Two Kraus representations: How to check if they're the same TPCPM?

[Ameno]

Hi

According to the Kraus representation theorem, a map $$\mathcal{E}: \text{End}(\mathcal{H}_A) \rightarrow \text{End}(\mathcal{H}_B)$$
is a trace-preserving completely positive map if and only if it can be written in an operator sum representation $$\mathcal{E}: \rho \mapsto \sum_k A_k \rho A_k^\dagger$$ with $$\sum_k A_k^\dagger A_k = \text{Id}$$
This operator sum representation is not unique. For example, $$\rho \mapsto (1-p)\rho + p \sigma_x \rho \sigma_x$$
and $$\rho \mapsto (1-2p)\rho + 2pP_+\rho P_+ + 2pP_-\rho P_-$$
where $$\sigma_x$$ is the Pauli x-operator and $$P_+, P_-$$ is the projector to the $$\sigma_z$$ eigenspace with eigenvalue +1, -1,
are two operator-sum representations of the same trace-preserving completely positive map.

My question is: Given two such operator-sum representations, what is the easiest way to find out whether the two representations give the same TPCPM or not?

 

[A. Neumaier]

According to the Kraus representation theorem, a map
$$\mathcal{E}: \text{End}(\mathcal{H}_A) \rightarrow \text{End}(\mathcal{H}_B)$$
is a trace-preserving completely positive map if and only if it can be written in an operator sum representation
$$\mathcal{E}: \rho \mapsto \sum_k A_k \rho A_k^\dagger$$
with
$$\sum_k A_k^\dagger A_k = \text{Id}$$
This operator sum representation is not unique. [...]

My question is: Given two such operator-sum representations, what is the easiest way to find out whether the two representations give the same TPCPM or not?

Working out the image of a suitable spanning set of matrices, I guess.

 

[Ameno]

Well, there are two ways I know how one can do this. One is what you have just written, the other one is to check if there is a unitary matrix s.t.
$$N_a = U_{\mu a}M_\mu$$
where the N's and M's are the operators of two operator-sum representations.
I find that both require a lot of time to work out in practice, so I wonder if there is a more efficient way to do that.

 

[A. Neumaier]

Well, there are two ways I know how one can do this. One is what you have just written, the other one is to check if there is a unitary matrix s.t.
$$N_a = U_{\mu a}M_\mu$$
where the N's and M's are the operators of two operator-sum representations.
I find that both require a lot of time to work out in practice, so I wonder if there is a more efficient way to do that.

You can also take the difference and simplify it to zero by expressing the operators involved in terms of a fixed set of generators for which you know all algebraic relations.

None of the methods is easy when the Kraus representations are arbitrary. But usually there is a preferred representation with a physical meaning, and when this is used consistentl;y, the question of equivalence doesn't arise.