Entanglement in the Dicke subspace
https://arxiv.org/pdf/2602.15800
Aabhas Gulati, Ion Nechita, Clément Pellegrini.

Projections with Respect to Bures Distance and Fidelity: Closed-Forms and Applications
https://arxiv.org/pdf/2602.14732
A. Afham, Marco Tomamichel.

Took a while, but such a fun project!

The main takeaway (for me) was that many of the 'natural' operations that is done in quantum information are actually projections w.r.t. fidelity/Bures distance. This showcases the importance of the Bures geometry.

Fin.

We study 'prior-channel' decomposition of CP maps, a unique decomposition of a CP map into a states and a channel.
This generalizes decomposing a PSD matrix into a density matrix and a scalar (via trace-normalization).
Geometry of this decomp. is discussed, & its relation to Choi isomorphism.

3. We provide an information-geometric underpinning to the Leifer-Spekkens state over time formalism, by showing their operations can be derived using Bures projections onto different sets.

More applications discussed in the paper!

As applications, we show that

1. Pretty good measurement is the fidelity/Bures projection of the ensemble to POVMs, providing new interpretation for PGM.
2. Petz map is the projection of a CP map constructed (rather naturally) from the channel-state pair to the set of reverse channels.

In the 'fixed-marginal' setting, is projection is given by a simple closed-form. If the marginal is Identity, then this recovers a standard 'partial-normalization' operation used widely in the literature.

We show that this partial normalization is not just easy to write down, but optimal!

We use the condition for the saturation of DPI for fidelity, along with properties of the matrix geometric mean, to derive a closed-form for certain projection problems.

For 'nice' problems, the projection is given by the 'Gamma map'.

The feasible sets are the PSD preimage of a channel and an output matrix. Examples include sets of
1. Bi- (& multi-)partite states with a given marginal on 1 of the spaces. Eg: Choi matrices of CPTP maps.
2. PSD decomposition of a given matrix. Eg: Set of all POVMs.

A meme indicating how various objects in quantum information (pretty good measurement, Petz recovery map, Leifer-Spekkens state over time, and others) are shown to be fidelity/Bures projections in our article.

New preprint out with @marcotomamichel.bsky.social !

Projections with Respect to Bures Distance and Fidelity: Closed-Forms and Applications
scirate.com/arxiv/2602.1...

We derive simple closed-form solutions for fidelity / Bures/purified distance projections to various sets of interest.

Liberté, égalité, fidelité

I realized I did not make clear that the e.vals have to be real and positive. Sorry!

But I did some further numerics and you are right in general. I generated X = AB with A, B >= 0 and used partial trace. X can have complex evals in general.

And yes, it's good to see these kinds of posts here!

Thanks for the reply! But the example you posted has complex e.vals which are conjugates of each other with +ve real part, hence the trace and det condition will be satisfied without the e.vals being (real) positive.

Q: Let X be a (non-Hermitian) matrix with positive eigenvalues (such X = AB, where A,B >= 0) and let Λ be a Completely positive map. Is Λ(X) guaranteed to be a matrix with positive eigenvalues?

That is, do CP maps take EVERY (including non-Herm) matrix with pos evals to matrices with pos evals?

Train your biceps with some dagger curls! Although \ddagger curls would be easier for balance.

Haha, took it almost verbatim from the Preliminaries of my thesis!

Here is a 'simple' 3-line proof for this statement!

(Although the simplicity hides behind the form and properties of the matrix geometric mean.)