Exact and asymptotic measures of multipartite pure-state entanglement (original) (raw)
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We formalize and extend an operational multipartite entanglement measure introduced in T. R. Oliveira, G. Rigolin, and M. C. de Oliveira, Phys. Rev. A 73, 010305(R) (2006) through the generalization of global entanglement (GE) [ D. A. Meyer and N. R. Wallach, J. Math. Phys. 43, 4273 (2002)]. Contrarily to GE the main feature of this new measure lies in the fact that we study the mean linear entropy of all possible partitions of a multipartite system. This allows the construction of an operational multipartite entanglement measure which is able to distinguish among different multipartite entangled states that GE failed to discriminate. Furthermore, it is also maximum at the critical point of the Ising chain in a transverse magnetic field being thus able to detect a quantum phase transition.
Entanglement and Its Multipartite Extensions
International Journal of Modern Physics B, 2013
The aspects of many particle systems as far as their entanglement is concerned is highlighted. To this end we briefly review the bipartite measures of entanglement and the entanglement of pairs both for systems of distinguishable and indistinguishable particles. The analysis of these quantities in macroscopic systems shows that close to quantum phase transitions, the entanglement of many particles typically dominates that of pairs. This leads to an analysis of a method to construct many-body entanglement measures. SL-invariant measures are a generalization to quantities as the concurrence, and can be obtained with a formalism containing two (actually three) orthogonal antilinear operators. The main drawback of this antilinear framework, namely to measure these quantities in the experiment, is resolved by a formula linking the antilinear formalism to an equivalent linear framework.
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Classification of nonasymptotic bipartite pure-state entanglement transformations
Physical Review A, 2002
Let {|ψ , |φ } be an incomparable pair of states (|ψ |φ), i.e., |ψ and |φ cannot be transformed to each other with probability one by local transformations and classical communication (LOCC). We show that incomparable states can be multiple-copy transformable, i.e., there can exist a k, such that |ψ ⊗k+1 → |φ ⊗k+1 , i.e., k + 1 copies of |ψ can be transformed to k + 1 copies of |φ with probability one by LOCC but |ψ ⊗n |φ ⊗n ∀n ≤ k. We call such states k-copy LOCC incomparable. We provide a necessary condition for a given pair of states to be k-copy LOCC incomparable for some k. We also show that there exist states that are neither k-copy LOCC incomparable for any k nor catalyzable even with multiple copies. We call such states strongly incomparable. We give a sufficient condition for strong incomparability. We demonstrate that the optimal probability of a conclusive transformation involving many copies, pmax |ψ ⊗m → |φ ⊗m can decrease exponentially with the number of source states m, even if the source state has more entropy of entanglement. We also show that the probability of a conclusive conversion might not be a monotonic function of the number of copies. Fascinating developments in quantum information theory [1] and quantum computing [2] during the past decade has led us to view entanglement as a valued physical resource. Consequently, recent studies have largely been devoted towards its quantification in appropriate limits (finite or asymptotic), optimal manipulation, and transformation properties under local operations and classical communication (LOCC) [3, 4, 5, 6, 7, 8]. Since the specific tasks that can be accomplished with entanglement as a resource is closely related to its transformation properties, it is of importance to know what transformations are allowed under LOCC. Suppose Alice and Bob share a pure state |ψ (source state), which they wish to convert to another entangled state |φ (target state) under LOCC. A necessary and sufficient condition for this transformation to be possible with certainty (denoted by |ψ → |φ) has been obtained by Nielsen [3]. If such a deterministic transformation is not possible but |ψ has at least as many Schmidt coefficients as |φ , then one
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Physical Review A, 2005
We demonstrate that even under positive partial transpose preserving operations in an asymptotic setting GHZ and W states are not reversibly interconvertible. We investigate the structure of minimal reversible entanglement generating set (MREGS) for tri-partite states under positive partial transpose (ppt) preserving operations. We demonstrate that the set consisting of W and EPR states alone cannot be an MREGS. In this context we prove the surprising result that the relative entropy of entanglement can be strictly sub-additive for certain pure tri-partite states which is crucial to keep open the possibility that the set of GHZ-state and EPR states together constitute an MREGS under ppt-preserving operations.
Statistical mechanics of multipartite entanglement
Journal of Physics A-mathematical and Theoretical, 2009
We characterize the multipartite entanglement of a system of n qubits in terms of the distribution function of the bipartite purity over all balanced bipartitions. We search for those (maximally multipartite entangled) states whose purity is minimum for all bipartitions and recast this optimization problem into a problem of statistical mechanics. PACS numbers: 03.67.Mn, 03.65.Ud, 89.75.−k, 03.67.−a (Some figures in this article are in colour only in the electronic version)
The Schmidt Measure as a Tool for Quantifying Multi-Particle Entanglement
2000
We present a measure of quantum entanglement which is capable of quantifying the degree of entanglement of a multi-partite quantum system. This measure, which is based on a generalization of the Schmidt rank of a pure state, is defined on the full state space and is shown to be an entanglement monotone, that is, it cannot increase under local quantum operations with classical communication and under mixing. For a large class of mixed states this measure of entanglement can be calculated exactly, and it provides a detailed classification of mixed states.
Physical Review A, 2008
We prove conjectures on the relative entropy of entanglement (REE) for two families of multipartite qubit states. Thus, analytic expressions of REE for these families of states can be given. The first family of states are composed of mixture of some permutation-invariant multi-qubit states. The results generalized to multi-qudit states are also shown to hold. The second family of states contain Dür's bound entangled states. Along the way, we have discussed the relation of REE to two other measures: robustness of entanglement and geometric measure of entanglement, slightly extending previous results.
An algebraic approach to the study of multipartite entanglement
Journal of Russian Laser Research
A simple algebraic approach to the study of multipartite entanglement for pure states is introduced together with a class of suitable functionals able to detect entanglement. On this basis, some known results are reproduced. Indeed, by investigating the properties of the introduced functionals, it is shown that a subset of such class is strictly connected to the purity. Moreover, a direct and basic solution to the problem of the simultaneous maximization of three appropriate functionals for three-qubit states is provided, confirming that the simultaneous maximization of the entanglement for all possible bipartitions is compatible only with the structure of GHZ-states.