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International Journal of Quantum Information

TOWARDS HYPERENTANGLED STATES OF TWO PHOTONS AND SIX QUBITS

Giuseppe Vallone, A. ROSSI, R. CECCARELLI, F. De Martini, Paolo Mataloni

Published Jan 1, 2009• 0 citations
Publisher

Abstract

We discuss two possible ways to increase the dimension of a 4-qubit hyperentangled state of two photons generated through the spontaneous parametric down conversion (SPDC) process. The two techniques are respectively based on the continuum of k-modes generated by a type I phase matched crystal and on the adoption of the time-energy entanglement as a further degree of freedom of the photons. The realization of an entangled state of two photons encoded in six (or even more) qubits, which can be transformed in a cluster state by suitable C-Phase gates, may represent a significant step towards the realization of efficient quantum computation based on the one-way model.

DOI
10.1142/s0219749909004797

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We discuss two possible ways to increase the dimension of a 4-qubit hyperentangled state of two photons generated through the spontaneous parametric down conversion (SPDC) process. The two techniques are respectively based on the continuum of k-modes generated by a type I phase matched crystal and on the adoption of the time-energy entanglement as a further degree of freedom of the photons. The realization of an entangled state of two photons encoded in six (or even more) qubits, which can be transformed in a cluster state by suitable C-Phase gates, may represent a significant step towards the realization of efficient quantum computation based on the one-way model.
A six-qubit hyperentangled state has been realized by entangling two photons in three degrees of freedom. These correspond to the polarization, the longitudinal momentum, and the indistinguishable emission produced by a two-crystal system operating with type I phase matching in the spontaneous parametric down conversion regime. The state has been characterized by a chained interferometric apparatus and its complete entangled nature has been tested by a witness criterion specifically presented for hyperentangled states. The experiment represents a realization of a genuine hyperentangled state with the maximum entanglement between the two particles allowed in the given ${2}^{3}\ifmmode\times\else\texttimes\fi{}{2}^{3}$ Hilbert space.
Six-qubit cluster states built on the simultaneous entanglement of two photons in three independent degrees of freedom, that is, polarization and a double longitudinal momentum, have been recently demonstrated. We present here the peculiar entanglement properties of the linear cluster state $|\mathrm{L͠C}{}_{6}\ensuremath{\rangle}$ related to the three degrees of freedom. This state has been adopted to realize various kinds of controlled not (cnot) gates, obtaining high values of the fidelity of the expected output states for all considered cases. Our results demonstrate that these states may represent a promising approach toward scalable quantum computation in a medium-term time scale. The future perspectives of a hybrid approach to one-way quantum computing based on multiple degrees of freedom and multiphoton cluster states are also discussed in the conclusion of this article.
Cluster states of two photons and four qubits, built on the double entanglement of two photons in the degrees of freedom of polarization and linear momentum, have been used in the realization of a complete set of basic operations of one-way quantum computation. Basic computation algorithms, namely, the Grover's search and the Deutsch's algorithm, have been realized by using these states. Hyperentangled states of increasing size are of paramount importance for the realization of even more complex algorithms and can be extended to a lager number of degrees of freedom of the photons. Some recent results obtained with entangled states of two photons and six qubits are presented.
Nowadays,the generation of multiphoton entangled states is almost realized by combining the coupled entangled photons emitted from spontaneous parametric down-conversion (SPDC) with the first-order term.In this case,one may focus mainly on the first-order term,and then avoid multipair emission events by restricting experimental parameters.On the other hand,for the higher-order terms in SPDC source,these emitted entangled photons have interesting features.For example,they are entangled maximally not only in photon number for the spatial modes,but also in polarization degree of freedom.In general,two photons,which are entangled in two or more degrees of freedom,are called hyperentangled pair of photons or hyperentangled state.We present a scheme to generate the four-photon hyperentangled state based on four indistinguishable photons emitted from SPDC source with the second-order term.Consider two SPDC sources with equal probability of emission of photons in respective spatial modes.With the passive linear optical devices,i.e., beam splitters,half wave plates,polarizing beam splitters,etc.,under the condition of registering a specified four-photon coincidence,we can obtain the four-photon hyperentangled state in which the photons are entangled in both polarization and spatial-mode degrees of freedom.Here,of course,for an arbitrary fourfold coincidence detection,one obtains a canonical four-photon Greenberger-Horne-Zeilinger (GHZ) state.Then we show the results of fourfold coincidence detections and the corresponding probabilities for the four-photon GHZ states,where the generation of the four-photon hyperentangled state is included as long as we are not to distinguish the two detectors located at the same locations. As a result,our scheme has two notable features.When we only consider the second-order emission,since it is not needed for us to distinguish between the two SPDC sources,the present scheme is simple and feasible.Also,based on the postselection with fourfold coincidence detection,our scheme is suitable for the normal first-order emission where we restrict the four photons emitted from the same source.In this sense,our scheme is efficient.In a word,we describe a method to generate the four-photon hyperentangled state with the second-order emission in SPDC source,which may contribute to the exploration of multipair entanglement with higher-order emissions from the SPDC source.
Hyperentanglement is a promising resource for achieving high capacity quantum communication. Here, we propose a compact scheme for the generation of path–frequency hyperentangled photon pairs via spontaneous parametric down-conversion (SPDC) processes, where six different paths and two different frequencies are covered. A two-dimensional periodical χ (2) nonlinear photonic crystal (NPC) is designed to satisfy type-I quasi-phase-matching conditions in the plane perpendicular to the incident pump beam, and a perfect phase match is achieved along the pump beam’s direction to ensure high conversion efficiency, with theoretically estimated photon flux up to 2.068 × 105 pairs⋅s−1⋅mm−2. We theoretically calculate the joint-spectral amplitude (JSA) of the generated photon pair and perform Schmidt decomposition on it, where the resulting entropy S of entanglement and effective Schmidt rank K reach 3.2789 and 6.4675, respectively. Our hyperentangled photon source scheme could provide new avenues for high-dimensional quantum communication and high-speed quantum information processing.