1

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 quantumcomputing.SE

glS
  • 15,488
  • 5
  • 42
  • 114
quantumorsch
  • 993

1 Answers1

3

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?

Yes, the Choi-Jamiolkowski isomorphism establishes a connection relating the constitution (state) of systems to evolution (channel) of other systems: It relates the evolution $\mathcal E(\rho)$ of a $d$-level system to the state $\sigma$ of a $d\times d$-level system. This can be operationally understood: You can use $\sigma$ to teleport the $d$-level system $\rho$ through it: This effectively applies $\mathcal E(\rho)$ to $\rho$ in case the teleportation projection yields the canonical maximally entangled state $\sum|i\rangle|i\rangle$.

You can find more details on the teleportation picture in this answer.

Norbert Schuch
  • 22,105