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Quantum Electronics

Qutrits in multiparticle systems

В. Н. Горбачев, Andrew I. Trubilko

Published Dec 31, 2007• 0 citations
Publisher

Abstract

The evolution of complex composite quantum systems which can be reduced to systems with a small dimensionality of the Hilbert space (of the qutrit, ququart type, etc.) is considered. In the case of the interaction of an ensemble of two-level atoms with light, the conditions are found under which a qutrit is produced from the light and atomic states. The properties and possible applications of a qutrit based of the Fock states of light in which two photons are distributed among three modes are discussed. It is shown that this state has the nonclassical photon statistics, is entangled and can be used as a quantum channel for the teleportation, dense coding, and key distribution.

DOI
10.1070/qe2007v037n12abeh013673

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The evolution of complex composite quantum systems which can be reduced to systems with a small dimensionality of the Hilbert space (of the qutrit, ququart type, etc.) is considered. In the case of the interaction of an ensemble of two-level atoms with light, the conditions are found under which a qutrit is produced from the light and atomic states. The properties and possible applications of a qutrit based of the Fock states of light in which two photons are distributed among three modes are discussed. It is shown that this state has the nonclassical photon statistics, is entangled and can be used as a quantum channel for the teleportation, dense coding, and key distribution.
Multiparticle entangled states generated via interaction between narrowband light and an ensemble of identical two-level atoms are considered. Depending on the initial photon statistics, correlation between atoms and photons can give rise to entangled states of these systems. It is found that the state of any pair of atoms interacting with weak single-mode squeezed light is inseparable and robust against decay. Optical schemes for preparing entangled states of atomic ensembles by projective measurement are described.
In this paper we discuss physical aspects of entanglement in biphoton-based qutrits and ququarts, taking into account the bosonic correlations due to the indistinguishability of photons. We show that two apparently different types of entanglement in these systems, 'entanglement of modes' and 'entanglement of photons', are in fact closely related to each other. The latter is to a remarkable extent connected with the degree of polarization of two-photon light. Specifically, we demonstrate that a beamsplitter and subsequent post-selection using a coincidence circuit converts 'entanglement of photons' of a qutrit into 'entanglement of modes' of a ququart.
Superposition of optical coherent states (SCS) [Formula: see text], possessing opposite phases, plays an important role as qubits in quantum information processing tasks like quantum computation, teleportation, key distribution, etc. and are of fundamental importance in testing quantum mechanics. Passage of such SCS from a 50:50 beam splitter leads to generation of entangled coherent states. Recently, ququats and qutrits defined in four- and three-dimensional Hilbert space, respectively, have attracted much attention as they offer advantage in secure quantum communication. However, practical utilization of these advantages essentially requires physical realization of quantum optical ququats and qutrits. Here, we define four new multi-photonic states (MPS) with [Formula: see text] (here, [Formula: see text] or 3 and [Formula: see text]) numbers of photon and show how the SCS can be used to encode ququat using these MPS as basis vectors of a four-dimensional Hilbert space. When these MPS fall upon a 50:50 beam splitter, the resulting states are bipartite four-component entangled coherent states (BFECS) equivalently representing the entangled ququats. We briefly discuss the photon statistical properties of such MPS and BFECS. We show that these MPS and BFECS can be synthesized using even coherent states as input to an interferometer. We give a simple linear optical protocol for almost perfect teleportation of a ququat encoded in SCS with the aid of BFECS as quantum channel. We also describe how these ququats can be used for realization of higher-dimensional BB84 protocol to increase the security of quantum key distribution. Finally, we discuss the possible advantages of using SCS as ququats and BFECS as quantum channel in different quantum information processing tasks.
Quantum entanglement, like other resources, is now considered to be a resource. It can be produced, concentrated if required, swapped, transported and consumed. During recent years, various schemes of quantum state teleportation have been proposed using different types of quantum channels. Not restricting to qubit based systems, qutrit states and channels have also been of considerable interest. In the present paper, we investigate the teleportation of an unknown single qutrit state, as well as a two qutrit state through a three qutrit quantum channel, along with the required operations to recover the state. This is further generalized to the case of teleportation of an n-qutrit system.
Quantum entanglement, like othre resources, is now coonsidered to be a resource which can be produced, concentrated if required, transported and consumed. After its inception [1] in 1933, various schemes of quantum state teleportation have been proposed using different types of channels. Not restricting to qubit based systems, qutrit states and channels have also been of considerable interest. In the present paper we investigate the teleportation of an unknown single qutrit state as well as two qutrit state through a three qutrit quantum channel along with the required operations to recover the state. This is further generalized to the case of teleportation of n-qutrit system.
We investigate, in a general form, entanglement of biphoton qutrits and ququarts, i.e. states formed in the processes of, correspondingly, degenerate and non-degenerate spontaneous parametric down-conversion. Indistinguishability of photons and, for ququarts, joint presence of the frequency and polarization entanglement are fully taken into account. In the case of qutrits, the most general three-parametric families of maximally entangled and non-entangled states are found, and anticorrelation of the degree of entanglement and polarization is shown to occur and to be characterized by a rather simple formula. Biphoton ququarts are shown to be two-qudits with the single-photon Hilbert space dimensionality d =4, which differentiates them significantly from the often used two-qubit model ( d =2). New expressions for entanglement quantifiers of biphoton ququarts are derived and discussed. Rather simple procedures for a direct measurement of the degree of entanglement are described for both qutrits and ququarts.