Related papers: An Introduction to Cartan's KAK Decomposition for …
We introduce a family of operations in quantum mechanics that one can regard as "universal quantum measurements" (UQMs). These measurements are applicable to all finite-dimensional quantum systems and entail the specification of only a…
The structure positive of unitary irreducible representations of the noncompact $u_q(2,1)$ quantum algebra that are related to a positive discrete series is examined. With the aid of projection operators for the $su_q(2)$ subalgebra, a…
Baranyai's theorem is a well-known theorem in the theory of hypergraphs. A corollary of this theorem says that one can partition the family of all $u$-subsets of an $n$-element set into ${n-1\choose u-1}$ sub-families such that each…
We prove a triangular decomposition theorem for the lower crystal lattice $\mathcal{O}_{t}^{A_{0}}(G)$ of the quantized function algebra $\mathcal{O}_{t}(G)$, where $G$ is a connected simply connected complex Lie group with Lie algebra…
When using cylindrical algebraic decomposition (CAD) to solve a problem with respect to a set of polynomials, it is likely not the signs of those polynomials that are of paramount importance but rather the truth values of certain quantifier…
Quantum computation in solid state quantum dots faces two significant challenges: Decoherence from interactions with the environment and the difficulty of generating local magnetic fields for the single qubit rotations. This paper presents…
This paper deals with quon algebras or deformed oscillator algebras, for which the deformation parameter is a root of unity. We show the interest of such algebras for fractional supersymmetric quantum mechanics, angular momentum theory and…
The Cylindrical Algebraic Decomposition (CAD) method is currently the only complete algorithm used in practice for solving real-algebraic problems. To ameliorate its doubly-exponential complexity, different exploration-guided adaptations…
We formalize the correspondence between quantum states and quantum operations isometrically, and harness its consequences. This correspondence was already implicit in the various proofs of the operator sum representation of Completely…
We conduct the first detailed analysis in quantum information of recently derived operator relations from the study of quantum one-way local operations and classical communications (LOCC). We show how operator structures such as operator…
We present a constructive method to translate small quantum circuits into their optical analogues, using linear components of present-day quantum optics technology only. These optical circuits perform precisely the computation that the…
Quantum compiling, a process that decomposes the quantum algorithm into a series of hardware-compatible commands or elementary gates, is of fundamental importance for quantum computing. We introduce an efficient algorithm based on deep…
Quantum compilation is the process of decomposing high-level quantum algorithms or arbitrary unitary operations into quantum circuits composed of a specific set of quantum gates. Neutral atom quantum computing platform is a quantum…
In these lectures we develop the projection operator method for quantum groups. Here the term "quantum groups" means q-deformed universal enveloping algebras of contragredient Lie (super)algebras of finite growth. Contains of the lectures…
The Cartan scheme $\cal X$ of a finite group $G$ with a $(B,N)$-pair is defined to be the coherent configuration associated with the action of $G$ on the right cosets of the Cartan subgroup $B\cap N$ by the right multiplications. It is…
The Pauli matrices are 2-by-2 matrices that are very useful in quantum computing. They can be used as elementary gates in quantum circuits but also to decompose any matrix of $\mathbb{C}^{2^n \times 2^n}$ as a linear combination of tensor…
We give a construction of a Dirac operator on a quantum group based on any simple Lie algebra of classical type. The Dirac operator is an element in the vector space $U_q(\g) \otimes \mathrm{cl}_q(\g)$ where the second tensor factor is a…
One of the principal obstacles on the way to quantum computers is the lack of distinguished basis in the space of unitary evolutions and thus the lack of the commonly accepted set of basic operations (universal gates). A natural choice,…
Nonlocal properties (globalness) of a non-separable unitary determine how the unitary affects the entanglement properties of a quantum state. We apply a given two-qubit unitary on a quadpartite system including two reference systems and…
A generalization of the differential geometry of forms and vector fields to the case of quantum Lie algebras is given. In an abstract formulation that incorporates many existing examples of differential geometry on quantum groups, we…