Related papers: Simple factorization of unitary transformations

Factorization of an $n\times n$ unitary matrix as a product of $n$ diagonal matrices containing only phases interlaced with $n-1$ orthogonal matrices each one generated by a real vector as well as an explicit form for the Weyl factorization…

We give a number of explicit matrix-algorithms for analysis/synthesis in multi-phase filtering; i.e., the operation on discrete-time signals which allow a separation into frequency-band components, one for each of the ranges of bands, say…

We introduce a novel parameterization of complex unitary matrices, which allows for the efficient photonic implementation of arbitrary linear discrete unitary operators. The proposed architecture is built on factorizing an $N \times N$…

Interferences in multi-path systems for single and multiple particles are theoretically analyzed. A holistic method is presented, which allows to construct the unitary transition matrix describing interferometers for any port number d and…

We discuss a refinement of triangular factorization for loops into SU(2).

This paper proposes new factorizations for computing the Neumann series. The factorizations are based on fast algorithms for small prime sizes series and the splitting of large sizes into several smaller ones. We propose a different basis…

Truncated Fourier, Gauss, Kummer and exponential sums can be used to factorize numbers: for a factor these sums equal unity in absolute value, whereas they nearly vanish for any other number. We show how this factorization algorithm can…

Unitary transformations are routinely modeled and implemented in the field of quantum optics. In contrast, nonunitary transformations that can involve loss and gain require a different approach. In this theory work, we present a universal…

Recent investigations suggest that the discrete linear unitary group $U(N)$ can be represented by interlacing a finite sequence of diagonal phase operations with an intervening unitary operator. However, despite rigorous numerical…

Sinkhorn proved that every entry-wise positive matrix can be made doubly stochastic by multiplying with two diagonal matrices. In this note we prove a recently conjectured analogue for unitary matrices: every unitary can be decomposed into…

The development of large-scale, programmable photonic circuits capable of performing generic matrix-vector multiplication is essential for both classical and quantum information processing. However, this goal is hindered by high losses,…

We present a constructive framework for designing transformations between structured light fields using birefringent optical elements, formulated in terms of SU(2) operations on polarization. Within this framework, transformations between…

In 1994, Reck et al. showed how to realize any unitary transformation on a single photon using a product of beamsplitters and phaseshifters. Here we show that any single beamsplitter that nontrivially mixes two modes, also densely generates…

We introduce a versatile numerical method for modeling light diffraction in periodically patterned photonic structures containing quadratically nonlinear non-centrosymmetric optical materials. Our approach extends the generalized source…

We analyze the process of two-particle scattering with unstable particle in an intermediate state. It was shown that the cross-section can be represented in the universal factorized form for an arbitrary set of particles. Phenomenological…

The dynamical symmetries of the Kratzer-type molecular potentials (generalized Kratzer molecular potentials) are studied by using the factorization method. The creation and annihilation (ladder) operators for the radial eigenfunctions…

Factorization of matrices of Laurent polynomials plays an important role in mathematics and engineering such as wavelet frame construction and filter bank design. Wavelet frames (a.k.a. framelets) are useful in applications such as signal…

Linear optical circuits of growing complexity are playing an increasing role in emerging photonic quantum technologies. Individual photonic devices are typically described by a unitary matrix containing amplitude and phase information, the…

A very simple and short proof of the polynomial matrix spectral factorization theorem (on the unit circle as well as on the real line) is presented, which relies on elementary complex analysis and linear algebra.

Linear optical operations are fundamental and significant for both quantum mechanics and classical technologies. We demonstrate a non-cascaded approach to perform arbitrary unitary and non-unitary linear operations for N-dimensional…