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arXiv:1201.0387 · Physical Review A

Optimal minimum-cost quantum measurements for imperfect detection

Erika Andersson

Published Jul 26, 2012• 6 citations
arXivHTMLPDF

Abstract

Knowledge of optimal quantum measurements is important for a wide range of situations, including quantum communication and quantum metrology. Quantum measurements are usually optimized with an ideal experimental realization in mind. Real devices and detectors are, however, imperfect. This has to be taken into account when optimizing quantum measurements. In this paper, we derive the optimal minimum-cost and minimum-error measurements for a general model of imperfect detection.

DOI
10.1103/physreva.86.012120

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Knowledge of optimal quantum measurements is important for a wide range of situations, including quantum communication and quantum metrology. Quantum measurements are usually optimized with an ideal experimental realization in mind. Real devices and detectors are, however, imperfect. This has to be taken into account when optimizing quantum measurements. In this paper, we derive the optimal minimum-cost and minimum-error measurements for a general model of imperfect detection.
Knowing about optimal quantum measurements is important for many applications in quantum information and quantum communication. However, deriving optimal quantum measurements is often difficult. We present a collection of results for minimum-cost quantum measurements, and give examples of how they can be used. Among other results, we show that a minimum-cost measurement for a set of given pure states is formally equivalent to a minimum-error measurement for mixed states of those same pure states. For pure symmetric states it turns out that for a certain class of cost matrices, the minimum-cost measurement is the square-root measurement. That is, the optimal minimum-cost measurement is in this case the same as the minimum-error measurement. Finally, we consider sequences of individual ``local" systems, and examine when the global minimum-cost measurement is a sequence of optimal local measurements. We also consider an example where the global minimum-cost measurement is, perhaps counter-intuitively, not a sequence of local measurements, and discuss how this is related to related to the Pusey-Barrett-Rudolph argument for the nature of the wave function.
The impact of measurement imperfections on quantum metrology protocols has not been approached in a systematic manner so far. In this work, we tackle this issue by generalising firstly the notion of quantum Fisher information to account for noisy detection, and propose tractable methods allowing for its approximate evaluation. We then show that in canonical scenarios involving N probes with local measurements undergoing readout noise, the optimal sensitivity depends crucially on the control operations allowed to counterbalance the measurement imperfections-with global control operations, the ideal sensitivity (e.g., the Heisenberg scaling) can always be recovered in the asymptotic N limit, while with local control operations the quantum-enhancement of sensitivity is constrained to a constant factor. We illustrate our findings with an example of NV-centre magnetometry, as well as schemes involving spin-1/2 probes with bit-flip errors affecting their two-outcome measurements, for which we find the input states and control unitary operations sufficient to attain the ultimate asymptotic precision.
Optimization of Quantum Measurements for Robustness Against Dark Counts: The DDD-Trace Optimality Framework
Quantum communication, while promising unparalleled security, faces significant practical challenges due to imperfections in quantum devices, particularly in single-photon detectors (SPDs). One of the key issues is the impact of dark counts, which cause erroneous detections and limit the effective range of quantum communication systems, such as in quantum key distribution (QKD). In this paper, we introduce a novel framework for addressing dark count errors through a new optimality criterion, termed DDD-trace optimality. This criterion aims to optimize quantum measurements to minimize the impact of dark counts without relying on hardware replacements. We propose an optimization scheme that transforms general measurements into DDD-trace optimal measurements, ensuring robustness against dark count effects. Our results suggest that, under certain conditions, this approach can improve the performance of quantum communication systems, especially in scenarios where the prior distribution of states is known. Although challenges remain in obtaining accurate prior distributions and ensuring compatibility with other strategies. This work lays the foundation for future research on optimizing quantum measurements to mitigate the effects of noise, including dark counts, in practice.
Security analyses in quantum key distribution (QKD) and other adversarial quantum tasks often assume perfect device models. However, real-world implementations often deviate from these models. Thus, it is important to develop security proofs that account for such deviations from ideality. In this work, we extend the idea of squashing maps to develop a general framework for analysing imperfect threshold detectors, treating uncharacterised device parameters such as dark counts and detection efficiencies as adversarially controlled within some ranges. This approach enables a rigorous worst-case analysis with exactly characterised devices, ensuring security proofs remain valid under realistic conditions. Our results strengthen the connection between theoretical security and practical implementations by introducing a flexible framework for integrating detector imperfections into adversarial quantum protocols.
This work considers the theory underlying a discrete-time quantum filter recently used in a quantum feedback experiment. It proves that this filter taking into account decoherence and measurement errors is optimal and stable. We present the general framework underlying this filter and show that it corresponds to a recursive expression of the least-square optimal estimation of the density operator in the presence of measurement imperfections. By measurement imperfections, we mean in a very general sense unread measurement performed by the environment (decoherence) and active measurement performed by non-ideal detectors. However, we assume to know precisely all the Kraus operators and also the detection error rates. Such recursive expressions combine well known methods from quantum filtering theory and classical probability theory (Bayes' law). We then demonstrate that such a recursive filter is always stable with respect to its initial condition: the fidelity between the optimal filter state (when the initial filter state coincides with the real quantum state) and the filter state (when the initial filter state is arbitrary) is a sub-martingale.