Geometric local invariants and pure three-qubit states

Journal Article · Wed Jun 15 04:00:00 EDT 2011 · Physical Review. A

DOI:https://doi.org/10.1103/PHYSREVA.83.062308· OSTI ID:21550058

Williamson, Mark S ^[1]; Ericsson, Marie; Johansson, Markus ^[2]; Sjoeqvist, Erik ^[1]; Sudbery, Anthony ^[3]; Vedral, Vlatko ^[1]; Wootters, William K ^[4]

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1. Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543 (Singapore)
2. Department of Quantum Chemistry, Uppsala University, Box 518, SE-751 20 Uppsala (Sweden)
3. Department of Mathematics, University of York, Heslington, York YO10 5DD (United Kingdom)
4. Department of Physics, Williams College, Williamstown, Massachusetts 01267 (United States)

We explore a geometric approach to generating local SU(2) and SL(2,C) invariants for a collection of qubits inspired by lattice gauge theory. Each local invariant or ''gauge'' invariant is associated with a distinct closed path (or plaquette) joining some or all of the qubits. In lattice gauge theory, the lattice points are the discrete space-time points, the transformations between the points of the lattice are defined by parallel transporters, and the gauge invariant observable associated with a particular closed path is given by the Wilson loop. In our approach the points of the lattice are qubits, the link transformations between the qubits are defined by the correlations between them, and the gauge invariant observable, the local invariants associated with a particular closed path, are also given by a Wilson looplike construction. The link transformations share many of the properties of parallel transporters, although they are not undone when one retraces one's steps through the lattice. This feature is used to generate many of the invariants. We consider a pure three-qubit state as a test case and find we can generate a complete set of algebraically independent local invariants in this way; however, the framework given here is applicable to generating local unitary invariants for mixed states composed of any number of d-level quantum systems. We give an operational interpretation of these invariants in terms of observables.

• OSTI ID: 21550058
• Journal Information: Physical Review. A, Journal Name: Physical Review. A Journal Issue: 6 Vol. 83; ISSN 1050-2947; ISSN PLRAAN
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
CORRELATIONS
GAUGE INVARIANCE
GEOMETRY
INFORMATION
INVARIANCE PRINCIPLES
LIE GROUPS
MATHEMATICS
MIXED STATES
QUANTUM INFORMATION
QUANTUM STATES
QUBITS
SL GROUPS
SPACE-TIME
SU GROUPS
SU-2 GROUPS
SYMMETRY GROUPS
TRANSFORMATIONS
WILSON LOOP