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arXiv:0808.3325 · Physical Review Letters

Shannon Dimensionality of Quantum Channels and Its Application to Photon Entanglement

J. B. Pors, Sumant S. R. Oemrawsingh, Andrea Aiello, Martin P. van Exter, E. R. Eliel, Gert W. ’t Hooft, J. P. Woerdman

Published Sep 18, 2008• 84 citations
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Abstract

We introduce the concept of Shannon dimensionality D as a new way to quantify bipartite entanglement as measured in an experiment. This is applied to orbital-angular-momentum entanglement of two photons, using two state analyzers composed of a rotatable angular-sector phase plate that is lens coupled to a single-mode fiber. We can deduce the value of D directly from the observed two-photon coincidence fringe. In our experiment, D varies between 2 and 6, depending on the experimental conditions. We predict how the Shannon dimensionality evolves when the number of angular sectors imprinted in the phase plate is increased and anticipate that D approximately 50 is experimentally within reach.

DOI
10.1103/physrevlett.101.120502

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We introduce the concept of Shannon dimensionality D as a new way to quantify bipartite entanglement as measured in an experiment. This is applied to orbital-angular-momentum entanglement of two photons, using two state analyzers composed of a rotatable angular-sector phase plate that is lens coupled to a single-mode fiber. We can deduce the value of D directly from the observed two-photon coincidence fringe. In our experiment, D varies between 2 and 6, depending on the experimental conditions. We predict how the Shannon dimensionality evolves when the number of angular sectors imprinted in the phase plate is increased and anticipate that D approximately 50 is experimentally within reach.
Most protocols for quantum communication rely on entanglement between qubits. In this contribution we report on an experiment on angular entanglement of two photons. We introduce a quantifier of the dimensionality of the entanglement, using concepts from classical information theory. This Shannon dimensionality gives the effective number of information channels. We exploit the orbital-angular-momentum degree of freedom of photons that emerge from a parametric down-conversion source. The number of entangled modes, known as the Schmidt number, is of order 40. The entanglement is analyzed with a pair of angular state projectors, each consisting of a phase plate, a single-mode fiber and a single-photon detector. The phase plates are rotated about their axes and we record the photon coincidence rate as a function of their independent orientations. The coincidence count rates depend only on the relative orientation of the analyzers. This is a key signature of entanglement.
We engineer high-dimensional orbital-angular-momentum entanglement of photon pairs that emerge from a parametric down-conversion source. By means of two angular state analysers, in essence composed of a rotatable multi-sector phase plate and a single-mode fibre, we perform selective projective measurements that maximize the Shannon dimensionality D of the measured entanglement. The multi-sector phase plates have a binary phase profile along the azimuthal coordinate, and the arc sector sizes are optimized so as to maximize D . We find that the maximum dimensionality increases linearly with the number of sectors N . The potential of our method is illustrated with an experiment for N = 4, yielding D = 16.5.
Quantum entanglement is a fundamental trait of quantum mechanics that causes the information about the properties of two (or more) objects to be inextricably linked. When a measurement on one of the objects is performed, the state of the other object is immediately altered, even when these objects are separated at arbitrary distances. In this thesis, we explore the rich properties of entanglement in a high-dimensional mode space. Experimentally, we have implemented the high dimensionality by use of the orbital-angular-momentum degree of freedom of entangled photon pairs. The emphasis is on the question how to quantify the dimensionality of the entanglement as measured in an experiment. We introduce the Shannon dimensionality as a useful quantifier of measured entanglement. Furthermore, we discuss various production methods of optical phase plates, which we use to manipulate the orbital-angular-momentum states of light. Finally, we present an experimental feasibility study on the potential of orbital-angular-momentum entanglement for free-space quantum communication through the atmosphere
We describe a simple experiment that is ideally suited to analyze the high-dimensional entanglement contained in the orbital angular momenta (OAM) of entangled photon pairs. For this purpose we use a two-photon interferometer with a built-in image rotator and measure the two-photon visibility versus rotation angle. Mode selection with apertures allows one to tune the dimensionality of the entanglement; mode selection with spiral phase plates and fibers allows detection of a single OAM mode. The experiment is analyzed in two different ways: either via the continuous two-photon amplitude function or via a discrete modal (Schmidt) decomposition of this function. The latter approach proves to be very fruitful, as it provides a complete characterization of the OAM entanglement.