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 the capacity, such a strong converse theorem enhances our understanding of the capacity as a very sharp dividing line between achievable and unachievable rates of communication. Here, we show that such a strong converse theorem holds for the classical capacity of all entanglement-breaking channels and all Hadamard channels (the complementary channels of the former). These results follow by bounding the success probability in terms of a "sandwiched" Rényi relative entropy, by showing that this quantity is subadditive for all entanglement-breaking and Hadamard channels, and by relating this quantity to the Holevo capacity. Prior results regarding strong converse theorems for particular covariant channels emerge as a special case of our results.

Since 1984, various optical quantum key distribution (QKD) protocols have been proposed and examined. In all of them, the rate of secret key generation decays exponentially with distance. A natural and fundamental question is then whether there are yet-to-be discovered optical QKD protocols (without quantum repeaters) that could circumvent this rate-distance tradeoff. This paper provides a major step towards answering this question. Here we show that the secret key agreement capacity of a lossy and noisy optical channel assisted by unlimited two-way public classical communication is limited by an upper bound that is solely a function of the channel loss, regardless of how much optical power the protocol may use. Our result has major implications for understanding the secret key agreement capacity of optical channels-a long-standing open problem in optical quantum information theory-and strongly suggests a real need for quantum repeaters to perform QKD at high rates over long distances.

We introduce information-theoretic definitions for noise and disturbance in quantum measurements and prove a state-independent noise-disturbance tradeoff relation that these quantities have to satisfy in any conceivable setup. Contrary to previous approaches, the information-theoretic quantities we define are invariant under relabelling of outcomes and allow for the possibility of using quantum or classical operations to 'correct' for the disturbance. We also show how our bound implies strong tradeoff relations for mean square deviations.Heisenberg's Uncertainty Principle (HUP) states, loosely speaking, that in quantum theory a measurement process cannot measure one observable accurately, such as the position, without causing a measurable disturbance to another incompatible observable, such as the momentum. Notwithstanding the crucial role played by Heisenberg's principle in modern science, it took a long time between its first exposition [1,2] and its rigorous formalisation in terms of noise and disturbance operators [3][4][5]. The statistical spreads of these operators are measurable quantities, and hence tradeoff relations satisfied by these spreads yield precise mathematical translations of Heisenberg's intuition [3,6,7], that have recently been experimentally tested in a number of scenarios [8][9][10][11][12][13][14]. The use of noise and disturbance operators allows for a detailed, state-dependent formulation of HUP, able to capture the idea of 'how accurate' a measurement is with respect to one dynamical variable and 'how delicate' the same measurement is with respect to another dynamical variable.In this paper, we will explore a different approach to HUP, focused not on the change per se in a system's dynamical variables, but on the loss of correlation introduced by this change. In doing so, we will make use of ideas from information theory such as a 'guessing strategy' and error correction, and our definitions will be given in terms of information-theoretic quantities like entropies and conditional entropies. While we focus on the noisedisturbance context here, our approach also yields tradeoff relations for joint measurements.In order to understand the difference between the present approach and the previous one, let us consider, for example, the case of noise. While the noise, in its conventional form of root-mean-square deviation, is a statistical measure of the distance between a given system observable and the quantity actually measured [6,15], here we will only be interested in how well one can infer (i.e., guess) the value of a system observable from a given measurement outcome. That is to say, we will look only at the degree of correlation between the measurement and the observable, irrespective of how the corresponding outcomes and values are numerically labelled.Analogously, when characterising the disturbance, we will consider the measurement process as a source of noise for the system, and the degree to which such noise can be corrected (for a given observable) will give us our definition of...

Abstract. The data processing inequality states that the quantum relative entropy between two states ρ and σ can never increase by applying the same quantum channel N to both states. This inequality can be strengthened with a remainder term in the form of a distance between ρ and the closest recovered state (R • N )(ρ), where R is a recovery map with the property that σ = (R • N )(σ). We show the existence of an explicit recovery map that is universal in the sense that it depends only on σ and the quantum channel N to be reversed. This result gives an alternate, information-theoretic characterization of the conditions for approximate quantum error correction.

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Abstract-Holevo, Schumacher, and Westmoreland's coding theorem guarantees the existence of codes that are capacityachieving for the task of sending classical data over a channel with classical inputs and quantum outputs. Although they demonstrated the existence of such codes, their proof does not provide an explicit construction of codes for this task. The aim of the present paper is to fill this gap by constructing nearexplicit "polar" codes that are capacity-achieving. The codes exploit the channel polarization phenomenon observed by Arikan for the case of classical channels. Channel polarization is an effect in which one can synthesize a set of channels, by "channel combining" and "channel splitting," in which a fraction of the synthesized channels are perfect for data transmission while the other fraction are completely useless for data transmission, with the good fraction equal to the capacity of the channel. The channel polarization effect then leads to a simple scheme for data transmission: send the information bits through the perfect channels and "frozen" bits through the useless ones. The main technical contributions of the present paper are threefold. First, we leverage several known results from the quantum information literature to demonstrate that the channel polarization effect occurs for channels with classical inputs and quantum outputs. We then construct linear polar codes based on this effect, and the encoding complexity is O (N log N ), where N is the blocklength of the code. We also demonstrate that a quantum successive cancellation decoder works well, in the sense that the word error rate decays exponentially with the blocklength of the code. For this last result, we exploit Sen's recent "non-commutative union bound" that holds for a sequence of projectors applied to a quantum state.

Finally, here is a modern, self-contained text on quantum information theory suitable for graduate-level courses. Developing the subject 'from the ground up' it covers classical results as well as major advances of the past decade. Beginning with an extensive overview of classical information theory suitable for the non-expert, the author then turns his attention to quantum mechanics for quantum information theory, and the important protocols of teleportation, super-dense coding and entanglement distribution. He develops all of the tools necessary for understanding important results in quantum information theory, including capacity theorems for classical, entanglement-assisted, private and quantum communication. The book also covers important recent developments such as superadditivity of private, coherent and Holevo information, and the superactivation of quantum capacity. This book will be warmly welcomed by the upcoming generation of quantum information theorists and the already established community of classical information theorists.

This paper defines the squashed entanglement of a quantum channel as the maximum squashed entanglement that can be registered by a sender and receiver at the input and output of a quantum channel, respectively. A new subadditivity inequality for the original squashed entanglement measure of Christandl and Winter leads to the conclusion that the squashed entanglement of a quantum channel is an additive function of a tensor product of any two quantum channels. More importantly, this new subadditivity inequality, along with prior results of Christandl, Winter, et al., establishes the squashed entanglement of a quantum channel as an upper bound on the quantum communication capacity of any channel assisted by unlimited forward and backward classical communication. A similar proof establishes this quantity as an upper bound on the private capacity of a quantum channel assisted by unlimited forward and backward public classical communication. This latter result is relevant as a limitation on rates achievable in quantum key distribution. As an important application, we determine that these capacities can never exceed log((1 + η)/(1 − η)) for a pure-loss bosonic channel for which a fraction η of the input photons make it to the output on average. The best known lower bound on these capacities is equal to log(1/(1 − η)). Thus, in the high-loss regime for which η 1, this new upper bound demonstrates that the protocols corresponding to the above lower bound are nearly optimal.