Are problems in Quantum Information Theory (un)decidable?

TL;DR

This paper explores the decidability of problems in quantum information theory, highlighting which are algorithmically solvable and which are inherently undecidable, especially in the context of entanglement and quantum channels.

Contribution

It reviews tools for determining decidability in quantum problems and applies them to identify undecidable cases, impacting the understanding of quantum information theory.

Findings

Certain entanglement problems are decidable via quantifier elimination.

Many asymptotic properties are likely undecidable or hard to decide.

Questions about surpassing fidelity thresholds can be undecidable.

Abstract

This note is intended to foster a discussion about the extent to which typical problems arising in quantum information theory are algorithmically decidable (in principle rather than in practice). Various problems in the context of entanglement theory and quantum channels turn out to be decidable via quantifier elimination as long as they admit a compact formulation without quantification over integers. For many asymptotically defined properties which have to hold for all or for one integer N, however, effective procedures seem to be difficult if not impossible to find. We review some of the main tools for (dis)proving decidability and apply them to problems in quantum information theory. We find that questions like "can we overcome fidelity 1/2 w.r.t. a two-qubit singlet state?" easily become undecidable. A closer look at such questions might rule out some of the "single-letter" formulas sought in quantum information theory.