Entanglement required in achieving entanglement-assisted channel capacities

Entanglement shared between the two ends of a quantum communication channel has been shown to be a useful resource in increasing both the quantum and classical capacities for these channels. The entanglement-assisted capacities were derived assuming an unlimited amount of shared entanglement per channel use. In this paper, bounds are derived on the minimum amount of entanglement required per use of a channel, in order to asymptotically achieve the capacity. This is achieved by introducing a class of entanglement-assisted quantum codes. Codes for classes of qubit channels are shown to achieve the quantum entanglement-assisted channel capacity when an amount of shared entanglement per channel given by, E = 1 - Q_E, is provided. It is also shown that for very noisy channels, as the capacities become small, the amount of required entanglement converges for the classical and quantum capacities.Comment: 9 pages, 2 figures, RevTex

Quantum Feedback Channels

In Shannon information theory the capacity of a memoryless communication channel cannot be increased by the use of feedback. In quantum information theory the no-cloning theorem means that noiseless copying and feedback of quantum information cannot be achieved. In this paper, quantum feedback is defined as the unlimited use of a noiseless quantum channel from receiver to sender. Given such quantum feedback, it is shown to provide no increase in the entanglement--assisted capacities of a memoryless quantum channel, in direct analogy to the classical case. It is also shown that in various cases of non-assisted capacities, feedback may increase the capacity of memoryless quantum channels.Comment: 5 pages, requires IEEEtran.cls, expanded, proofs added, references adde

Additive Extensions of a Quantum Channel

We study extensions of a quantum channel whose one-way capacities are described by a single-letter formula. This provides a simple technique for generating powerful upper bounds on the capacities of a general quantum channel. We apply this technique to two qubit channels of particular interest--the depolarizing channel and the channel with independent phase and amplitude noise. Our study of the latter demonstrates that the key rate of BB84 with one-way post-processing and quantum bit error rate q cannot exceed H(1/2-2q(1-q)) - H(2q(1-q)).Comment: 6 pages, one figur

Quantum Channel Capacities Per Unit Cost

Communication over a noisy channel is often conducted in a setting in which different input symbols to the channel incur a certain cost. For example, for bosonic quantum channels, the cost associated with an input state is the number of photons, which is proportional to the energy consumed. In such a setting, it is often useful to know the maximum amount of information that can be reliably transmitted per cost incurred. This is known as the capacity per unit cost. In this paper, we generalize the capacity per unit cost to various communication tasks involving a quantum channel such as classical communication, entanglement-assisted classical communication, private communication, and quantum communication. For each task, we define the corresponding capacity per unit cost and derive a formula for it analogous to that of the usual capacity. Furthermore, for the special and natural case in which there is a zero-cost state, we obtain expressions in terms of an optimized relative entropy involving the zero-cost state. For each communication task, we construct an explicit pulse-position-modulation coding scheme that achieves the capacity per unit cost. Finally, we compute capacities per unit cost for various bosonic Gaussian channels and introduce the notion of a blocklength constraint as a proposed solution to the long-standing issue of infinite capacities per unit cost. This motivates the idea of a blocklength-cost duality, on which we elaborate in depth.Comment: v3: 18 pages, 2 figure

Experimental detection of quantum channel capacities

We present an effcient experimental procedure that certifies non vanishing quantum capacities for qubit noisy channels. Our method is based on the use of a fixed bipartite entangled state, where the system qubit is sent to the channel input. A particular set of local measurements is performed at the channel output and the ancilla qubit mode, obtaining lower bounds to the quantum capacities for any unknown channel with no need of a quantum process tomography. The entangled qubits have a Bell state configuration and are encoded in photon polarization. The lower bounds are found by estimating the Shannon and von Neumann entropies at the output using an optimized basis, whose statistics is obtained by measuring only the three observables $\sigma_{x}\otimes\sigma_{x}$, $\sigma_{y}\otimes\sigma_{y}$ and $\sigma_{z}\otimes\sigma_{z}$.Comment: 5 pages and 3 figures in the principal article, and 4 pages in the supplementary materia

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

Capacities of Quantum Amplifier Channels

Quantum amplifier channels are at the core of several physical processes. Not only do they model the optical process of spontaneous parametric down-conversion, but the transformation corresponding to an amplifier channel also describes the physics of the dynamical Casimir effect in superconducting circuits, the Unruh effect, and Hawking radiation. Here we study the communication capabilities of quantum amplifier channels. Invoking a recently established minimum output-entropy theorem for single-mode phase-insensitive Gaussian channels, we determine capacities of quantum-limited amplifier channels in three different scenarios. First, we establish the capacities of quantum-limited amplifier channels for one of the most general communication tasks, characterized by the trade-off between classical communication, quantum communication, and entanglement generation or consumption. Second, we establish capacities of quantum-limited amplifier channels for the trade-off between public classical communication, private classical communication, and secret key generation. Third, we determine the capacity region for a broadcast channel induced by the quantum-limited amplifier channel, and we also show that a fully quantum strategy outperforms those achieved by classical coherent detection strategies. In all three scenarios, we find that the capacities significantly outperform communication rates achieved with a naive time-sharing strategy.Comment: 16 pages, 2 figures, accepted for publication in Physical Review