The usual definition of complete positivity (e.g. Stinespring (1955)) is that a linear map between the bounded operators on some Hilbert spaces $\phi:\mathcal{L}(\mathcal{H})\to\mathcal{L}(\mathcal{K})$ is $k$ positive if the map \begin{align} \left(\mathrm{id}_k\otimes\phi\right):\mathcal{L}(\mathbb{C}^{k})\otimes\mathcal{L}(\mathcal{H})\to\mathcal{L}(\mathbb{C}^{k})\otimes\mathcal{L}(\mathcal{K}), \end{align} you get by tensoring with the identity on $\mathcal{L}(\mathbb{C}^{k})$ is positive, and $\phi$ is completely positive if it is $k$ positive for all $k\in\mathbb{N}$.

My basic question is why do we consider all finite dimensional auxillary spaces $\mathbb{C}^k$, rather than infinite dimensional dimensional spaces? In particular from the point of view of quantum information theory we use complete positivity to ensure we can "act locally" on our system of interest whilst the global state remains positive. It seems natural to want this to happen even if the global state is infinite dimensional. Note that this is only an interesting question if $\mathcal{H}$ and $\mathcal{K}$ are infinite dimensional.

For convenience I will call maps $\phi$ such that \begin{align} \left(\mathrm{id}_{\mathcal{S}}\otimes\phi\right):\mathcal{L}(\mathcal{S})\otimes\mathcal{L}(\mathcal{H})\to\mathcal{L}(\mathcal{S})\otimes\mathcal{L}(\mathcal{K}), \end{align} is positive for every Hilbert space $\mathcal{S}$, where $\mathrm{id}_{\mathcal{S}}$ is the identity on $\mathcal{L}(\mathcal{S})$, extra completely positive. The question may make more sense if we restrict this to separable $\mathcal{S}$.

It is relevant that the Stinespring factorisation is possible if, and only if the map is completely positive in the usual sense.

My specific questions are

  1. Is there already a name for the extra completely positive maps in the literature? After some searching, and asking colleagues I have not found anything about them.
  2. Is there a $\phi$ which is completely positive but not extra completely positive? Conversely I would be very interested in a proof that all completely positive maps are extra completely positive.
  3. If the extra completely positive maps are a proper subset of the completely positive maps is there a nice characterisation of them (e.g. a "Stinespring-esque" factorisation)?

Note: Cross-posted to MO: https://mathoverflow.net/questions/327791/complete-positivity-with-infinite-dimensional-auxillary-spaces.

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    $\begingroup$ Where have you seen that complete positivity for maps on infinite dimensional spaces is defined only with finite dimensional ancillas? I highly doubt that. For finite dim. systems, of course, an ancilla of the same dimension is sufficient. Note that there is the concept of $k$-positivity, which is a map which is positive when tensoring a $k$-level system. This is different from complete positivity, so for infinite dim. systems, there will likely be examples where $k$-positivity for any finite $k$ is different. $\endgroup$ – Norbert Schuch Apr 10 '19 at 15:57
  • $\begingroup$ @NorbertSchuch The characterisation given by Stinespring in the 1955 paper I linked in my question defines complete positivity in exactly this way ($k$ positivity for all natural $k$) and the factorisation theorem proved in that paper holds for for arbitrary dimensions (indeed for arbitrary $C^*$ algebras. $\endgroup$ – or1426 Apr 10 '19 at 16:02
  • $\begingroup$ @NorbertSchuch Its too late to edit my other comment, but a good reference is Holevo, Statistical Structure of Quantum Theory (Springer 2001). He goes over this stuff in chapter 3. $\endgroup$ – or1426 Apr 10 '19 at 16:13
  • $\begingroup$ I'm not an expert, but I would argue that one can prove that if a positive map is positive with all finite-dimensional extensions, it is positive also with an infinite-dim. extension. My intuition would be that if the map is not positive for an infinite-dimensional extensions, then the output state must have some strictly negative eigenvalue. This means that one can truncate the infinite-dimensional input state to a finite-dimensional input state in a way where this eigenvalue is still negative. (This assumes that the infinite dimensional space is separable or sth the like, ... $\endgroup$ – Norbert Schuch Apr 12 '19 at 14:13
  • $\begingroup$ ... i.e. we can approximate any state there arbitrarily well by a sequence of states supported in finite-dimensional spaces. I'm fairly certain this can be formalized. $\endgroup$ – Norbert Schuch Apr 12 '19 at 14:13