# Cutting the quantum mustard

I had a relative to whom my parents referred, when I was little, as “that great-aunt of yours who walked into a glass door at your cousin’s birthday party.” I was a small child in a large family that mostly lived far away; little else distinguished this great-aunt from other relatives, in my experience. She’d intended to walk from my grandmother’s family room to the back patio. A glass door stood in the way, but she didn’t see it. So my great-aunt whammed into the glass; spent part of the party on the couch, nursing a nosebleed; and earned the epithet via which I identified her for years.

After 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 into a physicist, I came to appreciate her as one of my earliest instructors in necessary and sufficient conditions.

My great-aunt’s intended path satisfied one condition necessary for her to reach the patio: Nothing visible obstructed the path. But the path failed to satisfy a sufficient condition: The invisible obstruction—the glass door—had been neither slid nor swung open. Sufficient conditions, my great-aunt taught me, mustn’t be overlooked.

Her lesson underlies a paper I published this month, with coauthors from the Cambridge other than mine—Cambridge, England: David Arvidsson-Shukur and Jacob Chevalier Drori. The paper concerns, rather than pools and patios, quasiprobabilities, which I’ve blogged about many times [1,2,3,4,5,6,7]. Quasiprobabilities are quantum generalizations of probabilities. Probabilities describe everyday, classical phenomena, from Monopoly to March Madness to the weather in Massachusetts (and especially the weather in Massachusetts). Probabilities are real numbers (not dependent on the square-root of -1); they’re at least zero; and they compose in certain ways (the probability of sun or hail equals the probability of sun plus the probability of hail). Also, the probabilities that form a distribution, or a complete set, sum to one (if there’s a 70% chance of rain, there’s a 30% chance of no rain).

In contrast, quasiprobabilities can be negative and nonreal. We call such values nonclassical, as they’re unavailable to the probabilities that describe classical phenomena. Quasiprobabilities represent quantum states: Imagine some clump of particles in a quantum state described by some quasiprobability distribution. We can imagine measuring the clump however we please. We can calculate the possible outcomes’ probabilities from the quasiprobability distribution.

My favorite quasiprobability is an obscure fellow unbeknownst even to most quantum physicists: 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 quasiprobability has undergone a renaissance over the past few years: Experimentalists have measured it to infer particles’ quantum states in a new way. Also, colleagues and I have generalized the quasiprobability and discovered applications of the generalization across quantum physics, from quantum chaos to metrology (the study of how we can best measure things) to quantum thermodynamics to the foundations of quantum theory.

In some applications, nonclassical quasiprobabilities enable a system to achieve a quantum advantage—to usefully behave in a manner impossible for classical systems. Examples include metrology: Imagine wanting to measure a parameter that characterizes some piece of equipment. You’ll perform many trials of an experiment. In each trial, you’ll prepare a system (for instance, a photon) in some quantum state, send it through the equipment, and measure one or more observables of the system. Say that you follow the protocol described in this blog post. A Kirkwood-Dirac quasiprobability distribution describes the experiment.1 From each trial, you’ll obtain information about the unknown parameter. How much information can you obtain, on average over trials? Potentially more information if some quasiprobabilities are negative than if none are. The quasiprobabilities can be negative only if the state and observables fail to commute with each other. So noncommutation—a hallmark of quantum physics—underlies exceptional metrological results, as shown by Kirkwood-Dirac quasiprobabilities.

Exceptional results are useful, and we might aim to design experiments that achieve them. We can by designing experiments described by nonclassical Kirkwood-Dirac quasiprobabilities. When can the quasiprobabilities become nonclassical? Whenever the relevant quantum state and observables fail to commute, the quantum community used to believe. This belief turns out to mirror the expectation that one could access my grandmother’s back patio from the living room whenever no visible barriers obstructed the path. As a lack of visible barriers was necessary for patio access, noncommutation is necessary for Kirkwood-Dirac nonclassicality. But noncommutation doesn’t suffice, according to my paper with David and Jacob. We identified a sufficient condition, sliding back the metaphorical glass door on Kirkwood-Dirac nonclassicality. The condition depends on simple properties of the system, state, and observables. (Experts: Examples include the Hilbert space’s dimensionality.) We also quantified and upper-bounded the amount of nonclassicality that a Kirkwood-Dirac quasiprobability can contain.

From an engineering perspective, our results can inform the design of experiments intended to achieve certain quantum advantages. From a foundational perspective, the results help illuminate the sources of certain quantum advantages. To achieve certain advantages, noncommutation doesn’t cut the mustard—but we now know a condition that does.

For another take on our paper, check out this news article in Physics Today.

1Really, a generalized Kirkwood-Dirac quasiprobability. But that phrase contains a horrendous number of syllables, so I’ll elide the “generalized.”

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