# 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. Jun 19, 2019 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 ? Jun 19, 2019 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. Jun 19, 2019 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. Jun 19, 2019 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... Jun 19, 2019 at 16:49

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 $$\sum|i\rangle|i\rangle$$.