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arXiv:1910.06462 · IEEE Conference on Decision and Control (CDC)

Active versus Passive Coherent Equalization of Passive Linear Quantum Systems

Valery Ugrinovskii, M. R. James

Published Dec 1, 2019• 2 citations
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Abstract

The paper considers the problem of equalization of passive linear quantum systems. While our previous work was concerned with the analysis and synthesis of passive equalizers, in this paper we analyze coherent quantum equalizers whose annihilation (respectively, creation) operator dynamics in the Heisenberg picture are driven by both quadratures of the channel output field. We show that the characteristics of the input field must be taken into consideration when choosing the type of the equalizing filter. In particular, we show that for thermal fields allowing the filter to process both quadratures of the channel output may not improve mean square accuracy of the input field estimate, in comparison with passive filters. This situation changes when the input field is `squeezed'.

DOI
10.1109/cdc40024.2019.9029390

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The paper considers the problem of equalization of passive linear quantum systems. While our previous work was concerned with the analysis and synthesis of passive equalizers, in this paper we analyze coherent quantum equalizers whose annihilation (respectively, creation) operator dynamics in the Heisenberg picture are driven by both quadratures of the channel output field. We show that the characteristics of the input field must be taken into consideration when choosing the type of the equalizing filter. In particular, we show that for thermal fields allowing the filter to process both quadratures of the channel output may not improve mean square accuracy of the input field estimate, in comparison with passive filters. This situation changes when the input field is `squeezed'.
This paper considers a version of the Wiener filtering problem for equalization of passive linear quantum systems. We demonstrate that taking into consideration the quantum nature of the signals involved leads to features typically not encountered in classical equalization problems. Most significantly, finding a mean-square optimal quantum equalizing filter amounts to solving a nonconvex constrained optimization problem. We discuss two approaches to solving this problem, both involving a relaxation of the constraint. In both cases, unlike classical equalization, there is a threshold on the variance of the noise below which an improvement of the mean-square error cannot be guaranteed.
The coherent equalization problem consists in designing a quantum system acting as a mean-square near-optimal filter for a given quantum communication channel. The paper develops an improved method for the synthesis of transfer functions for such equalizing filters, based on a linear quantum system model of the channel. The method draws on a connection with the two-disk problem of H ∞ control for classical (i.e., non-quantum) linear uncertain systems. Compared with the previous methods, the proposed method applies to a broader class of linear quantum communication channels.
This paper introduces a H∞-like methodology of coherent filtering for equalization of passive linear quantum systems to help mitigate degrading effects of quantum communication channels. For such systems, which include a wide range of linear quantum optical devices and signals, we seek to find a near optimal equalizing filter which is itself a passive quantum system. The problem amounts to solving an optimization problem subject to constraints dictated by the requirement for the equalizer to be physically realizable. By formulating these constraints in the frequency domain, we show that the problem admits a convex H∞-like formulation. This allows us to derive a set of suboptimal coherent equalizers using J-spectral factorization. An additional semidefinite relaxation combined with the Nevanlinna–Pick interpolation is shown to lead to a tractable algorithm for the design of a near optimal coherent equalizer.
The coherent equalization problem involves designing a quantum system that serves as a mean-square near-optimal filter for a given quantum communication channel. Previous work on coherent equalization has typically assumed that the system parameters are known without uncertainty. In this work, we address the problem of coherent passive equalization with uncertainty in the Hamiltonian. Utilizing an S-procedure method, we remodel the passive quantum system with uncertainty as a scaled closed-loop state-space representation with parameter dependencies. We then propose a two-disk synthesis approach for coherent passive equalizers. Numerical examples based on a single optical cavity system and on two-cavity systems with beam splitters demonstrate the effectiveness of the proposed two-disk synthesis method for coherent equalization of passive quantum systems.
We consider a coherent-classical estimation scheme for a class of linear quantum systems. It comprises an estimator that is a mixed quantum-classical system without involving coherent feedback. The estimator yields a classical estimate of a variable for the quantum plant. We demonstrate that for a passive plant that can be characterized by annihilation operators only, such coherent-classical estimation provides no improvement over purely-classical estimation. An example is also given which shows that if the plant is not assumed to be an annihilation operator only quantum system, it is possible to get better estimates with such coherent-classical estimation compared with purely-classical estimation.