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This and more available at scirate.com/arxiv/2609.0... arxiv.org/abs/2609.03762

scirate.com

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For the invariant matrix projection, which generalizes the BW RGD and other optimization problems in quantum information, we should that the FP algorithm corresponds to RGD on the submanifold of invariant (symmetric) PSD matrices.

Our fix is simple, just project! We derive a closed-form BW projection onto the set of well-conditioned PSD matrices. We show the projection is non-expansive and given by eigenvalue clipping. Consequently, the projected RGD algorithm achieves dim. independent linear convergence at η = 1!

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The std. approach works well in practice but theoretical analyses present a dichotomy: unit step-size (η = 1) analyses carry a dimension-dependent factor; for dim.-independence, one must choose small step sizes of η = O(1/κ). Empirically the algorithm does not exhibit dim.-depend., even at η = 1.

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We present a Projected RGD algorithm for two problems, the BW barycenter and the invariant projection problem. The standard approach for these problems is a fixed-point algorithm that can be identified as an RGD on the BW manifold at unit step size. This approach works extremely well in practice.

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New preprint out! Paradoxical: both my first 'solo-authored' and first LLM-assisted paper. Projected Riemannian Gradient Descent for the Bures-Wasserstein Barycenter: Dimension-Independent Linear Convergence at Unit Step Size scirate.com/arxiv/2609.0... Also an ML paper with applications to QI?

arXiv bot (quant-ph) @krxiv-quant-ph.bsky.social · 18/02/2026

Reposted by Afham

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

arxiv.org

https://arxiv.org/abs/2602.15800

arXiv abstract link

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Reposted by Afham

arXiv bot (quant-ph) @krxiv-quant-ph.bsky.social · 17/02/2026

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

arxiv.org

https://arxiv.org/abs/2602.14732

arXiv abstract link

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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.

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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'.

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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.

Afham @afhamash.bsky.social · 08/05/2025

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.

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.

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Liberté, égalité, fidelité

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Afham @afhamash.bsky.social · 23/03/2025

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!

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Afham @afhamash.bsky.social · 23/03/2025

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.

Afham @afhamash.bsky.social · 21/03/2025

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?