Phase sensitivity of entanglement in the Quantum Phase Estimation algorithm

Abstract

Abstract We study entanglement in the steps before Quantum Fourier Transform (pre-QFT) part of the Quantum Phase Estimation and the Quantum Counting algorithms (QPEA—QCA) with the use of three entanglement detection tools. In particular we focus on the sensitivity of entanglement to the input value (the phase ϕ and the ratio of marked elements M N ) in some basic cases. One starts from numerical observations and deduce some general results in particular regarding the classes of entanglement. More precisely, when the second register of both algorithms (i.e. the register on which a specific unitary operator act, see section 2) is initialized in the non-entangled superposition of two (separable) eigenvectors, one proves that the QPEA and QCA curves of entanglement evolution are the same up to a scalar multiplication of the parameter ϕ. One demonstrates that a local minimum is obtained and corresponds to an EPR (Einstein-Podolsky-Rosen) state and finally one proves that, up to Stochastic Local Operation and Classical Communication (SLOCC), all states, except for a few values of ϕ, are equivalent to the product of a separable state and a generalized GHZ (Greenberger-Horen-Zeilinger) state , i.e. GHZ n + 1 = 1 2 ( 00 ... 0 + 11 ... 1 ) .