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Sets or pairs of incompatible observables, such as momentum and position, play a pivotal role in a wide range of distinctly quantum effects and applications, including quantum cryptography, the Heisenberg Uncertainty Principle, quantum state tomography, and Bell’s inequalities. In particular, in quantum physics, we are prohibited from precisely measuring the values of incompatible observables, a fact that is at the heart of the nature of the quantum state. In this talk, I will explore an assortment of strategies that simple-mindedly attempt to circumvent this prohibition. Motivated by these naïve strategies, we experimentally investigate the use of weak measurement and optimal quantum cloning to perform joint measurements on photons. The direct outcome of these measurements are, depending on the strategy, the wavefunction, the Dirac distribution, and the density matrix of the measured quantum system. Consequently, these naïve strategies provide new ways to characterize quantum systems and to understand the very entities that we are measuring, such as the wavefunction.

Tradeoffs in measurement and information are among the central themes of quantum mechanics. I will try to summarize in this talk a few of our experiments related to modern views of these topics. In particular, I will try to give an example or two of the power of "weak measurements," both for fundamental physics and for possible precision metrology. One example will involve revisiting the question of Heisenberg's famous principle, and an interpretation which is widespread but has now been experimentally shown to be incorrect. Then I will also discuss our recent work on a "quantum data compression" protocol which would allow a small-scale quantum memory to store all the extractable information from a larger ensemble of identically prepared systems. Finally, I will talk about our experiment entangling two optical beams to demonstrate "weak-value amplification," and the ongoing controversy about when if ever this technique could be useful in practice.

One of the most successful theories in physics until now is quantum mechanics. However, the physical origins of its mathematical structure are still under debate, and a "generalized" quantum theory to unify quantum mechanics and gravity is still missing. Recently, in an effort to better understand the mathematical structure of quantum mechanics, theories containing the essence of quantum mechanics, while also having a broader description of physical phenomena, have been proposed. These so-called "post-quantum theories" have only been recently tested at the lab. In this talk, I will present the results of our experimental test using single photons to probe one of these post-quantum theories; namely, hyper-complex quantum theories. Interestingly, in hyper-complex theories simple phases do not necessarily commute. To study this effect, we apply two physically different optical phases, one with a positive and one with a negative refractive index, to single photons inside of a Sagnac interferometer. Through our measurements we are able put bounds on this particular prediction of hyper-complex quantum theories.

The scientific journey from the first hints of quantum behaviour to the Bloch sphere in your textbook was a long and tortuous one. But using some of the technological and conceptual fruits of that journey, we show that an experiment can manifest the Bloch sphere via an analysis that doesn't require any quantum theory at all. Our technique is to fit experimental data to a generalised probabilistic theory, which allows us to infer both the dimension and shape of the state and measurement spaces of the system under study. We test our technique on an experiment measuring a variety of single-photon polarization states. As expected, the reconstructed state space closely resembles the Bloch sphere, and we are able to place small upper bounds on how much the true theory describing our experiment could possibly deviate from quantum mechanics.