无噪音量子码的一些性质

Some properties of noiseless quantum codes

量子码是描述量子系统的Hilbert空间H上一些密度算子之集,量子信道是C*-代数B(H)上的完全正保迹映射.本文揭示了无噪音量子码的一些性质,得到了强、弱保护码与无噪音码的关系.证明了:当量子码C在量子信道ε下的像包含在ε的不动点之集时,C为ε的无噪音码当且仅当C被ε弱保护;对于任一量子信道ε及量子码C,证明了C为ε的无噪音码当且仅当C为ε与ε~2的任意凸组合的无噪音码,还给出量子系统C^2C^2的一个量子信道的弱保护码与无噪音码.

A quantum code of a quantum systemSis an arbitrary non-empty subset Cof the set D（H）of all density operators on complex Hilbert space H that describes the system S,a quantum channel of Sis a completely positive and trace preserving linear mapεon B（H）.Some properties of noiseless quantum codes are researched and the relationships of strongly preserved code,weakly preserved code and noiseless codes are obtained.It is proved that when the image of a code C under a quantum channelεis contained in the fixed-point set of ε,C is a noiseless code ofεif and only if Cis weakly preserved byε.For an arbitrary quantum channelεand a quantum code C,it is also proved that C is a noiseless code ofεif and only if it is a noiseless code of any convex combination of ε and ε~2.Lastly,a weakly preserved code and a noiseless code of a quantum channel of the quantum system C^2C^2 are presented.