Entanglement Cost of Quantum Channels

Abstract

The entanglement cost of a quantum channel is the minimal rate at which entanglement (between sender and receiver) is needed in order to simulate many copies of a quantum channel in the presence of free classical communication. In this paper we show how to express this quantity as a regularised optimisation of the entanglement formation over states that can be generated between sender and receiver. Our formula is the channel analog of a well-known formula for the entanglement cost of quantum states in terms of the entanglement of formation; and shares a similar relation to the recently shattered hope for additivity. The entanglement cost of a quantum channel can be seen as the analog of the quantum reverse Shannon theorem in the case where free classical communication is allowed. The techniques used in the proof of our result are then also inspired by a recent proof of the quantum reverse Shannon theorem and feature the one-shot formalism for quantum information theory, the post-selection technique for quantum channels as well as Sion's minimax theorem. We discuss two applications of our result. First, we are able to link the security in the noisy-storage model to a problem of sending quantum rather than classical information through the adversary's storage device. This not only improves the range of parameters where security can be shown, but also allows us to prove security for storage devices for which no results were known before. Second, our result has consequences for the study of the strong converse quantum capacity. Here, we show that any coding scheme that sends quantum information through a quantum channel at a rate larger than the entanglement cost of the channel has an exponentially small fidelity.Comment: v3: error in proof of Lemma 13 corrected, corrected Figure 5, 24 pages, 5 figure

Amortization does not enhance the max-Rains information of a quantum channel

“…. Another function of a quantum channel is its entanglement cost [46], which we write as  ( ) E C and for which a definition is given in [46]. The following bounds and relations are known regarding these quantities:…”

Quantum cryptography beyond quantum key distribution

“…In fact, one can argue that the areas of limited-quantum storage models and entropic uncertainty relations have benefited a lot from each other, as research questions in one area have led to results in the other and vice versa. This fruitful co-existence is witnessed by a series of publications: [34,35,39,100,187].…”

Preface to the Second Edition

“…Berta et al (2013) proved that a quantity called the entanglement cost of a quantum channel is a strong converse rate for quantum communication. Morgan and Winter (2014) established what they called a "pretty strong converse" for the quantum capacity of degradable channels, meaning that there is a sharp transition in the fidelity from one to 1/2, when the rate of communication goes from below to above the quantum capacity (this is in the limit of many channel uses).…”