# Is tomography of the Choi state sufficient for channel tomography?

Given that there is an isomorphism between quantum states and quantum channels (the Choi-Jamiolkowski isomorphism) and given that state tomography is well-researched, why is quantum process or quantum channel tomography an interesting research topic?

Does it not follow trivially from state tomography and the Choi isomorphism? Specifically, let one do the following:

1. Use $$n$$ copies of the channel and send one half of a max-entangled state through each copy.
2. Performs state tomography on the $$n$$ i.i.d. output states.
3. Knowing the Choi state, one has a description of the channel

In 1), I am not sure if the setting of quantum channel tomography assumes access to the other half of the max-entangled state (that doesn't go into the channel) or not. Does this make a difference to the overall procedure?

Sure, process tomography is intimately related to state tomography, and one way to see it is via the Choi isomorphism, as you point out. See also eg the discussion in section IV of (Mohseni et al. 2007). Although it would rephrase your steps more precisely as just saying that you use as input the maximally entangled state, send part of it through the channel, and perform state tomography of the (global) output state. This automatically implies that you send the input multiple times as that's just how state tomography works in general.

But this only "trivially shows" that process tomography is possible if you can generate an entangled state, send part of it through the channel, and then perform tomographically complete measurements on the (generally entangled) outputs. Note that this approach would require to generate maximally entangled state of the form $$\sum_{i=1}^d |i,i\rangle$$ with $$d$$ the relevant state dimension. Not an easy task in general.

There's plenty of things that this argument tells you nothing about. For one thing, you can also clearly do process tomography without this kind of scheme and without the need to use an ancillary space and entangled states between ancillary and input space. What you really need to do is characterise the linear map that the channel is, so send any basis (or more generally, any copmlete and linearly independent set) of input density matrices, and do tomography of the associated outputs. This information is necessary and sufficient to obtain a "tomographically complete" description of the channel. Using the Choi is a "trick" to do this, that allows to use a single input state, at the cost of it having to be highly entangled, and thus having to work with larger spaces etc.

So, as to why "why is quantum process or quantum channel tomography an interesting research topic", that entirely depends on what one is interested in. There's an endless list of things one can study about this process that the above arguments tell you nothing about. What about the reconstruction efficiency in terms of required number of samples to have errors in a certain measure below a certain threshold with a certain probability? Which method is better from this point of view? Which set of input states is optimal from this point of view? What about robustness with respect to certain types of noise? Which methods will be better and which ones will be worse? What about all the problems associated with state tomography, such as the non-positivity of the estimated states you get when doing naive linear tomography? How do these transpose when you use these tomography methods for process tomography instead? How do you even quantify the quality of the estimated channel? There's many possible distances between channels that you can use, each one more suited for specific purposes. What about actual experimental implementations? What are the best experimental schemes to perform process tomography given the resources more easily available in different types of experimental platforms?

These are just some of the possible issues that came to mind on the spot (probably all of these have been worked out in the literature, I don't know). You could go on for a while. See for example the paper linked above for a more detailed overview.

• So just to clarify, the point is that while process tomography is reducible to state tomography, it isn't obvious that that's the best way to do process tomography? Feb 8 at 22:16
• no, the point is that there's a lot of possible ways to do process tomography, and a lot of possible questions to ask about and issues associated with each method. Passing through the Choi, that is to say, evolving an entangled state partly through a channel and doing tomography of the output, is just one possible way, which might or might not be advantageous depending on the goal and available resources (though generally speaking, I'd guess it's probably not a good method in most situations, due to its requiring to double the state dimension to do it)
– glS
Feb 9 at 10:10
• that process tomography is "reducible to state tomography" is kind of a trivial statement. Obviously you're going to measure something to assess the process, and doing tomography of some output state is the standard way to do it. But there's plenty of nontrivial aspects related to process tomography more generally
– glS
Feb 9 at 10:13