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Increasing the Quantum Number, Dimensionality and Complexity of Entanglement

Robert Fickler, Mario Krenn, Radek Łapkiewicz, Marcus Huber, William N. Plick, Sven Ramelow, Anton Zeilinger

Published Jan 1, 2015• 0 citations
Publisher

Abstract

Complex transverse spatial modes of photons are very interesting for both, foundations and applications of quantum information. In a first experiment, we generate two-dimensional bi-partite entanglement where each photon carries an orbital angular momentum of a quantum number up to 300. Secondly, we demonstrate (100 × 100)-dimensional entanglement, using again the spatial modes of photons and a novel nonlinear criterion which uses only information about two-dimensional subspace correlations. In a third test, we increase the complexity of the state by entangling ”vector photons” that show a transverse varying polarization. Thereby, we demonstrate a surprising property: Such photons can be both entangled and not entangled in polarization depending on their transverse spatial position.

DOI
10.1364/cleo_si.2015.stu1l.6

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Complex transverse spatial modes of photons are very interesting for both, foundations and applications of quantum information. In a first experiment, we generate two-dimensional bi-partite entanglement where each photon carries an orbital angular momentum of a quantum number up to 300. Secondly, we demonstrate (100 × 100)-dimensional entanglement, using again the spatial modes of photons and a novel nonlinear criterion which uses only information about two-dimensional subspace correlations. In a third test, we increase the complexity of the state by entangling ”vector photons” that show a transverse varying polarization. Thereby, we demonstrate a surprising property: Such photons can be both entangled and not entangled in polarization depending on their transverse spatial position.
Introduction In this chapter, we will be concerned with entangled states of light. Historically, quantum states of light, i.e., photons, have played a dominant role in the experimental study of quantum entanglement. For instance, the first Bell tests and the first realizations of quantum teleportation were performed using two-dimensional polarization-entangled twophoton states [3]. From an experimental point of view these states are readily prepared and manipulated using conventional polarization optics. Recently, however, both theoretical and experimental efforts have shifted towards entanglement in more than two modes , which is often referred to as high-dimensional entanglement. This is much richer than two-dimensional entanglement, as is, for instance, reflected in stronger violations of various measures of nonlocality which reveal the complexity of high-dimensional Hilbert spaces [4–7]. Also, from an applications perspective, it promises a larger channel capacity [8, 9] and improved security for quantum communication [10, 11]. There are several ways to implement high-dimensional two-photon entanglement. One can exploit the temporal [12], frequency [13], or spatial [14–18] degrees of freedom of the optical field. A promising way to implement high-dimensional two-photon entanglement is to use the orbital-angular-momentum (OAM) degree of freedom of light. Light beams that carry OAM have a helical wavefront that spirals around the optical axis during propagation [19]. The winding number m , which characterizes the OAM eigenmodes, can adopt any integer value between −∞ and ∞; since all coaxial OAM modes are orthogonal they can thus span a very high-dimensional Hilbert space.
Entangled quantum systems have properties that have fundamentally overthrown the classical worldview. Increasing the complexity of entangled states by expanding their dimensionality allows the implementation of novel fundamental tests of nature, and moreover also enables genuinely new protocols for quantum information processing. Here we present the creation of a (100 × 100)-dimensional entangled quantum system, using spatial modes of photons. For its verification we develop a novel nonlinear criterion which infers entanglement dimensionality of a global state by using only information about its subspace correlations. This allows very practical experimental implementation as well as highly efficient extraction of entanglement dimensionality information. Applications in quantum cryptography and other protocols are very promising.
Photons may have homogeneous polarization and may carry quantized orbital angular momentum (OAM). Photon entanglement has been realized in various degrees of freedom such as polarization and OAM. Using a pair of orthogonally polarized states carrying opposite-handedness quantized OAMs could create "quantized" vector polarization states with space-variant polarization structures. It is thus possible to extend the polarization degree of freedom from two dimensional space to indefinite dimensional discrete Hilbert space. We present a class of vector-polarization entangled Bell states, which use the spatial modes of the vector fields with space-variant polarization structure. We propose a scheme of creating the vector-polarization entangled Bell states using a Sagnac interferometer. We also design an analyzer for identifying the vector-polarization entangled Bell states. Such a class of entanglement is important for quantum information science and technology, and fundamental issues of quantum theory, due to its advantage of the increase in information capacity.
Entangled quantum systems have properties that have fundamentally overthrown the classical worldview. Increasing the complexity of entangled states by expanding their dimensionality allows the implementation of novel fundamental tests of nature, and moreover also enables genuinely new protocols for quantum information processing. Here we present the creation of a (100x100)-dimensional entangled quantum system, using spatial modes of photons. For its verification we develop a novel nonlinear criterion which infers entanglement dimensionality of a global state by using only information about its subspace correlations. This allows very practical experimental implementation as well as highly efficient extraction of entanglement dimensionality information. Applications in quantum cryptography and other protocols are very promising.
We report the efficient creation and detection of hybrid entanglement between one photon's polarization and another photon's complex transverse polarization pattern including various polarization singularities. The polarization measurement of the first photon triggers a polarization-sensitive imaging of its partner photon, the vector photon, using a single-photon-sensitive camera. Thereby, the vector photon's complex polarization pattern is reconstructed tomographically dependent on the type of polarization measurement performed on its partner. To evaluate the quality of the observed patterns, we introduce a fidelity-like measure which compares the reconstructed polarization with the theoretically expected one at every transverse spatial position. We find that the measured polarization patterns and the theory overlap by more than 90%. Moreover, we visualize the varying strengths of polarization entanglement for different transverse regions and demonstrate an interesting phenomenon: Each vector photon can be both entangled and not entangled in polarization with its partner photon. We give an intuitive, information-theoretical explanation for our results. Our results pave the way to study quantum properties of the rich field of such complex light modes with the potential to improve various applications in quantum informational tasks.
We put forward an approach to estimate the amount of bipartite spatial entanglement of down-converted photon states correlated in orbital angular momentum and the magnitude of the transverse (radial) wave vectors. Both degrees of freedom are properly considered in our framework, which only requires azimuthal local linear optical transformations and mode selection analysis with two fiber detectors. The coincidence distributions predicted by our approach give an excellent fit to the distributions measured in a recent experiment aimed to show the very high-dimensional transverse entanglement of twin photons from a down-conversion source. Our estimate for the Schmidt number is substantially lower but still confirms the presence of high-dimensional entanglement.