Elliptical orbits in the Bloch sphere
arXiv:quant-ph/0503208 · doi:10.1088/1464-4266/7/10/011
Abstract
As is well known, when an SU(2) operation acts on a two-level system, its Bloch vector rotates without change of magnitude. Considering a system composed of two two-level systems, it is proven that for a class of nonlocal interactions of the two subsystems including σ_i\otimesσ_j (with i,j \in {x,y,z}) and the Heisenberg interaction, the geometric description of the motion is particularly simple: each of the two Bloch vectors follows an elliptical orbit within the Bloch sphere. The utility of this result is demonstrated in two applications, the first of which bears on quantum control via quantum interfaces. By employing nonunitary control operations, we extend the idea of controllability to a set of points which are not necessarily connected by unitary transformations. The second application shows how the orbit of the coherence vector can be used to assess the entangling power of Heisenberg exchange interaction.
9 pages, 4 figures, few corrections, J. Opt. B: Quantum Semiclass. Opt. 7 (2005) S1-S6
References in corpus (18)
- Quantum Computation with Quantum Dots
- Universal Quantum Computation with the Exchange Interaction
- Electron Spin Resonance Transistors for Quantum Computing in Silicon-Germanium Heterostructures
- Time Optimal Control in Spin Systems
- Theory of Decoherence-Free Fault-Tolerant Universal Quantum Computation
- The Bloch Vector for N-Level Systems
- A geometric theory of non-local two-qubit operations
- Universal Fault-Tolerant Computation on Decoherence-Free Subspaces
- Nonlocal properties of two-qubit gates and mixed states and optimization of quantum computations
- Characterization of the Positivity of the Density Matrix in Terms of the Coherence Vector Representation
- Creating Decoherence-Free Subspaces with Strong and Fast Pulses
- Controllability properties for finite dimensional quantum Markovian master equations
- Reducing Constraints on Quantum Computer Design by Encoded Selective Recoupling
- Combined encoding and decoupling solution to problems of decoherence and design in solid-state quantum computing
- Universal quantum interfaces
- A multi-photon Stokes-parameter invariant for entangled states
- Tensor of coherences parameterization of multiqubit density operators for entanglement characterization
- Quantum control without access to the controlling interaction