Simultaneous Measurement of Complementary Observables with Compressive Sensing

Dick Gordon and several other readers mentioned the following paper to me. It aims at doing something that was thought to be impossible: i.e. measuring both position and momentum of a system at the same time. From afar, the Heisenberg principle says it cannot be done but as the authors mention:

...We strongly emphasize that our technique does not violate the uncertainty principle; at no point does a single detection event give precise information about both position and momentum. Instead, each detection event gives some information about both domains. Our approach economizes the use of this information... 

So if I understand correctly, the Heisenberg principle is somehow not applicable here because:

• the authors perform two detections (not one) and, in my view the next argument seems the most important assumption,
• the authors use the additional information about the sparsity of the event so that one can use the random projections of the masks. Had the event not been sparse, or in other words, had they had to detect many several positions and momentums, position measurements using random masks would not yield any position information because that information would not be sparse (and therefore algorithmic position reconstruction would fail).

Limits on these systems can therefore directly use current known limits of the reconstruction solvers (also known as the Donoho-Tanner sharp phase transition) and this with Hadamard matrices or even more economical measurement matrices. Suddenly, 10 years later, sharp phase transitions from combinatorics yield another interesting twist....If this interpretation holds, Quantum Mechanics and its many paradoxes might become less obscure. Without further ado (thanks Doug!), here is the paper:

Simultaneous Measurement of Complementary Observables with Compressive Sensing by Gregory A Howland, James Schneeloch, Daniel J Lum, John C. Howell

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.

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