Evolution of quantum states via Weyl expansion in dissipative channel**Project supported by the National Natural Science Foundation of China (Grant No. 11664017), the Outstanding Young Talent Program of Jiangxi Province, China (Grant No. 20171BCB23034), the Degree and Postgraduate Education Teaching Reform Project of Jiangxi Province, China (Grant No. JXYJG-2013-027), and the Science Fund of the Education Department of Jiangxi Province,

Abstract

Based on the Weyl expansion representation of Wigner operator and its invariant property under similar transformation, we derived the relationship between input state and output state after a unitary transformation including Wigner function and density operator. It is shown that they can be related by a transformation matrix corresponding to the unitary evolution. In addition, for any density operator going through a dissipative channel, the evolution formula of the Wigner function is also derived. As applications, we considered further the two-mode squeezed vacuum as inputs, and obtained the resulted Wigner function and density operator within normal ordering form. Our method is clear and concise, and can be easily extended to deal with other problems involved in quantum metrology, steering, and quantum information with continuous variable.