We reveal that the common eigenvector of two particles' center-of-mass coordinate and mass-weighted relative momentum is an entangled state. Its Schmidt decomposition exhibits that the entanglement involves squeezing which depends on the ratio of two particles' masses. The corresponding entangling operators are derived.
Bipartite entangled state |ζ⟩ and its applications in quantum communication
We analyse the entanglement involved in the common eigenvector | ζ ⟩ of the centre-of-mass coordinate and mass-weighted relative momentum of two particles. Its Schmidt decomposition exhibits that the | ζ ⟩ state maximally entangles two single-mode coordinate (or momentum) eigenstates, of which one is squeezed. The squeezing parameter is related to the ratio of the masses of two particles. The | ζ ⟩ state may describe the entangled state produced by a non-50/50 beamsplitter. Its applications in entanglement swapping and quantum dense coding are discussed. The corresponding entangling operators and swapping operators are derived.
We give the explicit form of the common eigenstate \ensuremath{\Vert}\ensuremath{\xi}〉 of the center-of-mass coordinate ${\mathrm{\ensuremath{\mu}}}_{1}$Q1+${\mathrm{\ensuremath{\mu}}}_{2}$Q2 and the mass-weighted relative momentum ${\mathrm{\ensuremath{\mu}}}_{2}$P1-${\mathrm{\ensuremath{\mu}}}_{1}$P2 of two particles, which is more complicated than the common eigenstate of the other pair of commutative operators Q2-Q1 and P1+P2. The orthonormal and completeness relation of \ensuremath{\Vert}\ensuremath{\xi}〉 are investigated easily by virtue of the technique of integration within an ordered product of operators. The normally ordered squeezing operator for ${\mathrm{\ensuremath{\mu}}}_{1}$Q1+${\mathrm{\ensuremath{\mu}}}_{2}$Q2 and ${\mathrm{\ensuremath{\mu}}}_{2}$P1-${\mathrm{\ensuremath{\mu}}}_{1}$P2 is also derived by using the \ensuremath{\Vert}\ensuremath{\xi}〉 representation.
We construct 2N operators for a N-particle system, namely one center-of-mass coordinate operator, N-1 relative coordinate operators, one total momentum operator and N-1 mass-weighted relative momentum operators, and give common eigenvectors of N compatible observables [Formula: see text], which are composed of N particles' coordinate [Formula: see text] and momentum [Formula: see text]. By compatible, we mean such observables can be simultaneously determined. Using the technique of integration within an ordered product of operators (IWOP), we prove that the common eigenvectors are complete and orthonormal, and hereby qualified for making up a representation. This new representation can be applied to solving some dynamic problems in quantum mechanics.