Abstract
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 that 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 nonorthogonal states, such as coherent states. Our derivation relies on two matrix inverses that, 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.
- Received 19 December 2020
- Accepted 1 April 2021
- Corrected 2 December 2021
DOI:https://doi.org/10.1103/PRXQuantum.2.020308
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.
Published by the American Physical Society
Physics Subject Headings (PhySH)
Corrections
2 December 2021
Correction: The subscripts to $\theta$ quantities in the sentence after Eq.(34) were erroneous and have been fixed.
Popular Summary
Anyone who has ever tried to diagonalize a matrix knows how difficult it can be—especially for larger matrices. In fact, Abel’s impossibility theorem states that no exact solution exists for general matrices of size and larger. In quantum metrology, the quantum Fisher information matrix (QFIM) is a central quantity that places fundamental limits on the measurement accuracy of quantum sensors. To compute exact solutions for the QFIM, one usually diagonalizes the density matrix (representing the quantum state) in an orthogonal basis—and this is difficult, as we know. But is this really necessary?
We show that one can not only avoid matrix diagonalization but also avoid expressing the density matrix in an orthogonal basis. The latter can be very convenient since the density matrix is often given in a nonorthogonal basis. We find a general solution to the QFIM that works for arbitrary nonorthogonal bases and only requires two matrices to be inverted for its computation. And anyone who has ever inverted a matrix knows that this works for matrices of any size using the same technique—a fundamental advantage over matrix diagonalization. We use our method to compute new exact results for the QFIM for the quantum imaging problem—the estimation of the location and intensity of discrete light sources, such as stars, based on quantum detection of the emitted light. Because of its generality, our method has the potential to be applied to many other problems of quantum metrology.
