# Correlation constraints and the Bloch geometry of two qubits

Simon Morelli<EMAIL_ADDRESS>BCAM - Basque Center for Applied
Mathematics, Mazarredo, 14. 48009 Bilbao, Spain  Christopher Eltschka
Institut für Theoretische Physik, Universität Regensburg, D-93040 Regensburg,
Germany  Marcus Huber  Atominstitut, Technische Universität Wien, 1020 Vienna,
Austria  Institute for Quantum Optics and Quantum Information - IQOQI Vienna,
Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria  Jens
Siewert<EMAIL_ADDRESS>Department of Physical Chemistry and EHU Quantum
Center, University of the Basque Country UPV/EHU, E-48080 Bilbao, Spain
IKERBASQUE Basque Foundation for Science, E-48013 Bilbao, Spain

###### Abstract

We present a novel inequality on the purity of a bipartite state depending
solely on the difference of the local Bloch vector lengths. For two qubits
this inequality is tight for all marginal states and so extends the previously
known solution for the 2-qubit marginal problem and opens a new research
avenue. We further use this inequality to construct a 3-dimensional Bloch
model of the 2-qubit quantum state space in terms of Bloch lengths, thus
providing a geometrically pleasing visualization of this difficult to access
high-dimensional state space. This allows to characterize quantum states
relying on a strongly reduced set of parameters alone and to investigate the
interplay between local properties of the marginal systems and global
properties encoded in the correlations.

##  I Introduction

The investigation of fundamental bounds in the form of entropy inequalities,
limiting the distribution of information within a multipartite system, marked
the birth of information theory as a field of study Shannon (1948). In the
same way, the exploration of fundamental limitations on the distribution of
quantum information within a system lies at the core of the rapidly evolving
field of quantum information theory. Analogous results to classical
information theory where found for the von Neumann entropy Araki (1970); Lieb
and Ruskai (1973); Pippenger (2003), but at the same time there remain many
open questions Linden and Winter (2005); Cadney  _et al._ (2012).

Moreover, these are not the only known constraints to the distribution of
quantum information within a system. Monogamy relations bound the shared
entanglement that a party can have with other parties and thus ultimately
limit the set of attainable quantum states Coffman  _et al._ (2000); Osborne
and Verstraete (2006); Eltschka  _et al._ (2009); Bai  _et al._ (2014);
Lancien  _et al._ (2016); Regula  _et al._ (2016); Allen and Meyer (2017); S.
(2017); Gour and Guo (2017); Eltschka (2018); Eltschka  _et al._ (2018).

The classical probability space is described geometrically by a simplex of the
corresponding dimension. While the latter geometrically is fairly regular, the
quantum state space becomes a highly complex object that admits no simple
description Bengtsson and Życzkowski (2017); Bengtsson  _et al._ (2012).
Ultimately the characterization of attainable quantum states is linked to the
famous quantum marginal problem, investigating the compatibility between local
and global eigenvalues of a state Klyachko (2006); Yu  _et al._ (2021).

In this work we present a novel constraint of the purity of any bipartite
quantum state, based solely on the difference of the local Bloch lengths,
i.e., the Hilbert-Schmidt distance of the marginal states to the normalized
identity. For two qubits this constraint is tight for all possible marginal
states, thus extending the previously known solution for the marginal problem
for two qubits Bravyi (2004). Further, the bound can be rewritten in terms of
a new entropy relation for three-qubit pure states.

We then use this novel inequality to construct a three dimensional model for
the state space of two qubits, where we use the two local Bloch lengths and
the correlation tensor length as the coordinates. This allows to characterize
important properties of states based on a very reduced set of parameters. In
the model we further identify regions of special interest, such as purely
entangled and purely separable regions and plot families of states, such as
the states maximizing the entanglement for a given purity.

##  II Purity constraint from sector-difference

An immediate consequence of the Schmidt decomposition is that for every pure
bipartite quantum state the marginal states are equal up to local unitaries.
This means that if the eigenvalues of the marginals are equal, there exists no
non-trivial constraint for the purity of the global state. However, if the
marginals have different eigenvalues no pure state is compatible with those
constraints. Therefore there exists a purity constraint based on the
difference of the local states. The following theorem gives an upper bound on
the purity of a bipartite quantum state based solely on the difference of the
local Bloch lengths. For qubits this bound is tight.

######  Theorem 1.

For any bipartite state $\rho_{AB}$ with equal local dimension $d$ it holds
that

$\displaystyle d\operatorname{Tr}(\rho^{2}_{AB})\leq
d-\sqrt{2d}\big{|}\|\mathbf{a}\|-\|\mathbf{b}\|\big{|}+\big{|}\|\mathbf{a}\|-\|\mathbf{b}\|\big{|}^{2},$
(1)

where $\mathbf{a}$ and $\mathbf{b}$ are the local Bloch vectors of
$\rho_{A}=\operatorname{Tr}_{B}(\rho_{AB})$ and
$\rho_{B}=\operatorname{Tr}_{A}(\rho_{AB})$, respectively, and $\|\cdot\|$
denotes the 2-norm.

###### Proof.

Denote by $\Delta=\big{|}\|\mathbf{a}\|-\|\mathbf{b}\|\big{|}$. Note that any
state with eigenvalues $\lambda_{1}=1-\Delta/\sqrt{2d}$,
$\lambda_{2}=\Delta\big{|}/\sqrt{2d}$ and $\lambda_{j}=0$ ($j>2$) satisfies
the inequality. Assume now that there exist a state $\rho$ such that
$d\operatorname{Tr}(\rho^{2})>d-\sqrt{2d}\Delta+\Delta^{2}$. The trace of
$\rho$ can only exceed the bound if its leading eigenvalue satisfies
$\lambda_{1}(\rho)>1-\Delta/\sqrt{2d}$. Therefore it must hold that
$\lambda_{2}(\rho)<\Delta/\sqrt{2d}$. Any such state can be written as

$\displaystyle\rho=\lambda_{1}|\psi\rangle\langle\psi|+\lambda_{2}\sigma$ (2)

Writing this state in the Schmidt basis of $|\psi\rangle$, we find

$\displaystyle\rho_{A}=\lambda_{1}\rho_{\psi}+\lambda_{2}\sigma_{A}$ (3)
$\displaystyle\rho_{B}=\lambda_{1}\rho_{\psi}+\lambda_{2}\sigma_{B},$ (4)

with equal marginals
$\rho_{\psi}=\operatorname{Tr}_{B}(|\psi\rangle\langle\psi|)=\operatorname{Tr}_{A}(|\psi\rangle\langle\psi|)$.
Now it follows

$\displaystyle\lambda_{2}(\rho)$
$\displaystyle<\frac{1}{\sqrt{2d}}\big{|}\|\mathbf{a}\|-\|\mathbf{b}\|\big{|}$
(5)
$\displaystyle=\tfrac{1}{\sqrt{2}d}\big{|}\|d\rho_{A}-\mathds{1}\|_{\mathrm{HS}}-\|d\rho_{B}-\mathds{1}\|_{\mathrm{HS}}\big{|}$
(6) $\displaystyle\leq\tfrac{1}{\sqrt{2}}\|\rho_{A}-\rho_{B}\|_{\mathrm{HS}}$
(7)
$\displaystyle=\tfrac{1}{\sqrt{2}}\lambda_{2}\|\sigma_{A}-\sigma_{B}\|_{\mathrm{HS}}$
(8) $\displaystyle\leq\lambda_{2}(\rho),$ (9)

where we have used $\|\sigma_{A}-\sigma_{B}\|_{\mathrm{HS}}\leq\sqrt{2}$,
which follows immediately from the positivity of the matrices. But this is a
contradiction, therefore the assumption that there exists a state $\rho$
violating the inequality is wrong. ∎

Figure 1: The maximal value of $\operatorname{Tr}(\rho^{2})$ plotted against
the difference $\Delta(\rho)=\big{|}\|\mathbf{a}\|-\|\mathbf{b}\|\big{|}$, in
blue the bound given by Eq. (1) and in orange the bound following the triangle
inequality of the linear entropy.

Remarkable is the fact that for qubits this inequality is not only tight with
respect to the difference $\big{|}\|\mathbf{a}\|-\|\mathbf{b}\|\big{|}$, but
actually for all values of $\|\mathbf{a}\|$ and $\|\mathbf{b}\|$, see Eq. (20)
for a family of states that saturate the inequality. But the local Bloch
length describes the state of the marginal qubits uniquely up to local
unitaries, which clearly do not change any of the quantities in the relation.
This means, that for every two marginal states, there exists a global state
compatible with them that saturates the inequality. So we give a more
extensive answer to the two-qubit marginal problem, cf. Ref. Bravyi (2004).
Instead of saying that there only exists a pure state compatible with the
marginals if their spectrum is equal, we are able to exactly quantify the
maximum purity of any global state compatible with any two given marginals (or
in fact only the difference of their local Bloch lengths).

A direct consequence of this inequality is that for any pure tripartite state
it holds that the three local Bloch vectors satisfy

$\displaystyle\|\mathbf{c}\|^{2}\leq
d-1-\sqrt{2d}\big{|}\|\mathbf{a}\|-\|\mathbf{b}\|\big{|}+\big{|}\|\mathbf{a}\|-\|\mathbf{b}\|\big{|}^{2}.$
(10)

This can be seen by simply using that for a pure three-party state it holds
that $\operatorname{Tr}(\rho_{AB}^{2})=\operatorname{Tr}(\rho_{C}^{2})$ and
inserting this into Eq. (1).

For qubits this relation can be simplified to

$\displaystyle\|\mathbf{a}\|+\|\mathbf{c}\|\leq 1+\|\mathbf{b}\|$ (11)

and is saturated by states of the form

$\displaystyle\left|\right.\\!{\psi}\\!\left.\right\rangle=\sqrt{\tfrac{x-y}{2}}\left|\right.\\!{001}\\!\left.\right\rangle+\sqrt{\tfrac{1+y}{2}}\left|\right.\\!{010}\\!\left.\right\rangle+\sqrt{\tfrac{1-x}{2}}\left|\right.\\!{100}\\!\left.\right\rangle.$
(12)

While conceptually different, this constraint can be compared to the famous
Coffman-Kundu-Wootters relation (CKW) Coffman  _et al._ (2000). The states
that saturate this relation for three qubits form a strict subset of the
states saturating CKW, thus showing the strength of this new constraint but
ultimately proving it weaker than CKW. At the same time Eq. (10) holds in all
finite dimensions, opening an exciting new avenue for the study of high-
dimensional multipartite quantum systems.

##  III Quantum state space

In what is known as the _Bloch representation_ , density operators of qubits
are parametrized by the Pauli matrices $\sigma_{x}$, $\sigma_{y}$,
$\sigma_{z}$ and the identity $\mathds{1}_{2}$

$\displaystyle\rho\ =\ \frac{1}{2}(\mathds{1}_{2}+x\ \sigma_{x}+y\
\sigma_{y}+z\ \sigma_{z}),$ (13)

where $x$, $y$ and $z$ are real numbers satisfying $x^{2}+y^{2}+z^{2}\leq 1$
Fano (1954, 1957). Hence, the state space $\mathcal{Q}_{2}$ of a qubit can be
identified with a three-dimensional ball of radius 1 about the origin, with a
one to one correspondence between points $(x,y,z)$ and states $\rho$.
Unfortunately, the Bloch ball falls short of giving an extensive comprehension
of the state space of quantum systems, not because of shortcomings of the
geometric model but because the system of a single qubit itself is too
elementary to exhibit all the complex features of a general quantum system.
Looking at higher-dimensional systems gives a richer and more diverse
structure of the QSS and its geometrical representation, e.g., Refs. Bengtsson
and Życzkowski (2017); Bloore (1976); Kimura (2003); Kurzyński (2011);
Sarbicki and Bengtsson (2012); Bengtsson  _et al._ (2012); Goyal  _et al._
(2016); Sharma and Ghosh (2021); Rau (2021); Eltschka  _et al._ (2021). As
interesting and complex as these single systems are, they still lack an
essential property: they are unable to display correlations between different
parties of a more complex quantum system. Steps in that direction were taken
in Ref. Gamel (2016); Wie (2020); Zyczkowski (2022).

By taking tensor products of the Pauli matrices, this matrix-basis for the
density operators of a single qubit can be extended to a product basis of the
composite space of multiple qubits. That is, every two-qubit state can be
expanded in the product Pauli basis

$\rho_{AB}=\frac{1}{4}\big{(}\mathds{1}_{4}+\sum_{i}a_{i}\sigma_{i}\otimes\mathds{1}_{2}+\sum_{j}b_{j}\mathds{1}_{2}\otimes\sigma_{j}+\sum_{i,j}t_{ij}\sigma_{i}\otimes\sigma_{j}\big{)},$
(14)

where $i,j=x,y,z$ and
$a_{i}=\langle\sigma_{i}\otimes\mathds{1}_{2}\rangle\equiv\operatorname{Tr}\left(\rho\
\sigma_{i}\otimes\mathds{1}_{2}\right)$,
$b_{j}=\langle\mathds{1}_{2}\otimes\sigma_{j}\rangle$ and
$t_{ij}=\langle\sigma_{i}\otimes\sigma_{j}\rangle$. The state space of a two-
qubit system is now characterized by 15 real parameters and can thus be
identified with a region in $\mathbb{R}^{15}$. While the condition
$\sum_{i}a_{i}^{2}+\sum_{j}b_{j}^{2}+\sum_{i,j}t_{ij}^{2}\leq 3$ is still
necessary to describe a state, it is not sufficient any longer, see Ref.
Englert and Metwally (2001); Gamel (2016) for necessary and sufficient
criteria.

##  IV A three-dimensional model for two qubits

We have seen that every two-qubit state can be expanded in the product Pauli
basis and thus be identified with some finite convex region in
$\mathbb{R}^{15}$. While points of this object are bijectively mapped to
quantum states, the high dimensionality limits its usefulness. Moreover, a
full description of the quantum state is often not desired, rather than
general contraints on physically relevant quantities, such as purities and
entropies. In the following we focus on a more restricted set of parameters to
construct a lower-dimensional model of the quantum state space of a two-qubit
system. We compress the local information of each subsystem into a single
coordinate. The state of a qubit is characterized uniquely up to local
unitaries by its purity, or alternatively, the length of its Bloch vector.
With this choice we obtain our first two coordinates

$\displaystyle x=\|\mathbf{a}\|:=\sqrt{\sum_{i}a_{i}^{2}},\quad
y=\|\mathbf{b}\|:=\sqrt{\sum_{j}b_{j}^{2}},$ (15)

as the local Bloch vectors lengths, i.e., the length of the Bloch vector of
the local state in the corresponding subsystem. Finally, the last coordinate
describes the strength of (classical and quantum) correlation between the
local states. As the third coordinate we choose

$\displaystyle z=\|\mathbf{T}_{AB}\|:=\sqrt{\sum_{i,j}t_{ij}^{2}},$ (16)

the remaining length of the global Bloch vector not encoded in the local Bloch
vectors, which we call the _correlation tensor norm_ $\|\mathbf{T}_{AB}\|$.

With these reduced coordinates we can describe the set of points
$(\|\mathbf{a}\|,\|\mathbf{b}\|,\|\mathbf{T}_{AB}\|)$ attainable by quantum
states.

Besides positivity of the entries, the set is completely characterized by the
inequalities

$\displaystyle\|\mathbf{T}_{AB}\|\geq$ $\displaystyle\
\|\mathbf{a}\|+\|\mathbf{b}\|-1$ (17)
$\displaystyle\|\mathbf{T}_{AB}\|^{2}\leq$ $\displaystyle\
3+\|\mathbf{a}\|^{2}+\|\mathbf{b}\|^{2}-4\|\mathbf{a}\|\|\mathbf{b}\|-4\big{|}\|\mathbf{a}\|-\|\mathbf{b}\|\big{|}.$
(18)

Figure 2: The model of the Bloch body $\mathcal{Q}_{2\times 2}$, depending on
the local Bloch lengths $\|\mathbf{a}\|$, $\|\mathbf{b}\|$ and the correlation
tensor length $\|\mathbf{T}_{AB}\|$. It is restricted from below by Eq. (17)
in blue and from above by Eq. (18) in orange.

Eq. (17) was proven in Ref. Morelli  _et al._ (2020) and Eq. (18) is
equivalent to Eq. (1). Eq. (17) cuts out the lower corner of the Bloch model,
depicted in blue in Fig. 2. It is saturated by states of the form

$\rho_{\text{lb}}(p,q)=p\ |0\rangle\langle 0|\otimes\mathds{1}/2+q\
\mathds{1}/2\otimes|0\rangle\langle 0|+(1-p-q)|00\rangle\langle 00|.$ (19)

Eq. (18) gives an upper bound, shown in orange in Fig. 2. It is saturated by
states of the form

$\displaystyle\rho_{\text{ub}}(p,q)=\frac{1}{2}\scalebox{0.8}{$\begin{pmatrix}pq&0&0&p\sqrt{q(1-q))}\\\
0&(1-p)&0&0\\\ 0&0&0&0\\\ p\sqrt{q(1-q))}&0&0&p(1-q)\end{pmatrix}$}.$ (20)

The coordinates are length preserving, the length of the global Bloch vector
is $\sqrt{\|\mathbf{a}\|^{2}+\|\mathbf{b}\|^{2}+\|\mathbf{T}_{AB}\|^{2}}$, so
the distance from the origin in our model remains the Bloch vector length. The
resulting shape is convex with flat and curved surfaces. Since for pure states
$\|\mathbf{a}\|=\|\mathbf{b}\|$, all pure states lie on a circular arc of
radius $\sqrt{3}$ spanning from $(0,0,\sqrt{3})$, representing maximally
entangled states, to $(1,1,1)$, representing product states. This forms the
ridge of the model shown in Fig. 2. The maximally mixed state
$\mathds{1}_{4}/4$ is mapped to the origin. The model has three ”artificial”
boundaries, since the norms we consider are always positive. In this sense,
the boundaries where one coordinate is zero are not considered surfaces of the
model.

The upper boundary consists of rank-$2$ states, they turn out to be exactly
the states maximizing the entanglement for given mixedness of the subsystems,
see section VI. The lower boundary consists of states of rank $3$. So all
boundaries are occupied only by states of reduced rank. The converse does not
hold, i.e., there are states of rank $2$ and rank $3$ that are mapped to the
interior. However, all the states that map to the interior of the ball with
radius $1/3$ around the origin are of full rank.

The classical probabilities of a two-bit system, corresponding to diagonal
states, form a 3-dimensional subset in the quantum states space. The classical
probability space is described by the simplex $\mathbf{S}_{3}$, also known as
a tetrahedron. In the parametrization with the Pauli matrices
$(\langle\sigma_{z}\otimes\mathds{1}_{2}\rangle,\langle\mathds{1}_{2}\otimes\sigma_{z}\rangle,\langle\sigma_{z}\otimes\sigma_{z}\rangle)$,
this tetrahedron corresponds to the convex hull of the points $(1,1,1)$,
$(-1,-1,1)$, $(-1,1,-1)$ and $(1,-1,-1)$. The intersection with the positive
octant is inscribed in our model, described by the convex hull of the points
$(1,1,1)$, $(1,0,0)$, $(0,1,0)$, $(0,0,1)$ and $(0,0,0)$. Looking closely at
Eq. (1) one notices a remarkable fact. The inequality is saturated everywhere
and it depends only on the difference of the local Bloch lengths. But since
the trace is invariant under unitary transformations, this means that all
boundary states with equal local Bloch length difference have the same
distance from the origin. In fact we notice that the states on the lower edge
$\tilde{\rho}_{\text{lt}}(p,0)=p\ |00\rangle\langle 00|+(1-p)\
|0\rangle\langle 0|\otimes\mathds{1}/2$ form exactly the states on the
boundary, if rotated by the entangling unitary

$\displaystyle
U(\theta)=\scalebox{0.8}{$\begin{pmatrix}\cos{\theta}&0&0&-\sin{\theta}\\\
0&1&0&0\\\ 0&0&1&0\\\ \sin{\theta}&0&0&\cos{\theta}\end{pmatrix}$}\ \ .$ (21)

Therefore our model is obtained by rotating the classical probability space
about the axis $(1,-1,0)$, visualizing the fact that all quantum states can be
obtained by an appropriate unitary transformation of a positive and normalized
diagonal matrix.

##  V Entangled and separable regions

One interesting question is to locate entangled states in this picture. The
$z$-axis shows the correlation between the two subsystems. This correlation is
the sum of the classical and the quantum correlations, see Ref. Adesso  _et
al._ (2016). Although states mapping to the same point in the model have the
same amount of correlation, the respective parts of classical and quantum
correlation might differ considerably. Therefore separable and entangled
states will not map into two disjoint regions. Nevertheless there can be
regions where only entangled or only separable states map into.

Figure 3: Entangled and separable region in our model. The purely entangled
region defined by Eq. (22) is shown in orange, being mapped into this region
is already a sufficient condition for entanglement. In blue there is the
separable ball of radius $1/\sqrt{3}$, that is extended to the purely
separable region defined by Eq. (24) shown in green. For every point in the
area in between there exists both an entangled and a separable state mapping
to that point. So this are in fact the strongest criteria for separability and
entanglement that can be found in this model.

It was shown that the condition
$S_{L}(\rho_{AB})<\max\\{S_{L}(\rho_{A}),S_{L}(\rho_{B})\\}$ implies that the
state is entangled Horodecki and Horodecki (1996). Using
$S_{L}(\rho_{A/B})=1-1/2(1+\|\mathbf{a/b}\|^{2})$ and
$S_{L}(\rho_{AB})=1-1/4(1+\|\mathbf{a}\|^{2}+\|\mathbf{b}\|^{2}+\|\mathbf{T}_{AB}\|^{2})$
this can be translated into the correlation picture Imai  _et al._ (2021)

$\displaystyle\|\mathbf{T}_{AB}\|^{2}>1-\big{|}\|\mathbf{a}\|^{2}-\|\mathbf{b}\|^{2}\big{|}\
\ .$ (22)

This is a sufficient condition for entanglement, but it is not necessary. For
example, the Werner states Werner (1989) for two qubits,
$\rho_{\text{W}}(p)=p|\phi^{+}\rangle\langle\phi^{+}|+(1-p)\mathds{1}/4$ are
entangled for every $p>1/3$, or $\|\mathbf{T}_{AB}\|>1/\sqrt{3}$. Since
$\|\mathbf{a}\|=\|\mathbf{b}\|=0$, the inequality would only detect
entanglement for $\|\mathbf{T}_{AB}\|>1$. But nonetheless it is the largest
possible entangled region as, e.g., the state
$\rho_{\text{cc}}=1/2(|00\rangle\langle 00|+|11\rangle\langle 11|$ is
separable with $\|\mathbf{a}\|=\|\mathbf{b}\|=0$ and $\|\mathbf{T}_{AB}\|=1$.
It was shown that separable states can achieve equality in Eq. (22) for all
values of $\|\mathbf{a}\|$ and $\|\mathbf{b}\|$ Imai  _et al._ (2021). In Fig.
3 this region is shown in orange.

A sufficient condition for separability was found by the authors of Refs.
Życzkowski  _et al._ (1998); Gurvits and Barnum (2002) in the form of a
separable ball around the maximally mixed state. A state is separable if

$\displaystyle\|\mathbf{a}\|^{2}+\|\mathbf{b}\|^{2}+\|\mathbf{T}_{AB}\|^{2}\leq\frac{1}{3}\
\ .$ (23)

This relation defines a ball of separable states of radius $1/\sqrt{3}$ around
the origin, shown in blue in Fig. 3.

But this is not the largest separable set. In fact, the inequality

$\displaystyle\|\mathbf{T}_{AB}\|\leq\begin{cases}\sqrt{\tfrac{1}{6}(2-3(\|\mathbf{a}\|+\|\mathbf{b}\|)^{2})}\quad&\|\mathbf{a}\|+\|\mathbf{b}\|\leq\tfrac{2}{3}\\\
1-\|\mathbf{a}\|-\|\mathbf{b}\|&\tfrac{2}{3}<\|\mathbf{a}\|+\|\mathbf{b}\|\leq
1\\\
\|\mathbf{a}\|+\|\mathbf{b}\|-1&1<\|\mathbf{a}\|+\|\mathbf{b}\|.\end{cases}$
(24)

is a necessary condition for the separability of a state and the strongest
such condition formulated in terms of $\|\mathbf{a}\|$, $\|\mathbf{b}\|$ and
$\|\mathbf{T}_{AB}\|$. This extended region is shown in green in Fig. 3.

##  VI Maximally entangled mixed states

When trying to maximize the entanglement of a state under certain restrictions
two families of states are of particular interest. The first family consists
of states that maximize the entanglement for a fixed global purity, referred
to as maximally entangled mixed states (MEMS) Ishizaka and Hiroshima (2000);
Munro  _et al._ (2001); Wei  _et al._ (2003). For a two-qubit system they are
of the form

$\displaystyle\rho_{\scalebox{0.5}{MEMS}}(x,\theta)=\begin{pmatrix}x+\theta/2&0&0&\theta/2\\\
0&1-2x-\theta&0&0\\\ 0&0&0&0\\\ \theta/2&0&0&x+\theta/2\end{pmatrix}\ \ .$
(25)

The entanglement, quantified by the Wootters concurrence Hill and Wootters
(1997), becomes $\mathcal{C}(\rho_{\scalebox{0.5}{MEMS}})=\theta$ for these
states. For $0\leq\mathcal{C}(\rho_{\scalebox{0.5}{MEMS}})\leq 2/3$ it is
maximized by $x=1/3-\theta/2$ and in the model the states have coordinates
$(1/3,1/3,\sqrt{2\theta^{2}+1/9})$. For
$2/3\leq\mathcal{C}(\rho_{\scalebox{0.5}{MEMS}})\leq 1$ the states that
maximize the concurrence are those with $x=0$ and have coordinates
$(1-\theta,1-\theta,\sqrt{1-4\theta+6\theta^{2}})$. In Fig. 4 they are
depicted in red, where three special cases are highlighted: the state
$\rho^{\prime}=\rho_{\scalebox{0.5}{MEMS}}(1/3,0)$ is the only unentangled
state of this family located at the boundary of the separable region. The
state $\rho^{\prime\prime}=\rho_{\scalebox{0.5}{MEMS}}(0,2/3)$ connects the
two different pieces and incidentally also sits on the boundary of the
entangled region. Finally the state
$\rho_{\scalebox{0.5}{MEMS}}(0,1)=|\phi^{+}\rangle\langle\phi^{+}|$ maximizes
the overall entanglement.

Figure 4: Maximally entangled states with respect to some purity constraint.
The states maximizing the concurrence for a given purity (MEMS) are shown in
red, they start at the separable region with the state $\rho^{\prime}$ and
continue in a straight line to the state $\rho^{\prime\prime}$. Here they bend
and continue (not in a straight line) until the maximally entangled state. The
states maximizing the concurrence for a given mixedness of the marginals
(MEMMS) correspond to the states on the boundary of our model and are shown in
orange.

The other family of interest are maximally entangled states with fixed
marginal mixedness (MEMMS) Adesso  _et al._ (2003); Baio  _et al._ (2019). For
two qubits they are of the form of Eq. (20) and therefore form exactly the
upper boundary of our model, i.e., saturating Eq. (18). They are shown in
orange in Fig. 4.

##  VII Conclusion

We have introduced a novel inequality that relates local with global Bloch
vector lengths that is provably tight for qubits and provides deeper insight
for qudits. By visualizing the three relevant quantities we provide an
instructive representation of physically relevant properties in higher-
dimensional state-spaces. Beyond aesthetic and didactic value, we believe that
such simple relations will provide an important tool in the study of complex
many-body systems, where complete information about the quantum state is
unattainable for reasons of both experimental and theoretical complexity.
While our inequality is not necessarily tight in higher dimensions it remains
a valuable tool, and strengthening the inequality is an open problem that will
further broaden the application of this research program.

##  VIII Acknowledgements

S.M. acknowledges the Ministry of Science and Innovation: PID2020-112948GB-I00
funded by MCIN/AEI/10.13039/501100011033 and by ”ERDF A way of making Europe”
and the funding from the Basque Government’s IKUR initiative on Quantum
Technologies (Department of Education). M.H. acknowledges funding from the
European Research Council (Consolidator grant “Cocoquest” 101043705) and the
European Comission through the grant ’HyperSpace’ (101070168). J.S. was
supported by Grant PID2021-126273NB-I00 funded by
MCIN/AEI/10.13039/501100011033 and by ”ERDF A way of making Europe” as well as
by the Basque Government through Grant No. IT1470-22.

## References

  * Shannon (1948) C. E. Shannon,  _A mathematical theory of communication,_Bell Sys. Tech. J. 27, 379 (1948).
  * Araki (1970) H. Araki,  _Entropy Inequalities,_Commun. Math. Phys. 18, 160 (1970).
  * Lieb and Ruskai (1973) E. H. Lieb and M. B. Ruskai,  _Proof of the strong subadditivity of quantum-mechanical entropy,_J. Math. Phys. 14, 1938 (1973).
  * Pippenger (2003) N. Pippenger,  _The Inequalities of Quantum Information Theory,_IEEE Trans. Inf. Theory 49, 773 (2003).
  * Linden and Winter (2005) N. Linden and A. Winter,  _A new inequality for the von Neumann entropy,_Commun. in Math. Phys. 259, 129 (2005), arXiv:quant-ph/0406162.
  * Cadney  _et al._ (2012) J. Cadney, N. Linden, and A. Winter,  _Infinitely Many Constrained Inequalities for the von Neumann Entropy,_IEEE Trans. Inf. Theory 58, 3657 (2012), arXiv:1107.0624.
  * Coffman  _et al._ (2000) V. Coffman, J. Kundu, and W. K. Wootters,  _Distributed entanglement,_Phys. Rev. A 61 (2000), 10.1103/physreva.61.052306, arXiv:quant-ph/9907047.
  * Osborne and Verstraete (2006) T. Osborne and F. Verstraete,  _General Monogamy Inequality for Bipartite Qubit Entanglement,_Phys. Rev. Lett. 96, 220503 (2006), arXiv:quant-ph/0502176.
  * Eltschka  _et al._ (2009) C. Eltschka, A. Osterloh, and J. Siewert,  _Possibility of generalized monogamy relations for multipartite entanglement beyond three qubits,_Phys. Rev. A 80, 032313 (2009), arXiv:0904.1034.
  * Bai  _et al._ (2014) Y.-K. Bai, Y.-F. Xu, and Z. D. Wang,  _General monogamy relation for the entanglement of formation in multiqubit systems,_Phys. Rev. Lett. 113, 100503 (2014), arXiv:1401.3205.
  * Lancien  _et al._ (2016) C. Lancien, S. Di Martino, M. Huber, M. Piani, G. Adesso, and A. Winter,  _Should Entanglement Measures be Monogamous or Faithful?_Phys. Rev. Lett. 117, 060501 (2016), arXiv:1604.02189.
  * Regula  _et al._ (2016) B. Regula, A. Osterloh, and G. Adesso,  _Strong monogamy inequalities for four qubits,_Phys. Rev. A 93, 052338 (2016), arXiv:1604.03419.
  * Allen and Meyer (2017) G.W. Allen and D.A. Meyer,  _Polynomial Monogamy Relations for Entanglement Negativity,_Phys. Rev. Lett. 118, 080402 (2017), arXiv:1604.03419.
  * S. (2017) Camalet S.,  _Monogamy inequality for any local quantum resource and entanglement,_Phys. Rev. Lett. 119, 110503 (2017), arXiv:1704.03199.
  * Gour and Guo (2017) G. Gour and Y. Guo,  _Monogamy of entanglement without inequalities,_Quantum 2, 81 (2017), arXiv:1710.03295.
  * Eltschka (2018) J. Eltschka, C.and Siewert,  _Distribution of entanglement and correlations in all finite dimensions,_Quantum 2, 64 (2018), arXiv:1708.09639.
  * Eltschka  _et al._ (2018) C. Eltschka, F. Huber, O. Gühne, and J. Siewert,  _Exponentially many entanglement and correlation constraints for multipartite quantum states,_Phys. Rev. A 98, 052317 (2018), arXiv:1807.09165.
  * Bengtsson and Życzkowski (2017) I. Bengtsson and K. Życzkowski,  _Geometry of Quantum States: An Introduction to Quantum Entanglement, 2nd edition_ (Cambridge University Press, 2017).
  * Bengtsson  _et al._ (2012) I. Bengtsson, S. Weis, and K. Życzkowski,  _Geometry of the Set of Mixed Quantum States: An Apophatic Approach_ (Springer Basel, 2012) p. 175.
  * Klyachko (2006) A. A. Klyachko,  _Quantum marginal problem and $N$-representability, _J. Phys.: Conf. Ser. 36, 72 (2006), arXiv:quant-ph/0511102.
  * Yu  _et al._ (2021) X.-D. Yu, T. Simnacher, N. Wyderka, H. C. Nguyen, and O. Gühne,  _A complete hierarchy for the pure state marginal problem in quantum mechanics,_Nat. Commun. 12 (2021), arXiv:2008.02124.
  * Bravyi (2004) S. Bravyi,  _Requirements for compatibility between local and multipartite quantum states,_ Quant. Inf. Comp. 4, 12 (2004), arXiv:quant-ph/0301014.
  * Fano (1954) U. Fano,  _A stokes-parameter technique for the treatment of polarization in quantum mechanics,_Phys. Rev. 93, 121 (1954).
  * Fano (1957) U. Fano,  _Description of states in quantum mechanics by density matrix and operator techniques,_Rev. Mod. Phys. 29, 74 (1957).
  * Bloore (1976) F. J. Bloore,  _Geometrical description of the convex sets of states for systems with spin-1/2 and spin-1,_J. Phys. A 9, 2059 (1976).
  * Kimura (2003) G. Kimura,  _The bloch vector for n-level systems,_Phys. Lett. A 314, 339 (2003), arXiv:quant-ph/0301152.
  * Kurzyński (2011) P. Kurzyński,  _Multi-bloch vector representation of the qutrit,_Quantum Inf. Comp. 11, 361 (2011), arXiv:0912.3155.
  * Sarbicki and Bengtsson (2012) G. Sarbicki and I. Bengtsson,  _Dissecting the qutrit,_J. Phys. A: Math. Theor. 46, 035306 (2012), arXiv:1208.2118.
  * Goyal  _et al._ (2016) S. Goyal, B. N. Simon, R. Singh, and S. Simon,  _Geometry of the generalized bloch sphere for qutrits,_J. Phys. A: Math. Theor. 49, 165203 (2016), arXiv:1111.4427.
  * Sharma and Ghosh (2021) G. Sharma and S. Ghosh,  _Four-dimensional Bloch sphere representation of qutrits using Heisenberg-Weyl Operators,_arXiv:2101.06408 (2021).
  * Rau (2021) A. R. P. Rau,  _Symmetries and geometries of qubits, and their uses,_Symmetry 13, 1732 (2021), arXiv:2103.14105.
  * Eltschka  _et al._ (2021) C. Eltschka, M. Huber, S. Morelli, and J. Siewert,  _The shape of higher-dimensional state space: Bloch-ball analog for a qutrit,_Quantum 5, 485 (2021), arXiv:2012.00587.
  * Gamel (2016) O. Gamel,  _Entangled Bloch spheres: Bloch matrix and two-qubit state space,_Phys. Rev. A 93 (2016), arXiv:1602.01548.
  * Wie (2020) C.-R. Wie,  _Two-qubit bloch sphere,_Physics 2, 383 (2020), arXiv:2003.01699.
  * Zyczkowski (2022) K. Zyczkowski,  _Quantum version of the euler’s problem: a geometric perspective,_arXiv:2212.03903 (2022).
  * Englert and Metwally (2001) B.-G. Englert and N. Metwally,  _Remarks on 2–q-bit states,_Appl. Phys. B 72, 35 (2001), arXiv:quant-ph/0007053.
  * Morelli  _et al._ (2020) S. Morelli, C. Klöckl, C. Eltschka, J. Siewert, and M. Huber,  _Dimensionally sharp inequalities for the linear entropy,_Linear Algebra Appl. 584, 294 (2020), arXiv:1903.11887.
  * Adesso  _et al._ (2016) G. Adesso, M. Cianciaruso, and T. R. Bromley,  _An introduction to quantum discord and non-classical correlations beyond entanglement,_arxiv:1611.01959 (2016).
  * Horodecki and Horodecki (1996) R. Horodecki and M. Horodecki,  _Information-theoretic aspects of inseparability of mixed states,_Phys. Rev. A 54, 1838 (1996).
  * Imai  _et al._ (2021) S. Imai, N. Wyderka, A. Ketterer, and O. Gühne,  _Bound Entanglement from Randomized Measurements,_Phys. Rev. Lett. 126 (2021), arXiv:2010.08372.
  * Werner (1989) R. F. Werner,  _Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model,_Phys. Rev. A 40, 4277 (1989).
  * Życzkowski  _et al._ (1998) K. Życzkowski, P. Horodecki, A. Sanpera, and M. Lewenstein,  _Volume of the set of separable states,_Phys. Rev. A 58, 883–892 (1998).
  * Gurvits and Barnum (2002) L. Gurvits and H. Barnum,  _Largest separable balls around the maximally mixed bipartite quantum state,_Phys. Rev. A 66 (2002), arXiv:quant-ph/0204159.
  * Ishizaka and Hiroshima (2000) S. Ishizaka and T. Hiroshima,  _Maximally entangled mixed states under nonlocal unitary operations in two qubits,_Phys. Rev. A 62 (2000).
  * Munro  _et al._ (2001) W. J. Munro, D. F. V. James, A. G. White, and P. G. Kwiat,  _Maximizing the entanglement of two mixed qubits,_Phys. Rev. A 64 (2001), arXiv:quant-ph/0103113.
  * Wei  _et al._ (2003) T.-C. Wei, K. Nemoto, P. M. Goldbart, P. G. Kwiat, W. J. Munro, and F. Verstraete,  _Maximal entanglement versus entropy for mixed quantum states,_Phys. Rev. A 67 (2003), arXiv:quant-ph/0208138.
  * Hill and Wootters (1997) S. A. Hill and W. K. Wootters,  _Entanglement of a pair of quantum bits,_Phys. Rev. Lett. 78, 5022 (1997).
  * Adesso  _et al._ (2003) G. Adesso, F. Illuminati, and S. De Siena,  _Characterizing entanglement with global and marginal entropic measures,_Phys. Rev. A 68, 062318 (2003), quant-ph/0307192.
  * Baio  _et al._ (2019) G. Baio, D. Chruściński, P. Horodecki, A. Messina, and G. Sarbicki,  _Bounds on the entanglement of two-qutrit systems from fixed marginals,_Phys. Rev. A 99 (2019), arXiv:1904.07650.

##  Appendix A Appendix

Every state for with

$\displaystyle\|\mathbf{T}_{AB}\|\leq\begin{cases}\sqrt{\tfrac{1}{6}(2-3(\|\mathbf{a}\|+\|\mathbf{b}\|)^{2})}\quad&\|\mathbf{a}\|+\|\mathbf{b}\|\leq\tfrac{2}{3}\\\
1-\|\mathbf{a}\|-\|\mathbf{b}\|&\tfrac{2}{3}<\|\mathbf{a}\|+\|\mathbf{b}\|\leq
1\\\
\|\mathbf{a}\|+\|\mathbf{b}\|-1&1<\|\mathbf{a}\|+\|\mathbf{b}\|.\end{cases}$
(26)

is separable.

To see this, note that up to local unitary rotations every state has the form

$\displaystyle\frac{1}{4}\scalebox{1.0}{$\begin{pmatrix}1+a_{Z}+b_{Z}+c_{ZZ}&c_{XZ}-ic_{YZ}&c_{ZX}-ic_{ZY}&c_{XX}-c_{YY}-ic_{XY}-ic_{YX}\\\
c_{XZ}+ic_{YZ}&1+a_{Z}-b_{Z}-c_{ZZ}&c_{XX}+c_{YY}-ic_{XY}+ic_{YX}&-c_{ZX}+ic_{ZY}\\\
c_{ZX}+ic_{ZY}&c_{XX}+c_{YY}+ic_{XY}-ic_{YX}&1-a_{Z}+b_{Z}-c_{ZZ}&-c_{XZ}+ic_{YZ}\\\
c_{XX}-c_{YY}+ic_{XY}+ic_{YX}&-c_{ZX}-ic_{ZY}&-c_{XZ}-ic_{YZ}&1-a_{Z}-b_{Z}+c_{ZZ}\end{pmatrix}$},$
(27)

where $a_{Z},b_{Z}\geq 0$.

Consider first the case $a_{Z}+b_{Z}>1$. Since a non-zero off-diagonal element
is a necessary condition for entanglement and $c_{ZZ}\geq a_{Z}+b_{Z}-1$,
states satisfying $\|\mathbf{T}_{AB}\|\leq\|\mathbf{a}\|+\|\mathbf{b}\|-1$ are
separable. Negativity under partial transpose is a necessary and sufficient
condition for entanglement of two qubits. Let $\rho^{\Gamma}$ be the partial
transpose of $\rho$. Since $|\rho_{12}|=|\rho_{34}|$ and
$|\rho_{13}|=|\rho_{24}|$ and both quantities remain unchanged under the
partial transposition, separability of a state is equivalent to the following
two inequalities

$\displaystyle(1+c_{ZZ})^{2}-(a_{Z}+b_{Z})^{2}$
$\displaystyle\geq(c_{XX}+c_{YY})^{2}+(c_{XY}-c_{YX})^{2}$
$\displaystyle(1-c_{ZZ})^{2}-(a_{Z}-b_{Z})^{2}$
$\displaystyle\geq(c_{XX}-c_{YY})^{2}+(c_{XY}+c_{YX})^{2}\ .$

Hence the stronger conditions

$\displaystyle(1+c_{ZZ})^{2}-(a_{Z}+b_{Z})^{2}$ $\displaystyle\geq
2(c_{XX}^{2}+c_{YY}^{2}+c_{XY}^{2}+c_{YX}^{2})$
$\displaystyle(1-c_{ZZ})^{2}-(a_{Z}-b_{Z})^{2}$ $\displaystyle\geq
2(c_{XX}^{2}+c_{YY}^{2}+c_{XY}^{2}+c_{YX}^{2})\ ,$

are sufficient for separability and finally also

$\displaystyle\|\mathbf{T}_{AB}\|^{2}\leq\tfrac{1}{2}\pm
c_{ZZ}+\tfrac{3}{2}c_{ZZ}^{2}-\tfrac{1}{2}(a_{Z}\pm b_{Z})^{2}$ (28)

for both signs. Choosing $c_{ZZ}$ negative we can just take the positive sign.
Minimizing over $c_{ZZ}$ gives $c_{ZZ}=-1/3$ for $a_{Z}+b_{Z}\leq 2/3$ and
$c_{ZZ}=a_{Z}+b_{Z}-1$ for $a_{Z}+b_{Z}>2/3$, resulting in
$\|\mathbf{T}_{AB}\|\leq\sqrt{\tfrac{1}{6}(2-3(\|\mathbf{a}\|+\|\mathbf{b}\|)^{2})}$
and $\|\mathbf{T}_{AB}\|\leq 1-\|\mathbf{a}\|-\|\mathbf{b}\|$ respectively.

To see that this is indeed the optimal bound, choose $c_{ZZ}=a_{Z}+b_{Z}-1$
and $c_{XX}=c_{YY}=\epsilon/2$. This state is entangled for every
$\epsilon\geq 0$. For the case $a_{Z}+b_{Z}\leq 2/3$ take $c_{ZZ}=1/3$ and
$c_{XX}=c_{YY}=1/9-(a_{Z}+b_{Z})^{2}/4+\epsilon\leq 1/9-(a_{Z}-b_{Z})^{2}/4$,
which is an entangled state for all $\epsilon>0$.