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arXiv:1010.0442 · Physical Review A

Measurement of damping and temperature: Precision bounds in Gaussian dissipative channels

Alex Monràs, Fabrizio Illuminati

Published Jan 24, 2011• 92 citations
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Abstract

We present a comprehensive analysis of the performance of different classes of Gaussian states in the estimation of Gaussian phase-insensitive dissipative channels. In particular, we investigate the optimal estimation of the damping constant and reservoir temperature. We show that, for two-mode squeezed vacuum probe states, the quantum-limited accuracy of both parameters can be achieved simultaneously. Moreover, we show that for both parameters two-mode squeezed vacuum states are more efficient than coherent, thermal, or single-mode squeezed states. This suggests that at high-energy regimes, two-mode squeezed vacuum states are optimal within the Gaussian setup. This optimality result indicates a stronger form of compatibility for the estimation of the two parameters. Indeed, not only the minimum variance can be achieved at fixed probe states, but also the optimal state is common to both parameters. Additionally, we explore numerically the performance of non-Gaussian states for particular parameter values to find that maximally entangled states within $d$-dimensional cutoff subspaces ($d\ensuremath{\leqslant}6$) perform better than any randomly sampled states with similar energy. However, we also find that states with very similar performance and energy exist with much less entanglement than the maximally entangled ones.

DOI
10.1103/physreva.83.012315

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We construct a practical method for finding optimal Gaussian probe states for the estimation of parameters encoded by Gaussian unitary channels. This method can be used for finding all optimal probe states, rather than focusing on the performance of specific states as shown in previous studies. As an example, we apply this method to find optimal probes for the channel that combines the phase-change and squeezing channels, and for generalized two-mode squeezing and mode-mixing channels. The method enables a comprehensive study of temperature effects in Gaussian parameter estimation. It has been shown that the precision in parameter estimation using single-mode states can be enhanced by increasing the temperature of the probe. We show that not only higher temperature, but also larger temperature differences between modes of a Gaussian probe state can enhance the estimation precision.
The central issue in quantum parameter estimation is to find out the optimal measurement setup that leads to the ultimate lower bound of an estimation error. We address here a question of whether a Gaussian measurement scheme can achieve the ultimate bound for phase estimation in single-mode Gaussian metrology that exploits single-mode Gaussian probe states in a Gaussian environment. We identify three types of optimal Gaussian measurement setups yielding the maximal Fisher information depending on displacement, squeezing, and thermalization of the probe state. We show that the homodyne measurement attains the ultimate bound for both displaced thermal probe states and squeezed vacuum probe states, whereas for the other single-mode Gaussian probe states, the optimized Gaussian measurement cannot be the optimal setup, although they are sometimes nearly optimal. We then demonstrate that the measurement on the basis of the product quadrature operators XP+PX, i.e., a non-Gaussian measurement, is required to be fully optimal.
Abstract The central issue in quantum parameter estimation is to find out the optimal measurement setup that leads to the ultimate lower bound of an estimation error. We address here a question of whether a Gaussian measurement scheme can achieve the ultimate bound for phase estimation in single-mode Gaussian metrology that exploits single-mode Gaussian probe states in a Gaussian environment. We identify three types of optimal Gaussian measurement setups yielding the maximal Fisher information depending on displacement, squeezing, and thermalization of the probe state. We show that the homodyne measurement attains the ultimate bound for both displaced thermal probe states and squeezed vacuum probe states, whereas for the other single-mode Gaussian probe states, the optimized Gaussian measurement cannot be the optimal setup, although they are sometimes nearly optimal. We then demonstrate that the measurement on the basis of the product quadrature operators X^P^+P^X^\hat X\hat P + \hat P\hat XX^P^+P^X^ X ̂ P ̂ + P ̂ X ̂ , i.e., a non-Gaussian measurement, is required to be fully optimal.
We study temperature estimation using quantum probes, including single-mode initial states and two-mode states generated via stimulated parametric down-conversion in a nonlinear crystal at finite temperature. We explore both transient and equilibrium regimes and compare the performance of Gaussian and non-Gaussian probe states for temperature estimation. In the nonequilibrium regime, we show that single-mode non-Gaussian probe states---such as Fock, odd cat, and Gottesman-Kitaev-Preskill states---can significantly enhance the speed of estimation, particularly at short interaction times. In the two-mode setting, entangled states such as the two-mode squeezed vacuum, NOON state, and entangled cat state can enable access to temperature information at earlier times. In the equilibrium regime, we analyze temperature estimation using two-mode squeezed thermal states, which outperform single-mode strategies. We evaluate practical measurement strategies and find that energy-based observables yield optimal precision, population difference observables provide near-optimal precision, while quadrature-based measurements are suboptimal. The precision gain arises from squeezing, which suppresses fluctuations in the population difference.
The major problem of multiparameter quantum estimation theory is to find an ultimate measurement scheme to go beyond the standard quantum limits that each quasi-classical estimation measurement is limited by. Although, in some specifics quantum protocols without environmental noise, the ultimate sensitivity of a multiparameter quantum estimation can beat the standard quantum limit. However, the presence of noise imposes limitations on the enhancement of precision due to the inevitable existence of environmental fluctuations. Here, we address the motivation behind the usage of Gaussian quantum resources and their advantages in reaching the standard quantum limits under realistic noise. In this context, our work aims to explore the ultimate limits of precision for the simultaneous estimation of a pair of parameters that characterize the displacement channel acting on Gaussian probes and subjected to open dynamics. More precisely, we focus on a general two-mode mixed squeezed displaced thermal state, after reducing it to various Gaussian probes states, like; a two-mode pure squeezed vacuum, two-mode pure displaced vacuum, two-mode mixed displaced thermal, two-mode mixed squeezed thermal. To study the ultimate estimation precision, we evaluate the upper and bottom bound of HCRB in various cases. We find that when the entangled states, two-mode pure squeezed vacuum and two-mode mixed squeezed thermal, are employed as probes states, the upper and bottom bound of HCRB beats the standard quantum limit in the presence of a noisy environment.
Target reflectivity estimation using a two-mode squeezed vacuum (TMSV) probe offers a theoretical advantage over classical schemes, but realizing this potential under the measurement constraints of microwave platforms remains a central challenge. In this work, we study the precision limits for target reflectivity estimation across different energy and loss regimes, while accounting for realistic measurement restrictions. We characterize the optimal measurements and identify a transition in their structure: above a specific reflectivity threshold, a parametric amplifier receiver is optimal, whereas below it, the optimal observables are two-mode squeezing generators. We then study the performance of Gaussian measurements. When restricted to standard local homodyne detection, the TMSV probe is highly non-optimal. However, we show that suitable non-local Gaussian measurements can closely approach the quantum Cramér-Rao bound at the large noise limit. These results demonstrate that near-optimal quantum target reflectivity estimation is achievable in various relevant noisy regimes, even under the restriction of Gaussian measurements.