# Does the Choi-Jamiolkowski isomorphism really establish a connection between kinematics and dynamics?

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

• Why do you think a channel is "evolution"? Time evolution to a fixed time is a channel (from a system to itself), but to capture time evolution for all times, you'd have a family of channels parametrized by time. For channels between different systems, I have a hard time seeing where the "dynamics" is. – ACuriousMind Jun 19 at 16:33
• @ACuriousMind Well if the problem is only semantics, then let us call the channel „discrete time evolution to a fixed time“ - why should that in any way be related to a quantum state ? – quantumorsch Jun 19 at 16:38
• @quantumorsch "why should that in any way be related to a quantum state ?" -- Because you can apply the evolution to half of an entangled state, and you obtain a state which -- as the Choi-Jamiolkowski isomorphism shows -- contains the full information about the evolution. – Norbert Schuch Jun 19 at 16:39
• But I struggle to understand what would be a good answer to the question. Would a plain "Yes, it does." do the job? If no, why not? If yes, I don't think this is a good question for this site. – Norbert Schuch Jun 19 at 16:40
• @NorbertSchuch „ Because you can apply the evolution to half of an entangled state, and you obtain a state which contains the full information about the evolution“. Sure, I understand the mathematical procedure; guess I was hoping for some further insight as to why this construction works or what it implies beyond the mere encoding of the evolution law in a state. Sometimes there is some additional/different operational perspective on the same math - that is what I was angling for... – quantumorsch Jun 19 at 16:49

## 1 Answer

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.