Characterization of multilevel quantum coherence without ideal measurements

Abstract

Coherent superpositions are one of the hallmarks of quantum mechanics and are vital for any quantum mechanical device to outperform the classically achievable. Generically, superpositions are verified in interference experiments, but despite their longstanding central role we know very little about how to extract the number of coherently superposed amplitudes from a general interference pattern. A fundamental issue is that performing a phasesensitive measurement is as challenging as creating a coherent superposition, so that assuming a perfectly implemented measurement for verification of quantum coherence is hard to justify. In order to overcome this issue, we construct a coherence certifier derived from simple statistical properties of an interference pattern, such that any imperfection in the measurement can never over-estimate the number of coherently superposed amplitudes. We numerically test how robust this measure is to under-estimating the coherence in the case of imperfect state preparation or measurement, and find it to be very resilient in both cases. arXiv:1901.08599v2 [quant-ph]

Certifying Multilevel Coherence in the Motional State of a Trapped Ion

“…satisfies the requisites [18]. It requires the pattern to be evaluated at only a few different free-evolution phases for good statistics, as it relies only on low-order moments.…”

“…It requires the pattern to be evaluated at only a few different free-evolution phases for good statistics, as it relies only on low-order moments. Obtaining a value of 𝐶 greater than 1 requires 2-coherence, greater than 5/4 = 1.25 requires 3-coherence, and greater than 179/96 ≈ 1.86 is necessary to certify 4-coherence in a Hilbert space of arbitrary dimension [18]. While the metric can certify the presence of coherence, this threshold system does not provide the completely tight bounds needed to unambiguously certify the absence of coherence above a given level.…”

“…This coherence metric is resilient in the face of imperfect realisation of coherent operations during the measurementmapping procedure, however the analysis of ref. [18] is valid only when the actual measurement is a projection onto a single basis state. In order to be used for coupled systems, it must first be shown that the same hierarchical structure of the threshold values applies for general measurements.…”

“…More recent schemes have sought to avoid these problems by ensuring that their coherence metrics provide a strict lower bound on the amount of higher-order coherence in a state, no matter how successfully intermediate operations were implemented [6]. Of particular interest is the ability to certify this coherence based only on an interference pattern [18], effectively enabling verification of multilevel coherence based on a projective measurement onto only one state. These schemes enable insight into coherence in systems inaccessible by measurement, without risking false positives from imperfect mapping operations.…”

“…In this work, we experimentally certify the existence of multiple superposition elements in the motional state of a single trapped ion, using an interference pattern method derived from, and extending, ref. [18]. The only available projective measurement can distinguish the state of the coupled internalstate qubit, but not the different motional states, making it a more general operation than originally considered.…”