Measuring Entangled Qutrits and Their Use for Quantum Bit Commitment

Abstract

We produce and holographically measure entangled qudits encoded in transverse spatial modes of single photons. With the novel use of a quantum state tomography method that only requires two-state superpositions, we achieve the most complete characterisation of entangled qutrits to date. Ideally, entangled qutrits provide better security than qubits in quantum bit-commitment: we model the sensitivity of this to mixture and show experimentally and theoretically that qutrits with even a small amount of decoherence cannot offer increased security over qubits. PACS numbers: 42.50.Dv, 03.65.Wj, 03.67.Dd, 03.67.Mn Many two-level quantum systems, or qubits, have been used to encode information [1]; using d-level systems, or qudits, enables access to larger Hilbert spaces, which can provide significant improvements over qubits such as increased channel capacity in quantum communication [2]. When entangled, qutrits (d=3) provide the best known levels of security in quantum bit-commitment and coinflipping protocols, which cannot be matched using qubitbased systems [3]. The ability to completely characterise entangled qudits is critical for applications. This is only possible using quantum state tomography [4,5].Entangled qudits have been realised in few physical systems, and only indirect measurements have been made of the quantum states of these systems. Qutrit entanglement has been generated between the arrival times of correlated photon pairs, where fringe measurements were used to infer features such as fidelities with specific entangled states and to estimate a potential Bell violation [6]. It is also possible to encode qudits in the transverse spatial modes of a photon, Fig. 1. There have been measurements demonstrating, but again not quantifying, spatial mode entanglement in parametric downconversion [7], including fringe measurements [8,9] and the violation of a two-qutrit Bell inequality [10,11].Here, we use quantum state tomography to completely characterise entangled, photonic qudits (both d = 2 and 3) encoded in transverse spatial modes, measuring the amount of entanglement and the degree of mixture. We show how to use the qutrit system in a quantum bitcommitment protocol and investigate the experimental requirements for achieving the best known security [3]. To illustrate these results, we first introduce and demonstrate two conceptually distinct ways of encoding information in transverse spatial modes, which differ in the behaviour of superposition states. This work constitutes the most complete characterisation of spatially-encoded qubits and qutrits and the first quantitative measurement of entangled qutrit states.The Gaussian spatial modes are a complete basis for describing the paraxial propagation of light [13]. Two orthonormal mode families are shown in Fig. 1(a): the Hermite-Gauss (HG rs ) and Laguerre-Gauss-Vortex (LGV pl ). These modes are self-similar under propagation; modes of the same order experience the same propagation-dependent phase shift, the Gouy phase shift. We define degenerate qudits...

Hybrid entanglement swapping of photons: Creating the orbital angular momentum Bell states and Greenberger-Horne-Zeilinger states

“…As an ideal candidate to represent higher-dimensional quantum al-phabet, OAM has been employed to test local realism with the violation of a two-photon, three-dimensional Bell inequality [19] and OAM-based quantum cryptographic scheme for key distribution has been reported [20]. Besides, trigger qutrits for point-to-point communication protocol and the complete characterization of entangled qutrits for quantum bit commitment are both demonstrated [21,22]. Recently, much attention was also focused on the interaction between spin and orbital angular momentum.…”

Tomographic reconstruction of time-bin-entangled qudits

“…QST was previously demonstrated with entangled qudits using OAM modes (up to d = 8) spatial modes (up to d = 3), and time-energy modes (up to d = 4) [16][17][18][19]. To the best of our knowledge, time-bin QST has been previously demonstrated for qubits only [20][21][22][23].…”

Generation of time-bin-entangled photons without temporal postselection

“…By this process highly pure entangled states are produced by encoding qubits in a particular degree of freedom (DOF) of the photons, such as polarization [6], linear and angular momentum [7,8,9] and energy-time [10]. More recently, the possibility of spanning a large Hilbert space by encoding two photons in more than one DOF at the same time was demonstrated by exploiting the so-called hyperentanglement [11,12,13,14].…”