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arXiv:0906.4716 · Journal of Physics A Mathematical and Theoretical

Algebraic characterization ofX-states in quantum information

A R P Rau

Published Sep 22, 2009• 123 citations
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Abstract

A class of two-qubit states called X-states are increasingly being used to discuss entanglement and other quantum correlations in the field of quantum information. Maximally entangled Bell states and "Werner" states are subsets of them. Apart from being so named because their density matrix looks like the letter X, there is not as yet any characterization of them. The su(2) X su(2) X u(1) subalgebra of the full su(4) algebra of two qubits is pointed out as the underlying invariance of this class of states. X-states are a seven-parameter family associated with this subalgebra of seven operators. This recognition provides a route to preparing such states and also a convenient algebraic procedure for analytically calculating their properties. At the same time, it points to other groups of seven-parameter states that, while not at first sight appearing similar, are also invariant under the same subalgebra. And it opens the way to analyzing invariant states of other subalgebras in bipartite systems.

DOI
10.1088/1751-8113/42/41/412002

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A class of two-qubit states called X-states are increasingly being used to discuss entanglement and other quantum correlations in the field of quantum information. Maximally entangled Bell states and "Werner" states are subsets of them. Apart from being so named because their density matrix looks like the letter X, there is not as yet any characterization of them. The su(2) X su(2) X u(1) subalgebra of the full su(4) algebra of two qubits is pointed out as the underlying invariance of this class of states. X-states are a seven-parameter family associated with this subalgebra of seven operators. This recognition provides a route to preparing such states and also a convenient algebraic procedure for analytically calculating their properties. At the same time, it points to other groups of seven-parameter states that, while not at first sight appearing similar, are also invariant under the same subalgebra. And it opens the way to analyzing invariant states of other subalgebras in bipartite systems.
Several families of states such as Werner states, Bell-diagonal states, and Dicke states are useful in understanding multipartite entanglement. Here we present a $({2}^{N+1}\ensuremath{-}1)$-parameter family of NNN-qubit ``XXX states'' that embraces all those families, generalizing previously defined states for two qubits. We also present the algebra of the operators that characterize the states and an iterative construction for this algebra, a subalgebra of su(2N)({2}^{N})(2N). We show how a variety of entanglement witnesses can detect entanglement in such states. Connections are also made to structures in projective geometry.
Quantum discord, a kind of quantum correlation, is defined as the difference between quantum mutual information and classical correlation in a bipartite system. In general, this correlation is different from entanglement, and quantum discord may be nonzero even for certain separable states. Even in the simple case of bipartite quantum systems, this different kind of quantum correlation has interesting and significant applications in quantum information processing. So far, quantum discord has been calculated explicitly only for a rather limited set of two-qubit quantum states and expressions for more general quantum states are not known. In this article, we derive explicit expressions for quantum discord for a larger class of two-qubit states, namely, a seven-parameter family of so called XXX states that have been of interest in a variety of contexts in the field. We also study the relation between quantum discord, classical correlation, and entanglement for a number of two-qubit states to demonstrate that they are independent measures of correlation with no simple relative ordering between them.
This dissertation describes the application of symmetry to multipartite quantum systems from the viewpoint of quantum information theory. It is written in the framework of abstract quantum information theory, i.e. without distinguishing the physical nature of the d-level systems considered. The results are mathematically rigorous and aim at investigating entanglement properties as well as operational properties of states of multipartite systems. The first chapter gathers the mathematical tools and concepts needed in the following chapters and can be omitted by the experienced reader. In chapter two we apply the concept of symmetry to multipartite systems and show how to construct families of symmetric states that can be described by few parameters only regardless of the size of single systems. Chapter three is devoted to the states of tripartite systems having Werner symmetry. We characterize their separability properties and analyze the strength of the known separability criteria. Furthermore we investigate their entanglement properties in terms of the relative entropy of entanglement, the entanglement monotone induced by the trace norm distance and the maximal violation of Bell inequalities. As a third aspect we analyze the possibility of embedding bipartite Werner states and explore the inner geometry of the manifold given by the statistical distance of two neighbouring tripartite Werner states. In the fourth chapter we introduce multipartite quantum data hiding as an application of multipartite Werner states. We prove the security of this protocol for any coalition and give various examples. The last chapter is concerned with the shared fidelity, its frustration and how it can be maximized in certain multipartite systems.
The entanglement properties of a two-qubit system are analyzed from the point of view of Lie algebras and geometry. We deal with $4\ifmmode\times\else\texttimes\fi{}4$ matrices belonging to a particular class having seven free parameters, constructed from the imposed symmetries on two-qubit states expanded in terms of a basis whose elements belong to a subalgebra of su(4). The entanglement or separability character of the state is determined and measured by analyzing the eigenvalues of an auxiliary matrix obtained from the original state after performing on it a nonunitary operation. In terms of its eigenvalues, we define two squared distances having a Minkowski metric, ${s}^{2}={t}^{2}\ensuremath{-}{\stackrel{P\vec}{V}}^{2}$, which become a measure of the entanglement. The Minkowski metric is a signature of that class of two-qubit states (matrices) and also of the auxiliary matrices to be constructed. The squared distance is invariant by any unitary transformation on the two-qubit state. We illustrate the theory with examples.
Entanglement is a key resource in the emerging field of Quantum Information. The strong correlations between systems described by an entangled state allow us to perform certain tasks more efficiently than it would be possible by using only classical resources. This is why the characterization of entanglement is one of the most important problems in Quantum Information. In this thesis, we analyze several aspects of entanglement. First, we introduce a new family of criteria to determine if a bipartite mixed state is entangled or not. This family consists of a sequence of tests that can be implemented efficiently, and has the property that all entangled states can be detected by some test in the sequence. Each test in the family can be stated as a semidefinite program, which is a class of convex optimization problems. The duality structure of these programs allows us to explicitly construct an entanglement witness that proves entanglement of a state, whenever the state fails one of the tests in the sequence. The entanglement witnesses constructed in this manner have well-defined algebraic properties that can be used to give a characterization of the interior of the set of all possible entanglement witnesses, as well as the set of strictly positive bihermitian forms and the set of strictly positive maps. We also study deterministic transformations of three-qubit pure state when only local operations and classical communication (LOCC) are allowed. We derive strong constraints that the operations and states involved must satisfy, and we apply these results to characterize the set of real states that can be obtained from the GHZ state by LOCC.
We derive a classification and a measure of classical- and quantum-correlation of multipartite qubit, qutrit, and in general, nnn-level systems, in terms of SU(n)(n)(n) representations of density matrices. We compare the measure for the case of bipartite correlation with concurrence and the entropy of entanglement. The characterization of correlation is in terms of the number of nonzero singular values of the correlation matrix, but that of mixed state entanglement requires additional invariant parameters in the density matrix. For the bipartite qubit case, the condition for mixed state entanglement is written explicitly in terms of the invariant paramters in the density matrix. For identical particle systems we analyze the effects of exchange symmetry on classical and quantum correlation.