Stratified manifold of quantum states, actions of the complex special linear group

Journal Article · Tue Jan 15 00:00:00 EST 2019 · Annals of Physics

DOI:https://doi.org/10.1016/J.AOP.2018.11.015· OSTI ID:22852375

Chruściński, D., E-mail: [e-mail] ^[1]; Ciaglia, F.M., E-mail: [e-mail] ^[2]; Ibort, A., E-mail: [e-mail] ^[3]; ICMAT, Instituto de Ciencias Matemáticas ^[4]; Marmo, G., E-mail: [e-mail]; Ventriglia, F., E-mail: [e-mail] ^[5]; INFN - Sezione di Napoli, Complesso Universitario di Monte S. Angelo, via Cintia, I-80126 Napoli ^[6]

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1. Institute of Physics, Faculty of Physics, Astronomy and Informatics Nicolaus Copernicus University, Grudzia̧dzka 5/7, 87-100 Toruń (Poland)
2. Max Planck Institute for Mathematics in the Sciences, Inselstrasse 22, 04103 Leipzig (Germany)
3. Departamento de Matemáticas, Universidad Carlos III de Madrid, Avda. de la Universidad 30, 28911, Leganés, Madrid (Spain)
4. (CSIC - UAM - UC3M - UCM), Nicolás Cabrera, 13-15, Campus de Cantoblanco, UAM 28049, Madrid (Spain)
5. Dipartimento di Fisica “Ettore Pancini”, Università di Napoli “Federico II”, Complesso Universitario di Monte S. Angelo, via Cintia, I-80126 Napoli (Italy)
6. (Italy)

We review the geometry of the space of quantum states S(H) of a finite-level quantum system with Hilbert space H from a group-theoretical point of view. This space carries two stratifications generated by the action of two different Lie groups, namely, the special unitary group SU(H) and its complexification SL(H), the complex special linear group. A stratum of the stratification generated by SU(H) is composed of isospectral states, that is, density operators with the same spectrum. A stratum of the stratification generated by SL(H) is composed of quantum states with the same rank. We prove that on every submanifold of isospectral quantum states there is also a canonical left action of SL(H) which is related with the canonical Kähler structure on isospectral quantum states. The fundamental vector fields of this SL(H)-action are divided into Hamiltonian and gradient vector fields. The former give rise to invertible maps on S that preserve the von Neumann entropy and the convex structure of S(H), while the latter give rise to invertible maps on S(H) that preserve the von Neumann entropy but not the convex structure of S(H). A similar decomposition is given for the fundamental vector fields of the SL(H)-action generating the stratification of S(H) into manifolds of quantum states with the same rank. However, in this case, the gradient vector fields preserve the rank but do not preserve entropy. Finally, some comments on multipartite quantum systems are made, and it is proved that the sets of product states of a multipartite quantum system are homogeneous manifolds for the local action of the complex special linear group associated with the partition.

• OSTI ID: 22852375
• Journal Information: Annals of Physics, Vol. 400; Other Information: © 2018 Elsevier Inc. All rights reserved.; Country of input: International Atomic Energy Agency (IAEA); ISSN 0003-4916
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
ENTROPY
GROUP THEORY
HAMILTONIANS
HILBERT SPACE
LIE GROUPS
QUANTUM ENTANGLEMENT
QUANTUM INFORMATION
QUANTUM STATES
QUANTUM SYSTEMS
SPECTRA
STRATIFICATION
VECTOR FIELDS