A Stabilizer Formalism for Infinitely Many Qubits

Abstract

The study of infinite dimensional quantum systems has been an active area of discussion in quantum information theory, particularly in settings where certain properties are shown to be not attainable by any finite dimensional system (such as nonlocal correlations). Similarly, the notion of stabilizer states has yielded interesting developments in areas like error correction, efficient simulation of quantum systems and its relation to graph states. However, the commonly used model of tensor products of finite dimensional Hilbert spaces is not sufficiently general to capture infinite dimensional stabilizer states. A more general framework quantum mechanical systems using C*-algebras has been instrumental in studying systems with an infinite number of discrete systems in quantum statistical mechanics and quantum field theory. We propose a framework in the C*-algebra model (specifically, the CAR algebra) for the stabilizer formalism that extends to infinitely many qubits. Importantly, the stabilizer states on the CAR algebra form a class of states that can attain unbounded entanglement and yet has a simple characterization through the group structure of its stabilizer. In this framework, we develop a theory for the states, operations and measurements needed to study open questions in quantum information.