Pierre Botteron, Anne Broadbent, Reda Chhaibi, Ion Nechita, Clément Pellegrini
{"title":"非局部盒代数与通信复杂性的崩溃","authors":"Pierre Botteron, Anne Broadbent, Reda Chhaibi, Ion Nechita, Clément Pellegrini","doi":"10.22331/q-2024-07-10-1402","DOIUrl":null,"url":null,"abstract":"Communication complexity quantifies how difficult it is for two distant computers to evaluate a function $f(X,Y)$, where the strings $X$ and $Y$ are distributed to the first and second computer respectively, under the constraint of exchanging as few bits as possible. Surprisingly, some nonlocal boxes, which are resources shared by the two computers, are so powerful that they allow to $collapse$ communication complexity, in the sense that any Boolean function f can be correctly estimated with the exchange of only one bit of communication. The Popescu-Rohrlich (PR) box is an example of such a collapsing resource, but a comprehensive description of the set of collapsing nonlocal boxes remains elusive.<br/>\n<br/> In this work, we carry out an algebraic study of the structure of wirings connecting nonlocal boxes, thus defining the notion of the \"product of boxes\" $P\\boxtimes Q$, and we show related associativity and commutativity results. This gives rise to the notion of the \"orbit of a box\", unveiling surprising geometrical properties about the alignment and parallelism of distilled boxes. The power of this new framework is that it allows us to prove previously-reported numerical observations concerning the best way to wire consecutive boxes, and to numerically and analytically recover recently-identified noisy $\\texttt{PR}$ boxes that collapse communication complexity for different types of noise models.","PeriodicalId":20807,"journal":{"name":"Quantum","volume":null,"pages":null},"PeriodicalIF":5.1000,"publicationDate":"2024-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Algebra of Nonlocal Boxes and the Collapse of Communication Complexity\",\"authors\":\"Pierre Botteron, Anne Broadbent, Reda Chhaibi, Ion Nechita, Clément Pellegrini\",\"doi\":\"10.22331/q-2024-07-10-1402\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Communication complexity quantifies how difficult it is for two distant computers to evaluate a function $f(X,Y)$, where the strings $X$ and $Y$ are distributed to the first and second computer respectively, under the constraint of exchanging as few bits as possible. Surprisingly, some nonlocal boxes, which are resources shared by the two computers, are so powerful that they allow to $collapse$ communication complexity, in the sense that any Boolean function f can be correctly estimated with the exchange of only one bit of communication. The Popescu-Rohrlich (PR) box is an example of such a collapsing resource, but a comprehensive description of the set of collapsing nonlocal boxes remains elusive.<br/>\\n<br/> In this work, we carry out an algebraic study of the structure of wirings connecting nonlocal boxes, thus defining the notion of the \\\"product of boxes\\\" $P\\\\boxtimes Q$, and we show related associativity and commutativity results. This gives rise to the notion of the \\\"orbit of a box\\\", unveiling surprising geometrical properties about the alignment and parallelism of distilled boxes. The power of this new framework is that it allows us to prove previously-reported numerical observations concerning the best way to wire consecutive boxes, and to numerically and analytically recover recently-identified noisy $\\\\texttt{PR}$ boxes that collapse communication complexity for different types of noise models.\",\"PeriodicalId\":20807,\"journal\":{\"name\":\"Quantum\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":5.1000,\"publicationDate\":\"2024-07-10\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Quantum\",\"FirstCategoryId\":\"101\",\"ListUrlMain\":\"https://doi.org/10.22331/q-2024-07-10-1402\",\"RegionNum\":2,\"RegionCategory\":\"物理与天体物理\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"PHYSICS, MULTIDISCIPLINARY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Quantum","FirstCategoryId":"101","ListUrlMain":"https://doi.org/10.22331/q-2024-07-10-1402","RegionNum":2,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"PHYSICS, MULTIDISCIPLINARY","Score":null,"Total":0}
Algebra of Nonlocal Boxes and the Collapse of Communication Complexity
Communication complexity quantifies how difficult it is for two distant computers to evaluate a function $f(X,Y)$, where the strings $X$ and $Y$ are distributed to the first and second computer respectively, under the constraint of exchanging as few bits as possible. Surprisingly, some nonlocal boxes, which are resources shared by the two computers, are so powerful that they allow to $collapse$ communication complexity, in the sense that any Boolean function f can be correctly estimated with the exchange of only one bit of communication. The Popescu-Rohrlich (PR) box is an example of such a collapsing resource, but a comprehensive description of the set of collapsing nonlocal boxes remains elusive.
In this work, we carry out an algebraic study of the structure of wirings connecting nonlocal boxes, thus defining the notion of the "product of boxes" $P\boxtimes Q$, and we show related associativity and commutativity results. This gives rise to the notion of the "orbit of a box", unveiling surprising geometrical properties about the alignment and parallelism of distilled boxes. The power of this new framework is that it allows us to prove previously-reported numerical observations concerning the best way to wire consecutive boxes, and to numerically and analytically recover recently-identified noisy $\texttt{PR}$ boxes that collapse communication complexity for different types of noise models.
QuantumPhysics and Astronomy-Physics and Astronomy (miscellaneous)
CiteScore
9.20
自引率
10.90%
发文量
241
审稿时长
16 weeks
期刊介绍:
Quantum is an open-access peer-reviewed journal for quantum science and related fields. Quantum is non-profit and community-run: an effort by researchers and for researchers to make science more open and publishing more transparent and efficient.