Observing Dirac's classical phase space analog to the quantum state

Abstract

In 1945, Dirac attempted to develop a "formal probability" distribution to describe quantum operators in terms of two non-commuting variables, such as position x and momentum p [Rev. Mod. Phys. 17, 195 (1945)]. The resulting quasi-probability distribution is a complete representation of the quantum state and can be observed directly in experiments. We measure Dirac's distribution for the quantum state of the transverse degree of freedom of a photon by weakly measuring transverse x so as to not randomize the subsequent p measurement. Further, we show that the distribution has the classical-like feature that it transforms (e.g., propagates) according to Bayes' law.

9 pages, 4 figures, Supplementary Material

References in corpus (16)

• Ultrasensitive Beam Deflection Measurement via Interferometric Weak Value Amplification
• Experimental joint weak measurement on a photon pair as a probe of Hardy's Paradox
• Complex weak values in quantum measurement
• Negativity and contextuality are equivalent notions of nonclassicality
• Weak values and the Leggett-Garg inequality in solid-state qubits
• Direct observation of Hardy's paradox by joint weak measurement with an entangled photon pair
• No extension of quantum theory can have improved predictive power
• A double-slit `which-way' experiment on the complementarity--uncertainty debate
• Frame representations of quantum mechanics and the necessity of negativity in quasi-probability representations
• State tomography via weak measurements
• Probing the negative Wigner function of a pulsed single photon point by point
• Modular values and weak values of quantum observables
• Nonnegative subtheories and quasiprobability representations of qubits
• Sequential measurement of conjugate variables as an alternative quantum state tomography
• Quantum Mechanics of Successive Measurements with Arbitrary Meter Coupling
• Positive phase space transformation incompatible with classical physics

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