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Bulletin of the American Physical Society

Some Statistical Measures and Analytical Results of Quantum Walks under Weak Measurement and Weak Values Regime

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

Published Mar 22, 2011• 0 citations
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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.

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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.
In the weak measurement regime, post-selection can result in the observation of anomalous weak values, seemingly contradicting the eigenvalue statistics observed when the measurement interaction is strong. Here, we investigate the dependence of meter statistics on measurement strength in a post-selected measurement. We find that the meter statistics in the intermediate regime between weak and strong measurements is nearly independent of measurement strength and show that, in this regime, the system performs a measurement of momentum on the meter. Our results show that post-selection is extremely sensitive to the action of the meter on the system, highlighting the difference between the weak measurement regime where this back-action is negligible and the average meter position is a valid measurement result, and the intermediate regime where the meter statistics should not be misinterpreted as a measurement outcome. The transition between weak values and eigenvalues is facilitated by a reversal of the roles of the system and the meter, where the post-selection acts as a readout of information about the meter. The intermediate measurement regime thus reveals that the apparent contradictions between anomalous weak values and eigenvalues originate from the non-classical effects of back-action on the post-selected meter statistics.
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.