This paper introduces a Quantum Correlation Matrix Memory (QCMM) and Enhanced QCMM (EQCMM), which are useful to work with quantum memories. A version of classical GramSchmidt orthogonalisation process in Dirac notation (called Quantum Orthogonalisation Process: QOP) is presented to convert a nonorthonormal quantum basis, i.e., a set of non-orthonormal quantum vectors (called qudits) to an ortho...

We prove that there does not exist an orthonormal basis {bn} for L(R) such that the sequences {μ(bn)}, {μ(c bn)}, and {∆(bn)∆( bn)} are bounded. A higher dimensional version of this result that involves generalized dispersions is also obtained. The main tool is a time-frequency localization inequality for orthonormal sequences in L(R). On the other hand, for d > 1 we construct a basis {bn} for ...

Square-integrable functions f ∈ L are those of length ‖f‖L2 = 〈f, f〉 L2 < ∞. A subset D ⊂ L is said to be dense, if any f ∈ L can be approximated by a sequence fn ∈ D. This means limn ‖f − fn‖L2 = 0. An orthonormal basis {Ωn} is a set of orthonormal vectors whose finite linear combinations are dense. A linear transformation T is continuous on L if ‖Tf‖L2 ≤ M‖f‖L2 for some constant M < ∞. A cont...

Exercise 1 Suppose that H = C3 and let V be the one-dimensional subspace spanned by |ψy = ?1/3(|0y+ ei/3|1y |2y). Provide an orthonormal basis for VK. Solution: We need to identify a pair of orthonormal states that are also orthogonal to |ψy. An easy first state to take is |ψK 1 y = 1 2 (|0y ei/3|1y). Any other vector of the form a(|0y+ ei/3|1y+ b|2ywill be orthogonal to |ψK 1 y. To also make i...

We prove that any Parseval wavelet frame is the projection of an orthonormal wavelet basis for a representation of the Baumslag-Solitar group BS(1, 2) = 〈u, t | utu = t〉. We give a precise description of this representation in some special cases, and show that for wavelet sets, it is related to symbolic dynamics (Theorem 3.14). We show that the structure of the representation depends on the ana...

We show that RIP frames, tight frames satisfying the restricted isometry property, give rise to nearly tight fusion frames which are nearly orthogonal and hence are nearly equi-isoclinic. We also show how to replace parts of the RIP frame with orthonormal sets while maintaining the restricted isometry property.

The supercell- based orthonormal basis method is proposed to investigate the modal properties of the Bragg fibers. A square lattice is constructed by the whole Bragg fiber which is considered as a supercell, and the periodical dielectric structure of the square lattice is decomposed using periodic functions (cosine). The modal electric field is expanded as the sum of the orthonormal set of Herm...

In this paper, an overview about the identification of dynamic systems using orthonormal basis function models, such as those based on Laguerre and Kautz functions, is presented. The mathematical foundations of these models as well as their advantages and limitations are discussed within the contexts of linear, robust, and nonlinear identification. The discussions comprise a broad bibliographic...

A real symmetric matrix is diagonalisable by a suitable orthonormal change of basis. The joint approximate diagonalisation problem is to find an orthonormal change of basis which simultaneously diagonalises, or approximately diagonalises, two or more real symmetric matrices. A number of important signal processing problems require the computation of a joint approximate diagonaliser. However, no...