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Physical Review A

Generation of qudits and entangled qudits

Yongmin Li, Kuanshou Zhang, Kunchi Peng

Published Jan 3, 2008• 36 citations
Publisher

Abstract

We propose schemes for the generation of $d$-level quantum states (qudits) and quantum entangled states of $N$ qudits by adopting the polarization degree of freedom of photons. Arbitrary qudits can be prepared by putting $d\ensuremath{-}1$ identical and independent photons (each with a given polarization) into a single spatial and temporal mode. Then, by using $N$ such qudits, linear optics, and proper projection measurements, quantum entangled states of $N$ qudits can be produced. The frequencies of the $N$ qudits can be identical or different according to the corresponding linear optics and projection measurements.

DOI
10.1103/physreva.77.015802

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We propose schemes for the generation of ddd-level quantum states (qudits) and quantum entangled states of NNN qudits by adopting the polarization degree of freedom of photons. Arbitrary qudits can be prepared by putting d\ensuremath−1d\ensuremath{-}1d\ensuremath−1 identical and independent photons (each with a given polarization) into a single spatial and temporal mode. Then, by using NNN such qudits, linear optics, and proper projection measurements, quantum entangled states of NNN qudits can be produced. The frequencies of the NNN qudits can be identical or different according to the corresponding linear optics and projection measurements.
Qudits, d-level quantum systems, have been shown to provide a better resource for quantum key distribution and other Quantum Information protocols. It is customary to generate photonic qudits using more than one degree of freedom of the same photon. In much the same way, multi-qubit states are generated using only a pair of photons and ingenious ways to manipulate more than one degree of freedom independently. In contrast to such costly implementations in terms of quantum resources, we present the controlled generation of two copies of two-qudit states using four photons and a single degree of freedom, transverse momentum. The degree of entanglement within each pair was inferred by exploiting the availability of two copies of the same state, without the need of a full tomographic reconstruction of the states, and both highly-entangled and separable states were generated. We show theoretically that the set of states obtainable using our setup is very diverse, ranging from maximally entangled states of qudits to separable states.
The generation of a high-dimensional quantum entangled state provides a crucial resource for achieving more complex and diverse quantum information processing tasks. In this paper, we present two deterministic protocols designed for the generation of a four-dimensional two-qudit Bell state and three-qudit Greenberger-Horne-Zeilinger (GHZ) state assisted by cross-Kerr nonlinearity, where the four-dimensional qudit is encoded in the hybrid polarization-spatial state of the single photon to conserve quantum resources and alleviate environmental noise. The proposed protocols for generating a four-dimensional two-qudit Bell state and three-qudit GHZ state are circuit concise, high success, robust fidelity, noise resistant, and resource saving. Moreover, the four-dimensional NNN-qudit (N>3N>3N>3) GHZ states can be generated with the help of the $N\ensuremath{-}1$ generating devices of the four-dimensional Bell state. Consequently, its efficiency and fidelity decay exponentially by a factor of $N\ensuremath{-}1$. Further, the arbitrary high-dimensional qudit-encoded entangled states can be achieved with linear-optics four-dimensional single-qudit XXX gates and ZZZ gates. These protocols offer a reference for further advancements in high-dimensional state generation technology and providing insights for the practical implementations of metrological enhancement, quantum simulation, high-dimensional quantum computation, and quantum error correction.
Hybrid entangled states, having entanglement between different degrees-of-freedom (DoF) of a particle pair, are of great interest for quantum information science and communication protocols. Among different DoFs, the hybrid entangled states encoded with polarization and orbital angular momentum (OAM) allow the generation of qubit-qudit entangled states, macroscopic entanglement with very high quanta of OAM and improvement in angular resolution in remote sensing. Till date, such hybrid entangled states are generated by using a high-fidelity polarization entangled state and subsequent imprinting of chosen amount of OAM using suitable mode converters such as spatial light modulator in complicated experimental schemes. Given that the entangled sources have feeble number of photons, loss of photons during imprinting of OAM using diffractive optical elements limits the use of such hybrid state for practical applications. Here we report, on a simple experimental scheme to generate hybrid entangled state in polarization and OAM through direct transfer of classical non-separable state of the pump beam in parametric down conversion process. As a proof of principle, using local non-separable pump state of OAM mode l=3, we have produced quantum hybrid entangled state with entanglement witness parameter of W-1.25 violating by 8 standard deviation. The generic scheme can be used to produce hybrid entangled state between two photons differing by any quantum number through proper choice of non-separable state of the pump beam.
We propose a method for the generation of a large variety of entangled states, encoded in the polarization degrees of freedom of NNN photons, within the same experimental setup. Starting with uncorrelated photons, emitted from NNN arbitrary single-photon sources, and using linear optical tools only, we demonstrate the creation of all symmetric states (e.g., GHZ and WWW states), as well as all symmetric and nonsymmetric total angular momentum eigenstates of the NNN-qubit compound.