Stabilizer formalism in ∞-qubit systems

Quantum error correction, in the sense of Knill and Laflamme, uses the language of quantum channels on spaces of operators to characterize when quantum information can be recovered after the action of noise. Utilizing the (anti-)commutation structure of the n-qubit Pauli group, Gottesman's stabilizer formalism algebraically reframes the recoverability of information in n-qubit systems. We introduce an infinite-order analogue of the Pauli group which acts naturally on a separable Hilbert space modeling a system of infinitely many qubits. Then, we recover the essential characterization of error correction via quantum channels in this ∞-qubit setting, deducing the usual Knill-Laflamme condition for correctability using Kraus' operator-sum decomposition. Finally, using the unitarity and commutation relations of the ∞-qubit Pauli group, we establish a stabilizer formalism for errors on ∞-qubit systems.