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Physical Review A

Secure information capacity of photons entangled in many dimensions

Jonathan Leach, Eliot Bolduc, Daniel J. Gauthier, Robert W. Boyd

Published Jun 28, 2012• 74 citations
Publisher

Abstract

We quantify precisely the maximum secure information capacity of photons entangled in high dimensions for entanglement in the orbital angular momentum and angular degrees of freedom. Our analysis takes careful account of the influence of experimental imperfections, such as nonunity detection efficiency, on the degree of Einstein-Podolsky-Rosen (EPR) entanglement and hence on the secure information capacity of the photon pairs. We find that there is is an optimal dimension that maximizes the secure information capacity whose value can be predicted analytically from the knowledge of only a few experimental parameters.

DOI
10.1103/physreva.85.060304

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We quantify precisely the maximum secure information capacity of photons entangled in high dimensions for entanglement in the orbital angular momentum and angular degrees of freedom. Our analysis takes careful account of the influence of experimental imperfections, such as nonunity detection efficiency, on the degree of Einstein-Podolsky-Rosen (EPR) entanglement and hence on the secure information capacity of the photon pairs. We find that there is is an optimal dimension that maximizes the secure information capacity whose value can be predicted analytically from the knowledge of only a few experimental parameters.
Orbital angular momentum (OAM) states of photons span a high-dimensional Hilbert space, and by using it, higher information capacity and enhanced security can be achieved for both classical communication and quantum key distribution. In this work, we propose a feasible basis-independent encoding scheme for OAM states and show its application in high-dimensional quantum secure direct communication, which enables direct transmission of meaningful messages, in addition to random secret key establishing. Imperfections of optical components may reduce the efficiency of the encoding scheme and, thus, the channel capacity. We discuss the optimal input distribution under realistic conditions to achieve maximal mutual information and an alternative solution using a different set of mutually unbiased bases in the case of a prime-dimensional quantum system.
The bandwidth of any communication system, classical or quantum, is limited by the number of orthogonal states in which the information can be encoded. Quantum key distribution systems available commercially rely on the two-dimensional polarisation state of photons. Quantum computation has also been largely designed on the basis of qubits. However, a photon is endowed with other degrees of freedom, such as orbital angular momentum (OAM). OAM is an attractive basis to be used for quantum information because it is discrete and theoretically infinite-dimensional. This promises a higher information capacity per photon which can lead to more complex quantum computation protocols and more security and robustness for quantum cryptography. Entanglement of OAM naturally arises from spontaneous parametric down-conversion (SPDC). However, any practical experiment utilising the innately high-dimensional entanglement of the orbital angular momentum (OAM) state space of photons is subject to the modal capacity of the detection system. Only a finite subset of this space is accessible experimentally. Given such a constraint, we show that the number of measured, entangled OAM modes in photon pairs generated by SPDC can be increased by tuning the phase-matching conditions in the SPDC process. We achieve this by tuning the orientation angle of the nonlinear crystal generating the entangled photons.
We present an experimental study of higher-dimensional quantum key distribution protocols based on mutually unbiased bases, implemented by means of photons carrying orbital angular momentum. We perform (d+1d+1d+1) mutually unbiased measurements in a classically simulated prepare-and-measure scheme and on a pair of entangled photons for dimensions ranging from d=2d=2d=2 to 5. In our analysis, we pay attention to the detection efficiency and photon pair creation probability. As security measures, we determine from experimental data the average error rate, the mutual information shared between the sender and receiver, and the secret key generation rate per photon. We demonstrate that increasing the dimension leads to an increased information capacity as well as higher key generation rates per photon. However, we find that the benefit of increasing the dimension is limited by practical implementation considerations, which in our case results in deleterious effects observed beyond a dimension of d=4d=4d=4.
Any practical experiment utilizing the innate DDD-dimensional entanglement of the orbital angular momentum (OAM) of photons is subject to the generation capacity of the entangled photon source and the modal capacity of the detection system. We report an experimental spontaneous parametric-down-conversion system able to generate and detect tunable high-dimensional OAM entanglement. By tuning the phase matching, we demonstrate a factor of 2 increase on the half-width of the OAM-correlation spectrum, from 10 to 20. In terms of quantum mutual information capacity, this is an increase from 3.183.183.18 to 4.954.954.95 bits/photon. Furthermore, we measure correlations in the conjugate variable, angular position, and obtain concurrence values 0.960.960.96 and 0.900.900.90. The good entanglement measures in both OAM and angular position bases indicate bipartite, DDD-dimensional entanglement where DDD is tunable.
Higher channel capacity and noise elimination are the key requirements for the implementation of long-distance quantum communication. As the additional degrees of freedom (DoF) of photons can be employed to achieve higher channel capacity and security beyond the polarizations DoF of photons, the photonic qubits are always employed as the flying qubits in quantum communication and quantum information processing. Here, exploiting the multiple DoFs of photons, we present an efficient quantum secure direct communication protocol based on the coding and manipulation of qubits on both the polarization and the orbital angular momentum of photons. Also, the numerical simulation is studied to further clarify the improvement of the channel capacity and the security. It is found that the channel capacity and the error rate (caused by eavesdropping) of the QSDC protocol which encoded on the polarization DoF and the OAM DoF is significantly higher than that of coding on only polarization DoF. We believe this work could provide more evidence for the applications of higher-dimensional qubits in quantum information science.
High-dimensional entangled states are of significant interest in quantum science as they increase the information content per photon and can remain entangled in the presence of significant noise. The authors develop the analytical theory and show experimentally that the noise tolerance of high-dimensional entanglement can be significantly increased by a modest increase in the size of the Hilbert space. For example, doubling the size of a Hilbert space with a local dimension of d = 300 leads to a reduction in the threshold detector efficiencies required for entanglement certification by two orders of magnitude. This work is developed in the context of spatial entanglement in the few-photon limit, but it can easily be translated to photonic states entangled in different degrees of freedom. The authors also demonstrate that knowledge of a single parameter, the signal-to-noise ratio, precisely links measures of entanglement to a range of experimental parameters quantifying noise in a quantum communication system, enabling accurate predictions of its performance. This work serves to answer a simple question: “Is high-dimensional photonic entanglement robust to noise?” Here, the authors show that the answer is more nuanced than a simple “yes” or “no” and involves a complex interplay between the noise characteristics of the state, channel, and detection system.