Lower bounds on the localisation length of balanced random quantum walks
We consider the dynamical properties of Quantum Walks defined on the d -dimensional cubic lattice, or the homogeneous tree of coordination number 2 d , with site-dependent random phases, further characterised by transition probabilities between neighbouring sites equal to 1/(2 d ). We show that the...
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| Veröffentlicht in: | Letters in mathematical physics 2019-09, Vol.109 (9), p.2133-2155 |
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| creator | Asch, Joachim Joye, Alain |
| description | We consider the dynamical properties of Quantum Walks defined on the
d
-dimensional cubic lattice, or the homogeneous tree of coordination number 2
d
, with site-dependent random phases, further characterised by transition probabilities between neighbouring sites equal to 1/(2
d
). We show that the localisation length for these Balanced Random Quantum Walks can be expressed as a combinatorial expression involving sums over weighted paths on the considered graph. This expression provides lower bounds on the localisation length by restriction to paths with weight 1, which allows us to prove the localisation length diverges on the tree as
d
2
. On the cubic lattice, the method yields the lower bound
1
/
ln
(
2
)
for all
d
and allows us to bound the localisation length from below by the correlation length of self-avoiding walks computed at 1/(2
d
). |
| doi_str_mv | 10.1007/s11005-019-01180-0 |
| format | Article |
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d
-dimensional cubic lattice, or the homogeneous tree of coordination number 2
d
, with site-dependent random phases, further characterised by transition probabilities between neighbouring sites equal to 1/(2
d
). We show that the localisation length for these Balanced Random Quantum Walks can be expressed as a combinatorial expression involving sums over weighted paths on the considered graph. This expression provides lower bounds on the localisation length by restriction to paths with weight 1, which allows us to prove the localisation length diverges on the tree as
d
2
. On the cubic lattice, the method yields the lower bound
1
/
ln
(
2
)
for all
d
and allows us to bound the localisation length from below by the correlation length of self-avoiding walks computed at 1/(2
d
).</description><identifier>ISSN: 0377-9017</identifier><identifier>EISSN: 1573-0530</identifier><identifier>DOI: 10.1007/s11005-019-01180-0</identifier><language>eng</language><publisher>Dordrecht: Springer Netherlands</publisher><subject>Combinatorial analysis ; Complex Systems ; Coordination numbers ; Cubic lattice ; Geometry ; Group Theory and Generalizations ; Localization ; Lower bounds ; Mathematical and Computational Physics ; Mathematical Physics ; Physics ; Physics and Astronomy ; Theoretical ; Transition probabilities</subject><ispartof>Letters in mathematical physics, 2019-09, Vol.109 (9), p.2133-2155</ispartof><rights>Springer Nature B.V. 2019</rights><rights>Copyright Springer Nature B.V. 2019</rights><rights>Distributed under a Creative Commons Attribution 4.0 International License</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c397t-1aed0d8f6b58f368eb416592453b3275d8c04465e08bdaaf26d7ec4eb0ebf6073</citedby><cites>FETCH-LOGICAL-c397t-1aed0d8f6b58f368eb416592453b3275d8c04465e08bdaaf26d7ec4eb0ebf6073</cites><orcidid>0000-0002-4830-6979 ; 0000-0002-8765-3025</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s11005-019-01180-0$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s11005-019-01180-0$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>230,314,780,784,885,27924,27925,41488,42557,51319</link.rule.ids><backlink>$$Uhttps://hal.science/hal-01953415$$DView record in HAL$$Hfree_for_read</backlink></links><search><creatorcontrib>Asch, Joachim</creatorcontrib><creatorcontrib>Joye, Alain</creatorcontrib><title>Lower bounds on the localisation length of balanced random quantum walks</title><title>Letters in mathematical physics</title><addtitle>Lett Math Phys</addtitle><description>We consider the dynamical properties of Quantum Walks defined on the
d
-dimensional cubic lattice, or the homogeneous tree of coordination number 2
d
, with site-dependent random phases, further characterised by transition probabilities between neighbouring sites equal to 1/(2
d
). We show that the localisation length for these Balanced Random Quantum Walks can be expressed as a combinatorial expression involving sums over weighted paths on the considered graph. This expression provides lower bounds on the localisation length by restriction to paths with weight 1, which allows us to prove the localisation length diverges on the tree as
d
2
. On the cubic lattice, the method yields the lower bound
1
/
ln
(
2
)
for all
d
and allows us to bound the localisation length from below by the correlation length of self-avoiding walks computed at 1/(2
d
).</description><subject>Combinatorial analysis</subject><subject>Complex Systems</subject><subject>Coordination numbers</subject><subject>Cubic lattice</subject><subject>Geometry</subject><subject>Group Theory and Generalizations</subject><subject>Localization</subject><subject>Lower bounds</subject><subject>Mathematical and Computational Physics</subject><subject>Mathematical Physics</subject><subject>Physics</subject><subject>Physics and Astronomy</subject><subject>Theoretical</subject><subject>Transition probabilities</subject><issn>0377-9017</issn><issn>1573-0530</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2019</creationdate><recordtype>article</recordtype><recordid>eNp9kMFLwzAYxYMoOKf_gKeAJw_VL03TtMcx1AkFL3oOSZOunV2yJa3D_97Mit48fDx4_N7j4yF0TeCOAPD7QKKwBEgZjxSQwAmaEcZpAozCKZoB5TwpgfBzdBHCBmIoZTBDq8odjMfKjVYH7CweWoN7V8u-C3LootEbux5a7BqsZC9tbTT20mq3xftR2mHc4oPs38MlOmtkH8zVj87R2-PD63KVVC9Pz8tFldS05ENCpNGgiyZXrGhoXhiVkZyVacaooilnuqghy3JmoFBayibNNTd1ZhQY1eTA6RzdTr2t7MXOd1vpP4WTnVgtKnH04gaMZoR9kMjeTOzOu_1owiA2bvQ2vifSlBdQEsLzSKUTVXsXgjfNby0BcVxXTOsem8X3ugJiiE6hEGG7Nv6v-p_UF4RKe8s</recordid><startdate>20190901</startdate><enddate>20190901</enddate><creator>Asch, Joachim</creator><creator>Joye, Alain</creator><general>Springer Netherlands</general><general>Springer Nature B.V</general><general>Springer Verlag</general><scope>AAYXX</scope><scope>CITATION</scope><scope>1XC</scope><scope>VOOES</scope><orcidid>https://orcid.org/0000-0002-4830-6979</orcidid><orcidid>https://orcid.org/0000-0002-8765-3025</orcidid></search><sort><creationdate>20190901</creationdate><title>Lower bounds on the localisation length of balanced random quantum walks</title><author>Asch, Joachim ; Joye, Alain</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c397t-1aed0d8f6b58f368eb416592453b3275d8c04465e08bdaaf26d7ec4eb0ebf6073</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2019</creationdate><topic>Combinatorial analysis</topic><topic>Complex Systems</topic><topic>Coordination numbers</topic><topic>Cubic lattice</topic><topic>Geometry</topic><topic>Group Theory and Generalizations</topic><topic>Localization</topic><topic>Lower bounds</topic><topic>Mathematical and Computational Physics</topic><topic>Mathematical Physics</topic><topic>Physics</topic><topic>Physics and Astronomy</topic><topic>Theoretical</topic><topic>Transition probabilities</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Asch, Joachim</creatorcontrib><creatorcontrib>Joye, Alain</creatorcontrib><collection>CrossRef</collection><collection>Hyper Article en Ligne (HAL)</collection><collection>Hyper Article en Ligne (HAL) (Open Access)</collection><jtitle>Letters in mathematical physics</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Asch, Joachim</au><au>Joye, Alain</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Lower bounds on the localisation length of balanced random quantum walks</atitle><jtitle>Letters in mathematical physics</jtitle><stitle>Lett Math Phys</stitle><date>2019-09-01</date><risdate>2019</risdate><volume>109</volume><issue>9</issue><spage>2133</spage><epage>2155</epage><pages>2133-2155</pages><issn>0377-9017</issn><eissn>1573-0530</eissn><abstract>We consider the dynamical properties of Quantum Walks defined on the
d
-dimensional cubic lattice, or the homogeneous tree of coordination number 2
d
, with site-dependent random phases, further characterised by transition probabilities between neighbouring sites equal to 1/(2
d
). We show that the localisation length for these Balanced Random Quantum Walks can be expressed as a combinatorial expression involving sums over weighted paths on the considered graph. This expression provides lower bounds on the localisation length by restriction to paths with weight 1, which allows us to prove the localisation length diverges on the tree as
d
2
. On the cubic lattice, the method yields the lower bound
1
/
ln
(
2
)
for all
d
and allows us to bound the localisation length from below by the correlation length of self-avoiding walks computed at 1/(2
d
).</abstract><cop>Dordrecht</cop><pub>Springer Netherlands</pub><doi>10.1007/s11005-019-01180-0</doi><tpages>23</tpages><orcidid>https://orcid.org/0000-0002-4830-6979</orcidid><orcidid>https://orcid.org/0000-0002-8765-3025</orcidid><oa>free_for_read</oa></addata></record> |
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| ispartof | Letters in mathematical physics, 2019-09, Vol.109 (9), p.2133-2155 |
| issn | 0377-9017 1573-0530 |
| language | eng |
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| source | SpringerLink Journals - AutoHoldings |
| subjects | Combinatorial analysis Complex Systems Coordination numbers Cubic lattice Geometry Group Theory and Generalizations Localization Lower bounds Mathematical and Computational Physics Mathematical Physics Physics Physics and Astronomy Theoretical Transition probabilities |
| title | Lower bounds on the localisation length of balanced random quantum walks |
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