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arXiv:2406.06713 · Quantum Science and Technology

Quasiprobability distributions with weak measurements

Gabriele Bizzarri, Stefano Gherardini, Mylenne Manrique, Fabio Bruni, Ilaria Gianani, Marco Barbieri

Published Jul 29, 2025• 4 citations
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Abstract

Abstract We discuss and experimentally demonstrate the role of quantum coherence in a sequence of two measurements collected at different times using weak measurements. For this purpose, we have realized a weak-sequential measurement protocol with photonic qubits, where the first measurement is carried out as a positive operator-valued measure, whereas the second one is a projective operation. We determine the quasiprobability distributions associated to this procedure using both the commensurate and the Margenau-Hill quasiprobabilities. By tuning the weak measurements, we obtain a quasidistribution that may or may not exhibit negative parts, depending on the suitability of a contextual model for describing the experiment. Our results show how quasidistributions may find application in inspecting quantum monitoring, when part of the initial quantum coherence needs to be preserved.

DOI
10.1088/2058-9565/adf573

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Abstract We discuss and experimentally demonstrate the role of quantum coherence in a sequence of two measurements collected at different times using weak measurements. For this purpose, we have realized a weak-sequential measurement protocol with photonic qubits, where the first measurement is carried out as a positive operator-valued measure, whereas the second one is a projective operation. We determine the quasiprobability distributions associated to this procedure using both the commensurate and the Margenau-Hill quasiprobabilities. By tuning the weak measurements, we obtain a quasidistribution that may or may not exhibit negative parts, depending on the suitability of a contextual model for describing the experiment. Our results show how quasidistributions may find application in inspecting quantum monitoring, when part of the initial quantum coherence needs to be preserved.
Negative probability values have been widely employed as an indicator of the nonclassicality of quantum systems. Known as a quasiprobability distribution, they are regarded as a useful tool that provides significant insight into the underlying fundamentals of quantum theory when compared to the classical statistics. However, in this approach, an operational interpretation of these negative values with respect to the definition of probability---the relative frequency of occurred event---is missing. An alternative approach is therefore considered where the quasiprobability operationally reveals the negativity of measured quantities. We here present an experimental realization of the operational quasiprobability, which consists of sequential measurements in time. To this end, we implement two sets of polarization measurements of single photons. We find that the measured negativity can be interpreted in the context of selecting measurements, and it reflects the nonclassical nature of photons. Our results suggest a new operational way to unravel the nonclassicality of photons in the context of measurement selection.
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Quantum measurement can be strong or weak, the former being important in readout and initialization of quantum objects and the latter being useful for monitoring and maneuvering quantum evolution. However, the boundary between weak and strong measurement is unclear. Here we show that a phase transition occurs in sequential quantum measurement, which unambiguously separates the weak and strong measurement by a critical value of measurement strength or duration. We formulate the probability distribution of the output of a sequence of quantum measurements as the Boltzmann distribution of an interacting spin model. The measurement results present phase transitions similar to those in the spin model. In particular the sequential commuting positive operator-valued measurement is mapped to a long-range Ising model, and a projective measurement emerges from sequential weak measurements when the strength or the number of measurements becomes above certain critical values, corresponding to the ferromagnetic phase transition of the spin model. This finding sheds insights on sequential quantum measurement, and also provides the theoretical foundation for constructing projective measurements from sequential weak measurements, which have applications in steering the quantum evolution and initializing quantum systems where strong measurement in a single shot is often not possible.
Measurements in the quantum domain can exceed classical notions. This concerns fundamental questions about the nature of the measurement process itself, as well as applications, such as their function as building blocks of quantum information processing protocols. In this paper we explore the notion of entanglement for detection devices in theory and experiment. A method is devised that allows one to determine nonlocal quantum coherence of positive-operator-valued measures via negative contributions in a joint distribution that fully describes the measurement apparatus under study. This approach is then applied to experimental data for detectors that ideally project onto Bell states. In particular, we describe the reconstruction of the aforementioned entanglement quasidistributions from raw data and compare the resulting negativities with those expected from theory. Therefore, our method provides a versatile toolbox for analyzing measurements regarding their quantum-correlation features for quantum science and quantum technology.
We discuss the role of contextuality within quantum fluctuation theorems, in the light of a recent no-go result by Perarnau-Llobet et al. We show that any fluctuation theorem reproducing the two-point-measurement scheme for classical states either admits a notion of work quasiprobability or fails to describe protocols exhibiting contextuality. Conversely, we describe a protocol that smoothly interpolates between the two-point-measurement work distribution for projective measurements and Allahverdyan's work quasiprobability for weak measurements, and show that the negativity of the latter is a direct signature of contextuality.
We generalize the operational quasiprobability involving sequential measurements proposed by Ryu et al. [Phys. Rev. A 88, 052123 (2013)] to a continuous-variable system. The quasiprobabilities in quantum optics are incommensurate, i.e., they represent a given physical observation in different mathematical forms from their classical counterparts, making it difficult to operationally interpret their negative values. Our operational quasiprobability is commensurate, enabling one to compare quantum and classical statistics on the same footing. We show that the operational quasiprobability can be negative against the hypothesis of macrorealism for various states of light. Quadrature variables of light are our examples of continuous variables. We also compare our approach to the Glauber-Sudarshan P function. In addition, we suggest an experimental scheme to sequentially measure the quadrature variables of light.