General Expressions for the Quantum Fisher Information Matrix with Applications to Discrete Quantum Imaging
General Expressions for the Quantum Fisher Information Matrix with Applications to Discrete Quantum Imaging
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DOI：
10.1103/prxquantum.2.020308
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发表时间：
2020-12
期刊：
影响因子：
9.7
通讯作者：
Lukas J. Fiderer;T. Tufarelli;S. Piano;G. Adesso
中科院分区：
文献类型：
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作者：
Lukas J. Fiderer;T. Tufarelli;S. Piano;G. Adesso
量子Fisher信息矩阵是多参数量子估计理论中的一个核心问题。它通常是具有挑战性的，以获得其解析表达式，因为大多数计算方法依赖于对角化的密度矩阵。在本文中，我们推导出一般表达式的量子Fisher信息矩阵，绕过矩阵对角化，不需要展开的运营商的一组正交态。此外，我们可以处理任意秩的密度矩阵。这里提出的方法大大简化了分析计算，例如，密度矩阵更自然地表示在非正交状态，如相干态。我们的推导依赖于两个矩阵的逆，在原则上，可以分析评估，即使当密度矩阵是不可对角化的封闭形式。我们证明了我们的方法的力量，在离散量子成像的及时领域获得新的结果：非相干点源的位置和强度的估计。我们找到了两个不同强度的点源的完整估计问题的解析表达式，并与三个点源的具体例子。我们希望我们的方法将成为量子计量学的标准。
The quantum Fisher information matrix is a central object in multiparameter quantum estimation theory. It is usually challenging to obtain analytical expressions for it because most calculation methods rely on the diagonalization of the density matrix. In this paper, we derive general expressions for the quantum Fisher information matrix which bypass matrix diagonalization and do not require the expansion of operators on an orthonormal set of states. Additionally, we can tackle density matrices of arbitrary rank. The methods presented here simplify analytical calculations considerably when, for example, the density matrix is more naturally expressed in terms of non-orthogonal states, such as coherent states. Our derivation relies on two matrix inverses which, in principle, can be evaluated analytically even when the density matrix is not diagonalizable in closed form. We demonstrate the power of our approach by deriving novel results in the timely field of discrete quantum imaging: the estimation of positions and intensities of incoherent point sources. We find analytical expressions for the full estimation problem of two point sources with different intensities, and for specific examples with three point sources. We expect that our method will become standard in quantum metrology.