dorsal/arxiv
View SchemaCommutativity and the third Reidemeister movement
| Authors | Philippe Leroux |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0207004 |
| URL | https://arxiv.org/abs/quant-ph/0207004 |
Abstract
In quantum information theory, for $a,b$ two positive operators living in $B(\mathcal{H})$, where $\mathcal{H}$ is a separable Hilbert space, the quantum fidelity is denoted by $a*b =(b^{1/2}ab^{1/2})^{1/2}$. One of the aim of this let ter is to interpret the quantum fidelity as an algebraic law. We remark that if $a,b,c$ are three positive operators whi ch commute pairwise, the law * becomes self-distributive, i.e. the third Reidemeister movement in knot theory is verif ied. We study the converse. Let three positive operators be given, does the fact that the third Reidemeister movement between them is possible implie that they commute pairwise ? Though in general we only conjecture it for the moment, we prove it in some par ticular but important cases. Should this movement be not possible, we interpret it as an obstruction to comm utativity. We give also new examples of quandle algebras and left distributive systems and study the generalisation of Ito maps.
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"abstract": "In quantum information theory, for $a,b$ two positive operators living in\n$B(\\mathcal{H})$, where $\\mathcal{H}$ is a separable Hilbert space, the quantum\nfidelity is denoted by $a*b =(b^{1/2}ab^{1/2})^{1/2}$. One of the aim of this\nlet ter is to interpret the quantum fidelity as an algebraic law. We remark\nthat if $a,b,c$ are three positive operators whi ch commute pairwise, the law *\nbecomes self-distributive, i.e. the third Reidemeister movement in knot theory\nis verif ied. We study the converse. Let three positive operators be given,\ndoes the fact that the third Reidemeister movement between them is possible\nimplie that they commute pairwise ? Though in general we only conjecture it for\nthe moment, we prove it in some par ticular but important cases. Should this\nmovement be not possible, we interpret it as an obstruction to comm utativity.\nWe give also new examples of quandle algebras and left distributive systems and\nstudy the generalisation of Ito maps.",
"arxiv_id": "quant-ph/0207004",
"authors": [
"Philippe Leroux"
],
"categories": [
"quant-ph"
],
"title": "Commutativity and the third Reidemeister movement",
"url": "https://arxiv.org/abs/quant-ph/0207004"
},
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