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arXiv:2603.16821

Measurement-Based Estimation of Causal Conditional Variances and Its Application to Macroscopic quantum phenomenon

Kosei Hatakeyama, Ryotaro Fukuzumi, Akira Matsumura, Daisuke Miki, Kazuhiro Yamamoto

quant-phgr-qcSubmitted Mar 17, 2026• Announced Mar 18, 2026• 0 citations
v1 · Mar 17, 2026v2 · Mar 18, 2026
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Abstract

We analytically investigate a quantum estimation method for a mechanical oscillator in a detuned cavity system based solely on homodyne measurement records, building on the framework developed by C.Meng et al. (Science Advances 8, 7585 (2022)). Estimation based only on measurement records is important because it enables state verification without assuming knowledge of the true system state. We construct a relative estimate operator from causal and anti-causal quantum Wiener filters and calculate its variance. The deviation from the causal conditional variance is defined as a reconstruction bias, whose magnitude is evaluated analytically. We show that, within experimentally relevant parameter regimes for typical quantum-state preparation, the reconstruction bias is sufficiently small to be neglected. As applications to state verification, we apply the method to proposals for macroscopic quantum entanglement mediated by electromagnetic interactions and for conditional momentum-squeezed states generated by homodyne detection, and clarify the conditions under which the bias remains negligible and when the reconstruction bias becomes significant.

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16 pages, 6 figures

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We analytically investigate a quantum estimation method for a mechanical oscillator in a detuned cavity system based solely on homodyne measurement records, building on the framework developed by C.Meng et al. (Science Advances 8, 7585 (2022)). Estimation based only on measurement records is important because it enables state verification without assuming knowledge of the true system state. We construct a relative estimate operator from causal and anti-causal quantum Wiener filters and calculate its variance. The deviation from the causal conditional variance is defined as a reconstruction bias, whose magnitude is evaluated analytically. We show that, within experimentally relevant parameter regimes for typical quantum-state preparation, the reconstruction bias is sufficiently small to be neglected. As applications to state verification, we apply the method to proposals for macroscopic quantum entanglement mediated by electromagnetic interactions and for conditional momentum-squeezed states generated by homodyne detection, and clarify the conditions under which the bias remains negligible and when the reconstruction bias becomes significant.
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The observables of a noisy quantum system can be estimated by appropriately filtering the records of their continuous measurement. Such filtering is relevant for state estimation, and if the filter is causal, also relevant for measurement-based feedback control. It is therefore imperative that a pair of conjugate observables estimated causally satisfy the Heisenberg uncertainty principle. In this article, we prove this fact---without assuming Markovian dynamics or Gaussian noises, in the presence or absence of feedback control of the system, and where in the feedback loop (inside or outside) the measurement record is accessed. Indeed, causal estimators using the in-loop measurement record can be as precise as those using the out-of-loop record. These results clarify the role of causal estimators to non-Markovian quantum systems, restore the equanimity of in-loop and out-of-loop measurements in their estimation and control, and simplify future experiments on measurement-based quantum feedback control.
Shadow estimation provides an efficient framework for estimating observable expectation values using randomized measurements. While originally developed for discrete-variable systems, its recent extensions to continuous-variable (CV) quantum systems face practical limitations due to idealized assumptions of continuous phase modulation and infinite measurement resolution. In this work, we develop a practical shadow estimation protocol for CV systems using discretized homodyne detection with a finite number of phase settings and quadrature bins. We construct an unbiased estimator for the quantum state and establish both sufficient conditions and necessary conditions for informational completeness within a truncated Fock space up to nmaxn_{\mathrm{max}}nmax​ photons. We further provide a comprehensive variance analysis, showing that the shadow norm scales as O(nmax4)\mathcal{O}(n_{\mathrm{max}}^4)O(nmax4​), improving upon previous O(nmax13/3)\mathcal{O}(n_{\mathrm{max}}^{13/3})O(nmax13/3​) bounds. Our work bridges the gap between theoretical shadow estimation and experimental implementations, enabling robust and scalable quantum state characterization in realistic CV systems.
We analyze the conditional quantum state of a mechanical mirror in an optomechanical system subject to continuous measurement, feedback control, and quantum filtering. We identify a parameter regime in which the mirror exhibits momentum squeezing beyond the standard quantum limit, achieved through an appropriate choice of the homodyne detection angle. In this regime, we show that optimal filtering effectively realizes a free-particle-like conditional state. When this mechanism is applied to a configuration consisting of two optomechanical systems, the resulting momentum squeezing significantly enhances the signal of gravity-induced entanglement. This enhancement arises because the momentum squeezing not only amplifies the distinction between the common and differential modes, but also, in the high-purity regime, increases the position uncertainty in accordance with the uncertainty principle, thereby enlarging the spatial extent of the quantum superposition. Our results provide insights into experimental strategies for probing the quantum nature of gravity using optomechanical platforms.
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What knowledge can be obtained from the record of a continuous measurement about the quantum state of the measured system at the beginning of the measurement? The task of quantum state retrodiction, the inverse of the more common state prediction, is rigorously addressed in quantum measurement theory through retrodictive positive operator-valued measures (POVMs). This introduction to this general framework presents its practical formulation for retrodicting Gaussian quantum states using continuous-time homodyne measurements and applies it to optomechanical systems. We identify and characterize achievable retrodictive POVMs in common optomechanical operating modes with resonant or off-resonant driving fields and specific choices of local oscillator frequencies in homodyne detection. In particular, we demonstrate the possibility of a near-ideal measurement of the quadrature of the mechanical oscillator, giving direct access to the position or momentum distribution of the oscillator at a given time. This forms the basis for complete quantum state tomography, albeit in a destructive manner.