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Computing partial traces and reduced density matrices

Author

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  • Jonas Maziero

    (Departamento de Física, Centro de Ciências Naturais e Exatas, Universidade Federal de Santa Maria, Avenida Roraima 1000, Santa Maria 97105-900, RS, Brazil2Instituto de Física, Facultad de Ingeniería, Universidad de la República, J. Herrera y Reissig 565, 11300, Montevideo, Uruguay)

Abstract

Taking partial traces (PTrs) for computing reduced density matrices, or related functions, is a ubiquitous procedure in the quantum mechanics of composite systems. In this paper, we present a thorough description of this function and analyze the number of elementary operations (ops) needed, under some possible alternative implementations, to compute it on a classical computer. As we note, it is worthwhile doing some analytical developments in order to avoid making null multiplications and sums, what can considerably reduce the ops. For instance, for a bipartite system ℋa⊗ℋb with dimensions da=dimℋa and db=dimℋb and for da,db≫1, while a direct use of PTr definition applied to ℋb requires 𝒪(da6db6) ops, its optimized implementation entails 𝒪(da2db) ops. In the sequence, we regard the computation of PTrs for general multipartite systems and describe Fortran code provided to implement it numerically. We also consider the calculation of reduced density matrices via Bloch’s parametrization with generalized Gell Mann’s matrices.

Suggested Citation

  • Jonas Maziero, 2017. "Computing partial traces and reduced density matrices," International Journal of Modern Physics C (IJMPC), World Scientific Publishing Co. Pte. Ltd., vol. 28(01), pages 1-17, January.
    • Handle: RePEc:wsi:ijmpcx:v:28:y:2017:i:01:n:s012918311750005x
    DOI: 10.1142/S012918311750005X
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