# Interpreting the Quantum Fisher Information

I recently saw this article in Nature Physics regarding Quantum Fisher Information (QFI):

http://www.nature.com/nphys/journal/v12/n8/full/nphys3700.html

Frankly I have not been following Quantum Information for a number of years now, and have never heard of even classical Fisher information.

What I was surprised to see was that there is a connection between the QFI and an energy integral over a response function (e.g. Optical conductivity).

I have the following questions:

1. What is the motivation for defining (Q)FI the way it is?
2. How do you interpret QFI, especially with regards to measuring entanglement?
3. Is there a prototypical model Hamiltonian/system that has nontrivial information encoded in its QFI?

My background in measure theory and the like is lacking, so please don't throw around too many definitions and feel free to be non-rigorous.

General intuition: The classical fisher information (CFI) is a measure of how quickly a probability distribution changes with respect to some parameter. While the quantum fisher information(QFI) is how quickly a quantum state (represented by a density matrix) changes with respect to some parameter. To define such a measure, one needs to a define a distance on the manifold of probability distributions or quantum states (Projective Hilbert Space). For a probability distribution such a metric can be fixed by a set of subtle mathematical assumptions but in general the direct expression for the fisher information is more illuminating:

$$F_c(\theta)=\sum_x P(x,\theta)[\frac{1}{P(x,\theta)}\frac{d}{d\theta}P(x,\theta)]^2$$

This is the average of the percent change of each $$P(x,\theta)$$ for a small amount of change in $$\theta$$. The percent change, $$\frac{1}{P(x,\theta)}\frac{d}{d\theta}P(x,\theta)$$ is squared because otherwise the average would be 0. The QFI can be constructed from this expression. First, notice that to obtain a probability distribution of a quantum state, you must measure in a particular basis, say $$R$$. Then by optimizing the CFI over that observable R we obtain the quantum fisher information.

$$F_q(\theta)=max_R F_c(\rho(\theta),R)$$

By thinking about the Brunes Metric and specifying how $$\rho$$ depends on $$\theta$$ one can relate this expression to one easy for calculations. If we define $$\rho(\theta)=e^{iQ\theta}\rho e^{-iQ\theta}$$, it's possible to derive:

$$F_q(Q,\theta=0)= 2\sum_{l,l'}\frac{(p_{l}-p_{l'})^{2}}{p_{l}+p_{l'}}\left|\left\right|^2$$ where $$\left| l\right>$$ are the eigenvectors of the density matrix and $$p_{l}$$ are the eigenvalues.

I don't know this proof so you'll have to look for articles if you interested. But it's not necessary for interpretation. Simply from the optimization of the CFI, you can see it is some quantification of how quickly a quantum state changes with respect to the parameter $$\theta$$.

Now the article you linked makes really only a few substitutions to connect the QFI to the linear response of a thermal state $$\rho=e^{-\beta H}/Z$$ to a perturbation $$Q$$ driven at various frequencies. Thinking about the QFI as quantifying how quickly a state changes with respect to some operation Hamiltonian $$Q$$ gives some intuition about why it might be related to response functions.

Entanglement Relating the QFI to entanglement is done in just a few steps.

1st) one uses the fact that the QFI of a pure state is 4 times the variance of $$Q$$: $$F_q(Q,\theta=0)=4(\left-\left^2)$$ (Easily proved from the above expression).

2nd) one uses the fact that the quantum fisher information is convex in the space of density matricies. That is if I have two pure states $$\rho_a=\left|a\right>\left and $$\rho_b=\left|b\right>\left the convex sum of them has a lower QFI:

$$F_q(p\rho_a+(1-p)\rho_b)

for $$0 and any $$Q$$. Thus for a mixed state, the QFI will be limited by the QFI of the eigenstate with maximum variance in Q.

The final step is to bound the variance of an unentangled state for a particular $$Q$$. Suppose I break my system into $$N$$ parts for which I want to quantify the entanglement between. For example, I could take the parts as the $$N$$ qubits in a quantum computer or the $$N$$ particles in a system. I will now choose $$Q$$ to be a sum of observable of the individual parts: $$Q=\sum_{i=1}^N Q_i$$. I will also assume these operators to be bounded and have norm $$|Q_i|=1$$(magnitude of the maximum eigenvalue of $$Q_i$$ is 1). For a untangled pure state,the statistics of the parts are independent so the maximum variance is $$N$$. Due to convexity, the maximum fisher information for a unentangled mixed state is then 4N. Therefore a fisher information $$F_q>4N$$ is a witness of entanglement.

There is also has been work to argue that $$\max_Q F_q(Q)/4N$$ is a approximation of the number of parts which must be mutually entangled. I can't find the work that does this but they do it by working with generalizations of the GHZ state

Is there a prototypical model Hamiltonian/system that has nontrivial information encoded in its QFI?

The QFI has general applicability to all quantum states (such as those produced by a sequence of Unitaries in a quantum computer) rather then equilibrium states of physical systems. Therefore I would claim that the prototypical states for discussing quantum fisher information are the GHZ state (the characteristic entangled state) and coherent states(the characteristic unentangled state). Since they are pure states, it's straight forward to calculate there variance and show that the GHZ state has maximum QFI $$F_q=4N^2$$ for the "maximum" observable, $$Q$$ and the coherent state has QFI $$F_q(Q)=4N$$. Again assuming $$Q=\sum_iQ_i$$ and $$|Q_i|=1$$.

I believe the review suggested above("Introduction to quantum Fisher information" (14 Aug 2010), by Petz and Ghinea) has a more careful/precise analysis of these states.

I have the following questions:

• What is the motivation for defining (Q)FI the way it is?
• How do you interpret QFI, especially with regards to measuring entanglement?
• Is there a prototypical model Hamiltonian/system that has nontrivial information encoded in its QFI?

First some of the background that you inquired about.

In mathematical statistics, the Fisher information (sometimes simply called information) is a way of measuring the amount of information that an observable random variable X carries about an unknown parameter θ of a distribution that models X. Formally, it is the variance of the score, or the expected value of the observed information. ...

Definition

The Fisher information is a way of measuring the amount of information that an observable random variable X carries about an unknown parameter θ upon which the probability of X depends. Let f(X; θ) be the probability density function (or probability mass function) for X conditional on the value of θ. This is also the likelihood function for θ. It describes the probability that we observe a given sample X, given a known value of θ. If f is sharply peaked with respect to changes in θ, it is easy to indicate the “correct” value of θ from the data, or equivalently, that the data X provides a lot of information about the parameter θ. If the likelihood f is flat and spread-out, then it would take many, many samples like X to estimate the actual “true” value of θ that would be obtained using the entire population being sampled. This suggests studying some kind of variance with respect to θ. ...

The optical conductivity is a material property, which links the current density to the electric field for general frequencies. In this sense, this linear response function is a generalization of the electrical conductivity, which is usually considered in the static limit, i.e., for a time-independent (or sufficiently slowly varying) electric field. While the static electrical conductivity is vanishingly small in insulators (such as Diamond or Porcelain), the optical conductivity always remains finite in some frequency intervals (above the optical gap in the case of insulators); the total optical weight can be inferred from sum rules. ...

From: "Introduction to quantum Fisher information" (14 Aug 2010), by Petz and Ghinea:

Introduction

Parameter estimation of probability distributions is one of the most basic tasks in information theory, and has been generalized to quantum regime since the description of quantum measurement is essentially probabilistic. ...

...

In the quantum formalism a probability measure is replaced by a positive matrix of trace 1. (Its eigenvalues form a probability measure, but to determine the so-called density matrix a basis of the eigenvectors is also deterministic.) If a parametrized family of density matrices Dθ is given, then there is a possibility for the quantum Fisher information. This quantity is not unique, the possibilities are determined by linear mappings. The analysis of the linear mappings is the main issue of the paper. ...

From: "Simple expression for the quantum Fisher information matrix" (12 Apr 2018), by Dominik Šafránek:

Abstract

Quantum Fisher information matrix (QFIM) is a cornerstone of modern quantum metrology and quantum information geometry. Apart from optimal estimation, it finds applications in description of quantum speed limits, quantum criticality, quantum phase transitions, coherence, entanglement, and irreversibility. We derive a surprisingly simple formula for this quantity, which, unlike previously known general expression, does not require diagonalization of the density matrix, and is provably at least as efficient. With a minor modification, this formula can be used to compute QFIM for any finite-dimensional density matrix. Because of its simplicity, it could also shed more light on the quantum information geometry in general.

• Usefulness:

A search for the terms 'quantum Fisher information matrix "optical conductivity" "Mott transition"' returns over 4500 results, one example paper is:

"Quantum Phase Transitions in quasi-one dimensional systems" (7 July 2010), by Thierry Giamarchi, where he writes:

Among the various systems, one dimensional (1D) and quasi-one dimensional (quasi-1D) systems are a fantastic playground for quantum phase transitions (QPTs), with rather unique properties. There are various reasons for that special behavior.

First, purely 1D systems are rather unique. Contrary to their higher- dimensional counterparts 1, interactions play a major role since in 1D particles cannot avoid the effects of interactions. This transforms any individual motion of the particles into a collective one. In addition to these very strong interaction effects, in 1D the quantum and thermal fluctuations are pushed to a maximum, and prevent the breaking of continuous symmetries, making simple mean-field physics inapplicable. The combination of these two effects leads to a very special universality class for interacting quantum systems, known as Luttinger liquids (LLs).

...

Although in principle one can realize coupled bosonic systems in condensed matter, e.g. using Josephson junction arrays, it is relatively difficult to obtain a good realization. Recently, cold atomic systems in optical lattices have provided a remarkable and very controlled realization on which many of the aspects discussed below can be tested in experiments, ...

Some people are passionate about this and their work, the full extent of the uses has yet to be discovered.

A more general search for '"optical conductivity" " quantum Fisher"' turned up this short paper:

"Preservation of quantum Fisher information and geometric phase of a single qubit system in a dissipative reservoir through the addition of qubits" (27 June 2017), by Youneng Guo, Qinglong Tian, Yunfei Mo, and K Zeng:

INTRODUCTION

Quantum Fisher information (QFI) which extends the classical Fisher Information plays a significant role in the fields of quantum metrology. QFI characterizes the sensitivity of the state with respect to changes in a parameter. According to the quantum estimation theory, the ultimate achievable parameter estimation precision is characterized by the QFI through the Cramér-Rao inequality. For an estimation parameter with a larger QFI value, the accuracy is more clearly achieved. Moreover, QFI may serve as a new resource in quantum information tasks to witness entanglement detection, non-Markovianity characterization, and uncertainty relations and so on.

However, any realistic quantum system of interest is unavoidably disturbed by surrounding environments which not only result in the loss of quantum coherence of interest systems but also degrease of parameter estimation precision. Hence, how to preserve and enhance the QFI becomes a key problem to be solved. In recent years, great attention has been paid to protecting and improving the QFI. Different protocols and strategies have been proposed and realized in the quantum metrology. ...

As you can see it's also applicable to quantum computing.

• I appreciate the references, but this doesn't really do anything more than a few Google searches. I don't think anyone would gain any understanding by reading your answer – KF Gauss Jun 4 '18 at 1:12
• Now that it's been bumped by the Community user perhaps the answer you seek will be forthcoming. – Rob Sep 7 '18 at 17:29