Abstract
The resources needed to conventionally characterize a quantum system are overwhelmingly large for high-dimensional systems. This obstacle may be overcome by abandoning traditional cornerstones of quantum measurement, such as general quantum states, strong projective measurement, and assumption-free characterization. Following this reasoning, we demonstrate an efficient technique for characterizing high-dimensional, spatial entanglement with one set of measurements. We recover sharp distributions with local, random filtering of the same ensemble in momentum followed by position—something the uncertainty principle forbids for projective measurements. Exploiting the expectation that entangled signals are highly correlated, we use fewer than 5000 measurements to characterize a 65,536-dimensional state. Finally, we use entropic inequalities to witness entanglement without a density matrix. Our method represents the sea change unfolding in quantum measurement, where methods influenced by the information theory and signal-processing communities replace unscalable, brute-force techniques—a progression previously followed by classical sensing.
- Received 23 September 2015
DOI:https://doi.org/10.1103/PhysRevX.6.021018
This article is available under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
Published by the American Physical Society
Physics Subject Headings (PhySH)
Popular Summary
A strange feature of quantum mechanics—Heisenberg’s uncertainty principle—forbids precise, predictive knowledge of both the position and the momentum of a quantum particle: A precise position measurement is believed to “collapse” the quantum wave function, randomizing its momentum. Here, we show that by using imprecise measurements in the form of random filters, we can avoid collapsing the quantum wave function while still extracting useful information. We are able to very efficiently characterize entanglement in large quantum systems and reduce measurement times from years to hours.
We demonstrate a technique for efficiently characterizing spatial entanglement in a 65,536-dimensional, two-photon quantum system using fewer than 5000 measurements. Our approach employs three nontraditional components. First, we use partially projective measurements taking the form of random filters. Remarkably, these measurements allow us to simultaneously obtain statistics about incompatible observables (in our case, the photons’ positions and momentums). Second, we use our prior knowledge that entangled photons should be strongly correlated to perform compressive sensing. Compressive sensing exploits a signal’s compressibility to sample dramatically below the Nyquist limit. Finally, we use an information-based entanglement criterion to witness quantum entanglement directly from joint distributions in position and momentum.
Our approach is representative of a current trend in quantum measurements in which intractable, brute-force techniques are being replaced with intelligent strategies—a path previously followed in classical sensing. Beyond practical advantages, these strategies often give an interesting perspective on fundamental phenomena such as the uncertainty principle in this investigation.