Superdense coding: Difference between revisions

Revision as of 10:55, 22 July 2006

Revision as of 11:03, 22 July 2006

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:<math>\begin{bmatrix} \frac{1}{n} & \; & \; \\ \; & \ddots \; \\ \; & \; & \frac{1}{n} \end{bmatrix}</math>

as its [[partial trace]]. Let the subsystems initially possessed by Alice and Bob be labeled 1 and 2, respectively. To transmit the message ''x'', Alice performsapplies thean operationappropriate channel

:<math>\Phi_x; (\cdot) = V_x^* ( \cdot ) V_xPhi_x</math>

on subsytem 1. On the combined system, this is effected by

:<math>(\Phi_xomega \otimesrightarrow Id)(\omega) = (V_x^*Phi_x \otimes IId)(\omega(V_x \otimes I)</math>

where ''Id'' denotes the identity map on subsystem 2. Alice then sends her subsystem to Bob, who performs a measurement on the combined system to recover the message. Let the ''effects'' of Bob's measurement be ''F_y''. The probability that Bob's measuring apparatus registers the message ''y'' is

:<math>\operatorname{Tr}\; (V_x^*\Phi_x \otimes IId)(\omega (V_x) \otimes I)cdot F_y .</math>

Therefore, to achieve the desired transmission, we require that

:<math>\operatorname{Tr}\; (V_x^*\Phi_x \otimes IId)(\omega (V_x) \otimes I)cdot F_y = \delta_{xy}</math>

where ''δ_xy'' is the Kronecker delta.