Minimally disturbing HeisenbergWeyl symmetric measurements using hardcore collisions of Schrödinger particles
Abstract
In a previous paper we have presented a general scheme for the implementation of symmetric generalized measurements (POVMs) on a quantum computer. This scheme is based on representation theory of groups and methods to decompose matrices that intertwine two representations. We extend this scheme in such a way that the measurement is minimally disturbing, i.e., it changes the state vector ∣Ψ⟩ of the system to √Π ∣Ψ⟩ where Π is the positive operator corresponding to the measured result. Using this method, we construct quantum circuits for measurements with HeisenbergWeyl symmetry. A continuous generalization leads to a scheme for optimal simultaneous measurements of position and momentum of a Schrödinger particle moving in one dimension such that the outcomes satisfy ∆x∆p ⩾ℏ. The particle to be measured collides with two probe particles, one for the position and the other for the momentum measurement. The position and momentum resolution can be tuned by the entangled joint state of the probe particles which is also generated by a collision with hardcore potential. The parameters of the POVM can then be controlled by the initial widths of the wave functions of the probe particles. We point out some formal similarities and differences to simultaneous measurements of quadrature amplitudes in quantum optics.
 Publication:

Journal of Mathematical Physics
 Pub Date:
 August 2006
 DOI:
 10.1063/1.2222080
 arXiv:
 arXiv:quantph/0507097
 Bibcode:
 2006JMP....47h2102J
 Keywords:

 03.65.Ta;
 03.65.Ud;
 42.50.Dv;
 03.67.Mn;
 Foundations of quantum mechanics;
 measurement theory;
 Entanglement and quantum nonlocality;
 Nonclassical states of the electromagnetic field including entangled photon states;
 quantum state engineering and measurements;
 Entanglement production characterization and manipulation;
 Quantum Physics
 EPrint:
 32 pages, latex, 6 figures, presentation improved, The implementation scheme for the continuousvariable measurement has been added. To appear in Journ. Math. Phys