We collect several foundational results regarding the interaction between locally compact spaces, probability spaces and algebras, commutative $C^*$-algebras von Neumann algebras equipped with traces, in “uncountable” setting wh

For the efficient identification of quantum states, we propose notion linear canonical wavelet transform in framework mechanics. Using machinery Dirac representation theory and integration within an ordered product operators, recast to a matrix element squeezing–displacing operator U(?,s)KM between analyzing vector ??| two-mode state |f? be transformed. We also derive inner relation inversion f...

From the Schmidt representation we have that, up to local gates, every 2-qubit state can be written as $[\phi\rangle=\lambda_1 [00\rangle+\lambda_2 [11\rangle$ with $\lambda_1$ and $\lambda_2$ real numbers. For 3-qubits states, it is known, PRL 2000 85, that gates 3-qubit [000\rangle+\lambda_2 e^{i \theta }[100\rangle+\lambda_3[101\rangle+\lambda_4[110\rangle+\lambda_5[111\rangle$ $\lambda_i\ge...

The characterization and the canonical representation of order-n phase type distributions (PH(n)) is an open research problem. This problem is solved for n = 2, since the equivalence of the acyclic and the general PH distributions has been proven for a long time. However, no canonical representations have been introduced for the general PH distribution class so far for n > 2. In this paper we s...

We prove that any minimal (maximal) strongly regular surface in the threedimensional Minkowski space locally admits canonical principal parameters. Using this result, we find a canonical representation of minimal strongly regular time-like surfaces, which makes more precise the Weierstrass representation and shows more precisely the correspondence between these surfaces and holomorphic function...

The loop representation plays an important role in canonical quantum gravity because loop variables allow a natural treatment of the constraints. In these lectures we give an elementary introduction to (i) the relevant history of loops in knot theory and gauge theory, (ii) the loop representation of Maxwell theory, and (iii) the loop representation of canonical quantum gravity.