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Chinese Physics Letters

Possibility of Quantum Teleportation and the Reduced Density Matrix

Hongbo Zhu, Zeng Jinyan

Published Jul 1, 2001• 3 citations
Publisher

Abstract

It is shown that only the maximally entangled two-particle (spin 1/2) states whose one-particle reduced density matrix is ρ(i) = (1/2)I2 can realize the teleportation of an arbitrary one-particle spin state. Based on this, to teleport an arbitrary k-particle spin state, one must prepare an N-particle entangled state whose k-particle (k k≥2) state, except for special states with only two components.

DOI
10.1088/0256-307x/18/7/306

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It is shown that only the maximally entangled two-particle (spin 1/2) states whose one-particle reduced density matrix is ρ(i) = (1/2)I2 can realize the teleportation of an arbitrary one-particle spin state. Based on this, to teleport an arbitrary k-particle spin state, one must prepare an N-particle entangled state whose k-particle (k k≥2) state, except for special states with only two components.
It is pointed out that the possibility of teleporting an arbitrary unknown one-particle spin state is crucially connected with the maximal entanglement of the Einstein-Podolsky-Rosen pair, whose one-particle reduced density matrix is $\ensuremath{\rho}(i)=\frac{1}{2}{\mathbf{I}}_{2}$ (i=1,2).(i=1,2).(i=1,2). It is shown that, to teleport an arbitrary k-particle spin state, one must prepare an ancillary N-particle (N> 2k)(N>~2k)(N> 2k) entangled state, whose k-particle reduced density matrix has the form (1/2k)I2k{(1/2}^{k}){\mathbf{I}}_{{2}^{k}}(1/2k)I2k​ (I2k({\mathbf{I}}_{{2}^{k}}(I2k​ is the ${2}^{k}\ifmmode\times\else\texttimes\fi{}{2}^{k}$ identity matrix). An alternative approach to constructing many-particle entangled states is developed by using ${R}_{x}(\ensuremath{\pi}),$ the collective rotation of $\ensuremath{\pi}$ around a given axis (say, x axis). The entangled states constructed by using ${R}_{x}(\ensuremath{\pi})$ operating on the basis of angular momentum uncoupling representation are just the GHZ states, which cannot be used for the teleportation of an arbitrary k (> 2)(>~2)(> 2) particle spin state. The entangled states constructed by using ${R}_{x}(\ensuremath{\pi})$ operating on the basis of angular momentum coupling representation turn out to be effective for the teleportation of an arbitrary multiparticle state. A formal extension of the scheme of Bennett et al. to deal with the teleportation of an arbitrary two (or more) particle spin state is discussed.
We propose two schemes for teleporting an arbitrary three-particle state. In the first scheme, a two-particle state and a three-particle entangled state (both non-maximally entangled states) are used as quantum channels, while in the second scheme, three non-maximally entangled particle pairs are employed as quantum channels. We show that teleportation can be successfully realized with certain probability if a receiver adopts some appropriate unitary transformations. Their success probabilities and the classical communication costs are different.
Recently M. Horodecki, P. Horodecki, and R. Horodecki [Phys. Rev. Lett. 80, 5239 (1998)] have introduced a set of density matrices of two spin-1 particles from which it is not possible to distill any maximally entangled states, even though the density matrices are entangled. Thus these density matrices do not allow reliable teleportation. However, it might nevertheless be the case that these states can be used for teleportation, not reliably, but still with fidelity greater than that which may be achieved with a classical scheme. We show that, at least for some of these density matrices, teleportation cannot be achieved with better than classical fidelity.