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Transmission of high dimensional entanglement through quantum channels is a significant area of interest in quantum information science. The certification of high dimensional entanglement is usually done through Schmidt numbers. Schmidt numbers quantify the entanglement dimension of quantum states. States with high Schmidt numbers provide a larger advantage in various quantum information processing tasks compared to quantum states with low Schmidt numbers. However, some quantum channels can reduce the Schmidt number of states. Here we present a comprehensive analysis of Schmidt number breaking channels which reduce the Schmidt number of bipartite composite systems. From a resource theoretic perspective, it becomes imperative to identify channels that preserve the Schmidt number. Based on our characterization we lay down prescriptions to identify such channels which are non-resource breaking, i.e., preserve the Schmidt number. Additionally, we introduce a new class of quantum channels, termed Schmidt number annihilating channels which reduce the Schmidt number of a quantum state that is a part of a larger composite system. Finally, we study the connection between entanglement breaking, Schmidt number breaking, and Schmidt number annihilating channels., Comment: 22 pages

The block classical Gram--Schmidt (BCGS) algorithm and its reorthogonalized variant are widely-used methods for computing the economic QR factorization of block columns $X$ due to their lower communication cost compared to other approaches such as modified Gram--Schmidt and Householder QR. To further reduce communication, i.e., synchronization, there has been a long ongoing search for a variant of reorthogonalized BCGS variant that achieves $O(u)$ loss of orthogonality while requiring only \emph{one} synchronization point per block column, where $u$ represents the unit roundoff. Utilizing Pythagorean inner products and delayed normalization techniques, we propose the first provably stable one-synchronization reorthogonalized BCGS variant, demonstrating that it has $O(u)$ loss of orthogonality under the condition $O(u) \kappa^2(X) \leq 1/2$, where $\kappa(\cdot)$ represents the condition number. By incorporating one additional synchronization point, we develop a two-synchronization reorthogonalized BCGS variant which maintains $O(u)$ loss of orthogonality under the improved condition $O(u) \kappa(X) \leq 1/2$. An adaptive strategy is then proposed to combine these two variants, ensuring $O(u)$ loss of orthogonality while using as few synchronization points as possible under the less restrictive condition $O(u) \kappa(X) \leq 1/2$. As an example of where this adaptive approach is beneficial, we show that using the adaptive orthogonalization variant, $s$-step GMRES achieves a backward error comparable to $s$-step GMRES with BCGSI+, also known as BCGS2, both theoretically and numerically, but requires fewer synchronization points., Comment: 25 pages, 8. figures

Schmidt decomposition is a powerful tool in quantum information. While Schmidt decomposition is universal for bipartite states, its not for multipartite states. In this article, we review properties of bipartite Schmidt decompositions and study which of them extend to multipartite states. In particular, Schmidt number (the number of non-zero terms in Schmidt decomposition) define an equivalence class using separable unitary transforms. We show that it is NP-complete to partition a multipartite state that attains the highest Schmidt number. In addition, we observe that purifications of a density matrix of a composite system preserves Schmidt decomposability., Comment: Added Theorem 9

Quantum states can be written in infinitely many ways depending on the choices of basis. Schmidt decomposition of a quantum state has a lot of properties useful in the study of entanglement. All bipartite states admit Schmidt decomposition, but this does not extend to multipartite systems. We obtain necessary and sufficient conditions for the existence of Schmidt decompositions of multipartite states. Moreover, we provide an efficient algorithm to obtain the decomposition for a Schmidt decomposable multipartite state.

The ability to efficiently characterize the spatial correlations of entangled states of light is critical for applications of many quantum technologies such as quantum imaging. Here, we demonstrate highly efficient theoretical and experimental characterization of the spatial Schmidt modes and the Schmidt spectrum of bright multiphoton entangled states of light produced from high-gain parametric down-conversion. In contrast to previous studies, we exploit the approximate quasihomogeneity and isotropy of the signal field and dramatically reduce the numerical computations involved in the experimental and theoretical characterization procedures. In our particular case where our experimental data sets consist of 5000 single-shot images of 256*256 pixels each, our method reduced the overall computation time by 2 orders of magnitude. This speed-up would be even more dramatic for larger input sizes. Consequently, we are able to rapidly characterize the Schmidt modes and Schmidt spectrum for a range of pump amplitudes and study their variation with increasing gain. Our results clearly reveal the broadening of the Schmidt modes and narrowing of the Schmidt spectrum for increasing gain with good agreement between theory and experiment., Comment: 8 pages, 4 figures

We study entanglement in a system of three coupled quantum harmonic oscillators. Specifically, we use the Schmidt decomposition to analyze how the entanglement is distributed among the three subsystems. The Schmidt decomposition is a powerful mathematical tool for characterizing bipartite entanglement in composite quantum systems. It allows to write a multipartite quantum state as a sum of product states between the subsystems, with coefficients known as Schmidt coefficients. We apply this decomposition to the general quantum state of three coupled oscillators and study how the Schmidt coefficients evolve as the interaction strengths between the oscillators are varied. This provides insight into how entanglement is shared between the different bipartitions of the overall three-particle system. Our results advance the fundamental understanding of multipartite entanglement in networked quantum systems. They also have implications for quantum information processing using multiple entangled nodes., Comment: 18 pages, 3 figures

Interest in communication-avoiding orthogonalization schemes for high-performance computing has been growing recently. This manuscript addresses open questions about the numerical stability of various block classical Gram-Schmidt variants that have been proposed in the past few years. An abstract framework is employed, the flexibility of which allows for new rigorous bounds on the loss of orthogonality in these variants. We first analyze a generalization of (reorthogonalized) block classical Gram-Schmidt and show that a "strong" intrablock orthogonalization routine is only needed for the very first block in order to maintain orthogonality on the level of the unit roundoff. Then, using this variant, which has four synchronization points per block column, we remove the synchronization points one at a time and analyze how each alteration affects the stability of the resulting method. Our analysis shows that the variant requiring only one synchronization per block column cannot be guaranteed to be stable in practice, as stability begins to degrade with the first reduction of synchronization points. Our analysis of block methods also provides new theoretical results for the single-column case. In particular, it is proven that DCGS2 from [Bielich, D. et al. Par. Comput. 112 (2022)] and CGS-2 from [\'{S}wirydowicz, K. et al, Num. Lin. Alg. Appl. 28 (2021)] are as stable as Householder QR. Numerical examples from the BlockStab toolbox are included throughout, to help compare variants and illustrate the effects of different choices of intraorthogonalization subroutines.

In this study, we enhance the understanding of entanglement transformations and their quantification by extending the concept of Schmidt vector from pure to mixed bipartite states, exploiting the lattice structure of majorization. The Schmidt vector of a bipartite mixed state is defined using two distinct methods: as a concave roof extension of Schmidt vectors of pure states, or equivalently, from the set of pure states that can be transformed into the mixed state through local operations and classical communication (LOCC). We demonstrate that the Schmidt vector fully characterizes separable and maximally entangled states. Furthermore, we prove that the Schmidt vector is monotonic and strongly monotonic under LOCC, giving necessary conditions for conversions between mixed states. Additionally, we extend the definition of the Schmidt rank from pure states to mixed states as the cardinality of the support of the Schmidt vector and show that it is equal to the Schmidt number introduced in previous work [Phys. Rev. A 61, 040301 (R), 2000]. Finally, we introduce a family of entanglement monotones by considering concave and symmetric functions applied to the Schmidt vector.

We study the character theory of diagonal products as well as alternating and elementary enrichments of groups. Our motivation is to show Hilbert-Schmidt stability for various diagonal products, including the classical family of B. H. Neumann groups. Additionally, we initiate a quantitative study of Hilbert-Schmidt stability for infinitely presented groups, through the notion of stability radius growth, and exhibit an uncountable family of Hilbert-Schmidt stable amenable groups with arbitrarily large such growth. Some applications to local Hilbert-Schmidt stability are also provided., Comment: 42 pages; includes a table of contents

We give sufficent conditions for a derivation of a $k$-algebra $A$ of finite type to be $\infty$-integrable in the sense of Hasse-Schmidt, when $A$ is a complete intersection, or when $A$ is reduced and $k$ is a regular ring. As a consequence, we prove that, if in addition $A$ contains a field, then the set of leaps of $A$ is finite along the minimal primes of certain Fitting ideal of $\Omega_{A/k}$.

The quantum spin Hall effect in non-magnetic and Mn-doped HgTe quantum well is strongly affected by Kondo scattering of edge electrons by holes localized on acceptors. A generalized eigenvalue method is usually employed for determining impurity binding energies from the multiband Kohn-Luttinger Hamiltonians in bulk samples and semiconductor quantum structures. Such an approach provides accurate values of the level positions but its applicability for determining the impurity localization radius can be questioned. As an alternative method we propose here the Gram-Schmidt ortogonalization procedure allowing to employ the standard eigenvalue algorithms and, thus, to determine both impurity level energies and the set of normalized eigenvectors. We apply this approach to singly-ionized acceptor states in HgTe quantum wells and obtain impurity level energies and localization radiuses even for states degenerate with the continuum of band states. Such information allows us to assess the energy of bound magnetic polarons in quantum wells doped with magnetic ions. We determine the polaron energies and discuss consequences of the resonant polaron formation on band transport in the bulk samples and quantum wells in the regimes of quantum Hall effects., Comment: 7 pages, 3 figures

We explore the Schmidt-Kennicutt (SK) relations and the star formation efficiency per free-fall time ($\eff$), mirroring observational studies, in numerical simulations of filamentary molecular clouds undergoing gravitational contraction. We find that {\it a)} collapsing clouds accurately replicate the observed SK relations for galactic clouds and {\it b)} the so-called efficiency per free-fall time ($\eff$) is small and constant in space and in time, with values similar to those found in local clouds. This constancy is a consequence of the similar radial scaling of the free-fall time and the internal mass in density structures with spherically-averaged density profiles near $r^{-2}$. We additionally show that {\it c)} the star formation rate (SFR) increases rapidly in time; {\it d)} the low values of $\eff$ are due to the different time periods over which $\tauff$ and $\tausf$ are evaluated, together with the fast increasing SFR, and {\it e)} the fact that star clusters are significantly denser than the gas clumps from which they form is a natural consequence of the fast increasing SFR, the continuous replenishment of the star-forming gas by the accretion flow, and the near $r^{-2}$ density profile generated by the collapse Finally, we argue that the interpretation of $\eff$ as an efficiency is problematic because its maximum value is not bounded by unity, and because the total gas mass in the clouds is not fixed, but rather depends on the environment where clouds are embedded. In summary, our results show that the SK relation, the typical observed values of $\eff$, and the mass density of clusters arise as a natural consequence of gravitational contraction., Comment: Submitted to MNRAS

This work considers the Fr\'echet derivative of the idealized forward map of two-dimensional electrical impedance tomography, i.e., the linear operator that maps a perturbation of the coefficient in the conductivity equation over a bounded two-dimensional domain to the linear approximation of the corresponding change in the Neumann-to-Dirichlet boundary map. It is proved that the Fr\'echet derivative is bounded from the space of square-integrable conductivity perturbations to the space of Hilbert--Schmidt operators on the mean-free $L^2$ functions on the domain boundary, if the background conductivity coefficient is constant and the considered simply-connected domain has a $C^{1,\alpha}$ boundary. This result provides a theoretical framework for analyzing linearization-based one-step reconstruction algorithms of electrical impedance tomography in an infinite-dimensional setting., Comment: 12 pages, 1 figure

We provide the Krull-Remak-Schmidt decomposition of group algebras of the form $k[G]$ where $k$ is a field, which includes fields with prime characteristic, and $G$ a finite abelian group. We achieved this by studying the geometric equivalence of $k[G]$ which we call circulant coordinate rings., Comment: 14 pages, 1 table

In this work we solve, for given bounded operators $B,C$ and Hilbert--Schmidt operator $M$ acting on potentially infinite-dimensional separable Hilbert spaces, the reduced rank approximation problem, $\text{min}\{\Vert{M-BXC}\Vert_{HS}:\ \text{dim}\ \text{ran}(X)\leq r\}$. This extends the result of Sondermann (Statistische Hefte, 1986) and Friedland and Torokhti (SIAM J. Matrix Analysis and Applications, 2007), which studies this problem in the case of matrices $M$, $B$, $C$, $X$, and the analysis involves the Moore-Penrose inverse. In classical approximation problems that can be solved by the singular value decomposition or Moore--Penrose inverse, the solution satisfies a minimal norm property. Friedland and Torokhti also state such a minimal norm property of the solution. However, we show that this minimal norm property does not hold in general and give a modified minimality property that is satisfied in general. We also show that the solution may be discontinuous in infinite-dimensional settings. Necessary and sufficient conditions for continuity of the solutions are given and continuous approximations are constructed when such conditions are not met. Finally, we study problems from signal processing, reduced rank regression and linear operator learning under a rank constraint. We show that our theoretical results enable us to explicitly find the solutions to these problems and to fully characterise their existence, uniqueness and minimality property.

Orbital Angular Momentum (OAM) modes are an important resource used in various branches of quantum science and technology due to their unique helical structure and countably infinite basis. Generating light that simultaneously carries high-order orbital angular momenta and exhibits quantum correlations is a challenging task. In this work, we present a theoretical approach to the generation of correlated Schmidt modes carrying OAM via parametric down-conversion (PDC) in cascaded nonlinear systems (nonlinear interferometers) pumped by Laguerre-Gaussian beams. We demonstrate how the number of generated modes and their population can be controlled by varying the pump parameters, the gain of the PDC process and the distance between the crystals. We investigate the angular displacement sensitivity of these interferometers and demonstrate that it can overcome the classical shot noise limit.

We explain the large variety of star formation laws in terms of one single, simple law that can be inferred from the definition of the star formation rate and basic algebra. The resulting equation, $\SFR = \eff\ \Mcollapsing/\tauff$, although it has been presented elsewhere, is interpreted in terms of clouds undergoing collapse { rather than being turbulence-supported, an idea that different groups have pursued this century}. Under such assumption, one can explain the constancy of $\eff$, the different intra-cloud correlations observed in Milky Way's molecular clouds, as well as the resolved and unresolved extragalactic relationships between SFR and a measurement of the mass in CO, HCN, and CO+HI. We also explain why the slope of the correlation changes when the orbital time $\tauorb$ is considered instead of the free-fall time, and why estimations of the free-fall time from extragalactic observations skew the correlation, providing a false sublinear correlation. We furthermore show that the apparent nearly linear correlation between the star formation rate and the dynamical equilibrium pressure in the midplane of the galaxies, $\PDE$, is just a consequence of $\PDE$ values being dominated by the variation of the column density of molecular gas. All in all, we argue that the star formation law is driven by the collapse of cold, dense gas, which happens to be primarily molecular in the present Universe, and that the role of stellar feedback is just to shut down the star formation process, not to shape the star formation law., Comment: 17 pages, one figure. Accepted by MNRAS

Block classical Gram-Schmidt (BCGS) is commonly used for orthogonalizing a set of vectors $X$ in distributed computing environments due to its favorable communication properties relative to other orthogonalization approaches, such as modified Gram-Schmidt or Householder. However, it is known that BCGS (as well as recently developed low-synchronization variants of BCGS) can suffer from a significant loss of orthogonality in finite-precision arithmetic, which can contribute to instability and inaccurate solutions in downstream applications such as $s$-step Krylov subspace methods. A common solution to improve the orthogonality among the vectors is reorthogonalization. Focusing on the "Pythagorean" variant of BCGS, introduced in [E. Carson, K. Lund, & M. Rozlo\v{z}n\'{i}k. SIAM J. Matrix Anal. Appl. 42(3), pp. 1365--1380, 2021], which guarantees an $O(\varepsilon)\kappa^2(X)$ bound on the loss of orthogonality as long as $O(\varepsilon)\kappa^2(X)<1$, where $\varepsilon$ denotes the unit roundoff, we introduce and analyze two reorthogonalized Pythagorean BCGS variants. These variants feature favorable communication properties, with asymptotically two synchronization points per block column, as well as an improved $O(\varepsilon)$ bound on the loss of orthogonality. Our bounds are derived in a general fashion to additionally allow for the analysis of mixed-precision variants. We verify our theoretical results with a panel of test matrices and experiments from a new version of the \texttt{BlockStab} toolbox.

Lyapunov-Schmidt reduction is a dimensionality reduction technique in nonlinear systems analysis that is commonly utilised in the study of bifurcation problems in high-dimensional systems. The method is a systematic procedure for reducing the dimensionality of systems of algebraic equations that have singular points, preserving essential features of their solution sets. In this article, we establish estimates for the region of validity of the reduction by leveraging recently derived bounds on the Implicit Function Theorem. We then apply these bounds to an illustrative example of a two-dimensional system with a pitchfork bifurcation., Comment: This is the final manuscript accepted for presentation at the 63rd IEEE Conference on Decision and Control, scheduled to be held in Milan, Italy, in December 2024

Higher Berry curvature (HBC) is the proposed generalization of Berry curvature to infinitely extended systems. Heuristically HBC captures the flow of local Berry curvature in a system. Here we provide a simple formula for computing the HBC for extended $d = 1$ systems at the level of wave functions using the Schmidt decomposition. We also find a corresponding formula for matrix product states (MPS), and show that for translationally invariant MPS this gives rise to a quantized invariant. We demonstrate our approach with an exactly solvable model and numerical calculations for generic models using iDMRG, Comment: 9 pages, 2 figures

In this paper, a positive operator is given. It is shown that the product of this positive operator and the convolution operator is a trace class Hilbert-Schmidt integral operator and has nonnegative eigenvalues. A formula is given for the trace of this product operator. It seems that this product operator is the closest trace class integral operator which has nonnegative eigenvalues and is related to the Weil distribution. A relation is given between the trace of the product operator and the Weil distribution.

In this paper we improve the best known constant for the discrepancy formulated in the Komlos Conjecture. The result is based on the improvement of the subgaussian bound for the random vector constructed in the Gram-Schmidt Random Walk algorithm. Moreover, we present detailed argument for the smoothed analysis of this random vector. The analysis concerns a modification of a given matrix in the conjecture by a Gaussian type perturbation. Our result improves the recent paper in this direction., Comment: 20 pages

Let $R$ be a commutative ring If $\mathcal{C}_1$ and $\mathcal{C}_2$ are $R$-linear triangulated categories then we can give an obvious triangulated structure on $\mathcal{C} = \mathcal{C}_1 \oplus \mathcal{C}_2$ where $Hom_\mathcal{C}(U, V) = 0$ if $U \in \mathcal{C}_i$ and $V \in \mathcal{C}_j$ with $i \neq j$. We say a $R$-linear triangulated category $\mathcal{C}$ is disconnected if $\mathcal{C} = \mathcal{C}_1 \oplus \mathcal{C}_2$ where $\mathcal{C}_i$ are non-zero triangulated subcategories of $\mathcal{C}$. Let $\mathcal{C}_i$ and $\mathcal{D}_j$ be connected triangulated $R$ categories with $i \in \Gamma$ and $j \in \Lambda$. Suppose there is an equivalence of triangulated $R$-categories \[ \Phi \colon \bigoplus_{i \in \Gamma}\mathcal{C}_i \xrightarrow{\cong} \bigoplus_{j \in \Lambda}\mathcal{D}_j \] Then we show that there is a bijective function $\pi \colon \Gamma \rightarrow \Lambda$ such that we have an equivalence $\mathcal{C}_i \cong \mathcal{D}_{\pi(i)} $ for all $i \in \Gamma$. We give several examples of connected triangulated categories and also of triangulated subcategories which decompose into utmost finitely many components.

Higher dimensional entangled states demonstrate significant advantages in quantum information processing tasks. Schmidt number is a quantity on the entanglement dimension of a bipartite state. Here we build families of k-positive maps from the symmetric information complete positive operator-valued measurements and mutually unbiased bases, and we also present the Schmidt number witnesses, correspondingly. At last, based on the witnesses obtained from mutually unbiased bases, we show the distance between a bipartite state and the set of states with Schmidt number less than k.

This paper considers the problem of interpreting orthogonalization model coefficients. We derive a causal economic interpretation of the Gram-Schmidt orthogonalization process and provide the conditions for its equivalence to total effects from a recursive Directed Acyclic Graph. We extend the Gram-Schmidt process to groups of simultaneous regressors common in economic data sets and derive its finite sample properties, finding its coefficients to be unbiased, stable, and more efficient than those from Ordinary Least Squares. Finally, we apply the estimator to childhood reading comprehension scores, controlling for such highly collinear characteristics as race, education, and income. The model expands Bohren et al.'s decomposition of systemic discrimination into channel-specific effects and improves its coefficient significance levels.

A deep understanding of quantum entanglement is vital for advancing quantum technologies. The strength of entanglement can be quantified by counting the degrees of freedom that are entangled, which results in a quantity called Schmidt number. A particular challenge is to identify the strength of entanglement in quantum states which remain positive under partial transpose (PPT), otherwise recognized as undistillable states. Finding PPT states with high Schmidt number has become a mathematical and computational challenge. In this work, we introduce efficient analytical tools for calculating the Schmidt number for a class of bipartite states, called generalized grid states. Our methods improve the best known bounds for PPT states with high Schmidt number. Most notably, we construct a Schmidt number three PPT state in five dimensional systems and a family of states with a Schmidt number of $(d+1)/2$ for odd $d$-dimensional systems, representing the best-known scaling of the Schmidt number in a local dimension. Additionally, these states possess intriguing geometrical properties, which we utilize to construct indecomposable entanglement witnesses., Comment: 14 pages, 6 figures, comments welcome