We give the first systematic study on the second order asymptotics of the operational task of coherence distillation with and without assistance. In the unassisted setting, we introduce a variant of randomness extraction framework where free incoherent operations are allowed before the incoherent measurement and the randomness extractors. We then show that the maximum number of random bits extractable from a given quantum state is precisely equal to the maximum number of coherent bits distillable from the same state. This relation enables us to derive tight second order expansions of both tasks in the independent and identically distributed setting. Remarkably, the incoherent operation classes that can empower coherence distillation for generic states all admit the same second order expansions, indicating their operational equivalence for coherence distillation in both asymptotic and large block length regimes. We then generalize the above line of research to the assisted setting, arising naturally in bipartite quantum systems where Bob distills coherence from the state at hand, aided by the benevolent Alice possessing the other system. More precisely, we introduce a new assisted incoherent randomness extraction task and establish an exact relation between this task and the assisted coherence distillation. It strengthens the one-shot relation in the unassisted setting and confirms that this cryptographic framework offers a new perspective to the study of quantum coherence distillation. Likewise, this relation yields second order characterizations to the assisted tasks. As by-products, we show the strong converse property of the tasks above from their second order expansions.

We give the first systematic study on the second order asymptotics of the operational task of coherence distillation with and without assistance. In the unassisted setting, we introduce a variant of randomness extraction framework where free incoherent operations are allowed before the incoherent measurement and the randomness extractors. We then show that the maximum number of random bits extractable from a given quantum state is precisely equal to the maximum number of coherent bits that can be distilled from the same state. This relation enables us to derive tight second order expansions of both tasks in the independent and identically distributed setting. Remarkably, the incoherent operation classes that can empower coherence distillation for generic states all admit the same second order expansions, indicating their operational equivalence for coherence distillation in both asymptotic and large block length regime. We then generalize the above line of research to the assisted setting, arising naturally in bipartite quantum systems where Bob distills coherence from the state at hand, aided by the benevolent Alice possessing the other system. More precisely, we introduce a new assisted incoherent randomness extraction task and establish an exact relation between this task and the assisted coherence distillation. This strengthens the one-shot relation in the unassisted setting and confirms that this cryptographic framework indeed offers a new perspective to the study of quantum coherence distillation. Likewise, this relation yields second order characterizations to the assisted tasks. As by-products, we show the strong converse property of the aforementioned tasks from their second order expansions.

We study classical communication over a noisy quantum channel when bipartite states are preshared between the sender and the receiver, and one of the following encoding strategies are available: i) local operations; ii) local operations and one-way classical communication; iii) local operations and global permutations. Our main result is a capacity formula for strategy iii). This formula's two endpoints are the capacity formula in strategy i) and the entanglement-assisted classical capacity. Interestingly, these capacities satisfy the strong converse property, and thus the formula serves as a sharp dividing line between achievable and unachievable rates of communication. We prove that the difference between the capacities by strategy i) and strategy iii) is upper bounded by the discord of formation of the preshared state. What's more, we show that strategy ii) has no advantage over strategy i) in the weak converse regime. As examples, we derive these capacities analytically by the above strategies for some fundamental quantum channels. In some cases, the capacity of strategy iii) is strictly larger than those of strategies i) and ii) whenever entanglement assistance is available. Our results witness the power of random permutation in entanglement-assisted classical communication.

We give a capacity formula for the classical communication over a noisy quantum channel, when local operations and global permutations allowed in the encoding and bipartite states preshared between the sender and the receiver. The two endpoints of this formula are the Holevo capacity (without entanglement assistance) and the entanglement assisted capacity (with unlimited entanglement assistance). What's more, we show that the capacity satisfies the strong converse property and thus the formula serves as a sharp dividing line between achievable and unachievable rates of communication. We prove that the difference between the assisted capacity and the Holevo capacity is upper bounded by the discord of formation of the preshared state. As examples, we derive analytically the classical capacity of various quantum channels of interests. Our result witnesses the power of random permutation in classical communication, whenever entanglement assistance is available.

The unextendibility or monogamy of entangled states is a key property of quantum entanglement. Unlike conventional ways of expressing entanglement monogamy via entanglement measure inequalities, we develop a state-dependent resource theory to quantify the unextendibility of bipartite entangled states. First, we introduce a family of entanglement measures called unextendible entanglement. Given a bipartite state ρAB, the key idea behind these measures is to minimize a divergence between ρABand any possibly reduced state ρAB' of an extension ρABB' of ρAB. These measures are intuitively motivated by the fact that the more a bipartite state is entangled, the less each of its individual systems can be entangled with a third party. Second, we show that the unextendible entanglement is an entanglement monotone under two-extendible operations, which include local operations and one-way classical communication as a special case. Unextendible entanglement has several other desirable properties, including normalization and faithfulness. As applications, we show that the unextendible entanglement provides efficiently computable benchmarks for the rate of perfect entanglement distillation, as well as for the overhead of entanglement distillation.