In this paper we study an application of a tool called the “discriminant of algebra”, which gives us an effective method to determine the complete reducibility of algebras. This tool can be applied to the analysis of open quantum systems and we study some examples of it.
The analysis of an open quantum system can be by far difficult if the dimension of the system Hilbert space is large or infinite. However, in some cases the dynamics on a finite-dimensional Hilbert space can be decomposed into a block-diagonal form, which simplifies the system structure. In this presentation, we will study several criteria for the complete reducibility and, in addition, a computational method for a basis of each simplified component to apply for the analysis of open quantum systems. An important point of these tools is that they are “effective” methods (one can complete the task in a finite number of steps).
We develop a general algebraic method for investigating the diagonalizability of a composite-system observable in the, so-called, noncorrelated basis of the Hilbert state space of a composite ("two-body") quantum system. The method consists in rewriting a composite-system obsrvable in linear form, for which one directly and uniquely obtains an answer concerning the diagonalizability. The method has been applied to the analysis of the concept of separability of the interaction part of the Hamiltonian of the composite system, "quantum system plus environment (apparatus)". The application leads to a new (more precise) operational definition of separability, which represents a necessary condition concerning the occurrence of decoherence in the, so-called, orthodox approach to the theory of decoherence.
Developed as a workhorse for classical simulations of quantum algorithms and quantum many-body systems, Tensor Network methods have entered the scientific mainstream in quantum physics. Among various types of tensor networks, Tensor Trains (commonly know as Matrix Product States in the quantum computing community) have already found applications in machine learning. These methods often rely on a powerful linear algebra tool called the Singular Value Decomposition (SVD). Several conditional GAN architectures for image denoising incorporate SVD as a single-cut decomposition step applied to generator feature maps. In this work we introduce TT-Net, which replaces the per-channel SVD denoising block with a two-cut tensor-train decomposition capable of accessing cross-channel information directly, a capability absent from contemporary alternatives. In a controlled comparison differing only in this decomposition mechanism, TT-Net outperforms SVD-Net on PSNR and SSIM across all three noise types tested (Gaussian, motion blur, and salt-and-pepper), supporting the hypothesis that cross-channel access improves denoising quality. Training-dynamics analysis further shows that TT-Net's adversarial loss term consistently saturates to a stagnant state across all three noise types, more so than SVD-Net's, while reconstruction quality continues to improve regardless, raising an open question about the adversarial component's contribution that this work identifies but does not resolve. Furthermore, for Gaussian noise our method outperforms both the EigenGAN and the state of the art Pix2pix method which does not assume any linear algebra decompositions and does not retain any linear algebra information. Our manuscript shows how quantum inspired tools can be used as practical real world feature filters for deep learning applications.
Decompositions of the Lie group of unitary matrices are very useful tools in the control and analysis of quantum dynamics. In this paper, we survey recent results on decompositions, concerning their significance in terms of symmetries, entanglement, and their applications to control and dynamics. Several decompositions can be obtained by recursively applying the Cartan classification of the symmetric spaces of the classical Lie groups. The emphasis is on a novel recursive procedure to decompose the unitary evolution of bipartite quantum systems of arbitrary dimensions, in simpler factors. This procedure makes transparent the contributions of the entangling and non entangling transformations
The purpose of this paper is to present general algebraic methods for describing quantum networks. These methods are motivated by new imperatives of quantum network analysis and design, in particular, feedback control. The basic tools in our methodology are a matrix representation of open quantum systems, and the concatenation and series products of two such systems. We show how these methods can be used to efficiently model open quantum systems, or networks of such systems. We also explain how quantum feedback networks can be designed using these methods.