The Rains relative entropy of a bipartite quantum state is the tightest known upper bound on its distillable entanglement – which has a crisp physical interpretation of entanglement as a resource – and it is efficiently computable by convex programming. It has not been known to be a selective entanglement monotone in its own right. In this work, we strengthen the interpretation of the Rains relative entropy by showing that it is monotone under the action of selective operations that completely preserve the positivity of the partial transpose, reasonably quantifying entanglement. That is, we prove that Rains relative entropy of an ensemble generated by such an operation does not exceed the Rains relative entropy of the initial state in expectation, giving rise to the smallest, most conservative known computable selective entanglement monotone. Additionally, we show that this is true not only for the original Rains relative entropy, but also for Rains relative entropies derived from various Rényi relative entropies. As an application of these findings, we prove, in both the non-asymptotic and asymptotic settings, that the probabilistic approximate distillable entanglement of a state is bounded from above by various Rains relative entropies.Full version available at https://arxiv.org/abs/2201.00835
The Rains relative entropy of a bipartite quantum state is the tightest known upper bound on its distillable entanglement – which has a crisp physical interpretation of entanglement as a resource – and it is efficiently computable by convex programming. It has not been known to be a selective entanglement monotone in its own right. In this work, we strengthen the interpretation of the Rains relative entropy by showing that it is monotone under the action of selective operations that completely preserve the positivity of the partial transpose, reasonably quantifying entanglement. That is, we prove that Rains relative entropy of an ensemble generated by such an operation does not exceed the Rains relative entropy of the initial state in expectation, giving rise to the smallest, most conservative known computable selective entanglement monotone. Additionally, we show that this is true not only for the original Rains relative entropy, but also for Rains relative entropies derived from various Rényi relative entropies. As an application of these findings, we prove, in both the non-asymptotic and asymptotic settings, that the probabilistic approximate distillable entanglement of a state is bounded from above by various Rains relative entropies.Full version available at https://arxiv.org/abs/2201.00835
Multipartite entanglement admits phenomena such as the activation of genuine multipartite entanglement (GME) and the existence of inequivalent classes of entanglement, and existing bipartite entanglement measures have no unique generalization to this regime. In this work, we define the Rains, monsoon, hurricane, and squall entanglement as generalizations of the bipartite Rains relative entropy, and we establish various properties of these entanglement measures. We also prove that the Rains entanglement is monotone under selective quantum operations that completely preserve the positivity of the partial transpose. We establish single-letter upper bounds on the one-shot and asymptotic rates at which a fixed pure state can be distilled from an arbitrary state in both the standard and probabilistic approximate distillation scenarios. Among the entanglement measures we define, the tightest upper bound on the one-shot pure-state distillation rate is in terms of the Rains entanglement. However, the activation of GME (or, equivalently, the tensor instability of biseparability) makes it unclear if the one-shot bound in terms of the Rains entanglement can be extended to a single-letter asymptotic bound. Instead, we establish upper bounds on the asymptotic pure-state distillation rate in terms of the hurricane and squall entanglement. Upper bounds on the GHZ- and W-distillable entanglement follow as a consequence. Additionally, we define the multipartite max-Rains entanglement, write it as a semidefinite program, and derive a dual program for it. Finally, we analyze these measures for quantum pairwise independent networks, and we establish a conditional gradient (Frank-Wolfe) algorithm for computing the Rains entanglement.
We introduce the genuine multipartite Rains entanglement (GMRE) as a measure of genuine multipartite entanglement that can be computed using semi-definite programming. Similar to the Rains relative entropy (its bipartite counterpart), the GMRE is monotone under selective quantum operations that completely preserve the positivity of the partial transpose, implying that it is a multipartite entanglement monotone. As a consequence, we show that the GMRE bounds both the one-shot standard and probabilistic approximate GHZ-distillable entanglement from above. We also develop a generalization of this quantity that incorporates other entropies, including quantum Renyi relative entropies.
Exact entanglement distillation converts a noisy bipartite state into a maximally entangled state with zero error. Under completely PPT-preserving operations, the additive min-Rains relative entropy provides a single-letter upper bound on the regularized distillation rate. An interesting problem in entanglement theory dating back to 2016 has been whether this bound is always tight. Here we resolve this question in the negative. The key is a tensor-stable rigidity absent from the min-Rains relaxation: every feasible exact-distillation effect must act as the identity on the support of the input state. We convert this constraint into a new single-letter upper bound using a generally non-Hermitian, range-supported witness. For a rank-three subspace, we show this upper bound lies strictly below the min-Rains relative entropy for every state with that support. Thus, the constraint discarded by the min-Rains relaxation remains relevant under arbitrary tensor powers. Our result rules out the min-Rains relative entropy as a closed-form formula for exact PPT distillable entanglement and reveals the subtle asymptotic structure of exact entanglement manipulation under PPT operations.
We study entanglement distillation under operations that remain completely positive under partial-transpose conjugation, a.k.a. PPT channels, in the standard quantum Shannon theory regime of asymptotically vanishing but non-zero error. Our main results are: (a) a regularised formula for the PPT distillable entanglement in terms of a measured-relative-entropy-like quantity; and (b) two different single-letter converses, one based on operator quadratic forms and the other on Hirschman's strengthening of the Hadamard three-line theorem. As an immediate application of (b), we show that the PPT distillable entanglement can be strictly smaller than the regularised Rains bound, thereby resolving an open problem in the theory of entanglement manipulation [Regula et al., NJP 21:103017, 2019]. A gap appears already for 3×3 Werner states: at antisymmetric weight 25/26, a certified upper bound of 0.62107 ebits lies below the (regularised) Rains bound, which equals 2625log25−log23≈0.64766 ebits. On a different note, (a) implies a faithful lower bound on the PPT distillable entanglement in terms of the entanglement negativity, which provides a quantitative counterpart to the qualitatively known fact that any NPT state is PPT distillable [Eggeling et al., PRL 87:257902, 2001].
Quantum catalysts can overcome otherwise impossible quantum state transformations without being consumed, and allowing them to become correlated with the output makes this assistance substantially more powerful. This raises a fundamental question for entanglement theory: which limitations on state manipulation remain when such correlated catalysts are freely available? We answer this question in the positive-partial-transpose (PPT) resource theory, which allows a substantially broader class of operations than local operations and classical communication (LOCC). We identify general conditions under which regularized relative-entropy measures become strongly superadditive, and use them to construct monotones that constrain correlated catalytic PPT transformations without any knowledge of the catalyst. In particular, we prove that the regularized PPT relative entropy is fully additive and strongly superadditive, resolving an open problem in entanglement theory. Most importantly, these constraints show that even arbitrary correlated catalysts cannot restore asymptotic reversibility: for an explicit state, the optimal entanglement distillation rate remains strictly smaller than the entanglement cost. Thus, substantial catalytic assistance does not remove some of the fundamental limitations of mixed-state entanglement manipulation.
The logarithmic negativity of a bipartite quantum state is a widely employed entanglement measure in quantum information theory due to the fact that it is easy to compute and serves as an upper bound on distillable entanglement. More recently, the $\ensuremath{\kappa}$ entanglement of a bipartite state was shown to be an entanglement measure that is both easily computable and has a precise information-theoretic meaning, being equal to the exact entanglement cost of a bipartite quantum state when the free operations are those that completely preserve the positivity of the partial transpose [Xin Wang and Mark M. Wilde, Phys. Rev. Lett. 125, 040502 (2020)]. In this paper, we provide a nontrivial link between these two entanglement measures by showing that they are the extremes of an ordered family of $\ensuremath{\alpha}$-logarithmic negativity entanglement measures, each of which is identified by a parameter $\ensuremath{\alpha}\ensuremath{\in}\left[1,\ensuremath{\infty}\right]$. In this family, the original logarithmic negativity is recovered as the smallest with $\ensuremath{\alpha}=1$, and the $\ensuremath{\kappa}$ entanglement is recovered as the largest with $\ensuremath{\alpha}=\ensuremath{\infty}$. We prove that the $\ensuremath{\alpha}$-logarithmic negativity satisfies the following properties: entanglement monotone, normalization, faithfulness, and subadditivity. We also prove that it is neither convex nor monogamous. Finally, we define the $\ensuremath{\alpha}$-logarithmic negativity of a quantum channel as a generalization of the notion for quantum states, and we show how to generalize many of the concepts to arbitrary resource theories.