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I would like to know if there is a "nice" way to parametrize the set of $n$-qubit CPTP map?

I guess I could parametrize it by using the channel state duality (to every CPTP I can attribute a density matrix). Then I can parametrize an arbitrary density matrix (an $n$-qubit density matrix can be written with $n-1$ eigenvalues $\in [0,1]$). Then I can map the whole density matrix space by applying an arbitrary unitary to it (and I can parametrize unitaries by saying $U=\exp(i H)$, where $H=\sum_{i} c_i P_i$, $P_i$ being Pauli and $c_i \in \mathbb{R}$ because the Hamiltonian is Hermitian).

However it would really be unpractical.

Are there some "nice" ways to represent an arbitrary $n$-qubit CPTP? I know that my question is a bit subjective (it depends on what we mean by nice).

If there are known nice representation for single and two-qubit CPTP, it would already be interesting for me.

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  • $\begingroup$ Do you want to parametrize it for numerical or analytical purposes? To get a random density matrix, you can just take a positive matrix ($X^\dagger X$) and normalize it. Note that for TP you also need a condition on the RDM (but this you can get by conjugating with the square root of the inverse of the original RDM). --- In any case, I would say this is a pretty elegant characterization. Otherwise, Stinespring dilations come to mind. $\endgroup$
    – Norbert Schuch
    Commented Aug 13 at 17:51
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    $\begingroup$ For single-qubit CPTP, there are canonical forms, see e.g. Sec. VI of arxiv.org/abs/math-ph/0611057 and references therein. $\endgroup$
    – Norbert Schuch
    Commented Aug 13 at 17:54
  • $\begingroup$ @NorbertSchuch thanks for the reference, I will take a look. What does the acronym "RDM" means in your first comment? $\endgroup$
    – Marco Fellous-Asiani
    Commented Aug 15 at 14:03
  • $\begingroup$ reduced density matrix. $\endgroup$
    – Norbert Schuch
    Commented Aug 15 at 18:41

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