2015/07/31 by Jacob Turner, Jason Morton
Computer Science · Mathematics · Physics and Astronomy · #Algebra over a field #Combinatorics #Computer science #Discrete mathematics #Equivalence (formal languages) #Mathematical Analysis and Transform Methods #Mathematics #Matrix Theory and Algorithms #Pure mathematics #Set (abstract data type) #Spectral Theory in Mathematical Physics #math-ph #math.MP #math.RT
paper · pdf · doi:10.3842/sigma.2017.028
published as SIGMA 13 (2017), 028, 20 pages
arxiv created 2017/05/02 · openalex publication_date 2017/05/02 · arxiv updated 2017/05/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We show that two density operators of mixed quantum states are in the same local unitary orbit if and only if they agree on polynomial invariants in a certain Noetherian ring for which degree bounds are known in the literature. This implicitly gives a finite complete set of invariants for local unitary equivalence. This is done by showing that local unitary equivalence of density operators is equivalent to local GL equivalence and then using techniques from algebraic geometry and geometric invariant theory. We also classify the SLOCC polynomial invariants and give a degree bound for generators of the invariant ring in the case of n-qubit pure states. Of course it is well known that polynomial invariants are not a complete set of invariants for SLOCC.