Quantum channels, complex Stiefel manifolds, and optimization

Abstract

Most general dynamics of an open quantum system is commonly represented by a quantum channel, which is a completely positive trace-preserving map (CPTP or Kraus map). Well-known representations of quantum channels are described by Choi matrices and by Kraus operator-sum representation (OSR). As was shown before, one can use Kraus OSR to parameterize quantum channels by points of a suitable quotient of some complex Stiefel manifold by the action of the unitary group. In this work, we establish a homeomorphism between the topological space of quantum channels and the quotient of the complex Stiefel manifold. This homeomorphism can be applied to various quantum optimization problems. As an example, we apply it to the analysis of extrema points for a wide variety of quantum control objective functionals defined on the complex Stiefel manifolds, including mean value, generation of quantum gates, thermodynamic quantities involving entropy, etc. Finally, a metric on the space of quantum channels induced by the Riemannian metric on the Stiefel manifold is defined, and we show that it is a generalization of the Bures angle between density matrices.

23 pages. The published version having a clarified presentation and an expanded Stiefel-induced metric part showing the Bures-angle generalization

References in corpus (17)

• Structure of states which satisfy strong subadditivity of quantum entropy with equality
• Vanishing quantum discord is necessary and sufficient for completely positive maps
• High fidelity quantum gates via dynamical decoupling
• An additive and operational entanglement measure: conditional entanglement of mutual information
• Control Landscapes for Observable Preparation with Open Quantum Systems
• Control landscapes for two-level open quantum systems
• Riemannian geometry and automatic differentiation for optimization problems of quantum physics and quantum technologies
• Optimization search effort over the control landscapes for open quantum systems with Kraus-map evolution
• Reachable sets for two-level open quantum systems driven by coherent and incoherent controls
• Unified analysis of terminal-time control in classical and quantum systems
• Optimal quantum channels
• Control of the von Neumann Entropy for an Open Two-Qubit System Using Coherent and Incoherent Drives
• Finite-Dimensional Stinespring Curves Can Approximate Any Dynamics
• Optimal form of the Kretschmann-Schlingemann-Werner theorem for energy-constrained quantum channels and operations
• On characteristics of mixed unitary channels being additive or multiplicative with respect to taking tensor products
• Progress on the Kretschmann-Schlingemann-Werner Conjecture
• Randomized Gradient Descents on Riemannian Manifolds: Almost Sure Convergence to Global Minima in and beyond Quantum Optimization