# Accelerating Structural Optimization through Fingerprinting Space
Integration on the Potential Energy Surface

Shuo Tao  Department of Physics, Rutgers University, Newark, NJ 07102, United
States of America  Xuecheng Shao  Department of Physics, Rutgers University,
Newark, NJ 07102, United States of America  Li Zhu  Department of Physics,
Rutgers University, Newark, NJ 07102, United States of America
<EMAIL_ADDRESS>
###### Abstract

Structural optimization has been a crucial component in computational
materials research, and structure predictions have relied heavily on this
technique in particular. In this study, we introduce a novel method that
enhances the efficiency of local optimization by integrating an extra
fingerprint space into the optimization process. Our approach utilizes a mixed
energy concept in the hyper potential energy surface (PES), combining real
energy and a newly introduced fingerprint energy derived from the symmetry of
local atomic environment. This method strategically guides the optimization
process toward high-symmetry, low-energy structures by leveraging the
intrinsic symmetry of atomic configurations. The effectiveness of our approach
was demonstrated through structural optimizations of silicon, silicon carbide,
and Lennard-Jones cluster systems. Our results show that the fingerprint space
biasing technique significantly enhances the performance and probability of
discovering energetically favorable, high-symmetry structures, as compared to
conventional optimizations. The proposed method is anticipated to streamline
the search for new materials and facilitate the discovery of novel,
energetically favorable configurations.

###### keywords:

structure prediction, local optimization, fingerprint

DFT, PES, PSO, GA, AI, GTOs, MLPs, EPs, GOM, SCM, PCA

The design and discovery of novel materials has been a cornerstone in the
modern technological advancements, where computational materials science
emerges as a critical pillar, driving forward the innovative quest for new
materials. Central to this computational endeavor is the prediction of the
structure of materials, a fundamental information in understanding their
properties and functionalities at the atomic scale. Structure prediction
methods, such as CALYPSO1, 2, USPEX3, XtalOpt4, basin hopping5, minima
hopping6, and random structure search7, therefore, play a critical role in
navigating this challenge, forming the backbone of material exploration and
discovery. To predict a new (meta-)stable material, we need to study its
thermodynamic stability with respect to numerous local minima on a high-
dimensional potential energy space (PES) in the first place. However, finding
the most thermodynamically stable structure of a large assembly of atoms is a
very difficult problem, because the number of minima on the PES of a large
system increases exponentially with the number of atoms 8, 9, 10.

To surmount this obstacle, several methods have been developed to improve the
capabilities of structure searching methods. One approach is to introduce a
bias into the search space, targeting the high symmetry structures that are
more likely to be the global minimum. It is extensively studied in both
crystal 11 and cluster 12 systems that structures with high symmetry tend to
have either very low or very high energy (Pauling’s “rule of parsimony”). This
may explain why most of the ground state structures that exist in nature have
high symmetry. It would be feasible if one could take advantage of this
symmetry bias during the local optimization process and hence reduce the
computation time to investigate the local minima residing in the high energy
funnels. For example, Shao et al. 13 introduced a symmetry tree graph coupled
with an artificial-intelligence-based symmetry selection strategy, markedly
simplifying the problem of crystal structure prediction by bypassing the
exploration of the intricate low-symmetry subspace within the entire search
space. A recent work 14 shows that by explicitly introducing a symmetry biased
penalty function to the PES, high-symmetry structures will be found much
faster than on an unbiased surface when performing structure predictions using
the minima hopping method.

While these global optimization methods vary in their specific structure
searching approaches, they all rely on the integrative calling of the local
optimization process. However, local optimization methods often involve
lengthy computation times and high levels of complexity. Numerous local energy
minima, many of which have high energy levels, can impede the optimization
process. This impediment is especially noticeable in strong covalent systems
like carbon and silicon. Once covalent bonds are formed, they pose significant
barriers to reconfiguration. Thus, even if more energetically favorable
structures are within reach, accessing them remains challenging, trapping the
system in a potentially energy-intensive state. This situation has sparked
interest in more efficient methods capable of accelerating this process. For
example, by starting a structure stochastically generated in a higher-
dimensional space (hyperspace), the structure can be relaxed in the additional
dimension, which has been shown to be effective in enhance the probability of
reaching low-energy configurations 15.

In this work, we introduce a new method to enhance the efficiency of local
optimization by introducing an extra symmetry space. Rather than starting the
structural optimizations initiated from a structure stochastically generated
in hyperspace, we propose a new implicit approach to infuse symmetry
information into the process of local optimization. This strategy, designed to
prevent configurations from stagnating in high-energy states, is realized
through the introduction of the fingerprint energy that servers as the
indicator of the symmetry of structures. This performance boost is anticipated
to be advantageous for PES exploration methods that rely on the local
optimization of structures. Therefore, the work provides a path toward the
objective of predicting the material structure of complex systems.

In our approach, we introduce the fingerprint space to define the mixed energy
($E_{\text{mixed}}$) on the hyper PES as follows:

$\displaystyle
E_{\text{mixed}}=\omega_{\text{fp}}E_{\text{fp}}+\omega_{\text{real}}E_{\text{real}},$
(1)

where $\omega_{\text{fp}}$ and $\omega_{\text{real}}$ are the mixing weights,
and $E_{\text{real}}$ represents the real energy on the normal PES, as
described by methods such as density functional theory (DFT), machine-learning
potentials, or empirical potentials. $E_{\text{fp}}$ is the fingerprint energy
on the fingerprint PES, which is defined based on the sum of the Euclidean
norm of all pairwise atomic fingerprint vector16 distances within the crystal
structure:

$\displaystyle
E_{\text{fp}}=\eta\sum_{i=1}^{N_{\text{at}}}\sum_{j>i}^{N_{\text{at}}}\left\|\mathbf{fp}_{i}-\mathbf{fp}_{j}\right\|^{2},$
(2)

where $N_{\text{at}}$ denotes the number of atoms in the unit cell, and
$\mathbf{fp}_{i}$ and $\mathbf{fp}_{j}$ are the atomic fingerprint vectors for
the atoms indexed by $i$ and $j$, respectively. The coefficient $\eta$ is a
scaling factor utilized to equate the units of our fingerprint energy to the
corresponding physical units. The atomic fingerprint vectors
($\mathbf{fp}_{i}$) are employed to characterize the chemical environment of
atom ($i$) within structures. These vectors are derived from the eigenvalues
of the localized Gaussian overlap matrix17, 16, a method proven to be both
efficient and reliable in distinguishing various atomic environments.16, 18,
19

The concept of symmetry in crystal structures is intimately linked to the
similarity of local environments of atoms within the structure. In high-
symmetry structures, atoms typically share similar local environments,
reflecting a uniformity and orderliness in their spatial arrangement.
Conversely, low-symmetry structures are characterized by a diversity in the
local environments of atoms, indicating a more irregular and varied atomic
arrangement. This contrast in local environments serves as a fundamental
indicator of the overall symmetry of a structure. The degree of similarity in
the local environments of different atoms can be effectively quantified to
gauge the symmetry of a structure. Here, atomic fingerprints emerges as an
efficient tool. By assessing the differences in atomic fingerprints across a
structure to obtain a direct measure of its symmetry, the fingerprint energy
becomes an effective metric. Lower fingerprint energy corresponds to higher
structural symmetry, indicating a more uniform distribution of atomic
environments throughout the crystal. A structure where all atomic environments
are identical, such as in a diamond structure, would exhibit a fingerprint
energy of zero, epitomizing perfect symmetry. Therefore, the process of
minimizing fingerprint energy through a tailored fingerprint force-field
becomes a strategic approach to optimizing crystal structures towards enhanced
symmetry. This methodology not only provides a clear path to achieving higher-
symmetry configurations but also offers a nuanced understanding of the
underlying symmetry in complex crystal structures.

In order to effectively introduce the fingerprint space during the
optimization process, a specific parameterization is incorporated into the
mixing strategy, ensuring that the transition towards physically realistic
structures is informed by both the exploratory freedom of hyperspace and the
the actual features of the target structure. In the early stages of structural
optimization, structures are free to explore in the fingerprint landscape,
unconstrained by the typical limitations of normal structural space. This
initial phase is crucial for avoiding potential traps and exploring a broader
range of conformational possibilities. However, as the optimization process
advances, it becomes imperative to guide these structures towards a final
configuration that resides entirely in the normal space. To smoothly
transition from the expansive exploration of hyperspace to the final,
physically realistic structures, the mixing weights are chosen to as follows:

$\displaystyle\begin{split}&\omega_{\text{fp}}=\mathrm{\Theta}(1-x_{\text{iter}});\\\
&\omega_{\text{real}}=0.5\mathrm{\Theta}(x_{\text{iter}}-1)\left(\sin\left(-\frac{\pi}{2}+9\pi
x_{\text{iter}}^{2}\right)+1\right),\end{split}$ (3)

where $\mathrm{\Theta}$ represents the Heaviside step function, and
$x_{\text{iter}}=n_{\text{iter}}/n_{\text{max}}$. Here, $n_{\text{iter}}$
denotes the current step of the structural relaxation, and $n_{\text{max}}$ is
a user-defined integer representing the maximum number of structural
relaxations before deactivating the fingerprint space.

To implement the structural optimization in the mixed space, it is important
to calculate the forces of each atom in guiding the structures to an
energetically favorable configuration. The atomic forces on the mixed PES can
be obtained from taking the partial derivative of the $3N$ Cartesian
coordinates (where $N$ is the total number of atoms in the system) with
respect to mixed energy,

$\displaystyle\bm{F}_{\text{mixed}}$
$\displaystyle=-\omega_{\text{fp}}\frac{\partial}{\partial\bm{x}}E_{\text{fp}}-\omega_{\text{real}}\frac{\partial}{\partial\bm{x}}E_{\text{real}}.$
(4)

The derivatives of the real energy will be obtained from the DFT or force
field calculations, while the derivatives of the fingerprint energy can be
defined as follows:

$\displaystyle\frac{\partial}{\partial x_{k}^{D}}E_{\text{fp}}$
$\displaystyle=2\eta\sum_{i=1}^{N_{\text{at}}}\sum_{j>i}^{N_{\text{at}}}\left(\mathbf{fp}_{i}-\mathbf{fp}_{j}\right)^{\intercal}\cdot\left(\frac{\partial\mathbf{fp}_{i}}{\partial
x_{k}^{D}}-\frac{\partial\mathbf{fp}_{j}}{\partial x_{k}^{D}}\right),$ (5)

where $x_{k}^{D}$ denotes the $k$-th Cartesian coordinate of the $D$-th atom,
and the derivatives of the atomic fingerprint are calculated using the
Hellmann-Feynman theorem (see details in Supporting Information). For the
crystal structures, the Cauchy stress tensor is calculated using finite
difference method, $\sigma_{\alpha\beta}=-\frac{1}{\Omega}\frac{\partial
E_{fp}}{\partial\epsilon_{\alpha\beta}}$, where $\epsilon_{\alpha\beta}$ are
the elements of the strain tensor. The stress tensor elements
$\sigma_{\alpha_{\beta}}$ are defined as the volume normalized negative strain
derivatives of the fingerprint energy.

Our methodology was implemented in Python 3, utilizing the Atomic Simulation
Environment (ASE) 20 to interface with various calculators to perform energy
and force calculations at the level of DFT or force fields. To demonstrate the
effectiveness of our approach, we performed structural optimizations on
randomly generated initial structures for both crystal and cluster systems.

For crystals, our focus was on silicon and silicon carbide systems, chosen for
their propensity to form complex allotropes and compounds. This leads to
intricate and challenging PES landscapes, making them suitable testbeds for
our method. This complexity arises from the intricate PES and small energy
variations relative to their global minima, primarily influenced by $sp^{2}$
and $sp^{3}$ hybridization 21, 22, 23, 24, 25. We generated initial random
structures (300 for each system) using the CALYPSO code 1, 2 without bias
towards specific space or point groups. For local optimization, we employed
the ASE built-in optimizer, FIRE (fast inertial relaxation engine) 26 for
simultaneous optimization of atomic positions and cell parameters. The energy
and force calculations for the real PES were performed using DFTB$+$ (Density
Functional Tight Binding method) 27 with pbc Slater-Koster parameterization
set 28, 29, 30 and 0.04 k-grid (equivalent to 0.04$\times 2\pi$Å-1 $k$-point
meshes). Figure 2 shows the comparison of the distribution of local minima
obtained through optimization on the real PES with those identified using the
mixed PES approach. We can find that the fingerprint space biasing technique
effectively shifts the energy landscape, promoting the emergence of high-
symmetry structures. For both the Si32 and Si16C16 systems, there is a
significant density of high-symmetry structures at lower enthalpy levels when
the optimization is guided by the fingerprint space, indicating the
probability of finding energetically more stable structures is significantly
enhanced. In addition to identifying lower-energy structures, our method also
successfully identified high-symmetry structures that were not found in the
optimization using the real PES. For example, the $I4/mmm$-Si32 structure
(FIG. 2(a)) and the $P4/mmm$-Si16C16 structure (FIG. 2(b)) were discovered
during the local optimization when relaxed on the mixed PES. This indicates
that our method can successfully identify high-symmetry structures that might
otherwise be missed or underrepresented in conventional optimizations.

We also performed structural optimizations on Lennard-Jones (LJ) and binary
Lennard-Jones (BLJ) clusters, which are commonly used as benchmark systems in
structure optimization methods due to their well-defined PES landscapes. Our
method was first tested on LJ38 and LJ75 clusters, known for their nontrivial
double-funnel energy landscapes 31. For the BLJ clusters, the parameters
$\sigma_{AA}=1.0$, $\sigma_{BB}=0.8$, $\epsilon_{AA}=1.0$,
$\epsilon_{BB}=0.64$ are chosen, with
$\sigma_{AB}=0.5\times\left(\sigma_{AA}+\sigma_{BB}\right)$ and
$\epsilon_{AB}=\sqrt{\epsilon_{AA}\epsilon_{BB}}$.32, 33 A total of 200 random
configurations were generated for each cluster using the CALYPSO code, and the
energy and force calculations were performed using the ASE built-in calculator
20. Figure 3 shows the comparison of the density of states (DOS) as a function
of the energy per atom for different cluster optimizations. We can find that
our method effectively shifts the energy landscape in a way that promotes the
emergence of low-energy configurations. For both the LJ and BLJ systems, there
is a significant density of low-energy structures when the search is guided by
the fingerprint space, indicating an enhanced probability of finding
energetically more stable structures.

To further investigate the efficiency of our method, we plot the trajectories
(FIG. 4) for LJ-38 and Si16C16 in the local optimization. Utilizing the FIRE
(fast inertial relaxation engine) 26 optimizer, we observe a notable
acceleration in the energy and force convergence when the relaxation process
is conducted within the mixed space. This is particularly evident in the case
of the silicon carbide system, where the convergence is markedly swift. For
the Si16C16 system, we encounter a funnel-like energy landscape which
typically induces sluggish convergence. In our case, the optimization in the
normal PES failed to converge within 5000 steps. However, the integration of
our fingerprint space dramatically alters this dynamic. It becomes evident
that within the initial 100 steps, the optimizer adeptly surpasses substantial
energy barriers. This capability is crucial, as it allows the system to access
lower energy minima in subsequent geometry relaxation steps. Our approach thus
serves as a testament to the power of incorporating hyper space strategies,
which promise to advance the field of computational material science.

In conclusion, we have developed a new method to utilize the intrinsic
symmetry of atomic structures to enhance the efficiency of local optimization.
By integrating a fingerprint energy derived from the symmetry of local atomic
environments into the potential energy surface, we have developed an approach
that guides the optimization process towards high-symmetry, low-energy
structures. This implicit symmetry bias, embedded in the hyperdimensional
optimization space, has been shown to accelerate the discovery of
energetically favorable configurations in both crystal and cluster systems,
thereby streamlining the search for new materials. The benchmark results
obtained from the application of our method to silicon, silicon carbide, and
Lennard-Jones cluster systems underscore its effectiveness. Notably, the
emergence of high-symmetry structures at lower energy levels and the rapid
convergence of energy and forces during optimization highlight the potential
to reveal previously underexplored regions of the potential energy landscape.
Overall, our method represents a step forward in the structure prediction
field, offering an efficient tool for the exploration of complex potential
energy surfaces. As we continue to refine and apply this approach to a wider
range of materials, we anticipate that it will become an important component
of the toolkit in computational material science.

(a)

(b)

Figure 1: Comparison of the distribution of local minima optimized on real PES
(red) and on fingerprint mixed PES (blue) for (a) Si32 crystal and (b) Si16C16
crystal at ambient conditions. Symmetry, plotted along the x-axis, is measured
by the number of symmetry operations that leave the structure invariant. The
opacity of the colored box indicates the relative abundance of meta-stable
structures corresponding to a given symmetry and enthalpy level.

(a)

(b)

Figure 2: (a) Top view (from c-axis) of the relaxed structure ($I4/mmm$-Si32)
in fingerprint space. The initial structure ($P4nc$-Si32) has 8 symmetry
operations, but after relaxation in fingerprint space, the structure exhibits
128 symmetry operations. Without fingerprint biasing, the initial structure
would be relaxed into a low symmetry ($P1$). (b) Top view (from c-axis) of the
relaxed structure ($P4/mmm$-Si16C16) in fingerprint space. While the initial
structure ($P\bar{4}2_{1}c$-Si16C16) has 8 symmetry operations, this
fingerprint-relaxed structure has 64 symmetry operations. Without fingerprint
biasing, the initial structure will be relaxed into $I\bar{4}2m$-Si16C16,
which has 16 symmetry operations. Brown and blue spheres represent C and Si
atoms, respectively. The blue cubic box denotes the unit cell.

Figure 3: Density of states (DOS) of the energy distribution of the optimized
random-generated clusters with (in salmon) or without (in blue) introducing
fingerprint space. For binary LJ clusters, we choose the chemical formula as
A13B20 in the BLJ-33 case and and A55B60 in the BLJ-115 case.

Figure 4: Trajectory plots for LJ-38 cluster (left) and Si16C16 crystal
(right) during local optimization process. The top panel shows the energy at
each local relaxation step. The bottom panel displays the maximum component of
the forces of all the atoms in the system at each local relaxation step. The
force convergence criteria (dashed line in the bottom panel) is set at
$1\times 10^{-3}$ in the corresponding units.

This work was supported by the National Science Foundation, Division of
Materials Research (NSF-DMR) under Grant No. 2226700, and startup funds of the
office of the Dean of SASN of Rutgers University-Newark. The authors
acknowledge the Office of Advanced Research Computing (OARC) at Rutgers, The
State University of New Jersey, for providing access to the Amarel cluster and
associated research computing resources that have contributed to the results
reported here.

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# Separability for mixed states with operator Schmidt rank two

Gemma De las Cuevas  Institute for Theoretical Physics, Technikerstr. 21a,
A-6020 Innsbruck, Austria  Tom Drescher  Department of Mathematics,
Technikerstr. 13, A-6020 Innsbruck, Austria  Tim Netzer  Department of
Mathematics, Technikerstr. 13, A-6020 Innsbruck, Austria

###### Abstract

The operator Schmidt rank is the minimum number of terms required to express a
state as a sum of elementary tensor factors. Here we provide a new proof of
the fact that any bipartite mixed state with operator Schmidt rank two is
separable, and can be written as a sum of two positive semidefinite matrices
per site. Our proof uses results from the theory of free spectrahedra and
operator systems, and illustrates the use of a connection between
decompositions of quantum states and decompositions of nonnegative matrices.
In the multipartite case, we prove that any Hermitian Matrix Product Density
Operator (MPDO) of bond dimension two is separable, and can be written as a
sum of at most four positive semidefinite matrices per site. This implies that
these states can only contain classical correlations, and very few of them, as
measured by the entanglement of purification. In contrast, MPDOs of bond
dimension three can contain an unbounded amount of classical correlations.

##  1 Introduction

Entanglement is an essential ingredient in many applications in quantum
information processing and quantum computation [NC00, HHHH09]. Mixed states
which are not entangled are called _separable_ , and they may contain only
classical correlations—in contrast to entangled states, which contain quantum
correlations. As many other problems in theoretical physics and elsewhere, the
problem of determining whether a state is entangled or not is NP-hard [Gur03,
Gha10]. This does not prevent the existence of multiple separability criteria
[HHHH09]. One example are criteria based on the rank of a bipartite mixed
state $0\leqslant\rho\in\mathcal{M}_{d_{1}}\otimes\mathcal{M}_{d_{2}}$ (where
$\mathcal{M}_{d}$ denotes the set of complex matrices of size $d\times d$, and
$A\geqslant 0$ denotes that $A$ is positive semidefinite) [KCKL00, HLVC00].
Another example is the realignment criterion [KW03, Rud03], which is based on
the operator Schmidt decomposition of the state.

Here, we focus on the _operator Schmidt rank_ of $\rho$, which is the minimum
$p$ so that

$\displaystyle\rho=\sum_{\alpha=1}^{p}A_{\alpha}\otimes B_{\alpha},$ (1)

where $A_{\alpha}\in\mathcal{M}_{d_{1}}$ and
$B_{\alpha}\in\mathcal{M}_{d_{2}}$, i.e. these matrices need not fulfill any
conditions of Hermiticitiy or positivity. Clearly, a product state (i.e. a
state without classical or quantum correlations) has $p=1$, but it is
generally not clear what kind of correlations a state with $p>1$ has.

The first main result of this paper is that, if a state has operator Schmidt
rank two, then it is separable. Moreover, it can be written as a sum of only
two positive semidefinite matrices at each site. Our proof is constructive and
gives a method to obtain the positive semidefinite matrices, as we will
explicitly show. To state our result formally, denote by $\textrm{PSD}_{d}$
the set of $d\times d$ positive semidefinite matrices.

######  Theorem 1 (Bipartite case).

Let $\rho$ be a positive semidefinite matrix
$0\leqslant\rho\in\mathcal{M}_{d_{1}}\otimes\mathcal{M}_{d_{2}}$ where
$d_{1},d_{2}$ are arbitrary, such that

$\rho=\sum_{\alpha=1}^{2}A_{\alpha}\otimes B_{\alpha}$

where $A_{\alpha}\in\mathcal{M}_{d_{1}}$ and
$B_{\alpha}\in\mathcal{M}_{d_{2}}$. Then $\rho$ is separable and can be
written as

$\rho=\sum_{\alpha=1}^{2}\sigma_{\alpha}\otimes\tau_{\alpha}$

where $\sigma_{\alpha}\in\mathrm{PSD}_{d_{1}}$ and
$\tau_{\alpha}\in\mathrm{PSD}_{d_{2}}$.

To the best of our knowledge, this result was first proven by Cariello in
[Car14, Lemma 4.5] (see also [Car17, Theorem 3.44] and this blog post [Joh]),
and extended to the case that one side is infinite dimensional in [Gie18,
Theorem 3.3]. (See also the related inequalities [Car14, Theorem 5.2, Theorem
5.3].) Here we provide a new proof for this result, using other techniques and
a new context. The techniques are based on free spectrahedra, which form a
relatively young field of study within convex algebraic geometry, and whose
connections with decompositions of quantum states are explored in [Net19]. In
particular, the core result of this paper is a relation between separable
states and minimal operator systems (Theorem 11), and its consequences
(Corollary 13), which we leverage to prove Theorem 1. (See Ref. [BN18] for
other recent connections between free spectrahedra and quantum information.)
The context of this work is a connection between decompositions of quantum
states and decompositions of nonnegative matrices presented in [DN19]. Theorem
1 is inspired by an analogous result for nonnegative matrices, and can be seen
as a generalisation thereof, as we will show in Section 4.3.

What happens in the multipartite case? For pure states, multipartite
entanglement features many novel properties with respect to the bipartite
case, see e.g. [SWGK18]. For a mixed state that describes the state of a spin
chain in one spatial dimension,
$0\leqslant\rho\in\mathcal{M}_{d_{1}}\otimes\mathcal{M}_{d_{2}}\otimes\cdots\otimes\mathcal{M}_{d_{n}}$,
the multipartite analogue of Eq. (1) is the Matrix Product Density Operator
(MPDO) form [VGRC04, ZV04],

$\displaystyle\rho=\sum_{\alpha_{1},\ldots,\alpha_{n-1}=1}^{D}A^{[1]}_{\alpha_{1}}\otimes
A^{[2]}_{\alpha_{1},\alpha_{2}}\otimes\cdots\otimes A^{[n]}_{\alpha_{n-1}},$

where $A^{[1]}_{\alpha}\in\mathcal{M}_{d_{1}}$,
$A^{[2]}_{\alpha,\beta}\in\mathcal{M}_{d_{2}}$, …,
$A^{[n]}_{\alpha}\in\mathcal{M}_{d_{n}}$. The minimal such $D$ is also called
the operator Schmidt rank of $\rho$. If, additionally, we require that each of
the local matrices is Hermitian, i.e.
$A^{[1]}_{\alpha}\in\mathrm{Her}_{d_{1}}$,
$A^{[2]}_{\alpha,\beta}\in\mathrm{Her}_{d_{2}}$, …,
$A^{[n]}_{\alpha}\in\mathrm{Her}_{d_{n}}$, where $\textrm{Her}_{d}$ denotes
the set of $d\times d$ Hermitian matrices, then the minimal such $D$ is called
the _Hermitian operator Schmidt rank_ of $\rho$ [DN19].

One of the problems of the MPDO form is that the amount of quantum or
classical correlations of the state (as measured by the entanglement of
purification [THLD02]) is not upper bounded by the operator Schmidt rank
[DSPGC13, DN19]. Instead, the amount of correlations is upper bounded by the
_purification rank_ of $\rho$, which is the smallest operator Schmidt rank of
a matrix $L$ that satisfies $\rho=LL^{\dagger}$, i.e. that makes explicit the
positivity of $\rho$ [DSPGC13, DN19]. Since the purification rank can be
arbitrarily larger than the operator Schmidt rank [DSPGC13], the latter
generally does not tell us anything about the correlations of the state. In
particular, there are families of states with operator Schmidt rank three
whose purification rank diverges [DSPGC13]. A similar thing happens for the
Hermitian operator Schmidt rank: while it is lower bounded by the operator
Schmidt rank, it does not allow us to upper bound the amount of correlations
of the state. The exception to this rule is again the trivial case of product
states, in which case the operator Schmidt rank, its Hermitian counterpart and
the purification rank are all one.

The second main result of this paper is that if a multipartite state has
Hermitian operator Schmidt rank two, then it is separable. Moreover, it admits
a separable decomposition of bond dimension two, which means that it can be
written as a sum of at most four positive semidefinite matrices per site.
Formally:

######  Theorem 2 (Multipartite case).

Let $\rho$ be a positive semidefinite matrix,
$0\leqslant\rho\in\mathcal{M}_{d_{1}}\otimes\mathcal{M}_{d_{2}}\otimes\cdots\otimes\mathcal{M}_{d_{n}}$
where $d_{1},d_{2},\ldots,d_{n}$ are arbitrary, such that

$\displaystyle\rho=\sum_{\alpha_{1},\ldots,\alpha_{n-1}=1}^{2}A^{[1]}_{\alpha_{1}}\otimes
A^{[2]}_{\alpha_{1},\alpha_{2}}\otimes\cdots\otimes A^{[n]}_{\alpha_{n-1}}$

where $A^{[1]}_{\alpha}\in\mathrm{Her}_{d_{1}}$,
$A^{[2]}_{\alpha,\beta}\in\mathrm{Her}_{d_{2}}$, …,
$A^{[n]}_{\alpha}\in\mathrm{Her}_{d_{n}}$. Then $\rho$ is separable and can be
written as

$\rho=\sum_{\alpha_{1},\ldots,\alpha_{n-1}=1}^{2}\sigma^{[1]}_{\alpha_{1}}\otimes\sigma^{[2]}_{\alpha_{1},\alpha_{2}}\otimes\cdots\otimes\sigma^{[n]}_{\alpha_{n-1}}$

where $\sigma^{[1]}_{\alpha}\in\mathrm{PSD}_{d_{1}}$,
$\sigma^{[2]}_{\alpha,\beta}\in\mathrm{PSD}_{d_{2}}$, $\ldots$,
$\sigma^{[n]}_{\alpha}\in\mathrm{PSD}_{d_{n}}$.

This situation is summarised in Table 1.

States | Product | Separable |
---|---|---|---
hosr | 1 | 2 | 3
puri-rank | 1 | 2 | unbounded
sep-rank | 1 | 2 | unbounded
Table 1: Multipartite mixed states $\rho$ with hermitian operator Schmidt rank
(cf. Definition 3) $\textrm{hosr}(\rho)=1$ are product states, and the
separable rank sep-rank (Definition 5) and purification rank puri-rank
(Definition 6) need also be 1. If $\textrm{hosr}(\rho)=2$ then $\rho$ must be
separable and the other two ranks also need be 2 (Theorem 2). If
$\textrm{hosr}(\rho)=3$ and $\rho$ is separable, then the other two ranks may
diverge with the system size [DSPGC13].

We remark that [Car14, Corollary 4.8] showed that if a state $\rho$ has tensor
rank 2, then it is separable. Recall that the tensor rank,
$\textrm{tsr}(\rho)$, is the minimal $D$ required to express $\rho$ as

$\rho=\sum_{\alpha=1}^{D}A_{\alpha}^{[1]}\otimes A_{\alpha}^{[2]}\otimes\ldots
A_{\alpha}^{[n]}.$

Theorem 2, in contrast, shows that if the Hermitian operator Schmidt rank of a
state $\rho$ is 2, then $\rho$ is separable and its separable rank is 2 (the
latter will be defined in Definition 5). Since both the hermitian operator
Schmidt rank (hosr) and the tensor rank (tsr) are lower bounded by the
operator Schmidt rank (osr),

$\textrm{osr}\leq\textrm{hosr},\quad\textrm{osr}\leq\textrm{tsr},$

our Theorem 2 does not follow from [Car14, Corollary 4.8] (For more relations
between the ranks of several types of tensor decompositions, see [DHN19]).
Note also that Theorem 2 does not only show that $\rho$ is separable but also
that the separable rank (see Definition 5) is 2.

This paper is structured as follows. In Section 2 we present the preliminary
material needed for this work. In Section 3 we prove the core result, Theorem
11. In Section 4 we focus on the bipartite case: we prove Theorem 1, provide a
method to obtain the separable decomposition, and explore several implications
of this result. In Section 5 we focus on the multipartite case: we prove
Theorem 2, and explore its implications. Finally, in Section 6 we conclude and
present an outlook.

##  2 Preliminaries

Here we present the preliminary material needed to prove the results of this
work. First we will define the relevant decompositions of mixed states
(Section 2.1), and finally define free spectrahedra and minimal operator
systems (Section 2.2).

We use the following notation. The set of positive semidefinite matrices,
complex Hermitian matrices, and complex matrices of size $r\times r$ is
denoted $\textrm{PSD}_{r}$, $\textrm{Her}_{r}$ and $\mathcal{M}_{r}$,
respectively. We denote that a matrix $A$ is positive semidefinite (psd) by
$A\geqslant 0$. The identity matrix is denoted by $I$. We will also often
ignore normalizations, as they do not modify any of the ranks [DN19]. For this
reason, we often refer to positive semidefinite matrices as states.

###  2.1 Decompositions of mixed states

Throughout this paper we consider a multipartite positive semidefinite matrix
$0\leqslant\rho\in\mathcal{M}_{d_{1}}\otimes\mathcal{M}_{d_{2}}\otimes\cdots\otimes\mathcal{M}_{d_{n}}$.
We now review the MPDO, the Hermitian MPDO form, the separable decomposition
and the local purification of $\rho$. They are all discussed and characterized
in Ref. [DN19].

######  Definition 3.

The _Matrix Product Density Operator (MPDO)_ form [VGRC04, ZV04] of $\rho$ is
given by

$\displaystyle\rho=\sum_{\alpha_{1},\ldots,\alpha_{n-1}=1}^{D}A^{[1]}_{\alpha_{1}}\otimes
A^{[2]}_{\alpha_{1},\alpha_{2}}\otimes\cdots\otimes A^{[n]}_{\alpha_{n-1}},$
(2)

where $A^{[l]}_{\alpha}\in\mathcal{M}_{d_{l}}$ for $l=1,n$, and
$A^{[l]}_{\alpha,\beta}\in\mathcal{M}_{d_{l}}$ for $1<l<n$. The minimum such
$D$ is called the operator Schmidt rank of $\rho$, denoted
$\textrm{osr}(\rho)$.

The MPDO form can also be defined for matrices which are not Hermitian or
positive semidefinite. In this case, it is simply called the Matrix Product
Operator form (since $\rho$ is no longer a density operator), and the
corresponding minimum dimension is also called the operator Schmidt rank.

We also need to consider a modified version of the MPDO form, in which the
local matrices are Hermitian.

######  Definition 4.

The _Hermitian Matrix Product Density Operator (Hermitian MPDO)_ form [DN19]
of $\rho$ is given by

$\displaystyle\rho=\sum_{\alpha_{1},\ldots,\alpha_{n-1}=1}^{D}A^{[1]}_{\alpha_{1}}\otimes
A^{[2]}_{\alpha_{1},\alpha_{2}}\otimes\cdots\otimes A^{[n]}_{\alpha_{n-1}},$
(3)

where $A^{[l]}_{\alpha}\in\mathrm{Her}_{d_{l}}$ for $l=1,n$, and
$A^{[l]}_{\alpha,\alpha^{\prime}}\in\mathrm{Her}_{d_{l}}$ for $1<l<n$. The
minimum such $D$ is called the Hermitian operator Schmidt rank of $\rho$,
denoted $\textrm{hosr}(\rho)$.

In this paper, we say that $\rho$ is _separable_ if it admits a separable
decomposition. That this is a meaningful definition is shown in [DN19].

######  Definition 5.

A _separable decomposition_ of $\rho$ is a form where

$\displaystyle\rho=\sum_{\alpha_{1},\ldots,\alpha_{n-1}=1}^{D}\sigma^{[1]}_{\alpha_{1}}\otimes\sigma^{[2]}_{\alpha_{1},\alpha_{2}}\otimes\cdots\otimes\sigma^{[n]}_{\alpha_{n-1}}$
(4)

where each of these matrices is psd, i.e.
$\sigma^{[1]}_{\alpha}\in\mathrm{PSD}_{d_{1}}$,
$\sigma^{[n]}_{\alpha}\in\mathrm{PSD}_{d_{n}}$, and
$\sigma^{[j]}_{\alpha,\beta}\in\mathrm{PSD}_{d_{j}}$ for $1<j<n$. The minimal
such $D$ is called the separable rank of $\rho$, denoted $\textrm{sep-
rank}(\rho)$.

We now review the local purification form.

######  Definition 6.

The _local purification form_ of $\rho$ is an expression $\rho=LL^{\dagger}$
where $L$ is in Matrix Product Operator form. The minimal operator Schmidt
rank of such an $L$ is called the purification rank of $\rho$, denoted
$\textrm{puri-rank}(\rho)$:

$\textrm{puri-rank}(\rho)=\min\\{\textrm{osr}(L)|\rho=LL^{\dagger}\\}.$

Finally we review some results about these ranks.

######  Proposition 7 ([DN19]).

Let

$0\leqslant\rho\in\mathcal{M}_{d_{1}}\otimes\mathcal{M}_{d_{2}}\otimes\cdots\otimes\mathcal{M}_{d_{n}}.$

Then

  * (i)

$\textrm{osr}(\rho)=1\iff\textrm{puri-rank}(\rho)=1\iff\textrm{sep-
rank}(\rho)=1.$

  * (ii)

$\textrm{osr}(\rho)\leq\textrm{puri-rank}(\rho)^{2}$, and this bound is tight
for pure states.

  * (iii)

If $\rho$ is separable, then $\textrm{osr}(\rho)\leq\textrm{sep-rank}(\rho)$,
and $\textrm{puri-rank}(\rho)\leq\textrm{sep-rank}(\rho)$.

It is also known that there is a separation between $\textrm{osr}(\rho)$ and
$\textrm{puri-rank}(\rho)$, and between $\textrm{puri-rank}(\rho)$ and
$\textrm{sep-rank}(\rho)$ [DN19].

###  2.2 Free spectrahedra

Here we review the definitions of (free) spectrahedra, which are discussed
more thoroughly in [Net19], as well as the definition of minimal operator
system.

######  Definition 8.

A spectrahedron is a set of the following form

$\textrm{S}(A_{1},\ldots,A_{m})=\\{(b_{1},\ldots,b_{m})\in\mathbb{R}^{m}|\sum_{i=1}^{m}b_{i}A_{i}\geqslant
0\\}.$

where $A_{1},\ldots A_{m}\in\mathrm{Her}_{s}$ are Hermitian matrices.

The free spectrahedron is its non-commutative generalization:

######  Definition 9.

For $A_{1},\ldots,A_{m}\in\mathrm{Her}_{s}$ and $r\in\mathbb{N}$ we define

$\displaystyle\textrm{FS}_{r}(A_{1},\ldots,A_{m}):=$
$\displaystyle\\{(B_{1},\ldots,B_{m})\in{\rm Her}_{r}^{m}\mid$ (5)
$\displaystyle\sum_{i=1}^{m}A_{i}\otimes B_{i}\geqslant 0\\}$

and call it the $r$-level of the free spectrahedron defined by
$A_{1},\ldots,A_{m}$. The collection

$\textrm{FS}(A_{1},\ldots,A_{m}):=\left(\textrm{FS}_{r}(A_{1},\ldots,A_{m})\right)_{r\geq
1}$

is called the free spectrahedron defined by $A_{1},\ldots,A_{m}$.

Note that the free spectrahedron at level 1 coincides with the spectrahedron.

Finally we review the definition of minimal operator system (see, for example,
Ref. [FNT16] for more details).

######  Definition 10.

Let $C\subseteq\mathbb{R}^{m}$ be a closed salient convex cone (i.e. a closed
cone that does not contain a nontrivial subspace) with nonempty interior.
Define

$C_{r}^{\min}:=\left\\{\sum_{i=1}^{n}c_{i}\otimes P_{i}\mid
n\in\mathbb{N},c_{i}\in C,P_{i}\in\mathrm{PSD}_{r}\right\\}$

Note that since the $c_{i}$ are from $\mathbb{R}^{m}$, elements from
$C_{r}^{\min}$ can be understood as $m$-tuples of Hermitian matrices of size
$r$ (exactly as elements from the $r$-level of a free spectrahedron).

The minimal operator system is defined as

$C^{\min}=\left(C_{r}^{\min}\right)_{r\geq 1}.$

##  3 Separable states and minimal operator systems

In this section we prove the core result of this paper, namely Theorem 11,
which is a relation between separable states and minimal operator systems. We
will extract several consequences thereof in Corollary 13, which will be
exploited later on to prove Theorem 1 and Theorem 2.

######  Theorem 11.

Let $P_{1},\ldots,P_{m}\in{\rm Her}_{s}$ be $\mathbb{R}$-linearly independent,
and $Q_{1},\ldots,Q_{m}\in{\rm Her}_{t}$ with

$\displaystyle\rho=\sum_{i=1}^{m}P_{i}\otimes Q_{i}\geqslant 0.$ (6)

Define

$C:={\rm FS}_{1}(P_{1},\ldots,P_{m})\subseteq\mathbb{R}^{m}$

and

$V:={\rm span}_{\mathbb{R}}\left\\{P_{1},\ldots,P_{m}\right\\}\subseteq{\rm
Her}_{s}.$

Then the following are equivalent:

  1. (i).

$(Q_{1},\ldots,Q_{m})\in C^{\min}_{t}$

  2. (ii).

$\rho$ admits a separable decomposition

$\displaystyle\rho=\sum_{j=1}^{n}\tilde{P}_{j}\otimes\tilde{Q}_{j}$ (7)

with $\tilde{P}_{j}\in V$ for all $j$.

###### Proof.

(i)$\Rightarrow$(ii): By definition of $C^{\min}$ there are
$v_{1},\ldots,v_{n}\in C$ and $H_{1},\ldots,H_{m}\in{\rm Her}_{t}$ with all
$H_{i}\geqslant 0$ and

$Q_{i}=\sum_{j=1}^{n}v_{ji}H_{j}$

for all $i$. Substituting in the expression of $\rho$ (Eq. (6)), we obtain

$\rho=\sum_{j=1}^{n}\left(\sum_{i=1}^{m}v_{ji}P_{i}\right)\otimes H_{j}.$

From $v_{j}\in C={\rm FS}_{1}(P_{1},\ldots,P_{m})$ it follows that
$\sum_{i}v_{ji}P_{i}\geqslant 0$ for all $j$, which proves (ii).

(ii) $\Rightarrow$ (i): Assume that $\rho$ is of form (7) with all
$\tilde{P}_{j},\tilde{Q}_{j}\geqslant 0$ and

$\tilde{P}_{j}=\sum_{i=1}^{m}v_{ji}P_{i}$

for some $v_{j}=(v_{j1},\ldots,v_{jm})\in\mathbb{R}^{m}$. Substituting in the
expression of $\rho$ [Eq. (7)] we obtain that

$\rho=\sum_{i=1}^{m}P_{i}\otimes\left(\sum_{j=1}^{n}v_{ji}\tilde{Q}_{j}\right).$

Comparing this with expression (6) and using that $P_{1},\ldots,P_{m}$ are
linearly independent, we obtain

$Q_{i}=\sum_{j=1}^{n}v_{ji}\tilde{Q}_{j}$

for all $i$, i.e.

$(Q_{1},\ldots,Q_{m})=\sum_{j=1}^{n}v_{j}\otimes\tilde{Q}_{j}\in
C^{\min}_{t}.$

∎

######  Remark 12.

In Theorem 11, “(ii) $\Rightarrow$(i)” fails without the assumption
$\tilde{P}_{j}\in V$ for all $j$. To see this, consider the following example
taken from [HKM13]. Consider

$P_{1}=\left(\begin{array}[]{ccc}1&0&0\\\ 0&1&0\\\ 0&0&1\end{array}\right),\
P_{2}=\left(\begin{array}[]{ccc}0&1&0\\\ 1&0&0\\\ 0&0&0\end{array}\right),\
P_{3}=\left(\begin{array}[]{ccc}0&0&1\\\ 0&0&0\\\ 1&0&0\end{array}\right)$
$Q_{1}=\left(\begin{array}[]{cc}1&0\\\ 0&1\end{array}\right),\
Q_{2}=\left(\begin{array}[]{cc}\tfrac{1}{2}&0\\\ 0&0\end{array}\right),\
Q_{3}=\left(\begin{array}[]{cc}0&\tfrac{3}{4}\\\
\tfrac{3}{4}&0\end{array}\right)$

and

$\rho:=\sum_{i=1}^{3}P_{i}\otimes Q_{i}$

One easily checks that $\rho\geqslant 0$, and since all appearing matrices are
symmetric, $\rho$ passes the separability test of partial transposition
[HHHH09]. This test is however sufficient for separability in the $3\times 2$
case, so $\rho$ is separable.

Now consider the Pauli matrices

$\sigma_{z}=\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right),\quad\sigma_{x}=\left(\begin{array}[]{cc}0&1\\\
1&0\end{array}\right),$

and note that

$C={\rm FS}_{1}(P_{1},P_{2},P_{3})={\rm
FS}_{1}(I_{2},\sigma_{z},\sigma_{x})\subseteq\mathbb{R}^{3}$

is the circular cone, i.e. the cone defined by the real numbers $a,b,c$ such
that $b^{2}+c^{2}\leq a^{2}$, and $a\geq 0$. However, on the other hand,

$I_{2}\otimes Q_{1}+\sigma_{z}\otimes Q_{2}+\sigma_{3}\otimes Q_{3}\ngeqslant
0,$

which implies that $(Q_{1},Q_{2},Q_{3})\notin C^{\min}_{2}.$

######  Corollary 13.

Let $P_{1},\ldots,P_{m}\in{\rm Her}_{s}$ be $\mathbb{R}$-linearly independent,
and set

$C:={\rm FS}_{1}(P_{1},\ldots,P_{m})\subseteq\mathbb{R}^{m}.$

  1. (i).

If ${\rm FS}(P_{1},\ldots,P_{m})=C^{\min}$ then for any choice of
$Q_{1},\ldots,Q_{m}\in{\rm Her}_{t}$ with

$\rho=\sum_{i=1}^{m}P_{i}\otimes Q_{i}\geqslant 0,$

the state $\rho$ is separable.

  2. (ii).

If $C$ is a simplex cone, then for any choice of $Q_{1},\ldots,Q_{m}\in{\rm
Her}_{t}$ with

$\rho=\sum_{i=1}^{m}P_{i}\otimes Q_{i}\geqslant 0,$

the state $\rho$ is separable. Furthermore, it admits a separable
decomposition with at most $m$ terms, that is, $\textrm{sep-rank}(\rho)\leq
m$.

  3. (iii).

If $m=2$, then for any choice of $Q_{1},Q_{2}\in{\rm Her}_{t}$ with

$\rho=P_{1}\otimes Q_{1}+P_{2}\otimes Q_{2}\geqslant 0,$

the state $\rho$ is separable and admits a separable decomposition with at
most $2$ terms, i.e. $\textrm{sep-rank}(\rho)\leq 2$.

###### Proof.

(i): The condition $\rho\geqslant 0$ just means

$(Q_{1},\ldots,Q_{m})\in{\rm FS}_{t}(P_{1},\ldots,P_{m})=C_{t}^{\min}.$

So the result follows from Theorem 11.

(ii): From [FNT16] we know that if $C$ is a simplex cone, then

$C^{\min}={\rm FS}(P_{1},\ldots,P_{m})$

is automatically true. So the result follows from (i). Note that the number of
terms in the constructed separable decomposition is $m$, since in the proof of
Theorem 11 “(i) $\Rightarrow$ (ii)” we can choose the $v_{i}$ as the extreme
rays of $C$, of which there are $m$ many.

(iii): For $C={\rm FS}_{1}(P_{1},P_{2})\subseteq\mathbb{R}^{2}$ there are
three possibilities. First, $C$ might be a simplex cone, so the claim follows
from (ii). Second, $C$ might be a single ray. In this case one easily checks
that $\rho$ is in fact a product state, so $\textrm{sep-rank}(\rho)=1$.
Finally $C=\\{0\\}$ implies $\rho=0$, so everything is trivial. ∎

##  4 Bipartite states with operator Schmidt rank two

In this section we focus on bipartite states which have operator Schmidt rank
two. We will first prove Theorem 1 (Section 4.1), and then provide a method to
obtain the separable decomposition of such bipartite states (Section 4.2). The
method closely follows the proof and thereby illustrates our construction.
Then we will put Theorem 1 in the broader context of decompositions of
nonnegative matrices, and derive some implications for the ranks (Section
4.3). Finally we will derive further implications for quantum channels and the
entanglement of purification (Section 4.4).

###  4.1 Proof of Theorem 1

In this section we prove Theorem 1. The key missing element is Lemma 14, which
tells us that in the bipartite case we can force the local matrices to be
Hermititian without increasing the number of terms.

######  Lemma 14 (Hermiticity in the bipartite case).

Let $P_{1},\ldots,P_{r}\in\mathcal{M}_{s}$ and
$Q_{1},\ldots,Q_{r}\in\mathcal{M}_{t}$ be two sets of linearly independent
matrices such that

$\rho=\sum_{\alpha=1}^{r}P_{\alpha}\otimes Q_{\alpha}$

is Hermitian. Then $\rho$ can be expressed as

$\rho=\sum_{\alpha=1}^{r}A_{\alpha}\otimes B_{\alpha}$

where $A_{\alpha}\in\mathrm{Her}_{s}$ and $B_{\alpha}\in\mathrm{Her}_{t}$.

###### Proof.

First note that a Hermitian decomposition
$\rho=\sum_{\alpha=1}^{r^{\prime}}A_{\alpha}\otimes B_{\alpha}$ (with each
$A_{\alpha}$ and $B_{\alpha}$ Hermitian, and possibly $r^{\prime}\geq r$) can
always be found, since
$\mathrm{Her}_{s}\otimes\mathrm{Her}_{t}=\mathrm{Her}_{st}$. This holds simply
because $\mathrm{Her}_{s}\otimes\mathrm{Her}_{t}\subseteq\mathrm{Her}_{st}$
and the two vector spaces have the same dimension, namely $s^{2}t^{2}$.

To see that $r^{\prime}=r$, assume that
$\rho=\sum_{\alpha=1}^{r^{\prime}}A_{\alpha}\otimes B_{\alpha}$ is a minimal
Hermitian decomposition, i.e. where $\\{A_{\alpha}\\}_{\alpha}$ are linearly
independent over $\mathbb{R}$, and so are $\\{B_{\alpha}\\}_{\alpha}$. But
this implies that they are also linearly independent over $\mathbb{C}$, since
consider $\lambda_{\alpha}\in\mathbb{C}$ and the equation

$\sum_{\alpha}\lambda_{\alpha}A_{\alpha}=0.$

Summing and subtracting its complex conjugate transpose of this equation, we
see that the real and imaginary part of each $\lambda_{\alpha}$ needs to
vanish. Thus $r=r^{\prime}$. ∎

Now we are ready to prove Theorem 1. Let us first state the theorem again.

######  Theorem (Theorem 1).

Let $\rho$ be a bipartite state
$0\leqslant\rho\in\mathcal{M}_{d_{1}}\otimes\mathcal{M}_{d_{2}}$ where
$d_{1},d_{2}$ are arbitrary, such that

$\rho=\sum_{\alpha=1}^{2}A_{\alpha}\otimes B_{\alpha}$

where $A_{\alpha}\in\mathcal{M}_{d_{1}}$ and
$B_{\alpha}\in\mathcal{M}_{d_{2}}$. Then $\rho$ is separable and can be
written as

$\rho=\sum_{\alpha=1}^{2}\sigma_{\alpha}\otimes\tau_{\alpha}$

where $\sigma_{\alpha}\in\mathrm{PSD}_{d_{1}}$ and
$\tau_{\alpha}\in\mathrm{PSD}_{d_{2}}$.

###### Proof.

By Lemma 14, $\rho$ can be written with Hermitian local matrices, namely as

$\rho=\sum_{\alpha=1}^{2}P_{\alpha}\otimes Q_{\alpha}$

where $P_{\alpha}\in\mathrm{Her}_{d_{1}}$ and
$Q_{\alpha}\in\mathrm{Her}_{d_{2}}$. By Corollary 13 (iii), there is a
separable decomposition with only two terms,

$\rho=\sum_{\alpha=1}^{2}\sigma_{\alpha}\otimes\tau_{\alpha}.$ ∎

###  4.2 Method to obtain the separable decomposition

We now provide an explicit procedure to obtain the separable decomposition of
a bipartite state with operator Schmidt rank two. We remark that another such
procedure was provided in [Car17, Theorem 3.44]. The procedure follows the
proof of Theorem 1, as the latter is constructive. In the following we focus
on presenting the method, rather than on justifying why it works. At the end
of this section, we will apply this method to an example.

First, the input and output to this method are as follows:

Input

A decomposition of a bipartite positive semidefinite matrix

$\rho=\sum_{\alpha=1}^{2}P_{\alpha}\otimes Q_{\alpha}\geqslant 0.$

where the $P_{\alpha}$’s and the $Q_{\alpha}$’s are linearly
independent.111This could be, for example, the operator Schmidt decomposition
of $\rho$.

Output

$\sigma_{\alpha}\geqslant 0$ and $\tau_{\alpha}\geqslant 0$ such that

$\displaystyle\rho=\sum_{\alpha=1}^{2}\sigma_{\alpha}\otimes\tau_{\alpha}.$
(8)

The method works as follows:

Step 1

Making the matrices Hermitian.

  1. 1.

Express $P_{\alpha}=P_{\alpha}^{0}+iP_{\alpha}^{1}$, where $P_{\alpha}^{0}$
and $iP_{\alpha}^{1}$ are the Hermitian and antihermitian part of
$P_{\alpha}$, respectively222Namely
$P_{\alpha}^{0}=(P_{\alpha}+P_{\alpha}^{\dagger})/2$ and
$P_{\alpha}^{1}=-i(P_{\alpha}-P_{\alpha}^{\dagger})/2$., and similarly for
$Q_{\alpha}$.

  2. 2.

Find two matrices of the set $\\{P_{1}^{0},P_{1}^{1},P_{2}^{0},P_{2}^{1}\\}$
which are $\mathbb{R}$-linearly independent, i.e. that are not a multiple of
each other.

  3. 3.

Express the other two matrices as a real linear combination of the two
linearly independent matrices. Substitute this in $\rho$ and obtain an
expression of the form

$\displaystyle\rho=A\otimes C+B\otimes D\geqslant 0$ (9)

where each matrix is Hermitian.

Step 2

Finding the extreme rays $v,u\in\mathbb{R}^{2}$:

$\displaystyle\textrm{S}(A,B)$ $\displaystyle=$
$\displaystyle\\{(x,y)\in\mathbb{R}^{2}|xA+yB\geqslant 0\\}$ $\displaystyle=$
$\displaystyle\textrm{convex-cone}\\{v,u\\}\subseteq\mathbb{R}^{2}.$

Denote the $i$th diagonal element of $C$, $D$ by $C_{ii}$, $D_{ii}$,
respectively. Note that $C_{ii}A+D_{ii}B\geqslant 0$ by construction. There
are two cases:

  1. Case 1:

There is an $i$ such that $C_{ii}A+D_{ii}B>0.$ Since this is positive
definite, there is an invertible matrix $P$ such that
$P^{\dagger}(C_{ii}A+D_{ii}B)P=I$. Set

$\tilde{A}=P^{\dagger}AP,\quad\tilde{B}=P^{\dagger}BP$

and let $\lambda_{\min},\lambda_{\max}$ be the smallest and largest eigenvalue
of

$C_{ii}\tilde{B}-D_{ii}\tilde{A}.$

The extreme rays are

$u=-(\lambda_{\min}C_{ii}+D_{ii},\lambda_{\min}D_{ii}-C_{ii}),$
$v=(\lambda_{\max}C_{ii}+D_{ii},\lambda_{\max}D_{ii}-C_{ii}).$

  2. Case 2:

For all $i$, $C_{ii}A+D_{ii}B\ngtr 0$. Then $(C_{ii},D_{ii})$ is an extremal
ray for any $i$. If there are two $i$’s whose corresponding vectors
$(C_{ii},D_{ii})$ are linearly independent, then we have already found the two
extremal rays. If all $(C_{ii},D_{ii})$ are linearly dependent, then we only
have one extremal ray. In this case, consider

$(C^{\prime}_{ii},D^{\prime}_{ii})=(C_{ii},D_{ii})\pm\varepsilon(-D_{ii},C_{ii}).$

For small $\varepsilon$, the new point will be in the interior of $S(A,B)$.
The kernel of $C^{\prime}_{ii}A+D^{\prime}_{ii}B$ is then precisely the joint
kernel of $A$ and $B$. After splitting it off, the new matrices
$A^{\prime},B^{\prime}$ will define the same spectrahedral cone and
$C^{\prime}_{ii}A^{\prime}+D^{\prime}_{ii}B^{\prime}>0$. Hence we are in Case
1.

Note that it is very unlikely that Case 2 of Step 2 happens, as the set of all
$(C,D)\in\mathbb{R}^{2}$ satisfying $C\cdot A+D\cdot B\geqslant 0$ but $C\cdot
A+D\cdot B\ngtr 0$ is a set of measure zero. In most cases there will be an
$i$ which provides an interior point.

Step 3

Finding the positive semidefinite matrices $H_{1},H_{2}$ such that $A,B$

$\displaystyle C=u_{1}H_{1}+v_{1}H_{2},$ $\displaystyle
D=u_{2}H_{1}+v_{2}H_{2},$

where $u=(u_{1},u_{2})$ and $v=(v_{1},v_{2})$. Since $u,v$ are linearly
independent, this system of equations has a unique solution.

Substituting in (9) we obtain that the positive semidefinite matrices of (8)
are

$\displaystyle\sigma_{1}=u_{1}A+u_{2}B,\qquad\qquad\sigma_{2}=v_{1}A+v_{2}B$
$\displaystyle\tau_{1}=H_{1},\quad\tau_{2}=H_{2}.$

We now illustrate this procedure with a simple example, namely a two-qubit
state. In this case the negativity of the partial transpose is a sufficient
criterion for separability. We use the standard conventions for qubits, where
$I=|0\rangle\langle 0|+|1\rangle\langle 1|$, $\sigma_{z}=|0\rangle\langle
0|-|1\rangle\langle 1|$, and $\sigma_{x}=|0\rangle\langle 1|+|0\rangle\langle
1|$.

######  Example 15 (Two qubits correlated in $\sigma_{x}$ basis).

Consider the two-qubit state

$\rho=\frac{1}{2}(I\otimes I+\sigma_{x}\otimes\sigma_{x}).$

This has $\textrm{osr}(\rho)=2$ and is thus separable. In this case the
separable decomposition can be found by inspection, namely

$\displaystyle\rho=\frac{1}{2}(|+,+\rangle\langle+,+|+|-,-\rangle\langle-,-|),$
(10)

where $|\pm\rangle=(|0\rangle\pm|1\rangle)/\sqrt{2}$, which indeed has
$\textrm{sep-rank}(\rho)=2$.

We now apply the method above to this state to obain the separable
decomposition. From now on we ignore normalization, i.e. we consider $2\rho$.

Step 1

The matrices are already Hermitian.

Step 2

The spectrahedron defined by $I,\sigma_{x}$ is

$S(I,\sigma_{x})=\\{(a,b)|aI+b\sigma_{x}\geqslant 0\\}$

The extreme rays are $u=(1,-1)$ and $v=(1,1)$.

Step 3

The positive semidefinite matrices $H_{1},H_{2}$ that satisfy

$\displaystyle I=H_{1}+H_{2},\quad\sigma_{x}=-H_{1}+H_{2}$

are $H_{1}=|-\rangle\langle-|$ and $H_{2}=|+\rangle\langle+|$. Thus

$\displaystyle\sigma_{1}=|-\rangle\langle-|,\quad\sigma_{2}=|+\rangle\langle+|,$
$\displaystyle\tau_{1}=I-\sigma_{x},\quad\tau_{2}=I+\sigma_{x}$

which is indeed the decomposition of (10).

$\diamond$

###  4.3 Context and implications of Theorem 1

In this section we put Theorem 1 in the context of Ref. [DN19], which presents
a relation between decompositions of quantum states and decompositions of
nonnegative matrices. Namely, Ref. [DN19] and [DHN19] show that for bipartite
positive semidefinite matrices which are diagonal in the computational basis,

$\displaystyle\rho=\sum_{i,j}M_{i,j}|i,j\rangle\langle i,j|,$ (11)

certain decompositions of $\rho$ correspond to decompositions of the
nonnegative matrix333That is, entrywise nonnegative. A Hermitian matrix with
nonnegative eigenvalues is called positive semidefinite.

$\displaystyle M=\sum_{i,j}M_{i,j}|i\rangle\langle j|.$ (12)

In particular, the operator Schmidt decomposition, the separable
decomposition, and the local purification form of $\rho$ correspond to the
singular value decomposition, the nonnegative factorisation [Yan91], and the
positive semidefinite factorisation [FMP+12] of $M$, respectively. (There are
also correspondences between the translationally invariant (t.i.) versions, by
which the t.i. separable decomposition and the t.i. local purification form of
$\rho$ correspond to the completely positive factorisation [BDSM15] and the
completely positive semidefinite transposed factorisation of $M$ [LP15],
respectively). Moreover, each factorisation has an associated rank, and these
correspondences imply that the operator Schmidt rank of $\rho$ equals the rank
of $M$, the separable rank of $\rho$ equals the nonnegative rank of $M$, and
the purification rank of $\rho$ equals the positive semidefinite rank of $M$.
In a formula, if $\rho$ is of the form (11) and $M$ of the form (12), then

$\displaystyle\textrm{osr}(\rho)$ $\displaystyle=$
$\displaystyle\textrm{rank}(M)$ $\displaystyle\textrm{sep-rank}(\rho)$
$\displaystyle=$ $\displaystyle\textrm{rank}_{+}(M)$ (13)
$\displaystyle\textrm{puri-rank}(\rho)$ $\displaystyle=$
$\displaystyle\textrm{psd-rank}(M)$

Now, it is known that:

######  Proposition 16 ([FGP+15]).

Let $M$ be a nonnegative matrix. Then
$\textrm{rank}(M)=2\implies\textrm{rank}_{+}(M)=\textrm{psd-rank}(M)=2.$

Theorem 1 can be seen as a generalization of Proposition 16 to the case that
$\rho$ is not diagonal in the computational basis. And, as we will see in
Section 5.3, Theorem 2 can be seen as a generalization of Proposition 16 to
the multipartite case. The precise implications of Theorem 1 for the ranks of
bipartite positive semidefinite matrices are the following.

######  Corollary 17 (of Theorem 1).

Let $\rho$ be a bipartite positive semidefinite matrix,
$0\leqslant\rho\in\mathcal{M}_{d_{1}}\otimes\mathcal{M}_{d_{2}}$ where
$d_{1},d_{2}$ are arbitrary. Then the following are equivalent:

  1. (i)

$\textrm{osr}(\rho)=2.$

  2. (ii)

$\textrm{hosr}(\rho)=2.$

  3. (iii)

$\textrm{sep-rank}(\rho)=2.$

Moreover, they imply that:

  1. (iv)

$\textrm{puri-rank}(\rho)=2$ and $\rho$ is separable.

###### Proof.

(i) implies (ii): is shown in Lemma 14.

(ii) implies (i): since $\textrm{osr}(\rho)\leq\textrm{hosr}(\rho)$ we only
have to see that $\textrm{osr}(\rho)=1$ cannot hold. If $\textrm{osr}(\rho)=1$
then $\rho=A\otimes B$ for some matrices $A,B$, but since $\rho\geqslant 0$,
$A$ and $B$ must both be positive semidefinite or both negative semidefinite.
In either case they are both Hermitian, which shows that
$\textrm{hosr}(\rho)=1$, and contradicts the assumption
$\textrm{hosr}(\rho)=2$.

(i) implies (iii): is shown in Theorem 1.

(iii) implies (i): From Proposition 7 (iii) we have that
$\textrm{osr}(\rho)\leq\textrm{sep-rank}(\rho)$. But $\textrm{osr}(\rho)=1$
cannot hold, since this would imply that $\textrm{sep-rank}(\rho)=1$ by
Proposition 7 (i), which contradicts the assumption that $\textrm{sep-
rank}(\rho)=2$.

(iii) implies (iv): From Proposition 7 (iii) we have that $\textrm{puri-
rank}(\rho)\leq\textrm{sep-rank}(\rho)$, thus we only have to see that
$\textrm{puri-rank}(\rho)=1$ cannot be true. By Proposition 7 (i) this implies
that $\textrm{sep-rank}(\rho)=1$, which contradicts the assumption that
$\textrm{sep-rank}(\rho)=2$. ∎

This leaves the following situation for bipartite positive semidefinite
matrices. Note that we only need to analyse osr, since
$\textrm{osr}(\rho)=\textrm{hosr}(\rho)$ by Lemma 14.

  * •

If $\textrm{osr}(\rho)=1$, then $\rho$ is a product state, and
$\textrm{hosr}(\rho)=\textrm{sep-rank}(\rho)=\textrm{puri-rank}(\rho)=1$ by
Proposition 7 (i).

  * •

If $\textrm{osr}(\rho)=2$ then $\rho$ is a separable state, and
$\textrm{hosr}(\rho)=\textrm{sep-rank}(\rho)=\textrm{puri-rank}(\rho)=2$ by
Corollary 17.

  * •

Ref. [GPT13] provides a family of nonnegative matrices
$\\{M_{t}\in\mathbb{R}^{t\times t}_{\geq 0}\\}_{t\geq 3}$, for which
$\textrm{rank}(M_{t})=3$ for all $t$ but $\textrm{psd-rank}(M_{t})$ diverges
with $t$. Defining the bipartite state

$\rho_{t}=\sum_{i,j=1}^{t}(M_{t})_{i,j}|i,j\rangle\langle i,j|,$

and using relations (13), this immediately implies that there is a family
$\\{\rho_{t}\\}$ for which $\textrm{osr}(\rho_{t})=3$ for all $t$ but
$\textrm{puri-rank}(\rho_{t})$ diverges. By Proposition 7 (iii), this implies
that $\textrm{sep-rank}(\rho_{t})$ also diverges with $t$.

This is summarised in Table 2, and this situation is to be compared with the
multipartite case, summarised in Table 3.

$\textrm{osr}(\rho)=\textrm{hosr}(\rho)$ | Bipartite state $\rho$
---|---
1 | Product state
2 | Separable state, and separable rank and purification rank are both 2
3 | If separable, purification rank can be unbounded,
| and thus separable rank can be unbounded too
Table 2: Overview of the properties of bipartite positive semidefinite
matrices $\rho$ with small operator Schmidt rank (osr). Recall that the
operator Schmidt rank equals the hermitian operator Schmidt rank (hosr) by
Lemma 14.

###  4.4 Further implications of Theorem 1

We now derive some further implications of Theorem 1, first concerning quantum
channels, and then the entanglement of purification.

First we focus on quantum channels, which are completely positive and trace
preserving maps. Recall that every completely positive map
$\mathcal{E}:\mathcal{M}_{d_{2}}\to\mathcal{M}_{d_{1}}$ is dual to a positive
semidefinite matrix
$0\leqslant\rho\in\mathcal{M}_{d_{1}}\otimes\mathcal{M}_{d_{2}}$ (sometimes
called the Choi matrix of $\mathcal{E}$) via the following relation [Jam72]

$\displaystyle\rho=\frac{1}{\sqrt{d_{1}d_{2}}}(\mathcal{E}\otimes\textrm{id})(\sum_{j,k=1}^{d_{2}}|j,j\rangle\langle
k,k|)$ (14)

where id is the identity map. Moreover, $\mathcal{E}$ is trace preserving if
and only if the Choi matrix satisfies that $\mathrm{tr_{1}}(\rho)=I/d_{2}$,
where $\mathrm{tr_{1}}$ indicates the trace over the first subsystem, and $I$
is the identity matrix of size $d_{2}$. A quantum channel
$\mathcal{E}:\mathcal{M}_{d_{2}}\to\mathcal{M}_{d_{1}}$ is called
_entanglement breaking_ if $(\mathcal{E}\otimes I)(\sigma)$ is separable for
all $0\leqslant\sigma\in\mathcal{M}_{d_{1}}\otimes\mathcal{M}_{d_{2}}$
[HSR03]. It is shown in Ref. [HSR03] that $\mathcal{E}$ is entanglement
breaking if and only if its Choi matrix $\rho$ is separable. Note that
determining whether a quantum channel is entanglement breaking is NP-hard
[Gha10].

######  Corollary 18 (of Theorem 1).

Let $d_{1},d_{2}\geq 2$ be arbitrary and
$\mathcal{E}:\mathcal{M}_{d_{2}}\to\mathcal{M}_{d_{1}}$ be a quantum channel
such that the rank of $\mathcal{E}$ (i.e. the dimension of the image) is 2.
Then $\mathcal{E}$ is entanglement breaking.

###### Proof.

If the dimension of the image of $\mathcal{E}$ is 2, then $\mathcal{E}$ can be
written as
$\mathcal{E}(X)=\sum_{\alpha=1}^{2}P_{\alpha}\textrm{tr}(Q_{\alpha}^{t}X)$,
where $t$ denotes transposition, and $P_{1},P_{2}$ are linearly independent,
as are $Q_{1},Q_{2}$. From (14) it is immediate to see that the Choi matrix is

$\rho=\frac{1}{\sqrt{d_{1}d_{2}}}\sum_{\alpha=1}^{2}P_{\alpha}\otimes
Q_{\alpha},$

i.e. it has operator Schmidt rank 2. By Corollary 13 (ii), $\rho$ is
separable, and thus $\mathcal{E}$ is entanglement breaking. ∎

Note that the statement of Corollary 18 also holds if $\mathcal{E}$ is not
trace preserving, since in that case the Choi matrix also has operator Schmidt
rank 2.

Now we turn to the implications of Theorem 1 for the correlations in the
system. We consider again a bipartite state, which we now denote by
$\rho_{AB}$. The entanglement of purification $E_{p}(\rho_{AB})$ [THLD02] is
defined as

$E_{p}(\rho_{AB})=\min_{\psi}\\{E(|\psi\rangle_{A,A^{\prime},B,B^{\prime}}))|\textrm{tr}_{A^{\prime}B^{\prime}}|\psi\rangle\langle\psi|=\rho\\}$

where the entropy of entanglement
$E(|\psi\rangle_{A,A^{\prime},B,B^{\prime}})$ is defined as

$E(|\psi\rangle_{A,A^{\prime},B,B^{\prime}})=S_{1}(\rho_{AA^{\prime}}),$

where
$\rho_{AA^{\prime}}=\textrm{tr}_{BB^{\prime}}|\psi\rangle_{AA^{\prime}BB^{\prime}}\langle\psi|$
and $S_{1}$ is the von Neumann entropy,
$S_{1}(\rho)=-\textrm{tr}(\rho\log_{d}\rho)$, where $\rho$ has size $d\times
d$. The entanglement of purification is a measure of the amount of classical
and quantum correlations of the system [THLD02]. It follows from Theorem 1
that if a state has operator Schmidt rank two, then its entanglement of
purification is very small:

######  Corollary 19 (of Theorem 1).

Let $0\leqslant\rho\in\mathcal{M}_{d_{1}}\otimes\mathcal{M}_{d_{2}}$ be such
that $\textrm{osr}(\rho)=2$. Then its entanglement of purification satisfies
$E_{p}(\rho)\leq\log_{d_{1}}2$.

###### Proof.

From Corollary 17 we have that $\textrm{osr}(\rho)=2$ implies $\textrm{puri-
rank}(\rho)=2$. From [DN19] we know that
$E_{p}(\rho)\leq\log_{d_{1}}(\textrm{puri-rank}(\rho))$, from which the result
follows. ∎

##  5 Multipartite states with Hermitian operator Schmidt rank two

In this section we focus on multipartite states with Hermitian operator
Schmidt rank two. First we prove Theorem 2 (Section 5.1), then analyse how to
impose Hermiticity in the multipartite case (Section 5.2), and finally discuss
the context and derive several implications of this result (Section 5.3).

###  5.1 Proof of Theorem 2

In this section we prove Theorem 2. The main difference between the bipartite
and the multipartite case (i.e. Theorem 1 and Theorem 2) is that in the
bipartite case we need not assume that the matrices are Hermitian. This is
because Lemma 14 allows to enforce the Hermiticity of the matrices without
increasing the number of terms, i.e. it shows that
$\textrm{osr}(\rho)=\textrm{hosr}(\rho)$. Yet, this need not be true in the
multipartite case, where we can only prove the looser relations between osr
and hosr presented in Section 5.2. In particular, they do not guarantee that
if $\textrm{osr}(\rho)=2$ then $\textrm{hosr}(\rho)=2$. Note that, ultimately,
we need Hermitian matrices in order to use Corollary 13 (iii).

Let us now state and prove Theorem 2.

######  Theorem (Theorem 2).

Let $\rho$ be a positive semidefinite matrix,
$0\leqslant\rho\in\mathcal{M}_{d_{1}}\otimes\mathcal{M}_{d_{2}}\otimes\cdots\otimes\mathcal{M}_{d_{n}}$
where $d_{1},d_{2},\ldots,d_{n}$ are arbitrary, such that

$\displaystyle\rho=\sum_{\alpha_{1},\ldots,\alpha_{n-1}=1}^{2}A^{[1]}_{\alpha_{1}}\otimes
A^{[2]}_{\alpha_{1},\alpha_{2}}\otimes\cdots\otimes A^{[n]}_{\alpha_{n-1}}$

where $A^{[1]}_{\alpha}\in\mathrm{Her}_{d_{1}}$,
$A^{[2]}_{\alpha,\beta}\in\mathrm{Her}_{d_{2}}$, …,
$A^{[n]}_{\alpha}\in\mathrm{Her}_{d_{n}}$. Then $\rho$ is separable and can be
written as

$\rho=\sum_{\alpha_{1},\ldots,\alpha_{n-1}=1}^{2}\sigma^{[1]}_{\alpha_{1}}\otimes\sigma^{[2]}_{\alpha_{1},\alpha_{2}}\otimes\cdots\otimes\sigma^{[n]}_{\alpha_{n-1}}$

where $\sigma^{[1]}_{\alpha}\in\mathrm{PSD}_{d_{1}}$,
$\sigma^{[2]}_{\alpha,\beta}\in\mathrm{PSD}_{d_{2}}$, $\ldots$,
$\sigma^{[n]}_{\alpha}\in\mathrm{PSD}_{d_{n}}$.

Note that we are assuming that $\sigma^{[1]}_{1}$ and $\sigma^{[1]}_{2}$ are
linearly independent, and similarly for the other sites, since otherwise the
sum over the corresponding index would only need to take one value.

###### Proof.

We prove the statement by induction on $n$. In the bipartite case we have

$\rho=\sum_{\alpha=1}^{2}A^{[1]}_{\alpha}\otimes A_{\alpha}^{[2]},$

where $A^{[1]}_{\alpha}$ and $A^{[1]}_{\alpha}$ are Hermitian. This is a
particular case of Theorem 1, and thus it follows that

$\rho=\sum_{\alpha=1}^{2}\sigma^{[1]}_{\alpha}\otimes\sigma_{\alpha}^{[2]},$

where $\sigma^{[1]}_{\alpha}$ and $\sigma^{[2]}_{\alpha}$ are positive
semidefinite. For the induction step we also need the following property: if
another state admits a representation with the same $A^{[1]}_{\alpha}$ but
different $A^{[2]}_{\alpha}$, then its separable decomposition can be chosen
with the same $\sigma^{[1]}_{\alpha}$, i.e. only $\sigma^{[2]}_{\alpha}$ is
modified. This follows directly from the proof of Theorem 1.

Now, for the induction step, assume that the statement holds for $n$ sites. We
first write the positive semidefinite matrix $\rho$ on $n+1$ sites as

$\displaystyle\rho=\sum_{\alpha_{n}=1}^{2}B_{\alpha_{n}}^{[1\ldots n]}\otimes
A^{[n+1]}_{\alpha_{n}}$ (15)

where

$B_{\alpha_{n}}^{[1\ldots
n]}=\sum_{\alpha_{1},\ldots,\alpha_{n-1}=1}^{2}A^{[1]}_{\alpha_{1}}\otimes
A^{[2]}_{\alpha_{1},\alpha_{2}}\otimes\cdots\otimes
A^{[n]}_{\alpha_{n-1},\alpha_{n}}.$

Note that $A^{[n+1]}_{1}$, $A^{[n+1]}_{2}$ are Hermitian, and so are
$B_{1}^{[1\ldots n]}$ and $B_{2}^{[1\ldots n]}$. Thus, applying Corollary 13
(iii) to $\rho$ in Eq. (15), we find that it is separable and can be written
as

$\displaystyle\rho=\sum_{\alpha_{n}=1}^{2}\sigma^{[1\ldots
n]}_{\alpha_{n}}\otimes\sigma_{\alpha_{n}}^{[n+1]}$ (16)

where $\sigma^{[1\ldots n]}_{\alpha_{n}}\in\mathrm{PSD}_{d_{1}\cdot
d_{2}\cdots d_{n}}$ and
$\sigma_{\alpha_{n}}^{[n+1]}\in\mathrm{PSD}_{d_{n+1}}$. Moreover,
$\sigma^{[1\ldots n]}_{\alpha_{n}}$ is a real linear combination of
$B_{1}^{[1\ldots n]}$ and $B_{2}^{[1\ldots n]}$,

$\displaystyle\sigma^{[1\ldots
n]}_{\alpha_{n}}=\sum_{j=1}^{2}\lambda_{\alpha_{n}j}B_{j}^{[1\ldots n]}$
$\displaystyle=\sum_{\alpha_{1},\ldots,\alpha_{n-1}}^{2}A^{[1]}_{\alpha_{1}}\otimes\cdots\otimes\left(\sum_{j=1}^{2}\lambda_{\alpha_{n}j}A^{[n]}_{\alpha_{n-1},j}\right)$
(17)

Thus, both $\sigma_{1}^{[1\ldots,n]}$ and $\sigma_{2}^{[1\ldots,n]}$ are
positive semidefinite, defined on $n$ sites, and of Hermitian operator Schmidt
rank $2$, as Eq. (17) shows. Note also that $\sigma_{1}^{[1\ldots,n]}$ and
$\sigma_{2}^{[1\ldots,n]}$ only differ at the $n$-th site. By the induction
hypothesis, both $\sigma_{\alpha_{n}}^{[1\ldots,n]}$ have separable rank 2
representations, using the same positive semidefinite matrices at the first
$n-1$ sites. Explicitly,

$\sigma_{\alpha_{n}}^{[1\ldots
n]}=\sum_{\alpha_{1},\ldots,\alpha_{n-1}=1}^{2}\sigma^{[1]}_{\alpha_{1}}\otimes\sigma^{[2]}_{\alpha_{1},\alpha_{2}}\otimes\cdots\otimes\sigma^{[n]}_{\alpha_{n-1},\alpha_{n}},$

where $\sigma^{[1]}_{\alpha}\in\mathrm{PSD}_{d_{1}}$,
$\sigma^{[2]}_{\alpha,\beta}\in\mathrm{PSD}_{d_{2}}\ldots$ Inserting this
expression in $\rho$ of Eq. (16) we obtain a separable decomposition of $\rho$
with separable rank 2.

Finally, note that if only the $A^{[n+1]}_{\alpha_{n}}$ in the initial
representation of $\rho$ are changed (Eq. (15)), then the separable
decomposition that we have just constructed will only differ at the last site.
∎

###  5.2 Imposing Hermiticity in the multipartite case

Here we analyse how to impose Hermiticity of the local matrices for
multipartite states, and then prove a modified version of Theorem 2 in which
the local matrices need not be Hermitian but linearly independent.

As we have seen in Lemma 14, in the bipartite case we can impose Hermiticity
of the local matrices without increasing the number of terms, that is, the
operator Schmidt rank and its Hermitian counterpart coincide,
$\textrm{hosr}(\rho)=\textrm{osr}(\rho)$. In the multipartite case we can only
give the following bound, which we do not know if it is tight.

######  Proposition 20 (Hermiticity in the multipartite case).

Let $\rho$ be a Hermitian matrix in
$\mathcal{M}_{d_{1}}\otimes\cdots\otimes\mathcal{M}_{d_{n}}$, where
$d_{1},\ldots,d_{n}$ are arbitrary. Then

$\textrm{osr}(\rho)\leq\textrm{hosr}(\rho)\leq 2^{n-1}\textrm{osr}(\rho),$

except if $n=2$, in which case $\textrm{hosr}(\rho)=\textrm{osr}(\rho)$.

###### Proof.

The statement for $n=2$ corresponds to Lemma 14. For general $n$, we consider
the MPDO form of $\rho$ [Eq. (2)], and express each matrix as a sum of its
Hermitian and anti-Hermitian part, namely as

$A_{\alpha}^{[l]}=\sum_{k=0}^{1}i^{k}A_{\alpha}^{[l]k}$

for $l=1,n$, and

$A_{\alpha,\beta}^{[l]}=\sum_{k=0}^{1}i^{k}A_{\alpha,\beta}^{[l]k}$

for $1<l<n$. Substituting in the MPDO form of $\rho$ we obtain

$\rho=\sum_{k_{1},\ldots,k_{n}=0}^{1}i^{\mathbf{k}}\sum_{\alpha_{1},\ldots,\alpha_{n-1}=1}^{D}A^{[1]k_{1}}_{\alpha_{1}}\otimes
A^{[2]k_{2}}_{\alpha_{1},\alpha_{2}}\otimes\cdots\otimes
A^{[n]k_{n}}_{\alpha_{n-1}},$

where $\mathbf{k}=k_{1}+\cdots+k_{n}$. Since $\rho$ is Hermitian, all terms
where $\mathbf{k}$ is odd must vanish. Thus the above expression contains
$2^{n-1}$ terms, all of which are are Hermitian. Thus $\textrm{hosr}(\rho)\leq
2^{n-1}\textrm{osr}(\rho)$. ∎

With the help of an additional condition we can prove an analogous result to
the bipartite case.444We thank an anonymous referee for suggesting this
argument. The following result is presented for finitely many matrices
$\rho_{i}$, because this is needed in the induction step of the proof.
Afterwards, we will prove a corollary for a single $\rho$.

######  Proposition 21.

Let

$\rho_{1},\ldots,\rho_{r}\in\mathcal{M}_{d_{1}}\otimes\mathcal{M}_{d_{2}}\otimes\cdots\otimes\mathcal{M}_{d_{n}}$

be Hermitian matrices such that

$\displaystyle\rho_{i}=\sum_{\alpha_{1},\ldots,\alpha_{n-1}=1}^{k}P^{[1]}_{\alpha_{1}}\otimes
P^{[2]}_{\alpha_{1},\alpha_{2}}\otimes\cdots\otimes
P_{\alpha_{n-2},\alpha_{n-1}}^{[n-1]}\otimes P^{[n]}_{\alpha_{n-1},i}$

where at each site the local matrices of $\rho_{i}$ are linearly independent.
Then $\rho_{1},\ldots,\rho_{r}$ can be written as

$\rho_{i}=\sum_{\alpha_{1},\ldots,\alpha_{n-1}=1}^{k}A^{[1]}_{\alpha_{1}}\otimes
A^{[2]}_{\alpha_{1},\alpha_{2}}\otimes\cdots\otimes
A_{\alpha_{n-2},\alpha_{n-1}}^{[n-1]}\otimes A^{[n]}_{\alpha_{n-1},i}$

where all local matrices are Hermitian.

###### Proof.

The proof is by induction on $n$. The case $n=2$ is precisely Lemma 14, where
the fact that the Hermitian matrices at site one can be chosen the same for
all $\rho_{i}$ follows from choosing a Hermitian basis of ${\rm
span}\\{P_{1}^{[1]},\ldots,P_{k}^{[1]}\\}$.

For a general $n$, we proceed as in the proof of Theorem 2. It is easy to see
that linear independence of the local matrices at each site is maintained
throughout the construction. ∎

######  Corollary 22.

Let
$0\leqslant\rho\in\mathcal{M}_{d_{1}}\otimes\mathcal{M}_{d_{2}}\otimes\cdots\otimes\mathcal{M}_{d_{n}}$
be positive semidefinite with

$\displaystyle\rho=\sum_{\alpha_{1},\ldots,\alpha_{n-1}=1}^{2}P^{[1]}_{\alpha_{1}}\otimes
P^{[2]}_{\alpha_{1},\alpha_{2}}\otimes\cdots\otimes
P_{\alpha_{n-2},\alpha_{n-1}}^{[n-1]}\otimes P^{[n]}_{\alpha_{n-1}}$

where at each site the local matrices are linearly independent. Then $\rho$ is
separable with $\textrm{sep-rank}(\rho)=2.$

###### Proof.

Clear from Proposition 21 and Theorem 2. ∎

###  5.3 Context and implications of Theorem 2

As mentioned in Section 4.3, Theorem 2 can be seen as a generalization of
Proposition 16 to the case that $\rho$ is a general quantum state (i.e. not
diagonal in the computational basis), and it is multipartite. In particular,
Theorem 2 implies the following relations among ranks for the multipartite
case.

######  Corollary 23 (of Theorem 2).

Let $\rho$ be a multipartite positive semidefinite matrix,

$0\leqslant\rho\in\mathcal{M}_{d_{1}}\otimes\mathcal{M}_{d_{2}}\otimes\cdots\otimes\mathcal{M}_{d_{n}},$

where $d_{1},d_{2},\ldots,d_{n}$ are arbitrary. Then the following are
equivalent:

  * (i)

$\textrm{hosr}(\rho)=2.$

  * (ii)

$\textrm{sep-rank}(\rho)=2$.

Moreover, they imply the following:

  * (iii)

$\textrm{puri-rank}(\rho)=2$ and $\rho$ is separable.

###### Proof.

(i) implies (ii): is shown in Theorem 2.

(ii) implies (i): From Proposition 7 we have that
$\textrm{hosr}(\rho)\leq\textrm{sep-rank}(\rho)$, and $\textrm{hosr}(\rho)=1$
implies $\textrm{sep-rank}(\rho)=1$, which contradicts the assumption.

(ii) implies (iii): From Proposition 7 (iii) we have that $\textrm{puri-
rank}(\rho)\leq\textrm{sep-rank}(\rho)$, and from Proposition 7 (i)
$\textrm{puri-rank}(\rho)=1$ implies that $\textrm{sep-rank}(\rho)=1$, which
contradicts the assumption. ∎

This leaves the following situation for multipartite positive semidefinite
matrices of small operator Schmidt rank (summarised in Table 3):

  * •

If $\textrm{osr}(\rho)=1$, then $\rho$ is a product state, and
$\textrm{hosr}(\rho)=\textrm{sep-rank}(\rho)=\textrm{puri-rank}(\rho)=1$ by
Proposition 7 (i).

  * •

If $\textrm{osr}(\rho)=2$ and $\textrm{hosr}(\rho)=2$, then $\rho$ is a
separable state, and $\textrm{osr}(\rho)=\textrm{sep-rank}(\rho)=\textrm{puri-
rank}(\rho)=2$ by Corollary 23.

  * •

If $\textrm{osr}(\rho)=2$ and $\textrm{hosr}(\rho)>2$, we cannot conclude
anything about the state. (Note that from Proposition 20 we only know that, if
$\textrm{osr}(\rho)=2$, then $2\leq\textrm{hosr}(\rho)\leq 2^{n}$, where $n$
is the number of sites, i.e.
$\rho\in\mathcal{M}_{d_{1}}\otimes\cdots\otimes\mathcal{M}_{d_{n}}$).

  * •

Ref. [DSPGC13] provides a family of states

$\\{\rho_{n}\in\mathcal{M}_{2}\otimes\cdots\otimes\mathcal{M}_{2}\>(n\textrm{
times})\\}_{n\geq 1}$

which are diagonal in the computational basis (thus, separable) for which
$\textrm{osr}(\rho_{n})=3$ for all $n$, and $\textrm{puri-rank}(\rho_{n})$
diverges with $n$. By Proposition 7 (iii), this implies that $\textrm{sep-
rank}(\rho_{n})$ also diverges with $n$.

In words, the case $\textrm{hosr}(\rho)=2$ behaves very similarly to the
trivial case $\textrm{hosr}(\rho)=1$, as one can determine the kind of state
$\rho$ is, and one can upper bound the amount correlations. On the other hand,
the separation between $\textrm{osr}(\rho)$ and $\textrm{puri-rank}(\rho)$
appears for a very small value of osr, namely $\textrm{osr}(\rho)=3$.

$\textrm{osr}(\rho)$ | $\textrm{hosr}(\rho)$ | Multipartite state $\rho$
---|---|---
1 | 1 | Product state
2 | 2 | Separable state, and separable rank and purification rank are both 2
2 | $>2$ | ?
3 | $\geq 3$ | If separable, purification rank can be unbounded,
|  | and thus separable rank can be unbounded too
Table 3: Overview of the properties of multipartite positive semidefinite
matrices $\rho$ with small operator Schmidt rank (osr) and hermitian operator
Schmidt rank (hosr).

##  6 Conclusions & Outlook

We have presented two main results. First, we have shown that any bipartite
positive semidefinite matrix of operator Schmidt rank two is separable, and
admits a separable decomposition with two positive semidefinite matrices per
site (Theorem 1). We have also provided a step-by-step method to obtain this
separable decomposition (Section 4.2), and have drawn some consequences of
this result concerning the relation among the ranks (Corollary 17, see also
the summary of Table 2), for quantum channels (Corollary 18), and for the
amount of correlations of the state (Corollary 19). While this result was
already known [Car14], we have provided a new proof thereof. In particular, we
have leveraged a relation between minimal operator systems and separable
states (Theorem 11) and its implications (Corollary 13).

Second, we have shown that any multipartite positive semidefinite matrix of
Hermitian operator Schmidt rank two is separable, and it has separable rank
two (Theorem 2). We have drawn consequences of this result for the ranks
(Corollary 23, see also the summary of Table 3).

This work leaves a number of interesting open questions. A first question
concerns the generalisation of these results to the case of operator Schmidt
rank three. Bipartite states
$\rho\in\mathcal{M}_{d}\otimes\mathcal{M}_{d^{\prime}}$ with operator Schmidt
rank three must be separable if $d=2$ or $d^{\prime}=2$ [Car14], but this is
no longer true in the case that $d>2$ and $d^{\prime}>2$. Further results for
these states are shown in [CMHW18, Corollary III.5]. It would be interesting
to analyse the consequences of these results for the multipartite case, and
compare them to Table 3.

A second question is to clarify the relation between the operator Schmidt rank
and the hermitian operator Schmidt rank in the multipartite case. In
particular, it would be interesting to find out whether Proposition 20 is
tight or can be improved, and to find direct relations between the hosr and
the separable rank and/or the purification rank.

On a broader perspective, it would be worth studying which of these results
extends to the translationally invariant (t.i.) case. Ref. [DN19] presents the
t.i. versions of the MPDO form, the separable form, and the local purification
form. For bipartite states which are diagonal in the computational basis, the
ranks associated to each of these decompositions correspond to well-studied
ranks of nonnegative matrices (such as the cp rank and the cpsd rank), as
mentioned in Section 4.3. Exploiting these relations may allow us to
generalize results about ranks in that case, too, and gain further insight
into the decompositions of t.i. quantum states.

More generally, this paper illustrates how fruitful the connection between
quantum states and (free) spectrahedra can be. The latter provide a novel set
of techniques that can shed further light into the decompositions of quantum
states. This is being explored in [Net19].

_Acknowledgements_. We thank an anonymous referee for suggesting Proposition
21 and bringing Refs. [Car17, Gie18] to our attention. We thank A.
Müller–Hermes, M. Studiński and N. Johnston for bringing Refs. [Car14, Car15]
to our attention. T. D. and T. N. are supported by the Austrian Science Fund
FWF through project P 29496-N35.

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