I understand the mathematical construction of the Choi-Jamiolkowski isomorphism aka channel-state duality. It all makes sense formally, yet I still struggle to grasp its physical (or quantum-informational) meaning. Does the isomorphism between quantum states and quantum channels in any sense establish some kind of connection relating the constitution (state) of systems to evolution (channel) of other systems?

Cross-posted on physics.SE

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    $\begingroup$ If you give me the state dual of a Clifford channel, I can teleport a new state through it in order to apply the channel to the new state. $\endgroup$ – Craig Gidney Jun 19 at 23:12

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