Displacement based quantum receivers have gained increasing attention because they can beat the standard quantum limit. The displacement value in quantum receivers can be calibrated by using number states as probes. Current estimating methods assumed that ideal number states can be prepared perfectly, which is impossible in practical implementations. In this work, we focus on the impacts of thermal noise when preparing the number states on the estimation performance of displacement operations. We first derive the probability density of photon numbers in a thermal noise contaminated number state after a displacement operation, which we call the noisy displaced number state (NDNS). Based on the derived probability density, we then analyze the performance of estimating displacement operations with NDNS probes by using the maximum likelihood estimation (MLE) method. Numerical results found that the thermal noise in preparing the number states has great influence on the MLE accuracy of the displacement operations.
Displacement based quantum receivers have gained increasing attention because they can beat the standard quantum limit. The displacement value in quantum receivers can be calibrated by using number states as probes. Current estimating methods assumed that ideal number states can be prepared perfectly, which is impossible in practical implementations. In this work, we focus on the impacts of thermal noise when preparing the number states on the estimation performance of displacement operations. We first derive the probability density of photon numbers in a thermal noise contaminated number state after a displacement operation, which we call the noisy displaced number state (NDNS). Based on the derived probability density, we then analyze the performance of estimating displacement operations with NDNS probes by using the maximum likelihood estimation (MLE) method. Numerical results found that the thermal noise in preparing the number states has great influence on the MLE accuracy of the displacement operations.
Displacement based quantum receivers can beat the standard quantum limit and have gained increasing attention in recent years. The displacement operation value can be estimated and calibrated by using Fock states as probes. In this Letter, we focus on the impacts of thermal noise when preparing the Fock probes on the estimation performance. We first derive the density matrix and the photon statistics of the noisy displaced number state with a uniformly distributed displacement phase. Based on the derived density matrix, we then analyze the performance of estimating displacement operations by using the maximum likelihood estimation (MLE) method. To decrease the computational complexity of the MLE, we further derive an approximated probability density of the photon numbers under small displacement. Besides, the quantum Cramér–Rao bound (QCRB) is also derived to analyze the performance limit of the displacement estimation. Numerical results found that, though thermal noise can greatly degrade the MLE accuracy of the displacement operations, the performance advantage of using Fock probes still persists over classical coherent probes. Besides, we demonstrate that the MLE method based on Fock probes can well approach the QCRB when the displacement strength N c ≤ ~1.
The performance of Kennedy receiver with threshold detection is analyzed in the presence of thermal noise. A lower bound is first derived for the error probability of the Kennedy receiver in the presence of thermal noise and it is shown that the performance of the Kennedy receiver is greatly affected by the thermal noise. Then we analyze the performance of the Kennedy receiver using the maximum a posteriori detection, which is proven to be equivalent to a threshold detection. We also optimize the displacement operation to improve further the receiver performance. Numerical results show that the Kennedy receiver with threshold detection can surpass the standard quantum limit achieved by homodyne detection when the average number of thermal photons is smaller than 0.1. The Kennedy receiver with threshold detection using optimized displacement can surpass the standard quantum limit when the average number of thermal photons is smaller than 0.2.
Near-optimal binary coherent state discrimination can be achieved by performing a displacement operation followed by photon detection. This displacement serves as a control parameter that can be tuned to optimize a chosen figure of merit, which makes such receivers attractive beyond state discrimination. Here, we turn this discrimination strategy into a complete continuous-variable quantum key distribution (CV-QKD) protocol based on a binary alphabet of noisy coherent states (displaced thermal states) and displacement receivers. We develop a strategy for estimating the channel parameters with the same displacement receiver, since standard quadrature-based methods do not apply. While a key map can be obtained with on/off detection, we demonstrate that reliable parameter estimation requires photon-number-resolving detection with sufficient resolution. Finite detector resolution underestimates the observed noise and constitutes a potential security loophole, which we quantify and show to be mitigable with modest resolution. We analyze the protocol's security against entangling-cloner collective attacks in the asymptotic regime, for both direct and reverse reconciliation, and show that it can generate secret keys even under noisy state preparation and imperfect detection. Interestingly, we find that the displacement that minimizes the discrimination error probability also performs well for key generation, thus providing a useful and analytically tractable tuning strategy for the CV-QKD receiver. We evaluate the protocol's performance over a range of transmittances and noise parameters, showing that the displacement receiver can outperform conventional homodyne detection in experimentally relevant regimes. Together, our results point to displacement receivers as a promising alternative for CV-QKD implementations.
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.
Photon-number-resolving detectors play a crucial role in quantum optics and quantum information processing. Their characterization is typically expressed through a positive operator-valued measure (POVM), making accurate estimation methods that reliably link the POVM to measurement outcomes essential. However, shorter measurement times result in fewer detection events, rendering photon counting measurements highly susceptible to shot noise. Consequently, statistical fluctuations limit estimation accuracy. In this study, we numerically investigate the impact of limited data on POVM estimation through simulations. We compare two estimation approaches: least-squares-based convex optimization and maximum likelihood estimation. By incorporating appropriately chosen constraints, convex optimization mitigates shot-noise-induced instabilities. In contrast, maximum likelihood estimation, although limited, achieves superior performance without relying on such additional constraints. To validate these findings, we conduct experiments that use a superconducting transition-edge sensor as a photon-number-resolving detector. The experimental results show strong agreement with numerical predictions, confirming the reliability of our simulation framework.
Thermal radiation played a pivotal role in the preliminary development of quantum physics where it helped resolve the apparent incongruity of the ultraviolet catastrophe. In contemporary physics, thermal state generation and manipulation finds new application in fields such as quantum imaging and quantum illumination and as a practical realization of Maxwell's demon. These applications often go hand in hand with photon subtraction operations which probabilistically amplify the mean photon number (MPN) of thermal light as a result of its super-Poissonian photon statistics. In this article, we introduce an operation for thermal states of light based on a generalized photon subtraction scheme. Displaced-photon subtraction (DPS) makes use of coherent state displacement followed by a subsequent anti-displacement in combination with single-photon detection to probe the MPN of a thermal state. We find regimes in which the output of a successful DPS is amplified, unchanged, or attenuated relative to the unconditioned output state. The regime of operation is controlled via the magnitude of the coherent displacement. A theoretical description of generalized photon subtraction of a displaced thermal state is derived via a two-mode moment-generating function (MGF) and used to describe generalized DPS. We perform a proof of principle experimental implementation of DPS for the case of a balanced beam splitter for which results demonstrate good agreement with the model.