Superadditivity of Private Information for Any Number of Uses of the Channel

Abstract

The quantum capacity of a quantum channel is always smaller than the capacity of the channel for private communication. Both quantities are given by the infinite regularization of the coherent and the private information, respectively, which makes their evaluation very difficult. Here, we construct a family of channels for which the private and coherent information can remain strictly superadditive for unbounded number of uses, thus demonstrating that the regularization is necessary. We prove this by showing that the coherent information is strictly larger than the private information of a smaller number of uses of the channel. This implies that even though the quantum capacity is upper bounded by the private capacity, the nonregularized quantities can be interleaved. Efficient information transmission is the cornerstone of all information processing tasks in our interconnected world. In the most basic scenario, two parties, linked by a fixed communication channel wish to exchange messages with each other. What is the maximum rate at which they can reliably transmit information?Classical information theory gives an exhaustive answer to this question [1]. There exists an efficient convex optimization algorithm which takes the description of a channel and calculates its capacity to convey information. This is the consequence of a particularly simple analytic expression for the classical capacity of a channel. Our world is inherently quantum and when we turn to the channels that transmit quantum information we are able to perform many novel information processing tasks which are impossible in the classical theory, such as establishing entanglement between sender and receiver. Presently, when confronted with the above question for the quantum channels, there is no known efficient algorithm that takes the description of an arbitrary channel and calculates its capacity. Different types of capacity of the quantum channel are defined as regularized quantities [2-9], which implies that in order to compute them it is necessary to perform an unbounded optimization over the number of the copies of the channel. In practice it means that to estimate the capacity for n uses of the channel the dimension of the state space which one has to optimize over may increase exponentially in n.Arguably, the biggest practical success of quantum information theory to date is the possibility of quantum key distribution (QKD) [10][11][12]. QKD allows two distant parties to agree on a secret key independent of any eavesdropper. The required assumptions are access to a quantum channel with positive private capacity and the validity of quantum physics. However, in practice one does not know the quantum channel exactly, and to characterize it one uses a public authentic classical channel. On the other hand, key distribution is a primitive that can only be implemented with classical resources if one is willing to constrain the power of the eavesdropper. Even though there exist practical QKD schemes which enable secure communication over large dis...

Memory effects can make the transmission capability of a communication channel uncomputable

“…Furthermore, our result connects with recent work regarding the different capacities of memoryless quantum channels [31,32], showing some evidence that these capacities might FIG. 1.…”

Approximate Degradable Quantum Channels

“…The private classical capacity of a quantum channel characterizes the highest possible rate at which classical information can be transmitted asymptotically reliably per channel use such that no information about the message leaks to the environment. Both of these quantities are mathematically characterized by a multi-letter expression, using regularization, that is complicated to evaluate -as a matter of fact, it is not even known to be computable [11,17]. In general, it is even difficult to derive good upper and lower bounds that can be evaluated efficiently for the two capacities.…”

Superadditivity in Trade-Off Capacities of Quantum Channels

“…The second inequality comes from our assumption P < and Eq. (27). The last inequality comes from Eq.…”

“…Subsequently, they are used in Ref. [27] to show the superadditivity of private information, with an alternative definition. Recently, they are also used in Ref.…”