Experimental realization of any discrete unitary operator

Key Points

• To establish a theoretical and algorithmic method for constructing any finite-dimensional discrete unitary operator and measuring any discrete Hermitian observable using optical setups.
• Designed a recursive algorithm that mathematically factorizes any N×N unitary matrix into a sequence of two-dimensional transformations.
• Mapped the decomposed mathematical operations directly to optical devices, specifically networks of two-port beam splitters.
• Demonstrated that any arbitrary discrete unitary operator can be realized physically in the laboratory through sequential beam splitter configurations.
• Showed that observables corresponding to any discrete Hermitian matrix can be measured, making higher-dimensional quantum experiments feasible for various radiation types, including photons and atoms.