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Multipartite states under local unitary transformations

Sergio A Albeverio, Cattaneo, L, Fei, S M, Wang, X H

Published Jan 1, 2005• 17 citations
OpenAlex

Abstract

The equivalence problem under local unitary transformation for $n$--partite pure states is reduced to the one for $(n-1)$--partite mixed states. In particular, a tripartite system $\\mathcal{H}_A\\otimes\\mathcal{H}_B\\otimes\\mathcal{H}_C$, where $\\mathcal{H}_j$ is a finite dimensional complex Hilbert space for $j=A,B,C$, 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 $\\mathcal{H}_A$ belongs to the class of generic mixed states.

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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.
Two pure states of a multi-partite system are alway are related by a unitary transformation acting on the Hilbert space of the hole system. This transformation involves multi-partite transformations. On the other hand some quantum information protocols such as the quantum teleportation and quantum dense coding are based on equivalence of some classes of states of bi-partite systems under the action of local (one-particle) unitary operations.In this paper we adress the question: “Under what conditions are the two states states, ϱ and σ of a multi-partite system locally unitary equivalent?” We present a set of conditions which have to be satisfied in order that the two states are locally unitary equivalent.In addition, we study whether it is possible to prepare a state of a multi-qudit system. Which is devided into two parts A and B, by unitary operations acting only on the systems A and B, separately.
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