# Domain Generalization by Learning from Privileged Medical Imaging
Information

###### Abstract

Learning the ability to generalize knowledge between similar contexts is
particularly important in medical imaging as data distributions can shift
substantially from one hospital to another, or even from one machine to
another. To strengthen generalization, most state-of-the-art techniques inject
knowledge of the data distribution shifts by enforcing constraints on learned
features or regularizing parameters. We offer an alternative approach:
Learning from Privileged Medical Imaging Information (LPMII). We show that
using some privileged information such as tumor shape or location leads to
stronger domain generalization ability than current state-of-the-art
techniques. This paper demonstrates that by using privileged information to
predict the severity of intra-layer retinal fluid in optical coherence
tomography scans, the classification accuracy of a deep learning model
operating on out-of-distribution data improves from $0.911$ to $0.934$. This
paper provides a strong starting point for using privileged information in
other medical problems requiring generalization.

Index Terms—  Deep learning, medical imaging, generalization, segmentation,
classification, privileged information, multi-task learning, optical coherence
tomography, OCT.

##  1 Introduction

Deep learning techniques often perform poorly in medical image applications,
where data distributions exhibit significant shifts from training data to
testing data due to variations in scanners, imaging protocols, etc. To
overcome this challenge of domain shift, domain generalization (DG) techniques
have emerged, involving training a model on multiple datasets, each
characterized by distinct distributions, with the ultimate goal of developing
a model capable of generalizing to unseen test datasets.

Numerous methods for achieving domain generalization have been proposed, and
these techniques typically involve incorporating knowledge of data
distribution shifts into the learning processes through strategies such as
data augmentation (introducing artificial variations in the training data),
imposing constraints on learned features, or applying regularization [1, 2].
However, extensive empirical evaluations of popular domain generalization
algorithms have shown that they consistently fail to outperform “do-nothing”
baselines like empirical risk minimization (ERM) specifically when fair model
selection strategies are employed [3, 4].

Deep neural network training, especially for classification tasks, typically
focuses on learning the most discriminative feature(s) in the data. However,
this can be problematic when these features vary across different contexts
leading to poor generalization performance. Humans, on the other hand, tend to
learn a broad range of contexts when grasping new concepts, making them more
adaptable to different data scenarios. Motivated by the above, this paper aims
to explore whether incorporating extra, task-specific information during
training can improve the ability of deep neural networks to adapt to different
data contexts. This additional information, known as “privileged information,”
is only available during training and not during testing. In medical imaging,
examples of privileged information could include details like tumor shape and
location, as well as patient or disease-specific background information found
in medical reports.

While the concept of privileged information has been explored before [5], the
application of privileged information for domain generalization has not been
previously investigated. This paper introduces a novel framework designed to
integrate privileged information into the learning process for domain
generalization tasks. We also assess the viability of this approach through an
empirical investigation focused on predicting the severity of intra-layer
retinal fluid in optical coherence tomography (OCT) scans.

##  2 Background

###  2.1 Domain Generalization

There are many techniques that aim to address the domain generalization
problem; however, they typically fall under two broad categories: context-free
and context-focused. Context-free techniques aim to regularize the learning in
some manner to automate the learning of generalizable features. The most
common approach is domain feature learning. These methods aim to work across
all variations of the domain generalization problem, as they require no
additional input, beyond the multiple training domain samples and labels. By
comparison, context-focused methods utilize additional knowledge about the
context of the problem to achieve generalization. Techniques such as self-
supervised learning and data augmentation use handcrafted transformations that
can enable a model to be invariant to those forms of transformations; thereby
aiding in generalization. For a more detailed breakdown of generalization
techniques see [2].

###  2.2 Learning using Privileged Information

As with the previous context-focused methods, learning from privileged
information [5] revolves around the addition of extra information about the
dataset that can be used to enhance the performance of a model in some way.
Intuitively, the additional contextual information guides the model training
process to learn more useful and relevant features for the desired task.

There are two similar techniques used for addressing generalization issues:
self-supervised learning and multi-task learning. Both of these use additional
training signals to aid the model in extracting stronger features. Self-
supervised learning tasks the model with understanding algorithmically applied
variations to data (akin to data augmentation) [6, 7], such as classifying a
rotation that has been applied to an image. Multi-task learning on the other
hand utilises existing and parallel labels to train the model alongside the
main task [8]. Our proposed method, using learning from privileged medical
imaging information (LPMII), is a targeted subset of multi-task learning;
where task-relevant medical knowledge is leveraged to give the model more
context about what features may or may not be important for generalization.
Although this type of approach has been proposed before, using privileged
medical information to improve performance with self-supervised learning [9]
[10] [11] [12], it has been used to improve in-domain (or i.i.d) performance
as opposed to targeting generalization across different data distributions.

##  3 Methodology

Domain Generalization aims to learn a robust predictive model
$f_{\theta}:\mathcal{X}\rightarrow\mathcal{Y}$ from $K$ source domain
datasets, $\\{\mathcal{S}^{k}\sim P_{XY}^{\mathcal{S}^{k}}\\}_{k=1}^{K}$, that
minimizes the prediction error when applied to an unseen target domain
$\mathcal{T}\sim P_{XY}^{\mathcal{T}}$. Here the target data distribution is
different from the source domains i.e., $P_{XY}^{\mathcal{T}}\neq
P_{XY}^{\mathcal{S}^{K}}\leavevmode\nobreak\ \forall k\in[1\dots K]$.

We consider a supervised learning scenario where we have multiple source
domains. Each source domain has $N_{k}$ input-target pairs
$\left\\{\left(x_{i},y_{i}\right)\right\\}_{i=1}^{N_{k}}$, where the inputs
(covariates) are noted by $x_{i}$ and the targets by $y_{i}$. Moreover, we
assume that each source domain also has privileged information $q_{i}\in Q$
that is related to the task. Privileged information is only available during
training, not during testing. Therefore, each source domain has $N_{k}$
triplets of the form
$\left\\{\left(x_{i},q_{i},y_{i}\right)\right\\}_{i=1}^{N_{k}}$.

Our proposed method, LPMII, leverages the privileged information to enhance
the learning process. Assume that the model $f_{\theta}$ consists of a
stochastic encoder (or feature extractor)
$g_{\phi}:\mathcal{X}\rightarrow\mathcal{Z}$ and a classifier head
$c_{\psi}:\mathcal{Z}\rightarrow\mathcal{Y}$. Since the privileged information
is relevant to the task and provides context, our goal is to capture that in
the feature representation, $Z$. To do this, we maximize the mutual
information between the learned features, $Z$, and the privileged information,
$Q$, on the source training data, $S$. The learning objective can be expressed
as follows:

$\underset{\phi,\psi}{\mathrm{arg\leavevmode\nobreak\
min}}\leavevmode\nobreak\ \underset{x,y\sim
P_{XY}^{\mathcal{S}}}{\mathbb{E}}\left[\mathcal{L}_{ce}\left(y,c_{\psi}\left(g_{\phi}(x)\right)\right)\right]-\alpha
I(Q,{Z})$ (1)

Here, the cross-entropy loss $\mathcal{L}_{ce}(\cdot)$ measures the prediction
error, while the mutual information $I(Q,Z)$ encourages the features to
contain the privileged information. The scalar $\alpha$ controls the trade-off
between them. We can compute the mutual information as:

$\displaystyle\mathrm{I}(Z;Q)$ $\displaystyle{}=\mathrm{H}(Q)-\mathrm{H}(Q\mid
Z)$ $\displaystyle{}=\mathrm{H}(Q)+\int_{q,z}p(q,z)\log{{p(q\mid z)}}.$

By ignoring the constant term $H(Q)$ and using a neural network
$r_{\gamma}(q\mid z)$ to approximate the intractable $p(q\mid z)$, we can
derive the following variational lower bound for $\mathrm{I}(Z;Q)$:

$\displaystyle\mathrm{I}(Z;Q)$
$\displaystyle{}\geq\operatorname{\mathbb{E}}_{x_{i},q_{i}\sim
P_{XQ}^{\mathcal{S}}}\left[\operatorname{\mathbb{E}}_{z\sim P_{z\mid
x}}\left[\log{{r_{\gamma}(q_{i}\mid z)}}\right]\right]$ (2)

Using a reparametrization scheme and defining $g_{\phi}$ as a combination of a
deterministic encoder, which produces $\left[g^{\mu},g^{\sigma}\right]$, and a
Gaussian random variable $\epsilon\sim p(\epsilon)$ i.e.,
$g_{\phi,\epsilon}(x)=g^{\mu}(x)+\epsilon\odot g^{\sigma}(x)$, we can rewrite
the lower bound as:

$\displaystyle\mathrm{I}(Z;Q)$
$\displaystyle{}\geq\operatorname{\mathbb{E}}_{x_{i},q_{i}\sim
P_{XQ}^{\mathcal{S}}}\left[\mathbb{E}_{p(\epsilon)}\left[\log{{r_{\gamma}(q_{i}\mid
g_{\phi,\epsilon}(x))}}\right]\right]$
$\displaystyle{}\geq-\operatorname{\mathbb{E}}_{x_{i},q_{i}\sim
P_{XQ}^{\mathcal{S}}}\left[\mathbb{E}_{p(\epsilon)}\left[\mathcal{L}_{ce}\left(q_{i},r_{\gamma}(g_{\phi}(x)\right)\right]\right]$
(3)

If the privileged information takes the form of a regression task, the cross-
entropy loss can then be replaced by an appropriate loss function such as the
mean squared error. A similar lower bound has been derived in [13]. The final
objective function is:

$\displaystyle\underset{\phi,\psi,\gamma}{\mathrm{arg\leavevmode\nobreak\
min}}\leavevmode\nobreak\ \underset{x,y,q\sim
P_{XYQ}^{\mathcal{S}}}{\mathbb{E}}$
$\displaystyle[\mathcal{L}_{ce}\left(y,c_{\psi}\left(g_{\phi}(x)\right)\right)$
$\displaystyle+\alpha\mathbb{E}_{p(\epsilon)}\left[\mathcal{L}_{ce}\left(q,r_{\gamma}(g_{\phi}(x)\right)\right]]$

The overall block diagram is depicted in Figure 1, where we use a ResNet-50 as
the encoder, $g_{\phi}$, and two MLP classifiers to represent $c_{\psi}$ and
$r_{\gamma}$.

Fig. 1: The model structure of learning from privileged medical information
(LPMII). The feature extracting variation encoder used was a ResNet50
structurally.

Experimental setup: To validate this approach a dataset with both multiple
domains and adequate additional information is required. The Retinal Optical
Coherence Tomography Fluid Challenge (Retouch) OCT [14] dataset has both
required components. This dataset contains three sets (24, 23, and 21 scans
each with 100 slices per scan approximately) of 3D scans of retinas captured
using scanners manufactured by three different vendors, each with segmentation
masks that indicate the presence of retinal fluids (i,e., Intraretinal fluid,
Subretinal fluid, and Pigment Epithelial Detachment). Each 3D scan contains
approximately 100 slices, which were separated and resized to $224$x$224$ for
training on the 2D ResNet50 network. Examples of the data can be seen in
figure 2. In this problem, we focus on the classification of the presence of
each type of inter-layer retinal fluid for each slice independently.

Fig. 2: A sample slice from a scan from each manufacturer (Cirrus, Spectralis,
and Topcon) within the Retouch dataset. While the underlying structural
information is similar, the noise patterns are significantly different leading
to domain shift.

For the privileged information source, we derived several ways to extract
information from the provided fluid segmentation masks. Learning from
privileged information requires a careful balance between the informativeness
and the availability of the data. As such, we use the segmentation masks only
to create simpler labels that expert radiologists could easily produce when
collecting data. We derived two forms of side information that can be used for
this task: (1) Total mass of the fluid build-up, and (2) “severity” score.

The most intuitive information that could be extracted from segmentation masks
is the mass of the fluid buildup, $q_{i}^{(mass,l)}$. This was calculated by
summing the positive pixels in the segmentation mask, $M_{i}$, for a given
fluid type, $l$.

$q_{i}^{(mass,l)}=\sum^{H,W}_{h=1,w=1}M_{h,w}^{(l)}$

Here, $H,W$ are the height and the width of a B-scan. The total mass can be
affected by minor errors in the annotation process. In many medical imaging
tasks, we observe inter and intra-annotator variations. Additionally, this
mass calculation is highly dependent on the resolution of the scan. To address
this, a severity estimation was used as well. This severity metric was
designed to map the total mass to a scale of 0 to 5 indicating the severity of
the fluid build-up, independent of the resolution of the scan. This was
achieved by dividing the mass by the total number of pixels in the scan,
giving a proportion of how much of the scan is fluid or non-fluid. Given the
actual range of values in the data is lower, with the largest amount of
normalized fluid being approximately 0.2, the proportion was multiplied by 5
to fully occupy the range of values between 0 and 1. An additional
multiplication is done to scale the values to the desired number of severity
categories, $N_{sc}$: in our experiments we used $N_{sc}=5$ to approximate the
0 to 5 severity rating system. This final value was then passed through the
ceiling function to discretize the values to integers.

$q^{(severity,l)}=\mathrm{ceiling}\left(\frac{5\cdot N_{sc}}{H\cdot
W}\sum^{H,W}_{h=1,w=1}M_{h,w}^{(l)}\right)$

The intuition for these metrics was to introduce some correlating information
about the fluid. To be able to predict the mass or severity of the fluid
build-up the model must be able to accurately identify what sections of the
scan are fluid or not, and thus cannot rely on any other correlating factors.
Figure 3 shows an example scan along with the segmentation mask.

Fig. 3: Visual representation of the two types of privileged information in
the Retouch OCT. X) A single slice of a retinal OCT scan. X+Y) The
segmentation mask overlaid on the original scan.

##  4 Results and Discussion

In this section, we present and discuss the results for the several different
variations on the learning from privileged medical information training
paradigm. Our method, LPMII was implemented in PyTorch and the code is
available at Github (available upon publication). There are two variations
that we will be comparing to each other plus the baseline method, ERM, which
has no additional domain generalization methods applied. Each method variation
was trained in a hold-one-out fashion, training on two domains and testing on
the third, with each domain acting as the test set three times. The out-of-
distribution test domain classification accuracies were then averaged together
to display the final result for each method.

Table 1: Average top-1 classification accuracy for different sources of privileged information. Privileged Information | Average Top-1 Accuracy
---|---
None (ERM) |  0.911 $\pm$ 0.01
Fluid Mass Regression |  0.902 $\pm$ 0.02
Severity Prediction |  0.934 $\pm$ 0.01

Additionally, we compare the performance of learning from privileged medical
imaging information (LPMII) to seven methods from the DomainBed framework,
which represents a wide variety of state-of-the-art domain generalization
techniques and mechanisms. The methods chosen are Invariant Risk Minimisation
(IRM) [15], Maximum Mean Discrepancy (MMD) [16], Domain Adversarial Neural
Networks (DANN) [17], Gradient Matching for Domain Generalization (Fish) [18],
Inter-domain Mixup (Mixup) [19], Style Agnostic Network (SagNet) [20], and
Self-supervised Contrastive Regularization (SelfReg) [7]. The DomainBed
framework provides a platform for fare comparison of domain generalization
techniques that also include proper model selection methodology [3].

Table 2: Average top-1 class-weighted classification accuracy on out-of-distribution data of domain generalization algorithms. Method | Average Top-1 Accuracy
---|---
ERM |  0.911 $\pm$ 0.01
IRM [15] |  0.896 $\pm$ 0.01
MMD [16] |  0.907 $\pm$ 0.02
DANN [17] |  0.880 $\pm$ 0.02
Fish [18] |  0.911 $\pm$ 0.01
Mixup [19] |  0.896 $\pm$ 0.01
SagNet [20] |  0.899 $\pm$ 0.01
SelfReg [7] |  0.893 $\pm$ 0.01
LPMII |  0.934 $\pm$ 0.01

The above table (table 1) shows that the severity prediction task manages to
improve the out-of-distribution accuracy by approximately $0.02$ for the
Retouch dataset. While this improvement is marginal, it is the first method
that has led to a noticeable and consistent improvement in accuracy.
Unfortunately, the other privileged information task does not fare as
strongly.

Fluid mass regression performs roughly equivalently to ERM though slightly
worse, and with a larger margin of error. Most likely this is due to the
network suffering from the requirement to output accurate values above the
bounds of 0 to 1, where neural networks typically perform the best.

While one of the side information methods failed to improve upon ERM,
achieving an improvement with the severity prediction indicates that training
a model with additional information about the ailment that needs to be
classified can be successful in improving generalization performance.

##  5 Conclusion

This article explores the novel use case of improving domain generalization by
learning from privileged medical imaging information. This method was designed
to introduce a stronger training signal that informs the model about what
types of features are more likely to be useful for the main task that will be
invariant to domain shifts. We show that by training a model to predict a
simple and fast to obtain severity indicator for a retinal layer fluid
detection task (from the Retouch dataset) the out-of-distribution binary
classification accuracy of the model was increased from $0.911$ to $0.934$.
Even though this is a strong preliminary result, it is likely that more
informative sources of privileged information could lead to even greater
improvements in performance. As such, significantly more testing on learning
from privileged information for domain generalization must be done on more
challenging medical imaging tasks on a wider variety of different imaging
modalities and ailments, with more sources of privileged information to fully
explore the potential of this technique.

The most prominent issue standing in the way of this method’s wide use is the
availability of privileged information. For natural imaging, capturing
additional data describing the images is rarely done as a matter of course.
Likewise, the types of additional information required to provide an adequate
training signal may not be as straightforward. For medical data, there is
almost always additional information available in the form of medical reports.
These reports often contain highly relevant and useful information pertaining
to the issue that is being examined in the scan, as such medical imaging is a
prime use case for this technique.

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# Sparse InSAR Data 3D Inpainting for Ground Deformation Detection Along the
Rail Corridor

###### Abstract

Monitoring of ground movement close to the rail corridor, such as that
associated with landslips caused by ground subsidence and/or uplift, is of
great interest for the detection and prevention of possible railway faults.
Interferometric synthetic-aperture radar (InSAR) data can be used to measure
ground deformation, but its use poses distinct challenges, as the data is
highly sparse and can be particularly noisy. Here we present a scheme for
processing and interpolating noisy, sparse InSAR data into a dense spatio-
temporal stack, helping suppress noise and opening up the possibility for
treatment with deep learning and other image processing methods.

Index Terms—  InSAR, Ground Deformation, Sparsity, Interpolation, Deep Image
Prior

##  1 Introduction

The rail network is one of the key elements of transport infrastructure in
most countries, facilitating the movement of people and goods throughout
regions. Rail service unavailability can not only critically disrupt transport
flows but also incur significant economic costs and even cause societal
problems [1]. While disruptions in rail service can occur due to a vast number
of factors, environmental or otherwise, one particularly severe can be the
blocking of railway lines by soil, rocks and earth. This often occurs when a
landslip or other ground movement causes a translational, rotational or other
failure on an slope (such as an embankment or cutting) along the rail
corridor.

Monitoring ground movement along the rail corridor, detecting anomalous
movements for a given region, and possibly forecasting ground movement that
may result in rail service disruption is therefore of direct interest to
organisations tasked with the operation and maintenance of rail networks [2].
In situ methods (e.g. GPS, manual inspections) for monitoring ground
deformation and rail network structural health are often labor and time
intensive, and can only feasibly be deployed in locations where a priori
knowledge of risk of fault or deformation exists.

Synthetic Aperture Radar Interferometry (InSAR) can provide ground deformation
data at millimetre accuracy over wide regions of interest and with a revision
frequency of only a few days. This makes it a prime candidate for detecting
landslides [3], thus complementing and enhancing railway structural health
monitoring [4]. Such InSAR ground deformation data have in the past been used
for monitoring the structural health of rail infrastructure [2] [4], detecting
(and even predicting) uplift and subsidence in the built environment [5] [6]
as well as possible volcanic activity [7].

Previous work has looked at detecting and predicting ground movement, working
with the complete time series data of a small number of hand-selected points
of interest [6], and also at detecting ground movement over larger areas by
converting the sparse point representation to a dense deformation image
suitable for deep learning processing [5]. In the later case however only the
average velocity, rather than the full time series data, was utilised.

Here we present a method for producing a temporal stack of dense ground
movement images by conditioning, denoising and inpainting the data, in the
hope of leveraging the above two approaches [6] [5]. This would open up the
possibility of employing various image processing methods, including deep
learning architectures such as those employed in [5]. We present a simple
proof of concept using a variance-based detector and demonstrate superior
results compared to working with the original sparse time series data. It
should be noted that existing deep learning-based techniques of ground
movement detection are applied to wrapped InSAR data, where appearing fringes
are the key features of object detection [8]. However, this requires a large
area of ground movement, whist a landslip at railway embankment may be
represented by only one InSAR data point.

##  2 DATASET

The Sentinel-1 satellite constellation provides SAR imagery of high resolution
at near global coverage, with a revisit time of 6 days. This imagery can be
used to form interferometric image pairs and ground motion data maps.

There are various techniques for producing InSAR ground motion data - most
fall in one of two categories, small baseline methods and persistent scatterer
methods. Small baseline is suitable for large area monitoring, but is
susceptible to including many incoherent points; persistent scatterer methods
typically produce more reliable measurements but result in a sparse
representation of the area of interest [5].

Here we employ data provided by SatSense Ltd., generated using their
proprietary RapidSAR pixel selection algorithm [9]. This results in
interferograms that largely avoid incoherent points and produce ground motion
measurements of enhanced accuracy and high SNR. These are sparse ground motion
products, relying on identifying strong solid reflectors in the scene, and
therefore contain unevenly spatially distributed measurement points.The built
environment can provide multiple suitable reflectors, leading to high
measurement density. Rural areas on the other hand present fewer such
opportunities, meaning the already sparse data become even sparser, containing
much fewer reliable data points in rural areas of high vegetation.

(a)

(b)

Fig. 1: Example of SatSense data from (a) Site A and (b) Site B. Fault sites
marked in red, datapoint colour indicates average displacement velocity
($mm/yr$). Note the difference in measurement density between rural and built-
up areas.

An example of this change in measurement density can be seen in Figure 1 -
most returns in rural areas are from the railway infrastructure itself, with
little else around it. Unfortunately these are the areas in which rail
corridor monitoring is of most interest, as monitoring is easier within urban
areas.

###  2.1 Case studies of Railway faults

These data are complimented by a database of known railway failures provided
by Network Rail. We have identified a couple of areas in which a known fault
due to ground movement has occurred within the time period investigated. The
first site (Site A) investigated is on the East Coast Line, near Oak Tree Farm
just north of Stamford, UK. A fault was identified in the area on April 22nd,
2020, involving loss of support due to translational failure on the cutting
slope, with similar faults having also occurred in the area on December 18th,
2019 and on June 13, 2018. The second location (Site B) is on the West Coast
Main Line, northeast of Abington, UK. Two faults were identified here on
January 7th and January 13th, 2016, involving a translational failure of the
natural slope by the rail tracks due to excessive rainfall.

##  3 PREPROCESSING & INTERPOLATION

###  3.1 Time Series Interpolation

Given an area of interest, our dataset will consist of a number of points,
unevenly distributed across the scene, each representing a time series of
displacement measurements from May 2015 to current day (July 2021 for the data
used in this paper). The timeframe selected is of course subject to each
scenario of interest.

The time series data may contain missing values (represented as NaN) at dates
when a particular datapoint was too incoherent so as to extract a meaningful
signal from it. Water cover, as well as thick vegetation can cause a datapoint
to be incoherent at certain dates; a change in the built environment, a
landslide or any other major change in the landscape can also cause a
previously coherent point to become permanently incoherent past a certain
date.

We begin by identifying datapoints that contain too many NaN measurements,
typically due to a point becoming permanently incoherent after significant
environmental change. These timeseries are deemed unusable and removed from
the dataset. For this study we experimentally set the cut-off threshold to
$15\%$ \- any time series that is not coherent over at least $85\%$ of its
total sample length is eliminated.

The Sentinel-1 constellation has a revisit time of 6 days, but this was only
achieved with the launch of Sentinel-1B in April 2016; prior to that the
revisit time of Sentinel-1A was at 12 days. Additionally, there are isolated
instances of SAR acquisition frames missing from the temporal sequence. All
selected time series are therefore amended to a regular sampling period of 6
days, with NaN-valued time frames inserted on these missing dates, e.g. as in
Figure 2.

Fig. 2: Sample time series data with temporally interpolated points (missing
satellite acquisitions) circled in red.

NaN values in the time series, whether pre-existing or added during the
temporal sampling correction, indicate missing data points, and then need to
be interpolated. In order to avoid unnecessary assumptions we here opt for a
simple linear interpolation between neighbouring frames in the time series.

Finally, we include a simple median denoising step here aimed at mitigating
the spiky noise in the time series data. The process applies a 2-D median
filter $Med_{3\times 3}(\cdot)$ that omits NaN values in the median
calculation. The time series data are then projected into a 3-dimensional
sparse image stack.

###  3.2 Spatio-temporal Interpolation

This stack now needs to be interpolated into a dense 3-dimensional volume. We
have utilised two inpainting methods to do so. Inpainting typically refers to
algorithms for filling in gaps in images, though they can be extended to more
general interpolation/extrapolation tasks as in this case.

Such techniques are often based on solving partial differential equations
(PDE) formulated over the image space, typically emulating physical analogues
such as plate deformation, spring motion, diffusion or fluid motion [10].
These have been found to produce interpolation results that tend to preserve
and propagate linear and textural features, a property highly desirable in the
context of image inpainting [11]. We have here opted for a spring model [12],
where pixels are connected to their neighbours (horizontally and vertically)
via springs of infinitesimal length.

Additionally, we have attempted inpainting using the Deep Image Prior (DIP)
framework [13]. DIP is a state-of-the-art method capable of addressing a
variety of inverse problems in imaging, from denoising to inpainting. It uses
a randomly initialised deep neural network as the prior in an inverse problem
formulation. Inverse problems are typically formulated as the minimisation of
(1) aiming to recover the image $x$ from corrupted observation $x_{0}$, where
$E(x;x_{0})$ is the data term and $R(x)$ is a regularisation image prior.

$min_{x}E(x;x_{0})+R(x)$ (1)

The DIP framework instead attempts to solve this problem by minimising

$min_{\theta}E(f_{\theta}(z);x_{0})$ (2)

where $f$ is a deep convolution neural network with parameters $\theta$ and
$z$ is a fixed noise input. This means the minimisation solution is searched
not on the image space but on the space of the deep network’s parameters.

The network used here is U-net architecture with a depth of 3. All depths have
128 feature channels, upsampling is by bilinear interpolation and the
activation layer is a leaky ReLU. We use the Adam optimiser, with an initial
learning rate of 0.001. As the dataset here is quite small, the process
terminates after 3000 epochs, otherwise obvious overfitting is observed.

##  4 EXPERIMENTAL RESULTS

(a)

(b)

(c)

Fig. 3: Site A (a) Raw data (random stack sample), corresponding (b) PDE 3D
interpolated, (c) DIP 3D interpolated data. Top Right: magnified area of known
faults.

(a)

(b)

(c)

Fig. 4: Binary detection maps of points in Site A exhibiting high variance
($>150mm^{2}$) derived from (a) untreated, sparse displacement data, (b)
proposed method using PDE inpainting and (c) proposed method using DIP
inpainting. Detections dilated for display purposes.

A simple demonstration of the benefits of working with the interpolated,
denser data as opposed to the original raw data can be seen in Figures 3 & 4.
The data as originally provided contains irregularities and noise that
effectively necessitates treatment as described in Section 3.1 prior to any
further processing. The data is however still very noisy at that point, with
individual points demonstrating high variance over time. Interpolating the
time series into dense image stacks can reduce this apparent variance in the
data, making significant movements easier to discern from the noise. Figure 4
shows binary detection maps of points in the scene that demonstrate variance
above an experimentally set threshold of $150mm^{2}$. Variance here is
calculated per point over the entire time series length / image stack depth.

While the spatio-temporally interpolated stacks contain far more points of
measurable variance compared to the sparse raw data stack, a considerably
smaller number of those exhibits high variance; additionally, these points (as
will be seen later) correlate with known faults in the area.

(a) 13/06/2018

(b) 23/12/2019

(c) 27/04/2020

(d) 13/06/2018

(e) 23/12/2019

(f) 27/04/2020

Fig. 5: Local Variance Detector results for Site A, first available frame
after fault dates. (a), (b) and (c) detector output on PDE interpolation, (d),
(e), (f) detector output on DIP interpolation. Fault area (aprox.) marked in
red, positive detections marked green. Variance threshold at $50mm^{2}$,
calculated over $5\times 5\times 5$ window.

(a)

(b)

(c)

(d)

Fig. 6: Local Variance Detector results for Site B. (a) raw data, (b) detector
output on PDE interpolation, (c) detector output on DIP interpolation. Fault
locations (approx.) marked in red,positive detections marked green. Variance
threshold at $5mm^{2}$, calculated over $5\times 5\times 5$ window. Frame
taken at 18th Jan. 2016.

For a more practical experiment, we redesigned our detector to compute local
variance in a $5x5$ neighbourhood over approximate month intervals, i.e. 5
frames in the temporal dimension. Figure 5 shows the binary thresholding
result of the variance map on 3 dates corresponding to the 3 known faults in
Site A. The frames shown are the first available after the date on which the
fault was reported, which may of course not necessarily coincide with an InSAR
acquisition date for the site. The fault area has been marked in a red circle,
and it can be seen that this simple detector is capable of capturing motion in
that region in most cases.

The DIP variant performs slightly better for the same variance threshold, and
detects movement that can reasonably be attributed with a fault (Figure 5 (e))
that the PDE variant misses. Note there may well be motion detected by our
method in parts of the image that does not correspond with a known fault,
simply because no fault resulted of it (especially when this movement did not
occur at a point near the rail corridor). Instead we are interested in seeing
whether our method can detect in the data ground movement that can reasonably
be correlated, spatially and temporally, to the failures known.

We repeat a similar experiment for Site B, shown in Figure 6. As both known
faults here occurred within a few days, they are marked on the same frame. We
also include an example of the detector on the raw data, demonstrating the
much higher number of detections, almost equal to the number of total
datapoints.

##  5 Conclusions and Future Work

We demonstrate here a scheme for treating noisy, sparse InSAR data and
interpolating them into a dense spatiotemporal image stack, along with a proof
of concept experiment for the detection of significant ground movement near
the rail corridor. We wish to extend this work in the near future to further
improve data conditioning, develop a novel motion detector using more advanced
deep learning methods and to conduct a larger scale field study for more
accurate correlation of ground movement and railway faults.

## References

  * [1] A. Woodburn, “Rail Network Resilience and Operational Responsiveness During Unplanned Disruption: A Rail Freight Case Study,” _Journal of Transport Geography_ , vol. 77, pp. 59–69, 2019.
  * [2] L. Chang, R. P. B. J. Dollevoet, and R. F. Hanssen, “Nationwide Railway Monitoring Using Satellite SAR Interferometry,” _IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing_ , vol. 10, no. 2, pp. 596–604, 2017.
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  * [4] M. North, T. Farewell, S. Hallett, and A. Bertelle, “Monitoring the Response of Roads and Railways to Seasonal Soil Movement with Persistent Scatterers Interferometry over Six UK Sites,” _Remote Sensing_ , vol. 9, no. 9, 2017\.
  * [5] N. Anantrasirichai, J. Biggs, K. Kelevitz, Z. Sadeghi, T. Wright, J. Thompson, A. M. Achim, and D. Bull, “Detecting Ground Deformation in the Built Environment Using Sparse Satellite InSAR Data With a Convolutional Neural Network,” _IEEE Transactions on Geoscience and Remote Sensing_ , vol. 59, no. 4, pp. 2940–2950, 2021.
  * [6] P. Hill, J. Biggs, V. Ponce-López, and D. Bull, “Time-Series Prediction Approaches to Forecasting Deformation in Sentinel-1 InSAR Data,” _Journal of Geophysical Research: Solid Earth_ , vol. 126, no. 3, p. e2020JB020176, 2021.
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  * [8] N. Anantrasirichai, J. Biggs, F. Albino, P. Hill, and D. Bull, “Application of Machine Learning to Classification of Volcanic Deformation in Routinely-Generated InSAR data,” _J. Geophys. Res.: Solid Earth_ , vol. 123, no. 8, pp. 6592–6606, August 2018.
  * [9] K. Spaans and A. Hooper, “InSAR Processing for Volcano Monitoring and Other Near-real Time Applications,” _Journal of Geophysical Research: Solid Earth_ , vol. 121, no. 4, pp. 2947–2960, 2016.
  * [10] S. Crema, M. Llena, A. Calsamiglia, J. Estrany, L. Marchi, D. Vericat, and M. Cavalli, “Can Inpainting Improve Digital Terrain Analysis? Comparing Techniques for Void Filling, Surface Reconstruction and Geomorphometric Analyses,” _Earth Surface Processes and Landforms_ , vol. 45, no. 3, pp. 736–755, 2020.
  * [11] T. F. Chan and J. Shen, “Nontexture Inpainting by Curvature-Driven Diffusions,” _Journal of Visual Communication and Image Representation_ , vol. 12, no. 4, pp. 436–449, 2001.
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# A unified approach to Local Quantum Uncertainty and Interferometric Power by
Metric Adjusted Skew Information

Paolo Gibilisco  Department of Economics and Finance, University of Rome “Tor
Vergata”, Via Columbia 2, Rome 00133, Italy<EMAIL_ADDRESS>Davide Girolami<EMAIL_ADDRESS>Politecnico di Torino, Corso Duca
degli Abruzzi 24, Torino 129, Italy Frank Hansen  Department of Mathematical
Sciences, University of Copenaghen, Universitetsparken 5 DK-2100 Copenhagen,
Denmark<EMAIL_ADDRESS>
###### Abstract

Local quantum uncertainty and interferometric power have been introduced by
Girolami et al. in GTA:2013 ; GSGTFSSOA:2014  as geometric quantifiers of
quantum correlations. The aim of the present paper is to discuss their
properties in a unified manner by means of the the metric adjusted skew
information defined by Hansen in Hansen:2006b .

##  I Introduction

One of the key traits of many-body quantum systems is that the full knowledge
of their global configurations does not imply full knowledge of their
constituents. The impossibility to reconstruct the local wave functions
$|\psi_{1}\rangle,\,|\psi_{2}\rangle$ (pure states) of two interacting quantum
particles from the wave function of the whole system,
$|\psi_{12}\rangle\neq|\psi_{1}\rangle\otimes|\psi_{2}\rangle$, is due to the
existence of entanglement ent . Investigating open quantum systems, whose
(mixed) states are described by density matrices
$\rho_{12}=\sum_{i}p_{i}|\psi_{i}\rangle_{12}\langle\psi_{i}|$, revealed that
the boundary between the classical and quantum worlds is more blurred than we
thought. There exists a genuinely quantum kind of correlation, quantum
discord, which manifests even in absence of entanglement, i.e. in separable
density matrices $\rho_{12}=\sum_{i}p_{i}\rho_{1,i}\otimes\rho_{2,i}$ oz ; hv
. The discovery triggered theoretical and experimental studies to understand
the physical meaning of quantum discord, and the potential use of it as a
resource for quantum technologies rev . Relying on the known interplay between
geometrical and physical properties of mixed states Uhlmann:1992 ; book , a
stream of works employed information geometry techniques to construct
quantifiers of quantum discord AyGibiliscoMatus: 2018 ; BogaertGirolami:2017 ;
CFTA:2018 ; GII:2009 ; FPA:2017 ; GibiliscoIsola:2011 . In particular, two of
the most popular ones are the Local Quantum Uncertainty (LQU) and the
Interferometric Power (IP) GTA:2013 ; GSGTFSSOA:2014 . A merit of these two
measures is that they admit an analytical form for $N$ qubit states across the
$1\,vs\,N-1$ qubit partition. Also, they have a clear-cut physical
interpretation. The lack of certainty about quantum measurement outcomes is
due to the fact that density matrices are changed by quantum operations. The
LQU evaluates the minimum uncertainty about the outcome of a local quantum
measurement, when performed on a bipartite system. It is proven that two-
particle density matrices display quantum discord if and only if they are not
“classical-quantum” states. That is, they are not (mixture of) eigenvalues of
local observables, $\rho_{12}\neq\sum_{i}p_{i}|i\rangle_{1}\langle
i|\otimes\rho_{2,i}$, or
$\rho_{12}\neq\sum_{i}p_{i}\rho_{1,i}\otimes|i\rangle_{2}\langle i|$, in which
$\\{|i\rangle\\}$ is an orthonormal basis. Indeed, this is the only case in
which one can identify a local measurement that does not change a bipartite
quantum state, whose spectral decomposition reads
$A_{1}=\sum_{i}\lambda_{i}|i\rangle_{1}\langle i|$, or
$A_{2}=\sum_{i}\lambda_{i}|i\rangle_{2}\langle i|$. The LQU was built as the
minimum of the Wigner-Yanase skew information, a well-known information
geometry measure WignerYanase:1963 , between a density matrix and a finite-
dimensional observable (Hermitian operator). It quantifies how much a density
matrix $\rho_{12}$ is different from being a zero-discord state. The IP was
concocted by following a similar line of thinking. Quantum discord implies a
non-classical sensitivity to local perturbations. This feature of quantum
particles, while apparently a limitation, translates into an advantage in the
context of quantum metrology metro . It was theoretically proven and
experimentally demonstrated that quantum systems sharing quantum discord are
more sensitive probes for interferometric phase estimation. The figure of
merit of such measurement protocols is the quantum Fisher information of the
state under scrutiny with respect to a local Hamiltonian (in Information
Geometry the QFI is known as the SLD or Bures-Uhlmann metric). The latter
generates a unitary evolution that imprints information about a physical
parameter on the quantum probe. The IP is the minimum quantum Fisher
information over all the possible local Hamiltonians, being zero if and only
if the probe states are classically correlated.

Here, we polish and extend the mathematical formalization of information-
geometric quantum correlation measures. We build a class of parent quantities
of the LQU (and consequently of the IP) in terms of the the metric adjusted
skew informations Hansen:2006b . In Sections 2,3, we review definition and
main properties of operator means. In Sections 4-6, we discuss information-
geometric quantities that capture complementarity between quantum states and
observables. In particular, we focus on the quantum $f$-covariances and the
quantum Fisher information. They quantify the inherent uncertainty about
quantum measurement outcomes. After having recalled the definition of metric
adjusted skew information (Section 7), we build a new quantum discord measure,
the metric adjusted local quantum uncertainty ($f$-LQU), in Section 8. Finally
we are able to show that LQU and IP are just two particular members of this
family allowing a unified treatment of their fundamental properties.

##  II Means for positive numbers

We use the notation ${\mathbb{R}}_{+}=(0,+\infty)$.

######  Definition 1.

A bivariate mean PetzTemesi:2005  is a function
$m\colon{\mathbb{R}}_{+}\times{\mathbb{R}}_{+}\to{\mathbb{R}}_{+}$ such that

  1. 1.

$m(x,x)=x.$

  2. 2.

$m(x,y)=m(y,x).$

  3. 3.

$x<y$ $\,\Rightarrow\,$ $x<m(x,y)<y.$

  4. 4.

$x<x^{\prime}$ and $y<y^{\prime}$ $\,\Rightarrow\,$
$m(x,y)<m(x^{\prime},y^{\prime}).$

  5. 5.

$m$ is continuous.

  6. 6.

$m$ is positively homogeneous; that is $m(tx,ty)=t\cdot m(x,y)$ for $t>0.$

We use the notation $\mathcal{M}_{num}$ for the set of bivariate means
described above.

######  Definition 2.

Let $\mathcal{F}_{num}$ denote the class of functions
$f\colon\mathbb{R}_{+}\to\mathbb{R}_{+}$ such that

  1. 1.

$f$ is continuous.

  2. 2.

$f$ is monotone increasing.

  3. 3.

$f(1)=1.$

  4. 4.

$tf(t^{-1})=f(t)$.

The following result is straightforward.

######  Proposition 1.

There is a bijection $f\mapsto m_{f}$ betwen ${\mathcal{F}}_{nu}$ and
${\mathcal{M}}_{nu}$ given by

$m_{f}(x,y)=yf(y^{-1}x)\qquad\text{and in reverse}\qquad f(t)=m(1,t)$

for positive numbers $x,y$ and $t.$

In Table 1 we have some examples of means.

Table 1:  $\begin{array}[]{|c|c|c|}\hline\cr{\rm Name}&{f}&{m_{f}}\\\
\hline\cr{\rm
arithmetic}&\displaystyle\frac{1+x}{2}&\displaystyle\frac{x+y}{2}\\\
\hline\cr{\rm
WYD},\beta\in(0,1)&\displaystyle\frac{x^{\beta}+x^{1-\beta}}{2}&\displaystyle\frac{x^{\beta}y^{1-\beta}+x^{1-\beta}y^{\beta}}{2}\\\
\hline\cr{\rm geometric}&\sqrt{x}&\sqrt{xy}\\\ \hline\cr{\rm
harmonic}&\displaystyle\frac{2x}{x+1}&\displaystyle\frac{2}{x^{-1}+y^{-1}}\\\
\hline\cr{\rm logarithmic}&\displaystyle\frac{x-1}{\log
x}&\displaystyle\frac{x-y}{\log x-\log y}\\\ \hline\cr\end{array}$

##  III Means for positive operators in the sense of Kubo-Ando

The celebrated Kubo-Ando theory of operator means KuboAndo79/80 ;
PetzTemesi:2005 ; GibiliscoHansenIsola:2009  may be viewed as the operator
version of the results of Section II.

######  Definition 3.

A bivariate mean $m$ for pairs of positive operators is a function

$(A,B)\to m(A,B),$

defined in and with values in positive definite operators on a Hilbert space,
that satisfies, mutatis mutandis, conditions $(1)$ to $(5)$ in Definition 1.
In addition, the transformer inequality

$Cm(A,B)C^{*}\leq m(CAC^{*},CBC^{*}),$

should also hold for positive definite $A,B$ and arbitrary $C.$

Note that the transformer inequality replaces condition $(6)$ in Definition 1.
We denote by $\displaystyle{\mathcal{M}}_{op}$ the set of matrix means.

######  Example 1.

The arithmetic, geometric and harmonic operator means are defined,
respectively, by setting

$\begin{array}[]{rcl}A\nabla B&=&\frac{1}{2}(A+B)\\\\[6.45831pt]
A\\#B&=&A^{1/2}\bigl{(}A^{-1/2}BA^{-1/2}\bigr{)}^{1/2}A^{1/2}\\\\[8.61108pt]
A{\rm!}B&=&2(A^{-1}+B^{-1})^{-1}.\end{array}$

We recall that a function $f\colon(0,\infty)\to\mathbb{R}$ is said to be
operator monotone (increasing) if

$A\leq B\quad\Rightarrow\quad f(A)\leq f(B)$

for positive definite matrices of arbitrary order. It then follows that the
inequality also holds for positive operators on an arbitrary Hilbert space. An
operator monotone function $f$ is said to be symmetric if $f(t)=tf(t^{-1})$
for $t>0$ and normalized if $f(1)=1.$

######  Definition 4.

${\mathcal{F}}_{op}$ is the class of functions
$f:{\mathbb{R}}_{+}\to{\mathbb{R}}_{+}$ such that

  1. 1.

$f$ is operator monotone increasing,

  2. 2.

$tf(t^{-1})=f(t)\qquad t>0,$

  3. 3.

$f(1)=1.$

The fundamental result, due to Kubo and Ando, is the following.

######  Theorem.

There is a bijection $f\mapsto m_{f}$ between ${\mathcal{M}}_{op}$ and
${\mathcal{F}}_{op}$ given by the formula

$m_{f}(A,B)=A^{1/2}f(A^{-1/2}BA^{-1/2})A^{1/2}.$

######  Remark 1.

The function in ${\mathcal{F}}_{op}$ are (operator) concave which makes the
operator case quite different from the numerical (commutative) case. For
example, there exist convex functions in $\mathcal{F}_{num},$ see GH:2017 .

If $\rho$ is a density matrix (a quantum state) and $A$ is a self-adjoint
matrix (a quantum observable), then the expectation of $A$ in the state $\rho$
is defined by setting

${\rm E}_{\rho}(A)={\rm Tr\hskip-1.99997pt}~{}(\rho A).$

##  IV The correspondence between Fisher information and metric adjusted skew
information

We introduce now a technical tool which is useful to establish some
fundamental relations between quantum covariance, quantum Fisher information
and the metric adjusted skew information.

######  Definition 5.

For $f\in{\mathcal{F}}_{op}$ we define $f(0)=\lim_{x\to 0}f(x).$ We say that a
function $f\in{\mathcal{F}}_{op}$ is regular if $f(0)\not=0,$ and non-regular
if $f(0)=0,$ cf. PetzSudar:1996 ; Hansen:2006b .

######  Definition 6.

A quantum Fisher information is extendable if its radial limit exists and it
is a Riemannian metric on the real projective space generated by the pure
states.

For the definition of the radial limit see PetzSudar:1996  where the following
fundamental result is proved.

######  Theorem.

An operator monotone function $f\in{\mathcal{F}}_{op}$ is regular, if and only
if $\langle\cdot,\cdot\rangle_{\rho,f}$ is extendable.

######  Remark 2.

The reader should be aware that there is no negative connotation associated
with the qualification “non-regular”. For example, a very important quantum
Fisher information in quantum physics (see FickSauermann:1990 ), namely the
Kubo-Mori metric related to the function $f(x)=(x-1)/\log x,$ is non-regular.

We introduce the sets of regular and non-regular functions

${\mathcal{F}}_{op}^{\,r}:=\\{f\in{\mathcal{F}}_{op}\mid
f(0)\not=0\\},\quad{\mathcal{F}}_{op}^{\,n}:=\\{f\in{\mathcal{F}}_{op}\mid
f(0)=0\\}$

and notice that trivially
${\mathcal{F}}_{op}={\mathcal{F}}_{op}^{\,r}\dot{\cup}{\mathcal{F}}_{op}^{\,n}$
.

######  Definition 7.

We introduce to $f\in{\mathcal{F}}_{op}^{\,r}$ the transform $\tilde{f}$ given
by

$\tilde{f}(x)=\frac{1}{2}\left[(x+1)-(x-1)^{2}\frac{f(0)}{f(x)}\right]$

for $x>0.$ We may also write ${\tilde{f}}={\mathcal{G}}(f),$ cf.
GibiliscoImparatoIsola:2007 ; GibiliscoHansenIsola:2009 .

The following result is taken from (GibiliscoHansenIsola:2009, , Theorem 5.1).

######  Theorem.

The correspondence $f\to\tilde{f}$ is a bijection between
${\mathcal{F}}_{op}^{\,r}$ and ${\mathcal{F}}_{op}^{\,n}\,.$

In Table 2 we have some examples (where $0<\beta<1$).

Table 2:  $\displaystyle\begin{array}[]{|c|c|}\hline\cr f&\tilde{f}\\\
\hline\cr\displaystyle\frac{1+x}{2}&\displaystyle\frac{2x}{x+1}\\\
\hline\cr\displaystyle\frac{(\sqrt{x}+1)^{2}}{4}&\sqrt{x}\\\
\hline\cr\displaystyle\beta(1-\beta)\frac{(x-1)^{2}}{(x^{\beta}-1)(x^{1-\beta}-1)}&\displaystyle\frac{x^{\beta}+x^{1-\beta}}{2}\\\
\hline\cr\end{array}$

##  V Quantum f-Covariance

The notion of quantum $f$-covariance has been introduced by Petz, see
Petz:2003 ; GibiliscoHiaiPetz:2009 . Any Kubo-Ando function
$m_{f}(x,y)=yf(y^{-1}x)$ for $x,y>0$ has a continuous extension to
$[0,+\infty)\times[0,+\infty)$ given by

$m_{f}(0,y)=f(0)y,\quad m_{f}(x,0)=f(0)x,\quad m_{f}(0,0)=0,\quad x,y>0.$

The operator $m_{f}(L_{\rho},R_{\rho})$ is well-defined by the spectral
theorem for any state, see (GibiliscoImparatoIsola:2007, , Proposition 11.1
page 11). To self-adjoint $A$ we set $A_{0}=A-({\rm Tr\hskip-1.99997pt}~{}\rho
A)I,$ where $I$ is the identity operator. Note that

${\rm Tr\hskip-1.99997pt}~{}\rho A_{0}={\rm Tr\hskip-1.99997pt}~{}\rho A-({\rm
Tr\hskip-1.99997pt}~{}\rho A){\rm Tr\hskip-1.99997pt}~{}\rho=0,$

if $\rho$ is a state.

######  Definition 8.

Given a state $\rho,$ a function $f\in\mathcal{F}_{op}$ and self-adjoint $A,B$
we define the quantum $f$-covariance by setting

${\rm Cov}_{\rho}^{f}(A,B)={\rm
Tr\hskip-1.99997pt}~{}B_{0}\,m_{f}(L_{\rho},R_{\rho})A_{0}$

and the corresponding quantum $f$-variance by ${\rm Var}_{\rho}^{f}(A)={\rm
Cov}_{\rho}^{f}(A,A).$

The $f$-variance is a positive semi-definite sesquilinear form and

$f\leq g\quad\Rightarrow\quad{\rm Var}_{\rho}^{f}(A)\leq{\rm
Var}_{\rho}^{g}(A).$ (1)

Note that for the standard covariance we have ${\rm Cov}_{\rho}(A,B)={\rm
Cov}_{\rho}^{SLD}(A,B),$ where the SLD or Bures-Uhlmann metric is the one
associated with the function $(1+x)/2$.

######  Proposition 2.

If $\rho$ is a pure state then ${\rm Var}_{\rho}^{f}(A)=2\,m_{f}(1,0)\cdot{\rm
Var}_{\rho}(A),$ cf. TothPetz:2013 .

######  Corollary.

If $\rho$ is a pure state and $f$ is non-regular, then ${\rm
Var}_{\rho}^{f}(A)=0.$

###### Proof.

If $f$ is non regular $m_{f}(1,0)=0$ QED

##  VI Quantum Fisher Information

The theory of quantum Fisher information is due to Petz and we recall here the
basic results. If ${\mathcal{N}}$ is a differentiable manifold we denote by
$T_{\rho}\mathcal{N}$ the tangent space to $\mathcal{N}$ at the point
$\rho\in{\mathcal{N}}$. Recall that there exists a natural identification of
$T_{\rho}{\mathcal{D}}^{1}_{n}$ with the space of self-adjoint traceless
matrices; namely, for any $\rho\in{\mathcal{D}}^{1}_{n}$

$T_{\rho}{\mathcal{D}}^{1}_{n}=\\{A\in M_{n}\mid
A=A^{*}\,,\,\hbox{Tr}\,A=0\\}.$

A stochastic map is a completely positive and trace preserving operator
$T:M_{n}\to M_{m}$. A monotone metric is a family of Riemannian metrics
$g=\\{g^{n}\\}$ on $\\{{\mathcal{D}}^{1}_{n}\\}$, $n\in\mathbb{N}$, such that

$g^{m}_{T(\rho)}(TX,TX)\leq g^{n}_{\rho}(X,X)$

holds for every stochastic map $T:M_{n}\to M_{m}$, every faithful state
$\rho\in{\mathcal{D}}^{1}_{n},$ and every $X\in
T_{\rho}{\mathcal{D}}^{1}_{n}$. Usually monotone metrics are normalized in
such a way that $[A,\rho]=0$ implies $g_{\rho}(A,A)={\rm
Tr}({\rho}^{-1}A^{2})$. A monotone metric is also called (an example of)
quantum Fisher information (QFI). This notation is inspired by Chentsov’s
uniqueness theorem for commutative monotone metrics Chentsov:1982 .

Define $L_{\rho}(A)=\rho A$ and $R_{\rho}(A)=A\rho$, and observe that
$L_{\rho}$ and $R_{\rho}$ are commuting positive superoperators on $M_{n}.$
For any $f\in{\mathcal{F}}_{op}$ one may also define the positive (non-linear)
superoperator $m_{f}(L_{\rho},R_{\rho})$. The fundamental theorem of monotone
metrics may be stated in the following way:

######  Theorem.

(See Petz:1996 ). There exists a bijective correspondence between monotone
metrics (quantum Fisher information(s)) on ${\mathcal{D}}^{1}_{n}$ and
functions $f\in{\mathcal{F}}_{op}$. The correspondence is given by the formula

$\langle A,B\rangle_{\rho,f}={\rm Tr}(A\cdot
m_{f}(L_{\rho},R_{\rho})^{-1}(B))$

for positive matrices $A$ and $B.$

##  VII Metric adjusted skew information

By using the general form of the quantum Fisher information it is possible to
greatly generalize the Wigner-Yanase information measure. To
$f\in\mathcal{F}_{op}$ the so-called Morosova function $c_{f}(x,y)$ is defined
by setting

$c_{f}(x,y)=\frac{1}{yf(xy^{-1})}=m_{f}(x,y)^{-1}\qquad x,y>0.$ (2)

The corresponding monotone symmetric metric $K_{\rho}$ is given by

$K_{\rho}^{f}(A,B)={\rm
Tr\hskip-1.99997pt}~{}A^{*}c_{f}\bigl{(}L_{\rho},R_{\rho}\bigr{)}B,$ (3)

where $L_{\rho}$ and $R_{\rho}$ denote left and right multiplication with
$\rho.$ Note that $K^{f}_{\rho}(A)$ is increasing in $c_{f}$ and thus
decreasing in $f.$ If furthermore $f$ is regular, the notion of metric
adjusted skew information (Hansen:2006b, , Definition 1.2) is defined by
setting

$I^{f}_{\rho}(A)=I^{f}(\rho,A)=\frac{f(0)}{2}K^{f}_{\rho}\bigl{(}i[\rho,A^{*}],i[\rho,A]\bigr{)},$
(4)

where $\rho>0.$ We use the second notation, $I^{f}(\rho,A),$ when the
expression of the state takes up too much space. We also tacitly extended the
metric adjusted skew information to arbitrary (non-self-adjoint) operators
$A.$ It is convex (Hansen:2006b, , Theorem 3.7) in the state variable $\rho$
and

$0\leq I^{f}_{\rho}(A)\leq{\rm Var}_{\rho}(A)$ (5)

with equality if $\rho$ is pure (Hansen:2006b, , Theorem 3.8), see also the
summery with interpretations in (CaiHansen:2010, , Theorem 1.2). Furthermore,
the notion of unbounded metric adjusted skew information for non-regular
functions in $\mathcal{F}_{op}$ is introduced in (CaiHansen:2010, , Theorem
5.1). For regular $f\in\mathcal{F}_{op}$ the metric adjusted skew information
may be written as

$I_{\rho}^{f}(A)={\rm Tr\hskip-1.99997pt}~{}\rho A^{2}-{\rm
Tr\hskip-1.99997pt}~{}A\,m_{\tilde{f}}(L_{\rho},R_{\rho})A,$

se (AudenaertCaiHansen:2008, , equation (7)). We thus obtain that the metric
adjusted skew information is decreasing in the transform $\tilde{f}$ for
arbitrary self-adjoint $A,$ that is

$\tilde{f}\leq\tilde{g}\quad\Rightarrow\quad I_{\rho}^{f}(A)\geq
I_{\rho}^{g}(A)\qquad\text{for}\quad f,g\in{\mathcal{F}}_{op}^{\,r}\,.$ (6)

We may also write

$\check{f}=\frac{f(0)}{f(t)}\qquad\text{and}\qquad\check{c}(x,y)=y^{-1}\check{f}(xy^{-1})$

and obtain

$I_{\rho}^{f}(A)=\frac{1}{2}{\rm
Tr\hskip-1.99997pt}~{}i[\rho,A^{*}]\check{c}\bigl{(}L_{\rho},R_{\rho}\bigr{)}i[\rho,A],$

cf. (AudenaertCaiHansen:2008, , equation (10)). It follows that the metric
adjusted skew information is increasing in $\check{f}$ for arbitrary $A.$ It
may be derived from (GibiliscoImparatoIsola:2007, , Proposition 6.3, page 11),
that the metric adjusted skew information can be expressed as the difference

$I^{f}_{\rho}(A)={\rm Var}_{\rho}(A)-{\rm Var}^{\tilde{f}}_{\rho}(A)$

with extension to the sesquilinear form

$I^{f}_{\rho}(A,B)={\rm Cov}_{\rho}(A,B)-{\rm Cov}^{\tilde{f}}_{\rho}(A,B).$

###  VII.1 Information inequalities

A function $f\colon\mathbb{R_{+}}\to\mathbb{R_{+}}$ is in $\mathcal{F}_{op}$
if and only if it allows a representation of the form

$f(t)=\frac{1+t}{2}\exp\int_{0}^{1}\frac{(\lambda^{2}-1)(1-t)^{2}}{(\lambda+t)(1+\lambda
t)(1+\lambda)^{2}}\,h_{f}(\lambda)\,d\lambda,$ (7)

where the weight function $h_{f}\colon[0,1]\to[0,1]$ is measurable. The
equivalence class containing $h_{f}$ is uniquely determined by $f,$ cf.
(AudenaertCaiHansen:2008, , Theorem 2.1). This representation gives rise to an
order relation in $\mathcal{F}_{op}.$

######  Definition 9.

Let $f,g\in\mathcal{F}_{op}.$ We say that $f$ is majorized by $g$ and write
$f\preceq g,$ if the function

$\varphi(t)=\frac{t+1}{2}\,\frac{f(t)}{g(t)}\qquad t>0$

is in $\mathcal{F}_{op}\,.$

The partial order relation $\preceq$ is stronger that the usual order relation
$\leq,$ and it renders $(\mathcal{F}_{op}\,,\preceq)$ into a lattice with

$f_{\text{min}}(t)=\frac{2t}{t+1}\qquad\text{and}\qquad
f_{\text{max}}(t)=\frac{t+1}{2}\qquad$ (8)

as respectively minimal element and maximal element. Furthermore,

$f\preceq g\quad\text{if and only if}\quad h_{f}\geq h_{g}\qquad\text{almost
everywhere},$ (9)

cf. (AudenaertCaiHansen:2008, , Theorem 2.4). The restriction of $\preceq$ to
the regular part of $\mathcal{F}_{op}$ induces a partial order relation
$\preceq$ on the set of metric adjusted skew informations.

######  Proposition 3.

The restriction of the order relation $\preceq$ renders the regular part of
$\mathcal{F}_{op}$ into a lattice. In addition, if one of two functions
$f,g\in\mathcal{F}_{op}$ is non-regular, then the minorant $f\wedge g$ is also
non-regular.

###### Proof.

Take $f\in\mathcal{F}_{op}$ with representative function $h_{f}$ as given in
(7). Then it follows that $f$ is regular if and only if the integral

$\int_{0}^{1}\frac{h_{f}(\lambda)}{\lambda}\,\hskip-1.99997pt\operatorname{\mathit{d}}\hskip-2.29996pt{}\lambda<\infty.$
(10)

Take now regular functions $f,g\in\mathcal{F}_{op\,.}$ We know that
$\bigl{(}\mathcal{F}_{op}\,,\preceq\bigr{)}$ is a lattice
(AudenaertCaiHansen:2008, , bottom of page 141), and that the representative
function in (7) for the minorant $f\wedge g$ is given by

$h_{f\wedge g}=\max\\{h_{f},h_{g}\\}\leq h_{f}+h_{g}$

showing that also $h_{f\wedge g}$ satisfies the integrability condition (10)
implying that $f\wedge g$ is regular. Since

$h_{f\vee g}=\min\\{h_{f},h_{g}\\}\leq h_{f}$

it also follows that the majorant is regular.

We now take functions $f,g\in\mathcal{F}_{op}$ with representative functions
$h_{f}$ and $h_{g}$ and assume that $f$ is non-regular. Since

$h_{f\wedge g}=\max\\{h_{f},h_{g}\\}\qquad\text{and thus}\qquad h_{f}\leq
h_{f\wedge g}$

we obtain that also the minorant $f\wedge g$ is non-regular. QED

###  VII.2 The Wigner-Yanase-Dyson skew informations

The Wigner-Yanase-Dyson skew information (with parameter $p)$ is defined by
setting

$I_{p}(\rho,A)=-\frac{1}{2}{\rm
Tr\hskip-1.99997pt}~{}[\rho^{p},A[[\rho^{1-p},A],\qquad 0<p<1.$

It is an example of a metric adjusted skew information and reduces to the
Wigner-Yanase skew information for $p=1/2\,.$ The representing function
$f_{p}$ of $I_{p}(\rho,A)$ is given by

$f_{p}(t)=p(1-p)\cdot\frac{(t-1)^{2}}{(t^{p}-1)(t^{1-p}-1)}\qquad 0<p<1,$

that is $I_{p}(\rho,A)=I^{f_{p}}_{\rho}(A).$ The weight-functions
$h_{p}(\lambda)$ in equation (7) corresponding to the representing functions
$f_{p}$ are given by

$h_{p}(\lambda)=\frac{1}{\pi}\arctan\frac{(\lambda^{p}+\lambda^{1-p})\sin
p\pi}{1-\lambda-(\lambda^{p}-\lambda^{1-p})\cos p\pi}\qquad 0<\lambda<1.$

It is a non-trivial result that the Wigner-Yanase-Dyson skew informations
$I_{p}(\rho,A)$ are increasing in the parameter $p$ for $0<p\leq 1/2$ and
decreasing in $p$ for $1/2\leq p<1$ with respect to the order relation
$\preceq,$ cf. (AudenaertCaiHansen:2008, , Theorem 2.8). The Wigner-Yanase
skew information is thus the maximal element among the Wigner-Yanase-Dyson
skew informations with respect to the order relation $\preceq.$

###  VII.3 The monotonous bridge

The family of metrics with representing functions

$f_{\alpha}(t)=t^{\alpha}\left(\frac{1+t}{2}\right)^{1-2\alpha}\qquad t>0,$

decrease monotonously (with respect to $\preceq)$ from the largest monotone
symmetric metric down to the Bures metric for $\alpha$ increasing from $0$ to
$1.$ They correspond the the constant weight functions
$h_{\alpha}(\lambda)=\alpha$ in equation (7). However, the only regular metric
in this bridge is the Bures metric $(\alpha=1).$ It is however possible to
construct a variant bridge by choosing the weight functions

$h_{p}(\lambda)=\left\\{\begin{array}[]{lrl}0,&\lambda&<1-p\\\\[4.30554pt]
p,&\lambda&\geq 1-p\end{array}\right.\qquad 0\leq p\leq 1$

in equation (7) instead of the constant weight functions. It is non-trivial
that these weight functions provide a monotonously decreasing bridge (with
respect to $\preceq)$ of monotone symmetric metrics between the smallest and
the largest (monotone symmetric) metric. The benefit of this variant bridge is
that all the constituent metrics are regular except for $p=1.$

##  VIII Metric adjusted local quantum uncertainty

We consider a bipartite system
$\mathcal{H}=\mathcal{H}_{1}\otimes\mathcal{H}_{2}$ of two finite dimensional
Hilbert spaces.

######  Definition 10.

Let $f\in\mathcal{F}_{\text{op}}$ be regular and take a vector
$\Lambda\in\mathbf{R}^{d}.$ We define the Metric Adjusted Local Quantum
Uncertainty (or $f$-LQU) by setting

$\mathcal{U}_{1}^{\Lambda,f}(\rho)=\inf\\{I^{f}_{\rho}(K_{1}\otimes 1_{2})\mid
K_{1}\text{ has spectrum $\Lambda$}\\},$ (11)

where $\rho_{12}$ is a bipartite state, and $K_{1}$ is the partial trace of an
observable $K$ on $\mathcal{H}.$

The minimum in the above definition is thus taken over local observables
$K_{1}\otimes 1_{2}\in B(\mathcal{H}_{1}\otimes H_{2})$ such that $K_{1}$ is
unitarily equivalent with the diagonal matrix $\text{diag}(\Lambda).$

######  Remark 3.

The metric adjusted LQU has been studied in the literature for specific
choices of $f.$

  * •

If $f(x)=f_{WY}(x)=\Bigl{(}\frac{1+\sqrt{x}}{2}\Bigr{)}^{2}$ then
${\mathcal{U}}_{1}^{\Lambda,f}$ coincide with the LQU introduced in (GTA:2013,
, equation 2).

  * •

If $f(x)=f_{SLD}(x)=\frac{1+x}{2}$ then $\mathcal{U}_{1}^{\Lambda,f}$
coincides with the Interferometric Power introduced in GSGTFSSOA:2014 .

######  Proposition 4.

For $f,g\in\mathcal{F}_{op}^{\,r}$ with $\tilde{g}\leq\tilde{f}$ we have the
inequality
${\mathcal{U}}^{\Lambda,f}_{1}(\rho_{12})\leq{\mathcal{U}}^{\Lambda,g}_{1}(\rho_{12}).$
In particular the LQU is smaller than the IP.

###### Proof.

Let $\tilde{K}_{1}$ be the local observable with spectrum $\Lambda$ minimizing
the metric adjusted skew information. Then

${\mathcal{U}}^{\Lambda,f}_{1}(\rho_{12})=I^{f}_{\rho_{12}}\bigl{(}\tilde{K}_{1}\otimes
1_{2}\bigr{)}\geq I^{g}_{\rho_{12}}\bigl{(}\tilde{K}_{1}\otimes
1_{2}\bigr{)}\geq{\mathcal{U}}^{\Lambda,g}_{1}(\rho_{12}),$

where we used the inequality in (6). QED

######  Corollary.

Let $g_{1}$ and $g_{2}$ be regular functions in $\mathcal{F}_{op}$ and set
$f=\tilde{g}_{1}\wedge\tilde{g}_{2}$ with respect to the lattice structure in
$\mathcal{F}_{op}\,.$ Then there is a regular function $g$ in
$\mathcal{F}_{op}$ such that $\tilde{g}=f=\tilde{g}_{1}\wedge\tilde{g}_{2}$
and

$\max\bigl{\\{}{\mathcal{U}}^{\Lambda,g_{1}}_{1}(\rho_{12}),\,{\mathcal{U}}^{\Lambda,g_{2}}_{1}(\rho_{12})\bigr{\\}}\leq{\mathcal{U}}^{\Lambda,g}_{1}(\rho_{12})$

for arbitrary $\rho_{12}\,.$

###### Proof.

The functions $\tilde{g}_{1}$ and $\tilde{g}_{2}$ are non-regular by Theorem
Theorem. By Proposition 3 we thus obtain that also the minorant $f$ is non-
regular. Therefore there exists, by the correspondence in Theorem Theorem, a
(unique) regular function $g$ in $\mathcal{F}_{op}$ such that $\tilde{g}=f.$
The assertion then follows by Proposition 4. QED

Following BogaertGirolami:2017  we prove that the metric adjusted LQU is a
measure of non-classical correlations, i.e. it meets the criteria which
identify discord-like quantifiers, see rev .

######  Theorem.

If the state $\rho$ is classical-quantum in the sense of Piani , then the
metric adjusted LQU vanishes, that is $\mathcal{U}_{1}^{\Lambda,f}(\rho)=0.$
Conversely, if the coordinates of $\Lambda$ are mutually different (thus
rendering the operator $K_{1}$ non-degenerate) and
$\mathcal{U}_{1}^{\Lambda}(\rho)=0,$ then $\rho$ is classical-quantum.

###### Proof.

We note that the metric adjusted skew information $I_{\rho}^{f}(A)$ for a
faithful state $\rho$ is vanishing if and only if $\rho$ and $A$ commute. If
$\rho$ is classical-quantum, then

$P_{1}(\rho)=\sum_{i}(P_{1,i}\otimes 1_{2})\rho(P_{1,i}\otimes 1_{2})=\rho$

for some von Neumann measurement $P$ given by a resolution $(P_{i})$ of the
identity $1_{1}$ in terms of one-dimensional projections. We may choose
$K_{1}$ diagonal with respect to this resolution, so $K_{1}\otimes 1_{2}$ and
$\rho$ commute and thus $\mathcal{U}_{1}^{\Lambda,f}(\rho)=0.$

If on the other hand the $f$-LQU $\mathcal{U}_{1}^{\Lambda,f}(\rho)=0,$ then
there exist a local observable $K_{1}\otimes 1_{2}$ such that
$[\rho,K_{1}\otimes 1_{2}]=0.$ By the spectral theorem we write

$K_{1}=\sum_{i}\lambda_{i}P_{1,i}=\sum_{i}\lambda_{i}|i\rangle_{1}\langle i|$

and since

$\rho(K_{1}\otimes 1_{2})=(K_{1}\otimes 1_{2})\rho$

we obtain by multiplying with $P_{1,i}\otimes 1_{2}$ from the left and
$P_{1,j}\otimes 1_{2}$ from the right the identity

$\lambda_{j}(P_{1,i}\otimes 1_{2})\rho(P_{1,j}\otimes
1_{2})=\lambda_{i}(P_{1,i}\otimes 1_{2})\rho(P_{1,j}\otimes 1_{2}).$

If $K_{1}$ is non-degenerate, it thus follows that

$(P_{1,i}\otimes 1_{2})\rho(P_{1,j}\otimes 1_{2})=0\qquad\text{for}\quad i\neq
j.$

By summing over all $j$ different from $i$ we obtain

$(P_{1,i}\otimes 1_{2})\rho((1_{1}-P_{1,i})\otimes 1_{2})=0,$

thus

$(P_{1,i}\otimes 1_{2})\rho=(P_{1,i}\otimes 1_{2})\rho(P_{1,i}\otimes 1_{2}),$

so $P_{1,i}\otimes 1_{2}$ and $\rho$ commute. It follows that

$P_{1}(\rho)=\sum_{i}(P_{1,i}\otimes 1_{2})\rho(P_{1,i}\otimes 1_{2})=\rho,$

so $\rho$ is left invariant under the von Neumann measurement $P$ given by
$(P_{i}).$ Therefore, $\rho$ is classical-quantum. QED

Recall that Luo and Zhang LuoZang:2008  proved that a state $\rho$ is
classical-quantum if and only if there exists a resolution $(P_{i})$ of the
identity $1_{1}$ such that

$\rho=\sum_{i}p_{i}P_{1,i}\otimes\rho_{i},$

where each $\rho_{i}$ is a state on $\mathcal{H}_{2}$ and $p_{i}\geq 0,$ and
the sum $\sum_{i}p_{i}=1.$ By (CaiHansen:2010, , Lemma 3.1) the inequality

$I_{\rho}^{f}(K_{1}\otimes 1_{2})\geq I_{\rho_{1}}^{f}(K_{1})$

is valid, where $\rho_{1}={\rm Tr\hskip-1.99997pt}~{}_{2}\,\rho_{12}\,.$
Consequently, we obtain that

$\mathcal{U}_{1}^{\Lambda,f}(\rho)\geq\inf_{K_{1}}\,I^{f}_{\rho_{1}}(K_{1})=\inf_{\sigma_{1}}\,I^{f}_{\sigma_{1}}\bigl{(}K_{1}\bigr{)},$
(12)

where the minimum is taken over states $\sigma_{1}$ on $\mathcal{H}_{1}$
unitarily equivalent with $\rho_{1}.$

######  Theorem.

The metric adjusted LQU is invariant under local unitary transformations.

###### Proof.

For the metric adjusted skew information and local unitary transformations we
have

$\begin{array}[]{l}\mathcal{U}_{1}^{\Lambda,f}\bigl{(}(U_{1}\otimes
U_{2})\rho_{12}(U_{1}\otimes U_{2})^{\dagger}\bigr{)}\\\\[6.45831pt]
=\min_{K_{1}}I^{f}\bigl{(}(U_{1}\otimes U_{2})\rho_{12}(U_{1}\otimes
U_{2})^{\dagger},K_{1}\otimes 1_{2}\bigr{)}\\\\[6.45831pt]
=\min_{K_{1}}I^{f}\bigl{(}\rho_{12},(U_{1}\otimes
U_{2})^{\dagger}(K_{1}\otimes 1_{2})(U_{1}\otimes
U_{2})\bigr{)}\\\\[6.45831pt]
=\min_{K_{1}}I^{f}\bigl{(}\rho_{12},(U_{1}^{\dagger}K_{1}U_{1}\otimes
1_{2}\bigr{)}=\mathcal{U}^{\Delta,f}_{1}(\rho_{12}),\end{array}$

where we used the definition in (11). QED

######  Theorem.

The metric adjusted LQU is contractive under completely positive trace-
preserving maps on the non-measured subsystem.

###### Proof.

Let $\tilde{K}_{1}$ be the local observable minimizing the metric adjusted
skew information. A completely positive trace preserving map $\Phi_{2}$ on
system 2 is obtained as an amplification followed by a partial trace
(Stinespring dilation): ${\rm
Tr\hskip-1.99997pt}~{}_{3}(U_{23}\rho_{23}U_{23}^{\dagger})=\Phi_{2}\,\rho_{2}$.
The metric adjusted LQU is invariant under local unitaries. Also, the metric
adjusted skew information is contractive under partial trace. Calling $d_{3}$
the dimension of the Hilbert space of the ancillary system 3, one has

$\displaystyle\mathcal{U}_{1}^{\Lambda,f}(\rho_{12})=$
$\displaystyle\,I^{f}\left(\rho_{12},\tilde{K}_{1}\otimes
1_{2}\right)=I^{f}\left(\rho_{12}\otimes\frac{1}{d_{3}}1_{3},\tilde{K}_{1}\otimes
1_{23}\right)$ $\displaystyle=$ $\displaystyle\,\ I^{f}\left((1_{1}\otimes
U_{23})\left(\rho_{12}\otimes\frac{1}{d_{3}}1_{3}\right)(1_{1}\otimes
U_{23}^{\dagger}),\tilde{K}_{1}\otimes 1_{23}\right)$ $\displaystyle\geq$
$\displaystyle\,I^{f}\left({\rm
Tr\hskip-1.99997pt}~{}_{3}\left\\{(1_{1}\otimes
U_{23})\left(\rho_{12}\otimes\frac{1}{d_{3}}1_{3}\right)(1_{1}\otimes
U_{23}^{\dagger})\right\\},\tilde{K}_{1}\otimes 1_{2}\right)$ $\displaystyle=$
$\displaystyle\,I^{f}\left((1_{1}\otimes\Phi_{2})\rho_{12},\tilde{K}_{1}\otimes
1_{2}\right)$ $\displaystyle\geq$
$\displaystyle\,\mathcal{U}_{1}^{\Lambda,f}\bigl{(}(1_{1}\otimes\Phi_{2})\rho_{12}\bigr{)},$

as desired. QED

######  Theorem.

The metric adjusted LQU reduces to an entanglement monotone for pure states.

###### Proof.

The metric adjusted $f$-LQU coincides with the standard variance on pure
states, that is

$I_{\rho}^{f}(A)={\rm Var}_{\rho}(A)={\rm Tr\hskip-1.99997pt}~{}\rho
A^{2}-({\rm Tr\hskip-1.99997pt}~{}\rho A)^{2}$

whenever $\rho$ is pure (Hansen:2006b, , Theorem 3.8). But in GTA:2013  it has
been proved that the minimum local variance is an entanglement monotone for
pure states. QED

##  IX Conclusion

In this work, we have built a unifying information-geometric framework to
quantify quantum correlations in terms of metric adjusted skew informations.
We extended the physically meaningful definition of LQU to a more general
class of information measures. Crucially, metric adjusted quantum correlation
quantifiers enjoy, by construction, a set of desirable properties which make
them robust information measures.
An important open question is whether information geometry methods may help
characterize many-body quantum correlations. In general, the very concept of
multipartite statistical dependence is not fully grasped in the quantum
scenario. In particular, we do not have axiomatically consistent and
operationally meaningful measures of genuine multipartite quantum discord.
Unfortunately, the LQU and IP cannot be straightforwardly generalized to
capture joint properties of more than two quantum particles. A promising
starting point could be to translate into the information-geometry language
the entropic multipartite correlation measures developed in weaving . We plan
to investigate the issue in future studies.

###### Acknowledgements.

This research is supported by a Rita Levi Montalcini Fellowship of the Italian
Ministry of Research and Education (MIUR), grant number 54$\\_$AI20GD01.

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