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Invariants for a Class of Nongeneric Three-qubit States

Bao-zhi Sun A, Shao-Ming Fei

Published Jan 1, 2006• 6 citations
OpenAlex

Abstract

We investigate the equivalence of quantum states under local unitary transformations. A complete set of invariants under local unitary transformations is presented for a class of non-generic three-qubit mixed states. It is shown that two such states in this class are locally equivalent if and only if all these invariants have equal values for them. PACS numbers: 03.67.-a, 02.20.Hj, 03.65.-w Key words: Invariant, quantum state, local unitary transformation Quantum entangled states are playing very important roles in quantum information processing and quantum computation [1]. As the properties of entanglement of a multipartite quantum system remain invariant under local unitary transformations on the subsystems, the entanglement can be characterized by all the invariants of the local unitary transformations. The trace norms of realigned or partial transposed density matrices in entanglement measure and separability criteria are some of these invariants [2]. Invariants are also relevant in the discussions of Bell inequalities [3, 4] and teleportation, etc. [5]. A complete

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Local unitary (LU) equivalent states are crucial for understanding the structure of quantum entanglement. We investigate the LU equivalence based on the LU Bargmann invariants (LUBIs). We consider for which states the LUBIs are sufficient to ensure the LU equivalence. We focus on a class of multipartite states with complete local commutativity, called CLC states. We first show that for multipartite CLC states, all the LUBIs are sufficient to ensure the local permutation equivalence, if their single-party marginals are diagonal and non-degenerate. By virtue of the fact that the CLC property remains unchanged under local operations, we verify that the LUBIs are also sufficient to ensure the LU equivalence of multipartite CLC states whose single-party marginals are non-degenerate. Furthermore, when the single-party marginals are degenerate, we propose concrete examples of two-qudit and three-qubit systems, showing that there are states which share all the LUBIs but are not LU equivalent.

quant-pharXiv:2609.22902•Sep 22, 2026•0 citations
We investigate the equivalence of quantum states under local unitary transformations. A complete set of invariants under local unitary transformations is presented for a class of non-generic three-qubit mixed states. It is shown that two such states in this class are locally equivalent if and only if all these invariants have equal values for them. PACS numbers: 03.67.-a, 02.20.Hj, 03.65.-w Key words: Invariant, quantum state, local unitary transformation Quantum entangled states are playing very important roles in quantum information processing and quantum computation [1]. As the properties of entanglement of a multipartite quantum system remain invariant under local unitary transformations on the subsystems, the entanglement can be characterized by all the invariants of the local unitary transformations. The trace norms of realigned or partial transposed density matrices in entanglement measure and separability criteria are some of these invariants [2]. Invariants are also relevant in the discussions of Bell inequalities [3, 4] and teleportation, etc. [5]. A complete
Local unitary (LU) equivalent states are crucial for understanding the structure of quantum entanglement. We investigate the LU equivalence based on the LU Bargmann invariants (LUBIs). We consider for which states the LUBIs are sufficient to ensure the LU equivalence. We focus on a class of multipartite states with complete local commutativity, called CLC states. We first show that for multipartite CLC states, all the LUBIs are sufficient to ensure the local permutation equivalence, if their single-party marginals are diagonal and non-degenerate. By virtue of the fact that the CLC property remains unchanged under local operations, we verify that the LUBIs are also sufficient to ensure the LU equivalence of multipartite CLC states whose single-party marginals are non-degenerate. Furthermore, when the single-party marginals are degenerate, we propose concrete examples of two-qudit and three-qubit systems, showing that there are states which share all the LUBIs but are not LU equivalent.