# Understanding Quantum Channel and Choi Jamiolkowski Notation

I am given the following $$\newcommand{\on}[1]{{\operatorname{#1}}}$$

Let $$|i\rangle$$, $$1 \leq i \leq \on{dim}\,\mathcal{H_A}, |s\rangle$$, $$1 \leq s \leq \on{dim}\,\mathcal{H_B}$$ be unitary bases. Write $$\Lambda = \sum_{i,j,s,t}\Lambda^{\begin{pmatrix}i \\ j\end{pmatrix}}_{ \begin{pmatrix}s \\t \end{pmatrix}}|i\rangle \langle j | \otimes|s\rangle\langle t|$$ so $$CJ(\Lambda)$$ Hermitian implies $$\Lambda^{\begin{pmatrix}i \\ j\end{pmatrix}}_{ \begin{pmatrix}s \\t \end{pmatrix}} = \overline{\Lambda}^{\begin{pmatrix}j \\ t\end{pmatrix}}_{ \begin{pmatrix}i \\s \end{pmatrix}}$$

I know the following

1. $$\Lambda$$ is a quantum channel such that $$\Lambda : \on{End}(\mathcal{H_A}) \xrightarrow{} \on{End}(\mathcal{H_B})$$ Where $$\on{End}(\mathcal{H_A})$$ refers to the endomorphism over Hilbert space A 

2. $$CJ$$ refers to the Choi-Jamiolkowki isomorphism which is of the form: $$CJ: \on{Hom}(\on{End}(\mathcal{H_A}),\on{End}(\mathcal{H_B)}) \rightarrow \on{End}(\mathcal{H_A^*}\otimes\mathcal{H_B})$$ where $$\on{Hom}$$ refers to homomorphism (I believe)

What does the subscript, superscript notation mean on the $$\Lambda$$s?

• I'm confused, are the binomial-factor-looking things on the $\Lambda$ a latex misprint? If not, where did you see this notation used? Knowing nothing about the context where you saw this, I have to say that I personally often use a similar kind of notation myself, simply because sometimes using upper and lower indices makes things a bit clearer. If you write eg $\Lambda=\sum \Lambda^{ij}_{st} |i\rangle\!\langle j|\otimes|s\rangle\!\langle t|$, you know that upper indices refer to the first system and lower indices to the second one. I haven't seen it used much in papers though – glS Apr 16 at 7:01
• I agree with @glS that the notation is presumably just a way to denote the matrix elements. I'd write $\Lambda_{is,jt}$, but there are several ways you could do it. I've never seen this way of doing it. I'd have liked to confirm that by checking the stated Hermitian relation, but I don't understand it. Each of the 4 indices ought to run over the same range. That would mean $(i,j)$ have the same range (top row of the Hermitian relation), as should $(j,t)$, $(s,i)$ and $(t,s)$. So all indices have the same range. But there's supposed to be two distinct ranges. – DaftWullie Apr 16 at 7:50
• It is somewhat a Latex misprint, the binomial factor looking things are supposed to look more like superscripts and subscripts of the $\Lambda$ but still in the binomial factor looking form, just smaller. That makes sense though that it would just be a way of denoting matrix elements. I was confused in that i and s may have completely different dimensions, so then listing them one on top of the other was a little confusing for me. – john smith Apr 16 at 16:06
• @johnsmith Where have you seen this notation? – Norbert Schuch Apr 16 at 16:54