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Chinese Physics Letters

The generation of Entangled Qudits and their Application in Probabilistic Superdense Coding

Lin Qing

Published Mar 31, 2009• 7 citations
Publisher

Abstract

A scheme of the generation of entangled qutrits is presented, and then is generalized to entangled ququads and entangled qudits. With the entangled qutrits, an experimental scheme of probability superdense coding with only linear optical elements is proposed. It is shown that this scheme will be suitable for the entangled ququads, even for the entangled qudits if some nonlinearity is used. This scheme is feasible in the laboratory with the current experimental technology.

DOI
10.1088/0256-307x/26/4/040301

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A scheme of the generation of entangled qutrits is presented, and then is generalized to entangled ququads and entangled qudits. With the entangled qutrits, an experimental scheme of probability superdense coding with only linear optical elements is proposed. It is shown that this scheme will be suitable for the entangled ququads, even for the entangled qudits if some nonlinearity is used. This scheme is feasible in the laboratory with the current experimental technology.
We discuss the properties of probabilistic coding of two qubits to one qutrit and generalize the scheme to higher dimensions. We show that the protocol preserves the entanglement between the qubits to be encoded and the environment and can also be applied to mixed states. We present a protocol that enables encoding of nnn qudits to one qudit of dimension smaller than the Hilbert space of the original system and then allows probabilistic but error-free decoding of any subset of kkk qudits. We give a formula for the probability of successful decoding.
We explore the possibility of performing superdense coding with nonmaximally entangled states as a resource. Using this we find that one can send two classical bits in a probabilistic manner by sending a qubit. We generalize our scheme to higher dimensions and show that one can communicate 2log2d2\phantom{\rule{0.2em}{0ex}}{\mathrm{log}}_{2}d2log2​d classical bits by sending a ddd-dimensional quantum state with a certain probability of success. The success probability in superdense coding is related to the success probability of distinguishing nonorthogonal states. The optimal average success probabilities are explicitly calculated. We consider the possibility of sending 2log2d2\phantom{\rule{0.2em}{0ex}}{\mathrm{log}}_{2}d2log2​d classical bits with a shared resource of a higher dimensional entangled state $(D\ifmmode\times\else\texttimes\fi{}D,D>d)$. It is found that more entanglement does not necessarily lead to higher success probability. This also answers the question as to why we need log2d{\mathrm{log}}_{2}dlog2​d ebits to send 2log2d2\phantom{\rule{0.2em}{0ex}}{\mathrm{log}}_{2}d2log2​d classical bits in a deterministic fashion.
In this paper, we propose a linear-optical quantum computation scheme that starts from high-dimensionally entangled qudits but performs quantum computation on qubits defined in their subspaces. This approach enables us to increase the success probability of linear-optical fusions from 50%50\%50% to 1−1/d1-1/d1−1/d for qudits of dimension ddd without relying on existing approaches using ancilla photons and quantum codes. In particular, we show that a 2D cluster state of qubits can be generated from 5-qudit 1D cluster states using pairwise fusion gates while preserving their boosted success probability. We numerically demonstrate that this approach can improve the loss tolerance of a percolation-based scheme while keeping the photon number per resource state constant.
One of the challenges of photon-based quantum teleportation is that both a source of entangled photons and an entangled basis measurement are required. For qubits, one can perform a probabilistic entangled basis measurement using linear optics, making the scheme efficient. However, for photonic qudits, an equivalent scheme remains difficult to devise. In this paper, we generalize the probabilistic photonic qubit teleportation protocol to qudits. The method relies on producing permutation entangled states nondeterministically which are superpositions of permutations of the spatial and qudit states of the photons. Our scheme nondeterministically teleports a photonic qudit using only entangled photon sources, linear optics, and photon detectors, and should be experimentally realizable for small qudit dimensions.