# Stereographic Geometry of Coherence and Which-path Information

Yusef Maleki<EMAIL_ADDRESS>Department of Physics and Astronomy,
Texas A&M University, College Station, Texas 77843-4242

###### Abstract

Recently, it was shown that quantum entanglement is an indispensable part of
the duality behavior of light. Here, we report a surprisingly intimate
connection between the stereographic projection and the duality–entanglement
nature of a single photon. We show that, the duality–entanglement relation
[Optica 5, 942 (2018)], naturally emerges from the stereographic projection
geometry. We demonstrate that this geometry is complementarity sensitive; in
the sense that, it is sensitive to the particle nature, wave nature, and
entanglement nature of a single photon.

The attribution of wave nature to particles by de Broglie was one of the most
profound counterintuitive weirdnesses of quantum physics De Broglie (1923).
Later, this bizarre nature of a single quantum was introduced by Bohr in a
more general setting, as the complementarity principle; where, a quantum
object possesses properties which are equally real but mutually exclusive Bohr
(1928).

The duality nature of a single quantum becomes manifest by considering a
photon subjected to an interferometer that can demonstrate wave or particle
properties Scully  _et al._ (1991). The particleness is determined by the
amount of information one attains on the path distingishablity of photon
Scully  _et al._ (1991); Dürr  _et al._ (1998). Alternatively, the waveness
can be characterized by the visibility of the interference pattern on the
screen Scully  _et al._ (1991); Dürr  _et al._ (1998).

Richard Feynman considers the wave-particle duality incorporating the
underlying mystery of quantum mechanics. More precisely, he says: ”In reality,
it contains the only mystery” Feynman  _et al._ (1965).

The ’mystery’ pointed to by Feynman, remains as a long–standing subject of
dedicated investigations Jacques  _et al._ (2008); Chang  _et al._ (2008);
Arndt  _et al._ (1999); Rab  _et al._ (2017); Menzel  _et al._ (2012); Peruzzo
_et al._ (2012), providing an underlying platform for most of the interesting
features and far–reaching application of quantum mechanics Jacques  _et al._
(2007); Kim  _et al._ (2000); Kaiser  _et al._ (2012); Jeong  _et al._ (2013);
Bagan  _et al._ (2016); Winter and Yang (2016); Cheng and Hall (2015);
Lostaglio  _et al._ (2015); Baumgratz  _et al._ (2014). However, despite
dedicated efforts, the quantitive description of the wave-particle duality was
not found until 1979; where, Wootters and Zurek quantified the wave-particle
duality nature of a single quantum Wootters and Zurek (1979). Later, the
duality nature of a quantum system was explicitly formulated as an inequality,
where the visibility of the interference pattern $V$ and the which–path
distinguishability $D$ of a single quantum, in a two-dimensional Hilbert
space, was shown to satisfy $V^{2}+D^{2}\leq 1$ Wootters and Zurek (1979);
Englert (1996).

Recently, it was shown that analysis of the vector mode coherence in Young
double slit experiment uncovers the significant role of the entanglement on
the duality nature of the light modes Qian  _et al._ (2018, 2016).
Accordingly, for the double slit experiment, the duality relation can be
expressed through $V^{2}+D^{2}+C^{2}=1$ Qian  _et al._ (2018, 2016). The
concurrence, $C$, quantifies the entanglement of the two vector modes involved
in the double slit experiment.

On the other hand, a pure two–dimensional quantum state could be expressed via
complex variables $\alpha$ and $\beta$ as
$|\psi\rangle=\alpha|0\rangle+\beta|1\rangle$, enabling the state to be
represented as a point on the Bloch sphere. The Bloch sphere can be obtained
from Hopf fibraion of $S^{3}$ Lee  _et al._ (2002). In which, $S^{3}$ is the
unit sphere embedded in $\mathbb{R}^{4}$, the fibre $S^{1}$ is the global
$U(1)$ phase of the state and $S^{2}$ the base manifold. Similarely, two–qubit
states, are characterized by $S^{7}$ sphere embedded in $\mathbb{R}^{8}$,
providing fibration geometry in the quaternionic skew-field Mosseri and
Dandoloff (2001); Najarbashi  _et al._ (2007). In this platform, the product
states are mapped into a 2-dimensional planar subspace, enabling entangled and
separable subspaces be distiguishable through this map Mosseri and Dandoloff
(2001).

In this lettter, we show that the duality–entanglement relation
$V^{2}+D^{2}+C^{2}=1$, naturally emerges from the stereographic projection Lee
_et al._ (2002); Najarbashi  _et al._ (2007) obtained from fibration of the
$S^{7}$ sphere geometry. In this setting, there is no need to specify the
experimental setup, as the emergence of the relation is a generic property of
the geometry. The full geometry of the duality is captured by $S^{4}$ sphere,
obtained from Hopf fibration of $S^{7}$. This enables stratification of the
duality relation through entanglement, providing a full geometric picture of
the scenario. This surprisingly intimate connection between the $S^{7}$ Hopf
fibration and the complementarity nature of a single photon is shown to be
complementarity sensitive; in the sense that, it is sensitive to the
particleness, waveness, and entangledness of a single photon.

We consider a generic scenario, where a single photon could be correlated with
some other system (be it an atom, photon, environment, etc). In this case , we
assume the two-state photon Hibert space to be encoded by the basis
$|0\rangle$ and $|1\rangle$ (note that these basis can be two orthonormal
polarization degrees of freedom, photon number states, etc). Thus, the generic
form of such a state can be given as

$|\psi\rangle=\mu|0\rangle|\chi_{1}\rangle+\nu|1\rangle|\chi_{2}\rangle,$ (1)

where $|\chi_{1}\rangle$ and $|\chi_{2}\rangle$ are associated to any
correlated system. We can write $|\chi_{1}\rangle=\sum
a_{i.j,...k}|a_{i}\rangle|b_{j}\rangle...|c_{k}\rangle$, incorporating any
involved degrees of freedom. Similarly, $|\chi_{2}\rangle=\sum
b_{i.j,...k}|a_{i}\rangle|b_{j}\rangle...|c_{k}\rangle$. Where,
$|\chi_{1}\rangle$ and $|\chi_{2}\rangle$ are not necessarily orthogonal;
however, they can always be considered to span a two dimentsional vector space
Maleki and Zheltikov (2019, 2018); Maleki and Maleki (2018). Thus, they can
always be mapped into a two dimensional space such that
$|\chi_{1}\rangle=a|e\rangle+b|f\rangle$ and
$|\chi_{2}\rangle=c|e\rangle+d|f\rangle$ Maleki and Zheltikov (2019, 2018);
Maleki and Maleki (2018). Here, $|e\rangle$ and $|f\rangle$ form orthonormal
basis of the two dimensional Hilbert space. The coefficients are determined by
decomposition of the vectors $|\chi_{1}\rangle$ and $|\chi_{2}\rangle$ in the
$|e\rangle$ and $|f\rangle$ directions Maleki and Zheltikov (2019, 2018);
Maleki and Maleki (2018). Thus, the most general form of the scenario can be
written as

$|\psi\rangle=\alpha_{0}|0\rangle|e\rangle+\alpha_{1}|0\rangle|f\rangle+\alpha_{2}|1\rangle|e\rangle+\alpha_{3}|1\rangle|f\rangle.$
(2)

In the two dimensional Hilbert space framework of the photon spanned by the
bases $\\{|0\rangle,|1\rangle\\}$, one can assign Pauli matrices
$\sigma_{x}=|0\rangle\langle 1|+|1\rangle\langle 0|$,
$\sigma_{y}=-i|0\rangle\langle 1|+i|1\rangle\langle 0|$ and
$\sigma_{z}=|0\rangle\langle 0|-|1\rangle\langle 1|$ to describe the system.
The duality nature of a correlated photon reduces to considering the state in
Eq. 2. The wave nature of the photon is determined by the fring visibility
Qian and Agarwal (2019)

$V=\frac{p_{D}^{max}-p_{D}^{min}}{p_{D}^{max}+p_{D}^{min}}.$ (3)

To determine the visibility we can measure the observable
$\frac{1}{2}(1+\sigma_{x})$ such that
$p_{D}=\langle\psi|\frac{1}{2}(1+\sigma_{x})|\psi\rangle$ Englert (1996). This
results in $p_{D}=\frac{1}{2}(1+2|\gamma|\cos\varphi)$, where
$\gamma=|\gamma|e^{i\varphi}=\bar{\alpha}_{2}\alpha_{0}+\bar{\alpha}_{3}\alpha_{1}$.
Here, $\bar{\alpha}_{i}$ is the complex conjugate of ${\alpha}_{i}$. Hence,
the visibility of a single photon is given by

$V=2|\bar{\alpha}_{2}\alpha_{0}+\bar{\alpha}_{3}\alpha_{1}|.$ (4)

On the other hand, the particle nature of the photon is related to our priori
knowledge on the predictability of the photon being in the state $|0\rangle$
or $|1\rangle$. Therefore, the particle nature of the photon can be quantified
as Qian  _et al._ (2018, 2016)

$D=\frac{|p_{0}-p_{1}|}{p_{0}+p_{1}},$ (5)

where, $p_{0}$ and $p_{1}$ are the probabilities of the photons being detected
in the states $|0\rangle$ and $|1\rangle$, respectively.

From Eq. 2 we have, $p_{0}=|\alpha_{0}|^{2}+|\alpha_{1}|^{2}$ and
$p_{1}=|\alpha_{2}|^{2}+|\alpha_{3}|^{2}$. Thus, the particleness is given by

$D=|(|\alpha_{0}|^{2}+|\alpha_{1}|^{2})-(|\alpha_{2}|^{2}+|\alpha_{3}|^{2})|.$
(6)

The relations above can be understood by considering the reduced density
matrix of the photon

$\displaystyle\rho_{ph}=$
$\displaystyle\left(\begin{array}[]{cc}|\alpha_{0}|^{2}+|\alpha_{1}|^{2}&(\bar{\alpha}_{2}\alpha_{0}+\bar{\alpha}_{3}\alpha_{1})\\\
({\alpha}_{2}\bar{\alpha}_{0}+{\alpha}_{3}\bar{\alpha}_{1})&|\alpha_{2}|^{2}+|\alpha_{3}|^{2}\\\
\end{array}\right).$

Accordingly, the waveness is determined by the coherence terms of the density
matrix and the particlness is determined by our knowledge on the probabilities
of finding the system in each basis, as expected.

It is remarkable that, since Eq.2 is expressed as a two-qubit superposition,
one may consider the photon state to be encoded in the second subsystem with
the bases $\\{|e\rangle,|f\rangle\\}$. In this scenario, $V$ and $D$ can be
determined through the reduced density matrix of the second subsystem; where,
$V=2|\bar{\alpha}_{1}\alpha_{0}+\bar{\alpha}_{2}\alpha_{1}|$, and
$D=|(|\alpha_{0}|^{2}+|\alpha_{2}|^{2})-(|\alpha_{1}|^{2}+|\alpha_{3}|^{2})|$.

Now, to develop a stereographic projection platform for the complementarity
concept of the photon, we introduce the quaternification map $\mathcal{F}$,
such that it maps any $|\psi\rangle\in H^{\mathbb{C}}_{2}\otimes
H^{\mathbb{C}}_{2}$ to $|\psi\rangle_{q}\in H^{\mathbb{Q}}_{2}$ Najarbashi
_et al._ (2007); Najarbashi and Seifi (2017)

$\mathcal{F}(|\psi\rangle):=|\psi\rangle_{q}=\ q_{1}|0\rangle_{q}+\
q_{2}|1\rangle_{q}=\left(\begin{array}[]{c}q_{1}\\\ q_{2}\\\
\end{array}\right).$ (7)

Where, $|0\rangle_{q}$ and $|1\rangle_{q}$ are the basis of spinors of the
quaternionic states. $q_{1}$ and $q_{2}$ are quaternion numbers defined as

$q_{1}=\alpha_{0}+\alpha_{1}\hat{e}_{2},\qquad
q_{2}=\alpha_{2}+\alpha_{3}\hat{e}_{2},$ (8)

satisfying the normalization condition $|q_{1}|^{2}+|q_{2}|^{2}=1$.

Recall that, quaternions are usually presented by four real numbers
$x_{0},x_{1},x_{2},x_{3}$, such that
$q=x_{0}\hat{e}_{0}+x_{1}\hat{e}_{1}+x_{2}\hat{e}_{2}+x_{3}\hat{e}_{3},$ with
$\hat{e}_{0}=1$, and
$\hat{e}_{1}^{2}=\hat{e}_{2}^{2}=\hat{e}_{3}^{2}=\hat{e}_{1}\hat{e}_{2}\hat{e}_{3}=-1$
as the imaginary units of the quaternionic algebra. Therefore, defining the
complex numbers $z_{1}=x_{0}\hat{e}_{0}+x_{1}\hat{e}_{1}$ and
$z_{2}=x_{2}\hat{e}_{0}+x_{3}\hat{e}_{1}$, one has $q=z_{1}+z_{2}\hat{e}_{2}$.

The quaternionic stereographic projection $\mathcal{P}$, maps the points on
$S^{7}$ to the extended quaternionic space
$\mathbb{\tilde{Q}}\equiv\mathbb{Q}\cup\\{\infty\\}$ through Mosseri and
Dandoloff (2001); Najarbashi  _et al._ (2007)

$\mathcal{P}:(q_{1},q_{2})\rightarrow
Q={q_{1}q_{2}^{-1}}=\dfrac{1}{|q_{2}|^{2}}(\pi_{1}+\pi_{2}\hat{e}_{2}),$ (9)

Where, $\pi_{1}=(\bar{\alpha}_{2}\alpha_{0}+\bar{\alpha}_{3}\alpha_{1})$ and
$\pi_{2}=(\alpha_{1}\alpha_{2}-\alpha_{0}\alpha_{3}).$

Interestingly, comparing with Eq. 4, $\pi_{1}$ is associated to the coherence
term and the visibility can be given by $V=2|\pi_{1}|$. Moreover, the term
$\pi_{2}$ is related to the concurrence measure of the two–qubit entanglement
Wootters (1998) through $C=2|\pi_{2}|$. The concurrence measure of the state
(2) is defined through
$C=|\langle\psi|J(\sigma_{y}\otimes\sigma_{y})|\psi\rangle|$, where the
operator J is an antilinear operator, such that
$\langle\psi|J=\langle\psi^{*}|$ Wootters (1998).

The second map is an inverse stereographic projection from
$\mathbb{\tilde{Q}}$ onto the unit sphere $S^{4}$ defined by Najarbashi  _et
al._ (2007); Mosseri and Dandoloff (2001)

$\mathcal{Q}:Q\rightarrow\\{\\{x_{i}\\},i=0,1,2,3,4\\}\ \text{with}\
\sum^{4}_{i=0}\ x^{2}_{i}=1,$ (10)

where, the coordinates $x_{i}$s are given by

$\displaystyle x_{0}$ $\displaystyle=\langle\psi|(\sigma_{z}\otimes
I_{a})|\psi\rangle=|q_{1}|^{2}-|q_{2}|^{2},$ (11) $\displaystyle x_{1}$
$\displaystyle=\langle\psi|(\sigma_{x}\otimes
I_{a})|\psi\rangle=2\text{Re}(\pi_{1}),$ (12) $\displaystyle x_{2}$
$\displaystyle=\langle\psi|(\sigma_{y}\otimes
I_{a})|\psi\rangle=2\text{Im}(\pi_{1}),$ (13) $\displaystyle x_{3}$
$\displaystyle=\text{Im}[\langle\psi|J(\sigma_{y}\otimes\sigma_{y})|\psi\rangle]=2\text{Re}(\pi_{2}),$
(14) $\displaystyle x_{4}$
$\displaystyle=\text{Re}[\langle\psi|J(\sigma_{y}\otimes\sigma_{y})|\psi\rangle]=2\text{Im}(\pi_{2}).$
(15)

Surprisingly, the coordinates of the stereographic conformal mapping provide

$\displaystyle D^{2}$ $\displaystyle=x_{0}^{2},$ (16) $\displaystyle V^{2}$
$\displaystyle=x_{1}^{2}+x_{2}^{2},$ (17) $\displaystyle C^{2}$
$\displaystyle=x_{3}^{2}+x_{4}^{2}.$ (18)

Since, cooerdinates of the $S^{4}$ hypersphere satisfy $\sum^{4}_{i=0}\
x^{2}_{i}=1$, the relation $D^{2}+V^{2}+C^{2}=1$ emerges naturally. This
provides a fully geomerical proof for the duality–entanglement relation based
on the Hopf fibration of $S^{7}$. The analysis reveals that the
duality–entanglement relation is a characteristics of the state-vectors
geometry in the Hilbert space, which is independant of the platform by which
two-qubit state is physically realized. Therefore, the relation is valid
beyond the scope of the vector mode coherence in Young double slit experiment
Qian  _et al._ (2018), and holds for any two–qubit pure quntum state.

According to Eq. 9, product states are mapped into the pure complex subspace
in the quaternion field though the quaternionic stereographic projection
$\mathcal{P}$. In other words, this map provides a one to one correspondence
between the points on complex plane $\mathbb{\tilde{C}}$ and separable single
photon states. Also, when the wave nature of the photon vanishes
($\pi_{1}=0$), $\mathcal{P}$ reduces to subset of the quaterinionic field that
is only spanned by the $\hat{e}_{2}$ and $\hat{e}_{3}$ coordinates of the
system, such that

$Q=\dfrac{1}{|q_{2}|^{2}}(\text{Re}(\pi_{2})\hat{e}_{2}+\text{Im}(\pi_{2})\hat{e}_{3})=\dfrac{1}{2|q_{2}|^{2}}(x_{3}\hat{e}_{2}+x_{4}\hat{e}_{3}).$

Similarly, states with no particle nature ($D=0$) are mapped into the unite
quaternion $|Q|=1$ subspace of $\mathbb{\tilde{Q}}$. States with only wave
nature ($V=1$) are mapped into the unite circle in the subspace spanned by
$\\{\hat{e}_{0},\hat{e}_{1}\\}$, and the ones with only particle nature
($D=1$) are mapped onto the $Q=\\{0,\infty\\}$ subspace of
$\mathbb{\tilde{Q}}$. Finally, states with only entangled nature ($C=1$) are
mapped into the unite circle in the subspace spanned by
$\\{\hat{e}_{2},\hat{e}_{3}\\}$, and product state are never mapped to this
subspace. This proves that Hopf fibration is entangledness, waveness and
particlness sensitive.

Separable states of the photon are mapped into $\mathbb{\tilde{C}}$.
Geometrically, this means that $S^{7}$ is reduced to the two-dimensional
planer subspace of $\mathbb{\tilde{Q}}$, i.e., $S^{2}$ sphere described by
$x^{2}_{0}+x^{2}_{1}+x^{2}_{2}=1$. From Eqs. 11–15 it is clear that on the
base sphere $S^{4}$ of the Hopf map, only $x_{3}$ and $x_{4}$ components
depend on the observable of the second subsystem of Eq. 2. Thus, the second
qubit lives on the $S^{3}$ fiber of the geometry. This analysis suggests that
for a pure photon on the $S^{2}$ sphere we always have $V^{2}+D^{2}=1$. To
consider this from a different perspective, we note that the pure state of a
photon can be expressed as

$|\psi\rangle=\cos(\theta/2)|0\rangle+e^{i\varphi}\sin(\theta/2)|1\rangle.$
(19)

The waveness of the photon in this scenario is given by
$V=2\sin(\theta/2)\cos(\theta/2)$ and the particleness by
$D=|\cos(\theta/2)^{2}-\sin(\theta/2)^{2}|$. This immediately gives
$V^{2}+D^{2}=1$. For the mixed photon state with the density matrix
$\rho_{ph}$ we can obtain Qian and Agarwal (2019)

$V^{2}+D^{2}=2Tr(\rho_{ph}^{2})-1,$ (20)

where the right–hand–side quantifies degree of pureness of the photon. For a
maximally mixed state $Tr(\rho^{2})=1/2$, one has $V^{2}+D^{2}=0$.

Figure 1:  The geometry of the unit ball $B^{3}$, sliced as sells of radius
$\sqrt{1-C^{2}}$. The boundary of the ball gives the $S^{2}$ sphere of the
pure states, shown as the red (largest) shell. The inner shells represent
entangled photon states, and the smaller shells correspond to the larger
entanglement. The center of the ball gives the maximally entangled states. The
north and the south poles are associated with the pure particle nature of the
photon with $D=1$. The points on the $x_{0}$ axis (shown in blue in the left
panel) are the states with no wave nature. The great disc on $x_{0}=0$
corresponds to the states with not particle natures; where the boundary of the
disc gives the great circle of the geometry corresponding to states having
only the wave nature.

For the maximally entangled states, we have $x_{0}=x_{1}=x_{2}=0$, where, both
particleness and waveness nature of the photon vanishes, and the photon is
left with the only entangledness. These states are given by the circle
$x^{2}_{3}+x^{2}_{4}=1$ on the $S^{4}$ sphere, parametrized by real and
complex parts of $2\pi_{2}$.

The relation $V^{2}+D^{2}\leq 1$ can be understood by expressing it through
Eqs. 6–4 as $x^{2}_{0}+x^{2}_{1}+x^{2}_{2}\leq 1$. This demonstrates a unit
ball $B^{3}$ of the radius 1. From $V^{2}+D^{2}=1-C^{2}$, separable states
cover the $S^{2}$ boundary of the ball (the Bloch sphere of the photon).
States with the same entanglement can be represented in concentric spherical
shells of radius $\sqrt{1-C^{2}}$ around the center, determined through
$x^{2}_{0}+x^{2}_{1}+x^{2}_{2}=1-C^{2}$, with the maximally entangled states
in the center of the ball (see Fig.1).

The north pole ($x_{0}=1$) is given by
$|\psi\rangle_{Q=\infty}=|0\rangle_{ph}\otimes(\alpha_{0}|g_{0}\rangle+\alpha_{1}|e\rangle)$,
where both $C$ and $V$ vanish; and, the only surviving part of the
duality–entanglement scenario is the particleness with $D=1$. States with no
quantum coherence are mapped into the points on the $x_{0}$ axis (see Fig.1).
By decreasing $D$ along $x_{0}$, we attain states with entanglement determined
via $1-x_{0}^{2}$. On this axis, state 2 can be expressed in the Schmidt form
$|q_{1}||0\rangle|\tilde{e}\rangle+|q_{2}||1\rangle|\tilde{f}\rangle$, where
$|\tilde{e}\rangle$ and $|\tilde{f}\rangle$ are two orthonormal basis for
second subsystem. When the particleness becomes zero, the geometry reduces to
the great disc of the ball at $x_{0}=0$. The boundary of the disc is the
circle $x^{2}_{1}+x^{2}_{2}=1$, where the photon has no entanglement or the
particleness; rather, it only has the wave nature.

In conclusion, coherence and which-path information duality, serving as one of
the most fundamental characteristics of the quantum mechanics, has remained as
a long–standing subject of continued investigations. In this work, we have
reported a surprisingly intimate correspondence between the Hopf fibration
geometry of $S^{7}$ and the duality nature of a single quantum. It was shown
that this geometry is complementarity sensitive; in the sense that, it is
sensitive to the particleness, waveness, and entangledness nature of a single
photon.

Even though we have considered a photon which could be correlated with some
other systems, the results presented in this letter is valid for any pure two-
qubit quantum state living in the Hilbert space
$\mathcal{H}_{2}\otimes\mathcal{H}_{2}$.

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