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Conference on Lasers and Electro-Optics Pacific Rim (CLEO-PR)

On the Constraints in Convex Optimization to Estimate POVM Elements of a Photon-Number-Resolving Detector from Coherent-State Inputs

Akio Yoshizawa, Daiji Fukuda

Published Jul 31, 2022• 0 citations
Publisher

Abstract

Smoothing regulation in constrained convex optimization is numerically studied to well estimate POVM elements of a photon-number-resolving detector from coherent-state inputs. Smoothing itself should be optimized with the detection efficiency in mind.
DOI
10.1109/cleo-pr62338.2022.10432202

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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.
The problem addressed is to design a detector which is maximally sensitive to specific quantum states. Here we concentrate on quantum state detection using the worst-case a posteriori probability of detection as the design criterion. This objective is equivalent to asking the question: if the detector declares that a specific state is present, what is the probability of that state actually being present? We show that maximizing this worst-case probability (maximizing the smallest possible value of this probability) is a quasiconvex optimization over the matrices of the POVM (positive operator valued measure) which characterize the measurement apparatus. We also show that with a given POVM, the optimization is quasiconvex in the matrix which characterizes the Kraus operator sum representation (OSR) in a fixed basis. We use Lagrange Duality Theory to establish the optimality conditions for both deterministic and randomized detection. We also examine the special case of detecting a single pure state. Numerical aspects of using convex optimization for quantum state detection are also discussed.
Quantum enhancements of precision in metrology can be compromised by system imperfections. These may be mitigated by appropriate optimization of the input state to render it robust, at the expense of making the state difficult to prepare. In this paper, we identify the major sources of imperfection of an optical sensor: input state preparation inefficiency, sensor losses, and detector inefficiency. The second of these has received much attention; we show that it is the least damaging to surpassing the standard quantum limit in a optical interferometric sensor. Further, we show that photonic states that can be prepared in the laboratory using feasible resources allow a measurement strategy using photon-number-resolving detectors that not only attain the Heisenberg limit for phase estimation in the absence of losses, but also deliver close to the maximum possible precision in realistic scenarios including losses and inefficiencies. In particular, we give bounds for the tradeoff between the three sources of imperfection that will allow true quantum-enhanced optical metrology