We study entanglement distillation and dilution of states and channels in the single-shot regime. With the help of a recently introduced conversion distance, we provide compact closed-form expressions for the dilution and distillation of pure states and show how this can be used to efficiently calculate these quantities on multiple copies of pure states. These closed-form expressions also allow us to obtain second-order asymptotics. We then prove that the ε-single-shot entanglement cost of mixed states is given exactly in terms of an expression containing a suitably smoothed version of the conditional max-entropy. For pure states, this expression reduces to the smoothed max-entropy of the reduced state, for which we provide a closed-form expression. Analogously, we provide a closed-form expression for the smoothed min-entropy and connect it to the ε-single-shot distillable entanglement. Based on these results, we bound the single-shot entanglement cost of channels. We then turn to the one-way entanglement distillation of states and channels and provide bounds in terms of a quantity we denote coherent information of entanglement.

We introduce an axiomatic approach to entropies and relative entropies that relies only on minimal information-theoretic axioms, namely monotonicity under mixing and data-processing as well as additivity for product distributions. We find that these axioms induce sufficient structure to establish continuity in the interior of the probability simplex and meaningful upper and lower bounds, e.g., we find that every relative entropy satisfying these axioms must lie between the Rényi divergences of order 0 and ∞. We further show simple conditions for positive definiteness of such relative entropies and a characterisation in terms of a variant of relative trumping. Our main result is a one-to-one correspondence between entropies and relative entropies.

The von Neumann entropy is a central concept in physics and information theory, having a number of compelling physical interpretations. There is a certain perspective that the most fundamental notion in quantum mechanics is that of a quantum channel, as quantum states, unitary evolutions, measurements, and discarding of quantum systems can each be regarded as certain kinds of quantum channels. Thus, an important goal is to define a consistent and meaningful notion of the entropy of a quantum channel. Motivated by the fact that the entropy of a state ρ can be formulated as the difference of the number of physical qubits and the “relative entropy distance” between ρ and the maximally mixed state, here we define the entropy of a channel N as the difference of the number of physical qubits of the channel output with the “relative entropy distance” between N and the completely depolarizing channel. We establish that this definition satisfies all of the axioms, recently put forward in [Dour, IEEE Trans. Inf. Theory 65, 5880 (2019)], required for a channel entropy function. The task of quantum channel merging, in which the goal is for the receiver to merge his share of the channel with the environment's share, gives a compelling operational interpretation of the entropy of a channel. We define Rényi and min-entropies of a channel and establish that they satisfy the axioms required for a channel entropy function. Among other results, we also establish that a smoothed version of the min-entropy of a channel satisfies the asymptotic equipartition property.

We extend the definition of the conditional min-entropy from bipartite quantum states to bipartite quantum channels. We show that many of the properties of the conditional min-entropy carry over to the extended version, including an operational interpretation as a guessing probability when one of the subsystems is classical. We then show that the extended conditional min-entropy can be used to fully characterize when two bipartite quantum channels are related to each other via a superchannel (also known as supermap or a comb) that is acting on one of the subsystems. This relation is a pre-order that extends the definition of “quantum majorization” from bipartite states to bipartite channels, and can also be characterized with semidefinite programming. As a special case, our characterization provides necessary and sufficient conditions for when a set of quantum channels is related to another set of channels via a single superchannel. We discuss the applications of our results to channel discrimination, and to resource theories of quantum processes. Along the way we study channel divergences, entropy functions of quantum channels, and noise models of superchannels, including random unitary superchannels and doubly-stochastic superchannels. For the latter we give a physical meaning as being completely-uniformity preserving.

We find an upper bound for the minimum output entropy of a unital quantum channel, and obtain an exact formula for general qubit channels. Our techniques incorporate the Rényi entropies, particularly, with Rényi parameter α = 2. Moreover, since our upper bound is additive under tensor product, we get as a corollary an upper bound for the classical capacity of unital quantum channels. Interestingly, our upper bound for the classical capacity depends only on the operator norm of matrix representations of channels on the space of traceless Hermitian operators, and is tight in the sense that it gives the precise quantity of classical capacity of the Werner-Holevo channel. As an example, we study quantum channels with operator sum representation that is made of the discrete Weyl operators (generalized Pauli operators), and explain how our formula works in this case. Finally, we find new examples for which the minimum output Rényi 2-entropy is additive.

We show that the minimum von Neumann entropy output of a quantum channel is locally additive. Hastings' counterexample for the additivity conjecture makes this result quite surprising. In particular, it indicates that the nonadditivity of the minimum entropy output is a global effect of quantum channels.

We demonstrate a new construction for perfect quantum secret sharing (QSS) schemes based on imperfect “ramp” secret sharing combined with classical encryption, in which the individual parties' shares are split into quantum and classical components, allowing the former to be of lower dimension than the secret itself. We show that such schemes can be performed with smaller quantum components and lower overall quantum communication than required for existing methods. We further demonstrate that one may combine both imperfect quantum and imperfect classical secret sharing to produce an overall perfect QSS scheme, and that examples of such schemes (which we construct) can have the smallest quantum and classical share components possible for their access structures, something provably not achievable using perfect underlying schemes. Our construction has significant potential for being adapted to other QSS schemes based on stabilizer codes.