Asymptotic compatibility between local-operations-and-classical-communication conversion and recovery

Abstract

Recently, entanglement concentration was explicitly shown to be irreversible. However, it is still not clear what kind of states can be reversibly converted in the asymptotic setting by LOCC when neither the initial nor the target state is maximally entangled. We derive the necessary and sufficient condition for the reversibility of LOCC conversions between two bipartite pure entangled states in the asymptotic setting. In addition, we show that conversion can be achieved perfectly with only local unitary operation under such condition except for special cases. Interestingly, our result implies that an error-free reversible conversion is asymptotically possible even between states whose copies can never be locally unitarily equivalent with any finite numbers of copies, although such a conversion is impossible in the finite setting. In fact, we show such an example. Moreover, we establish how to overcome the irreversibility of LOCC conversion in two ways. As for the first method, we evaluate how many copies of the initial state is to be lost to overcome the irreversibility of LOCC conversion. The second method is to add a supplementary state appropriately, which also works for LU conversion unlike the first method. Especially, for the qubit system, any non-maximally pure entangled state can be a universal resource for the asymptotic reversibility when copies of the state is sufficiently many. More interestingly, our analysis implies that far-from-maximally entangled states can be better than nearly maximally entangled states as this type of resource. This fact brings new insight to the resource theory of state conversion.

Beyond the thermodynamic limit: finite-size corrections to state interconversion rates

“…In fact, it diverges when the error approaches zero, preventing a perfect reversible cycle. However, pairs of states with equal ratios of relative entropy and relative entropy variance with respect to the thermal state (such that ν = 1) are reversibly interconvertible up to second-order asymptotic corrections, mirroring a recent result in entanglement theory [39]. Thus, ν can be interpreted as the irreversibility parameter that quantifies the amount of infidelity of an approximate cyclic process.…”

Semi-Finite Length Analysis for Information Theoretic Tasks

“…Lemma 11 (Leftover Hash: [36], [48], [47]): Let F be the uniform random variable on a set of universal 2 hash family F . Then, for P AE ∈ P(A × E) and R E ∈ P(E), we have 1 Corollary 1 implies the second inequality in (22). 1 Technically, RE must be such that supp(PE) ⊂ supp(RE).…”

“…Indeed, strong large deviation was employed for information theory in the papers [13], [18], [19]. While the papers [13], [18], [19] employed saddle point approximation in addition to strong large deviation, in a similar way to the papers [20], [21], [22], [23] in other topics, we directly use the formula for strong large deviation to calculate higher order asymptotics so that we do not employ saddle point approximation.…”

“…Since the first inequality in (22) follows from (5), we show the second inequality in (22). Then, we define Hε min (P AE |R E ) := max…”

One-Shot Yield-Cost Relations in General Quantum Resource Theories

“…(See also Refs. [10][11][12] for related discussions of second-order asymptotics in the context of entanglement theory.) However, in the non-asymptotic regime, the relation between the distillable resource and the resource cost is more subtle, especially when errors incurred in the transformations are also taken into account.…”