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ON CLASSES OF LOCAL UNITARY TRANSFORMATIONS

Naihuan Jing

Published Aug 4, 2013• 0 citations
OpenAlex

Abstract

Abstract. We give a one-to-one correspondence between classes of density matrices under local unitary invariance and the double cosets of unitary groups. We show that the interrelationship among classes of local unitary equivalent multi-partite mixed states is independent from the actual values of the eigenvalues and only depends on the multiplicities of the eigenvalues. The interpretation in terms of homogeneous spaces of unitary groups is also discussed. PACS numbers: 03.67.Mn, 02.20.Hj, 02.10.Ox

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Abstract. We give a one-to-one correspondence between classes of density matrices under local unitary invariance and the double cosets of unitary groups. We show that the interrelationship among classes of local unitary equivalent multi-partite mixed states is independent from the actual values of the eigenvalues and only depends on the multiplicities of the eigenvalues. The interpretation in terms of homogeneous spaces of unitary groups is also discussed. PACS numbers: 03.67.Mn, 02.20.Hj, 02.10.Ox
The equivalence problem under local unitary transformation for nnn--partite pure states is reduced to the one for (n−1)(n-1)(n−1)--partite mixed states. In particular, a tripartite system mathcalHAotimesmathcalHBotimesmathcalHC\\mathcal{H}_A\\otimes\\mathcal{H}_B\\otimes\\mathcal{H}_CmathcalHA​otimesmathcalHB​otimesmathcalHC​, where mathcalHj\\mathcal{H}_jmathcalHj​ is a finite dimensional complex Hilbert space for j=A,B,Cj=A,B,Cj=A,B,C, is considered and a set of invariants under local transformations is introduced, which is complete for the set of states whose partial trace with respect to mathcalHA\\mathcal{H}_AmathcalHA​ belongs to the class of generic mixed states.