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Register a new userMeasure of quantum Fisher information flow in multi-parameter scenario
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We generalize the quantum Fisher information flow proposed by Lu textit{et al}. [Phys. Rev. A textbf{82}, 042103 (2010)] to the multi-parameter scenario from the information geometry perspective. A measure named the textit{intrinsic density flow} (IDF) is defined with the time-variation of the intrinsic density of quantum states (IDQS). IDQS measures the local distinguishability of quantum states in state manifolds. The validity of IDF is clarified with its vanishing under the parameter-independent unitary evolution and outward-flow (negativity) under the completely positive-divisible map. The temporary backflow (positivity) of IDF is thus an essential signature of the non-Markovian dynamics. Specific for the time-local master equation, the IDF decomposes according to the channels, and the positive decay rate indicates the inwards flow of the sub-IDF. As time-dependent scalar fields equipped on the state space, the distribution of IDQS and IDF comprehensively illustrates the distortion of state space induced by its environment. As example, a typical qubit model is given.
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This review aims at gathering the most relevant quantum multi-parameter estimation methods that go beyond the direct use of the Quantum Fisher Information concept. We discuss in detail the Holevo Cramer-Rao bound, the Quantum Local Asymptotic Normality approach as well as Bayesian methods. Even though the fundamental concepts in the field have been laid out more than forty years ago, a number of important results have appeared much more recently. Moreover, the field drew increased attention recently thanks to advances in practical quantum metrology proposals and implementations that often involve estimation of multiple parameters simultaneously. Since these topics are spread in the literature and often served in a very formal mathematical language, one of the main goals of this review is to provide a largely self-contained work that allows the reader to follow most of the derivations and get an intuitive understanding of the interrelations between different concepts using a set of simple yet representative examples involving qubit and Gaussian shift models.
Recently, there is a growing interest in study quantum mechanics from the information geometry perspective, where a quantum state is depicted with a point in the projective Hilbert space. By taking quantum Fisher information (QFI) as the metric of projective Hilbert spaces, estimating a small parameter shift is equivalent to distinguishing neighboring quantum states along a given curve. Henceforth, information geometry plays a significant role in the single parameter estimation. However, the absence of high dimensional measures limits its applications in studying the multi-parameter systems. In this paper, we will discuss the physical implications of the volume element of QFI. It measures the intrinsic density of quantum states (IDQS) in projective Hilbert spaces, which is, then, a measure to define the (over) completeness relation of a class of quantum states. As an application, IDQS can be used in quantum measurement and multi-parameter estimation. We find the density of distinguishable states (DDS) for a set of efficient estimators is measured by the invariant volume of the classical Fisher information, which is the classical counterpart of QFI and serves as the metric of statistical manifolds. Correspondingly, a determinant form of quantum Cram{e}r-Rao inequality is proposed to quantify the ability to infer the IDQS via quantum measurement. As a result, we find a gap between IDQS and maximal DDS over the measurements. The gap has tight connections with the uncertainty relationship. Exemplified with the three-level system with two parameters, we find the maximal DDS attained via the emph{vertex measurements} (MvDDS) equals the square root of the quantum geometric tensors determinant. It indicates the square gap between IDQS and MvDDS is proportional to the square of Berry curvature.
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.
In estimating an unknown parameter of a quantum state the quantum Fisher information (QFI) is a pivotal quantity, which depends on the state and its derivate with respect to the unknown parameter. We prove the continuity property for the QFI in the sense that two close states with close first derivatives have close QFIs. This property is completely general and irrespective of dynamics or how states acquire their parameter dependence and also the form of parameter dependence---indeed this continuity is basically a feature of the classical Fisher information that in the case of the QFI naturally carries over from the manifold of probability distributions onto the manifold of density matrices. We demonstrate that in the special case where the dependence of the states on the unknown parameter comes from one dynamical map (quantum channel), the continuity holds in its reduced form with respect to the initial states. In addition, we show that when one initial state evolves through two different quantum channels, the continuity relation applies in its general form. A situation in which such scenario can occur is an open-system metrology where one of the maps represents the ideal dynamics whereas the other map represents the real (noisy) dynamics. In the making of our main result, we also introduce a regularized representation for the symmetric logarithmic derivative which works for general states even with incomplete rank, and its features continuity similarly to the QFI.
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.
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