Multiparticle reduced density matrix and a useful kind of entangled state for quantum teleportation

Abstract

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 $\ensuremath{\rho}(i)=\frac{1}{2}{\mathbf{I}}_{2}$ $(i=1,2).$ It is shown that, to teleport an arbitrary k-particle spin state, one must prepare an ancillary N-particle $(N>~2k)$ entangled state, whose k-particle reduced density matrix has the form ${(1/2}^{k}){\mathbf{I}}_{{2}^{k}}$ $({\mathbf{I}}_{{2}^{k}}$ 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 $(>~2)$ 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.