Tripartite entanglement and matrix inversion quantum algorithm

The role of entanglement is discussed in the Harrow-Hassidim-Lloyd (HHL) algorithm. We compute all tripartite entanglement at every steps of the HHL algorithm. The tripartite entanglement is generated at the first quantum phase estimation (QPE) step. However, it turns out that amount of the generated entanglement is not maximal except very rare cases. In the second rotation step some tripartite entanglement is used. Thus, the net tripartite entanglement is diminished. At the final inverse-QPE step the matrix inversion task is completed at the price of complete annihilation of the entanglement. An implication of this result is briefly discussed.