Efficient Approximation of Quantum Channel Capacities

We propose an iterative method for approximating the capacity of classical-quantum channels with a discrete input alphabet and a finite dimensional output, possibly under additional constraints on the input distribution. Based on duality of convex programming, we derive explicit upper and lower bounds for the capacity. To provide an $\varepsilon$-close estimate to the capacity, the presented algorithm requires $O(\tfrac{(N \vee M) M^3 \log(N)^{1/2}}{\varepsilon})$, where $N$ denotes the input alphabet size and $M$ the output dimension. We then generalize the method for the task of approximating the capacity of classical-quantum channels with a bounded continuous input alphabet and a finite dimensional output. For channels with a finite dimensional quantum mechanical input and output, the idea of a universal encoder allows us to approximate the Holevo capacity using the same method. In particular, we show that the problem of approximating the Holevo capacity can be reduced to a multidimensional integration problem. For families of quantum channels fulfilling a certain assumption we show that the complexity to derive an $\varepsilon$-close solution to the Holevo capacity is subexponential or even polynomial in the problem size. We provide several examples to illustrate the performance of the approximation scheme in practice.Comment: 36 pages, 1 figur

Algorithmic Superactivation of Asymptotic Quantum Capacity of Zero-Capacity Quantum Channels

The superactivation of zero-capacity quantum channels makes it possible to use two zero-capacity quantum channels with a positive joint capacity for their output. Currently, we have no theoretical background to describe all possible combinations of superactive zero-capacity channels; hence, there may be many other possible combinations. In practice, to discover such superactive zero-capacity channel-pairs, we must analyze an extremely large set of possible quantum states, channel models, and channel probabilities. There is still no extremely efficient algorithmic tool for this purpose. This paper shows an efficient algorithmical method of finding such combinations. Our method can be a very valuable tool for improving the results of fault-tolerant quantum computation and possible communication techniques over very noisy quantum channels.Comment: 35 pages, 17 figures, Journal-ref: Information Sciences (Elsevier, 2012), presented in part at Quantum Information Processing 2012 (QIP2012), v2: minor changes, v3: published version; Information Sciences, Elsevier, ISSN: 0020-0255; 201

Remote State Preparation

Quantum teleportation uses prior entanglement and forward classical communication to transmit one instance of an unknown quantum state. Remote state preparation (RSP) has the same goal, but the sender knows classically what state is to be transmitted. We show that the asymptotic classical communication cost of RSP is one bit per qubit - half that of teleportation - and becomes even less when transmitting part of a known entangled state. We explore the tradeoff between entanglement and classical communication required for RSP, and discuss RSP capacities of general quantum channels.Comment: 4 pages including 1 epsf figure; v3 has an additional author and discusses relation to work of Devetak and Berger (quant-ph/0102123); v4 improves low-entanglement protocols without back communication to perform as well as low-entanglement protocols with back communication; v5 (journal version) has a few small change

Communicating over adversarial quantum channels using quantum list codes

We study quantum communication in the presence of adversarial noise. In this setting, communicating with perfect fidelity requires using a quantum code of bounded minimum distance, for which the best known rates are given by the quantum Gilbert-Varshamov (QGV) bound. By asking only for arbitrarily high fidelity and allowing the sender and reciever to use a secret key with length logarithmic in the number of qubits sent, we achieve a dramatic improvement over the QGV rates. In fact, we find protocols that achieve arbitrarily high fidelity at noise levels for which perfect fidelity is impossible. To achieve such communication rates, we introduce fully quantum list codes, which may be of independent interest.Comment: 6 pages. Discussion expanded and more details provided in proofs. Far less unclear than previous versio

Characterizing the performance of continuous-variable Gaussian quantum gates

The required set of operations for universal continuous-variable quantum computation can be divided into two primary categories: Gaussian and non-Gaussian operations. Furthermore, any Gaussian operation can be decomposed as a sequence of phase-space displacements and symplectic transformations. Although Gaussian operations are ubiquitous in quantum optics, their experimental realizations generally are approximations of the ideal Gaussian unitaries. In this work, we study different performance criteria to analyze how well these experimental approximations simulate the ideal Gaussian unitaries. In particular, we find that none of these experimental approximations converge uniformly to the ideal Gaussian unitaries. However, convergence occurs in the strong sense, or if the discrimination strategy is energy bounded, then the convergence is uniform in the Shirokov-Winter energy-constrained diamond norm and we give explicit bounds in this latter case. We indicate how these energy-constrained bounds can be used for experimental implementations of these Gaussian unitaries in order to achieve any desired accuracy.Comment: v3: 26 pages, 10 figures, final version accepted for publication in Physical Review Researc