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Frontiers in Quantum Science and Technology

Joint observables induced by indirect measurements in cavity QED

Kalle Raikisto, Kimmo Luoma

Published Oct 17, 2025• 0 citations
Publisher

Abstract

A fundamental feature of quantum mechanics is that there are observable pairs that cannot be measured jointly, such as observables corresponding to position and momentum or spin direction measurements. However, unsharp versions of non-jointly measurable observables may become jointly measurable. In this study, we investigate the joint measurability of time-continuous observables emerging from indirect time-continuous measurements. In particular, we study a paradigmatic situation where a qubit is interacting with a mode of light in a cavity, and the light escaping the cavity is continuously monitored. We find that the properties of the observables can be tuned by changing the type of the monitoring scheme or by tuning the initial state of the cavity. In particular, we demonstrate that heterodyne measurements are a joint measurement of a noisy homodyne measurement of a pair of canonical quadratures. Moreover, we investigate the purity of the induced qubit observables as a function of the noise.

DOI
10.3389/frqst.2025.1601795

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A fundamental feature of quantum mechanics is that there are observable pairs that cannot be measured jointly, such as observables corresponding to position and momentum or spin direction measurements. However, unsharp versions of non-jointly measurable observables may become jointly measurable. In this study, we investigate the joint measurability of time-continuous observables emerging from indirect time-continuous measurements. In particular, we study a paradigmatic situation where a qubit is interacting with a mode of light in a cavity, and the light escaping the cavity is continuously monitored. We find that the properties of the observables can be tuned by changing the type of the monitoring scheme or by tuning the initial state of the cavity. In particular, we demonstrate that heterodyne measurements are a joint measurement of a noisy homodyne measurement of a pair of canonical quadratures. Moreover, we investigate the purity of the induced qubit observables as a function of the noise.
A fundamental feature of quantum mechanics is that there are observables which can be measured jointly only when some noise is added to them. Their sharp versions are said to be incompatible. In this work we investigate time-continuous joint qubit observables induced by a indirect time-continuous measurements. In particular we study a paradigmatic situation where a qubit is interacting with a mode of light in a cavity and the light escaping the cavity is continuously monitored. We find that the properties of the qubit observables can be tuned by changing the type of the monitoring scheme or by tuning the initial state of the cavity. We observe that homodyning two orthogonal quadratures produces an optimal pair of biased jointly measurable qubit observables.
Joint measurements of multiple qubits open new possibilities for quantum information processing. Here we present an approach based on homodyne detection to realize such measurements in the dispersive regime of cavity or circuit QED. By changing details of the measurement, the readout can be tuned from extracting only single-qubit to extracting only multiqubit properties. We obtain a reduced stochastic master equation describing this measurement and its effect on the qubits. As an example, we present results showing parity measurements of two qubits. In this situation, measurement of an initially unentangled state can yield, with near unit probability, a state of significant concurrence.
Joint measurements of qubit observables have recently been studied in conjunction with quantum information processing tasks such as cloning. Considerations of such joint measurements have until now been restricted to a certain class of observables that can be characterized by a form of covariance. Here we investigate conditions for the joint measurability of arbitrary pairs of qubit observables. For pairs of noncommuting sharp qubit observables, a notion of approximate joint measurement is introduced. Optimal approximate joint measurements are shown to lie in the class of covariant joint measurements. The marginal observables found to be optimal approximators are generally not among the coarse-grainings of the observables to be approximated. This yields scope for the improvement of existing joint measurement schemes. Both the quality of the approximations and the intrinsic unsharpness of the approximators are shown to be subject to Heisenberg-type uncertainty relations.
Continuous weak measurements of quantum systems are of great relevance in quantum foundations and applications. They can be achieved by probing the quantum system of interest by repeatedly measuring an auxiliary system weakly coupled to it. Here we study two qubits coupled in different points to a common one-dimensional electromagnetic field and simultaneously monitored in the right- and left-propagating output channels by homodyne detection. Using a collision-model description, we derive an analytical Stochastic Master Equation (SME) governing the resulting diffusive quantum trajectories, including the interference between the two measurement channels. For nonlinear functions of the quantum state, such as entropies, averages over quantum trajectories generally differ from the corresponding quantities evaluated on the unconditional state. Through this mechanism, we show that continuous monitoring generates entanglement, absent in the unconditional dynamics, and enhances quantum magic in the qubit pair during the decay. Both resources can be tuned through the optical phase accumulated between the qubits and the phases of homodyne local oscillators. Our results establish continuous homodyne monitoring of multiple emitters as a tunable mechanism for generating quantum resources.
We discuss recent developments in measurement protocols that generate quantum entanglement between two remote qubits, focusing on the theory of joint continuous detection of their spontaneous emission. We consider a device geometry similar to that used in well-known Bell state measurements, which we analyze using a conceptually transparent model of stochastic quantum trajectories; we use this to review photodetection, the most straightforward case, and then generalize to the diffusive trajectories from homodyne and heterodyne detection as well. Such quadrature measurement schemes are a realistic two-qubit extension of existing circuit QED experiments, which obtain quantum trajectories by homodyning or heterodyning a superconducting qubit’s spontaneous emission, or an adaptation of existing optical measurement schemes to obtain jump trajectories from emitters. We mention key results, presented from within a single theoretical framework, and draw connections to concepts in the wider literature on entanglement generation by measurement (such as path information erasure and entanglement swapping). The photon which-path information acquisition, and therefore the two-qubit entanglement yield, is tunable under the homodyne detection scheme we discuss, at best generating equivalent average entanglement dynamics as in the comparable photodetection case. In addition to deriving this known equivalence, we extend past analyses in our characterization of the measurement dynamics: we include derivations of bounds on the fastest possible evolution toward a Bell state under joint homodyne measurement dynamics and characterize the maximal entanglement yield possible using inefficient (lossy) measurements.
We study a model of a qubit in interaction with the electromagnetic field. By means of homodyne detection, the field-quadrature A t + A * t is observed continuously in time. Due to the interaction, information about the initial state of the qubit is transferred to the field, thus influencing the homodyne measurement results. We construct random variables (pointers) on the probability space of homodyne measurement outcomes having distributions close to the initial distributions of σ x and σ z . Using variational calculus, we find the pointers that are optimal. These optimal pointers are very close to hitting the bound imposed by Heisenberg's uncertainty relation on joint measurement of two non-commuting observables. We close the paper by giving the probability densities of the pointers.