Teleportation of general finite dimensional quantum systems

Teleportation of finite dimensional quantum states by a non-local entangled state is studied. For a generally given entangled state, an explicit equation that governs the teleportation is presented. Detailed examples and the roles played by the dimensions of the Hilbert spaces related to the sender, receiver and the auxiliary space are discussed. PACS numbers: 03.67.-a, 89.70.+c, 03.65.-w AMS classification codes: 81P68 68P30 SFB 256; SFB 237; BiBoS; CERFIM (Locarno); Acc.Arch., USI (Mendrisio) e-mail: [email protected] Institute of Physics, Chinese Academy of Science, Beijing e-mail: [email protected] 1 The discovery of quantum teleportation protocols belongs to the most important results of quantum information theory. Quantum teleportation has been introduced in [1] and discussed by a number of authors for both spin2 states and arbitrary quantum states, see e.g. [2-11]. By means of a classical and a distributed quantum communication channel, realized by a non-local entangled state chosen in a special way (e.g., an EPR-pair of particles), the teleportation process allows to transmit an unknown quantum state from a sender (traditionally) named “Alice” to a receiver “Bob” that are spatially separated. For teleportation of N -dimensional quantum states, the teleportation problem has been discussed in [7] in the case where the dimensions of the Hilbert spaces associated with the sender, receiver and the auxiliary space are all equal to N = 2, for a given m ∈ IN . The relations among quantum teleportation, dense coding, orthonormal bases of maximally entangled vectors and unitary operators with respect to the Hilbert-Schmidt scalar product, and depolarizing operations are investigated in [8]. The details of protocols for teleportation vary with the shared entangled state and joint measurements at Alice or Bob. In this note we study the general properties of teleportation for finite dimensional quantum states without the assumption on equality for the dimensions of the Hilbert spaces involved. We give a teleportation protocol for generally given entangled states and a constraint equation that governs the teleportation. The solutions of the constraint equation give the unitary transformations of teleportation protocols. Detailed examples and the roles played by the dimensions of the Hilbert spaces are discussed. Let H1 be a Hilbert space with dimensions N1. Let ei, i = 1, ..., N1, be an orthogonal basis in H1, so that ei is an N1-dimensional column vector with entry 1 for the i-th component and 0 for the other components. Alice has a general quantum state on the Hilbert space H1 of the form Ψ0 = 