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Lindblad equation using Kraus operators and a CPTP map

This is a derivation found in [1] but with details from [2]. I am only emphasizing the details that I found interesting, because it is a very short derivation originally and the way it hides some mysteries is very enticing. Most consider a postulate of quantum mechanics that a completly positive trace preserving (CPTP) map, famous for the propriety of mapping physical systems to other physical systems, can be described by Kraus operators, specifically a map from one density matrix to another. Considering said map (we make implicit the Choi-Kraus’ theorem), then $$\hat{\rho}^{\prime}=\sum_{\mu}\hat{K}_{\mu}\hat{\rho}\hat{K}_{\mu}^{\dagger}\qquad\text{with }\quad\sum_{\mu=0}K_{\mu}^{\dagger}K_{\mu}=\mathbb{I}.$$ Lets now check that if $\hat{\rho}$ obeys the proprieties of a density matrix, then so will $\hat{\rho}^{\prime}$. That is, $\hat{\rho}\prime$ is hermitian, positive semi-definite, and unit trace. The unit trace comes from the fact that it is a CPTP map, but to show this noti...
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Study of a gauge theory with a pure $Z(N)$ lattice in 4D

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This is a solution attempt at the last problem from the 2025 Rudolf Ortvay competition proposed by Dávid Pesznyák. I want to develop my skills in this particular subject, so I paraphrase the problem here as a discussion. Firstly, there is many interest in describing our world with gauge theories for technical reasons such as renormalization of observables, but consider first that such theories involve many symmetries that are natural to rationalize. Question the fact that electrons all have the same charge or that systems with conservation of energy are far simpler to describe. Those can be viewed as symmetries (assumptions) that in many cases describe our world very well up to a given generalization. A common way to achieve a gauge theory is with a path integral formulation, which can be viewed through the lens of the lattice field theory method, making sense of infinite-dimensional integrals with a small spacetime lattice of length $\Lambda$. The gauge field $U_{\mu}\left(x\righ...
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Renormalization group on neural scaling laws (part 1)

Neural scaling laws are until now not fully explained. Networks that generalize to other data require validation with completely unseen sets of data. In many cases the graph of these validation losses follow a power law, that is, the behavior is scale independent. Given the history of successful application of renormalization group (RG) theory to describe a variety of scale-free and critical phenomena, here we investigate how RG techniques can provide a systematic framework for describing scale-free aspects of loss functions. Most of these tricks with RG theory rely on the following simple behavior of the validation loss $$\lambda^{c}L\left(n,d\right)=L\left(\lambda^{a}n,\lambda^{b}d\right),$$ which is completely empirical (Kaplan et al. 2020). If we assume the parametrization of the loss is differentiable, differenciating with respect to $\lambda$ gives the following PDE $$\left(c-an\frac{\partial}{\partial n}-bd\frac{\partial}{\partial d}\right)L\left(n,d\right)=0,$$ which has...
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