Generalized remote state preparation: Trading cbits, qubits and ebits in quantum communication

We consider the problem of communicating quantum states by simultaneously making use of a noiseless classical channel, a noiseless quantum channel and shared entanglement. We specifically study the version of the problem in which the sender is given knowledge of the state to be communicated. In this setting, a trade-off arises between the three resources, some portions of which have been investigated previously in the contexts of the quantum-classical trade-off in data compression, remote state preparation and superdense coding of quantum states, each of which amounts to allowing just two out of these three resources. We present a formula for the triple resource trade-off that reduces its calculation to evaluating the data compression trade-off formula. In the process, we also construct protocols achieving all the optimal points. These turn out to be achievable by trade-off coding and suitable time-sharing between optimal protocols for cases involving two resources out of the three mentioned above.Comment: 15 pages, 2 figures, 1 tabl

Trade-off capacities of the quantum Hadamard channels

Coding theorems in quantum Shannon theory express the ultimate rates at which a sender can transmit information over a noisy quantum channel. More often than not, the known formulas expressing these transmission rates are intractable, requiring an optimization over an infinite number of uses of the channel. Researchers have rarely found quantum channels with a tractable classical or quantum capacity, but when such a finding occurs, it demonstrates a complete understanding of that channel's capabilities for transmitting classical or quantum information. Here, we show that the three-dimensional capacity region for entanglement-assisted transmission of classical and quantum information is tractable for the Hadamard class of channels. Examples of Hadamard channels include generalized dephasing channels, cloning channels, and the Unruh channel. The generalized dephasing channels and the cloning channels are natural processes that occur in quantum systems through the loss of quantum coherence or stimulated emission, respectively. The Unruh channel is a noisy process that occurs in relativistic quantum information theory as a result of the Unruh effect and bears a strong relationship to the cloning channels. We give exact formulas for the entanglement-assisted classical and quantum communication capacity regions of these channels. The coding strategy for each of these examples is superior to a naive time-sharing strategy, and we introduce a measure to determine this improvement.Comment: 27 pages, 6 figures, some slight refinements and submitted to Physical Review

Optimal superdense coding of entangled states

We present a one-shot method for preparing pure entangled states between a sender and a receiver at a minimal cost of entanglement and quantum communication. In the case of preparing unentangled states, an earlier paper showed that a 2n-qubit quantum state could be communicated to a receiver by physically transmitting only n+o(n) qubits in addition to consuming n ebits of entanglement and some shared randomness. When the states to be prepared are entangled, we find that there is a reduction in the number of qubits that need to be transmitted, interpolating between no communication at all for maximally entangled states and the earlier two-for-one result of the unentangled case, all without the use of any shared randomness. We also present two applications of our result: a direct proof of the achievability of the optimal superdense coding protocol for entangled states produced by a memoryless source, and a demonstration that the quantum identification capacity of an ebit is two qubits.Comment: Final Version. Several technical issues clarifie

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

Entanglement-assisted communication of classical and quantum information

We consider the problem of transmitting classical and quantum information reliably over an entanglement-assisted quantum channel. Our main result is a capacity theorem that gives a three-dimensional achievable rate region. Points in the region are rate triples, consisting of the classical communication rate, the quantum communication rate, and the entanglement consumption rate of a particular coding scheme. The crucial protocol in achieving the boundary points of the capacity region is a protocol that we name the classically-enhanced father protocol. The classically-enhanced father protocol is more general than other protocols in the family tree of quantum Shannon theoretic protocols, in the sense that several previously known quantum protocols are now child protocols of it. The classically-enhanced father protocol also shows an improvement over a time-sharing strategy for the case of a qubit dephasing channel--this result justifies the need for simultaneous coding of classical and quantum information over an entanglement-assisted quantum channel. Our capacity theorem is of a multi-letter nature (requiring a limit over many uses of the channel), but it reduces to a single-letter characterization for at least three channels: the completely depolarizing channel, the quantum erasure channel, and the qubit dephasing channel.Comment: 23 pages, 5 figures, 1 table, simplification of capacity region--it now has the simple interpretation as the unit resource capacity region translated along the classically-enhanced father trade-off curv

The quantum dynamic capacity formula of a quantum channel

The dynamic capacity theorem characterizes the reliable communication rates of a quantum channel when combined with the noiseless resources of classical communication, quantum communication, and entanglement. In prior work, we proved the converse part of this theorem by making contact with many previous results in the quantum Shannon theory literature. In this work, we prove the theorem with an "ab initio" approach, using only the most basic tools in the quantum information theorist's toolkit: the Alicki-Fannes' inequality, the chain rule for quantum mutual information, elementary properties of quantum entropy, and the quantum data processing inequality. The result is a simplified proof of the theorem that should be more accessible to those unfamiliar with the quantum Shannon theory literature. We also demonstrate that the "quantum dynamic capacity formula" characterizes the Pareto optimal trade-off surface for the full dynamic capacity region. Additivity of this formula simplifies the computation of the trade-off surface, and we prove that its additivity holds for the quantum Hadamard channels and the quantum erasure channel. We then determine exact expressions for and plot the dynamic capacity region of the quantum dephasing channel, an example from the Hadamard class, and the quantum erasure channel.Comment: 24 pages, 3 figures; v2 has improved structure and minor corrections; v3 has correction regarding the optimizatio

An introduction to quantum game theory

The application of the methods of quantum mechanics to game theory provides us with the ability to achieve results not otherwise possible. Both linear superpositions of actions and entanglement between the players' moves can be exploited. We provide an introduction to quantum game theory and review the current status of the subject.Comment: 8 pages, RevTeX; v2 minor changes to the text in light of referees comments, references added/update

Superadditivity in trade-off capacities of quantum channels

In this article, we investigate the additivity phenomenon in the dynamic capacity of a quantum channel for trading classical communication, quantum communication and entanglement. Understanding such additivity property is important if we want to optimally use a quantum channel for general communication purpose. However, in a lot of cases, the channel one will be using only has an additive single or double resource capacity, and it is largely unknown if this could lead to an superadditive double or triple resource capacity. For example, if a channel has an additive classical and quantum capacity, can the classical-quantum capacity be superadditive? In this work, we answer such questions affirmatively. We give proof-of-principle requirements for these channels to exist. In most cases, we can provide an explicit construction of these quantum channels. The existence of these superadditive phenomena is surprising in contrast to the result that the additivity of both classical-entanglement and classical-quantum capacity regions imply the additivity of the triple capacity region.Comment: 15 pages. v2: typo correcte

Entanglement generation with a quantum channel and a shared state

We introduce a new protocol, the channel-state coding protocol, to quantum Shannon theory. This protocol generates entanglement between a sender and receiver by coding for a noisy quantum channel with the aid of a noisy shared state. The mother and father protocols arise as special cases of the channel-state coding protocol, where the channel is noiseless or the state is a noiseless maximally entangled state, respectively. The channel-state coding protocol paves the way for formulating entanglement-assisted quantum error-correcting codes that are robust to noise in shared entanglement. Finally, the channel-state coding protocol leads to a Smith-Yard superactivation, where we can generate entanglement using a zero-capacity erasure channel and a non-distillable bound entangled state.Comment: 5 pages, 3 figure

Asymptotic Compressibility of Entanglement and Classical Communication in Distributed Quantum Computation

We consider implementations of a bipartite unitary on many pairs of unknown input states by local operation and classical communication assisted by shared entanglement. We investigate to what extent the entanglement cost and the classical communication cost can be compressed by allowing nonzero but vanishing error in the asymptotic limit of infinite pairs. We show that a lower bound on the minimal entanglement cost, the forward classical communication cost, and the backward classical communication cost per pair is given by the Schmidt strength of the unitary. We also prove that an upper bound on these three kinds of the cost is given by the amount of randomness that is required to partially decouple a tripartite quantum state associated with the unitary. In the proof, we construct a protocol in which quantum state merging is used. For generalized Clifford operators, we show that the lower bound and the upper bound coincide. We then apply our result to the problem of distributed compression of tripartite quantum states, and derive a lower and an upper bound on the optimal quantum communication rate required therein.Comment: Section II and VIII adde