dorsal/arxiv
View SchemaMarkovian Entanglement Networks
| Authors | Pierfrancesco La Mura, Lukasz Swiatczak |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0702072 |
| URL | https://arxiv.org/abs/quant-ph/0702072 |
Abstract
Graphical models of probabilistic dependencies have been extensively investigated in the context of classical uncertainty. However, in some domains (most notably, in computational physics and quantum computing) the nature of the relevant uncertainty is non-classical, and the laws of classical probability theory are superseded by those of quantum mechanics. In this paper we introduce Markovian Entanglement Networks (MEN), a novel class of graphical representations of quantum-mechanical dependencies in the context of such non-classical systems. MEN are the quantum-mechanical analogue of Markovian Networks, a family of undirected graphical representations which, in the classical domain, exploit a notion of conditional independence among subsystems. After defining a notion of conditional independence appropriate to our domain (conditional separability), we prove that the conditional separabilities induced by a quantum-mechanical wave function are effectively reflected in the graphical structure of MEN. Specifically, we show that for any wave function there exists a MEN which is a perfect map of its conditional separabilities. Next, we show how the graphical structure of MEN can be used to effectively classify the pure states of three-qubit systems. We also demonstrate that, in large systems, exploiting conditional independencies may dramatically reduce the computational burden of various inference tasks. In principle, the graph-theoretic representation of conditional independencies afforded by MEN may not only facilitate the classical simulation of quantum systems, but also provide a guide to the efficient design and complexity analysis of quantum algorithms and circuits.
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"abstract": "Graphical models of probabilistic dependencies have been extensively\ninvestigated in the context of classical uncertainty. However, in some domains\n(most notably, in computational physics and quantum computing) the nature of\nthe relevant uncertainty is non-classical, and the laws of classical\nprobability theory are superseded by those of quantum mechanics. In this paper\nwe introduce Markovian Entanglement Networks (MEN), a novel class of graphical\nrepresentations of quantum-mechanical dependencies in the context of such\nnon-classical systems. MEN are the quantum-mechanical analogue of Markovian\nNetworks, a family of undirected graphical representations which, in the\nclassical domain, exploit a notion of conditional independence among\nsubsystems.\n After defining a notion of conditional independence appropriate to our domain\n(conditional separability), we prove that the conditional separabilities\ninduced by a quantum-mechanical wave function are effectively reflected in the\ngraphical structure of MEN. Specifically, we show that for any wave function\nthere exists a MEN which is a perfect map of its conditional separabilities.\nNext, we show how the graphical structure of MEN can be used to effectively\nclassify the pure states of three-qubit systems. We also demonstrate that, in\nlarge systems, exploiting conditional independencies may dramatically reduce\nthe computational burden of various inference tasks. In principle, the\ngraph-theoretic representation of conditional independencies afforded by MEN\nmay not only facilitate the classical simulation of quantum systems, but also\nprovide a guide to the efficient design and complexity analysis of quantum\nalgorithms and circuits.",
"arxiv_id": "quant-ph/0702072",
"authors": [
"Pierfrancesco La Mura",
"Lukasz Swiatczak"
],
"categories": [
"quant-ph",
"cs.AI"
],
"title": "Markovian Entanglement Networks",
"url": "https://arxiv.org/abs/quant-ph/0702072"
},
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