dorsal/arxiv
View SchemaA classification of incomparable states
| Authors | Spmshubhro Bandyopadhyay, Vwani P. Roychowdhury, Ujjwal Sen |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0103131 |
| URL | https://arxiv.org/abs/quant-ph/0103131 |
| DOI | 10.1103/PhysRevA.65.052315 |
| Journal | Phys. Rev. A 65, 052315 (2002) |
Abstract
Let (\{| \psi> ,| \phi>}) be an incomparable pair of states ((| \psi \nleftrightarrow | \phi>)), \emph, i.e., (| \psi>) and (| \phi>) 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, \emph, i.e., there can exist a \emph{k}, such that (| \psi> ^{\otimes k+1}\to | \phi> ^{\otimes k+1}), i.e., (k+1) copies of (| \psi>) can be transformed to (k+1) copies of (| \phi>) with probability one by LOCC but (| \psi> ^{\otimes n}\nleftrightarrow | \phi> ^{\otimes n} \forall n\leq k). We call such states \emph{k}-copy LOCC incomparable. We provide a necessary condition for a given pair of states to be \emph{k}-copy LOCC incomparable for some \emph{k}. We also show that there exist states that are neither \emph{k}-copy LOCC incomparable for any \emph{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, (p_{max}(| \psi> ^{\otimes m}\to | \phi> ^{\otimes m})) can decrease exponentially with the number of source states (m), even if the source state has \emph{more} entropy of entanglement.
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"abstract": "Let (\\{| \\psi\u003e ,| \\phi\u003e}) be an incomparable pair of states ((| \\psi\n\\nleftrightarrow | \\phi\u003e)), \\emph, i.e., (| \\psi\u003e) and (| \\phi\u003e) cannot be\ntransformed to each other with probability one by local transformations and\nclassical communication (LOCC). We show that incomparable states can be\nmultiple-copy transformable, \\emph, i.e., there can exist a \\emph{k}, such that\n(| \\psi\u003e ^{\\otimes k+1}\\to | \\phi\u003e ^{\\otimes k+1}), i.e., (k+1) copies of (|\n\\psi\u003e) can be transformed to (k+1) copies of (| \\phi\u003e) with probability one by\nLOCC but (| \\psi\u003e ^{\\otimes n}\\nleftrightarrow | \\phi\u003e ^{\\otimes n} \\forall\nn\\leq k). We call such states \\emph{k}-copy LOCC incomparable. We provide a\nnecessary condition for a given pair of states to be \\emph{k}-copy LOCC\nincomparable for some \\emph{k}. We also show that there exist states that are\nneither \\emph{k}-copy LOCC incomparable for any \\emph{k} nor catalyzable even\nwith multiple copies. We call such states strongly incomparable. We give a\nsufficient condition for strong incomparability.\n We demonstrate that the optimal probability of a conclusive transformation\ninvolving many copies, (p_{max}(| \\psi\u003e ^{\\otimes m}\\to | \\phi\u003e ^{\\otimes m}))\ncan decrease exponentially with the number of source states (m), even if the\nsource state has \\emph{more} entropy of entanglement.",
"arxiv_id": "quant-ph/0103131",
"authors": [
"Spmshubhro Bandyopadhyay",
"Vwani P. Roychowdhury",
"Ujjwal Sen"
],
"categories": [
"quant-ph"
],
"doi": "10.1103/PhysRevA.65.052315",
"journal_ref": "Phys. Rev. A 65, 052315 (2002)",
"title": "A classification of incomparable states",
"url": "https://arxiv.org/abs/quant-ph/0103131"
},
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