Strong complementarity and non-locality in categorical quantum mechanics

Bob Coecke, Ross Duncan, Aleks Kissinger, Quanlong Wang

Research output: Chapter in Book/Report/Conference proceedingChapter

26 Citations (Scopus)

Abstract

Categorical quantum mechanics studies quantum theory in the framework of dagger-compact closed categories. Using this framework, we establish a tight relationship between two key quantum theoretical notions: non-locality and complementarity. In particular, we establish a direct connection between Mermin-type non-locality scenarios, which we generalise to an arbitrary number of parties, using systems of arbitrary dimension, and performing arbitrary measurements, and a new stronger notion of complementarity which we introduce here. Our derivation of the fact that strong complementarity is a necessary condition for a Mermin scenario provides a crisp operational interpretation for strong complementarity. We also provide a complete classification of strongly complementary observables for quantum theory, something which has not yet been achieved for ordinary complementarity. Since our main results are expressed in the (diagrammatic) language of dagger-compact categories, they can be applied outside of quantum theory, in any setting which supports the purely algebraic notion of strongly complementary observables. We have therefore introduced a method for discussing non-locality in a wide variety of models in addition to quantum theory. The diagrammatic calculus substantially simplifies (and sometimes even trivialises) many of the derivations, and provides new insights. In particular, the diagrammatic computation of correlations clearly shows how local measurements interact to yield a global overall effect. In other words, we depict non-locality.
LanguageEnglish
Title of host publicationLogic in Computer Science (LICS), 2012 27th Annual IEEE Symposium on
Place of PublicationPiscataway, NJ
PublisherIEEE
Pages245-254
Number of pages10
ISBN (Print)9781467322638
DOIs
Publication statusPublished - 2012
Event27th Annual IEEE Symposium on Logic in Computer Science. (LiCS2012) - , Croatia
Duration: 25 Jun 2012 → …

Conference

Conference27th Annual IEEE Symposium on Logic in Computer Science. (LiCS2012)
CountryCroatia
Period25/06/12 → …

Fingerprint

Quantum theory
quantum theory
quantum mechanics
derivation
calculus

Cite this

Coecke, B., Duncan, R., Kissinger, A., & Wang, Q. (2012). Strong complementarity and non-locality in categorical quantum mechanics. In Logic in Computer Science (LICS), 2012 27th Annual IEEE Symposium on (pp. 245-254). Piscataway, NJ: IEEE. https://doi.org/10.1109/LICS.2012.35
Coecke, Bob ; Duncan, Ross ; Kissinger, Aleks ; Wang, Quanlong. / Strong complementarity and non-locality in categorical quantum mechanics. Logic in Computer Science (LICS), 2012 27th Annual IEEE Symposium on . Piscataway, NJ : IEEE, 2012. pp. 245-254
@inbook{fe725b8d1a554dd7be6f6cf0221fe54a,
title = "Strong complementarity and non-locality in categorical quantum mechanics",
abstract = "Categorical quantum mechanics studies quantum theory in the framework of dagger-compact closed categories. Using this framework, we establish a tight relationship between two key quantum theoretical notions: non-locality and complementarity. In particular, we establish a direct connection between Mermin-type non-locality scenarios, which we generalise to an arbitrary number of parties, using systems of arbitrary dimension, and performing arbitrary measurements, and a new stronger notion of complementarity which we introduce here. Our derivation of the fact that strong complementarity is a necessary condition for a Mermin scenario provides a crisp operational interpretation for strong complementarity. We also provide a complete classification of strongly complementary observables for quantum theory, something which has not yet been achieved for ordinary complementarity. Since our main results are expressed in the (diagrammatic) language of dagger-compact categories, they can be applied outside of quantum theory, in any setting which supports the purely algebraic notion of strongly complementary observables. We have therefore introduced a method for discussing non-locality in a wide variety of models in addition to quantum theory. The diagrammatic calculus substantially simplifies (and sometimes even trivialises) many of the derivations, and provides new insights. In particular, the diagrammatic computation of correlations clearly shows how local measurements interact to yield a global overall effect. In other words, we depict non-locality.",
author = "Bob Coecke and Ross Duncan and Aleks Kissinger and Quanlong Wang",
note = "Proceedings of the 27th Annual IEEE Symposium on Logic in Computer Science. (LiCS2012) pp 245--254 IEEE Computer Society Press",
year = "2012",
doi = "10.1109/LICS.2012.35",
language = "English",
isbn = "9781467322638",
pages = "245--254",
booktitle = "Logic in Computer Science (LICS), 2012 27th Annual IEEE Symposium on",
publisher = "IEEE",

}

Coecke, B, Duncan, R, Kissinger, A & Wang, Q 2012, Strong complementarity and non-locality in categorical quantum mechanics. in Logic in Computer Science (LICS), 2012 27th Annual IEEE Symposium on . IEEE, Piscataway, NJ, pp. 245-254, 27th Annual IEEE Symposium on Logic in Computer Science. (LiCS2012), Croatia, 25/06/12. https://doi.org/10.1109/LICS.2012.35

Strong complementarity and non-locality in categorical quantum mechanics. / Coecke, Bob; Duncan, Ross; Kissinger, Aleks; Wang, Quanlong.

Logic in Computer Science (LICS), 2012 27th Annual IEEE Symposium on . Piscataway, NJ : IEEE, 2012. p. 245-254.

Research output: Chapter in Book/Report/Conference proceedingChapter

TY - CHAP

T1 - Strong complementarity and non-locality in categorical quantum mechanics

AU - Coecke, Bob

AU - Duncan, Ross

AU - Kissinger, Aleks

AU - Wang, Quanlong

N1 - Proceedings of the 27th Annual IEEE Symposium on Logic in Computer Science. (LiCS2012) pp 245--254 IEEE Computer Society Press

PY - 2012

Y1 - 2012

N2 - Categorical quantum mechanics studies quantum theory in the framework of dagger-compact closed categories. Using this framework, we establish a tight relationship between two key quantum theoretical notions: non-locality and complementarity. In particular, we establish a direct connection between Mermin-type non-locality scenarios, which we generalise to an arbitrary number of parties, using systems of arbitrary dimension, and performing arbitrary measurements, and a new stronger notion of complementarity which we introduce here. Our derivation of the fact that strong complementarity is a necessary condition for a Mermin scenario provides a crisp operational interpretation for strong complementarity. We also provide a complete classification of strongly complementary observables for quantum theory, something which has not yet been achieved for ordinary complementarity. Since our main results are expressed in the (diagrammatic) language of dagger-compact categories, they can be applied outside of quantum theory, in any setting which supports the purely algebraic notion of strongly complementary observables. We have therefore introduced a method for discussing non-locality in a wide variety of models in addition to quantum theory. The diagrammatic calculus substantially simplifies (and sometimes even trivialises) many of the derivations, and provides new insights. In particular, the diagrammatic computation of correlations clearly shows how local measurements interact to yield a global overall effect. In other words, we depict non-locality.

AB - Categorical quantum mechanics studies quantum theory in the framework of dagger-compact closed categories. Using this framework, we establish a tight relationship between two key quantum theoretical notions: non-locality and complementarity. In particular, we establish a direct connection between Mermin-type non-locality scenarios, which we generalise to an arbitrary number of parties, using systems of arbitrary dimension, and performing arbitrary measurements, and a new stronger notion of complementarity which we introduce here. Our derivation of the fact that strong complementarity is a necessary condition for a Mermin scenario provides a crisp operational interpretation for strong complementarity. We also provide a complete classification of strongly complementary observables for quantum theory, something which has not yet been achieved for ordinary complementarity. Since our main results are expressed in the (diagrammatic) language of dagger-compact categories, they can be applied outside of quantum theory, in any setting which supports the purely algebraic notion of strongly complementary observables. We have therefore introduced a method for discussing non-locality in a wide variety of models in addition to quantum theory. The diagrammatic calculus substantially simplifies (and sometimes even trivialises) many of the derivations, and provides new insights. In particular, the diagrammatic computation of correlations clearly shows how local measurements interact to yield a global overall effect. In other words, we depict non-locality.

UR - http://ieeexplore.ieee.org/xpl/mostRecentIssue.jsp?punumber=6275587

U2 - 10.1109/LICS.2012.35

DO - 10.1109/LICS.2012.35

M3 - Chapter

SN - 9781467322638

SP - 245

EP - 254

BT - Logic in Computer Science (LICS), 2012 27th Annual IEEE Symposium on

PB - IEEE

CY - Piscataway, NJ

ER -

Coecke B, Duncan R, Kissinger A, Wang Q. Strong complementarity and non-locality in categorical quantum mechanics. In Logic in Computer Science (LICS), 2012 27th Annual IEEE Symposium on . Piscataway, NJ: IEEE. 2012. p. 245-254 https://doi.org/10.1109/LICS.2012.35