Teleportation for Septuagenarians

To Jurge, Hebert and Tom

Abstract

An introduction to the theory of teleportation.

Appendix: Averaging

Known quantum state are, in principle, easy to teleport: All one needs is broadcast its preparation protocol. Quantum teleportation deals with the case that $|{\psi }\rangle$ is unknown. In order to evaluate different protocols, it is natural to assume that $|{\psi }\rangle$ is uniformly distributed under the unitary group U(d).

To compute averages over $|{\psi }\rangle$ the following is handy

(A.1)

To see this note, first, that the average is invariant under unitary transformations of ${{\mathbf { C}}}$ and ${{\mathbf { D}}}$ and so the result must be a linear combination of $Tr\ {{\mathbf { CD}}}$ and ${Tr\,{{\mathbf { C}}}\ Tr\,{{\mathbf { D}}}}$. To see that they come with equal weight write $|{\psi }\rangle =(\psi _1,\dots ,\psi _d)$ and then note that the correlator

$$\begin{aligned} \mathbb {E}(\bar{\psi }_j\psi _k\bar{\psi }_m\psi _n)\propto \delta _{jk}\delta _{mn}+\delta _{jn}\delta _{mk} \end{aligned}$$

(A.2)

by Wick theorem, or directly by symmetry under the exchange $j\leftrightarrow m$ and phase averaging. It follows that

(A.3)

The constant of proportionality is determined by considering the special case .