Separability and Fourier representations of density matrices

Pittenger, A. O.; Rubin, Morton H.

doi:10.1103/physreva.62.032313

Cited by 62 publications

(91 citation statements)

References 22 publications

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“…Recently, several generalizations have been consider for d = 3 [15] and higher [3,17]. Another natural choice of basis has G jk = |j k| for some orthonormal basis |j .…”

Section: Representations In Basesmentioning

confidence: 99%

Entanglement Breaking Channels

2003

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This paper studies the class of stochastic maps, or channels, for which (I ⊗ Φ)(Γ) is always separable (even for entangled Γ). Such maps are called entanglement breaking, and can always be written in the form Φ(ρ) = k R k Tr F k ρ where each R k is a density matrix and F k > 0. If, in addition, Φ is trace-preserving, the {F k } must form a positive operator valued measure (POVM). Some special classes of these maps are considered and other characterizations given.Since the set of entanglement-breaking trace-preserving maps is convex, it can be characterized by its extreme points. The only extreme points of the set of completely positive trace preserving maps which are also entanglement breaking are those known as classical quantum or CQ. However, for d ≥ 3, the set of entanglement breaking maps has additional extreme points which are not extreme CQ maps.

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“…Recently, several generalizations have been consider for d = 3 [15] and higher [3,17]. Another natural choice of basis has G jk = |j k| for some orthonormal basis |j .…”

Section: Representations In Basesmentioning

confidence: 99%

Entanglement Breaking Channels

2003

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“…The main goal will be to characterize the distance of an entangled state to the set of separable states. Related questions were addressed in the papers of Zyczkowski et al [1][2][3], Pittenger et al [4] and Witte et al [5] (see also Ozawa [6]), although here we attack the problem from a different perspective.…”

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confidence: 99%

On the geometry of entangled states

Verstraete

Dehaene

Moor

2002

Journal of Modern Optics

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The basic question that is addressed in this paper is finding the closest separable state for a given entangled state, measured with the Hilbert Schmidt distance. While this problem is in general very hard, we show that the following strongly related problem can be solved: find the Hilbert Schmidt distance of an entangled state to the set of all partially transposed states. We prove that this latter distance can be expressed as a function of the negative eigenvalues of the partial transpose of the entangled state, and show how it is related to the distance of a state to the set of positive partially transposed states (PPT-states). We illustrate this by calculating the closest biseparable state to the W-state, and give a simple and very general proof for the fact that the set of W-type states is not of measure zero. Next we show that all surfaces with states whose partial transposes have constant minimal negative eigenvalue are similar to the boundary of PPT states. We illustrate this with some examples on bipartite qubit states, where contours of constant negativity are plotted on two-dimensional intersections of the complete state space. 03.65.Bz

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“…After submission we became aware of a recent preprint by A. Pittenger and M. Rubin [22] where the ideas of this paper have been further developed. In particular it is shown there that if N is prime, a projector onto a maximally entangled state with full Schmidt rank can be measured with N + 1 local measurements, so our bound can be reached for this case.…”

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confidence: 99%

Detection of entanglement with few local measurements

et al. 2002

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We introduce a general method for the experimental detection of entanglement by performing only few local measurements, assuming some prior knowledge of the density matrix. The idea is based on the minimal decomposition of witness operators into a pseudo-mixture of local operators. We discuss an experimentally relevant case of two qubits, and show an example how bound entanglement can be detected with few local measurements. 03.67.Dd, 03.67.Hk, A central aim in the physics of quantum information is to create and detect entanglement -the resource that allows to realize various quantum protocols. Recently, much progress has been achieved experimentally in creating entangled states [1]. In every real experiment noise and imperfections are present so that the generated states, although intended to be entangled, may in fact be separable. Therefore, it is important to find efficient experimental methods to test whether a given imperfect state ρ is indeed entangled.Obviously, the ultimate goal of entanglement detection is to characterize entanglement quantitatively, and identify regions in the parameter space which allow to maximize entanglement for a particular quantum information processing task. The first step towards this ambitious goal is to detect whether a given state is entangled or not.The question of direct detection of quantum entanglement has been recently addressed in Refs. [2][3][4]. In [3,4] the authors study the case of mixed states and find efficient ways to estimate the entanglement of an unknown state. Their method is based on structural approximations of some linear maps followed by a spectrum estimation. Although experimentally viable the method is not very easy to implement and it requires further modifications in order to be performed by local measurements [5]. Here, we approach the same problem from a different perspective. We use special observables, the so-called witness operators [6,7] and their optimal decomposition into a sum of local projectors. Note that in this way we answer an open question posed recently in [8], where non-local measurements of entanglement witnesses were studied.The construction of a witness for a given arbitrary state is, in general, a formidable task. It can, however, be accomplished in typical experimental situations where one has some a priori information about the density matrix. This is always the case when the experiment is aimed at producing a certain state, rather than checking properties of an a priori unknown state. We discuss two experimentally relevant situations in this paper, namely the generation of a definite pure entangled state of two parties, and the generation of a specific bound entangled edge state. In both cases our method can be applied in arbitrary dimensions.Having constructed a witness, its measurement can be performed locally, since every observable can be decomposed in terms of a product basis in the operator space. Here we propose two ways of optimizing such local measurements. The first one consists in looking for the optimal number of l...

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Separability and Fourier representations of density matrices

Cited by 62 publications

References 22 publications

Entanglement Breaking Channels

Entanglement Breaking Channels

On the geometry of entangled states

Detection of entanglement with few local measurements

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