Reduction of unitary operators, quantum graphs and quantum channels

TL;DR

The paper introduces a rigorous framework for reducing a finite-dimensional unitary operator $U$ to a subspace via a unitary connection $\Omega$, proving existence and unitarity of the reduced map $\Omega\cdot U$ and linking the construction to Kraus operators and quantum channels. It develops multiple equivalent representations (block-operator forms, series, and Kraus decompositions) and establishes key structural properties, including sequential reductions and reciprocity with inverses. The reduction framework is then applied to quantum graphs, where reduced scattering matrices obey $S' = \Omega\cdot S$ and more general formulas, enabling construction of larger graph scatters from components. The authors extend the approach to non-unitary reductions and quantum channels, deriving conditions for existence, series-based constructions, and unital/TP properties, and discuss potential quantum-computation implementations, concluding that unitary reductions for black-box gates are challenging while channel reductions admit practical realizations. Overall, the work provides a unified mathematical method for composing and simplifying quantum networks and channels, with direct implications for scattering theory on graphs and for quantum information processing.

Abstract

Given a unitary operator in a finite dimensional complex Hilbert space, its unitary reduction to a subspace is defined. The application to quantum graphs is discussed. It is shown how the reduction allows to generate the scattering matrices of new quantum graphs from assembling of simpler graphs. The reduction of quantum channels is also defined. The implementation of the quantum gates corresponding to the reduced unitary operator is investigated, although no explicit construction is presented. The situation is different for the reduction of quantum channels for which explicit implementations are given.

Figures (8)

• Figure 1: Schematics of the reduction.
• Figure 2: A unitary network with two nodes. Some of the connections may be absent.
• Figure 3: Reduction of a graph. The external lines $4$ and $5$ are reduced and the new line is assigned a length $l$, so that the phase is $\omega = e^{ikl}$. The lines $1$, $2$ and $3$ remain external so they carry a label $E$, while $4$ and $5$ become internal and carry a label $I$.
• Figure 4: A unitary reduction implemented through a beam of stationary coherent light.
• Figure 5: A setting for unitary reduction in quantum circuits.