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arXiv:quant-ph/0107016 · Fortschritte der Physik

On the Local Unitary Equivalence of States of Multi-partite Systems

Mário Ziman, Peter Štelmachovič, Vladimír Bužek

Published Oct 1, 2001• 9 citations

Abstract

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.

DOI: 10.1002/1521-3978(200110)49:10/11<1123::aid-prop1123>3.0.co;2-q

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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.

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.

Understanding multipartite entanglement is vital, as it underpins a wide range of phenomena across physics. The study of transformations of states via Local Operations assisted by Classical Communication (LOCC) allows one to quantitatively analyse entanglement, as it induces a partial order in the Hilbert space. However, it has been shown that, for systems with fixed local dimensions, this order is generically trivial, which prevents relating multipartite states to each other with respect to any entanglement measure. In order to obtain a non-trivial partial ordering, we study a physically motivated extension of LOCC: multi-state LOCC. Here, one considers simultaneous LOCC transformations acting on a finite number of entangled pure states. We study both multipartite and bipartite multi-state transformations. In the multipartite case, we demonstrate that one can change the stochastic LOCC (SLOCC) class of the individual initial states by only applying Local Unitaries (LUs). We show that, by transferring entanglement from one state to the other, one can perform state conversions not possible in the single copy case; provide examples of multipartite entanglement catalysis; and demonstrate improved probabilistic protocols. In the bipartite case, we identify numerous non-trivial LU transformations and show that the source entanglement is not additive. These results demonstrate that multi-state LOCC has a much richer landscape than single-state LOCC.

We derive the necessary and sufficient conditions for the local unitary (LU) equivalence of two general (pure or mixed)

n

-qubit states and determine the local unitary operators connecting them. Almost all relevant information is contained in the one-qubit reduced matrices of the multiqubit states under investigation. Our technique relies on identifying ab initio all local symmetries and the corresponding local cyclic unitary operators. To derive the above conditions we use the reference forms of the multiqubit states whose definition requires the diagonalization of the one-qubit reduced matrices. Based on those conditions, we propose a straightforward protocol to decide whether or not two

n

-qubit states are LU equivalent.

In this brief report, we consider the equivalence between two sets of m + 1 bipartite quantum states under local unitary transformations. For pure states, this problem corresponds to the matrix algebra question of whether two degree m matrix polynomials are unitarily equivalent; i.e. UAiV† = Bi for 0 ≤ i ≤ m where U and V are unitary and (Ai, Bi) are arbitrary pairs of rectangular matrices. We present a randomized polynomial-time algorithm that solves this problem with an arbitrarily high success probability and outputs transforming matrices U and V.