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arXiv:1506.07139 · Physical Review A

Optimal encoding capacity of a linear optical quantum channel

J. A. Smith, Dmitry B. Uskov, Lev Kaplan

Published Aug 10, 2015• 1 citation
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Abstract

Here, we study the classical information capacity of a quantum channel, assuming linear optical encoding, as a function of available photons and optical modes. We present a formula for general channel capacity and show that this capacity is achieved without requiring the use of entangling operations typically required for scalable universal quantum computation, e.g., Knill-Laflamme-Milburn measurement-assisted transformations. As an example, we provide an explicit encoding scheme using the resources required of standard dense coding using two dual-rail qubits (two photons in four modes). In this case, our protocol encodes one additional bit of information. Greater gains are expected for larger systems.

DOI
10.1103/physreva.92.022324

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Here, we study the classical information capacity of a quantum channel, assuming linear optical encoding, as a function of available photons and optical modes. We present a formula for general channel capacity and show that this capacity is achieved without requiring the use of entangling operations typically required for scalable universal quantum computation, e.g., Knill-Laflamme-Milburn measurement-assisted transformations. As an example, we provide an explicit encoding scheme using the resources required of standard dense coding using two dual-rail qubits (two photons in four modes). In this case, our protocol encodes one additional bit of information. Greater gains are expected for larger systems.
Entanglement can effectively increase communication channel capacity as evidenced by dense coding that predicts a capacity gain of 1 bit when compared to entanglement-free protocols. However, dense coding relies on Bell states and when implemented using photons the capacity gain is bounded by 0.5850.5850.585 bits due to one's inability to discriminate between the four optically encoded Bell states. In this paper we study the following question: Are there alternative entanglement-assisted protocols that rely only on linear optics, coincidence photon counting and separable single photon input states and at the same time provide a greater capacity gain than 0.5850.5850.585 bits. We show that besides the Bell states there is a class of bipartite four-mode two-photon entangled states that facilitate an increase in channel capacity. We also discuss how the proposed scheme can be generalized to the case of two-photon NNN-mode entangled states for N=6,8N=6,8N=6,8.
We consider the fundamental protocol of dense coding of classical information assuming that noise affects both the forward and backward communication lines between Alice and Bob. Assuming that this noise is described by the same quantum channel, we define its dense coding capacity by optimizing over all adaptive strategies that Alice can implement, while Bob encodes the information by means of Pauli operators. Exploiting techniques of channel simulation and protocol stretching, we are able to establish the dense coding capacity of Pauli channels in arbitrary finite dimension, with simple formulas for depolarizing and dephasing qubit channels.
Entanglement can effectively increase communication channel capacity as evidenced by dense coding that predicts a capacity gain of 1bit1\phantom{\rule{4.pt}{0ex}}\text{bit}1bit when compared to entanglement-free protocols. However, dense coding relies on Bell states and when implemented using photons the capacity gain is bounded by 0.585bits0.585\phantom{\rule{4.pt}{0ex}}\text{bits}0.585bits due to one's inability to discriminate between the four optically encoded Bell states. In this paper we study the following question: Are there alternative entanglement-assisted protocols that rely only on linear optics, coincidence photon counting, and separable single-photon input states and at the same time provide a greater capacity gain than 0.585bits0.585\phantom{\rule{4.pt}{0ex}}\text{bits}0.585bits? We show that besides the Bell states there is a class of bipartite four-mode two-photon entangled states that facilitate an increase in channel capacity. We also discuss how the proposed scheme can be generalized to the case of two-photon NNN-mode entangled states for N=6,8N=6,8N=6,8.
We present a scheme for linear optical quantum computing using time-bin-encoded qubits in a single spatial mode. We show methods for single-qubit operations and heralded controlled-phase (cphase) gates, providing a sufficient set of operations for universal quantum computing with the Knill-Laflamme-Milburn [Nature (London) 409, 46 (2001)] scheme. Our protocol is suited to currently available photonic devices and ideally allows arbitrary numbers of qubits to be encoded in the same spatial mode, demonstrating the potential for time-frequency modes to dramatically increase the quantum information capacity of fixed spatial resources. As a test of our scheme, we demonstrate the first entirely single spatial mode implementation of a two-qubit quantum gate and show its operation with an average fidelity of 0.84±0.07.
Current advancements in communication equipment demand the investigation of classical information transfer over quantum channels, by encompassing realistic scenarios in finite dimensions. To address this issue, we develop a framework for analyzing classical capacities of quantum channels where the set of states used for encoding information is restricted based on various physical properties. Specifically, we provide expressions for the classical capacities of noiseless and noisy quantum channels when the average energy of the encoded ensemble or the energy of each of the constituent states in the ensemble is bounded. In the case of qubit energy-preserving dephasing channels, we demonstrate that a nonuniform probability distribution based on the energy constraint maximizes capacity, while we derive the compact form of the capacity for equiprobable messages. We suggest an energy-constrained dense coding (DC) protocol that we prove to be optimal in the two-qubit situation and obtain a closed-form expression for the DC capacity. Additionally, we demonstrate a no-go result, which states that when the dimension of the sender and the receiver is two, no energy-preserving operation can offer any quantum advantage for energy-constrained entanglement-assisted capacity. We exhibit that, in the energy-constrained situation, classical-quantum noisy channels can show improved capabilities under entanglement assistance, a phenomenon that is unattainable in the unrestricted scenario.
One of the primary goals of information theory is to provide limits on the amount of information one can send through communication channels, and to understand encoding methods that achieve such limits. The discovery of dense coding showed that quantum channels can transmit twice as much classical information than is possible with a classical channel. This optimal rate requires using quantum systems maximally entangled between sender and receiver, and then the sender can encode her messages using unitary operations. The use of nonmaximally entangled systems has been intensively studied in recent years, and it has appeared as though unitary encoding might well be optimal in all cases. Indeed, this optimality of unitary operations has been found to hold under far more general conditions. Nonetheless, we here present numerical evidence supported by analytical arguments that there exist circumstances where more classical information can be encoded using nonunitary, rather than unitary, operations.