Energy-time entangled qutrits: Bell tests and quantum communication

We have developed a scheme to generate, control, transmit and measure entangled photonic qutrits (two photons each of dimension d = 3). A Bell test of this source has previously been reported elsewhere [1], therefore, here we focus on how the control of the system is realized. Motivated by these results, we outline how the scheme can be used for two specific quantum protocols, namely key distribution and coin tossing and discuss some of their advantages and disadvantages. Performing quantum communication with high-dimensional systems would appear to be an obvious and straightforward extension to many of the qubit protocols that have driven quantum information science in recent years. There have been theoretical proposals for Quantum Key Distribution (QKD) [2] with greater security [3] as well as specific proposals for qutrits such as quantum coin tossing [4]. Fundamentally, high-dimensional Bell inequalities have revealed greater violations of non-locality [5] while increasing the dimensions of the entangled systems has been shown to facilitate closing the detection loop-hole [6] for such tests. Optics has been able to provide a realistic test-bed for these proposals with several different schemes being realized for generating high-dimensional entanglement [7, 8] as well as the scheme presented here [1, 9]. This source of entangled qutrits is analogous to the energy-time qubit arrangement of Franson [10]. The experimental scheme has been developed for telecom wavelengths and with proven long distance quantum communication architecture [11, 12] to optimise the usefulness of this high dimensional entanglement resource. We have already performed a Bell-type test on this system [1] where we determined a means of inferring a violation from the interference fringe visibility. We found a net visibility of Vnet= 0.979± 0.006 corresponding to a violation of the VBell limit by 34σ . From a practical perspective, it is interesting to discuss how the symmetry constraints were met and controlled in this set-up to achieve the violation. In doing this we also explore some of the subspaces of these high-dimensional states and we show how these correspond to the states required for quantum coin tossing. We also outline how we can use this scheme for QKD. We also briefly highlight some of the positive as well as negative aspects of these higher dimensional states in the case of these proposed protocols. The experimental set-up is illustrated in the schematic of Fig.1. We use energy-time entangled photon pairs created at telecom wavelengths, via a PPLN waveguide (courtesy of the Uni. of Nice) [13], and two three-arm interferometers to generate and analyze entangled qutrits. For each interferometer we can define a phase vector consisting of the two independent phases, e.g. the relative phases between the short-medium (m) and short-long (l), path-lengths. Coincidence measurements at the outputs of the interferometers project onto entangled qutrit states defined when the photons take the same path, i.e. short-short, medium-medium or long-long in each of Alice and Bob’s interferometers. In complete analogy with the Franson qubit arrangement [10], we use a continuous wave (CW) laser where the long coherence length does not provide a well defined creation time for the photon pairs. The coherence length of the photon pairs is much longer than the path-length differences in the interferometers, while the coherence length of the single photons is much less, such that no single-photon interference takes place. We have set the path-length differences, l −m ≈ m− s, corresponding to a time difference of 1.2 ns. An arrival-time-difference histogram with five peaks, due to all the possible path combinations, like that on the right in Fig.1 is generated for each detector combination. Therefore, when we select events in the central peak, i.e.. ∆t = tA − tB ≈ 0, these correspond to the post-selection of states of the form |ψ j,k〉 ∝ cs|ss〉+ cme i(αm+βm+χmjk)|mm〉+ cle i(αl+βl+χ ljk)|ll〉. (1)