dorsal/arxiv
View SchemaOn the geometry of entangled states
| Authors | Frank Verstraete, Jeroen Dehaene, Bart De Moor |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0107155 |
| URL | https://arxiv.org/abs/quant-ph/0107155 |
| DOI | 10.1080/09500340110115488 |
| Journal | J. Mod. Opt.: 49, 1277 (2002) |
Abstract
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.
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"abstract": "The basic question that is addressed in this paper is finding the closest\nseparable state for a given entangled state, measured with the Hilbert Schmidt\ndistance. While this problem is in general very hard, we show that the\nfollowing strongly related problem can be solved: find the Hilbert Schmidt\ndistance of an entangled state to the set of all partially transposed states.\nWe prove that this latter distance can be expressed as a function of the\nnegative eigenvalues of the partial transpose of the entangled state, and show\nhow it is related to the distance of a state to the set of positive partially\ntransposed states (PPT-states). We illustrate this by calculating the closest\nbiseparable state to the W-state, and give a simple and very general proof for\nthe fact that the set of W-type states is not of measure zero. Next we show\nthat all surfaces with states whose partial transposes have constant minimal\nnegative eigenvalue are similar to the boundary of PPT states. We illustrate\nthis with some examples on bipartite qubit states, where contours of constant\nnegativity are plotted on two-dimensional intersections of the complete state\nspace.",
"arxiv_id": "quant-ph/0107155",
"authors": [
"Frank Verstraete",
"Jeroen Dehaene",
"Bart De Moor"
],
"categories": [
"quant-ph"
],
"doi": "10.1080/09500340110115488",
"journal_ref": "J. Mod. Opt.: 49, 1277 (2002)",
"title": "On the geometry of entangled states",
"url": "https://arxiv.org/abs/quant-ph/0107155"
},
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