# Four-dimensional Bloch sphere representation of qutrits using Heisenberg-
Weyl Operators

Gautam Sharma<EMAIL_ADDRESS>Optics and Quantum Information Group,
Institute of Mathematical Sciences,
HBNI, CIT Campus, Taramani, Chennai 600113, India  Sibasish Ghosh
<EMAIL_ADDRESS>Optics and Quantum Information Group, Institute of
Mathematical Sciences,
HBNI, CIT Campus, Taramani, Chennai 600113, India

###### Abstract

In the Bloch sphere based representation of qudits with dimensions greater
than two, the Heisenberg-Weyl operator basis is not preferred because of
presence of complex Bloch vector components. We try to address this issue and
parametrize a qutrit using the Heisenberg-Weyl operators by identifying eight
real parameters and separate them as four weight and four angular parameters
each. The four weight parameters correspond to the weights in front of the
four mutually unbiased bases sets formed by the eigenbases of Heisenberg-Weyl
observables and they form a four-dimensional unit radius Bloch sphere. The
analysis of purity, rank, orthogonality, mutual unbiasedness conditions and
the distance between two qutrit states inside the sphere indicates that it is
a natural extension of the qubit Bloch sphere. However, unlike the qubit Bloch
sphere, the four-dimensional sphere is not a solid sphere. We also analyze the
two and three-dimensional sections which gives a non-convex but closed
structure for physical qutrit states inside the sphere. Significantly, we have
applied our representation to find mutually unbiased bases (MUBs), to
characterize the unital maps in three dimensions and also to characterize the
ensemble generated using Hilbert-Schmidt and Bures metrics. Lastly, we also
give a basic idea of how to extend this idea in higher dimensions.

##  I Introduction

A density matrix representing the state of a finite-dimensional quantum system
of dimension $d$, is a $d\crossproduct d$ matrix, which satisfies the
conditions of positive semi-definiteness, hermiticity and trace one. In
general, it is difficult to study the properties of a density matrix directly.
Parametrizations of the density matrix provides a simple method to study the
properties of the density matrix and to use the density matrix in solving
various problems of physics.

One of the most famous representations of density matrices is by using the
Bloch vector parametrization. The Bloch vector parametrization is extremely
popular because of its simplicity in representing a qubit and its various
applications, see refs Bertlmann and Krammer (2008); Fano (1983); Brüning  _et
al._ (2012). A qubit can be uniquely represented by a three-dimensional vector
in the Bloch sphere so that every point inside the Bloch sphere corresponds to
a physical qubit state. This lends a simple method to not only represent the
qubit states but also to identify the dynamics of the qubit. For example, all
rotations of the Bloch sphere correspond to a unitary operation. However, such
an extension of all the beautiful properties of qubit Bloch sphere is not
completely possible for higher dimensional states.

It is known that $d^{2}-1$ parameters are needed to characterize a $d$-level
density matrix $\rho$ Bertlmann and Krammer (2008). Most of the works till now
have used the Gell-Mann operator basis to characterize the qudits as they
admit real numbers as the Bloch vector elements. However, the structure of the
$d^{2}-1$ dimensional solid corresponding to valid quantum states is extremely
complex, even in three dimensions Kimura (2003); Kimura and Kossakowski
(2004); Bertlmann and Krammer (2008); Kryszewski and Zachcial (2006); Mendaš
(2006); Goyal  _et al._ (2016); Eltschka  _et al._ (2021). Moreover, all the
points inside the $d^{2}-1$ dimensional sphere do not represent physical
quantum states. A shortcoming of this feature is that all the rotations in the
sphere do not represent a unitary operation, which is a prominent feature in
the qubit Bloch sphere. To resolve this issue, there have been efforts to
develop a qubit based parametrization in higher dimensions Kurzyński (2011);
Kurzyński  _et al._ (2016), but without much success. Moreover, the structure
corresponding to physical states using the Gell-Mann operator based
representation, is asymmetric with respect to different axes. Also, in the
Gell-Mann operator based representation the relation between Bloch vectors
corresponding to orthonormal kets is not very useful geometrically. That is,
it is not very useful for constructing an orthonormal basis starting from a
pure qutrit state, by using the eight-D Bloch sphere. Therefore, these
features which are very prominent and useful in the qubit Bloch sphere are not
present for qudits, when we use Gell-mann basis based representation.

On contrary, the Heisenberg-Weyl(HW) operators have received much less
attention because they are not hermitian and thereby require complex numbers
as Bloch vector components Vourdas (2004); Asadian  _et al._ (2016). As such
it becomes difficult to study the parameters and put them to use. However, as
the HW operators do provide an alternative way to represent a quantum state,
it is worthwhile to study them despite the presence of complex coefficients as
there can be certain tasks where the HW operator based representation could be
more suitable.

In this work, we try to address the issue of complex Bloch vector components
while using the Weyl operator basis to parametrize a qutrit. We do so by
identifying 4 weight and 4 angular parameters and construct a four-
dimensional(four-D) Bloch sphere from the weight parameters. We also obtain
the constraints on the weight and angular parameters, which give a physical
qutrit state density matrix. It is found that not all the points inside the
four-D sphere correspond to a positive semidefinite matrix. But still, the new
Bloch sphere has several features that looks like an extension of the qubit
Bloch sphere. Namely, 1) the length of the Bloch vector determines the purity
of the state, 2) the rank of a randomly chosen qutrit state can be guessed to
a certain extent 3) the conditions for two orthogonal or mutually unbiased
vectors are quite similar to qubit Bloch sphere under some restriction and 4)
the Hilber-Schmidt distance between qutrit states is equivalent to a factor
time the Euclidean distance in the Bloch sphere for some states. It is also
conjectured that the orthonormal basis kets lie on the same Bloch vector or on
the antipodal points, similar to what we observe in the qubit Bloch sphere.
Further, we apply our representation to 1) identify mutually unbiased bases
(MUBs) in 3 dimensions from the geometry of four-D Bloch sphere 2) to
characterize the unital map acting on qutrit states and 3) to find the
representation of ensembles generated from Hilbert-Schmidt and Bures metric.
We also suggest a method to find a similar Bloch sphere representation in
higher dimensions than three.

The paper is organized as follows. First, we review the HW operator expansion
of a qudit in Sec.II. Then, in Sec.III we present the four-D Bloch sphere and
obtain the constraints on the parameters from one, two, and three-dimensional
sections in Sec.IV. In Sec.V, we have studied a few features of the new Bloch
sphere. Sec.VI describe applications of our representation. After that, in
Sec.VII we describe a way to use a similar approach in higher dimensions than
three. Finally, we conclude in sec.VIII with a summary and future works
possible based on our work.

##  II Expanding a Qudit in the Heisenberg Weyl operator basis

HW operators are unitary operators with several useful properties which makes
them useful in several applications Cotfas and Dragoman (2012); Baumgartner
_et al._ (2006); Asadian  _et al._ (2015); Bennett  _et al._ (1993). One of
the most significant applications of the HW operators has been in the
construction of the discrete Wigner function in finite dimensions Ruskai
(2003); Gibbons  _et al._ (2004). These operators are constructed from the
generalized Pauli operators $X$ and $Z$, which are also referred to as boost
and shift operators respectively. They can be defined by their action on a
pure state in the computational basis as

$\displaystyle X\ket{i}=\ket{i+1\text{ mod d}},$ $\displaystyle
Z\ket{i}=\omega^{i}\ket{i}.$

where $\omega=e^{\frac{i2\pi}{d}}$ is the $d^{th}$ root of unity. Now, the HW
displacement operators are defined as

$\displaystyle U_{pq}=\omega^{\frac{-pq}{2}}Z^{p}X^{q}.$ (1)

where $p$ and $q$ are integers modulo $d$. It is known Bertlmann and Krammer
(2008); Asadian  _et al._ (2016) that a qudit can be represented using the HW
operator basis. These operators satisfy the following properties that are
necessary to parametrize a physical state.

  1. 1.

There are $d^{2}$ matrices including the Identity matrix and $d^{2}-1$ other
traceless matrices.

  2. 2.

The $d^{2}$ matrices are unitary and form an orthonormal basis, i.e.,
$\Tr(U_{pq}^{\dagger}U_{p^{\prime}q^{\prime}})=d\delta_{pp^{\prime}}\delta_{qq^{\prime}}$.

Therefore, we can decompose a bounded density matrix operator using the HW
operator basis in the following form

$\displaystyle\rho=\frac{1}{d}\sum_{p=0,q=0}^{d-1}b_{pq}U_{pq}=\frac{1}{d}\big{(}\mathbb{I}+\sum_{p=0,q=0,p\neq
0\neq q}^{d-1}b_{pq}U_{pq}\big{)},$ (2)

where $b_{00}=1$ and $b_{pq}=\Tr{\rho U_{pq}^{\dagger}}$ form the Bloch vector
components. However, the $b_{pq}$’s are complex in general because of $U_{pq}$
not being hermitian. Hence, we need to find $d^{2}-1$ complex numbers to
completely characterize a state. Moreover, it is not possible to define what
is a “Bloch” sphere for these complex components. A solution was suggested in
Asadian  _et al._ (2016) by introducing a hermitian generalization of the HW
operators to make the Bloch vector components real. In the next section we
suggest an alternate approach to address this issue.

##  III four-D Bloch sphere representation of a qutrit

In this section, we propose a four-D unit radius Bloch sphere representation
of a qutrit state. We will now demonstrate the procedure to represent a qutrit
state by our technique. For convenience, it is helpful to write the HW
operators in three dimensions using Eq.(1), as following

$\displaystyle U_{00}=\begin{bmatrix}1&0&0\\\ 0&1&0\\\
0&0&1\end{bmatrix},U_{01}=\begin{bmatrix}0&1&0\\\ 0&0&1\\\
1&0&0\end{bmatrix},U_{10}=\begin{bmatrix}1&0&0\\\ 0&\omega&0\\\
0&0&\omega^{2}\end{bmatrix},$ $\displaystyle
U_{11}=\begin{bmatrix}0&\omega^{-\frac{1}{2}}&0\\\ 0&0&\omega^{\frac{1}{2}}\\\
-1&0&0\end{bmatrix},U_{02}=\begin{bmatrix}0&0&1\\\ 1&0&0\\\
0&1&0\end{bmatrix},U_{20}=\begin{bmatrix}1&0&0\\\ 0&\omega^{2}&0\\\
0&0&\omega\end{bmatrix},$ $\displaystyle
U_{12}=\begin{bmatrix}0&0&\omega^{2}\\\ 1&0&0\\\
0&\omega&0\end{bmatrix},U_{21}=\begin{bmatrix}0&\omega^{2}&0\\\ 0&0&\omega\\\
1&0&0\end{bmatrix},U_{22}=\begin{bmatrix}0&0&\omega\\\ 1&0&0\\\
0&\omega^{2}&0\end{bmatrix}.$ (3)

From Eq.(2) and Eq.(III) , a qutrit can be expanded as

$\displaystyle\rho=$
$\displaystyle\frac{1}{3}\left(U_{00}+b_{01}U_{01}+b_{10}U_{10}+b_{11}U_{11}+b_{02}U_{02}+b_{20}U_{20}+\right.$
$\displaystyle\left.b_{12}U_{12}+b_{21}U_{21}+b_{22}U_{22}\right).$ (4)

Using the form of HW matrices in Eq.(III), we find that the coefficients
$b_{pq}=\Tr{\rho U_{pq}^{\dagger}}$ must obey following relations

$\displaystyle b_{01}=n_{1}e^{i\theta_{1}},b_{02}=n_{1}e^{-i\theta_{1}},$
$\displaystyle b_{10}=n_{2}e^{i\theta_{2}},b_{20}=n_{2}e^{-i\theta_{2}},$
$\displaystyle
b_{12}=n_{3}e^{i\theta_{3}},b_{21}=n_{3}e^{-i(\theta_{3}-\frac{2\pi}{3})},$
$\displaystyle
b_{22}=n_{4}e^{i\theta_{4}},b_{11}=n_{4}e^{-i(\theta_{4}-\frac{\pi}{3})}.$ (5)

Thus, we can rewrite the expansion of $\rho$ as

$\displaystyle\rho=$
$\displaystyle\frac{1}{3}\left(U_{00}+n_{1}(e^{i\theta_{1}}U_{01}+e^{-i\theta_{1}}U_{02})+n_{2}(e^{i\theta_{2}}U_{10}+e^{-i\theta_{2}}U_{20})+\right.$
$\displaystyle\left.n_{3}(e^{i\theta_{3}}U_{12}+e^{-i(\theta_{3}-\frac{2\pi}{3})}U_{21})+n_{4}(e^{i\theta_{4}}U_{11}+e^{-i(\theta_{4}-\frac{\pi}{3})}U_{22})\right).$
(6)

Therefore, we have obtained eight real parameters that can completely
characterize a qutrit. The unique property of these parameters is that they
consist of four weight parameters $n_{i}$’s and and four angular parameters
$\theta_{i}$’s, with $-1\leq n_{i}\leq 1$ and $0\leq\theta_{i}\leq\pi$. We
define $\vec{n}=\\{n_{1},n_{2},n_{3},n_{4}\\}$ as the four-D Bloch vector. The
$n_{i}^{\prime}s$ are the weight elements corresponding to each commuting pair
of HW operators. We will now obtain the constraints on the Bloch vector
parameters $n_{i}$ and the angular parameters $\theta_{i}$.

Note 1- An implication of using the four-D Bloch vector representation is that
more than one state lie at the same point in the sphere. The states lying on
the same point are distinguished only by the angular parameters $\theta_{i}$.
These states are equivalent under the action of some unitary operators. It
would be interesting to identify these unitary operators.

Note 2- One can also characterize a qutrit state by allowing $0\leq n_{i}\leq
1$ and $0\leq\theta_{i}\leq 2\pi$ range of values, but then we will not obtain
a sphere structure.

Such a structure might also be worth studying because the weight parameters
have proper meaning when they are positive valued. However, we do not study
this in the current work.

##  IV Constraints on the Bloch vector and angular parameters

The matrix form of a qutrit can be written in terms of $n_{i}$s and
$\theta_{i}$s as following

$\displaystyle\rho=\frac{1}{3}\begin{pmatrix}1+2n_{2}\cos(\theta_{2})&n_{1}e^{i\theta_{1}}+n_{3}e^{-i\theta_{3}}+n_{4}e^{-i\theta_{4}}&n_{1}e^{-i\theta_{1}}+n_{3}e^{i(\theta_{3}-\frac{2\pi}{3})}+n_{4}e^{i(\theta_{4}+\frac{2\pi}{3})}\\\
\\\
n_{1}e^{-i\theta_{1}}+n_{3}e^{i\theta_{3}}+n_{4}e^{i\theta_{4}}&1-n_{2}\cos(\theta_{2})-\sqrt{3}n_{2}\sin(\theta_{2})&n_{1}e^{i\theta_{1}}+n_{3}e^{-i(\theta_{3}+\frac{2\pi}{3})}+n_{4}e^{-i(\theta_{4}-\frac{2\pi}{3})}\\\
\\\
n_{1}e^{i\theta_{1}}+n_{3}e^{-i(\theta_{3}-\frac{2\pi}{3})}+n_{4}e^{-i(\theta_{4}+\frac{2\pi}{3})}&n_{1}e^{-i\theta_{1}}+n_{3}e^{i(\theta_{3}+\frac{2\pi}{3})}+n_{4}e^{i(\theta_{4}-\frac{2\pi}{3})}&1-n_{2}\cos(\theta_{2})+\sqrt{3}n_{2}\sin(\theta_{2})\end{pmatrix}$
(7)

It is clear that, $\rho$ is hermitian, which is guaranteed by the choice of
expansion coefficients. Moreover, $\Tr[\rho]=1$ as the HW matrices are
traceless except for $U_{00}=\mathcal{I}$. The only condition that remains to
be satisfied is the positive semi-definiteness of $\rho$, i.e. $x_{i}\geq 0$,
where $x_{i}$’s are the eigenvalues of $\rho$. In order to do this, we
construct the characteristic polynomial Det($x\mathbb{I}-\rho$), of the
density matrix $\rho$. The necessary and sufficient condition for the
eigenvalues $x_{i}$ to be positive semi-definite is that the coefficients
$a_{i}$’s of the characteristic polynomial are also positive semi-definite
Kimura (2003). The characteristic polynomial has the following form

$\displaystyle\text{Det}(x\mathbb{I}-\rho)=\prod_{i=1}^{N}(x-x_{i})=\sum_{j=0}^{N}(-1)^{j}a_{j}x^{N-j}=0.$
(8)

Notice that $a_{0}=1$ by definition. Now, we apply Newton’s formulas to find
the values of other coefficients $a_{i}$’s(for details please see ref. Kimura
(2003)). Newton’s formulas relate the coefficients $a_{i}$ and the eigenvalues
$x_{i}$ as

$\displaystyle la_{l}=\sum_{k=1}^{l}C_{N,k}a_{l-k},(1\leq l\leq N),$

where $C_{N,k}=\sum_{i=1}^{N}x^{k}$. Using the results directly from ref.
Kimura (2003), we get the following expressions for $a_{i}$’s in terms of
$\rho$

$\displaystyle a_{0}=1$ $\displaystyle a_{1}=\Tr\rho,$ $\displaystyle
a_{2}=\frac{(1-\Tr\rho^{2})}{2!},$ $\displaystyle
a_{3}=\frac{1-3\Tr\rho^{2}+2\Tr\rho^{3}}{3!}.$ (9)

Ensuring that the terms in Eq.(IV) are non-negative results in positive semi-
definite density matrix $\rho$. $a_{1}=\Tr\rho=1>0$ is trivially satisfied.
Where as, $a_{2}=\frac{(1-\Tr\rho^{2})}{2!}\geq 0$ imposes the following
constraint

$\displaystyle 2!a_{2}=1-\Tr[\rho^{2}]$
$\displaystyle=1-\frac{1}{3}\Bigg{(}1+2\bigg{(}n_{1}^{2}+n_{2}^{2}+n_{3}^{2}+n_{4}^{2}\bigg{)}\Bigg{)}\geq
0$ $\displaystyle\implies n_{1}^{2}+n_{2}^{2}+n_{3}^{2}+n_{4}^{2}\leq 1.$ (10)

This constraint simply states that the physical states must lie inside a
four-D sphere of radius one. The only condition remaining to be satisfied now
is $a_{3}\geq 0$, which has the following form

$\displaystyle 3!a_{3}=1-3\Tr\rho^{2}+2\Tr\rho^{3}\geq 0$
$\displaystyle\implies\frac{2}{9}\Bigg{(}1-3\bigg{(}n_{1}^{2}+n_{2}^{2}+n_{3}^{2}+n_{4}^{2}\bigg{)}+2\bigg{(}n_{1}^{3}\cos(3\theta_{1})+n_{2}^{3}\cos(3\theta_{2})+n_{3}^{3}\cos(3\theta_{3})+n_{4}^{3}\cos(3\theta_{4})\bigg{)}-6n_{1}n_{3}n_{4}\cos(\theta_{1}-\theta_{3}-\theta_{4})$
$\displaystyle+6n_{1}n_{2}n_{3}\cos(\theta_{1}-\theta_{2}+\theta_{3}-\frac{\pi}{3})+6n_{2}n_{3}n_{4}\cos(\theta_{2}+\theta_{3}-\theta_{4}+\frac{\pi}{3})+6n_{1}n_{2}n_{4}\cos(\theta_{1}+\theta_{2}+\theta_{4}+\frac{\pi}{3})\Bigg{)}\geq
0.$ (11)

In the above form, it is difficult to picture the set of valid states inside
the sphere. We take the one, two, and three-dimensional sections passing
through the centre to get a better understanding of the allowed space inside
the four-D sphere.

###  IV.1 One-dimensional sections

One dimensional sections(one sections) passing through center can be obtained
by setting three out of four $n_{i}$’s as zero, in Eq.(IV). We find that the
expressions of one sections of $a_{3}$ are the same with respect to all
$n_{i}$’s with the following form

$\displaystyle
3!a_{3}^{one}=\frac{2}{9}\Bigg{(}1-3n_{i}^{2}+2n_{i}^{3}\cos[\theta_{i}]\Bigg{)}\geq
0.$ (12)

Therefore, these one-dimensional sections are symmetric with respect to the
four axes. The non-negativity constraints are:-

  * •

$a_{3}\geq 0$ $\forall$ $\theta_{i}$ for $-\frac{1}{2}\leq
n_{i}\leq\frac{1}{2}$.

  * •

For $0.5<n_{2}\leq 1$,
$\theta\in[0,\zeta]\cup[\frac{2\pi}{3}-\zeta,\frac{2\pi}{3}+\zeta]$.

  * •

For $-1\leq n_{2}<-0.5$,
$\theta\in[\frac{\pi}{3}-\zeta,\frac{\pi}{3}+\zeta]\cup[\pi-\zeta,\pi]$.

where $\zeta=\arccos(\frac{-1}{2|n_{2}|})-\frac{2\pi}{3}$. It can be also
observed (see Fig.1) that the range of allowed values of $\theta_{i}$ is
gradually reducing as we move away from the origin along the $n_{i}$, axis
after $|n_{i}|\geq 0.5$.

Figure 1: (Color online)The shaded region depicts the allowed values of
$n_{i}$ and $\theta_{i}$ for a physical state lying on the $n_{i}$ axis. As
can be easily seen that for $-0.5\leq n_{i}\leq 0.5$, all values of
$\theta_{i}$ correspond to a physical density matrix.

###  IV.2 Two-dimensional sections

A two-dimensional section(two section) centered at the origin can be obtained
by setting two out of four $n_{i}$’s to be zero in Eq.(IV). There are 6
possible two sections which can be constructed in the four-D sphere. The
positivity constraint for all the two-dimensional sections have the following
same form

$\displaystyle
3!a_{3}^{two}=\frac{2}{9}\bigg{(}1-3n_{i}^{2}-3n_{j}^{2}+2n_{i}^{3}\cos(3\theta_{i})+2n_{j}^{3}\cos(3\theta_{j})\bigg{)}\geq
0.$ (13)

Similar to the one-dimensional sections, we see that the two dimensional
sections are also symmetric with respect to the four axes. We point out that
this is unlike the Gell-Mann basis based Bloch vector representation of a
qutrit, where there are 4 different types of such two sections Kimura (2003),
which are asymmetric with respect to the axes.

Now, we are interested in obtaining the region in the two-dimensional section
which corresponds to physical qutrit states, i.e. there exist values of
$\theta_{i}$ and $\theta_{j}$ so that the inequality in Eq.(13) is satisfied.
This can be easily found by maximizing $a_{3}^{two}$ with respect $\theta_{i}$
and $\theta_{j}$, which gives the following inequality see Fig.2.

$\displaystyle{a_{3}}_{max}^{two}\geq
0\implies\frac{2}{9}\bigg{(}1-3n_{i}^{2}-3n_{j}^{2}+2|n_{i}^{3}|+2|n_{j}^{3}|\bigg{)}\geq
0.$ (14)

Figure 2: (Color online)The shaded region depicts the allowed values of
$n_{i}$ and $n_{j}$ for a physical state lying on the two-dimensional section
built on $n_{i}$ and $n_{j}$ axes.

(a) $\phi=\pi/4$ (b) $\phi=\cos^{-1}0.2$

(c) $\phi=0$ (d) $\phi=\pi/2$

Figure 3: (Color online)The solid region in the plots gives the allowed values
of $\theta_{i}$ and $\theta_{j}$ against r. Here $\phi$ is the angle the
direction makes with $n_{i}$.

Further, it is informative to see what are the allowed values of $\theta_{i}$
and $\theta_{j}$ in different directions in the two-dimensional section as we
move away from the center in the four-D sphere. To do this, we replace with
$n_{i}=r\cos(\phi)$ and $n_{j}=r\sin(\phi)$ in Eq.(13), so that

$\displaystyle
3!a_{3}^{two}=\frac{2}{9}\Bigg{(}1-3r^{2}+2r^{3}\bigg{(}\cos^{3}(\phi)\cos(3\theta_{i})+\sin^{3}(\phi)\cos(3\theta_{j})\bigg{)}\Bigg{)}\geq
0.$

We note that when $r\leq 0.5$, the above inequality is satisfied for all
values of $\theta_{i}$ and $\theta_{j}$ $\in$ {0,$\pi$}. Moreover, to find
what are the allowed values of $\theta_{i}$ and $\theta_{j}$ as we move away
from the origin in a radial direction, we do a 3D plot of the allowed region
between $r$, $\theta_{i}$ and $\theta_{j}$ by fixing the directions determined
by $\phi$ in fig.3, where $\phi$ is the angle the direction makes with
$n_{i}$.

###  IV.3 Three-dimensional sections

Next, we consider the three-dimensional sections(three sections) centered at
the origin inside the four-D sphere. There are four such three-dimensional
sections possible which can be obtained by setting one of the $n_{i}$’s as
zero in Eq.(IV). However, unlike the one and two dimensional sections the
three-dimensional sections are all different, with following expressions.

$\displaystyle
3!a_{3}^{three1}=\frac{2}{9}\Bigg{(}1-3\bigg{(}n_{1}^{2}+n_{2}^{2}+n_{3}^{2}\bigg{)}+2\bigg{(}n_{1}^{3}\cos(3\theta_{1})+n_{2}^{3}\cos(3\theta_{2})+n_{3}^{3}\cos(3\theta_{3})\bigg{)}+6n_{1}n_{2}n_{3}\cos(\theta_{1}-\theta_{2}+\theta_{3}-\frac{\pi}{3})\Bigg{)}.$
$\displaystyle
3!a_{3}^{three2}=\frac{2}{9}\Bigg{(}1-3\bigg{(}n_{1}^{2}+n_{2}^{2}+n_{4}^{2}\bigg{)}+2\bigg{(}n_{1}^{3}\cos(3\theta_{1})+n_{2}^{3}\cos(3\theta_{2})+n_{4}^{3}\cos(3\theta_{4})\bigg{)}+6n_{1}n_{2}n_{4}\cos(\theta_{1}+\theta_{2}+\theta_{4}+\frac{\pi}{3})\Bigg{)}.$
$\displaystyle
3!a_{3}^{three3}=\frac{2}{9}\Bigg{(}1-3\bigg{(}n_{1}^{2}+n_{3}^{2}+n_{4}^{2}\bigg{)}+2\bigg{(}n_{1}^{3}\cos(3\theta_{1})+n_{3}^{3}\cos(3\theta_{3})+n_{4}^{3}\cos(3\theta_{4})\bigg{)}-6n_{1}n_{3}n_{4}\cos(\theta_{1}-\theta_{3}-\theta_{4})\Bigg{)}.$
$\displaystyle
3!a_{3}^{three4}=\frac{2}{9}\Bigg{(}1-3\bigg{(}n_{2}^{2}+n_{3}^{2}+n_{4}^{2}\bigg{)}+2\bigg{(}n_{2}^{3}\cos(3\theta_{2})+n_{3}^{3}\cos(3\theta_{3})+n_{4}^{3}\cos(3\theta_{4})\bigg{)}+6n_{2}n_{3}n_{4}\cos(\theta_{2}+\theta_{3}-\theta_{4}+\frac{\pi}{3})\Bigg{)}.$
(15)

Figure 4: (Color online) Two different views of the three section
$a_{3}^{three1}$ showing that the three section is closely approximated by an
octahedron. The other three sections are also approximated by an octahedron,
which we have not shown here for brevity. The colourscale used is only for a
better clarity of the plots.

It seems that these three-dimensional sections are not symmetric with respect
to the axes as they have different forms in Eq.(IV.3). Therefore we need to
find the regions for which the expressions in Eq.(IV.3) are non-negative. To
find the non-negative regions of the three-dimensional sections means to find
out whether for a given triple of $n_{i},n_{j}$ and $n_{k}$, a corresponding
$\theta_{i},\theta_{j}$ and $\theta_{k}$ exists which gives a non-negative
value of terms in Eq.(IV.3). It is difficult to do so analytically, i.e., to
maximize the expressions of $a_{3}^{three}$ in Eq.(IV.3) with respect to
$\theta$ parameters only. Instead we numerically plot the regions which
satisfy $a_{3}^{three1}\geq 0$ only. As we can see from the fig.4 that the
different faces have different forms, we conclude that the three dimensional
sections are not symmetric with respect to the axes.

It can be seen from fig.4 that the three section $a_{3}^{three1}$ is of the
form with bulges on the faces of an octahedron. The other three sections also
are of a similar form but not the same, which we have not put in the paper for
lack of space and brevity.

A few remarks from the study of the one, two and three sections are in order.

  1. 1.

It is possible to approximately construct the three sections from the
knowledge of the two sections, which is not the case in the representation
using Gell-Mann operator based representation. This is because by rotating the
two sections along one of the axes, one will obtain an octahedron which is a
very good approximation to all the three sections.

  2. 2.

It looks like from the numerical plots, that the three sections structure is
not convex. This could be because of the presence of complex coefficients.

  3. 3.

It is also clearly visible how the one section arise from the two sections and
the two sections from the three sections.

Adding to the remarks from the study of one, two and three dimensional
sections is the following result.

Lemma 1\- The four dimensional Bloch sphere is not a solid sphere.

An implication of this result is that a rotation in the Bloch sphere does not
always correspond to a unitary operation, unlike the qubit Bloch sphere.
However, a very interesting finding is that inside the Bloch sphere of radius
$r\leq 0.5,a_{3}$ is positive for all the values of angular parameters
$\theta_{i}$. This can be proven by using the polar coordinate forms of
$n_{i}$’s in Eq.(IV), i.e. we replace with
$n1=r\cos\alpha_{1},n2=r\sin\alpha_{1}\cos\alpha_{2},n3=r\sin\alpha_{1}\sin\alpha_{2}\cos\alpha_{3},n4=r\sin\alpha_{1}\sin\alpha_{2}\sin\alpha_{3}\cos\alpha_{4}$
in Eq.(IV), where $\alpha_{1},\alpha_{2},\alpha_{3}$ and $\alpha_{4}$ are the
polar angles. Then $a_{3}$ can be written in the following simple form

$\displaystyle
3!a_{3}=1-3r^{2}+2r^{3}F(\theta_{1},\theta_{2},\theta_{3},\theta_{4},\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}).$
(16)

where $F$ is a function of $\theta_{i}$’s and $\alpha_{i}$’s. It can be easily
shown that $F$ can have values in the range [-1,1]. And, for all values of
$F$, $a_{3}$ is monotonically decreasing with $r$. If $a_{3}\geq 0$ at a
certain point $\\{r_{max},\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}\\}$, for
some theta values $\\{\theta_{1},\theta_{2},\theta_{3},\theta_{4}\\}$, then
$a_{3}\geq 0$ for all points on $\\{r\leq
r_{max},\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}\\}$ for the same theta
values. We state the following result from this analysis

Lemma 2\- If in a certain direction the point at distance $r=r_{max}$
represents a physical state, then all the points in the same direction with
$r\leq r_{max}$ represent a physical state.

Now, we set $r=0.5$ in Eq.(16), and then minimize it with respect to all the
angles $\alpha_{i}$’s and $\theta_{i}$’s, which gives a minimum value of zero.
Therefore, for $r<0.5$, $a_{3}$ must have values greater than zero for all
values of $\theta_{i}$’s. Therefore, we can present the following Lemma.

Lemma 3-All the points inside sphere of radius $r\leq 0.5$ are physical states
for all the angular parameter values of $\theta_{i}$’s.

This result is significant, as all the points lying inside the sphere of
radius 0.5 correspond to physical states.

##  V Features of the Bloch sphere

In this section, we discuss several features of the Bloch sphere which have
surprising resemblance with the qubit Bloch sphere representation.

###  V.1 Mixed and Pure states

$\Tr[\rho^{2}]$ is the measure of the purity of a density matrix operator.
Using Eq.(7) we can write

$\displaystyle\Tr[\rho^{2}]=\frac{1}{3}\Bigg{(}1+2\bigg{(}n_{1}^{2}+n_{2}^{2}+n_{3}^{2}+n_{4}^{2}\bigg{)}\Bigg{)}.$
(17)

Thus, we find that the length of the Bloch vector determines the purity of the
qutrit state. Further, $\Tr[\rho^{2}]=1$ for
$n_{1}^{2}+n_{2}^{2}+n_{3}^{2}+n_{4}^{2}=1$, i.e., the pure states lie on the
surface of the unit sphere. Also, $\Tr[\rho^{2}]=0$ only when
$n_{1}^{2}+n_{2}^{2}+n_{3}^{2}+n_{4}^{2}=0$, i.e., the maximally mixed state
lies at the center of the sphere. Also, the purity increases as we move away
from the center of the sphere.

###  V.2 Rank of a Qutrit state

A closely related concept to purity/mixedness is the rank of a physical state.
From the qubit Bloch sphere it is very easy to determine that the rank 1
states lie on the surface while all the remaining states are of rank 2. In our
representation of qutrit states, all the pure states lie on the surface, hence
they have rank 1. The challenging task is to determine where the rank 2 and 3
states lie inside the four-D sphere. To do this we note that if the
determinant of a qutrit is greater than zero, i.e., $Det(\rho)>0$, then the
state must be of rank 3. From the Eq.(7), we find that $Det(\rho)=a_{3}$. But
we saw in Lemma 3 that for $r<0.5$, $a3=Det(\rho)>0$. Hence all the states
inside the four-D sphere with radius $r=0.5$ are rank 3 states. However, the
states on the surface of this sphere with $r=0.5$ can be both rank 2 and 3.
Further, when $0.5\leq r<1$ the states can be both of rank 2 and rank 3
depending on the choice of the angular parameters $\theta_{i}$. To summarize

  1. 1.

$r=1\implies\rank(\rho)=1.$

  2. 2.

$0.5\leq r<1\implies\rank(\rho)=2\hskip 2.84544pt\text{and}\hskip 2.84544pt3.$

  3. 3.

$r<0.5\implies\rank(\rho)=3.$

A representative figure of the qutrit state space depicting the location of
rank 1, 2 and 3 states can be seen in Fig.(5). This, helps us to identify the
rank of the qutrit states by simply looking at where they lie inside the
four-D sphere.

Figure 5: (Color online)A representative figure depicting the position of the
Rank-1, 2, and 3 states and the qutrit state space containing a solid ball of
radius 1/2.

###  V.3 Orthogonal states

Let us consider two pure states $\rho_{a}$ and $\rho_{b}$ and expand it in the
form of Eq.(III)

$\displaystyle\rho_{a}=$
$\displaystyle\frac{1}{3}\left(U_{00}+n_{a1}(e^{i\theta_{a1}}U_{01}+e^{-i\theta_{a1}}U_{02})+n_{a2}(e^{i\theta_{a2}}U_{10}+e^{-i\theta_{a2}}U_{20})+\right.$
$\displaystyle\left.n_{a3}(e^{i\theta_{a3}}U_{12}+e^{-i(\theta_{a3}-\frac{2\pi}{3})}U_{21})+n_{a4}(e^{i\theta_{a4}}U_{11}+e^{-i(\theta_{a4}-\frac{\pi}{3})}U_{22})\right).$
$\displaystyle\rho_{b}=$
$\displaystyle\frac{1}{3}\left(U_{00}+n_{b1}(e^{i\theta_{b1}}U_{01}+e^{-i\theta_{b1}}U_{02})+n_{b2}(e^{i\theta_{b2}}U_{10}+e^{-i\theta_{b2}}U_{20})+\right.$
$\displaystyle\left.n_{b3}(e^{i\theta_{b3}}U_{12}+e^{-i(\theta_{b3}-\frac{2\pi}{3})}U_{21})+n_{b4}(e^{i\theta_{b4}}U_{11}+e^{-i(\theta_{b4}-\frac{\pi}{3})}U_{22})\right).$

The states $\rho_{a}$ and $\rho_{b}$ have Bloch vectors
$\vec{n_{a}}=\\{n_{a1},n_{a2},n_{a3},n_{a4}\\}$ and
$\vec{n_{b}}=\\{n_{b1},n_{b2},n_{b3},n_{b4}\\}$ respectively. The
orthogonality condition can simply be checked by $\Tr[\rho_{a}\rho_{b}]=0$,
which reduces to

$\displaystyle\Tr[\rho_{a}\rho_{b}]$
$\displaystyle=\frac{1}{3}\bigg{(}1+2n_{a1}n_{b1}\cos(\theta_{a1}-\theta_{b1})+2n_{a2}n_{b2}\cos(\theta_{a2}-\theta_{b2})$
$\displaystyle+2n_{a3}n_{b3}\cos(\theta_{a3}-\theta_{b3})+2n_{a4}n_{b4}\cos(\theta_{a4}-\theta_{b4})\bigg{)}=0.$
(18)

This condition is richer than the orthogonality condition that we know for the
two orthogonal qubit states. We know that Bloch vectors for two orthogonal
qubit states obey, $\vec{b1}\cdot\vec{b2}=-1=\cos(\frac{2\pi}{2})$. The
orthogonality condition in Eq.(V.3) reduces to
$\vec{n_{a}}.\vec{n_{b}}=\cos(\frac{2\pi}{3})$, whenever
$\theta_{ai}=\theta_{bi}$. However, it is also needed that the points
corresponding to $\vec{n_{a}}$ and $\vec{n_{b}}$ with an overlap of
$\cos(\frac{2\pi}{3})$ correspond to a positive semidefinite matrix for
$\theta_{ai}=\theta_{bi}$, which we were unable to identify. Thus, it is
similar to the orthogonality constraint for qubit Bloch vectors. For
$\theta_{ai}\neq\theta_{bi}$, the orthogonality condition is far more complex.

It can be easily shown that eigenbases of the four sets of non-commuting HW
operators lie at antipodal points on the Bloch sphere. For example, the
eigenkets of $Z$ lies on points with Bloch vectors
$\\{n_{1},n_{2},n_{3},n_{4}\\}:\\{0,1,0,0\\}$ and $\\{0,-1,0,0\\}$. This is
very similar to what we observe in the qubit Bloch sphere, where the
orthonormal basis kets lie at the antipodal points. Although, all the
orthonormal kets do not lie on the antipodal points in the four-D sphere. We
conjecture the following regarding the Bloch vectors belonging to an
orthonormal basis.

Conjecture 1\- The kets belonging to an orthonormal basis lie at the antipodal
points or at the same point on the surface of the Bloch sphere.

The motivation of this conjecture comes from the observation (see Appendix A)
that the orthonormal basis kets which are mutually unbiased to the
computational basis, lie on the same point in the four-D sphere.

###  V.4 Mutually unbiased states

By doing a similar analysis as for orthogonal states, we can also obtain the
relation between Bloch vectors corresponding to two mutually unbiased state
vectors. For two mutually unbiased state vectors in 3 dimension, $\rho_{a}$
and $\rho_{b}$, $\Tr[\rho_{a}\rho_{b}]=\frac{1}{3}$. By using the result from
previous subsection we get

$\displaystyle\frac{1}{3}\bigg{(}1+2n_{a1}n_{b1}\cos(\theta_{a1}-\theta_{b1})+2n_{a2}n_{b2}\cos(\theta_{a2}-\theta_{b2})$
$\displaystyle+2n_{a3}n_{b3}\cos(\theta_{a3}-\theta_{b3})+2n_{a4}n_{b4}\cos(\theta_{a4}-\theta_{b4})\bigg{)}=\frac{1}{3}$
$\displaystyle\bigg{(}n_{a1}n_{b1}\cos(\theta_{a1}-\theta_{b1})+n_{a2}n_{b2}\cos(\theta_{a2}-\theta_{b2})$
$\displaystyle+n_{a3}n_{b3}\cos(\theta_{a3}-\theta_{b3})+n_{a4}n_{b4}\cos(\theta_{a4}-\theta_{b4})\bigg{)}=0.$
(19)

Whenever $\cos(\theta_{ai}-\theta_{bi})=\text{const}$, Eq.(V.4) reduces to
$\vec{n_{a}}.\vec{n_{b}}=0$, i.e. the Bloch vectors corresponding to mutually
unbiased state vectors are orthogonal to each other, exactly what we get for
mutually unbiased qubits. Here we require that for orthogonal Bloch vectors
the angular parameters obey the relation
$\cos(\theta_{ai}-\theta_{bi})=\text{const}$. For
$\cos(\theta_{ai}-\theta_{bi})\neq\text{const}$, the condition is more
complex.

###  V.5 Distance between density matrices

Let us consider two states
$\rho_{1}:n_{1},n_{2},n_{3},n_{4},\theta_{1},\theta_{2},\theta_{3},\theta_{4}$
and $\rho_{2}:m_{1},m_{2},m_{3},m_{4},\phi_{1},\phi_{2},\phi_{3},\phi_{4}$. By
using the form of Eq.(7), we obtain the Hilbert-Schmidt distance between them
as Wilde (2013)

$\displaystyle
D_{HS}(\rho_{1},\rho_{2})=\Big{(}\Tr(\rho_{1}-\rho_{2})^{2}\Big{)}^{\frac{1}{2}}$
(20)
$\displaystyle=\sqrt{\frac{2}{3}}\Big{(}n_{1}^{2}+n_{2}^{2}+n_{3}^{2}+n_{4}^{2}+m_{1}^{2}+m_{2}^{2}+m_{3}^{2}+m_{4}^{2}$
$\displaystyle-2n_{1}m_{1}\cos(\theta_{1}-\phi_{1})-2n_{2}m_{2}\cos(\theta_{2}-\phi_{2})$
$\displaystyle-2n_{3}m_{3}\cos(\theta_{3}-\phi_{3})-2n_{4}m_{4}\cos(\theta_{4}-\phi_{4})\Big{)}^{\frac{1}{2}}.$
(21)

When $\theta_{i}=\phi_{i}$, the Hilbert-Schmidt distance reduces to the
euclidean distance(times a const.) in the four-D sphere, i.e.
$D_{HS}(\rho_{1},\rho_{2})=\sqrt{\frac{2}{3}}\sqrt{\sum_{i}(n_{i}-m_{i})^{2}}$.
This is analogous to the fact that the Hilbert-Schmidt distance between two
density matrices is proportional to the euclidean distance between them in the
qubit Bloch sphere Wilde (2013).

##  VI Applications

###  VI.1 Employing the Bloch sphere geometry to find MUBs in three dimensions

It is known that in prime or power of prime dimension $d=p^{n}$,where $p$ is a
prime number and $n$ is an integer greater than zero, there exist a maximum of
$d+1$ MUBs. For the qubit, the existence of 3 MUBs can be very easily
explained through qubit Bloch sphere, but such an explanation is difficult in
higher dimensions. In this section, we show that the qutrit Bloch sphere
geometry restricts the maximum number of MUBs to four.

MUBs in 2 dimensions\- The qubit Bloch sphere is a three dimensional sphere,
in which the Bloch vectors corresponding to orthonormal basis kets lie on the
antipodal points on the sphere, i.e. they lie along the line passing through
the center. Also, the Bloch vectors corresponding to mutually unbiased kets
are orthogonal to each other Sharma  _et al._ (2021). As, there can be only
three such orthogonal lines passing through the center, which explains why
there are only three possible mutually unbiased bases in dimension 2.

MUBs in 3 dimensions\- To find the qutrit MUBs, we first fix one of the
orthonormal basis to be the eigenbasis of HW operator $Z$ or the computational
basis. The eigenkets of $Z$ have the following Bloch vector and angular
parameters

$\displaystyle\ket{0}:n_{2}=1,\theta_{2}=0,n_{1}=n_{3}=n_{4}=0,\\{\theta_{1},\theta_{3},\theta_{4}\\}-\text{any
value},$
$\displaystyle\ket{1}:n_{2}=-1,\theta_{2}=\frac{\pi}{3},n_{1}=n_{3}=n_{4}=0,\\{\theta_{1},\theta_{3},\theta_{4}\\}-\text{any
value},$
$\displaystyle\ket{2}:n_{2}=1,\theta_{2}=\frac{2\pi}{3},n_{1}=n_{3}=n_{4}=0,\\{\theta_{1},\theta_{3},\theta_{4}\\}-\text{any
value}.$

Now any qutrit ket which is mutually unbiased to all the computational basis
kets must have $n_{2}=0$ and $\theta_{2}$ can have any arbitrary value, which
can be also deduced from Eq.(V.4). Hence, we need to find the remaining
mutually unbiased kets using the remaining weight parameters $n_{1},n_{3}$ and
$n_{4}$ and the angular parameters $\theta_{1},\theta_{3},\theta_{4}$.

We fix one of the ket $\vec{n_{\theta 0}}$, parametrized as

$\displaystyle\vec{n_{\theta
0}}:n_{1}=m_{1},n_{3}=m_{3},n_{4}=m_{4},\theta_{1}=\phi_{1},\theta_{3}=\phi_{3}\hskip
5.69054pt\text{and}\hskip 5.69054pt\theta_{4}=\phi_{4}.$

As shown in Appendix A, we find that an orthonormal basis $N=\\{\vec{n_{\theta
0}},\vec{n_{\theta 1}},\vec{n_{\theta 2}}\\}$ including $\vec{n_{\theta 0}}$
are parametrized as

$\displaystyle\vec{n_{\theta
0}}:n_{1}=m_{1},n_{3}=m_{3},n_{4}=m_{4},\theta_{1}=\phi_{1},\theta_{3}=\phi_{3}\hskip
5.69054pt\text{and}\hskip 5.69054pt\theta_{4}=\phi_{4},$
$\displaystyle\vec{n_{\theta
1}}:n_{1}=m_{1},n_{3}=m_{3},n_{4}=m_{4},\theta_{1}=\phi_{1}^{\prime},\theta_{3}=\phi_{3}^{\prime}\hskip
5.69054pt\text{and}\hskip 5.69054pt\theta_{4}=\phi_{4}^{\prime},$
$\displaystyle\vec{n_{\theta
2}}:n_{1}=m_{1},n_{3}=m_{3},n_{4}=m_{4},\theta_{1}=\phi_{1}^{\prime\prime},\theta_{3}=\phi_{3}^{\prime\prime}\hskip
5.69054pt\text{and}\hskip 5.69054pt\theta_{4}=\phi_{4}^{\prime\prime}.$

We have derived the values of angles $\phi^{{}^{\prime}}_{1}$,
$\phi^{{}^{\prime}}_{3}$, $\phi^{{}^{\prime}}_{4}$,
$\phi^{{}^{\prime\prime}}_{1}$, $\phi^{{}^{\prime\prime}}_{3}$ and
$\phi^{{}^{\prime\prime}}_{4}$ relative to $\phi_{1}$, $\phi_{3}$ and
$\phi_{4}$ in the Appendix.A. It should be noted that all the kets of the
orthonormal basis lie on the same point $\vec{n}=\\{m_{1},0,m_{3},m_{4}\\}$ in
the four-D sphere. Moreover, from Appendix A we note that the vectors mutually
unbiased to the kets of orthonormal basis $N$ lie on the points
$\vec{p}=\\{m_{3},m_{4},m_{1}\\}$ and $\vec{q}=\\{m_{4},m_{1},m_{3}\\}$. For
points $\vec{p}$ and $\vec{q}$ again we can form an orthonormal bases ${P}$
and ${Q}$ respectively, by varying the angular parameters only. The three
orthonormal bases lie at the end points of the Bloch vectors, with Bloch
vectors which can be obtained by cyclic permutations of
$\\{n_{1},n_{3},n_{4}\\}$, which we write down for brevity.

$\displaystyle{N}:n_{1}=m_{1},n_{3}=m_{3}\hskip 4.26773pt\text{and}\hskip
4.26773ptn_{4}=m_{4},$ $\displaystyle{P}:n_{1}=m_{4},n_{3}=m_{1}\hskip
4.26773pt\text{and}\hskip 4.26773ptn_{4}=m_{3},$
$\displaystyle{Q}:n_{1}=m_{3},n_{3}=m_{4}\hskip 4.26773pt\text{and}\hskip
4.26773ptn_{4}=m_{1}.$ (22)

Therefore, there can be only three orthonormal bases are mutually unbiased to
each other, so that all of them are mutually unbiased to the computational
basis. Hence, we have obtained from the geometry of the Bloch hyperesphere
that there are maximum four mutually unbiased bases in three dimension.

###  VI.2 Characterization of Unital Maps

In this section, we characterize the unital maps acting on the qutrit states.
Unital maps are quantum operations that preserve the identity matrix or the
maximally mixed density matrix. It is known that the unital maps acting on a
qubit density matrix is characterized by a convex tetrahedronKing and Ruskai
(2001); Beth Ruskai  _et al._ (2002).

To analyze the unital channels acting on a qutrit density matrix
$\rho=\frac{1}{3}\sum_{p,q}^{2}b_{pq}U_{pq}$ with Bloch vector $\vec{b}_{pq}$
(see Eq.(2)), we note that a linear quantum map can be written in the form of
an affine transformation acting on the $d^{2}-1=8$ dimensional Bloch vector.
Thus, every linear qutrit quantum map $\Phi:\mathcal{C}^{3\crossproduct
3}\rightarrow\mathcal{C}^{3\crossproduct 3}$ can be represented using a
$9\crossproduct 9$ matrix $\mathcal{L}=\begin{pmatrix}1&0\\\ l&L\end{pmatrix}$
acting on the column vector {1, $\vec{b_{pq}}$}, where $L$ is an
$8\crossproduct 8$ matrix and $l$ is a column vector containing eight
elements. The action of the quantum channel
$\rho\rightarrow\Phi(\rho)=\frac{1}{3}\sum_{p,q}^{2}b_{pq}^{\prime}U_{pq}$ can
be written as

$\displaystyle\vec{b}\rightarrow\vec{b^{\prime}}=L\vec{b}+\vec{l}.$

By observing Eq.(III), it can be seen that to make sure that
$\vec{b^{\prime}}$ corresponds to a hermitian density matrix
$\mathcal{E}(\rho)$, it is necessary that 1) $L$ is a diagonal matrix with
eigenvalues {$\lambda_{01},\lambda_{02},...,\lambda_{22}$} and 2) the
eigenvalues must be of the following form

$\displaystyle\lambda_{01}=\lambda_{1}e^{i\phi_{1}},\lambda_{02}=\lambda_{1}e^{-i\phi_{1}}$
$\displaystyle\lambda_{10}=\lambda_{2}e^{i\phi_{2}},\lambda_{20}=\lambda_{2}e^{-i\phi_{2}}$
$\displaystyle\lambda_{12}=\lambda_{3}e^{i\phi_{3}},\lambda_{21}=\lambda_{3}e^{-i\phi_{3}}$
$\displaystyle\lambda_{22}=\lambda_{4}e^{i\phi_{4}},\lambda_{11}=\lambda_{4}e^{-i\phi_{4}}.$

Next, we note that to preserve the identity matrix, $\vec{l}=\vec{0}$, so that
$\mathcal{L}=\begin{pmatrix}1&0\\\ 0&L\end{pmatrix}$. Now, to do the complete
characterization of the map $\mathcal{L}$ we impose the complete positivity
requirement via Choi’s theorem which requires that Choi Matrix
$\mathbf{C}=(\mathbb{I}\otimes\Phi)(\sum_{ij}E_{ij}\otimes E_{ij})$ is
positive semidefinite. To simplify the problem, we find the eigenvalues only
when the angles $\phi_{i}=0,\forall i\in\\{0,3\\}$. The constraints on the
parameters {$\lambda_{i}$}’s are given by

$\displaystyle 1+2\lambda_{1}-\lambda_{2}-\lambda_{3}-\lambda_{4}\geq 0,$
$\displaystyle 1-\lambda_{1}+2\lambda_{2}-\lambda_{3}-\lambda_{4}\geq 0,$
$\displaystyle 1-\lambda_{1}-\lambda_{2}+2\lambda_{3}-\lambda_{4}\geq 0,$
$\displaystyle 1-\lambda_{1}-\lambda_{2}-\lambda_{3}+2\lambda_{4}\geq 0,$
$\displaystyle 1+2\lambda_{1}+2\lambda_{2}+2\lambda_{3}+2\lambda_{4}\geq 0.$
(23)

The above constraint give a convex polygon space with five vertices

$\displaystyle v_{1}=\\{1,1,1,1\\},$ $\displaystyle
v_{2}=\\{1,\frac{-1}{2},\frac{-1}{2},\frac{-1}{2}\\},v_{3}=\\{\frac{-1}{2},1,\frac{-1}{2},\frac{-1}{2}\\},$
$\displaystyle
v_{4}=\\{\frac{-1}{2},\frac{-1}{2},1,\frac{-1}{2}\\},v_{5}=\\{\frac{-1}{2},\frac{-1}{2},\frac{-1}{2},1\\}.$

It is an irregular polygon with 8 edges, out of which 4 edges have euclidean
length $\sqrt{\frac{9}{2}}$ and 4 other edges have euclidean length
$\sqrt{\frac{27}{4}}$.

It is insightful to visualize the effect of the action of the channel on a
state in the four-D sphere. The parameters {$\lambda_{i}$} reduce the length
of each Bloch vector component from $n_{i}$ to $\lambda_{i}n_{i}$, thus
bringing the state closer to the origin. Now notice that, we have obtained the
constraints in Eq.(VI.2) assuming that all $\phi_{i}=0$, i.e., there is no
change in the angular parameters $\theta_{i}$. The allowed values of
$\phi_{i}$’s will therefore depend on how much the lengths of the Bloch vector
components have been reduced by $\lambda_{i}$.

###  VI.3 Characterization of Randomly Generated Density Matrices

In this section, we characterize the structure of the state space of randomly
generated density matrices, using the four-D Bloch sphere. Specifically, we
show the representation of ensembles generated by Hilbert-Schmidt and Bures
metrics Życzkowski  _et al._ (2011); Życzkowski and Sommers (2001). The
infinitesimal Hilbert-Schmidt(Eq.(20)) distance between $\rho$ and
$\delta\rho$, has a very simple form given as
$d_{HS}^{2}=\Tr[(\delta(\rho))^{2}]$. In $n$-dimensions, the probability
distribution induced by this metric, derived by Hall Hall (1998) is given by

$\displaystyle
P_{HS}(\lambda_{1},...,\lambda_{n})=C_{N}\delta(1-\sum_{i=1}^{n}\lambda_{i})\prod_{j<k}^{n}(\lambda_{j}-\lambda_{k})^{2},$
(24)

where $\lambda_{i}$’s are the eigenvalues of $\rho$ and $C_{N}$ is determined
by the normalization.

For mixed quantum states there is another useful distance measure known as the
Bures distanceBures (1969); Uhlmann (1992)

$\displaystyle
D_{Bures}(\rho_{1},\rho_{2})=\sqrt{2(1-\Tr[(\sqrt{\rho_{1}}\rho_{2}\sqrt{\rho_{1}})^{1/2}])}.$

Similar to the Hilbert-schmidt case, there exists the infinitesimal form Bures
metric derived by Hubner Hübner (1992)

$\displaystyle
d_{Bures}^{2}=\frac{1}{2}\sum_{j,k=1}^{n}\frac{|\bra{j}\delta_{\rho}\ket{k}|^{2}}{\lambda_{j}+\lambda_{k}},$

where again $\lambda_{k}$ and $\ket{k}$ are respectively the eigenvalues and
eigenvectors of $\rho$. For this metric also, the probability distribution was
derived by Hall Hall (1998), which is given by

$\displaystyle
P_{Bures}(\lambda_{1},...,\lambda_{n})=C_{N}^{\prime}\frac{\delta(1-\sum_{i=1}^{n}\lambda_{i})}{(\lambda_{1},...,\lambda_{n})^{1/2}}\prod_{j<k}^{n}\frac{(\lambda_{j}-\lambda_{k})^{2}}{\lambda_{j}+\lambda_{k}},$
(25)

where $C_{N}^{\prime}$ is again determined by the normalization. In Eqns.(24)
and (25), we have the probability distributions defined on the simplex of
eigenvalues. However, we want to see how this probability distribution picks
out the states from the bloch sphere. For a two-dimensional state
$\rho=\frac{1}{2}(I+\vec{r}\cdot\sigma)$, we can translate the eigenvalues to
Bloch sphere parameters using the simple formulas $\lambda_{1}=\frac{1+r}{2}$
and $\lambda_{2}=\frac{1-r}{2}$, where $\lambda_{1},\lambda_{2}$ are the two
eigenvalues of $\rho$. By substituting these in Eqns.(24) and (25), we get the
following probability distributions in terms of Bloch sphere parameters Hall
(1998)

$\displaystyle P_{HS}(\vec{r})=\frac{3}{4\pi},$ (26) $\displaystyle
P_{B}(\vec{r})=\frac{4}{\pi\sqrt{1-r^{2}}}.$ (27)

We can see that both probability distributions are dependent only on the
radial parameter $r$. While the HS distribution is uniform over the Bloch
sphere while the Bures distribution is sharply peaked at the surface of the
Bloch sphere.

Next, we derive the form of these probability distributions with respect to
our representation of qutrit states. For a qutrit state $\rho_{3}$, its
eigenvalues $\lambda_{1},\lambda_{2}$ and $\lambda_{3}$ can be written
directly in terms of the Bloch sphere parameters $n_{i}$’s and angular
parameters $\theta_{i}$’s. However, a direct approach will lead to cumbersome
calculations. Instead, we write the eigenvalues $\lambda_{i}$’s in terms of
the characteristic equation coefficients $a_{i}$’s from Eq.(8) and substitute
in the Eqns.(24) and (25) which gives us the following

$\displaystyle P_{HS}(\vec{r},\zeta_{i},\theta_{i})$
$\displaystyle=C_{HS}\frac{\left((r-1)^{2}(2r+1)-27\text{Det}(\rho)\right)\left((r+1)^{2}(2r-1)+27\text{Det}(\rho)\right)}{27r^{3}},$
(28) $\displaystyle P_{B}(\vec{r},\zeta_{i},\theta_{i})$
$\displaystyle=C_{B}\frac{\left((r-1)^{2}(2r+1)-27\text{Det}(\rho)\right)\left((r+1)^{2}(2r-1)+27\text{Det}(\rho)\right)}{27r^{3}\left(\frac{1-r^{2}}{3}-\text{Det}(\rho)\right)\sqrt{\text{Det}(\rho)}}.$
(29)

where we have switched to polar representation with
$n_{1}=r\cos(\zeta_{1}),n_{2}=r\sin(\zeta_{1})\cos(\zeta_{2}),n_{3}=r\sin(\zeta_{1})\sin(\zeta_{2})\cos(\zeta_{3})$
and $n_{4}=r\sin(\zeta_{1})\sin(\zeta_{2})\sin(\zeta_{3})$. Also, $C_{HS}$ and
$C_{B}$ are constants and are determined by the normalization. In this form
these probability distributions don’t give much information about the states
in the Bloch sphere because of dependence on the angular parameters
$\theta_{i}$’s which are not a part of the four-D sphere. We can obtain a
distribution for a subset of states by fixing the $\theta_{i}$’s and then
analyze the probability distributions. After fixing all the $\theta_{i}$’s
values(say all zero), we get Det$(\rho)=f(r,\zeta_{1},\zeta_{2},\zeta_{3})$.
The distributions in Eqns.(28) and (29) are not invariant with respect to
unitary operations unlike in the qubit scenario. This is a signature of the
fact that all points inside the four-D Bloch sphere don’t represent physical
states.

After some algebraic calculations, it is found that the HS distribution in
Eq.(28) is always positive irrespective of Det($\rho)$ being positive or
negative. Where as the Bures distribution in Eq.(29) is positive iff
Det($\rho)\geq 0$, hence picks out the closed structure of the qutrit states
inside the Bloch sphere. Moreover, the HS distribution in non-decreasing with
respect to the radial parameter $r$, everywhere. Where as, the Bures
distribution is non-decreasing with respect to $r$ in the region where the
Det($\rho)\geq 0$. It can also be seen that the Bures distribution is sharply
peaked whenever the denominator
$(\frac{1-r^{2}}{3}-\text{Det}(\rho))\text{Det}(\rho)$ vanishes.While
$\text{Det}(\rho)=0$ for rank-2 or rank-1 states, the
$\frac{1-r^{2}}{3}-\text{Det}(\rho)$ term can vanishes only at the surface of
the Bloch sphere or beyond.

Thus if we fix the $\theta_{i}$’s, both these distributions are localized
closer to the surface of the Bloch sphere. For the HS distribution this is
unlike to what happens in the qubit scenario where it is uniform all over the
sphere. Whereas, the Bures distribution is sharply peaked near or at the
surface of the Bloch sphere. It is similar to the behavior of the Bures
distribution in the qubit scenario, where the Bures distribution is sharply
peaked on the surface. These results are matching with the plots presented in
Fig.2 of Ref.Życzkowski and Sommers (2001), which depicts the plots in the
simplex of eigenvalues.

As an example, we fix the all $\theta_{i}=0$’ and all polar angles
$\zeta_{i}$’s as $\zeta_{1}=\pi/3,\zeta_{2}=0,\zeta_{3}=\pi/7$, to see the
dependence on the radial parameter $r$, and obtain the following

$\displaystyle P_{HS}(r)=C_{HS}\frac{6-\sqrt{3}}{72}r^{3}$ $\displaystyle
P_{B}(r)=C_{B}\frac{162(6-\sqrt{3})r^{3}}{(\sqrt{4-12r^{2}+6.19r^{3}})(-32+24r^{2}+6.19r^{3})}.$
(30)

We see that in the chosen direction, HS distribution is peaked on the surface
of the Bloch sphere and it is everywhere positive. While the Bures
distribution is sharply peaked at $r\approx 0.73$ and while is negative for
$r>0.73$. It simply tells that for the chosen $\theta_{i}$’s there are no more
physical states beyond $r\approx 0.73$ in the chosen direction and also that
there is a rank 2 state at $r\approx 0.73$. The other singularity of the Bures
distribution lies at $r\approx 1.02$, but $P_{B}(r)$ is negative after
$r=0.73$ and hence we ignore it.

In section B of the Appendix, we also do the analysis of HS and Bures
distributions when the qutrit states are represented using Gell-Mann
operators. There also we make similar observations, i.e., 1) The HS
distributions is always positive where as the Bures distribution is positive
iff Det$(\rho)\geq 0$. 2) HS distribution is non-decreasing with respect to
the radial parameter and hence the states are localized on the surface of the
convex structure of the states and 3) Bures distribution is non-decreasing for
Det$(\rho)\geq 0$ and it also blows up at the surface of the Bloch sphere or
for the rank-2 states.

##  VII Bloch vector for qudits

This approach of separating the weight parameters and angular parameters, in
the HW operator based representation can also be extended in higher
dimensions. If we consider a qudit of dimension $d$, the number of HW
operators are $d^{2}-1$ (excluding the identity matrix), among them there will
be $d+1$ sets of commuting HW operators containing $d-1$ HW operators each.
Each set will have one weight parameter associated with it as $w_{i}$ and
$d-2$ angular parameters, so that there are total $d^{2}-1$ real parameters.
In this way one can create a $d+1$ dimensional Bloch sphere built from $d+1$
weight parameters. However, as dimensions increase this analysis will only
become more complex.

##  VIII Conclusion

To conclude, we have used the HW operator basis to represent a qutrit state.
In doing so, we identified eight independent parameters consisting of four
weight and four angular parameters and constructed a four-D Bloch sphere
representation of the qutrit. We have obtained the constraints which must be
satisfied for the parametrization to represent a physical qutrit. This
representation seems like a natural extension of the qubit Bloch sphere
because of the following properties.

  1. 1.

The one and two-dimensional sections are symmetric with respect to the axes.

  2. 2.

Purity of a state depends on the length of the Bloch vector.

  3. 3.

Rank of the states can be identified to some extent by looking at the distance
from the origin of the four-D sphere.

  4. 4.

The conditions of orthogonality and mutual unbiasedness of two Bloch vectors
has lot of similarity with conditions for the qubit Bloch vectors.

  5. 5.

Hilbert-Schmidt distance between two qutrit states(with same angular parameter
values $\theta_{i}$’s) is proportional to the Euclidean distance in the four-D
sphere.

We have applied our Bloch vector representation to show that there can be a
maximum of four MUBs in three dimensions. The characterization of unital maps
acting on qutrits is also demonstrated using our representation. We also did a
characterization of randomly generated density matrices, when the probability
distributions are induced by Hilbert-Schmidt and Bures distances. Lastly, we
have mentioned the basic steps required to extend this representation in
dimensions greater than three.

One of the future works based on extending this work could be to identify the
structure of the allowed set of points which represent a physical qutrit
state. Another significant work would be to generalize this representation in
higher dimensions than three.

As we have shown in this paper that the geometry of the Bloch sphere limits
the existence of the number of MUBs in qubits and qutrits. This approach can
be used to study the existence of MUBs in 6 dimensions, where the maximum
number of MUBs is not known yet Grassl (2004); Raynal  _et al._ (2011);
Bengtsson  _et al._ (2007). An extension to the characterization of unital
maps would be to characterize qutrit entanglement breaking channels similar to
qubit entanglement breaking channels Ruskai (2003). Similar to
characterization of ensembles generated by HS and Bures metric, another
interesting study could be to identify the form of Fubini-Study metric and the
corresponding volume element Bengtsson  _et al._ (2002). Such an analysis
could be useful for sampling of pure qutrit states and averaging over them.

Our four-D sphere representation could also have significant applications in
studying the dynamics of qudit states and to find the constants of motion in
$d$-level systems. It can also be used to detect entanglement of bipartite
systems and identifying the reachable states in open system dynamics. We hope
that this approach leads to better insight in the study of qudit systems and
their dynamics.

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##  Appendix A Finding the orthonormal bases mutually unbiased to the
computational basis

Here we will show, how to construct an orthonormal basis given a qutrit ket
parametrized by $\vec{n_{\theta
0}}:n_{1}=m_{1},n_{2}=0,n_{3}=m_{3},n_{4}=m_{4},\theta_{1}=\phi_{1},\theta_{2}=\text{any
number},\theta_{3}=\phi_{3}\hskip 5.69054pt\text{and}\hskip
5.69054pt\theta_{4}=\phi_{4}$, so that the orthonormal basis is mutually
unbiased with respect to to the computational basis. Since, we are looking for
pure states mutually unbiased to the computational basis, we can also
represent $\vec{n_{\theta 0}}$:{$\alpha=\delta,\beta=\gamma$} using only two
parameters as

$\displaystyle\vec{n_{\theta 0}}\equiv\frac{1}{\sqrt{3}}\begin{pmatrix}1\\\
e^{i\delta}\\\
e^{i\gamma}\end{pmatrix}\rightarrow\frac{1}{3}\begin{pmatrix}1&e^{-i\delta}&e^{-i\gamma}\\\
e^{i\delta}&1&e^{-i(\gamma-\delta)}\\\
e^{i\gamma}&e^{i(\gamma-\delta)}&1\end{pmatrix}.$

The corresponding Bloch vector parameters can be obtained on comparing the
Bloch vector representation with the above two parameter representation.

$\displaystyle
n_{1}=\frac{1}{3}\sqrt{3+2\cos(\gamma-2\delta)+2\cos(2\gamma-\delta)+2\cos(\gamma+\delta)},$
$\displaystyle
n_{3}=\frac{1}{3}\sqrt{3+2\cos(\gamma-2\delta-\frac{2\pi}{3})+2\cos(2\gamma-\delta+\frac{2\pi}{3})+2\cos(\gamma+\delta-\frac{2\pi}{3})},$
$\displaystyle
n_{4}=\frac{1}{3}\sqrt{3+2\cos(\gamma-2\delta+\frac{2\pi}{3})+2\cos(2\gamma-\delta-\frac{2\pi}{3})+2\cos(\gamma+\delta+\frac{2\pi}{3})},$
$\displaystyle
e^{i\theta_{1}}=\frac{e^{-i\delta}+e^{-i(\gamma-\delta)}+e^{i\gamma}}{3n_{1}},e^{i\theta_{3}}=\frac{e^{i\delta}+\omega
e^{-i\gamma}+\omega^{2}e^{i(\gamma-\delta)}}{3n_{3}},e^{i\theta_{4}}=\frac{e^{i\delta}+\omega^{2}e^{-i\gamma}+\omega
e^{i(\gamma-\delta)}}{3n_{4}},$ (31)

where $\omega=e^{i\frac{2\pi}{3}}$ is the cube root of unity. In this
representation, a ket orthogonal to $\vec{n_{\theta 0}}$ must have parameters
$\vec{n_{1}}$:$\\{\alpha=\delta+\frac{2\pi}{3},\beta\gamma-\frac{2\pi}{3}\\}$
or
$\vec{n_{2}}$:$\\{\alpha=\delta-\frac{2\pi}{3},\beta=\gamma+\frac{2\pi}{3}\\}$.
It is straightforward to observe from Eq.(A) that for orthonormal vectors the
weight parameters $n_{i}$ are the sane, only the angular parameters
$\theta_{i}$ vary.

Moreover, the orthonormal bases ${P}$ and ${Q}$ which are mutually unbiased to
basis ${N}$ (and to each other) are obtained by changing the angular
parameters to $\alpha=\delta+\frac{2\pi}{3},\beta=\gamma$ and
$\alpha=\delta+\frac{4\pi}{3},\beta=\gamma$ respectively, and then forming an
orthonormal basis from them. It can be seen that on moving between the MUBs
${N}$, ${P}$ and ${Q}$, the weight parameters permute cyclically. Thus,
concluding the proof.

##  Appendix B Random Density Matrices in Gell-Mann operator repreentation

Using the Gell-Mann operator basis also one can write a qutrit state in the
following way Bertlmann and Krammer (2008)

$\displaystyle\rho=\frac{1}{3}(\mathbb{I}+\sum_{i=1}^{d^{2}-1}g_{i}\Lambda_{i}),$
(32)

where $\Lambda_{i}$ are the Gell-Mann operators in three dimensions and
$g_{i}=\Tr(\Lambda_{i}\rho)$ form the components of the eight-
dimensional(eight-D) Bloch vector $\vec{g}$. The eight Gell-Mann operators in
three dimensions contain diagonal, symmetric and anti-symmetric matrices, but
for simplicity we denote all of them with $\Lambda_{i}$. Using the similar
trick as in the case of Weyl operator representation we can get the HS and
Bures distribution in terms of the Bloch vector parameters $g_{i}$ as
following

$\displaystyle P_{HSG}(\vec{r_{g}},\gamma_{i})$
$\displaystyle=\frac{C_{HSG}}{r^{7}}\left(\frac{1}{729}(r^{2}-3)^{2}(4r^{2}-3)+(2-2r^{2}-27\text{Det}(\rho))\text{Det}(\rho)\right),$
(33) $\displaystyle P_{BG}(\vec{r_{g}},\gamma_{i})$
$\displaystyle=\frac{C_{BG}}{r^{7}(3-r^{2}-9\text{Det}(\rho))\sqrt{\text{Det}(\rho)}}\Big{(}\frac{1}{729}(r^{2}-3)^{2}(4r^{2}-3)$
$\displaystyle+(2-2r^{2}-27\text{Det}(\rho))\text{Det}(\rho)\Big{)},$ (34)

where we have switched to polar representation with $r_{g}$ being the radial
distance in the eight-D Bloch sphere and $\gamma_{i}$’s are the seven polar
angles. $C_{HSG}$ and $C_{BG}$ are constants determined by the normalization.

As in the Weyl representation, here also the HS distribution is always
positive inside the eight-D Bloch sphere irrespective of Det$(\rho)$ being
positive of negative. Also, it is non-decreasing with respect to $r_{g}$. Thus
the states chosen are localized at the surface of the Bloch sphere.

The Bures distribution also behaves similar to the Weyl representation. It is
positive iff Det$(\rho)\geq 0$ and also it is non-decreasing for
Det$(\rho)\geq 0$.The singularity in $P_{BG}(\vec{r_{g}},\gamma_{i})$ occurs
either at $(3-r^{2}-9\text{Det}(\rho))=0$ or when $\sqrt{\text{Det}(\rho)}=0$.
The first condition is only possible at or beyond the surface of the eight-D
sphere. Where as, $\text{Det}(\rho)=0$ can happen for rank-1 or rank-2 states,
i.e., at the surface of the structure formed by the qutrit states. Thus,
$H_{BG}$ is sharply localized at the surface of the convex structure formed by
the qutrit states.


# Distinct Fermi Surface Topology and Nodeless Superconducting Gap in
(Tl0.58Rb0.42)Fe1.72Se2 Superconductor

Daixiang Mou1, Shanyu Liu1, Xiaowen Jia1, Junfeng He1, Yingying Peng1, Lin
Zhao1, Li Yu1, Guodong Liu1, Shaolong He1, Xiaoli Dong1, Jun Zhang1, Hangdong
Wang2, Chiheng Dong2, Minghu Fang2, Xiaoyang Wang3, Qinjun Peng3, Zhimin
Wang3, Shenjin Zhang3, Feng Yang3, Zuyan Xu3, Chuangtian Chen3 and X. J.
Zhou1,∗
1Beijing National Laboratory for Condensed Matter Physics, Institute of
Physics, Chinese Academy of Sciences, Beijing 100190, China
2Department of Physics, Zhejiang University, Hangzhou 310027, China
3Technical Institute of Physics and Chemistry, Chinese Academy of Sciences,
Beijing 100190, China

(January 24, 2011)

###### Abstract

High resolution angle-resolved photoemission measurements have been carried
out to study the electronic structure and superconducting gap of the
(Tl0.58Rb0.42)Fe1.72Se2 superconductor with a Tc=32 K. The Fermi surface
topology consists of two electron-like Fermi surface sheets around $\Gamma$
point which is distinct from that in all other iron-based compounds reported
so far. The Fermi surface around the M point shows a nearly isotropic
superconducting gap of $\sim$12 meV. The large Fermi surface near the $\Gamma$
point also shows a nearly isotropic superconducting gap of $\sim$15 meV while
no superconducting gap opening is clearly observed for the inner tiny Fermi
surface. Our observed new Fermi surface topology and its associated
superconducting gap will provide key insights and constraints in understanding
superconductivity mechanism in the iron-based superconductors.

###### pacs:

74.70.-b, 74.25.Jb, 79.60.-i, 71.20.-b

The discovery of the Fe-based superconductorsKamihara ; ZARenSm ; RotterSC ;
MKWu11 ; CQJin111  has attracted much attention because it represents the
second class of high temperature superconductors in addition to the copper-
oxide (cuprate) superconductorsBednorz . It is important to explore whether
the high-Tc superconductivity mechanism in this new Fe-based system is
conventional, parallel to that in cuprates, or along a totally new
routeNPReview . Different from the cuprates where the low-energy electronic
structure is dominated by Cu 3d${}_{x^{2}-y^{2}}$ orbital, the electronic
structure of the Fe-based compounds involves all five Fe 3d-orbitals forming
multiple Fermi surface sheets: hole-like Fermi surface sheets around
$\Gamma$(0,0) and electron-like ones around M($\pi$,$\pi$)DJSingh1111 ; Kuroki
. It has been proposed that the interband scattering between the hole-like
bands near $\Gamma$ and electron-like bands near M gives rise to electron
pairing and superconductivityKuroki ; FeSCMagnetic . An alternative picture is
also proposed based on the interaction of local Fe magnetic momentJ1J2Picture
.

The latest discovery of superconductivity with a Tc above 30 K in a new
AxFe2-ySe2 (A=K, Tl, Cs, Rb and etc.) systemJGGuo ; Switzerland ; MHFang ;
GFChen  is surprising that provides new perspectives in understanding Fe-based
compounds. First, it may involve Fe vacancies in the FeSe layerMHFang ; GFChen
; ZWang . This is against the general belief that a perfect Fe-sublattice is
essential for superconductivity in the Fe-based compounds, similar to that
perfect CuO2 plane is considered essential for cuprate superconductors.
Second, the superconductivity of AxFe2-ySe2 is realized in a close proximity
to an antiferromagnetic semiconducting (insulating) phaseMHFang ; GFChen ;
QMSi . This is in strong contrast to other Fe-based compounds where the parent
compounds are spin-ordered metalsDongSDW ; RotterParent ; PCDai ; BFSNeutron3
. Third, the intercalation of A=(K,Tl,Cs,Rb and etc.) in AxFe2-ySe2 is
expected to introduce large number of electrons into the system; this usually
would lead to suppression or disappearance of superconductivity like in
heavily electron-doped Ba(Fe,Co)2As2 systemCo122 . The existence of
superconductivity in AxFe2-ySe2 at such a high Tc (over 30 K) with so high
electron doping is unexpected. Most interestingly, band structure
calculationsLJZhang ; IRShein ; XWYan  and electronic structure
measurementsYZhang ; TQian  all suggest that high electron doping in
AxFe2-ySe2 may lead to the disappearance of the hole-like Fermi surface sheets
around $\Gamma$. This would render it impossible for the electron scattering
between the hole-like bands near $\Gamma$ and electron-like bands near M that
is considered to be important for the electron pairing in Fe-based
superconductors by some theoriestsKuroki ; FeSCMagnetic . Therefore,
investigations of the AxFe2-ySe2 system would be important in searching for
common ingredients underlying the physical properties, especially the
superconductivity of the Fe-based superconductors.

Figure 1: Fermi surface of (Tl0.58Rb0.42)Fe1.72Se2 superconductor (Tc=32 K).
(a). Spectral weight distribution integrated within [-20meV,10meV] energy
window near the Fermi level as a function of kx and ky measured using
h$\nu$=21.2 eV light source. Two Fermi surface sheets are observed around
$\Gamma$ point which are marked as $\alpha$ for the inner small sheet and
$\beta$ for the outer large one. Near the M($\pi$,$\pi$) point, one Fermi
surface sheet is clearly observed which is marked as $\gamma$. (b) Three-
dimensional image of Fig. 1a. (c). Fermi surface mapping measured using
h$\nu$=40.8 eV light source Although the signal is relatively weak, one can
see traces of two Fermi surface sheets around $\Gamma$ and one around M.

In this paper, we report observation of a distinct Fermi surface topology and
nearly isotropic nodeless superconducting gap in (Tl0.58Rb0.42)Fe1.72Se2
superconductor (Tc=32 K) from high resolution angle-resolved photoemission
(ARPES) measurements. We have observed an electron-like Fermi surface sheet
near M($\pi$,$\pi$) and two electron-like Fermi surface sheets near
$\Gamma$(0,0). This Fermi surface topology is distinct from the hole-like
Fermi surface sheets near the $\Gamma$ point found in other Fe-based
compoundsDJSingh1111 ; Kuroki  or disappearance of hole-like Fermi surface
sheets near $\Gamma$ in AxFe2-ySe2 compoundsLJZhang ; IRShein ; XWYan ; YZhang
; TQian . We observe nearly isotropic superconducting gap around the Fermi
surface sheets near $\Gamma$ ($\sim$15 meV) and M ($\sim$12 meV); no gap node
is observed in both Fermi surface sheets. These rich information on this new
Fe-based superconductor will provide key insights on the superconductivity
mechanism in the Fe-based superconductors.

High resolution angle-resolved photoemission measurements were carried out on
our lab system equipped with a Scienta R4000 electron energy analyzerGDLiu .
We use Helium discharge lamp as the light source which can provide photon
energies of h$\upsilon$= 21.218 eV (Helium I) and 40.8 eV (Helium II). The
energy resolution was set at 10 meV for the Fermi surface mapping (Fig. 1a)
and band structure measurements (Fig. 2) and at 4 meV for the superconducting
gap measurements (Figs. 3 and 4). The angular resolution is $\sim$0.3 degree.
The Fermi level is referenced by measuring on a clean polycrystalline gold
that is electrically connected to the sample. The (Tl,Rb)Fe2-ySe2 crystals
were grown by the Bridgeman methodMHFang . Their composition determined by
using an Energy Dispersive X-ray Spectrometer (EDXS) measurement is
(Tl0.58Rb0.42)Fe1.72Se2. The crystals have a sharp superconducting transition
at Tc(onset)=32 K with a transition width of $\sim$1 K. The crystal was
cleaved in situ and measured in vacuum with a base pressure better than
5$\times$10-11 Torr.

Figure 2: Band structure and photoemission spectra of (Tl0.58Rb0.42)Fe1.72Se2
measured along two high symmetry cuts. (a). Band structure along the Cut 1
crossing the $\Gamma$ point; the location of the cut is shown on top of Fig.
2a. The $\alpha$ band and two Fermi crossings of the $\beta$ band ($\beta_{L}$
and $\beta_{R}$) are marked. (b). Corresponding EDC second derivative image of
Fig. 2a. (c). Band structure along the Cut 2 crossing M point; the location of
the cut is shown on top of Fig. 2c. The two Fermi crossings of the $\gamma$
band ($\gamma_{L}$ and $\gamma_{R}$) are marked. (d). Corresponding EDC second
derivative image of Fig. 2c. (e). EDCs corresponding to Fig. 2a for the Cut 1.
(f). EDCs corresponding to Fig. 2c for the Cut 2.

Fig. 1 shows Fermi surface mapping of the (Tl0.58Rb0.42)Fe1.72Se2
superconductor covering multiple Brillouin zones. The band structure along two
typical high symmetry cuts are shown in Fig. 2. An electron-like Fermi surface
is clearly observed around M($\pi$,$\pi$) (Fig. 1a, Figs. 2c and 2d). This
Fermi surface (denoted as $\gamma$ hereafter) is nearly circular with a Fermi
momentum (kF) of 0.35 in a unit of $\pi$/a (lattice constant a=3.896 $\AA$).
The Fermi surface near the $\Gamma$ point consists of two sheets. The inner
tiny pocket (denoted as $\alpha$) is electron-like with a band bottom barely
touching the Fermi level ( Figs. 2a and 2b for the Cut 1). The outer larger
Fermi surface sheet around $\Gamma$ (denoted as $\beta$) (Fig. 1a) is
electron-like (Figs. 2a and 2b) with a Fermi momentum of 0.35 $\pi$/a.

The observation of two electron-like Fermi surface sheets, $\alpha$ and
$\beta$, around $\Gamma$ in (Tl0.58Rb0.42)Fe1.72Se2 is distinct from that
observed in other Fe-based compounds where hole-like pockets are expected
around the $\Gamma$ pointDJSingh1111 ; Kuroki . It is also different from the
band structure calculationsLJZhang ; IRShein ; XWYan ; YZhang ; TQian  and
previous ARPES measurementsYZhang ; TQian  on AxFe2-ySe2 that suggest
disappearance of hole-like Fermi surface sheets near $\Gamma$ because of the
lifted chemical potential due to a large amount of electron doping.

Figure 3: Temperature dependence of energy bands and superconducting gap near
$\Gamma$ and M points. (a). Photoemission images along the Cut B (bottom-right
inset). (b). Photoemission spectra at the Fermi crossing of $\gamma$ Fermi
surface and their corresponding symmetrized spectra (c) measured at different
temperatures. (d). Temperature dependence of the superconducting gap. The
dashed line is a BCS gap form. (e). Photoemission images along the Cut A
(bottom-right inset) at different temperatures. The original EDCs at the
$\Gamma$ point and at the Fermi crossing of $\beta$ Fermi surface measured at
different temperatures are shown in (f) and their corresponding symmetrized
EDCs are shown in (g).

One immediate question is on the origin of the electron-like $\beta$ band
around $\Gamma$. The first possibility is whether it could be a surface state.
While surface state on some Fe-based compounds like the “1111” system was
observed beforeHYLiu , it has not been observed in the “11”-type Fe(Se,Te)
systemFeSTARPES . The second possibility is whether the $\beta$ band can be
caused by the folding of the electron-like $\gamma$ Fermi surface near M. It
is noted that the Fermi surface size, the band dispersion, and the band width
of the $\beta$ band at $\Gamma$ is similar to that of the $\gamma$ band near
M. A band folding picture would give a reasonable account for such a
similarity if there exists a ($\pi$,$\pi$) modulation in the system that can
be either structural or magnetic. An obvious issue with this scenario is that,
in this case, one should also expect the folding of the $\alpha$ band near
$\Gamma$ onto the M point; but such a folding is not observed at the M point
(Fig. 1a and Fig. 2c). The third possibility is whether the measured $\beta$
sheet is a Fermi surface at a special kz cut. Although the Fermi surface at
$\Gamma$ is absent in TlFe2Se2 from the band structure calculationsLJZhang ,
there is a 3-dimensional Fermi pocket that is present near the zone center at
kz=$\pi$$/$c when x is close to 1 in KxFe2Se2IRShein  and CsxFe2Se2XWYan . We
note that the electron doping in (Tl0.58Rb0.42)Fe1.72Se2 is lower than that of
(K,Cs)Fe2Se2. Also we observed similar $\beta$ Fermi surface at different
photon energies (Fig. 1a and Fig. 1c) which corresponds to different kz. The
final resolution of this possibility needs further detailed photon energy
dependent measurements.

The clear identification of various Fermi surface sheets makes it possible to
investigate the superconducting gap in this new superconductor. We start first
by examining the superconducting gap near the M point. Fig. 3a shows the
photoemission images along the Cut B near M (its location shown in the bottom-
right inset of Fig. 3) at different temperatures. The corresponding
photoemission spectra (energy distribution curves, EDCs) on the Fermi momentum
at different temperatures are shown in Fig. 3b. To visually inspect possible
gap opening and remove the effect of Fermi distribution function near the
Fermi level, we have symmetrized these original EDCs to get spectra in Fig.
3c, following the procedure that is commonly used in high temperature cuprate
superconductorsMNorman . For the $\gamma$ pocket near M, there is a clear gap
opening at low temperature (15 K), as indicated by an obvious dip at the Fermi
energy in the symmetrized EDCs (Fig. 3c). With increasing temperature, the dip
at EF is gradually filled up and is almost fully filled above Tc=32 K. The gap
size at different temperatures is extracted from the peak position of the
symmetrized EDCs or fitted by the phenomenological formulaMNorman  (Fig. 3d);
it is $\sim$11 meV at 15 K. The temperature dependence of the gap size roughly
follows the BCS-type form (Fig. 3d). Similar temperature dependent
measurements of the superconducting gap were also carried out along the
$\Gamma$-M cut near $\Gamma$ (Figs. 3e-g). The Fermi crossing on the $\beta$
Fermi surface also displays a clear superconducting gap in the superconducting
state which is closed above Tc (lower curves in Figs. 3f and 3g). For the
peculiar tiny $\alpha$ pocket near $\Gamma$, we do not find signature of clear
superconducting gap opening below Tc (upper curves in Figs. 3f and 3g).

Now we come to the momentum-dependent measurements of the superconducting gap.
For this purpose we took high resolution Fermi surface mapping (energy
resolution of 4 meV) of the $\gamma$ pocket at M (Fig. 4a) and the $\beta$
pocket at $\Gamma$ (Fig. 4b). Fig. 4c shows photoemission spectra around the
$\gamma$ Fermi surface measured in the superconducting state (T= 15 K); the
corresponding symmetrized photoemission spectra are shown in Fig. 4d. The
extracted superconducting gap (Fig. 4g) is nearly isotropic with a size of
(12$\pm$2) meV. The superconducting gap around the $\beta$ Fermi surface near
$\Gamma$ is also nearly isotropic with a size of (15$\pm$2) meV (Figs. 4e, 4f
and 4g).

Figure 4: Momentum dependent superconducting gap along the $\gamma$ and the
$\beta$ Fermi surface sheets measured at T=15 K. Fermi surface mapping near M
(a) and near $\Gamma$ (b) and the corresponding Fermi crossings marked by red
circles. (c). EDCs along the $\gamma$ Fermi surface and their corresponding
symmetrized EDCs (d). (e). EDCs along the $\beta$ Fermi surface and their
corresponding symmetrized EDCs (f). (g). Momentum dependence of the
superconducting gap along the $\gamma$ Fermi surface sheet (red circles) and
along the $\beta$ Fermi surface sheet (blue circles).

The observation of a distinct Fermi surface topology in
(Tl0.58Rb0.42)Fe1.72Se2 has important implications to the understanding of
superconductivity in Fe-based superconductors. The realization of high Tc in
this new superconductor with a distinct Fermi surface topology is helpful to
sort out key electronic structure ingredient that is responsible for
superconductivity. With the electron-like $\beta$ Fermi surface present in
(Tl0.58Rb0.42)Fe1.72Se2, the possibility of electron scattering between the
$\Gamma$ Fermi surface sheet(s) and the M Fermi surface sheet(s), proposed by
some theories to account for superconductivity in the Fe-based
superconductorKuroki ; FeSCMagnetic , cannot be ruled out. However, the
electron scattering between two electron-like bands may have different effect
on the electron pairing from that between an electron-like band and a hole-
like band. The nearly isotropic superconducting gap on the $\beta$ and
$\gamma$ Fermi surface sheets, together with the absence of gap nodes, appears
to favor s-wave superconducting gap symmetry in (Tl0.58Rb0.42)Fe1.72Se2. This
is similar to that in (Ba0.6K0.4)Fe2As2122Gap  and NdFeAsO0.9F0.1Kaminski1111
. The gap size of 12 (for $\gamma$) and 15 meV (for $\beta$) gives a ratio of
2$\Delta$/kTc=9 and 11, respectively, which is significantly larger than the
traditional BCS weak-coupling value of 3.52 for an s-wave gap. This indicates
that this new superconductor is at least in the strong coupling regime in
terms of the BCS picture. These will put strong constraints on various
proposed gap symmetries and the underlying pairing mechanisms for the iron-
based superconductors.

In summary, we have identified a distinct Fermi surface topology in the new
(Tl0.58Rb0.42)Fe1.72Se2 superconductor that is different from all other Fe-
based superconductors reported so far. Near the $\Gamma$ point, two electron-
like Fermi surface sheets are observed that are different from the band
structure calculations and previous ARPES measurement results. We observed
nearly isotropic superconducting gap around the Fermi surface sheets near
$\Gamma$ and M without gap nodes. These rich information will shed more light
on the nature of superconductivity in the Fe-based superconductors.

XJZ and MHF thank the funding support from NSFC (Grant No. 10734120 and
10974175) and the MOST of China (973 program No: 2011CB921703 and
2011CBA00103).

∗Corresponding author<EMAIL_ADDRESS>
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