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On the local unitary equivalence of states of multipartite systems
| Content Provider | Semantic Scholar |
|---|---|
| Author | Ziman, Mário Stelmachovic, Peter Bužek, Vladimı́r |
| Abstract | Quantum entanglement is the key ingredient in quantum information processing [1]. Bipartite entanglement is quite “well” understood by now, but the investigation of the multi-partite case is just beginning. Several quantum information protocols such as quantum teleportation and quantum dense coding are based on the equivalence of some classes of states of bi-partite systems under the action of local (one-particle) unitary operations [1,2]. So the question of local unitary equivalence of multi-partite states of quantum systems is of importance. Recently several authors [4–6] have studied the so-called polynomial invariants of local unitaries. In the present paper we discuss the problem of local unitary equivalence. We present a set of necessary conditions which the two multi-partite states have to fulfill in order to declare them locally unitary equivalent. The state of a quantum system is usually represented by a density matrix. We know that the elements of the density matrix depend on the choice of the basis in Hilbert space, H, of the system. Physically the basis can be thought of as a complete set of observables. For example, in the case of spin-1/2 particle, the choice of the basis in Hilbert space corresponds to the direction of the Stern-Gerlach experimental setup [3]. A local unitary transformation corresponds to a change of a basis in each of the subsystems. Such unitary transformations are known as passive transformations, because there is no physical action performed on the quantum system itself. These transformations simply reflect the choice of our point of view rather than any specific manipulation of the physical system. It reflects the fact that two locally-unitary-equivalent states have the same matrix form, only the choice basis of subsystems is different. An active unitary transformation corresponds to the most general dynamical map of the states for isolated systems. It means that under the dynamics described by local unitaries each of the subsystems evolves independently. Physically it corresponds to the absence of interactions between the subsystems. This property allows us to prepare from a given state of the multi-partite system some other state only by using local unitaries (here we assume that the subsystems can be separated by spacelike intervals). From a mathematical point of view there is no difference between the active and passive unitary transformations. In this paper we address the question: “Under what conditions are the two states states, % and σ, of a multipartite 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 divided into two parts A and B by unitaries acting only on the systems A and B, separately. In Section II we present examples of some invariants together with some applications for pure states. A criterion about the local unitary equivalence for mixed states is given in Section III. In Section IV we investigate the connection of bipartite and multi-partite entanglement via local unitaries for system of N qudits. |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | http://cds.cern.ch/record/507300/files/0107016.pdf |
| Language | English |
| Access Restriction | Open |
| Subject Keyword | Class Density matrix Dynamical system HL7PublishingSubSection |
| Content Type | Text |
| Resource Type | Article |
