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arXiv:1502.05326 · Physical Review Letters

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

David Elkouss, Sergii Strelchuk

Published Jul 20, 2015• 39 citations
arXivHTMLPDF

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.

DOI
10.1103/physrevlett.115.040501

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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.
The quantum capacity of a quantum channel is always smaller than the capacity of the channel for private communication. However, both quantities are given by the infinite regularization of respectively the coherent and the private information. Here, we construct a family of channels for which the private and coherent information can remain strictly superadditive for unbounded number of uses. 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. It turns out that even though the quantum capacity is upper bounded by the private capacity, the non-regularized quantities can be interleaved. From an operational point of view, the private capacity can be used for gauging the practical value of quantum channels for secure communication and, consequently, for key distribution. We thus show that in order to evaluate the interest a channel for this task it is necessary to optimize the private information over an unlimited number of uses of the channel.
Quantum communication channels differ from their classical counterparts because their capacities can be superadditive. The principle of monogamy of entanglement suggests that superadditive improvements in the transmission capacity of a channel should reduce the amount of information loss to the environment. We challenge this intuition by demonstrating that the coherent and private information of a channel and its complement can be simultaneously superadditive for arbitrarily many channel uses. To quantify the limits of this effect, we consider the notion of max (resp. total) private information of a channel, which represents the maximum (resp. sum) of the private information of the channel itself and its complement, and study its relationship with the coherent information of the individual direct and complementary channels. We show that these quantities can obey different interleaving sequences of inequalities for a varying number of channel uses.
Quantum channel capacity is a fundamental quantity in order to understand how well quantum information can be transmitted or corrected when subjected to noise. However, it is generally not known how to compute such quantities since the quantum channel coherent information is not additive for all channels, implying that it must be maximized over an unbounded number of channel uses. This leads to the phenomenon known as superadditivity, which refers to the fact that the regularized coherent information of nnn channel uses exceeds one-shot coherent information. In this article, we study how the gain in quantum capacity of qudit depolarizing channels relates to the dimension of the considered systems. We make use of an argument based on the no-cloning bound in order to prove that the possible superadditive effects decrease as a function of the dimension for such family of channels. In addition, we prove that the capacity of the qudit depolarizing channel coincides with the coherent information when $d\ensuremath{\rightarrow}\ensuremath{\infty}$. We also discuss the private classical capacity and obtain similar results. We conclude that when high-dimensional qudits experiencing depolarizing noise are considered, the coherent information of the channel is not only an achievable rate, but essentially the maximum possible rate for any quantum block code.
The quantum capacity of a noisy quantum channel determines the maximal rate at which we can code reliably over asymptotically many uses of the channel, and it characterizes the channel's ultimate ability to transmit quantum information coherently. In this paper, we derive single-letter upper bounds on the quantum and private capacities of quantum channels. The quantum capacity of a quantum channel is always no larger than the quantum capacity of its extended channels since the extensions of the channel can be considered as assistance from the environment. By optimizing the parametrized extended channels with specific structures such as the flag structure, we obtain new upper bounds on the quantum capacity of the original quantum channel. Furthermore, we extend our approach to estimating the fundamental limits of private communication and one-way entanglement distillation. As notable applications, we establish improved upper bounds to the quantum and private capacities for fundamental quantum channels of great interest in quantum information, some of which are also the sources of noise in superconducting quantum computing. In particular, our upper bounds on the quantum capacities of the depolarizing channel and the generalized amplitude damping channel are strictly better than previously best-known bounds.
Superadditivity of quantum capacity of communication channels is one of the most interesting findings of the field. Yard and Smith, finding a relation between the private capacity and the assisted quantum capacity, showed a striking example of superadditivity, i.e. two channels of zero quantum capacity could achieve a positive quantum capacity when used together [1]. The four dimensional channels they used are a 50% erasure channel (therefore zero quantum capacity, due to no-cloning theorem) and a Horodecki channel (again zero quantum capacity due to incapability of sharing free entanglement). In this work we present the more general cases of superadditivity. Directly calculating the lower bounds of joint quantum capacities without using the relation between private capacity and assisted quantum capacity, we examine scenarios considering erasure channels of arbitrary probabilities and different Horodecki channels, and discuss the roles of degradability and anti-degradability as well as the role of the private capacity in superadditivity. We also derive an upper bound for the joint quantum capacity for the superactivation case.
The quantum capacity of a noisy quantum channel determines the maximal rate at which we can code reliably over asymptotically many uses of the channel, and it characterizes the channel's ultimate ability to transmit quantum information coherently. In this paper, we derive single-letter upper bounds on the quantum and private capacities of quantum channels. The quantum capacity of a quantum channel is always no larger than the quantum capacity of its extended channels, since the extensions of the channel can be considered as assistance from the environment. By optimizing the parametrized extended channels with specific structures such as the flag structure, we obtain new upper bounds on the quantum capacity of the original quantum channel. Furthermore, we extend our approach to estimating the fundamental limits of private communication and one-way entanglement distillation. As notable applications, we establish improved upper bounds to the quantum and private capacities for fundamental quantum channels of interest in quantum information, some of which are also the sources of noise in superconducting quantum computing. In particular, our upper bounds on the quantum capacities of the depolarizing channel and the generalized amplitude damping channel are strictly better than previously best-known bounds for certain regimes.