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The Quantum Fisher Information (QFI) plays a crucial role in quantum information theory and in many practical applications such as quantum metrology. However, computing the QFI is generally a computationally demanding task. In this work we analyze a lower bound on the QFI which we call the sub-Quantum Fisher Information (sub-QFI). The bound can be efficiently estimated on a quantum computer for an $n$-qubit state using $2n$ qubits. The sub-QFI is based on the super-fidelity, an upper bound on Uhlmanns fidelity. We analyze the sub-QFI in the context of unitary families, where we derive several crucial properties including its geometrical interpretation. In particular, we prove that the QFI and the sub-QFI are maximized for the same optimal state, which implies that the sub-QFI is faithful to the QFI in the sense that both quantities share the same global extrema. Based on this faithfulness, the sub-QFI acts as an efficiently computable surrogate for the QFI for quantum sensing and quantum metrology applications. Finally, we provide additional meaning to the sub-QFI as a measure of coherence, asymmetry, and purity loss.
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The dynamics of two variants of quantum Fisher information under decoherence are investigated from a geometrical point of view. We first derive the explicit formulas of these two quantities for a single qubit in terms of the Bloch vector. Moreover, we obtain analytical results for them under three different decoherence channels, which are expressed as affine transformation matrices. Using the hierarchy equation method, we numerically study the dynamics of both the two information in a dissipative model and compare the numerical results with the analytical ones obtained by applying the rotating-wave approximation. We further express the two information quantities in terms of the Bloch vector for a qudit, by expanding the density matrix and Hermitian operators in a common set of generators of the Lie algebra $mathfrak{su}(d)$. By calculating the dynamical quantum Fisher information, we find that the collisional dephasing significantly diminishes the precision of phase parameter with the Ramsey interferometry.
Quantum Fisher information, as an intrinsic quantity for quantum states, is a central concept in quantum detection and estimation. When quantum measurements are performed on quantum states, classical probability distributions arise, which in turn lead to classical Fisher information. In this article, we exploit the classical Fisher information induced by quantum measurements, and reveal a rich hierarchical structure of such measurement-induced Fisher information. We establish a general framework for the distribution and transfer of the Fisher information. In particular, we illustrate three extremal distribution types of the Fisher information: the locally owned type, the locally inaccessible type, and the fully shared type. Furthermore, we indicate the significant role played by the distribution and flow of the Fisher information in some physical problems, e.g., the non-Markovianity of open quantum processes, the environment-assisted metrology, the cloning and broadcasting, etc.
It is well known that a suggestive relation exists that links Schrodingers equation (SE) to the information-optimizing principle based on Fishers information measure (FIM). The connection entails the existence of a Legendre transform structure underlying the SE. Here we show that appeal to this structure leads to a first order differential equation for the SEs eigenvalues that, in certain cases, can be used to obtain the eigenvalues without explicitly solving SE. Complying with the above mentioned equation constitutes a necessary condition to be satisfied by an energy eigenvalue. We show that the general solution is unique.
In recent proposals for achieving optical super-resolution, variants of the Quantum Fisher Information (QFI) quantify the attainable precision. We find that claims about a strong enhancement of the resolution resulting from coherence effects are questionable because they refer to very small subsets of the data without proper normalization. When the QFI is normalized, accounting for the strength of the signal, there is no advantage of coherent sources over incoherent ones. Our findings have a bearing on further studies of the achievable precision of optical instruments.
We provide a detailed analysis of the question: how many measurement settings or outcomes are needed in order to identify a quantum system which is constrained by prior information? We show that if the prior information restricts the system to a set of lower dimensionality, then topological obstructions can increase the required number of outcomes by a factor of two over the number of real parameters needed to characterize the system. Conversely, we show that almost every measurement becomes informationally complete with respect to the constrained set if the number of outcomes exceeds twice the Minkowski dimension of the set. We apply the obtained results to determine the minimal number of outcomes of measurements which are informationally complete with respect to states with rank constraints. In particular, we show that 4d-4 measurement outcomes (POVM elements) is enough in order to identify all pure states in a d-dimensional Hilbert space, and that the minimal number is at most 2 log_2(d) smaller than this upper bound.
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