Kaplansky's second test problem on similarity is solved affirmatively for operators with property (J) in type I_n von Neumann algebras, for all type I_n with n≤3, and for Banach-algebra elements with essentially finite-dimensional commutant.
Bargmann invariants and local unitary equivalence
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abstract
In this paper, we study the local unitary equivalence of quantum states on $\mathbb{C}^n\otimes\mathbb{C}^n$, which is an important notion in quantum information theory. For the case $n=2$, the local unitary orbits of two-qubit states are completely determined by their local unitary Bargmann invariants. We show that the local unitary Bargmann invariants do not form a complete set of invariants of local unitary orbits for $n\geqslant 3$, which negatively answers a problem proposed by L. Zhang and B. Xie.
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Kaplansky's second test problem in operator algebras
Kaplansky's second test problem on similarity is solved affirmatively for operators with property (J) in type I_n von Neumann algebras, for all type I_n with n≤3, and for Banach-algebra elements with essentially finite-dimensional commutant.