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Bulletin of the American Physical Society

Decoherence and Quantum Walks under the regime of weak measurements and weak values

Debabrata Ghoshal, Marco O. Lanzagorta, Salvador Elías Venegas-Andraca

Published Feb 28, 2012• 0 citations
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Quantum walks have been studied under several regimes. Motivated by experimental results on quantum weak measurements and weak values as well as by the need to develop new insights for quantum algorithm development, we are extending our knowledge by studying the behavior of quantum walks under the regime of quantum weak measurements and weak values of pre- and postselected measurements (QWWM hereinafter). In particular, we investigate the limiting position probability distribution and several statistical measures (such as standard deviation) of a QWWM on an infinite line, and compare such results with corresponding classical and quantum walks position probability distributions and statistical measures, stressing the differences provided by weak measurements and weak values with respect to results computed by using canonical observables. We start by producing a concise introduction to quantum weak values and quantum weak measurements. We then introduce definitions as well as both analytical and numerical results for a QWWM under Hadamard evolution and extend our analysis to quantum evolution ruled by general unitary operators. Moreover, we propose a definition and focus on the properties of mixing time of QWWM on an infinite line, followed by a comparison of known corresponding results for classical and quantum walks mixing times. We finish this paper by presenting a plausible experimental implementation of a QWWM.
Our previous work brought some interesting results of the discrete Quantum Walks in the regime of Weak Measurement (QWWM or QWWV). Using the knowledge of such results of QWWM, we are now exploring the search algorithms and investigating the factors associated with such walk. The study of such factors like dimensionality, connectivity of the dataset and the strength of disorder or percolation are already studied by others in the context of general quantum walks. It is our interest to show the similarities and/or differences of such factors of general quantum walks with QWWV. The subject of decoherence in quantum walks is another challenging research topic at present. We are also exploring the topic of decoherence in QWWM or QWWV.
The question of when the Kirkwood-Dirac quasiprobability serves as the most appropriate description for quantum measurements has remained unresolved, particularly across different measurement strengths. While known to generate anomalous weak values in the weak measurement regime and to reduce to classical probabilities under projective measurement, the physical mechanism governing its continuous transformation has been lacking. Here we demonstrate that the KD quasiprobability provides a general framework for all measurement regimes by identifying pointer-induced decoherence as the universal mechanism controlling this transition. We show that the decoherence factor F(t) simultaneously quantifies the loss of quantum coherence and interpolates the measurement strength from weak to strong. Within this framework, the KD quasiprobability naturally deforms from its full complex form-governing weak values-to the real, non-negative Wigner formula describing projective measurements, while maintaining informational completeness throughout the transition. Our work resolves the fundamental question of the KD distribution's applicability by establishing it as the universal framework that seamlessly connects all quantum measurement regimes through a physically transparent decoherence pathway.