Abstract A simple argument is presented that explicitly shows how to construct an arbitrary quantum gate acting on orbital angular momentum (OAM) of single photons. The scheme can be applied to implement subspace multiplexing, where a single high-dimensional OAM qudit represents effectively a stack of multiple independent lower-dimensional qudits. A special subclass of unitaries composed of single-photon controlled gates is studied in detail and notable examples of the general approach are discussed. The generalization of the simple argument leads to the parallelization scheme, which results in the savings of resources. The presented schemes utilize only conventional optical elements and apply not only to single photons but also to classical light.
Abstract The construction of subalgebras of the set of bilinear products of fermion operators which allow the formation of physically relevant solvable models is investigated. The subalgebras are assumed to be semisimple, hermitean closed and compatible with angular momentum. One obtains two classes of subalgebras, i) the wellknown set of spherical tensors of rank zero and ii) a new class of subalgebras which contain the angular momentum operators. The construction procedure for subalgebras of the second kind is presented in detail (hereby, one starts with the abstract type of the subalgebra and defines the physical model space by the carrier space of a representation) and a number of examples, which provide models not known from the literature, is given.
Photons can carry spin angular momentum (SAM) and orbital angular momentum (OAM), which can be used to realize a qubit system and a high-dimension system, respectively. This spin-orbital system is very suitable for implementing one-dimensional discrete-time quantum random walks. We propose a simple scheme of quantum walks on the spin-orbital angular momentum space of photons, where photons walk on the infinite OAM space controlled by their SAM. By employing the recent invention of an optical device, the so-called ``
q-plate,'' our scheme is more simple and efficient than others because there is no Mach-Zehnder interferometer in the scheme.
We study, classify, and explore the mathematical properties of finite-dimensional Lie algebras occurring in the quantum dynamics of single-mode and self-interacting bosonic systems. These Lie algebras are contained in the real skew-hermitian Weyl algebra
A^1, defined as the real subalgebra of the Weyl algebra
A1 consisting of all skew-hermitian polynomials. A central aspect of our analysis is the choice of basis for
A^1, which is composed of skew-symmetric combinations of two elements of the Weyl algebra called monomials, namely strings of creation and annihilation operators combined with their hermitian conjugate. Motivated by the quest for analytical solutions in quantum optimal control and dynamics, we aim at answering the following three fundamental questions: (i) What are the finite-dimensional Lie subalgebras in
A^1 generated by monomials alone? (ii)~What are the finite-dimensional Lie subalgebras in
A^1 that contain the free Hamiltonian? (iii) What are the non-abelian and finite-dimensional Lie subalgebras that can be faithfully realized in
A^1? We answer the first question by providing all possible realizations of all finite-dimensional non-abelian Lie algebras that are generated by monomials alone. We answer the second question by proving that any non-abelian and finite-dimensional subalgebra of
A^1 that contains a free Hamiltonian term must be a subalgebra of the Schrödinger algebra. We partially answer the third question by classifying all nilpotent and non-solvable Lie algebras that can be realized in
A^1, and comment on the remaining cases. Finally, we also discuss the implications of our results for quantum control theory. Our work constitutes an important stepping stone to understanding quantum dynamics of bosonic systems in full generality.
The conventional method of constructing wave functions of pure angular momentum for a composite system is based on the idea of coupling the angular momenta of the constituents. Instead of this synthetic method, an analytical approach is here introduced. It is pointed out that an arbitrary trial wave function for the total system in a unique way must be resolvable into orthogonal components of pure angular momentum associated with different quantum numbers. A particular component can be selected by means of certain 'projection operators', which annihilate all components except the one desired. Such a projection operator is simply a product of commuting factors, each one of which annihilates a specific eigenfunction to the angular momentum under consideration. Physically this idea is of importance, since one can now start out from a rough model wave function based essentially on qualitative arguments, for instance of the independent-particle type, and then obtain a mathematically and physically much better trial function by selecting the particular component of the original function which has the correct symmetry type desired. The projection operators are studied in some detail both as products and in expanded form. The case of degenerate subspaces of the same angular momentum quantum numbers is discussed, and the problem of constructing an orthogonal subset of functions is solved by a simple elimination procedure; the connection with the conventional seniority idea is briefly discussed.
Quantum communication in a higher dimensional system provides a higher capacity, more secure, and smaller latency compared to that of quantum communications based on 2-dimensional system or qubit. This paper studies a simple design of quantum communications based on orbital angular momentum (OAM). We consider two system models for quantum communications based on OAM, i.e., (i) a general structure of a simple OAM-based quantum communication system and (ii) a simple quantum error correction coding scheme for 3-dimensional system or quantum ternary digit (qutrit). We theoretically evaluate the possible construction of OAM-based quantum communications supported by qutrit-based error corrections. We found that OAM-based quantum communications are possibly supported by the derived controlled-NOT (CNOT) gate and stabilizers for qutrit bit flip error corrections. The qutrit stabilizers have shown that they can completely correct all qutrit single bit-flip errors. The results are expected to open new development on qudit-based quantum communications and error corrections.