An Observable-Algebra Approach to Quantum Coherence in Anyonic Systems

Abstract

Physical restrictions on operations can make quantum coherence inaccessible. Quantifying off-diagonal density matrix elements in a chosen basis alone does not reveal which information is hidden by each restriction. To distinguish their effects, we describe physical restrictions through subalgebras of accessible observables. The associated trace-preserving conditional expectations unify two reconstruction principles: each gives both the maximum-entropy state compatible with the accessible data and the closest state in the corresponding subalgebra in relative entropy. We apply this framework to anyonic systems to separate the contributions from local-charge superselection and non-Abelian fusion rules. We prove that the charge-line severing map is the conditional expectation onto the local-product subalgebra. It is completely positive and quantum-trace-preserving, and uniquely reconstructs a state in this space from all local-product expectation values. The hierarchy of measurement restrictions then separates total bipartite coherence into three nonnegative contributions: coherence between local charges, inaccessible fusion information, and conventional coherence. The first two decompose the anyonic charge entanglement and distinguish the effects of local-charge superselection from the remaining fusion information. Finally, we relate basis-independent measures of anyonic coherence to entanglement. For pure anyonic states with vacuum total charge, the optimized total and correlated coherence equal the total and conventional relative-entropy entanglement, respectively.

The paper asks how to distinguish information hidden by local-charge superselection from information hidden by non-Abelian fusion, rather than identifying coherence solely with off-diagonal density-matrix elements. It describes accessible measurements by finite-dimensional observable subalgebras. Their trace-preserving conditional expectations give both the unique maximum-entropy state compatible with accessible data and the unique relative-entropy minimizer within the subalgebra. Information loss equals the entropy increase, and nested subalgebras yield additive, nonnegative losses. These statements are established using expectation-value preservation and relative-entropy identities, including singular states and the anyonic quantum trace. For bipartite anyons, charge-line severing is identified with conditional expectation onto the local-product algebra, establishing complete positivity and quantum-trace preservation. A hierarchy of local-charge-preserving, local-product, and local-diagonal algebras gives $C=C_{\rm sec}+C_{\rm fus}+C_{\rm conv}$, with $S_{\rm ACE}=C_{\rm sec}+C_{\rm fus}$. Fusion information vanishes for Abelian anyons but can persist in globally diagonal states: two Fibonacci anyons fusing to vacuum have $C_{\rm fus}=2\log_2 d_\tau$ despite zero fusion-basis coherence. For vacuum-total-charge pure states, sectorwise Schmidt bases minimize total and correlated coherence, recovering total and conventional relative-entropy entanglement, respectively. The charge contributions are $H(p)$ and $2\sum_a p_a\log_2d_a$. The framework assumes accessible observables form unital subalgebras; local-product reconstruction does not additionally impose known global charge. The optimized entanglement equalities are restricted to vacuum pure states. A discussion expresses topological entanglement entropy through optimized microscopic regional coherences, but this optimization does not remove spurious conditional-mutual-information contributions.