Transmitting qudits through larger quantum channels

By a quantum channel is intended [1, 2] a completely positive and trace-precerving linear map Φ : σ(H) → σ(H), where σ(H) is a set of states in a Hilbert space H with dimH = n ≤ +∞. Even for a finite dimension n < +∞ there is a number of difficult problems concerning the construction of optimal transmission of the information through a quantum channel. The infinite dimensional case we are especially interested in, has additional particular features [3]. In the present paper we discuss how a set of states on the finite dimensional Hilbert space can be transmitted through a quantum channel associated with the infinite dimensional Hilbert space. The paper is organized as follow. In Sec. II the notion of subchannel is introduced. In Sec. III we study the phase and the amplitude damping channels within this context. Sec. IV is for conclusions.