Paper Info
Title
The compositional structure of multipartite quantum entanglement
Abstract
Multipartite quantum states constitute a (if not the) key resource for quantum computations and protocols. However obtaining a generic, structural understanding of entanglement in N-qubit systems is a long-standing open problem in quantum computer science. Here we show that multipartite quantum entanglement admits a compositional structure, and hence is subject to modern computer science methods. Recall that two N-qubit states are SLOCC-equivalent if they can be inter-converted by stochastic local (quantum) operations and classical communication. There are only two SLOCC-equivalence classes of genuinely entangled 3-qubit states, the GHZ-class and the W-class, and we show that these exactly correspond with two kinds of internal commutative Frobenius algebras on C2 in the symmetric monoidal category of Hilbert spaces and linear maps, namely 'special' ones and 'anti-special' ones. Within the graphical language of symmetric monoidal categories, the distinction between 'special' and 'anti-special' is purely topological, in terms of 'connected' vs. 'disconnected'. These GHZ and W Frobenius algebras form the primitives of a graphical calculus which is expressive enough to generate and reason about representatives of arbitrary N-qubit states. This calculus refines the graphical calculus of complementary observables in [5, ICALP'08], which has already shown itself to have many applications and admit automation. Our result also induces a generalised graph state paradigm for measurement-based quantum computing.
Year
DOI
Venue
2010
10.1007/978-3-642-14162-1_25
ICALP (2)
Keywords
Field
DocType
measurement-based quantum computing,symmetric monoidal category,n-qubit state,graphical calculus,quantum computer science,multipartite quantum entanglement,n-qubit system,multipartite quantum state,arbitrary n-qubit state,quantum computation,compositional structure,quantum entanglement,category theory,frobenius algebra,quantum computer,graphical model,quantum physics,quantum algebra
Quantum no-deleting theorem,Discrete mathematics,Combinatorics,Categorical quantum mechanics,Algebra,Quantum process,Computer science,Quantum topology,Quantum algorithm,No-communication theorem,Quantum operation,Quantum capacity
Conference
Volume
ISSN
ISBN
6199
0302-9743
3-642-14161-7
Citations 
PageRank 
References 
22
2.27
12
Authors
2
Name
Order
Citations
PageRank
Bob Coecke1912104.22
Aleks Kissinger217122.32