Universal adjointation of isometry operations using transformation of quantum supermaps

Identification of possible transformations of quantum objects including quantum states and quantum operations is indispensable in developing quantum algorithms. Universal transformations, defined as input-independent transformations, appear in various quantum applications. Such is the case for universal transformations of unitary operations. However, extending these transformations to non-unitary operations is nontrivial and largely unresolved. Addressing this, we introduce isometry adjointation protocols that convert an input isometry operation into its adjoint operation, which include both unitary operation and quantum state transformations. The paper details the construction of parallel and sequential isometry adjointation protocols, derived from unitary inversion protocols using quantum combs, and achieving optimal approximation error. This error is shown to be independent of the output dimension of the isometry operation. In particular, we explicitly obtain an asymptotically optimal parallel protocol achieving an approximation error $\epsilon = \Theta(d^2/n)$, where $d$ is the input dimension of the isometry operation and $n$ is the number of calls of the isometry operation. The research also extends to isometry inversion and universal error detection, employing semidefinite programming to assess optimal performances. The findings suggest that the optimal performance of general protocols in isometry adjointation and universal error detection is not dependent on the output dimension, and that indefinite causal order protocols offer advantages over sequential ones in isometry inversion and universal error detection.Comment: 65 pages, 6 figures. All codes are available at https://github.com/sy3104/isometry_adjointatio

Shift relations and reducibility of some Fuchsian differential equations of order 2,...,6 with three singular points

A Fuchsian differential equation of order six with nine free exponents as parameters and with three singular points is presented. This equation has various symmetries, which specify the accessory parameter as a polynomial of the local exponents. For some shifts of exponents, the shift operators are found, which lead to reducibility conditions of the equation. By several specializations of the parameters and successive factorizations of the equation, it produces several new equations and also known ones. For each equation, shift operators are studied, and when the equation is reducible, we observe the way of factorization and discuss the relation with the shift operators.Comment: 146 pages, 1 figur

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A STUDY OF A FUCHSIAN SYSTEM OF RANK 8 IN 3 VARIABLES AND THE ORDINARY DIFFERENTIAL EQUATIONS AS ITS RESTRICTIONS

A Fuchsian system of rank 8 in 3 variables with 4 parameters is found. The singular locus consists of six planes and a cubic surface. The restriction of the system onto the intersection of two singular planes is an ordinary differential equation of order four with three singular points. A middle convolution of this equation turns out to be the tensor product of two Gauss hypergeometric equations, and another middle convolution sends this equation to the Dotsenko-Fateev equation. Local solutions of these ordinary differential equations are found. Their coefficients are sums of products of the Gamma functions. These sums can be expressed as special values of the generalized hypergeometric series ₄F₃ at 1

Role of Protein Phosphatase 2A in Osteoblast Differentiation and Function

The reversible phosphorylation of proteins plays hugely important roles in a variety of cellular processes, such as differentiation, proliferation, and apoptosis. These processes are strictly controlled by protein kinases (phosphorylation) and phosphatases (de-phosphorylation). Here we provide a brief history of the study of protein phosphorylation, including a summary of different types of protein kinases and phosphatases. One of the most physiologically important serine/threonine phosphatases is PP2A. This review provides a description of the phenotypes of various PP2A transgenic mice and further focuses on the known functions of PP2A in bone formation, including its role in osteoblast differentiation and function. A reduction in PP2A promotes bone formation and osteoblast differentiation through the regulation of bone-related transcription factors such as Osterix. Interestingly, downregulation of PP2A also stimulates adipocyte differentiation from undifferentiated mesenchymal cells under the appropriate adipogenic differentiation conditions. In osteoblasts, PP2A is also involved in the ability to control osteoclastogenesis as well as in the proliferation and metastasis of osteosarcoma cells. Thus, PP2A is considered to be a comprehensive factor in controlling the differentiation and function of cells derived from mesenchymal cells such as osteoblasts and adipocytes

Galaxy clustering constraints on deviations from Newtonian gravity at cosmological scales II: Perturbative and numerical analyses of power spectrum and bispectrum

We explore observational constraints on possible deviations from Newtonian gravity by means of large-scale clustering of galaxies. We measure the power spectrum and the bispectrum of Sloan Digital Sky Survey galaxies and compare the result with predictions in an empirical model of modified gravity. Our model assumes an additional Yukawa-like term with two parameters that characterize the amplitude and the length scale of the modified gravity. The model predictions are calculated using two methods; the second-order perturbation theory and direct N-body simulations. These methods allow us to study non-linear evolution of large-scale structure. Using the simulation results, we find that perturbation theory provides reliable estimates for the power spectrum and the bispectrum in the modified Newtonian model. We also construct mock galaxy catalogues from the simulations, and derive constraints on the amplitude and the length scale of deviations from Newtonian gravity. The resulting constraints from power spectrum are consistent with those obtained in our earlier work, indicating the validity of the previous empirical modeling of gravitational nonlinearity in the modified Newtonian model. If linear biasing is adopted, the bispectrum of the SDSS galaxies yields constraints very similar to those from the power spectrum. If we allow for the nonlinear biasing instead, we find that the ratio of the quadratic to linear biasing coefficients, b_2/b_1, should satisfy -0.4 < b_2/b_1<0.3 in the modified Newtonian model.Comment: 12 pages, 7 figure