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arXiv:quant-ph/0012086 · Physical Review A

Entangled coherent states: Teleportation and decoherence

S. J. van Enk, Osamu Hirota

Published Jul 1, 2001• 516 citations
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Abstract

When a superposition $(|\ensuremath{\alpha}〉\ensuremath{-}|\ensuremath{-}\ensuremath{\alpha}〉)$ of two coherent states with opposite phase falls upon a 50-50 beam splitter, the resulting state is entangled. Remarkably, the amount of entanglement is exactly 1 ebit, irrespective of \ensuremath{\alpha}, as was recently discovered by Hirota and Sasaki [LANL e-print quant-ph/0101018]. Here we discuss decoherence properties of such states and give a simple protocol that teleports one qubit encoded in Schr\"odinger cat states.

DOI
10.1103/physreva.64.022313

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When a superposition (∣\ensuremathα〉\ensuremath−∣\ensuremath−\ensuremathα〉)(|\ensuremath{\alpha}〉\ensuremath{-}|\ensuremath{-}\ensuremath{\alpha}〉)(∣\ensuremathα〉\ensuremath−∣\ensuremath−\ensuremathα〉) of two coherent states with opposite phase falls upon a 50-50 beam splitter, the resulting state is entangled. Remarkably, the amount of entanglement is exactly 1 ebit, irrespective of \ensuremath{\alpha}, as was recently discovered by Hirota and Sasaki [LANL e-print quant-ph/0101018]. Here we discuss decoherence properties of such states and give a simple protocol that teleports one qubit encoded in Schr"odinger cat states.
A scheme of teleporting a superposition of coherent states |α⟩ and | − α⟩ using a beam splitter and two phase shifters was proposed by van Enk and Hirota (2001 Phys. Rev. A 64 022313). The authors concluded that the probability for successful teleportation is 1/2. In this paper, it is shown that the authors' scheme can be altered slightly so as to obtain an almost perfect teleportation for an appreciable value of |α| 2 . For |α| 2 = 5, the minimum of average fidelity, which is the minimum of the sum of the product of probability of occurrence of any case, and the corresponding fidelity is less than 1 by a quantity ∼10 −4 . We also discuss the effect of decoherence on teleportation fidelity. We find that if no photons are counted in both final outputs, the minimum assured fidelity is still non-zero except when there is no decoherence and the information is an even coherent state. For non-zero photon counts, minimum assured fidelity decreases with an increase in |α| 2 for low noise. For high noise, however, it increases, attains a maximum value and then decreases with |α| 2 . The average fidelity depends appreciably on the information for low values of |α| 2 only.
Superposition of optical coherent states (SCS) Ket(plus/minus alpha), possessing opposite phases, plays an important role as qubits in quantum information processing tasks like quantum computation, teleportation, cryptography etc. and are of fundamental importance in testing quantum mechanics. Recently, ququats and qutrits defined in four and three dimensional (D) Hilbert space, respectively, have attracted much more attention as they present advantage in secure quantum communication and also in researches on the foundation of quantum mechanics. Here, we show that superposition of four non-orthogonal coherent states Ket(plus/minus alpha) and Ket(plus/minus i alpha), that are 90 degrees out of phase, can be employed for encoding one ququat defined in a 4D Hilbert space spanned by four newly defined multi-photonic states, Ket(alpha subscript j) with 4n+j numbers of photons, where, j= 0, 1, 2, 3. We propose a scheme which generates states Ket(alpha subscript j). When these states fall on a 50-50 beam splitter, the result is bipartite four-component entangled coherent state which represents equivalently an entangled ququat. Finally, we propose a linear optical scheme that gives almost perfect teleportation (minimum average fidelity greater than or equal to 0.99) of single ququat encoded in SCS using entangled ququat based on coherent states with almost perfect success rate for coherent amplitude mod(alpha) greater than or equal to 3.2. The present theoretical investigation and teleportation protocol is important in the sense that ququat allows the quantum information to be encoded and processed much more compactly and efficiently using fewer coupled quantum system.
In the present paper, we will review the methods to produce superposition of entangled coherent state using polarizing beam splitter and Kerr non linearity. These coherent states have many attractive features and can be used in various schemes. Entanglement, refers to the superposition of a multiparticle system and explains a new type of correlations between any two subsystems of the quantum system, which is not existing in the classical physics. The present paper deals with the use of these states in quantum teleportation, entanglement diversion and entanglement swapping schemes. Entanglement diversion and entanglement-swapping refers to a scheme which may entangle those particles which had never interacted before. In the swapping scheme, two pairs of entangled state are taken. One particle from each pair is subjected to a Bell-state-measurement. This would result in projection of the other two outgoing particles in an entangled pair. Quantum Teleportation of two mode and three modes states is also studied with perfect fidelity. Minimum assured fidelity which is defined as the minimum of the fidelity for any unknown quantum information of the states is also discussed. It is also shown how the success rate of teleportation of a superposition of odd and even coherent states can be increased from 50% to almost 100%. The scheme suggested by van Enk and Hirota was modified by Prakash, Chandra, Prakash and Shivani in 2007. We find that an almost teleportation, diversion and swapping is possible by simply separating vacuum state from the even state. The present paper also deals with study of effect of decoherence and noise on these states and the effect of noise on fidelity and minimum assured fidelity. It is also discussed that these schemes can also be applied to the process of entanglement diversion and entanglement swapping.