Classical Computer Assisted Analysis of Small Multiqudit Systems

Abstract

Quantum computation model is regarded as a model which can overcome barriers in calculations efficiency of problems which appear in modern science. In spite of hardware development, in particular a recent emergence of several different physical installations of the pioneering quantum machines, the contemporary and numerical analysis of problems concerning quantum computing is very important. In the first part of this article, some useful computing techniques for quantum registers processed by a quantum circuits are presented. Applied classical parallel computational techniques are utilised to shorten the whole computational time. New methods of processing state vectors for qudits and density matrices are presented, indicating which operations may be performed in parallel in the context of the implementation of local unitary operations. There is also shown, how to use the reduction operation in parallel implementation of the von Neumann measurement by performing local measurements on a system of qudits. In addition to the purely technical results as described above, the paper includes also a bunch of purely theoretical results which substitute a solid mathematical ground for the computations performed with the help of the computational routines as described in Section III. In particular, a discussion concerning general multi-qudit quantum states through the prism of Entropy and Negativity measures of entanglement included in has been presented. Additionally, the notion of the total entanglement has been introduced. For certain classes of popular multi-qudit states, the introduced deficits of entanglement defined with the use of von Neumann Entropy and Negativity have been discussed. In particular, by the use of Gram matrix technique, the corresponding deficits of entanglement in the analysed states have been computed in an explicite way. Additionally, some new results on AME states for some multiqudit systems are also included in Section V. The last part of the article presents some numerical experiments on multi-qudit entanglement and determination of total entanglement values for convex combinations of GHZ and W states. Some details concerning technical nature of results are included in the attached Appendix A.