Simultaneous Measurement of Complementary Observables with Compressive Sensing

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Simultaneous Measurement of Complementary Observables with Compressive Sensing

Gregory A. Howland, James Schneeloch, Daniel J. Lum, and John C. Howell

Phys. Rev. Lett. 112, 253602 – Published 26 June 2014

Abstract

The more information a measurement provides about a quantum system’s position statistics, the less information a subsequent measurement can provide about the system’s momentum statistics. This information trade-off is embodied in the entropic formulation of the uncertainty principle. Traditionally, uncertainly relations correspond to resolution limits; increasing a detector’s position sensitivity decreases its momentum sensitivity and vice versa. However, this is not required in general; for example, position information can instead be extracted at the cost of noise in momentum. Using random, partial projections in position followed by strong measurements in momentum, we efficiently determine the transverse-position and transverse-momentum distributions of an unknown optical field with a single set of measurements. The momentum distribution is directly imaged, while the position distribution is recovered using compressive sensing. At no point do we violate uncertainty relations; rather, we economize the use of information we obtain.

Received 18 March 2014

DOI:https://doi.org/10.1103/PhysRevLett.112.253602

Authors & Affiliations

Gregory A. Howland^*, James Schneeloch, Daniel J. Lum, and John C. Howell

Department of Physics and Astronomy, University of Rochester, 500 Wilson Boulevard, Rochester, New York 14618, USA

^*ghowland@pas.rochester.edu

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Issue

Vol. 112, Iss. 25 — 27 June 2014

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Images

Figure 1
Experimental setup for simultaneous position and momentum imaging. A state is prepared by illuminating an object mask at $z = z_{0}$ with a plane wave from an attenuated HeNe laser. The field is imaged at $z = z_{x}$ where it is sequentially filtered by a series of $M$ $256 \times 256$ -pixel, random binary filters $F_{i}$ . Each filter partially projects the state by blocking about half of the position elements. A cooled CCD array in the focal plane $z = z_{k}$ of a Fourier-transforming lens records the momentum distribution for each filtered state. Position information is mapped to the total optical power passing each filter; this measures the correlation between the position intensity distribution at $z_{x}$ with the current filter. Because the filters do not strongly localize the photons’ position, the momentum distribution is directly recovered by averaging the CCD images. The position distribution is reconstructed using compressive sensing techniques such that an $N$ -pixel position image requires $M ≪ N$ filters.
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Figure 2
Discrete Fourier transform of a 256-pixel, one-dimensional random binary pattern. Panel (a) gives a random, binary one-dimensional filter function where a value 1 is fully transmitting and a value 0 is fully obstructing. Panel (b) shows the relative power spectrum of its Fourier transform, where the zero-momentum term is scaled to unity. The noise floor is a factor $\sqrt{2 / N}$ weaker in amplitude in both its mean and standard deviation. The same relationship holds for a two-dimensional filter.
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Figure 3
Partial projections of $ψ$ . The figure simulates the effect of filtering a triple slit object with a $128 \times 128$ -pixel random binary pattern. The momentum images are given as relative powers, where their maximum value is scaled to unity. A square-root color mapping emphasizes weaker momentum values. The filtered momentum distribution is a slightly noisy version of the true momentum distribution.
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Figure 4
Recovered position and momentum images for four objects. A double slit, triple slit, and $ℏ$ were reconstructed at $N = 256 \times 256$ resolution from only $M = 6553$ filters; the university logo used $M = 32768$ filters. No additional post-processing has been performed; position images are those returned by the reconstruction algorithm and momentum images are the recorded single or mean CCD images.
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Figure 5
Simulated momentum MSE: The simulated momentum MSEs of the four objects used in the experiment are given as a function of the number of random patterns $M$ . The shaded region encloses one standard deviation above and below the average MSE for 100 trials. The MSE rapidly approaches a small constant value as the second term of Eq. (4) vanishes. Even for a single pattern, the MSE is at least of order $10^{- 7}$ .
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