Integrity bases for local invariants of composite quantum systems

Abstract

Unitary group branchings appropriate to the calculation of local invariants of density matrices of composite quantum systems are formulated using the method of $S$-function plethysms. From this, the generating function for the number of invariants at each degree in the density matrix can be computed. For the case of two two-level systems the generating function is $F(q) = 1 + q + 4q^{2} + 6 q^{3} + 16 q^{4} + 23 q^{5} + 52 q^{6} + 77 q^{7} + 150 q^{8} + 224 q^{9} + 396 q^{10} + 583 q^{11}+ O(q^{12})$. Factorisation of such series leads in principle to the identification of an integrity basis of algebraically independent invariants. This note replaces Appendix B of our paper\cite{us} J Phys {\bf A33} (2000) 1895-1914 (\texttt{quant-ph/0001076}) which is incorrect.