Invariant-algebraic approach to problems of quantum optics

Ravinder Rupchand Puri

V. P. Karassiov

A general symmetry method for the discovery and analysis of hidden coherent structures in quantum models with an invariant Hamiltonian is proposed. The method is based on the mathematical concept of dual algebraic pairs, which unifies the actions of invariance groups and algebras of dynamic symmetry. Its efficiency is illustrated by the examples of biphotons and the invariant treatment of unpolarized light quasi-spin clusters, and “optical atoms” in nonlinear quantum-optical models.

Frank Vollmer, Deshui Yu

Giovanni Rodari, Tommaso Francalanci, Eugenio Caruccio +12 more

Over the past few years, various methods have been developed to engineer and to exploit the dynamics of photonic quantum states as they evolve through linear optical networks. Recent theoretical works have shown that the underlying Lie algebraic structure plays a crucial role in the description of linear optical Hamiltonians, as such formalism identifies intrinsic symmetries within photonic systems subject to linear optical dynamics. Here, we experimentally investigate the role of Lie algebras applied to the context of Boson sampling, a pivotal model to the current understanding of computational complexity regimes in photonic quantum information. Performing experiments of increasing complexity, realized within a fully reconfigurable photonic circuit, we show that sampling experiments do indeed fulfill the constraints implied by a Lie algebraic structure. In addition, we provide a comprehensive picture about how the concept of Lie algebraic invariant can be interpreted from the point of view of n th -order correlation functions in quantum optics. Our work shows how Lie algebraic invariants can be used as a benchmark tool for the correctness of an underlying linear optical dynamics within typical interferometric architectures for photonic quantum information. This opens avenues for the use of algebraic-inspired methods as analysis tools for photon-based quantum computing protocols.

Nobuo Nagai, Hirofumi Sanada, Takashi Yahagi

Heaviside proposed the telegrapher's equations as an alternative to Maxwell equations and obtained the solution of the telegrapher's equations by the operator method. Indeed, the solution was a one-dimensional wave function. One-dimensional wave equations are partial differential equations and are conventionally solved as initialvalue and boundary-value problems in mathematics. Heaviside's solution was found to be related to the algebraic invariants studied in mathematics in the 19th century. From algebraic invariants, lossless circuits were obtained as Junitary matrix. These research findings are still under study. To utilize the research achievements effectively, the application of lossless circuits obtained from algebraic invariants to quantum physics is one of the issues to be studied in the future.

I. R. Senitzky