Concise review of the basic properties of unitary matrix integrals. They are studied with the help of the three matrix models: the ordinary unitary model, Brezin-Gross-Witten model and the Harish-Charndra-Itzykson-Zuber model. Especial attention is paid to the tricky sides of the story, from De Wit-t’Hooft anomaly in unitary integrals to the problem of correlators with Itzykson-Zuber measure. O...

The unitary braiding operators describing topological entanglements can be viewed as universal quantum gates for quantum computation. With the help of the Brylinski’s theorem, the unitary solutions of the quantum Yang–Baxter equation can be also related to universal quantum gates. This paper derives unitary solutions of the quantum Yang–Baxter equation via Yang–Baxterization from the solutions ...

A unitary family is a family of unitary operators U(x) acting on a finite dimensional hermitian vector space, depending analytically on a real parameter x. It is monotone if 1 i U (x)U(x) is a positive operator for each x. We prove a number of results generalizing standard theorems on the spectral theory of a single unitary operator U0, which correspond to the ’commutative’ case U(x) = eU0. Als...

We consider the problem of completion of a matrix with a specified lower triangular part to a unitary matrix. In this paper we obtain the necessary and sufficient conditions of existence of a unitary completion without any additional constraints and give a general formula for this completion. The paper is mainly focused on matrices with the specified lower triangular part of a special form. For...