Inspired by a well-known characterization of the Jones index of an inclusion of II$_1$ factors due to Pimsner and Popa, we define an index-type invariant for inclusions of finite-dimensional operator systems. We will compute examples of this invariant, and explain how it generalizes the Lovasz theta invariant to general matricial systems in a manner that is closely related to the quantum Lovasz theta invariant defined by Duan, Severini, and Winter.