I had a relative whom my parents called “your great-aunt who came through a glass door at your cousin’s birthday party” when I was little. I was a small child in a large family that mostly lived far away; In my experience, this great-aunt was hardly any different from other relatives. She had planned to go from my grandmother’s family room to the back terrace. There was a glass door in the way, but she couldn’t see it. So my great aunt beaten up into the glass; spent part of the party on the couch nursing a nosebleed; and earned the nickname by which I identified them for years.

Growing up, I came to know this great-aunt as a kind, gentle woman who adored her family and was adored in return. After growing up to be a physicist, I came to appreciate her as one of my first teachers under the necessary and sufficient conditions.

My great-aunt’s intended path met one condition that was necessary to reach the patio: nothing visible was blocking the path. But the path did not meet a sufficient condition: the invisible obstacle – the glass door – was neither pushed open nor swung open. Sufficient conditions, as my great-aunt taught me, must not be overlooked.

Your lesson is based on an article I published this month with Cambridge co-authors other than mine – Cambridge, England: David Arvidsson-Shukur and Jacob Chevalier Drori. The paper is more about quasi-probabilities than pools and patios that I’ve blogged about many times [1,2,3,4,5,6,7] . Quasi-probabilities are quantum generalizations of probabilities. Probabilities describe everyday, classic phenomena, from Monopoly to March Madness to the weather in Massachusetts (and especially the weather in Massachusetts). Probabilities are real numbers (regardless of the square root of -1); they are at least zero; and they are composed in a certain way (the probability of sun or hail is equal to the probability of sun plus the probability of hail). Also, the probabilities that make up a distribution or a complete set add up to one (a 70% chance of rain has a 30% chance that no rain will occur).

In contrast, quasi-probabilities can be negative and non-real. We call such values not classicbecause they are not available for the probabilities that describe classical phenomena. Quasi-probabilities represent quantum states: Imagine a lump of particles in a quantum state, which is described by a quasi-probability distribution. We can imagine measuring the lump at will. We can calculate the probabilities of the possible outcomes from the quasi-probability distribution.

Not from my grandmother’s house, although I wouldn’t mind.

My favorite quasi-probability is an obscure companion that even most quantum physicists are unfamiliar with: the Kirkwood-Dirac distribution. John Kirkwood defined it in 1933, and Paul Dirac defined it independently in 1945. Then quantum physicists forgot about it for decades. But the quasi-probability has experienced a renaissance in recent years: Experimentalists have measured it in order to be able to infer the quantum states of particles in a new way. Also, colleagues and I generalized quasi-probability and discovered applications of generalization across quantum physics, from quantum chaos to metrology (the study of how best to measure things) to quantum thermodynamics to the fundamentals of quantum theory.

In some applications, non-classical quasi-probabilities allow a system to achieve a quantum advantage – useful behavior that is impossible for classical systems. Examples are metrology: Imagine you want to measure a parameter that characterizes a device. You will do many attempts at an experiment. In each experiment you prepare a system (for example a photon) in a quantum state, send it through the equipment and measure one or more observables of the system. Suppose you follow the protocol outlined in this blog post. A Kirkwood-Dirac quasi-probability distribution describes the experiment.1 You will get information about the unknown parameter from each attempt. How much information can you get on average about attempts? Potentially more information if some quasi-probabilities are negative than if none. The quasi-probabilities can only be negative if the state and the observable do not commute with one another. Non-commutation – a trademark of quantum physics – is based on extraordinary measurement results, as the quasi-probabilities of Kirkwood-Dirac show.

Exceptional results are useful, and we could try designing experiments to achieve them. We can design experiments that are described by non-classical Kirkwood-Dirac quasi-probabilities. When can the quasi-probabilities become non-classic? Whenever the relevant quantum state and the observables do not commute, the quantum community believed. This belief reflects the expectation that one could reach my grandmother’s back terrace from the living room if there were no visible barriers blocking the way. Since no visible barriers were required to access the terrace, the non-commutation is required for the non-classicism of Kirkwood-Dirac. But according to my work with David and Jacob, non-commutation is not enough. We identified a sufficient state by sliding the metaphorical glass door back to Kirkwood-Dirac non-classical. The condition depends on simple properties of the system, the state and the observables. (Experts: Examples are the dimensionality of Hilbert space.) We have also quantified the amount of non-classicity that a Kirkwood-Dirac quasi-probability can contain and limited it upwards.

From a technical point of view, our results can be incorporated into the design of experiments aimed at achieving certain quantum advantages. From a fundamental perspective, the results help shed light on the sources of certain quantum benefits. In order to achieve certain benefits, non-commutation doesn’t cut the mustard – but we now know one condition that does.

For another version of ours paper, Check out these News articles in Physics Today.

1Really, a generalized Kirkwood-Dirac quasi-probability. But this sentence contains an awful number of syllables, so I’ll leave out the “generalized”.


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