No second law of entanglement manipulation after all

We prove that the theory of entanglement manipulation is asymptotically irreversible under all non-entangling operations, showing from first principles that reversible entanglement transformations require the generation of entanglement in the process. Entanglement is thus shown to be the first example of a quantum resource that does not become reversible under the maximal set of free operations, that is, under all resource non-generating maps. Our result stands in stark contrast with the reversibility of quantum and classical thermodynamics, and implies that no direct counterpart to the second law of thermodynamics can be established for entanglement -- in other words, there exists no unique measure of entanglement governing all axiomatically possible state-to-state transformations. This completes the solution of a long-standing open problem [Problem 20 in arXiv:quant-ph/0504166]. We strengthen the result further to show that reversible entanglement manipulation requires the creation of exponentially large amounts of entanglement according to monotones such as the negativity. Our findings can also be extended to the setting of point-to-point quantum communication, where we show that there exist channels whose parallel simulation entanglement cost exceeds their quantum capacity, even under the most general quantum processes that preserve entanglement-breaking channels. The main technical tool we introduce is the tempered logarithmic negativity, a single-letter lower bound on the entanglement cost that can be efficiently computed via a semi-definite program.Comment: 16+30 pages, 3 figures. v2: minor clarification

Distillable entanglement under dually non-entangling operations

Computing the exact rate at which entanglement can be distilled from noisy quantum states is one of the longest-standing questions in quantum information. We give an exact solution for entanglement distillation under the set of dually non-entangling (DNE) operations -- a relaxation of the typically considered local operations and classical communication, comprising all channels which preserve the sets of separable states and measurements. We show that the DNE distillable entanglement coincides with a modified version of the regularised relative entropy of entanglement in which the arguments are measured with a separable measurement. Ours is only the second known regularised formula for the distillable entanglement under any class of free operations in entanglement theory, after that given by Devetak and Winter for one-way LOCCs. An immediate consequence of our finding is that, under DNE, entanglement can be distilled from any entangled state. As our second main result, we construct a general upper bound on the DNE distillable entanglement, using which we prove that the separably measured relative entropy of entanglement can be strictly smaller than the regularisation of the standard relative entropy of entanglement. This solves an open problem in [Li/Winter, CMP 326, 63 (2014)].Comment: 7+26 page

Computable lower bounds on the entanglement cost of quantum channels

A class of lower bounds for the entanglement cost of any quantum state was recently introduced in [arXiv:2111.02438] in the form of entanglement monotones known as the tempered robustness and tempered negativity. Here we extend their definitions to point-to-point quantum channels, establishing a lower bound for the asymptotic entanglement cost of any channel, whether finite or infinite dimensional. This leads, in particular, to a bound that is computable as a semidefinite program and that can outperform previously known lower bounds, including ones based on quantum relative entropy. In the course of our proof we establish a useful link between the robustness of entanglement of quantum states and quantum channels, which requires several technical developments such as showing the lower semicontinuity of the robustness of entanglement of a channel in the weak*-operator topology on bounded linear maps between spaces of trace class operators.Comment: 24 pages. Technical companion paper to [arXiv:2111.02438], now published as [Nat. Phys. 19, 184-189 (2023)]. In v2, which is close to the published version, we improved the presentation and corrected a few typo

Reversibility of quantum resources through probabilistic protocols

Among the most fundamental questions in the manipulation of quantum resources such as entanglement is the possibility of reversibly transforming all resource states. The most important consequence of this would be the identification of a unique entropic resource measure that exactly quantifies the limits of achievable transformation rates. Remarkably, previous results claimed that such asymptotic reversibility holds true in very general settings; however, recently those findings have been found to be incomplete, casting doubt on the conjecture. Here we show that it is indeed possible to reversibly interconvert all states in general quantum resource theories, as long as one allows protocols that may only succeed probabilistically. Although such transformations have some chance of failure, we show that their success probability can be ensured to be bounded away from zero, even in the asymptotic limit of infinitely many manipulated copies. As in previously conjectured approaches, the achievability here is realised through operations that are asymptotically resource non-generating. Our methods are based on connecting the transformation rates under probabilistic protocols with strong converse rates for deterministic transformations. We strengthen this connection into an exact equivalence in the case of entanglement distillation.Comment: 6+10 page

Exact solution for the quantum and private capacities of bosonic dephasing channels

The capacities of noisy quantum channels capture the ultimate rates of information transmission across quantum communication lines, and the quantum capacity plays a key role in determining the overhead of fault-tolerant quantum computation platforms. In the case of bosonic systems, central to many applications, no closed formulas for these capacities were known for bosonic dephasing channels, a key class of non-Gaussian channels modelling, e.g., noise affecting superconducting circuits or fiber-optic communication channels. Here we provide the first exact calculation of the quantum, private, two-way assisted quantum, and secret-key agreement capacities of all bosonic dephasing channels. We prove that that they are equal to the relative entropy of the distribution underlying the channel to the uniform distribution. Our result solves a problem that has been open for over a decade, having been posed originally by [Jiang & Chen, Quantum and Nonlinear Optics 244, 2010].Comment: 10+20 pages, 6 figures. v2 is close to the published versio

Entanglement-Breaking Indices

The entanglement is the one of the fundamental features which distinguish the quantum world from the classical one. Thanks to Bell's theorem, we know that it is a genuinely new effect, having no classical counterpart. From the point of view of the quantum information theory, the entanglement has to be regarded primarily as a computational resource. Indeed, it allows us to perform astonishing tasks (such as quantum teleportation or cryptography) which would be impossible in a world subjected to the classical laws. However, just like all the physical resources, also the entanglement is subjected to deterioration. Actually, one of the main issues physicists have to face in dealing with quantum computation tasks from an experimental point of view is the control of the noise interfering with non-classical correlations in a bipartite quantum system. Through this thesis, we examine the following experimental setup. Alice and Bob share a pair of entangled systems, but Alice's half of the global system suffers a noise caused by an uncontrolled interaction with an external environment. This open evolution of a quantum system (in a time-discretized approach) is called a quantum channel. Our central goal is to classify the amount of noise introduced by a given quantum channel only by means of its action on the entanglement. The first instance of this plan is the understanding of those channels that never break the entanglement between Alice and Bob, no matter how weak it is (provided that it exists). These channels are called universal entanglement-preserving. Our first contribution is the rigorous proof that the only universal entanglement-preserving channels are the unitary evolutions, taking place when Alice's subsystem is kept perfectly isolated. This means that even if the interaction of Alice's subsystem with the surrounding world is very feeble, nevertheless it can destroy some form of weak entanglement between Alice and Bob. Conceptually, this result clarifies the context of our investigations. From the physical point of view, it is natural to consider the repeated applications of a given quantum channel on Alice's half of the global system. For the purpose of classifying this noise by means of the damages it produces on the entanglement, we introduce some functionals (defined on the set of quantum channels), called entanglement--breaking indices. The most important ones are the direct n-index and the filtered N-index. The $n$--index is the minimum number of times we have to apply a given channel in order to produce the complete destruction of the entanglement. On the other hand, Alice could play an active role against the noise repeatedly affecting her subsystem, by choosing to apply some quantum channels (called filters) between consecutive actions of the noise. Then, the filtered N-index is by definition the minimum number of iterations of the noise, such that there is no filtering strategy by which Alice can hope to save her entanglement with Bob. However, every non-unitary filter creates some entanglement between Alice's subsystem and an external environment, lowering the level of quantum correlations between Alice and Bob. Therefore, we initially make the intuitive conjecture that the optimal filtering strategy is obtained by means of unitary operations only. However, once again the quantum entanglement has a surprise in store for us. Indeed, we provide an explicit counterexample, showing that this conjecture is in general false. Actually, the two-dimensional case seems to exhibit an anomalous behaviour, in the sense that our counterexample (exceptionally) does not work. In this respect, we collect a series of clues pointing out that the conjecture we presented could retain its validity for channels acting on two-dimensional quantum systems (called qubits). Next, we turn our attention to the study of the those channels (called entanglement-saving) which introduce so few noise in the system, that their direct n-index takes an infinite value. In other words, the complete destruction of the entanglement is never reached, regardless of the number of iterations. But also within the class of entanglement-saving channels, there are still two possibilities. Indeed, in the limit of an infinite number of applications of the channel, the amount of entanglement can tend to zero or remain well above a finite threshold. The latter case corresponds to the so-called asymptotically entanglement-saving channels. One of our main contributions is the complete characterization of these channels. It turns out that a quantum channel is asymptotically entanglement-saving if and only if it admits two non-commuting phase points. A phase point is (by definition) an input matrix whose transformation under the action of the channel is simply the multiplication by a phase. On the other hand, much effort is devoted to gain understanding of the entanglement-saving channels. Although the intrinsic difficulties of coping with quantum systems of arbitrary dimension, rather surprisingly we achieve our goal almost everywhere (that is, apart from a set of measure zero). Indeed, we find that almost everywhere the entanglement-saving property coincides with the presence of a positive semidefinite fixed point for the channel or for some of its powers. Moreover, it is shown that the restriction we pose is irrelevant for the case of two-level systems (qubits). As a consequence, we completely characterize the entanglement-saving qubit channels. In order to give an operational meaning to our abstract results, we provide also a concrete model and a sequence of operations reproducing it