# Towards establishing a connection between two-level quantum systems and
physical spaces

V. G. Valle  Depto. de Física, ICE, Universidade Federal de Juiz de Fora, MG,
Brazil  L. L. Brugger  Depto. de Física, ICE, Universidade Federal de Juiz de
Fora, MG, Brazil  B. F. Rizzuti<EMAIL_ADDRESS>Depto. de Física,
ICE, Universidade Federal de Juiz de Fora, MG, Brazil  Cristhiano Duarte
International Institute of Physics, Federal University of Rio Grande do Norte,
59078-970, Natal, Brazil  School of Physics and Astronomy, University of
Leeds, Leeds LS2 9JT, United Kingdom

###### Abstract

This work seeks to make explicit the operational connection between the
preparation of two-level quantum systems with their corresponding description
(as states) in a Hilbert space. This may sound outdated, but we show there is
more to this connection than common sense may lead us to believe. To bridge
these two separated realms—the actual laboratory and the space of states—we
rely on a paradigmatic mathematical object: the Hopf fibration. We illustrate
how this connection works in practice with a simple optical setup. Remarkably,
this optical setup also reflects the necessity of using two charts to cover a
sphere. Put another way, our experimental realization reflects the bi-
dimensionality of a sphere seen as a smooth manifold.

Keywords:Two-level quantum systems, Bloch sphere, Hopf fibration.

##  I Introduction

_Quantum phenomena do not occur in a Hilbert space, they occur in the
laboratory._ Asher Peres’s famous reminder Peres (2010) clearly stands out
among the many ways of dispelling misconceptions about the philosophical
content of quantum mechanics. Granted, it is difficult to pinpoint why it is
so; is it because of the apparent simplicity and putative form in which it is
formulated, or is it because of the hidden profoundness of his interjection?

Putting the explanation of its popularity aside for the moment, the fact we
want to emphasise is that Peres’s observation accomplishes two things at once.
It not only brings to the surface the idea that quantum theory may have a
descriptive aspect Leifer and Spekkens (2013), but it also dispels the common
mistake of equating that descriptive characteristic to the underlying physical
reality it is supposed to describe Leifer (2014). Put another way, quantum
theory is so far the best available framework to describe the microscopic
world—its predictive power is undeniable Colbeck and Renner (2011).
Nonetheless, and here is the caveat, we must stress that it consists of a
normative set of assumptions and rules designed to describe and deal with
phenomena having no classical explanation.

That is to say that unless we put forward a realist interpretation of quantum
mechanics—with its inherent difficulties and that falls short in explaining
satisfactorily all the striking quantum phenomena Aharonov et al. (2013)—the
(subjective) take-home message from Peres’s remark is that quantum theory is a
descriptive theory, that its mathematical framework merely formalises a
probabilistic description of certain phenomena that happen in the physical
space—very much like in the same spirit of strongly subjectivist
interpretations of (classical) probability theory Vineberg (2016); Kemeny
(1955); Anscombe and Aumann (1963); Savage (1954); de Finetti (1979); Neumann
and Morgenstern (1944).

That said, what cannot be forgotten, though, is that there is an intricate
connection between the physical space where, in Peres’s language, quantum
phenomena happen and the Hilbert space where our (probabilistic) description
of those phenomena lives in. That is exactly what we set out to do in this
contribution. Our introductory contribution intends to fill the gap that is
left behind by the ordinary, textbook approaches to quantum physics, which
does not bridge the physical space to the Hilbert space—as if the former was
just an auxiliary way to motivate and introduce the mathematical machinery of
the latter. Using a universal gadget for $SU(2)$ polarization optics Simon and
Mukunda (1989), together with the Hopf fibration Bengtsson and Życzkowski
(2017), we will establish a connection between indistinguishable states in
$\mathbb{S}^{3}$ and $\mathbb{S}^{2}\subset\mathds{R}^{3}$, and interpret this
connection beyond the mere mathematical construct.

This work closely follows the scholarship pursued by some of the authors,
where we set out to recast, and sometimes reconstruct, mathematical objects
that are accepted as ‘givens’ directly from our surrounding physical world
Vasconcelos Júnior et al. (2018); Gaio et al. (2019); Rizzuti et al. (2020,
2022). In all of those works in this line of inquiry, we have so far kept in
mind that our primary target audience is composed of students caught in the
crossfire of advanced technical texts and ordinary approaches that, for one
reason or another, rush through part of the subtleties involved with the
foundations of physics—although we also feel that experienced researchers will
also find our works as rewarding auxiliary texts where we introduce novelties
while dispelling some missteps.

In this sense, we subdivide this work as follows. In sec. II, we motivate the
connection between the physical space and the Hilbert space through a
paradigmatic example we will employ recurrently throughout the text. In sec.
III we flesh out the mathematical machinery we will use to bridge those two
spaces. In sec. IV and sec. V we argue that mathematical machinery has a
deeper physical meaning. The last section contains our concluding remarks.

##  II Motivation

Our motivation for this approach stems from the fact that two-level pure
quantum systems qu _b_ its are represented by normalised vectors
$\ket{\psi}\in\mathds{C}^{2}$, and usually parametrised by Nielsen and Chuang
(2010)

$\ket{\psi}=\cos\frac{\theta}{2}\ket{0}+e^{i\varphi}\sin\frac{\theta}{2}\ket{1}.$
(1)

Here, $\mathcal{B}=\\{\ket{0},\ket{1}\\}$ is the canonical basis of
$\mathds{C}^{2}$ and $\ket{\psi}$ lies in the so-called Bloch sphere Simon and
Mukunda (1992); Arecchi et al. (1972). The parameters $\theta$ and $\varphi$
are borrowed from spherical coordinates, so as usual: $\theta\in[0,\pi]$ and
$\varphi\in[0,2\pi]$. We name it $\mathfrak{S}$ Grossi et al. (2023). This
sphere provides a visual, geometrical and intuitive way to analyze qubits:
every point on the sphere represents a pure state, the north and south poles
are but $\ket{0}$ and $\ket{1}$ and more generally, antipodes represents a
pair of orthogonal vectors. One possible way to explore the geometrical
connection between $\mathfrak{S}$ and $\mathds{R}^{3}$ can be established by
looking to the density matrix $\rho$ constructed from (1). A direct
calculation shows that

$\rho=\ket{\psi}\bra{\psi}=\frac{1}{2}\left(I+\hat{r}\cdot\vec{\sigma}\right),$
(2)

where $\vec{\sigma}=(\sigma_{1},\sigma_{2},\sigma_{3})$ is the collection of
the Pauli matrices. As usual, $\sigma_{k}$, $k=1,2,3$ and we use the
representation,

$\sigma_{1}=\begin{pmatrix}0&1\\\
1&0\end{pmatrix},\,\sigma_{2}=\begin{pmatrix}0&-i\\\
i&0\end{pmatrix},\,\sigma_{3}=\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}.$ (3)

Moreover, $\hat{r}\in\mathds{R}^{3}$ in (2), known as the Bloch vector, so
obtained is just the unit vector pointing in an arbitrary radial direction,

$\hat{r}=(\sin{\theta}\cos{\varphi},\sin{\theta}\sin{\varphi},\cos{\theta}).$
(4)

Beyond a mere mathematical construction, we will delve into the physical
content of the relationship between the physical space—where quantum phenomena
are carried out—and the Bloch sphere—where the states involved in those
experiments are represented in.

We emphasise that, for simplicity, throughout the text, we consider only
quantum systems described by pure states. In doing so, given any vector

$\ket{\psi}=a\ket{0}+b\ket{1}$ (5)

with complex coefficients $a=x_{1}+ix_{2}$, $b=x_{3}+ix_{4}$,
$x_{\mu}\in\mathds{R}$, $\mu=1,2,3,4$, we demand its normalization,
$\braket{\psi}{\psi}=1$. Thus, we have

$x^{2}_{1}+x^{2}_{2}+x^{2}_{3}+x^{2}_{4}=1.$ (6)

That way, $\mathfrak{S}$ is (topologically) identified with the compact sphere
$S^{3}\subset\mathds{R}^{4}$ Munkres (1974).

Leveraging on an operational procedure, what actually defines a state is its
corresponding experimental preparation Jauch (1973). Consequently, we
anticipate that the parameters $\theta$ and $\varphi$ in equation (1) possess
both physical and geometrical meaning in terms of a list of procedures in a
laboratory.

Qubits quantum systems are not just toy models or convenient mathematical
abstractions belonging exclusively to textbooks. Two-level quantum systems
abound in nature [Cris], can be easily created in the laboratory Childs and
Chuang (2000); Harvey (2022); Unrau and Bimberg (2014), and are also
fundamental building blocks of paradigmatic applications of quantum
information and quantum foundations to tasks with no classical analogue Hensen
et al. (2015); Bennett et al. (1993); Walborn et al. (2010); Bennett and
Brassard (2014); Ekert (1991) or that provide significant speed-ups and gains
in relation to their classical counterparts Arute et al. (2019); Acín and
Masanes (2016); Somma et al. (2008); Brandao and Svore (2017). However,
circling back to Peres’s provocation, it is not always the case that a direct
a physical interpretation of $\theta$ and $\varphi$ —see eq.(1)— in these
contexts is readily discerned.

Let us give a concrete example of what we mean by state from our standpoint.
In a recent paper Grossi et al. (2023), to give a new purpose to the Stern-
Gerlach experiments, we utilized that traditional setup Stern (1921); Gerlach
and Stern (1921, 1922a, 1922b) as a standard example of how to prepare a qubit
as well as to provide a physical meaning to the pair $\theta$ and $\varphi$.
These parameters are connected to the orientation of the (non-homogeneous)
magnetic field through which the beam of silver atoms passes through. When the
beam of atoms is directed into a region permeated by a magnetic field oriented
as defined in $\mathds{R}^{3}$ by $(\theta,\varphi)$, it splits into two
branches. Upon selecting one of them, the state of the system is represented
by equation (1).

Clearly, the parametrization for pure states on $\mathfrak{S}$ is not unique.
In fact, we may consider the entire equivalence class of vectors differing by
a constant global phase factor and they are indistinguishable in the sense of
providing the same expectation values to any observable. We can rephrase it by
saying that pure states can also be represented by orbits of the action of the
group $\mathcal{U}(1)$ on $\mathfrak{S}$ defined by standard multiplication of
vectors by a global phase factor (for more details on group actions and
related concepts, see Appendix A). With more details, the orbits are
equivalence classes of vectors in $\mathfrak{S}$ differing by a phase, say,
$e^{i\alpha}\in\mathcal{U}(1)$. The geometry of indistinguishable states is
illuminated by the Hopf fibration Bengtsson and Życzkowski (2017). Far from
being only a mathematical construct, this fibration explores the intrinsic
connection between the dimension of the Hilbert space $\mathds{C}^{2}$ where
qubits’s representation live and the dimension of the physical space
$\mathds{R}^{3}$ where the system is prepared.—in ref.Grossi et al. (2023),
the authors partially address this relationship.

Strengthening this relationship, our primary objective is to uncover the
_operational_ meaning of the parameters $(\theta,\varphi)$ for concrete
examples. This requires finding an experimental prescription that associates
these parameters with physical significance, specifically for polarized
photons.

To achieve this, we need to take several steps. To begin with, we delve into
the details of the Hopf fibration—this more mathematical argument will clear
the path for what is to come. Next, we will present a kind of universal device
for polarization optics, similarly to what has been proposed in Simon and
Mukunda (1989). This device comprises two half-wave and two quarter-wave
plates coaxially mounted, allowing for the generation of any $SU(2)$
polarization transformation. Following that, we directly apply this device to
obtain the parametrization (1), providing a straightforward interpretation of
$\theta$ and $\varphi$ in terms of adjustable parameters within the plates.

##  III The relation between physical space and space of states: Hopf
fibration

Our starting point comes from the geometry of indistinguishable states on
$\mathfrak{S}$. Two state vectors $\ket{\psi}$ and $\ket{\psi^{\prime}}$ that
exclusively differ by a global phase factor

$\ket{\psi^{\prime}}=e^{i\alpha}\ket{\psi},\,\alpha\in\mathds{R}$ (7)

are said to be indistinguishable. Such indistinguishability results from the
fact that both states provide the same expectation values of an observable:

$\bra{\psi^{\prime}}\mathcal{A}\ket{\psi^{\prime}}=\bra{\psi}\mathcal{A}\ket{\psi},$
(8)

where $\mathcal{A}$ denotes a self-adjoint operator representing a given
physical observable. In this way, (pure) states can be seen as the equivalence
classes, or orbits Nakahara (1990), of the action of the group
$\mathcal{U}(1)$ on $\mathfrak{S}$,111The details of some set theory concepts
and action of groups on the latter are presented in the Appendix A. defined by

$\ket{\psi}\mapsto e^{i\alpha}\ket{\psi}.$ (9)

Given a representative $\ket{\psi}=a\ket{0}+b\ket{1}$, we label its unique
corresponding orbit $[(a,b)]\subset\mathds{C}^{2}/\\{(0,0)\\}$. Each orbit, in
turn, can be identified by the complex number given by the ratio

$\displaystyle h:\mathds{C}^{2}/\\{(0,0)\\}$
$\displaystyle\rightarrow\mathds{C}$ $\displaystyle[(a,b)]$
$\displaystyle\mapsto h\left([(a,b)]\right)=\frac{b}{a},\,a\neq 0.$ (10)

Clearly, $h$ is well-defined for equivalence classes, as its action is
independent of the class representative - see Appendix A.

Since the map $h$ takes its values on the space $\mathds{C}$, which, in turn,
can be bijectively mapped onto the sphere $S^{2}\subset\mathds{R}^{3}$, we
have a clue for connecting $S^{2}$ to $S^{3}\subset\mathds{R}^{4}$. First, let
us consider the equatorial (bijective) stereographic projection

$\displaystyle T:S^{2}\subset\mathds{R}^{3}$
$\displaystyle\longrightarrow\mathds{C}$ $\displaystyle(x_{1},x_{2},x_{3})$
$\displaystyle\longmapsto T(x_{1},x_{2},x_{3})=X+iY=Re^{i\theta}.$ (11)

$(x_{1},x_{2},x_{3})$ are the coordinates of a point on the sphere $S^{2}$,
while $X$ and $Y$ are the real and imaginary (resp.) parts of a complex number
with magnitude $R$ and phase $\theta$. The inverse map is given by

$\displaystyle T^{-1}(z=X+iY)=\quad\qquad\qquad\qquad\qquad\qquad\qquad$
$\displaystyle=\left(\frac{2X}{X^{2}+Y^{2}+1},\frac{2Y}{X^{2}+Y^{2}+1},\frac{X^{2}+Y^{2}-1}{X^{2}+Y^{2}+1}\right).$
(12)

The anticipated connection between the space of states and the physical space
is established through the following composition $T^{-1}\circ
h:\mathds{C}^{2}/\\{(0,0)\\}\subset S^{3}\rightarrow S^{2}$, explicitly given
by

$(T^{-1}\circ
h)\left([(a,b)]\right)=\left(2\mbox{Re}(ba^{*}),2\mbox{Im}(ba^{*}),|b|^{2}-|a|^{2}\right).$
(13)

We emphasize that eq. (13) is a consistent operation for equivalence classes,
as the three entries on its r.h.s. depend on combinations of the form
$z_{1}z_{2}^{*}$, which renders it independent of the phase factor for
different elements within the class. The details of this specific calculation
may be found in Grossi et al. (2023).

The triple $(T^{-1}\circ h,S^{3},S^{2})$ defines the Hopf Fibration de
Carvalho and Siejakowski (2021). We interpret the composition $T^{-1}\circ h$
as a projection $\pi$ from $S^{3}$ onto the base space $S^{2}$. On one hand,
indistinguishable states are projected onto the same point $P$ in $S^{2}$. On
the other hand, the inverse image $\pi^{-1}(P)$ contains subsets that
correspond to the orbits resulting from the action of $\mathcal{U}(1)$ on
$S^{3}$, which are great circles: $\pi^{-1}(P)\cong S^{1}$—for the action of a
group over a set, see Appendix A. In geometric terms, we summarize this as
$S^{1}\hookrightarrow S^{3}\to S^{2}$. The representation of such construction
is depicted in the Fig. 1.

Figure 1: Geometrical representation of the Hopf fibration.

This mathematical characterization implies that qubit states within
$\mathfrak{S}\cong S^{3}\subset\mathds{C}^{2}$ can be prepared in the physical
space represented by $S^{2}\subset\mathds{R}^{3}$.

In ref.Grossi et al. (2023), focusing exclusively on the Stern-Gerlach
experimental setup, the authors have established a similar connection between
the state space and the physical space. There they give a prescription on how
an ideal Stern-Gerlach apparatus, with a magnetic field pointing in a specific
direction

$\hat{r}=(\sin\theta\cos\varphi,\sin\theta\sin\varphi,\cos\theta)$ (14)

can be controlled to prepare the generic qubit state given by eq. (1).

Granted, in ref.Grossi et al. (2023), the authors have also shown how to turn
dimension witnesses on their heads so they could be used to bound the
dimension of the physical space—instead of bounding a target quantum system’s
dimension—but the core of their analysis resided on the particularities of the
Stern-Gerlach experiment. In this sense, their approach may leave room for
criticism for being apparently too system-dependent. The question, then, is:
what implications can we draw for other quantum systems? Put another way, does
the Hopf Fibration suggest a ‘universal’ approach to preparing states in
$\mathds{C}^{2}$ within a laboratory embedded in physical space represented by
$\mathds{R}^{3}$?

Consider another simple two-level quantum system involving a beam of light
passing through a tourmaline crystal, borrowed from Dirac’s seminal work Dirac
(2010). In this experiment, it is observed that when the beam is polarized at
an angle $\alpha$ to the optic axis of the crystal, only a fraction,
$\sin^{2}\alpha$, passes through according to classical electrodynamics. In
terms of a single photon, this implies a probability of $\sin^{2}\alpha$ for
it to be found on the back of the crystal, while $\cos^{2}\alpha$ represents
the probability of absorption.

However oversimplified is this description, the experimental scenario we
borrowed from Dirac’s study suggests that photons can be described by a state
vector in $\mathds{C}^{2}$, which we will use in a while. This is nothing new,
but we needed a concrete and easy-to-digest case to work with. That is exactly
what we will do.

The polarization of an electromagnetic wave can be described in the laboratory
through the vector structure of the corresponding electric field. Consider the
simplest case of a monochromatic wave with frequency $\omega$ traveling in the
$z$ direction. The complete description of polarization is provided by the
electric field

$\vec{E}(t)=\left(E_{0x}\cos(\omega t-\delta_{x}),E_{0y}\cos(\omega
t-\delta_{y})\right).$ (15)

In this case, we observe the most general form of elliptical polarization.
However, we should emphasise that there is a missing dimension. In the
following sense: the preparation of the polarized beam of light is confined to
a plane, a two-dimensional space where $\vec{E}(t)$ resides. Yet, in line with
the prescription of the Hopf Fibration, one would anticipate a three-
dimensional background. How can we reconcile these two perspectives? We will
address this issue in the forthcoming sections.

##  IV Building a universal gadget for polarization of states

Expanding our previous discussion, we start with the assumption that a beam of
polarized photons can be represented by a state in
$\mathfrak{S}\subset\mathds{C}^{2}$. As we seek to describe a preparation of
states that not only align with the parametrization in eq. (1), but that also
addresses the missing (background) dimension in the space of state preparation
hinted out at the end of the previous section, we turn to a tool known as the
‘universal gadget for polarization optics’ Simon and Mukunda (1989).

The universal gadget consists of two half-wave and two quarter-wave plates
that are coaxially aligned—see fig. 2. This setup is capable of realizing
every $SU(2)$ polarization transformation. This particular group represents
linear transformations on field vector components of light beams, keeping the
intensity untouched. Thus, $SU(2)$ is key in polarization optics. Leveraging
on operational grounds, how can we realize such transformations in a
laboratory? Putting aside the technical and experimental intricacies inherent
to the realization of this setup in the actual laboratories, we turn our
attention to the mathematical structure of this tool. One may use quarter-wave
and half-wave plates, which introduce (mathematically) rotations of $\pi/2$
and $\pi$, respectively, as particular instances of $SU(2)$ elements. The
gadget generalizes these transformations for any $SU(2)$ polarization. For
that, one rotates the plates around the common axis, obtaining, any particular
$SU(2)$ polarized state. The corresponding angular positions of the plates not
only realize uniquely the group polarization transformation as well as prepare
arbitrary states, represented by a vector on the Bloch sphere.

Figure 2: Schematic drawing for the preparation of a state $\ket{\psi_{out}}$
from an arbitrary initial state $\ket{\psi_{in}}$. Note that $\ket{\psi_{in}}$
and $\ket{\psi_{out}}$ represent physical entities actually prepared in the
laboratory.

Both half ($H$) and quarter-wave ($Q$) plates are represented by the
conjugation

$\displaystyle H_{\varphi}$
$\displaystyle=\vartheta(\varphi)i\sigma_{3}\vartheta(\varphi)^{-1}$ (16)
$\displaystyle Q_{\varphi}$
$\displaystyle=\vartheta(\varphi)\begin{pmatrix}e^{i\pi/4}&0\\\
0&e^{-i\pi/4}\end{pmatrix}\vartheta(\varphi)^{-1}$ (17)

where $\vartheta(\varphi)$ is the $SO(2)$ element

$\vartheta(\varphi)=e^{-i\varphi\sigma_{2}}=\begin{pmatrix}\cos\varphi&-\sin\varphi\\\
\sin\varphi&\cos\varphi\end{pmatrix}.$ (18)

The main result from ref. Simon and Mukunda (1989), that we are using here, is
that any element $U(\xi,\eta,\zeta)$ representing an arbitrary $SU(2)$
polarization transformation can be written as the composition

$\displaystyle
U(\xi,\eta,\zeta)=Q_{\frac{\xi}{2}+\frac{\pi}{4}}H_{\frac{\xi}{2}+\frac{\eta}{4}\mp\frac{\pi}{4}}Q_{\frac{\xi}{2}-\frac{\pi}{4}}H_{\frac{\xi-\zeta}{4}\pm\frac{\pi}{4}}.$
(19)

The parameters $\xi$, $\eta$, $\zeta$ are the known Euler angles, and one can
re-write222Different decompositions of unitary operations are allowed, see for
example refs. Nielsen and Chuang (2010); Tung (1985). eq.(19) in a neater
format:

$U(\xi,\eta,\zeta)=e^{-\frac{i}{2}\xi\sigma_{2}}e^{\frac{i}{2}\eta\sigma_{3}}e^{-\frac{i}{2}\zeta\sigma_{2}}.$
(20)

According to ref. Simon and Mukunda (1989), while $\xi$ and $\zeta$ range in
$[0,2\pi]$, $\eta$ is restricted to $[0,\pi]$. As we will argue in the next
section, this latter restriction will strongly impact the functioning of our
universal gadget, we will need to input at least two states through it to
cover the entirety of the Bloch sphere—additionally, the same restriction on
$\eta$ will also unexpectedly reflect the fact that spheres are two-
dimensional objects.

These parameters are viewed as adjustable values that an experimentalist can
handle within the plates to prepare arbitrary $SU(2)$ polarized states.

In the next section, we will show how we can prepare the whole Bloch sphere as
an outcome of the universal gadget when we input through it a particular set
of states.

##  V Preparing arbitrary SU(2) state polarization

Interpreting the Euler angles as accessible parameters for state preparation
enables us to apply $U(\xi,\eta,\zeta)$ to a specific initial state
$\ket{\psi_{in}}$ and examine the resulting outcome of this preparation. We
express this as follows,

$\ket{\psi_{out}}=U(\xi,\eta,\zeta)\ket{\psi_{in}}.$ (21)

This can be visualized as allowing the incident state to pass through the
universal gadget, as illustrated in Figure 2.

As our starting point, we select $\ket{\psi_{in}}=\ket{0}$. We also set 333The
motivation for such choice stems from the fact that any direction in the
three-dimensional space $\mathds{R}^{3}$ can be defined uniquely by the two
angles $\theta$ and $\varphi$. Thus, by fixing $\xi=0$, we are left with two
parameters, $\eta$ and $\zeta$, to be identified in one way or the other with
$\theta$ and $\varphi$. $\xi=0$. That way, we calculate $\ket{\psi_{out}}$ in
the canonical representation of the basis $\\{\ket{0},\ket{1}\\}$.

$\displaystyle\ket{\psi_{out}}$
$\displaystyle=U(0,\eta,\zeta)\ket{0}=e^{\frac{i}{2}\eta\sigma_{3}}e^{-\frac{i}{2}\zeta\sigma_{2}}\ket{0}$
$\displaystyle=\begin{pmatrix}e^{i\frac{\eta}{2}}&0\\\
0&e^{-i\frac{\eta}{2}}\end{pmatrix}\begin{pmatrix}\cos\zeta/2&-\sin\zeta/2\\\
\sin\zeta/2&\cos\zeta/2\end{pmatrix}\begin{pmatrix}1\\\ 0\end{pmatrix}$
$\displaystyle=\begin{pmatrix}e^{i\frac{\eta}{2}}\cos\zeta/2\\\
e^{-i\frac{\eta}{2}}\sin\zeta/2\end{pmatrix}.$ (22)

Because global phase factors are statistically irrelevant, we can factor out
$e^{i\frac{\eta}{2}}$, resulting in the equivalent state

$\ket{\psi^{\prime}_{out}}=\cos\frac{\zeta}{2}\ket{0}+e^{-i\eta}\sin\frac{\zeta}{2}\ket{1}.$
(23)

A comparison between eq. (1) and eq. (23) indicates that our universal gadget
can, indeed, generate any state on the Bloch sphere’s Western hemisphere. We
are half-way through the claimed universality of such gadget, as we still need
to cover its Eastern hemisphere.

To do so, we similarly select a second initial state $\ket{\psi_{in}}=\ket{1}$
and set $\xi=0$. With an analogous calculation, we obtain

$\ket{\psi^{\prime\prime}_{out}}=-\sin\frac{\zeta}{2}\ket{0}+e^{-i\eta}\cos\frac{\zeta}{2}\ket{1}.$
(24)

We notice that $\braket{\psi^{\prime\prime}_{out}}{\psi^{\prime}_{out}}=0$,
which allows us to conclude that the $\ket{\psi^{\prime}_{out}}$ and
$\ket{\psi^{\prime\prime}_{out}}$ are antipodes on $\mathfrak{S}$. Since the
former lies on the Western hemisphere, the latter is in fact in the Eastern
hemisphere, sweeping all the points on the Bloch sphere.

##  VI Conclusion

In this paper, we have investigated the intricate relationship between the
state space of two-level quantum systems and the physical space in which these
systems are prepared. The necessity of such presentation arises when we adopt
a universal character of the Hopf fibration in the following sense. By one
side the mathematical construct indicates that any two-level quantum system
could be prepared in a three-dimensional physical space. A concrete example of
this connection was explored in Grossi et al. (2023), where the authors
showed, through a modern rereading of the Stern-Gerlach experiment, how the
orientation of the magnetic field in an arbitrary direction in space prepares
a qubit state represented by (1). On the other hand, it seems that there is a
missing dimension when we think of polarized light as another example of qubit
being described by the two-dimensional - and not three - plane where its
electric field oscillates. With more details:

1\. Using the Hopf fibration, we establish a connection between
indistinguishable pure states in $S^{3}$ and $S^{2}\subset\mathds{R}^{3}$. Our
interpretation demonstrates that this connection is more than a mere
mathematical construct. Rather, we view the fibration as a description of how
states are prepared, with adjustable parameters in a laboratory defining the
Bloch vector (14). The corresponding inverse image leads to the states
parameterized by (1).

2\. While the fibration does not discriminate among specific two-level quantum
systems, we have chosen to focus on an illustrative example in our work:
polarized light. Initially, one might think that the plane of oscillation of
the electric field fully specifies polarization. However, this notion
contradicts the structure revealed by the fibration, which implies a three-
dimensional space of preparation. To reconcile this contradiction, we employed
a universal gadget for $SU(2)$ polarization optics. Our investigation
demonstrated that the adjustable parameters within this device define a state
in the Bloch sphere. Consequently, we have successfully ascribed a practical,
operational interpretation to the parametrization (1), aligning with our
objectives.

## Acknowledgements

BFR and VGV are in debt with Prof. Wallon A. T. Nogueira, Profa. Giovana
Trevisan Nogueira and Prof. Rodrigo A. Dias for helpful discussions and
recommendations for completing this work. CD wishes to thank the hospitality
of both the Universidade Federal de Pernambuco and the Universidade Federal de
Juiz de Fora, where part of this work was realized. LLB would like to express
his gratitude to the Universidade Federal de Juiz de Fora for his master’s
scholarship.

This work has also been supported by Programa Institucional de Bolsas de
Iniciação Científica – VI VIC/Universidade Federal de Juiz de Fora –
2022/2023, project number 51268.

##  Appendix A Basic set theory and the action of groups on arbitrary sets

In this appendix, we briefly introduce a selection of topics concerning set
theory: equivalence relations, equivalence classes, the action of a group on a
set and related concepts. Although the points we address here are not new, our
intention is to make the manuscript as self-consistent as possible. For an in-
depth description, we direct the reader to Jauch (1973); Nakahara (1990).

We start our overview by reviewing an alternative way of turning elements of a
set into equivalent elements. Presumably, the first and foremost notion of
equivalence is the one commonly connected with the concept of equality.
Disguised by common sense, equality is a binary relation—a relation between
two objects—which is reflexive (an object is equal to itself), symmetric (if
an object is equal to another, then the latter is also equal to the former)
and transitive (if an object is equal to a second, and this second to a third,
then the original and the final objects also are so). What is crucial here is
that equality, as we know it, is only a special case of a larger class of
binary relations intended to capture and classify distinctive characteristics
of a collection of objects. The next definitions rigorously address these
points.

###### Definition A.1

(Equivalence Relations). Let $\mathds{X}$ be a non-empty set and
$\sim\subset\mathds{X}\times\mathds{X}$ a relation. We say that $\sim$ is an
_equivalence relation_ whenever the three properties below hold true,

i. $\sim$ is reflexive: $x\sim x$, $\forall x\in\mathds{X}$.

ii. $\sim$ is symmetric: $x\sim y\Rightarrow y\sim x$, $\forall
x,y\in\mathds{X}$

iii. $\sim$ is transitive: if $x\sim y$ and $y\sim z$, then $x\sim z$,
$\forall x,y,z\in\mathds{X}$.

Remark. Any relation over a given set $\mathds{X}$ is a subset $R$ of
$\mathds{X}\times\mathds{X}$. In this sense, $R$ is a collection of ordered
pairs $(x,y)$ where $x,y\in\mathds{X}$. Mainly because we want to express the
relational aspect of $R$ and also retain the parallel with the notion of
equality, it is usual to write $xRy$ instead of $(x,y)\in R$—the reader will
certainly appreciate this change in notation.

###### Definition A.2

(Equivalence Classes). Let $\sim$ be an equivalence relation over
$\mathds{X}$. Given an element $y$ we define the following subsets of
$\mathds{X}$,

$[y]:=\\{x\in\mathds{X}\,|\,x\sim y\\}.$ (25)

Because $y\sim y$, $y\in[y]$, for all elements of $\mathds{X}$. Thus, these
subsets are well-defined and we name them _equivalence classes_. In
particular, for a given $y\in\mathds{X}$, the subset $[y]$ is called the
_equivalence class of $y$_.

One of the main consequences of establishing an equivalence relation on a set
resides in the theorem below. It guarantees that equivalence classes provide a
partition of the set they are defined over—that is, a cover of the entire set
formed by disjoint subsets.

###### Theorem A.1

(Partition via Equivalence Classes). If $\sim$ is an equivalence relation on
$\mathds{X}$, then

$\displaystyle 1)\,x\sim y\Rightarrow[x]=[y],$ (26) $\displaystyle 2)\,x\nsim
y\Rightarrow[x]\cap[y]=\emptyset,$ (27) $\displaystyle
3)\,\mathds{X}=\bigcup_{x\in\mathds{X}}[x].$ (28)

Remark. A useful notation to the disjoint union in eq. (28) is
$\mathds{X}/\sim$, as it denotes that the set has been partitioned across the
many equivalence classes defined by the equivalence relation. For both an
intuitive interpretation and formal demonstration, we direct the reader to
Vasconcelos Júnior et al. (2018).

Now we turn our attention to group actions on sets. Following the natural
steps of introducing sets and relations, we could start with functions (or
mappings), which are a special type of relation. However, we would like to do
so bearing in mind that the elements of the set should transform guided by a
group structure, retaining not only the symmetry properties but also the
special transformations the latter usually conveys. Hence, we define the

###### Definition A.3

(Action of a Group). Let $\mathcal{G}$ be a group and $\mathds{X}$ be a non-
empty set. The _action_ of $\mathcal{G}$ on $\mathds{X}$ is a map
$\varphi:\mathcal{G}\times\mathds{X}\rightarrow\mathds{X}$, satisfying the
following conditions,

i. $\varphi(e,x)=x$, $\forall\,\,x\in\mathds{X}$, where $e$ stands for the
group identity element.

ii. For each $g\in\mathcal{G}$,
$\varphi(g,\cdot):\mathds{X}\rightarrow\mathds{X}$ is a bijection.

iii. $\varphi(g_{1},\varphi(g_{2},x))=\varphi(g_{1}g_{2},x)$,
$\forall\,\,x\in\mathds{x}$ and $\forall\,\,g_{1},g_{2}\in\mathcal{G}$.

Now, in $\mathds{X}\times\mathds{X}$ we introduce the following relation,

$x\sim y\Leftrightarrow y=\varphi(g,x),\mbox{ for some }g\in\mathcal{G}.$ (29)

We affirm that $\sim$ is an equivalence relation. In fact, $\sim$ is reflexive
because i: $\varphi(e,x)=x$. The symmetry comes from

$y=\varphi(g,x)\Rightarrow x=\varphi(g^{-1},y).$

Finally, the transitivity can also be promptly deduced. It suffices to note
that if $y=\varphi(g_{1},x)$ and $z=\varphi(g_{2},y)$, then
$z=\varphi(g_{3},x)$, where $g_{3}=g_{2}g_{1}$.

With the equivalence relation defined by the action, we define the equivalence
classes by

$[x]:=\\{y\in\mathds{X}\,|\,y=\varphi(g,x)\mbox{ for some
}g\in\mathcal{G}\\}.$ (30)

Due to its special role, an equivalence class of such type is called an orbit
through $x$.

Crucially, according to the theorem A.1 above, $\mathds{X}$ is partitioned by
its disjoint orbits. All in all, that is to say, starting from the action of a
group over a set, we can partition that set across the orbits of each of its
elements.

Next, we will present some examples of interest not only to our previous
discussion per se, but also to quantum mechanics in general.

Example 1. For our first example, we consider $\mathds{X}=\mathds{C}^{2}$ and
$\mathcal{G}=\mathcal{U}(1)=\\{e^{i\alpha}\,{\color[rgb]{0.9,0,0}\,}\alpha\in\mathds{R}\\}$.
Define the map

$\displaystyle\varphi:\mathcal{U}(1)\times\mathds{C}^{2}$
$\displaystyle\rightarrow\mathds{C}^{2}$
$\displaystyle(e^{i\alpha},\ket{\psi})$
$\displaystyle\mapsto\ket{\psi}^{\prime}=e^{i\alpha}\ket{\psi}.$

To check that $\varphi$ is indeed an action, we have

i. $\mathcal{U}(1)\ni e=1\Rightarrow\varphi(e,\ket{\psi})=\ket{\psi}$.

ii. Given $\alpha\in\mathds{R}$,
$\varphi(e^{i\alpha},\cdot):\mathds{C}^{2}\rightarrow\mathds{C}^{2}$ is a
bijection. In effect,

$e^{i\alpha}\ket{\psi_{1}}=e^{i\alpha}\ket{\psi_{2}}\Rightarrow\ket{\psi_{1}}=\ket{\psi_{2}}$

which guarantees that the map is injective. In turn, we point out that any
$\ket{\psi}\in\mathds{C}^{2}$ is reached by $e^{-i\alpha}\ket{\psi}$ under
$\varphi(e^{i\alpha},\cdot)$. Thus, the map is surjective as well.

iii. At last,

$\varphi(e^{i\alpha_{1}},e^{i\alpha_{2}}\ket{\psi})=e^{i(\alpha_{1}+\alpha_{2})}\ket{\psi}=\varphi(e^{i\alpha_{1}}e^{i\alpha_{2}},\ket{\psi}),$

which concludes the proof.

The importance of the case in point lies on the fact that the orbits consists
of indistinguishable states for two-level quantum systems as well as fibers in
the Hopf fibration, as previously discussed in section III.

Example 2. Our second example is given by $\mathds{X}=\mathds{R}^{n}$ and
$\mathcal{G}=\mathds{R}^{*}=\mathds{R}\smallsetminus\\{0\\}$, where the group
product is the usual multiplication of non-zero real numbers. Define

$\displaystyle\varphi:\mathds{R}^{*}\times\mathds{R}^{n}$
$\displaystyle\rightarrow\mathds{R}^{n}$ $\displaystyle(\lambda,\vec{v})$
$\displaystyle\mapsto\lambda\vec{v}.$

It is not difficult to conclude that $\varphi$ is an action. In fact, we have,

i. $\mathds{R}^{*}\ni e=1\Rightarrow\varphi(1,\vec{v})=\vec{v}$.

ii. Given $\lambda\in\mathds{R}^{*}$, $\varphi(\lambda,\cdot)$ is a bijection
for

$\lambda\vec{v}_{1}=\lambda\vec{v}_{2}\Rightarrow\vec{v}_{1}=\vec{v}_{2},$

which shows the injection. Moreover, any $\vec{v}\in\mathds{R}^{n}$ can be
obtained by applying $\varphi(\lambda,\cdot)$ to $\lambda^{-1}\vec{v}$. Hence,
the map is also surjective.

iii. Finally,
$\varphi(\lambda_{1},\lambda_{2}\vec{v})=\varphi(\lambda_{1}\lambda_{2}\vec{v})$.

The classes, or orbits, here are straight lines crossing the origin, although
$\vec{0}\neq[\vec{V}]$, for every non-zero vector $\vec{v}$ in
$\mathds{R}^{n}$. Their union forms what is called the projective space, named
$\mathds{R}\mathds{P}^{n-1}$. The generalization for $\mathds{C}^{n}$ and,
accordingly, to $\mathds{C}\mathds{P}^{n-1}$ is straightforward. The value of
the latter to quantum mechanics stems from its intrinsic connection to
entanglement Bengtsson et al. (2002).

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