# Revisiting weak values through non-normality

Lorena Ballesteros Ferraz<EMAIL_ADDRESS>naXys, Namur Institute
for Complex Systems, University of Namur, Belgium  NISM, Namur Institute of
Structured Matter, University of Namur, Belgium  Research Unit Lasers and
Spectroscopies (UR-LLS), Department of Physics, University of Namur, Belgium
Riccardo Muolo  naXys, Namur Institute for Complex Systems, University of
Namur, Belgium  Department of Mathematics, University of Namur, Belgium  Yves
Caudano  naXys, Namur Institute for Complex Systems, University of Namur,
Belgium  NISM, Namur Institute of Structured Matter, University of Namur,
Belgium  Research Unit Lasers and Spectroscopies (UR-LLS), Department of
Physics, University of Namur, Belgium  Timoteo Carletti  naXys, Namur
Institute for Complex Systems, University of Namur, Belgium  Department of
Mathematics, University of Namur, Belgium

(August 30, 2024)

###### Abstract

Quantum measurement is one of the most fascinating and discussed phenomena in
quantum physics, due to the impact on the system of the measurement action and
the resulting interpretation issues. Scholars proposed weak measurements to
amplify measured signals by exploiting a quantity called a weak value, but
also to overcome philosophical difficulties related to the system perturbation
induced by the measurement process. The method finds many applications and
raises many philosophical questions as well, especially about the proper
interpretation of the observations. In this paper, we show that any weak value
can be expressed as the expectation value of a suitable non-normal operator.
We propose a preliminary explanation of their anomalous and amplification
behavior based on the theory of non-normal matrices and their link with non-
normality: the weak value is different from an eigenvalue when the operator
involved in the expectation value is non-normal. Our study paves the way for a
deeper understanding of the measurement phenomenon, helps the design of
experiments, and it is a call for collaboration to researchers in both fields
to unravel new quantum phenomena induced by non-normality.

##  I Introduction

Weak values play a role similar to expectation values in certain types of
quantum measurements. However, they can be anomalous, meaning that they lie
outside the range of eigenvalues of the measured operator, which represents a
determined quantum property. Hence they can become complex numbers and be
unbounded [1, 2, 3]. Anomalous weak values are used to amplify signals [4], to
measure complex properties such as wave functions [5] or expectation values of
non-Hermitian operators [6], and to study fundamental quantum phenomena, like
paradoxes [7, 8, 9, 10]. Anomalous weak values evidence an intrinsic property
of non-classicality, called contextuality [11, 12]. A particular class of
anomalous weak values is associated to amplification. This phenomenon occurs
when the modulus of the weak value is larger than the modulus of all the
observable eigenvalues. Such weak values are said to be in the amplifying
range.

Weak values were defined in the context of the weak von Neumann measurement
protocol with post-selection [13]. This procedure, schematically shown in Fig.
1, is called a weak measurement. The latter is performed by using the system
under scrutiny and a meter (or ancilla), each one represented by a horizontal
line in the scheme depicted in Fig. 1. The measurement protocol proceeds in
four steps. Firstly, the pre-selection step chooses the system initial state
${|{\psi_{i}}\rangle}$ (see the blue rectangle in Fig. 1). Similarly, the
known state ${|{\alpha}\rangle}$ defines the initial reference point of the
ancilla. In practice, these states are typically obtained via projective
measurements. Secondly, a reversible system–ancilla interaction is implemented
through a unitary operator $\hat{U}=e^{i\gamma\hat{O}\otimes\hat{P}}$ (green
panel in Fig. 1). The observable of interest $\hat{O}$ operates onto the
system space, while the momentum operator $\hat{P}$, called the pointer of the
protocol, belongs to the meter space; $\gamma$ is the interaction strength
(for simplicity, we set the normalized Plank constant $\hbar=1$ in the unitary
operator). This step entangles the system and the ancilla. For a weak
measurement, the interaction strength should be small, enabling the expansion
of the unitary operator in a Taylor series up to the first order. In this way,
the measurement only slightly modifies the system state, limiting the
measurement-induced perturbation. In return, the extracted information in a
single measurement is very limited as well. The third step, post-selection,
consists of a projective measurement imposed on the system, followed by
filtering to set into a pre-determined final state ${|{\psi_{f}}\rangle}$ the
system at the end of the protocol (see the yellow panel in Fig. 1).
Eventually, information about the pre- and post-selected system is extracted
by reading out the ancilla wave function conditioned on successful post-
selection (see the pink panel in Fig. 1).

Figure 1: Scheme of a weak measurement with post-selection. In the first step,
pre-selection, the initial state of the system is set to
${|{\psi_{i}}\rangle}$ (blue). Then, the system and ancilla interact through a
unitary operator $U=e^{-i\gamma\hat{O}\otimes\hat{P}}$ (green). This
interaction should be weak, meaning that the interaction strength $\gamma$
should be small. After the weak interaction, a projective measurement is
executed in the system and the final state is chosen to be
${|{\psi_{f}}\rangle}$ (yellow). If the post-selection is successful, the
ancilla position is measured by applying a projective measurement in the
ancilla (pink). The expectation value of the position is shifted by a quantity
that is proportional to the weak value $O_{w}$ of the system’s observable
$\hat{O}$ for the pre- and post-selected system states.

At the end of the protocol, the ancilla wave function is shifted in position
representation by a quantity proportional to the real part of the weak value,
$O_{w}=\frac{{\langle{\psi_{f}}|}\hat{O}{|{\psi_{i}}\rangle}}{\braket{\psi_{f}}{\psi_{i}}}$,
where ${|{\psi_{i}}\rangle}$ and ${|{\psi_{f}}\rangle}$ are the pre- and post-
selected states, while the ancilla wave function in momentum representation is
shifted by a quantity proportional to the imaginary part of the weak value.
Since the interaction strength is weak, many measurements must be performed
and averaged to determine the weak value precisely. In the absence of post-
selection, the von Neumann protocol describes standard measurements for
arbitrary interaction strengths, where the measuring device –represented by
the ancilla– is treated quantumly. Thus, in quantum measurements involving
weak interaction and post-selection, the weak value substitutes the
expectation value of the observable in the average wave function shift of the
meter.

The significance of weak values in physics extends beyond weak measurements.
They appear in strong quantum measurements [14], and in homodyne measurements
[15]. In general measurements, the real and the imaginary components of the
weak value are linked to the optimal conditional estimate of the observable
and the inaccuracy, respectively [16, 17, 18, 19]. They also contribute
intrinsically to dynamical phenomena [17]. While the real and imaginary parts
generate the meter shifts in typical weak measurements, the modulus and
argument of weak values also hold significance [20, 21, 22], particularly as
the argument characterizes geometric phases within the quantum state manifold.
In this work, special attention will be paid to the modulus of weak values.

The weak value of a generic observable $\hat{O}$, can always be expressed as
the expectation value of an operator, defined in terms of the pre-selected
state, the observable and the post-selected state,
$O_{w}=\text{Tr}\left[\hat{\Pi}_{i}\hat{A}\right]$, with
$\hat{A}=\frac{\hat{O}\hat{\Pi}_{i}}{\text{Tr}\left[\hat{\Pi}_{f}\hat{\Pi}_{i}\right]}$
and $\hat{\Pi}_{i}={|{\psi_{i}}\rangle}{\langle{\psi_{i}}|}$, resp.
$\hat{\Pi}_{f}={|{\psi_{f}}\rangle}{\langle{\psi_{f}}|}$, the projector of the
pre-selected, resp. post-selected, state. This operator is not necessarily
Hermitian. Most non-Hermitian operators are also non-normal, meaning that
$\hat{A}\hat{A}^{\dagger}\neq\hat{A}^{\dagger}\hat{A}$. The choice of
definition of $\hat{A}$ is arbitrary to some extent, as the weak value could
also be defined in terms of an operator involving the post-selected state
$\hat{A}^{\prime}$,
$O_{w}=\text{Tr}\left[\hat{\Pi}_{f}\hat{A}^{\prime}\right]$, with
$\hat{A}^{\prime}=\frac{\hat{\Pi}_{f}\hat{O}}{\text{Tr}\left[\hat{\Pi}_{f}\hat{\Pi}_{i}\right]}$.
In the framework of matrices, relevant to this paper, the above definition of
non-normality implies that the matrix cannot be diagonalized through an
orthonormal transformation [23]. Non-normality is a stronger relation of
asymmetry than simple non-Hermicity: in fact, all non-normal matrices are non-
Hermitian, but there is a class of non-Hermitian matrices which are normal,
the circulant matrices [24]. Such stronger asymmetry can be thought of as a
hierarchical structure of the matrix. This clearly emerges when we consider
networks: in such context, the adjacency matrix represents the way in which
the nodes (i.e, the units of the network) are connected to each other; when
the structure of the connections is hierarchical and directed, the adjacency
matrix is non-normal [25, 26]. The latter framework results particularly
interesting for applications, as real-world networks are non-normal [27].

A first main result of this work is to show that the operators $\hat{A}$ and
$\hat{A}^{\prime}$ must be non-normal for the weak value to become anomalous.
When imposing an arbitrary post-selection, the weakly measured quantity
changes from an expectation value of a Hermitian operator representing the
observable of interest, to an expectation value of a specific non-normal
operator. Interpreting weak values from this point of view might involve
studying the energetics of the protocol, especially the post-selection
process. Possible links with open systems and non-Hermitian quantum physics
[28] might appear, as post-selection involves a discarding process.

Non-normality of matrices and operators has been studied in numerical analysis
by Trefethen and collaborators [23, 29], triggered by the effects observed in
fluid dynamics [30]. In fact, such property makes the matrix more sensible to
perturbations, which results in an amplifying effect of the latter, with
dramatic consequences on the dynamics. For example, in fluid dynamics, it
gives rise to a premature emergence of turbulence [23], while in epidemics it
may lead to a lower threshold for an outbreak [31]. Effects of non-normality
have also been studied in ecology [32, 33], Turing pattern formation [34],
chemical reaction networks [35], and synchronization dynamics [24]. Such a
change in the behavior of the system can be ascribed to the possible emergence
of an initial transient growth, whose intensity is proportional to the non-
normality of the system [27]. The degree of non-normality is estimated by the
spectral properties of the matrix, resulting in different metrics [23, 27].
Those metrics will be used in the following to assess the strength of the
anomalous weak value. In particular, we will be interested in using the
Henrici departure from normality.

In this paper, we demonstrate that non-normality is a necessary condition for
obtaining a weak value that is different from an eigenvalue of the observable.
We will show that, in order to have a large weak value, the operator $\hat{A}$
and $\hat{A}^{\prime}$ must be non-normal; hence, the latter property is
necessary to obtain amplification. Furthermore, by comparing the modulus of
the weak value and the Henrici departure from normality with varying pre- and
post-selected states, we make a direct connection between the degree of weak-
value amplification and the degree of non-normality of the operator $\hat{A}$.
As a next step, we consider a varying observable, which is the other component
involved in the weak value. Ordinarily, experimenters select pre- and post-
selected states that are nearly orthogonal in order to achieve amplification.
However, there are cases in which we have some freedom in the choice of the
observable. In this study, we demonstrate the critical role that this choice
plays in the amplification yield. By varying the observable, we discovered
that the maximum weak value always occurs at the arithmetic average of the
maximum Henrici departure from normality of both matrices $\hat{A}$ and
$\hat{A}^{\prime}$. Furthermore, the weak value tends toward infinity when the
points at which the matrices $\hat{A}$ and $\hat{A}^{\prime}$ are degenerate
and nilpotent coincide.

The paper is organized as follows: in the next section, we formally express
the weak value as the expectation value of a non-normal operator. In section
III, we discuss our numerical results, showing the correlation between the
non-normality of the operator and the amplification of the weak value. Then,
in section IV, we study the correlation between the Henrici departure from
normality and the weak value when varying the analyzed observable, showing
that the anomalous weak value is still correlated to non-normality. Finally,
section V is left for the discussion of the results and conclusions.

##  II Weak values as expectation values of non-normal operators

The aim of this section is to show the importance of non-normality for the
emergence of weak values. To achieve this goal, we rewrite the weak value as
the expectation value of a non-normal operator, $\hat{A}$. Then, we prove that
when the latter operator is normal, the weak value is simply an eigenvalue of
the observable, $\hat{O}$. This shows that weak values are directly linked to
non-normal operators.

Let us consider a system that is initialized on the pure quantum state
${|{\psi_{i}}\rangle}$. The projector representing this state is
$\hat{\Pi}_{i}={|{\psi_{i}}\rangle}{\langle{\psi_{i}}|}$. The post-selected
projector is $\hat{\Pi}_{f}={|{\psi_{f}}\rangle}{\langle{\psi_{f}}|}$ and the
observable of interest, namely a Hermitian operator, is $\hat{O}$.

The weak value of the system can be expressed as

$O_{w}=\frac{{\langle{\psi_{f}}|}\hat{O}{|{\psi_{i}}\rangle}}{\braket{\psi_{f}}{\psi_{i}}}=\frac{\text{Tr}\left[\hat{\Pi}_{f}\hat{O}\hat{\Pi}_{i}\right]}{\text{Tr}\left[\hat{\Pi}_{f}\hat{\Pi}_{i}\right]}\,.$
(1)

A simple algebraic manipulation of the previous formula expresses the weak
value in terms of the expectation value of the operator $\hat{A}$,

$O_{w}={\langle{\psi_{f}}|}\hat{A}{|{\psi_{f}}\rangle}\,,$ (2)

where the non-normal operator $\hat{A}$ is defined in terms of the operator
$\hat{O}$ and the pre- and post-selected states as follows:

$\hat{A}=\frac{\hat{O}\hat{\Pi}_{i}}{\text{Tr}\left[\hat{\Pi}_{f}\hat{\Pi}_{i}\right]}\,.$
(3)

To measure the non-normality of a matrix $M$, we use the Henrici departure
from normality, $d_{f}$, defined as

$d_{f}\left(M\right)=\sqrt{||M||^{2}_{F}-\sum_{i=1}^{n}|\lambda_{i}|^{2}},$
(4)

where $\lambda_{i}$ are the eigenvalues and
$||M||_{F}=\sqrt{\sum_{i,j=1}^{n}|m_{ij}|^{2}}$, the Frobenius norm of the
matrix $M$ whose elements are $m_{ij}$ [23]. The Henrici departure from
normality of the matrix $\hat{A}$ is thus,

$d_{f}\left(\hat{A}\right)=\frac{\sqrt{{\langle{\psi_{i}}|}\hat{O}^{2}{|{\psi_{i}}\rangle}-{\langle{\psi_{i}}|}\hat{O}{|{\psi_{i}}\rangle}^{2}}}{|\braket{\psi_{f}}{\psi_{i}}|^{2}}=\frac{\Delta_{i}\hat{O}}{|\braket{\psi_{f}}{\psi_{i}}|^{2}},$
(5)

where $\Delta_{i}\hat{O}$ is the uncertainty of the observable $\hat{O}$ in
the initial state. The details on the derivation of this equation can be found
in Appendix A. The Henrici departure from normality of $\hat{A}$ vanishes and
the matrix is normal, only when the expectation value of $\hat{O}^{2}$ in the
initial state is equal to the square of the expectation of $\hat{O}$ in the
initial state, namely

${\langle{\psi_{i}}|}\hat{O}^{2}{|{\psi_{i}}\rangle}={\langle{\psi_{i}}|}\hat{O}{|{\psi_{i}}\rangle}^{2}\,.$
(6)

However the Cauchy-Schwarz inequality implies that,

${\langle{\psi_{i}}|}\hat{O}^{2}{|{\psi_{i}}\rangle}\geq{\langle{\psi_{i}}|}\hat{O}{|{\psi_{i}}\rangle}^{2},$
(7)

where the equality holds true only when ${|{\psi_{i}}\rangle}$ is an
eigenvector of $\hat{O}$,
$\hat{O}{|{\psi_{i}}\rangle}=\lambda{|{\psi_{i}}\rangle}$. In that case, the
operator is normal and the weak value is simply the eigenvalue $\lambda$.

Obviously, defining the operator $\hat{A}$ in terms of the final or the
initial state is an arbitrary choice. Hence the weak value can also be defined
as the expectation value in the initial state,

$O_{w}^{\prime}={\langle{\psi_{i}}|}\hat{A}^{\prime}{|{\psi_{i}}\rangle}\,,$
(8)

of the operator $\hat{A}^{\prime}$ given by

$\hat{A}^{\prime}=\frac{\hat{\Pi}_{f}\hat{O}}{\text{Tr}\left[\hat{\Pi}_{f}\hat{\Pi}_{i}\right]}.$
(9)

Weak values are called anomalous when the imaginary part is different from $0$
or, otherwise, when their value lies outside the range of the spectrum of the
Hermitian operator $\hat{O}$ [11, 36, 37], namely

$\imaginary O_{w}\neq 0\hskip 14.22636pt\text{or}\hskip
14.22636ptO_{w}>\text{max}\left(\langle\hat{O}\rangle\right)=\text{max}\left(\text{Tr}\left(\hat{\rho}\hat{O}\right)\right)=\text{max}\left(\lambda_{i}\right)\hskip
14.22636pt\text{or}\hskip
14.22636ptO_{w}<\text{min}\left(\langle\hat{O}\rangle\right)=\text{min}\left(\text{Tr}\left(\hat{\rho}\hat{O}\right)\right)=\text{min}\left(\lambda_{i}\right)\,,$
(10)

where $\lambda_{i}$ are the eigenvalues of the operator $\hat{O}$ and
$\hat{\rho}$ is an arbitrary (not necessarily pure) quantum state represented
here as a density operator. Amplifying weak values correspond to
$|O_{w}|>\max|\lambda_{i}|$.

If $\hat{A}$ or $\hat{A}^{\prime}$ are normal, the weak value cannot be
anomalous. Moreover, the weak value is one of the eigenvalues of the
observable $\hat{O}$. The operator $\hat{A}$, resp. $\hat{A}^{\prime}$, have
both eigenvalue zero with multiplicity $N-1$ and one eigenvalue equal to an
expectation of the observable in the initial state
${\langle{\psi_{i}}|}\hat{O}{|{\psi_{i}}\rangle}$, resp. final state
${\langle{\psi_{f}}|}\hat{O}{|{\psi_{f}}\rangle}$. Another interesting case
arises when the expectation value is equal to $0$. In such a setting, the weak
value is not necessarily equal to zero, but the non-normal matrix has all
eigenvalues equal to zero and it is degenerate, hence it is a nilpotent matrix
111The only matrix with a zero spectrum that is not nilpotent is the zero
matrix, which is not degenerate, since it has full geometric multiplicity. Our
case does not fall in the latter, our matrix being non-normal..

We have thus shown that weak values can be different from an eigenvalue only
if both operators $\hat{A}$ and $\hat{A}^{\prime}$ are non-normal. Since the
anomalous properties are fundamental in all applications of weak values, our
setting becomes interesting when the involved operators are non-normal.

Beyond this paper specific focus on weak values, we would like to stress that
our results also establish a truly general connection between quantum
fluctuations and non-normality. Indeed, assuming identical initial and final
pure states $\hat{\Pi}_{f}=\hat{\Pi}_{i}$, the weak value (1) is simply the
expectation value $\langle\hat{O}\rangle_{i}$ of the observable in the
considered quantum state $\hat{\Pi}_{i}$. We can as well express it as the
average of the non-normal operator $\hat{A}=\hat{O}\hat{\Pi}_{i}$ in the
initial state, given simply by the trace $\Tr(\hat{O}\hat{\Pi}_{i})$. Then,
Henrici’s departure from non-normality becomes exactly equal to the
uncertainty of the observable in the quantum state
$d_{f}(\hat{O}\hat{\Pi}_{i})=\Delta_{i}\hat{O}$. It appears thus that the non-
normality of the operator $\hat{O}\hat{\Pi}_{i}$ is a measure of the quantum
fluctuations around the observable expectation value evaluated through the
trace of the same operator $\hat{O}\hat{\Pi}_{i}$. When the state
$\hat{\Pi}_{i}$ is an eigenstate of the observable $\hat{O}$, the operator is
normal and the quantum uncertainty is zero. Indeed, measurements of a quantum
system in an eigenstate of the probed observable yield the associated
eigenvalue with probability 1. This general link between quantum uncertainties
and non-normality, and their relationship with Henrici’s departure from
normality in particular, does not seem to have been recognized in practice, to
the best of our knowledge.

##  III Correlation between non-normality and amplifying weak values

Non-normality, as we have shown in the previous sections, is necessary to
obtain weak values different from an eigenvalue of the observable, and thus
also to have amplifying and complex weak values. The goal of this section is
to study the relation between the level of amplification of the modulus of the
weak value, and the non-normality of the matrix $\hat{A}$, defined in Eq.(3).
We show that there is a direct dependence between the two. The larger the
amplification, the larger the level of non-normality. To measure the non-
normality of a matrix, we use the Henrici departure from normality, $d_{f}$,
defined in Eq. (4). For sake of clarity, we consider the case of two-level
systems. A general qubit state ${|{\psi_{a}}\rangle}$ can be expressed as

$\displaystyle{|{\psi_{a}}\rangle}=\begin{pmatrix}\cos{\theta_{a}}\\\
e^{i\xi_{a}}\sin{\theta_{a}}\end{pmatrix},$ (11)

where $0\leq\theta_{a}\leq\frac{\pi}{2}$ and $0\leq\xi_{a}\leq 2\pi$.

The pre- and post-selected states, ${|{\psi_{i}}\rangle}$ and
${|{\psi_{f}}\rangle}$, are described similarly to Eq.(11). Each state, pre-
and post-selected, depends on two free parameters and the measured operator
depends on four free parameters. Consequently, the full description of the
process depends on eight free parameters. In this section, we will vary the
pre- and post-selected states and fix the observable, while in the next
section, we will do the opposite. We restrict the parametric freedom of the
states by imposing the absence of phase of the initial and final states,
$\xi_{i}=0$ and $\xi_{f}=0$. We refer the interested reader to Appendix A.1
for the analysis of the general case.

To elucidate the correlation between amplifying weak values and non-normality,
the modulus of the weak value is qualitatively and quantitatively compared to
the Henrici departure from normality. The chosen observables to study are the
Pauli matrices and a linear combination of them,
$\hat{O}=\frac{1}{\sqrt{3}}\left(\hat{\sigma}_{x}+\hat{\sigma}_{y}+\hat{\sigma}_{z}\right)$,
where the Pauli matrices have been defined in Appendix E. Our choice has been
motivated by the important role the latter play in quantum physics by
describing the spin [38, 39, 40].

Figure 2: a) Color map of the modulus of the weak value of $\hat{\sigma}_{x}$
in terms of the polar angles of the pre- and post-selected states
$(\theta_{i},\theta_{f})$, imposing $\xi_{i}=0$, $\xi_{f}=0$. b) Color map of
the Henrici departure from normality of the non-normal matrix $\hat{A}$
associated to the weak value $\sigma_{x,w}$. a,b) The red curve corresponds to
the border of the area in which the modulus of the weak value is larger than
1.001. The black curve corresponds to the border of the area in which the
modulus of the weak value is twice the maximum possible expectation weak
value. c) Square modulus of the weak value as a function of the Henrici
departure from normality for anomalous weak values, $|\sigma_{x,w}|>1$,
obtained by varying $\theta_{f}$ between $0$ and $\frac{\pi}{2}$, while
$\theta_{i}$ is fixed (colored dots in the legend).  Figure 3: a) Color map of
the modulus of the weak value of $\hat{\sigma}_{y}$ in terms of the polar
angles of the pre- and post-selected states $(\theta_{i},\theta_{f})$,
imposing $\xi_{i}=0$, $\xi_{f}=0$. b) Color map of the Henrici departure from
normality of the non-normal matrix associated to the weak value, $\hat{A}$.
a,b) The red curve corresponds to the border of the area in which weak value
amplification occurs. The black curve corresponds to the border of the area in
which the modulus of the weak value is twice the maximum possible expectation
value. c) Square modulus of the amplified weak value as function of the
Henrici departure from normality, for various $0<\theta_{f}<\frac{\pi}{2}$.

Let us consider the Pauli matrix $\sigma_{x}$. Then by using the previous
definitions we get

$d_{f}\left(\hat{A}_{x}\right)=\frac{|\cos(2\theta_{i})|}{|\braket{\psi_{f}}{\psi_{i}}|^{2}}\,,$
(12)

where $\hat{A}_{x}$ denotes the operator $\hat{A}$ built from
$\hat{\sigma}_{x}$, and

$|\sigma_{x,w}|^{2}=\frac{\sin^{2}(\theta_{f}+\theta_{i})}{|\braket{\psi_{f}}{\psi_{i}}|^{2}}\,,$
(13)

with $|\braket{\psi_{f}}{\psi_{i}}|^{2}=\cos^{2}(\theta_{f}-\theta_{i})$. The
level curves of the latter are reported in Fig. 2 as a function of the angles
$\theta_{i}$ and $\theta_{f}$. The red line define the $1$-level, where the
modulus of the weak value equals 1, while the black line denotes the
$2$-level. In the region beyond the $1$-level, weak value amplification occurs
(as the eigenvalues of Pauli matrices are $\pm 1$). We can observe a very good
agreement among the results shown in the two top panels. Since we are
considering pre- and post-selected states with real coefficients, the
$1$-level curve also define the region beyond which weak values of
$\hat{\sigma}_{x}$ are anomalous. In other words, all real, anomalous weak
values of Pauli matrices provide amplification.

To strengthen this claim we can express $|\sigma_{x,w}|^{2}$ in terms of
$d_{f}\left(\hat{A}_{x}\right)$ by eliminating, e.g., the variable
$\theta_{f}$ and considering thus $\theta_{i}$ as a free parameter. We can
thus obtain

$|\hat{\sigma}_{x,w}|^{2}=\frac{1}{1+\tan^{2}\theta_{i}}\frac{1}{1+\tan^{2}\theta_{f}}\frac{(\tan\theta_{f}+\tan\theta_{i})^{2}}{|\cos(2\theta_{i})|}d_{f}\,,$
(14)

where $\tan\theta_{f}$ can be expressed as a function of $d_{f}$ by using
(12). The explicit formula can be found in Appendix A.1. This relation is
shown in panel c) of Fig. 2 for several values of the parameter $\theta_{i}$
and by restricting $d_{f}$ to the range corresponding to anomalous weak
values, i.e., $|\hat{\sigma}_{x,w}|^{2}>1$.

In conclusion in Fig. 2, we have considered the case for the observable
$\hat{\sigma}_{x}$ and observed that the Henrici departure from normality
(panel a) and the pattern of the modulus of the weak value (panel b) exhibit a
good correlation in the region of anomalous weak values, especially in the
region where $|O_{w}|>2$, namely associated to yellow values. Near the
boundary between anomalous and non-anomalous weak values (red curves), some
differences are appreciable: in particular, the value of the minimum of the
Henrici departure from normality is smaller than the modulus of the weak value
(the former is associated with a darker blue than the latter). On the other
hand, considering the region associated to values that are not anomalous,
i.e., the region bounded by the red curve and containing the point
$(\pi/4,\pi,4)$, the patterns are completely different, we can indeed
appreciate the presence of minima in the top-left and bottom-right corners for
the weak value (b) and in the center in the case of the Henrici departure from
normality (a). In Fig. 2c), we observed that the squared modulus of the weak
value increases as a function of the Henrici departure from normality. The
dependence appears linear once the weak value reaches large values (red and
orange curves). The point of interception with the axis
$\left(|\sigma_{x,w}|^{2}=0\right)$ depends on the initial angle,
$\theta_{i}$. However, when the weak value does not reach large values (green
and yellow curves), the dependence is closer to quadratic. The analytical
formulas can be found in appendix A.1. The behaviour of these functions is
thus very rich.

A similar analysis can be performed by using as observables the remaining
Pauli matrices and a combination of them. In the following Figs. 3, 4, 5, we
report the dependence of the modulus of the weak value of $\hat{\sigma}_{y}$,
$\hat{\sigma}_{z}$ and
$\hat{O}=\frac{1}{\sqrt{3}}\left(\hat{\sigma}_{x}+\hat{\sigma}_{y}+\hat{\sigma}_{z}\right)$
respectively, as a function of the parameters $\theta_{i}$ and $\theta_{f}$ by
using adapted color maps to emphasize the level curves. In the same figures,
we also show the Henrici departure from normality as a function of the same
parameters. As in the case of the observable $\hat{\sigma}_{x}$, we can
observe (see top panels in all the figures) a very good agreement between the
square of the modulus of the weak value and the Henrici departure from
normality. We have also shown the region in which the modulus of the weak
value is larger than the absolute value of the largest eigenvalue of the
observable (red curve) and the one for which the modulus is at least twice the
maximum eigenvalue (black curve). Outside the regions bounded by the latter
curves, i.e., the yellow regions, the weak value is amplifying. Inside the
regions bounded by the red curve, i.e., the blue ones, the modulus of the weak
value lies in the range of eigenvalues of the studied operator (but the weak
value can still be anomalous, i.e. if it is a complex number).

Figure 4: a) Color map of the modulus of the weak value of $\hat{\sigma}_{z}$
in terms of the polar angles of the pre- and post-selected states
$(\theta_{i},\theta_{f})$, imposing $\xi_{i}=0$, $\xi_{f}=0$. b) Color map of
the Henrici departure from normality of the non-normal matrix associated to
the weak value. c) Square modulus of the weak value as a function of the
Henrici departure from normality, with $0<\theta_{f}<\frac{\pi}{2}$. Figure 5:
a) Color map of the modulus of the weak value of the operator
$\hat{O}=\frac{1}{\sqrt{3}}\left(\hat{\sigma}_{y}+\hat{\sigma}_{x}+\hat{\sigma}_{z}\right)$
in terms of the polar angles of the pre- and post-selected states
$(\theta_{i},\theta_{f})$, imposing $\xi_{i}=0$, $\xi_{f}=0$. b) Color map of
the Henrici departure from normality of the non-normal matrix associated with
the weak value in terms of the polar angles of the pre- and post-selected
states $(\theta_{i},\theta_{f})$. a,b) The red curve corresponds to the border
of the area in which the weak value amplification occurs. The black curve
corresponds to the border of the area in which the modulus of the weak value
is twice the maximum possible expectation value. c) Square modulus of the weak
value as a function of the Henrici departure from normality for amplified
anomalous weak values, $|O_{w}|>1$, with $0<\theta_{f}<\frac{\pi}{2}$.

In Fig. 3, the chosen observable is $\hat{\sigma}_{y}$ and we report again the
weak value and the Henrici departure from normality as a function of the
angles by using a color code match. In this case too, we have a perfect
agreement of the behavior of the two quantities in the parameter region
associated to anomalous weak values. In the complementary region, a central
minimum appears in the case of the weak value that is absent in the Henrici
departure from normality plot. The results presented in Fig. 3c), show that
the square of the modulus of the weak value depends linearly on the Henrici
departure from normality, irrespective of the magnitude of the maximum weak
value for the specific case. Furthermore, the slope and the point of
interception with the axis $\left(|\sigma_{y,w}|^{2}=0\right)$ are independent
on the value of $\theta_{i}$. In appendix B, one can find the analytical
formula showing that the modulus of weak value to the square depends linearly
on the Henrici departure from normality, when both phases are null. Note that
in Fig. 3b), all non-zero weak values are anomalous because they are purely
imaginary numbers (anomalousness is not equivalent with amplification in this
case).

In Fig. 4, where the chosen observable is $\hat{\sigma}_{z}$, the patterns of
both quantities do not exhibit similarities. This difference with respect to
the other cases is due to the fact that no amplification occurs for this
observable with the chosen pre- and post selected states. The weak value is
real and never anomalous. In Fig. 4c), we have plotted the dependence of the
modulus of the weak value on the Henrici departure from normality for many
value of the angles. As one can see, a large weak value does not imply a large
Henrici departure from normality, indeed there is a parabola-like behavior.
The analytical formulas can also be found in appendix B.

In Fig. 5, we show the results for the observable
$\hat{O}=\frac{1}{\sqrt{3}}(\hat{\sigma}_{x}+\hat{\sigma}_{y}+\hat{\sigma}_{z})$.
The pattern of the modulus of the weak value and the Henrici departure from
normality show a good correlation in the region of weak value amplification,
especially where $|O_{w}|>2$ (black curve). Some differences can be
appreciated near the boundary (red curves) separating the regions where
amplification occurs or not, as the weak value and the departure from
normality exhibit different patterns near that area. In the complementary
region, i.e., inside the region bounded by the red curves and containing the
point $(\pi/4,\pi/4)$, large differences can be appreciated: the level sets
for the weak value show a ring-like shape absent in the Henrici departure from
normality case. In Fig. 5c), there is a linear-like dependence of the modulus
square of the weak value on the Henrici departure from normality. However, in
this case, both the point of interception of the axis
$\left(|O_{w}|^{2}=0\right)$ and the slope depend on $\theta_{i}$. When the
modulus of the weak value does not reach large values (green and blue lines),
the dependence is quadratic and not linear. For this observable and the chosen
pre- and post-selected states, the weak value is a complex number whenever
$\theta_{i}\neq\theta_{f}$, and thus anomalous everywhere but on the
descending diagonal of Fig. 5b).

For the sake of completeness, we have also computed the weak value and the
Henrici departure from normality for a three-level system, whose general state
is

${|{\psi_{a}}\rangle}=\left(\cos{\theta_{a}},e^{i\chi_{1,a}}\cos{\alpha_{a}}\sin{\theta_{a}},e^{i\chi_{2,a}}\sin{\alpha_{a}}\sin{\theta_{a}}\right)^{\top}\,.$
(15)

The chosen three-level operator is the Gell-Mann matrix
$\hat{O}=\hat{\lambda}_{5}$ (see Appendix F). The Gell-Mann matrices are the
traceless generators of the Lie group $\textrm{SU}(3)$ that generalize the
Pauli matrices, the traceless genarators of $\textrm{SU}(2)$. In Fig. 6, the
weak value and the Henrici departure from normality have been depicted by
using a color map scheme. As one can see, the plots match pretty well, with
the exception of the angles located in the top-right corner, for which the
Henrici departure from normality assumes larger values than the squared
modulus of the weak value, in particular in the strip contained between the
red and black curves. The square of the modulus of the weak value depends
linearly on the Henrici departure from normality. The slope and the point of
interception $\left(|\lambda_{5,w}|^{2}=0\right)$ depend on the pre-selected
polar angle.

Figure 6: a) Color map of the modulus of the weak value of the operator
$\hat{O}=\hat{\lambda}_{5}$, $\chi_{1,i}=\frac{\pi}{7}$,
$\chi_{2,i}=\frac{\pi}{21}$, $\alpha_{i}=\frac{\pi}{8}$,
$\chi_{1,f}=\frac{\pi}{4}$, $\chi_{2,f}=0$, $\alpha_{f}=\frac{\pi}{3}$ in
terms of the polar angles of the pre- and post-selected states ($\theta_{i}$
and $\theta_{f}$), imposing $\xi_{i}=0$, $\xi_{f}=0$. b) Color map of the
Henrici departure from normality of the non-normal matrix associated to the
weak value, $\hat{A}$. a,b) The red curve corresponds to the border of the
area in which the weak value is in the amplification range. The black curve
corresponds to the border of the area in which the modulus of the weak value
is twice the maximum possible expectation value. c) Square modulus of the weak
value as function of the Henrici departure from normality for amplifying weak
values, $|O_{w}|>1$, having $0<\theta_{f}<\frac{\pi}{2}$.

In conclusion, we have shown that the amplification degree of the weak value
depends on the non-normality of the matrix $\hat{A}$ when weak value
amplification occurs (so that the weak value is necessarily anomalous, even if
a real number). When one feature increases, the other does too. However, this
does not happen when amplified weak values are not present. In general, the
behaviour is complex, and there are cases in which some regions of discordance
are present, as shown in appendix A.1.

##  IV A reformulation of the problem with a varying observable

In the previous section, we examined the relationship between the Henrici
departure from normality and the weak value by varying the pre- and post-
selected states. However, in some cases, we may have the freedom to select the
observable for a given experiment. Here, we investigate how varying the
observable impacts non-normality and the modulus of weak values, for given
pre- and post selected states. To narrow our focus, we restrict our analysis
to two-level systems. Specifically, we consider an observable that depends on
two parameters,

$\hat{O}=\sin{\theta}\cos{\phi}\ \hat{\sigma}_{x}+\sin{\theta}\sin{\phi}\
\hat{\sigma}_{y}+\cos{\theta}\ \hat{\sigma}_{z}\,,$ (16)

where $0\leq\theta\leq\frac{\pi}{2}$, $0\leq\phi\leq 2\pi$, and
$\hat{\sigma}_{i}$ are the Pauli matrices (Appendix E). In the present
analysis, we investigate the relationship between the weak value and the
Henrici departure from normality for the two matrices: $\hat{A}$ and
$\hat{A}^{\prime}$, defined in Eq.(3) and Eq.(9), respectively. Because the
denominators of the latter matrices do not vary by modifying the operator, we
decided to compare, in this section, the normalized Henrici departure from
normality – $d_{f,n}\left(\hat{A}\right)$ and
$d_{f,n}\left(\hat{A}^{\prime}\right)$ – with the modulus of the numerator of
the weak value, i.e., $|{\langle{\psi_{f}}|}\hat{O}{|{\psi_{i}}\rangle}|$.

The results reported in Fig.7a) show a clear trend. Since the operator
$\hat{A}^{\prime}$ does not depend on the initial state, the normalized
Henrici departure from normality $d_{f,n}\left(\hat{A}^{\prime}\right)$ does
not vary when we change the pre-selected state, with a fixed final state
$\theta_{f}=0$; obviously, this is not the case for
$d_{f,n}\left(\hat{A}\right)$, as the different black curves clearly show.
When the initial state is completely orthogonal to the final one, i.e.,
$\theta_{i}=\frac{\pi}{2}$, the two Henrici indexes and the weak value
coincide. Starting from this value, the two indexes differ from each other but
evolve similarly, and the weak value also follows a similar trend in between
both departures from normality, in most of the range of values of $\theta$.

Figure 7: a) For different initial states ($\theta_{i}$), modulus of the
numerator of the weak value (red), normalized Henrici departure from normality
of $\hat{A}^{\prime}$ (blue), and normalized Henrici departure from normality
of $\hat{A}$ (black), all as a function of $\theta$ in the range of weak value
amplification. b) For different initial states ($\theta_{i}$), derivative of
the modulus of the numerator of the weak value
$|\braket{\psi_{f}}{\hat{O}}{\psi_{i}}|$ (red), the normalized Henrici
departure from normality of $\hat{A}^{\prime}$ (blue), and the normalized
Henrici departure from normality of $\hat{A}$ (black), all as a function of
$\theta$ in the range of weak value amplification. The chosen parameters are:
$\theta_{f}=\xi_{i}=\xi_{f}=0$, $\phi=\frac{\pi}{12}$, while $\theta_{i}$
varies from from $1.5446$ to $1.3352$. $\theta_{i}$ decreases in the direction
of the green arrow.

In order to understand this behavior, we study the derivative of the three
functions (see Fig. 7b)). As one can appreciate, the derivative of the weak
value with respect to $\theta$ is always between the ones of the two
normalized Henrici departures from normality, for the studied system.
Consequently, the variation of the weak value is not only determined by the
orthogonality of the pre- and post-selected states, as seen when varying
$\theta_{i}$, but also by the normalized non-normality of the operators
$\hat{A}$ and $\hat{A}^{\prime}$.

Figure 8: a) For different initial states ($\theta_{i}$), modulus of the
numerator of the weak value (red), normalized Henrici departure from normality
of $\hat{A}^{\prime}$ (blue), and normalized Henrici departure from normality
of $\hat{A}$ (black), all as a function of $\theta$ in the range of weak value
amplification. b) For different initial states ($\theta_{i}$), derivative of
the numerator of the modulus of the weak value
$|\braket{\psi_{f}}{\hat{O}}{\psi_{i}}|$ (red), the normalized Henrici
departure from normality of $\hat{A}^{\prime}$ (blue), and the normalized
Henrici departure from normality of $\hat{A}$ (black), all as a function of
$\theta$ in the range of weak value amplification. The chosen parameters are:
$\theta_{f}=\xi_{i}=\xi_{f}=0$, $\phi=3\frac{\pi}{2}$, while $\theta_{i}$
varies from from $1.5446$ to $1.3352$. $\theta_{i}$ decreases in the direction
of the green arrow.

In some cases, the behavior of numerator of the weak value and the normalized
Henrici departures from normality can be vastly different, as shown in Fig. 8.
For the chosen parameters in this scenario, the numerator of the weak value
does not fall between the two normalized Henrici departures from normality of
$\hat{A}$ and $\hat{A}^{\prime}$. Moreover, as $\theta_{i}$ decreases, the
curve of the weak value moves down in the plot, whereas the normalized Henrici
departure from normality of $\hat{A}$ moves in the opposite direction (see
opposite orientations of the green arrows). It is worth noting that this type
of behavior commonly occurs near the boundary of the amplification region of
the weak values. Fig. 9 shows the modulus of the weak value plotted against
$\theta$. In the second case, the modulus of the weak value does not even
reach twice the largest eigenvalue of the observable, whereas in the first
case, the anomalous weak value is significantly larger.

Figure 9: a) Modulus of the weak value of $\hat{O}$ in terms of $\theta$, for
the cases of Fig. 7. The chosen parameters are:
$\theta_{f}=\xi_{i}=\xi_{f}=0$, $\phi=\frac{\pi}{12}$, while $\theta_{i}$
varies from from $1.5446$ to $1.3352$. b) Modulus of the weak value of
$\hat{O}$ in terms of $\theta$, for the cases of Fig. 8. The chosen parameters
are: $\theta_{f}=\xi_{i}=\xi_{f}=0$, $\phi=3\frac{\pi}{2}$, while $\theta_{i}$
varies from from $1.5446$ to $1.3352$. $\theta_{i}$ decreases in the direction
of the green arrow.

The intricate relationship between the weak value and the Henrici departure
from normality of $\hat{A}$ and $\hat{A}^{\prime}$ in two-level systems is an
area that warrants further investigation. Detailed studies can uncover more
about this behavior. The interested readers can find the expressions of the
weak value, the Henrici departure from normality, and their derivatives in
Appendix D, with the assumption that $\xi_{i}=\xi_{f}=0$.

As already stated, the operators $\hat{A}$, resp. $\hat{A}^{\prime}$, have an
eigenvalue equal to zero and the other one equal to
$\frac{{\langle{\psi_{i}}|}\hat{O}{|{\psi_{i}}\rangle}}{\braket{\psi_{f}}{\psi_{i}}}$,
resp.
$\frac{{\langle{\psi_{f}}|}\hat{O}{|{\psi_{f}}\rangle}}{\braket{\psi_{f}}{\psi_{i}}}$.
For the parameter value at which the expectation value of the operator is also
equal to $0$, the matrix is nilpotent. This is an interesting observation
because it is well known that nilpotent matrices and operators affect the
system dynamics [41, 42]. Furthermore, the $0$ eigenvalue is now degenerate,
as one can observe from Fig. 10, where the largest eigenvalue in absolute
value and the angle between eigenvectors, calculated from the Fubini-Study
distance, have been plotted for different initial states. As one can
appreciate, for any value of the initial state, when the largest eigenvalue in
absolute value is $0$ (nilpotent matrix), the angle between the eigenvectors
is null, and thus the matrix becomes degenerate.

Figure 10:  For any given polar initial angle, a completely degenerate
nilpotent point exists. The largest eigenvalue in absolute value of $\hat{A}$,
$\alpha_{2}$, has been plotted in terms of $\theta$ and the angle between the
eigenvectors ($|\beta_{1}\rangle$ and $|\beta_{2}\rangle$) of $\hat{A}$,
$\beta_{1,2}=\arccos{\left|\langle\beta_{1}|\beta_{2}\rangle\right|}$, the
Fubini-Study distance, in terms of $\theta$. The chosen parameters are
$\phi=\frac{\pi}{12}$, $\theta_{f}=0$, $\xi_{f}=0$, $\xi_{i}=0$. The polar
angle of the initial state varies between $\frac{\pi}{2}$ and
$\frac{5\pi}{12}$.

Furthermore, the point of the maximum modulus of the numerator of the weak
value in the amplification regime (when the modulus of the weak value is
largest than the maximum absolute value of the eigenvalues of the operator) is
totally determined by the two points where the normalized Henrici departure
from normality of the matrices $\hat{A}$ and $\hat{A}^{\prime}$ reach their
maximum values. Indeed, the maximum modulus of the numerator of the weak value
is always attained at the average of the two points where the normalized
Henrici departure from normality reaches its maximum for the two matrices (it
is also point in which the largest eigenvalue in absolute value reaches its
minimum), Fig. 11. More details of these results can be found in appendix C.

Figure 11: Value of $\theta$ from the observable $\hat{O}$ corresponding to
the minimum of the largest eigenvalue in absolute value of $\hat{A}$,
$\alpha_{2}$ (green) and $\hat{A}^{\prime}$, $\alpha^{\prime}_{2}$ (black),
and to the maximum modulus of the weak value (blue) as a function of
$\theta_{i}$, the polar angle of the pre-selected state, for anomalous weak
values in the amplification range. The average of the $\theta$ corresponding
to the minima of the maximum of the absolute value of eigenvalue of both
matrices (orange) has also been included. The chosen parameters are
$\phi=\frac{\pi}{4}$, $\theta_{f}=0$, $\xi_{f}=0$, $\xi_{i}=0$.

As the angle $\theta_{i}$ increases, the parameter range in which weak value
amplification occurs becomes more pronounced. From $\theta_{i}=\frac{\pi}{4}$,
the degeneracy point, where both eigenvalues of $\hat{A}$ and
$\hat{A}^{\prime}$ are equal to zero, is located inside the amplification
region, Fig.12. Until this point, the maximum modulus of the weak value is
smaller than twice the largest eigenvalue of $\hat{O}$. However, as
$\theta_{i}$ increases beyond $\frac{\pi}{4}$, the maximum modulus of the weak
value begins to rise much more rapidly. The closer the degeneracy points of
the eigenvalues of the operators $\hat{A}$ and $\hat{A}^{\prime}$, the larger
the maximum of the modulus of the weak value. Actually, the maximum of the
modulus of the weak value tends to infinity when the degeneracy points are
identical (i.e., located at the same value of $\theta$). In summary, the
closer the degeneracy points of the eigenvalues of $\hat{A}$ and
$\hat{A}^{\prime}$ are to each other, the larger the maximum modulus of the
weak value becomes. Ultimately, the maximum modulus of the weak value tends to
infinity when the points of zero eigenvalues occur at the same angle $\theta$.

Figure 12: Largest value of the modulus of the weak value (orange), smallest
value of the largest eigenvalue in absolute value of $\hat{A}$, $\alpha_{2}$
(blue) and $\hat{A}^{\prime}$, $\alpha^{\prime}_{2}$(green), as a function of
the initial state ($\theta_{i}$). The extremal values are computed over the
range of all allowed values for the angle $\theta$ associated to the varying
observable $\hat{O}$. The chosen parameters are $\phi=\frac{\pi}{4}$,
$\theta_{f}=0$, $\xi_{f}=0$, $\xi_{i}=0$.

##  V Discussion and conclusions

In this paper, we have shown the connection between non-normality and
anomalous weak values, with particular focus on the amplification regime.
Unlike expectation values, weak values can be complex or even outside the
range of eigenvalues of the operator under scrutiny, i.e., larger than the
maximum eigenvalue or smaller than the minimum eigenvalue of the measured
operator, $\hat{O}$. In the latter cases, we say that the weak value is
anomalous. When such anomalous value has a modulus larger than the largest
modulus of the eigenvalues, we reach the amplification regime.

Firstly, we proved that weak values can be expressed as the expectation value
of matrix, $\hat{A}$, Eq.(2) and Eq.(3), which is typically non-normal. Then,
we have shown that, indeed, such matrix must be non-normal to obtain a weak
value different from an eigenvalue of the observable, i.e., non-normality is a
necessary condition to be in the anomalous regime. There are two arbitrary
ways to define such non-normal matrix: either in terms of the initial, i.e.,
pre-selected, state or of the final, i.e., post-selected, state. Both
matrices, called here $\hat{A}$ and $\hat{A}^{\prime}$, must be non-normal in
order to yiel an anomalous weak value. The non-normality is linked to the
quantum fluctuations of the observable in the pre- or post-selected states.
Interestingly, the quantum uncertainty of an arbitrary observable of any
system in a given quantum state is exactly equal to Henrici’s departure from
normality of a specific operator, naturally built from the product of the
observable operator with the quantum state projector. This result highlights
the importance of non-normality in quantum systems, far beyond the specific
context of weak measurements.

We showed for two-level systems that the two matrices $\hat{A}$ and
$\hat{A}^{\prime}$ can become simultaneously nilpotent, i.e., all the
eigenvalues are null and both matrices are degenerate. For a given observable,
this point is found by searching appropriate pre- and post-selected states. In
the case of 2-level systems, the matrices are completely degenerate.

We have derived the analytical expressions for the weak value and the Henrici
departure from normality in the relevant case of Pauli matrices and we have
numerically compared the modulus of weak values and non-normality when varying
the pre- and post-selected states. We have found a good correlation between
such quantities once parameters allow for anomalous weak values firmly in the
amplification regime.

Eventually, we have also analyzed the relation between the normalized Henrici
departure from normality of both non-normal matrices $\hat{A}$ and
$\hat{A}^{\prime}$ when considering a $1$-parameter family of observables in
the weak-value amplification regime. In this case, the point of the maximum
modulus of the numerator of the weak value is reached at the arithmetic
average of the two points where the Henrici departure from normality of the
matrices $\hat{A}$ and $\hat{A}^{\prime}$ are maximal.

Non-Hermitian Hamiltonians describe the evolution of systems that are not
isolated [28]. Even though in weak measurements the Hamiltonian of the weak
process is Hermitian, post-selection seems to introduce non-normality in the
process. It is interesting to observe the analogy with networks, where it is
conjectured that non-normality (namely the presence of source and sink nodes)
is correlated to the fact that the modeled system is an open one [26]. In weak
measurements, it is clear that post-selection plays an important role by
transforming an expectation values into weak values that can become anomalous.
A reconsideration of weak measurements by viewing them as open rather than
closed processes appears necessary.

Moreover, the fact that a non-normal matrix cannot be diagonalized through an
orthonormal transformation means that the information encoded in the system
cannot be completely disentangled. This is particularly interesting thinking
about the present framework, as when post-selection is executed on the system,
all the ancilla wave-function shifts are projected on a common state and thus
interfere. This interference seems to mix the information of the different
shifts in a disentangled manner. To do so, the expectation value of the
operator is transformed into a weak value that, as we have seen in this paper,
is strongly correlated to the non-normality of the operators $\hat{A}$ and
$\hat{A}^{\prime}$ in the amplification regime.

Lastly, let us recall that, in other fields, it has been observed that non-
normal operators enhance the possibility of phase transitions to occur [34,
24]. Since phase transitions can be observed in post-selected measurements
[43], we believe that revisiting quantum systems from this new perspective
could lead to the appearance of new, unexploited quantum phenomena linked to
anomalous weak values.

#### Authors contributions

Initial Idea and Conceptualization: L.B.F.; Methodology: L.B.F., R.M., T.C.,
Y.C.; Formal Analysis and Investigation: L.B.F., R.M., Y.C., T.C.; Numerical
Implementation: L.B.F.; Writing - original draft preparation: L.B.F., R.M;
Writing - review and editing: L.B.F., R.M., T.C., Y.C.

#### Acknowledgments

R.M. is supported by a FRIA-FNRS PhD fellowship, Grant FC 33443, funded by the
Walloon region. Y.C. is a research associate of the Fund for Scientific
Research (F.R.S.-FNRS). This research was supported by the Action de Recherche
Concertée WeaM at the University of Namur (19/23-001).

##  Appendix A Calculation of the expression of the Henrici departure from
normality

Let us consider the non-normal operator $\hat{A}$ defined in Eq.(3). The
square of this operator is

$\hat{A}^{2}=\frac{1}{\text{Tr}\left[\hat{\Pi}_{f}\hat{\Pi}_{i}\right]^{2}}\hat{O}{|{\psi_{i}}\rangle}{\langle{\psi_{i}}|}\hat{O}{|{\psi_{i}}\rangle}{\langle{\psi_{i}}|}=\frac{{\langle{\psi_{i}}|}\hat{O}{|{\psi_{i}}\rangle}}{|\braket{\psi_{f}}{\psi_{i}}|^{2}}\hat{A},$
(17)

which implies that the $N^{\textrm{th}}$ power of the operator $\hat{A}$ is

$\hat{A}^{N}=\frac{{\langle{\psi_{i}}|}\hat{O}{|{\psi_{i}}\rangle}^{N-1}}{|\braket{\psi_{f}}{\psi_{i}}|^{2\left(N-1\right)}}\hat{A}.$
(18)

The matrix $\hat{A}$ is almost idempotent, as $\hat{A}^{2}=\alpha\hat{A}$.
Consequently, we can redefine a matrix $\tilde{\hat{A}}$ that is idempotent,
i.e. $\tilde{\hat{A}}^{2}=\tilde{\hat{A}}$,

$\tilde{\hat{A}}=\frac{\hat{O}\hat{\Pi}_{i}|\braket{\psi_{f}}{\psi_{i}}|^{2}}{{\langle{\psi_{i}}|}\hat{O}{|{\psi_{i}}\rangle}}.$
(19)

The eigenvalues of an idempotent matrix are either $0$ or $1$. As the trace of
the matrix $\tilde{\hat{A}}$ is $1$, one eigenvalue should be $1$ and the rest
$0$. Consequently, one eigenvalue of the matrix $\hat{A}$ is
$\frac{{\langle{\psi_{i}}|}\hat{O}{|{\psi_{i}}\rangle}}{\braket{\psi_{f}}{\psi_{i}}^{2}}$
and the rest are 0. Obviously, when the expectation value of the observable in
the initial state is $0$, all eigenvalues are $0$. The Frobenious norm of the
matrix $\hat{A}$ is

$||\hat{A}||_{F}^{2}=\frac{1}{|\braket{\psi_{f}}{\psi_{i}}|^{4}}\text{Tr}\left[\left(\hat{O}\hat{\Pi}_{i}\right)\left(\hat{O}\hat{\Pi}_{i}\right)^{\dagger}\right]=\frac{1}{|\braket{\psi_{f}}{\psi_{i}}|^{4}}\text{Tr}\left[\hat{O}\hat{\Pi}_{i}\hat{\Pi}_{i}\hat{O}\right]=\frac{1}{|\braket{\psi_{f}}{\psi_{i}}|^{4}}\text{Tr}\left[\hat{O}\hat{\Pi}_{i}\hat{O}\right]=\frac{{\langle{\psi_{i}}|}\hat{O}^{2}{|{\psi_{i}}\rangle}}{|\braket{\psi_{f}}{\psi_{i}}|^{4}}\,.$
(20)

The Henrici departure from normality of the matrix $\hat{A}$ is

$d_{f}\left(\hat{A}\right)=\sqrt{\frac{{\langle{\psi_{i}}|}\hat{O}^{2}{|{\psi_{i}}\rangle}}{|\braket{\psi_{f}}{\psi_{i}}|^{4}}-\frac{{\langle{\psi_{i}}|}\hat{O}{|{\psi_{i}}\rangle}^{2}}{|\braket{\psi_{f}}{\psi_{i}}|^{4}}}=\frac{\Delta_{i}\hat{O}}{|\braket{\psi_{f}}{\psi_{i}}|^{2}}\,,$
(21)

which is proportional to the uncertainty of the operator $\hat{O}$ in the
initial state ${|{\psi_{i}}\rangle}$. Following the same procedure, one can
obtain the Henrici departure from normality of the operator
$\hat{A}^{\prime}$,

$d_{f}\left(\hat{A}^{\prime}\right)=\sqrt{\frac{{\langle{\psi_{f}}|}\hat{O}^{2}{|{\psi_{f}}\rangle}}{|\braket{\psi_{f}}{\psi_{i}}|^{4}}-\frac{{\langle{\psi_{f}}|}\hat{O}{|{\psi_{f}}\rangle}^{2}}{|\braket{\psi_{f}}{\psi_{i}}|^{4}}}=\frac{\Delta_{f}\hat{O}}{|\braket{\psi_{f}}{\psi_{i}}|^{2}}\,,$
(22)

which is proportional to the uncertainty of the operator $\hat{O}$ in the
final state ${|{\psi_{f}}\rangle}$.

###  A.1 Some analytical formulas for $\sigma_{x}$

The aim of this section is to present the analytical expressions relating
$O_{w}$ and $d_{f}$ for the cases numerically studied in the main text. Let us
assume the pre- and post-selected states to be given in the general form

$\displaystyle{|{\psi_{a}}\rangle}=\begin{pmatrix}\cos{\theta_{a}}\\\
e^{i\xi_{a}}\sin{\theta_{a}}\end{pmatrix},$

where for $a\in\\{i,f\\}$ we assume $0\leq\theta_{a}\leq\frac{\pi}{2}$ and
$0\leq\xi_{a}\leq 2\pi$. For sake of definitiveness, let us consider the case
of the $x$-Pauli matrix. Because $\hat{\sigma}_{x}^{2}=1$ and the
normalization of the initial state, we get from Eq. (21)

$d_{f}\left(\hat{A}_{x}\right)=\frac{\sqrt{1-\sin^{2}(2\theta_{i})\cos^{2}\xi_{i}}}{|\braket{\psi_{f}}{\psi_{i}}|^{2}}\,,$
(23)

where

$|\braket{\psi_{f}}{\psi_{i}}|^{2}=\cos^{2}\theta_{f}\cos^{2}\theta_{i}+\sin^{2}\theta_{f}\sin^{2}\theta_{i}+2\cos(\xi_{i}-\xi_{f})\cos\theta_{f}\cos\theta_{i}\sin\theta_{f}\sin\theta_{i}\,.$

To compute the weak value we use the very first definition (1) to obtain

$|\sigma_{x,w}|^{2}=\frac{\sin^{2}\theta_{f}\cos^{2}\theta_{i}+\cos^{2}\theta_{f}\sin^{2}\theta_{i}+2\cos(\xi_{i}+\xi_{f})\cos\theta_{f}\cos\theta_{i}\sin\theta_{f}\sin\theta_{i}}{|\braket{\psi_{f}}{\psi_{i}}|^{2}}\,.$
(24)

Both $d_{f}\left(\hat{A}_{x}\right)$ and $|\sigma_{x,w}|^{2}$ depend on the
angles, $\theta_{f}$ and $\theta_{i}$, and the phases $\xi_{i}$ and $\xi_{f}$.
In the following, we will assume the last three quantities as fixed parameters
and thus consider $X=d_{f}\left(\hat{A}_{x}\right)$ and $Y=|\sigma_{x,w}|^{2}$
to represent a curve parametrized by $\theta_{f}$ in the plane $(X,Y)$. In
Fig. 13, we report three typical behaviors of such a curve. In the left panel,
we assume $\xi_{i}=\xi_{f}=0$, and one clearly observes a positive correlation
between $d_{f}(\hat{A}_{x})$ and $|\hat{\sigma}_{x,w}|^{2}$: they both
increases as $\theta_{f}$ varies from $\tilde{\theta}_{f}$, i.e., the values
at which $|\hat{\sigma}_{x,w}|^{2}=1$ to $\theta_{f}=0$, corresponding here to
the maximum of both $d_{f}(\hat{A}_{x})$ and $|\hat{\sigma}_{x,w}|^{2}$. The
results shown in the middle panel have been obtained by setting
$\xi_{i}=\pi/5$ and still $\xi_{f}=0$ and we can observe the same behavior
than the one presented in the left panel. A similar behavior persists once we
increase $\xi_{i}$, up to a point at which a new behavior emerges (see right
panel), at which the relation between $d_{f}(\hat{A}_{x})$ and
$|\hat{\sigma}_{x,w}|^{2}$ is no longer monotonic. In this panel, we can
indeed observe the existence of a value $\hat{\theta}_{f}$ at which $d_{f}$
attains its maximum and this value is in the region of anomalous weak values.

Figure 13: We show the curves $d_{f}(\hat{A}_{x})$ and
$|\hat{\sigma}_{x,w}|^{2}$ as function of $\theta_{f}$ in the range
$[0,\tilde{\theta}_{f}]$, where the upper bound is determined by the condition
$|\hat{\sigma}_{x,w}|^{2}(\tilde{\theta}_{f})=1$ and it is marked with a red
diamond. The value $\theta_{f}=0$ is marked with a red square, while the value
$\hat{\theta}_{f}$ at which $d_{f}$ attains its maximum is marked by a red
circle, which denotes once the latter falls in the domain of anomalous weak
value. The remaining parameters have been fixed to $\theta_{i}=5\pi/12$,
$\xi_{i}=0$ and $\xi_{f}=0$ (left panel), $\theta_{i}=5\pi/12$,
$\xi_{i}=\pi/5$ and $\xi_{f}=0$ (middle panel), $\theta_{i}=5\pi/12$,
$\xi_{i}=4\pi/5$ and $\xi_{f}=0$ (right panel).

The value $\tilde{\theta}_{f}$ can be found by setting equation (24) equal to
$1$ and solving the resulting relation for $\tan\theta_{f}$. A straightforward
computation returns, if $\tan^{2}\theta_{i}\neq 1$,

$\tan\tilde{\theta}_{f}=\tan(2\theta_{i})\sin\xi_{i}\sin\xi_{f}\pm\sqrt{\tan^{2}(2\theta_{i})\sin^{2}\xi_{i}\sin^{2}\xi_{f}+1}\,,$

and we have to select the angle lying in $[0,\pi/2]$.

To determine the angle $\hat{\theta}_{f}$ that maximizes $d_{f}$, one can
observe that the numerator of Eq. (23) does not depend on $\theta_{f}$ and
thus this is equivalent to minimizing the denominator. By doing so, we obtain

$\tan(2\hat{\theta}_{f})=\tan(2\theta_{i})\cos(\xi_{i}-\xi_{f})\,,$

if $\cos(2\theta_{i})\neq 0$ and $\hat{\theta}_{f}=\pi/4+k\pi/2$ if if
$\cos(2\theta_{i})=0$.

Having those two angles, $\hat{\theta}_{f}$ and $\tilde{\theta}_{i}$, we can
obtain conditions on $\xi_{f}$, $\xi_{i}$ and $\theta_{i}$ to determine if
there is or not a monotonic dependence between $d_{f}(\hat{A}_{x})$ and
$|\hat{\sigma}_{x,w}|^{2}$.

To make some analytical progress we can explicitly express the dependence of
$|\sigma_{x,w}|^{2}$ on $d_{f}(\hat{A}_{x})$, again by removing $\theta_{f}$.
To do so, we first express (23) in terms of $\tan^{2}\theta_{f}$

$d_{f}\left(\hat{A}_{x}\right)=\sqrt{1-\sin^{2}(2\theta_{i})\cos^{2}\xi_{i}}\frac{(1+\tan^{2}\theta_{i})(1+\tan^{2}\theta_{f})}{1+\tan^{2}\theta_{f}\tan^{2}\theta_{i}+2\cos(\xi_{i}-\xi_{f})\tan\theta_{i}\tan\theta_{f}}\,,$

and we solve the second degree equation in $\tan\theta_{f}$ to express the
latter as a function of $d_{f}$ and the remaining variables

$\tan\theta_{f}=\frac{d_{f}\cos(\xi_{i}-\xi_{f})\tan\theta_{i}\pm\sqrt{d_{f}^{2}\cos^{2}(\xi_{i}-\xi_{f})\tan^{2}\theta_{i}-[d_{f}\tan^{2}\theta_{i}-(1+\tan^{2}\theta_{i})S][d_{f}-(1+\tan^{2}\theta_{i})S]}}{d_{f}\tan^{2}\theta_{i}-(1+\tan^{2}\theta_{i})S}\,,$
(25)

where $S=\sqrt{1-\sin^{2}(2\theta_{i})\cos^{2}\xi_{i}}$.

We hence explicitly rewrite Eq. (24) in terms of $\tan\theta_{f}$

$|\hat{\sigma}_{x,w}|^{2}=\frac{1}{1+\tan^{2}\theta_{i}}\frac{1}{1+\tan^{2}\theta_{f}}\frac{\tan^{2}\theta_{f}+\tan^{2}\theta_{i}+2\cos(\xi_{i}+\xi_{f})\tan\theta_{f}\tan\theta_{i}}{\sqrt{1-\sin^{2}(2\theta_{i})\cos^{2}\xi_{i}}}d_{f}\,,$

and eventually use the expression (25) for $\tan\theta_{f}$.

We can observe that the curves given parametrically by Eqs. (23) and (24)
shown in Fig. 13 correspond to the branch of $\tan\theta_{f}$ Eq. (25) with
the positive sign in front of the square root.

Figure 14: We show the curve $|\hat{\sigma}_{x,w}|^{2}$ as a function of
$d_{f}(\hat{A}_{x})$, in the range of weak value amplification. The black line
corresponds to the root with the positive sign while the blue one to the
negative sign. The red diamond marks the value of $d_{f}$ for which
$|\hat{\sigma}_{x,w}|^{2}=1$. The red square identifies the value of $d_{f}$
associated to $\theta_{f}=0$, while the red circle determines the largest
value of $d_{f}$. The remaining parameters have been fixed to
$\theta_{i}=5\pi/12$, $\xi_{i}=0$ and $\xi_{f}=0$ (left panel),
$\theta_{i}=5\pi/12$, $\xi_{i}=\pi/5$ and $\xi_{f}=0$ (middle panel),
$\theta_{i}=5\pi/12$, $\xi_{i}=4\pi/5$ and $\xi_{f}=0$ (right panel).

In Fig. 15 we propose a global view of $|\sigma_{x,w}|$ (top panels) and
$d_{f}(\hat{A}_{x})$ (bottom panels) as a function of $\theta_{i}$ and
$\theta_{f}$ for three sets of values of $(\xi_{i},\xi_{f})$, $(0,0)$ (left
panel), $(\pi/5,0)$ (middle panel) and $(4\pi/5,0)$ (right panel). We can
observe that as $\xi_{i}$ increases, so does the region corresponding to
strong anomalous weak values, e.g., larger than $2$.

Figure 15: We show in the plane $(\theta_{i},\theta_{f})$ the level curves of
the functions $|\hat{\sigma}_{x,w}|$ (top panels) and $d_{f}(\hat{A}_{x})$
(bottom panels). The remaining parameters have been fixed to
$(\xi_{i},\xi_{f})=(0,0)$ (left panel), $(\xi_{i},\xi_{f})=(\pi/5,0)$ (middle
panel) and $(\xi_{i},\xi_{f})=(4\pi/5,0)$ (right panel).

On the other hand (see Fig. 16), increasing $\xi_{f}$ reduces the region of
anomalous weak values.

Figure 16: We show in the plane $(\theta_{i},\theta_{f})$ the level curves of
the function $|\hat{\sigma}_{x,w}|$ for two sets of parameters
$(\xi_{i},\xi_{f})=(4\pi/5,\pi/5)$ (left panel),
$(\xi_{i},\xi_{f})=(4\pi/5,3\pi/5)$ (right panel).

##  Appendix B Analytical calculations with null phases

Let us consider now the Pauli matrix $\hat{\sigma}_{y}$. In this case, the
Henrici departure from normality of the matrix $\hat{A}$ is, assuming
$\xi_{i}=\xi_{f}=0$,

$df\left(\hat{A}_{y}\right)=\frac{1}{\cos^{2}\left(\theta_{f}-\theta_{i}\right)},$
(26)

while the weak value is

$\sigma_{y,w}=i\tan{\left(\theta_{f}-\theta_{i}\right)}.$ (27)

The Henrici departure from normality can be expressed in terms of the weak
value as

$|\sigma_{y,w}|^{2}=\frac{1}{\cos^{2}\left(\theta_{f}-\theta_{i}\right)}-1=df\left(\hat{A}_{y}\right)-1.$
(28)

A comparison between Fig.17 and Fig.3 reveals a remarkable agreement between
the figures generated from the numerical and analytical studies.

Figure 17: Henrici departure from normality of $\hat{A}_{y}$ in terms of
$\theta_{i}$ and $\theta_{f}$, weak value of $\hat{\sigma}_{y}$ in terms of
$\theta_{i}$ and $\theta_{f}$, and weak value of $\hat{\sigma}_{y}$ in terms
of the Henrici departure in the anomalous regime from normality for different
values of $\theta_{i}$ from $0$ to $\frac{\pi}{4}$.

Finally, we also studied the Pauli matrix $\hat{\sigma}_{z}$, assuming
$\xi_{i}=\xi_{f}=0$. The Henrici departure from normality is in that case,

$d_{f}\left(\hat{A}_{z}\right)=\sin^{2}{2\theta_{i}}\left[1+\tan^{2}{\left(\theta_{f}-\theta_{i}\right)}\right],$
(29)

while the weak value is

$|\sigma_{z,w}|=|\cos{2\theta_{i}}-\tan{\left(\theta_{f}-\theta_{i}\right)}\sin{2\theta_{i}}|.$
(30)

The weak value can be expressed in terms of the Henrici departure from
normality as,

$|\sigma_{z,w}|^{2}=\Bigg{|}\cos{2\theta_{i}}\pm\sin{2\theta_{i}}\sqrt{-1\pm\frac{df\left(\hat{A}_{z}\right)}{\sin{2\theta_{i}}}}\Bigg{|}^{2}.$
(31)

Figures 18 and 4 depict the results of the theoretical and numerical
calculations, and a striking similarity between the two figures is immediately
apparent.

Figure 18: Henrici departure from normality of $\hat{A}_{z}$ in terms of
$\theta_{i}$ and $\theta_{f}$, weak value of $\hat{\sigma}_{z}$ in terms of
$\theta_{i}$ and $\theta_{f}$, and weak value of $\hat{\sigma}_{z}$ in terms
of the Henrici departure from normality for different values of $\theta_{i}$
from $0$ to $\frac{\pi}{4}$.

##  Appendix C Study of the point of maximum weak value and maximum Henrici
departure from normality

In this section, we will show some analytical results about the value of the
observable parameter ($\theta$) in which the weak value presents a maximum and
the ones in which both matrices ($\hat{A}$ and $\hat{A}^{\prime}$) present a
Henrici departure from normality that is the largest. Let us consider, without
loss of generality, the specific example of Fig.11. The studied observable is

$\hat{O}=\begin{pmatrix}\cos{\theta}&\frac{\left(1-i\right)}{\sqrt{2}}\sin{\theta}\\\
\frac{\left(1+i\right)}{\sqrt{2}}\sin{\theta}&-\cos{\theta}\end{pmatrix}.$
(32)

The pre- and post-selected states are

${|{\psi_{i}}\rangle}=\begin{pmatrix}\cos{\theta_{i}}\\\
\sin{\theta_{i}}\end{pmatrix},\hskip
28.45274pt{|{\psi_{f}}\rangle}=\begin{pmatrix}1\\\ 0\end{pmatrix}.$ (33)

Both matrices $\hat{A}$ and $\hat{A}^{\prime}$ have an eigenvalue that is
equal to $0$. The other eigenvalues are

$\displaystyle\alpha_{\hat{A}}$ $\displaystyle=$ $\displaystyle\cos{\theta},$
(34) $\displaystyle\alpha_{\hat{A}^{\prime}}$ $\displaystyle=$
$\displaystyle\cos{\theta}\cos{2\theta_{i}}+\sqrt{2}\cos{\theta_{i}}\sin{\theta}\sin{\theta_{i}}.$

The Henrici departures from normality of $\hat{A}$, and $\hat{A}^{\prime}$ are

$\displaystyle d_{f}\left(\hat{A}\right)$ $\displaystyle=$
$\displaystyle|\sin{\theta}|$ (35) $\displaystyle
d_{f}\left(\hat{A}^{\prime}\right)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{8}}\sqrt{\left(5-\cos{4\theta_{i}}-\cos{2\theta}\left(1+3\cos{4\theta_{i}}\right)-2\sqrt{2}\sin{2\theta}\sin{4\theta_{i}}\right)}.$

The weak value is

$|O_{w}|=\sqrt{\cos^{2}{\theta}+\sqrt{2}\cos{\theta}\sin{\theta}\tan{\theta_{i}}+\sin^{2}{\theta}\tan^{2}{\theta_{i}}}.$
(36)

At $\theta=0$, the weak value is equal to $1$, independently on the value of
the initial polar angle, $\theta_{i}$. When $0\leq\theta\leq\frac{\pi}{4}$,
the weak value is only anomalous for a section of values of $\theta$,
actually, with
$0\leq\theta\leq\arctan{\frac{\sqrt{2}\tan{\theta_{i}}}{1-\tan^{2}{\theta_{i}}}}$.

The Henrici departure from normality is the largest when the eigenvalue
$\alpha$ is the smallest in absolute value. In the case of $\alpha_{\hat{A}}$,
it always decreases in the range of $\theta$ between $0$ and $\frac{\pi}{2}$.
Consequently, the maximum value of the Henrici departure from normality in the
range of anomalous weak values would be at
$\theta=\arctan{\frac{\sqrt{2}\tan{\theta_{i}}}{1-\tan^{2}{\theta_{i}}}}$.

The eigenvalue of $\hat{A}^{\prime}$ that is different from zero,
$\alpha_{\hat{A}^{\prime}}$ increases in absolute value when increasing
$\theta$ in the range
$0\leq\theta\leq\arctan{\frac{\sqrt{2}\tan{\theta_{i}}}{1-\tan^{2}{\theta_{i}}}}$.
The point of the largest Henrici departure from normality in the anomalous
regime is at $\theta=0$, for all values of $0\leq\theta_{i}\leq\frac{\pi}{4}$.

The value of $\theta$ for which the weak value is maximum is exactly at the
average of both points of the largest value of Henrici departure from
normality,
$\theta=\frac{1}{2}\arctan{\frac{\sqrt{2}\tan{\theta_{i}}}{1-\tan^{2}{\theta_{i}}}}$.

When moving to $\frac{\pi}{4}\leq\theta_{i}\leq\frac{\pi}{2}$, the weak value
is anomalous for all values of $\theta$ in the range
$0\leq\theta\leq\frac{\pi}{2}$. The largest Henrici departure from normality
of $\hat{A}$ in the anomalous regime is at $\theta=\frac{\pi}{2}$ for all
values of $\theta_{i}$, where the matrix is nilpotent, both eigenvalues are
$0$.

The Henrici departure from normality of $\hat{A}^{\prime}$ reaches its maximum
when the matrix is nilpotent, at
$\theta=\arctan{\left(-\sqrt{2}\cot{2\theta_{i}}\right)}$ which is in the
anomalous regime when $\frac{\pi}{4}\leq\theta_{i}\leq\frac{\pi}{2}$.

The anomalous weak value reaches its maximum at
$\theta=\frac{1}{2}\arctan{\left(-\sqrt{2}\cot{2\theta_{i}}\right)}$.

Here, we have explained in detail how the value of $\theta$ in which the weak
value is maximum is at the average between the one in which the Henrici
departure from normality of $\hat{A}$ and $\hat{A}^{\prime}$ are maximum for a
specific case. However, this is valid for any two-dimensional quantum system.

##  Appendix D Study of the derivative of the normalized Henrici departure
from normality and the numerator of the weak value

The numerator of the weak value of the operator defined in Eq. 16 is, assuming
$\xi_{i}=\xi_{f}=0$,

$O_{w}=\sqrt{\left(\cos{\theta}\cos{\left(\theta_{f}+\theta_{i}\right)}+\cos{\phi}\sin{\theta}\sin{\left(\theta_{f}+\theta_{i}\right)}\right)^{2}+\sin^{2}{\theta}\sin^{2}{\left(\theta_{f}-\theta_{i}\right)}\sin^{2}{\phi}}.$
(37)

The derivative with respect to $\theta$ is

$\frac{dO_{w}}{d\theta}=\frac{-\sin{2\theta}\left(\cos{\left[2\left(\theta_{f}-\theta_{i}\right)\right]}+3\cos{\left[2\left(\theta_{f}+\theta_{i}\right)\right]}-2\cos{2\phi}\sin{2\theta_{f}}\sin{2\theta_{i}}\right)+4\cos{2\theta}\cos{\phi}\sin{\left[2\left(\theta_{f}+\theta_{i}\right)\right]}}{8\sqrt{\left(\cos{\theta}\cos{\left(\theta_{f}+\theta_{i}\right)}+\cos{\phi}\sin{\theta}\sin{\left(\theta_{f}+\theta_{i}\right)}\right)^{2}+\sin^{2}{\theta}\sin^{2}{\left(\theta_{f}-\theta_{i}\right)}\sin^{2}{\phi}}}.$
(38)

The Henrici departure of $\hat{A}$ is

$df_{n}\left(\hat{A}\right)=\frac{1}{2}\sqrt{\left(3+2\cos{4\theta_{f}}\cos^{2}{\phi}-\cos{2\phi}\right)\sin^{2}{\theta}+4\cos^{2}{\theta}\sin^{2}{2\theta_{f}}-2\cos{\phi}\sin{2\theta}\sin{4\theta_{f}}}\
.$ (39)

The derivative of the normalized Henrici departure from normality of $\hat{A}$
is

$\frac{df_{n}\left(\hat{A}\right)}{d\theta}=\frac{-4\cos{2\theta}\cos{\phi}\sin{4\theta_{f}}+\sin{2\theta}\left(\cos{4\theta_{f}}\left(3+\cos{2\phi}\right)+2\sin^{2}{\phi}\right)}{4\sqrt{\left(3+2\cos{4\theta_{f}}\cos^{2}{\phi}-\cos{2\phi}\right)\sin^{2}{\theta}+4\cos^{2}{\theta}\sin^{2}{2\theta_{f}}-2\cos{\phi}\sin{2\theta}\sin{4\theta_{f}}}}.$
(40)

The Henrici departure of $\hat{A}^{\prime}$ is

$df_{n}\left(\hat{A}^{\prime}\right)=\frac{1}{2}\sqrt{\left(3+2\cos{4\theta_{i}}\cos^{2}{\phi}-\cos{2\phi}\right)\sin^{2}{\theta}+4\cos^{2}{\theta}\sin^{2}{2\theta_{i}}-2\cos{\phi}\sin{2\theta}\sin{4\theta_{i}}}\
.$ (41)

The derivative of the normalized Henrici departure from normality of
$\hat{A}^{\prime}$ is

$\frac{df_{n}\left(\hat{A}^{\prime}\right)}{d\theta}=\frac{-4\cos{2\theta}\cos{\phi}\sin{4\theta_{i}}+\sin{2\theta}\left(\cos{4\theta_{i}}\left(3+\cos{2\phi}\right)+2\sin^{2}{\phi}\right)}{4\sqrt{\left(3+2\cos{4\theta_{i}}\cos^{2}{\phi}-\cos{2\phi}\right)\sin^{2}{\theta}+4\cos^{2}{\theta}\sin^{2}{2\theta_{i}}-2\cos{\phi}\sin{2\theta}\sin{4\theta_{i}}}}.$
(42)

##  Appendix E Pauli matrices

The Pauli matrices are:

$\displaystyle\hat{\sigma}_{x}=\begin{pmatrix}0&1\\\ 1&0\end{pmatrix}\hskip
28.45274pt\hat{\sigma}_{y}=\begin{pmatrix}0&-i\\\ i&0\end{pmatrix}\hskip
28.45274pt\hat{\sigma}_{z}=\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}$ (43)

##  Appendix F Gell-Mann matrices

The Gell-Mann matrices are:

$\displaystyle\hat{\lambda}_{1}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}0&1&0\\\ 1&0&0\\\ 0&0&0\end{pmatrix}\hskip
28.45274pt\hat{\lambda}_{2}=\begin{pmatrix}0&-i&0\\\ i&0&0\\\
0&0&0\end{pmatrix}\hskip 28.45274pt\hat{\lambda}_{3}=\begin{pmatrix}1&0&0\\\
0&-1&0\\\ 0&0&0\end{pmatrix}$ (44) $\displaystyle\hat{\lambda}_{4}$
$\displaystyle=$ $\displaystyle\begin{pmatrix}0&0&1\\\ 0&0&0\\\
1&0&0\end{pmatrix}\hskip 28.45274pt\hat{\lambda}_{5}=\begin{pmatrix}0&0&-i\\\
0&0&0\\\ i&0&0\end{pmatrix}$ (45) $\displaystyle\hat{\lambda}_{6}$
$\displaystyle=$ $\displaystyle\begin{pmatrix}0&0&0\\\ 0&0&1\\\
0&1&0\end{pmatrix}\hskip 28.45274pt\hat{\lambda}_{7}=\begin{pmatrix}0&0&0\\\
0&0&-i\\\ 0&i&0\end{pmatrix}\hskip
28.45274pt\hat{\lambda}_{8}=\frac{1}{\sqrt{3}}\begin{pmatrix}1&0&0\\\ 0&1&0\\\
0&0&-2\end{pmatrix}\hskip 28.45274pt$ (46)

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