The Quantum Formalization Drive

Quantum information characteristics, such as quantum mutual information, loss, noise and coherent information are explicitly calculated for Bosonic attenuation/amplification channel with input Gaussian state. The coherent information is shown to be negative for the values of the attenuation coefficient $k<1/\sqrt{2}$.

The classical asymptotic equipartition property is the statement that, in the limit of a large number of identical repetitions of a random experiment, the output sequence is virtually certain to come from the typical set, each member of which is almost equally likely. In this paper, we prove a fully quantum generalization of this property, where both the…

We propose a general method for studying properties of quantum channels acting on an n-partite system, whose action is invariant under permutations of the subsystems. Our main result is that, in order to prove that a certain property holds for any arbitrary input, it is sufficient to consider the special case where the input is a particular de Finetti-type…

A strong converse theorem for the classical capacity of a quantum channel states that the probability of correctly decoding a classical message converges exponentially fast to zero in the limit of many channel uses if the rate of communication exceeds the classical capacity of the channel. Along with a corresponding achievability statement for rates below…

We are interested in the properties and relations of entanglement measures. Especially, we focus on the squashed entanglement and relative entropy of entanglement, as well as their analogues and variants. Our first result is a monogamy-like inequality involving the relative entropy of entanglement and its one-way LOCC variant. The proof is accomplished by…

Quantum f-divergences are a quantum generalization of the classical notion of f-divergences, and are a special case of Petz' quasi-entropies. Many well known distinguishability measures of quantum states are given by, or derived from, f-divergences; special examples include the quantum relative entropy, the Renyi relative entropies, and the Chernoff and…

We prove a lower bound on the relative entropy between two finite-dimensional states in terms of their entropy difference and the dimension of the underlying space. The inequality is tight in the sense that equality can be attained for any prescribed value of the entropy difference, both for quantum and classical systems. We outline implications for…

We derive one-shot upper bounds for quantum noisy channel codes. We do so by regarding a channel code as a bipartite operation with an encoder belonging to the sender and a decoder belonging to the receiver, and imposing constraints on the bipartite operation. We investigate the power of codes whose bipartite operation is non-signalling from Alice to Bob,…

We study a protocol in which many parties use quantum communication to transfer a shared state to a receiver without communicating with each other. This protocol is a multiparty version of the fully quantum Slepian-Wolf protocol for two senders and arises through the repeated application of the two-sender protocol. We describe bounds on the achievable rate…

The quantum capacity of a memoryless channel is often used as a single figure of merit to characterize its ability to transmit quantum information coherently. The capacity determines the maximal rate at which we can code reliably over asymptotically many uses of the channel. We argue that this asymptotic treatment is insufficient to the point of being…

A quantum channel is conjugate degradable if the channel's environment can be simulated up to complex conjugation using the channel's output. For all such channels, the quantum capacity can be evaluated using a single-letter formula. In this article we introduce conjugate degradability and establish a number of its basic properties. We then use it to…

We consider quantum channels with two senders and one receiver. For an arbitrary such channel, we give multi-letter characterizations of two different two-dimensional capacity regions. The first region is comprised of the rates at which it is possible for one sender to send classical information, while the other sends quantum information. The second region…

A qubit stored or transmitted in the presence of noise loses purity. If many copies of the same unknown pure state $\sigma = |\varphi\rangle\langle\varphi|$ have each passed through the same noisy channel, one may try to undo the noise collectively: feed the $N$ noisy copies into a quantum device and ask for $M$ output qubits that are as close as possible to $M$ per…

This paper is devoted to systematic study of properties of the quantum entropy and of the Holevo capacity considered as a function of a set of quantum states. The properties of restriction of the quantum entropy to arbitrary set of states are discussed. For some types of sets necessary and sufficient conditions of boundedness and of continuity of…

Degradable quantum channels are among the only channels whose quantum and private classical capacities are known. As such, determining the structure of these channels is a pressing open question in quantum information theory. We give a comprehensive review of what is currently known about the structure of degradable quantum channels, including a number of…

Channel capacity describes the size of the nearly ideal channels, which can be obtained from many uses of a given channel, using an optimal error correcting code. In this paper we collect and compare minor and major variations in the mathematically precise statements of this idea which have been put forward in the literature. We show that all the variations…

Coding theorems in quantum Shannon theory express the ultimate rates at which a sender can transmit information over a noisy quantum channel. More often than not, the known formulas expressing these transmission rates are intractable, requiring an optimization over an infinite number of uses of the channel. Researchers have rarely found quantum channels…

We give here a self contained and elementary introduction to the Conley-Zehnder index for a path of symplectic matrices. We start from the definition of the index as the degree of a map into the circle for a path starting at the identity and ending at a matrix for which 1 is not an eigenvalue. We prove some properties which characterize this index using…