For the case of pure states, there is a unique entanglement measure which completely specifies whether an entangled state can be converted into another [2]. One of the beautiful consequences of this result is its connection to thermodynamics and the second law, as studied by many authors [3,4,5].

Entanglement under asymptotically non-entangling operations is a fungible resource: any two entangled states can be reversibly interconverted in the asymptotic limit(n→ ∞),as long as the ratio of copies of each of them matches the ratio of the respective regularized relative entropies of entanglement.

The apparent universal applicability of thermodynamics suggested a deeper mathematical and structural foundation. There is a long history of examinations of the foundations of the theory, of particular note the work of Giles [6] and Lieb and Yngvason [7] on the second law. They state the second law as a complete order for equilibrium thermodynamical states that determines which state transformations are possible by means of an adiabatic process. From simple, abstract, axioms it can be shown that this order is uniquely determined by the entropy S: the thermodynamical state A can be transformed into B if, and only if, S(A) ≤ S(B). This is exactly the same structural law found for the manipulation of pure states under LOCC: a state A can be converted into B by LOCC if, and only, if E(A) ≥ E(B).

But how about the general case of mixed quantum states? Then, perhaps surprisingly, there exist bound entangled states, which contain quantum correlations and thus require a non-zero rate of pure state entanglement for their creation by LOCC, but from which no pure state entanglement can be extracted at all [8]. As a consequence no unique measure of entanglement exists in the general case and no unambiguous and rigorous direct connection to the second law appears possible.

Recent work by Brandão and Plenio [9] from Imperial College shows that such a conclusion might be too pessimistic. They analysed the manipulation of entangled states under operations that go beyond local operations and classical communication, yet are not capable of generating any entanglement. In this setting they proved that there is a total order for entangled states, shared by any number of parties, with a unique measure given by the so-called regularized relative entropy of entanglement, an entanglement measure that had been introduced ten years before [10]. The situation is therefore rigorously analogous to the formulation by Lieb and Yngvason of the second law of thermodynamics.

The main insight of their approach is to link the problem to a different subfield of quantum information theory: quantum hypothesis testing. There one is interested in finding optimal strategies for discriminating two or more quantum states, in the regime of many realizations, and in determining optimal distinguishability rates. A natural problem in the context of entanglement is the determination of the optimal rate for discriminating a given entangled state from any non-entangled state. The key technical tool for their approach is to indentify such a rate as the regularized relative entropy of entanglement. This then allows the establishment of the total order for entanglement in simpler terms.

Their discovery promises to give a fresh look into the way we understand and study entanglement and its relation to thermodynamics.

[1] M.B. Plenio and S. Virmani, Quant. Inf. Comp. 7, 1 (2007).

[2] C.H. Bennett, H.J. Bernstein, S. Popescu and B. Schumacher, Phys. Rev. A 53, 2046 (1996).

[3] S. Popescu and D. Rohrlich, Phys. Rev. A 56, R3319 (1997).

[4] P. Horodecki, R. Horodecki and M. Horodecki, Acta Phys. Slov. 48, 141 (1998).

[5] M.B. Plenio and V. Vedral, Contemp. Phys. 39, 431 (1998).

[6] R. Giles, Mathematical Foundations of Thermodynamics, Pergamon, Oxford, 1964.

[7] E.H. Lieb and J. Yngvason, Phys. Rept. 310, 1 (1999).

[8] M. Horodecki, P. Horodecki and R. Horodecki, Phys. Rev. Lett. 80, 5239 (1998).

[9] F.G.S.L. Brandão and M.B. Plenio, Nature Physics 4, 873-877 (2008).

[10] V. Vedral and M.B. Plenio, Phys. Rev. A 57, 1619 (1998)