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arXiv:quant-ph/9801076 · Physical Review Letters

Nonlocal Parameters for Multiparticle Density Matrices

Noah Linden, Sandu Popescu, Anthony Sudbery

Published Jul 12, 1999• 146 citations
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Abstract

As far as entanglement is concerned, two density matrices of $n$ particles are completely equivalent if one can be transformed into the other by local unitary transformations. We present two methods to find whether or not two generic density matrices of arbitrary numbers of spin-1/2 particles are equivalent. Both methods describe density matrices in terms of a finite number of invariant parameters.

DOI
10.1103/physrevlett.83.243

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As far as entanglement is concerned, two density matrices of nnn particles are completely equivalent if one can be transformed into the other by local unitary transformations. We present two methods to find whether or not two generic density matrices of arbitrary numbers of spin-1/2 particles are equivalent. Both methods describe density matrices in terms of a finite number of invariant parameters.
We build, using group-theoretic methods, a genaral framework for approaching multi-particle entanglement. As far as entanglement is concerned, two states of n spin-1/2 particles are equivalent if they are on the same orbit of the group of local rotations (U(2)n). We give a method for finding the number of parameters needed to describe inequivalent n spin-1/2 particle states. We also describe how entanglement of states on a given orbit may be characterized by the stability group of the action of the group of local rotations on any point on the orbit.
Research on quantum states often focuses on the correlation between nonlocal effects and local unitary invariants, among which local unitary equivalence plays a significant role in quantum state classification and resource theories. This paper focuses on the local unitary equivalence of multipartite quantum states in quantum information theory, aiming to determine a complete set of invariants that identify their local unitary orbits; these invariants are crucial for deriving polynomial invariants and describing the physical properties preserved under local unitary transformations. The study deeply explores the characterization of local unitary equivalence and the method of detecting entanglement using local unitary Bargmann invariants. Taking two-qubit systems as an example, it verifies the measurability of invariants that determine equivalence and establishes a connection between Makhlin fundamental invariants (a complete set of 18 local unitary invariants for two-qubit states) and local unitary Bargmann invariants. These Bargmann invariants, related to the traces of products of density operators and marginal states, can be measured through cycle tests (an extended form of swap tests).
Local unitary equivalence is an important ingredient for quantifying and classifying entanglement. Verifying whether or not two quantum states are local unitary equivalent is a crucial problem, where only the case of multipartite pure states is solved. For mixed states, however, the verification of local unitary equivalence is still a challenging problem. In this paper, based on the coefficient matrices of generalized Bloch representations of quantum states, we find a variety of local unitary invariants for arbitrary-dimensional bipartite quantum states. These invariants are operational and can be used as necessary conditions for verifying the local unitary equivalence of two quantum states. Furthermore, we extend the construction to the arbitrary-dimensional multipartite case. We finally apply these invariants to estimate concurrence, a vital entanglement measure, showing the practicability of local unitary invariants in characterizing entanglement.