dorsal/arxiv
View SchemaInfinitesimal local operations and differential conditions for entanglement monotones
| Authors | Ognyan Oreshkov, Todd A. Brun |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0506181 |
| URL | https://arxiv.org/abs/quant-ph/0506181 |
| DOI | 10.1103/PhysRevA.73.042314 |
| Journal | Phys. Rev. A 73, 042314 (2006) |
Abstract
Much of the theory of entanglement concerns the transformations that are possible to a state under local operations with classical communication (LOCC); however, this set of operations is complicated and difficult to describe mathematically. An idea which has proven very useful is that of the {\it entanglement monotone}: a function of the state which is invariant under local unitary transformations and always decreases (or increases) on average after any local operation. In this paper we look on LOCC as the set of operations generated by {\it infinitesimal local operations}, operations which can be performed locally and which leave the state little changed. We show that a necessary and sufficient condition for a function of the state to be an entanglement monotone under local operations that do not involve information loss is that the function be a monotone under infinitesimal local operations. We then derive necessary and sufficient differential conditions for a function of the state to be an entanglement monotone. We first derive two conditions for local operations without information loss, and then show that they can be extended to more general operations by adding the requirement of {\it convexity}. We then demonstrate that a number of known entanglement monotones satisfy these differential criteria. Finally, as an application, we use the differential conditions to construct a new polynomial entanglement monotone for three-qubit pure states. It is our hope that this approach will avoid some of the difficulties in the theory of multipartite and mixed-state entanglement.
{
"annotation_id": "205d6874-8249-446c-89eb-f2ae88e7c06c",
"date_created": "2026-03-02T18:02:16.992000Z",
"date_modified": "2026-03-02T18:02:16.992000Z",
"file_hash": "3c5abd9ada016edeec91d48497c26233864e75edebc5c38645550219356e964c",
"private": false,
"record": {
"abstract": "Much of the theory of entanglement concerns the transformations that are\npossible to a state under local operations with classical communication (LOCC);\nhowever, this set of operations is complicated and difficult to describe\nmathematically. An idea which has proven very useful is that of the {\\it\nentanglement monotone}: a function of the state which is invariant under local\nunitary transformations and always decreases (or increases) on average after\nany local operation. In this paper we look on LOCC as the set of operations\ngenerated by {\\it infinitesimal local operations}, operations which can be\nperformed locally and which leave the state little changed. We show that a\nnecessary and sufficient condition for a function of the state to be an\nentanglement monotone under local operations that do not involve information\nloss is that the function be a monotone under infinitesimal local operations.\nWe then derive necessary and sufficient differential conditions for a function\nof the state to be an entanglement monotone. We first derive two conditions for\nlocal operations without information loss, and then show that they can be\nextended to more general operations by adding the requirement of {\\it\nconvexity}. We then demonstrate that a number of known entanglement monotones\nsatisfy these differential criteria. Finally, as an application, we use the\ndifferential conditions to construct a new polynomial entanglement monotone for\nthree-qubit pure states. It is our hope that this approach will avoid some of\nthe difficulties in the theory of multipartite and mixed-state entanglement.",
"arxiv_id": "quant-ph/0506181",
"authors": [
"Ognyan Oreshkov",
"Todd A. Brun"
],
"categories": [
"quant-ph"
],
"doi": "10.1103/PhysRevA.73.042314",
"journal_ref": "Phys. Rev. A 73, 042314 (2006)",
"title": "Infinitesimal local operations and differential conditions for entanglement monotones",
"url": "https://arxiv.org/abs/quant-ph/0506181"
},
"schema_id": "dorsal/arxiv",
"source": {
"execution_id": "5f891d79-aa65-455e-bfab-f6c57938e193",
"id": "arXiv Dataset IDs",
"type": "Model",
"variant": "snapshot-2026-03-01",
"version": "0.1.0"
},
"user_id": 1000002
}