L. Accardi, W. Freudenberg, M. Ohya's Quantum Bio-Informatics II: From Quantum Information to PDF

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Extra resources for Quantum Bio-Informatics II: From Quantum Information to Bio-Informatics : Tokyo University of Science, Japan 12 - 16 March 2008 (Qp–pq: Quantum Probability and White Noise Analysis)

Example text

Note that fga",-;;2 is unitary even if is not unitary. There always exists a unitary key 1 vnf Wa == (fga",-;;2)* recovering Ix) from 1i:~O), and in this sense, ideal quantum teleport at ion linear map. ra is an 3. Teleportation Map of Complete m-Ievel Teleportation A specific type of teleport at ion we call m-Ievel teleport at ion is proposed in this section. We consider the following form of state vector as an input. m Ix(m)) = L Ci lei) , (8) i=l where {lei)} are the basis made by the eigenstates of "'a and the coefficients Ci=l, ..

Generally, this map is non-linear. In the relation of A~(p) = p", a unitary key to recover the input [x) from the output [x,,) exists only if fg,,/J(x[ 1'0" [x) in Eq. (4) is unitary. It is when the map A~ is linear. For example, with setting fof and fog" to be unitary, the existence of key is assured. ) The idea by Kossakowski and Ohya is to make the teleport at ion linear map that the existence of unitary key is always assured. In order to do this, they considered the dual map T" satisfying trAT,,(p) = trT,,(A)p for 22 t" any linear operator A.

We may also take cost c (u) = O~l (u) of = {u : [u[ :S I} and any of the above bequest functions 5 lel. we have (v (u, e) , p) = v (u, if) . p linear in u: _(_;;'\ V U, q) = _ Vm _ +U _ _ Vm x q, _ (_ ;;'\ _ P1 which is maximized by the unit if' = if X pi [if x [u[ :S 1. p. O" z) e and so (8 (e), p) = ).. (1 - Z2)pz - )"z (xPx + YPy). 3. The qubit HJB equation r With the customary abuse of notation we write 5 (t, e) = 5 (t, r') for = -if and 8 (e) = ).. (-zx, -zy, 1 - z2) in terms of = (x, y, z).

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