# Top quark measurements by the D0 collaboration

, for the D0 Collaboration
Stony Brook University
E-mail

###### Abstract:

The 2019 European Physical Society Prize for High Energy and Particle Physics
was awarded to the CDF and D0 collaborations for ”the discovery of the top
quark and the detailed measurement of its properties”. This paper is based
upon the talk accepting the prize on behalf of the D0 Collaboration.

In 1981, Fermilab Director Leon Lederman issued a call for an experiment to be
located in the D0 interaction region for ”a small (9 m cube), simple and
clever” experiment. About 20 expressions of interest were narrowed to 12
proposals that were then considered by the Physics Advisory Committee in
summer 1983. All proposals were rejected in favor of a new, completely
unspecified experiment, charged to build a detector that would be ”no worse
than those proposed.” The collaboration that then formed prepared a design [1]
that emphasized hermetic, finely segmented, uranium liquid-argon calorimetry
and a large-coverage muon detection system with stations before and after
solid iron magnetized toroids. The tracking detectors, including a transition
radiation detector, were placed within the 75 cm inner radius of the
calorimetry in a region with no magnetic field. It was deemed that it would be
too expensive to make a bypass for the 400 GeV Main Ring accelerator, used to
feed the fixed target experiments and to create the antiprotons for the
Tevatron, as had been done at CDF. So it was left to penetrate the D0 liquid
argon calorimeter, thus requiring that data taking be suspended when particles
were in the Main Ring.

D0 started its Run I operation in 1992, five years after CDF had its first run
(CDF’s head start was mitigated by the rapid increase in the Tevatron
luminosity). The second D0 physics publication in 1994 [2] set the last-ever
lower limit on the mass of the top quark at 131 GeV. For these masses top
decays to $Wb$ 100% of the time, so that $t\overline{t}$ events are
categorized according to the decays of the two $W$ bosons (to leptons or
quarks), resulting in $t\overline{t}$ classifications of dilepton, single
lepton, or all jets. That paper presented a striking event (Fig. 1) with a
high $p_{T}$ electron and muon, missing transverse momentum and two jets, for
which the background was very small. The most likely top mass from this single
event was 145 GeV.

In 1994, CDF published evidence for the top quark with a mass near 175 GeV. At
the time, D0 had similar sensitivity but fewer events. As a result of these
indications, D0 updated its analysis strategy to focus on the higher mass
region. In the absence of a magnetic field and the ability to tag $b$-quarks
by their finite lifetime, D0 relied upon the topological variables $H_{T}$,
the scalar sum of the transverse momenta of jets and leptons (see Fig. 1), and
Aplanarity (the tendency for the object momenta to be distributed uniformly in
3-space).

Figure 1: (left) Event display of an event with an electron ($p_{T}=100$ GeV),
muon (200 GeV), missing transverse momentum (100 GeV) and two jets of 25 and
20 GeV. (right) The $H_{T}$ distributions expected for top pair signal and
backgrounds for (a) dilepton and (b) single lepton events.

After realigning some magnets in summer 1994, the Tevatron instanteous
luminosity grew rapidly so that by early 1995 it became apparent that there
was now enough data to make a discovery. CDF and D0 had previously reached an
agreement with Director John Peoples that when one of the collaborations
announced that they were ready to submit for publication, the other
collaboration would have one week to finalize its analysis for a joint
submission. On Feb. 18, CDF started the clock. D0 had a manuscript in hand but
chose to spend a bit more time to check the systematic uncertainties and
significance calculations. The papers [3] were simultaneously submitted to
Phys. Rev. Letters on Feb. 25.

The D0 analysis, based on 50 pb-1 of $p\overline{p}$ collision data at 1.8 TeV
considered the dilepton channels ($ee$, $e\mu$ and $\mu\mu$), and single
lepton channels ($e$ or $\mu$) with and without tagging of $b$-jets by a decay
muon. The separation of signal and background for the D0 single lepton sample
for both tight and loose selection cuts, and the two-dimensional histogram of
the three jet mass vs. two jet mass for background, expected top pair signal
and data are shown in Fig. 2. An embargo was placed on the results until the
public announcement on Mar. 2 (Fig. 3), delayed since several of the
principals were at a conference in Brazil reporting on the now-obsolete
evidence results. Nevertheless several newspapers picked up the rumors of an
impending discovery and attempted scoops.

Figure 2:  (left) Number of events vs. fitted top quark mass for the single
lepton selection, for both tight and loose selection cuts. (right) Two jet
mass vs. three jet mass for the top quark decaying to three jets in the single
lepton selection. The data shows a clear contribution from the $t\rightarrow
W+b$-jet contribution.  Figure 3:  The audience at the Mar. 2, 1995 top
discovery seminars. We don’t know whether the speaker was CDF or D0, but note
that no one is glued to their phone or laptop!

Run 1 continued until 1996 with a total accumulation of about 100 pb-1 and led
to 16 top quark publications that focused mainly on measurements of the
$t\overline{t}$ cross section and top quark mass. These papers also included
the first measurement of the $t$ and $\overline{t}$ spin correlation and
searches for the electroweak single top production process and for
$t\overline{t}$ resonances.

The Tevatron had a long shutdown from 1996 to 2001 to complete the Main
Injector, remove the Main Ring, increase the collision energy to 1.96 TeV,
install helical orbits to separate $p$ and $\overline{p}$, and other
improvements. During this shutdown major upgrades were completed for the D0
detector [4]. A 2T solenoid magnet was introduced just inside the central
calorimeter. The tracking system was completely rebuilt with a four (later
five) layer silicon strip barrel and disk detector and an eight layer
scintillating fiber tracker detector read out by precursors of solid state
photomultipliers. The calorimetry was extended to include preshower detectors
on the front faces of the cryostats and the shaping electronics were replaced
to allow operation with bunch spacing interval reduction from 3 $\mu$s to 396
ns. New scintillator pixel panels were installed before and after the muon
toroids to improve muon triggering, and new small cell drift tube position
detectors were installed in the forward regions. Triggering was improved with
new track and preshower-based triggers and a new level of triggering was
introduced. The software was rewritten entirely in C++. The upgraded detector
was capable of collecting data at rates up to 1000 Hz. These improvements and
the 100-fold increase in integrated luminosity during Run II enabled large
improvements in physics reach and precision for top quark studies.

There have been 91 top quark publications [5] to date based on the D0 Run 2
data set. There is insufficient space in this report to discuss these results
in detail, but a recent review can be found in Ref. [6]. The topics
investigated include:

  * •

$t\overline{t}$ differential and inclusive cross section measurements in
dilepton, single lepton (including $\tau$+jets) and all jets channels;

  * •

observation of single top quark production via the electroweak interaction;

  * •

observation of the distinct s- and t-channel single top production processes
(see Fig. 4);

  * •

observation of spin correlations in $t\overline{t}$ production;

  * •

measurement of top quark polarization in dilepton and lepton+jets production;

  * •

forward-backward asymmetries in top quark production and for the decay leptons
(see Fig. 5);

  * •

search for Lorentz non-invariance in $t\overline{t}$ production and decay;

  * •

top quark mass measuremens using Monte Carlo templates (see Fig. 6);

  * •

top quark pole mass measurements using inclusive and differential
$t\overline{t}$ cross sections (see Fig. 6);

  * •

differentiation between the standard model value of 2/3e for the top quark
charge and 4/3e;

  * •

measurement of the width of the top quark state;

  * •

comparison of the mass of $t$ and $\overline{t}$ and test of CPT;

  * •

measurement of the helicity fractions of $W$ bosons in top decay;

  * •

measurements of anomalous top quark tensor and axial vector couplings;

  * •

measurement of the ratio ($B(t\rightarrow Wb$))/($B(t\rightarrow W\Sigma
q_{i}$))

  * •

measurement of the CKM parameter $V_{tb}$ (see Fig. 5);

  * •

study of color flow in $t\overline{t}$ events;

  * •

search for fourth generation top quarks;

  * •

search for flavor changing neutral currents in top decay;

  * •

search for charged Higgs bosons in top decay;

  * •

search for narrow $t\overline{t}$ resonances;

  * •

search for admixtures of spin 0 and spin 1/2 top quarks;

  * •

search for $W^{\prime}$ bosons in $W^{\prime}\rightarrow t\overline{b}$;

  * •

search for anomalous $Wtb$ couplings in single top production

Figure 4:  Three diagrams contribute to single top quark production through
the electroweak interaction. Only the s-channel and t-channel processes have
been observed at the Tevatron. (The LHC experiments observe only the t-channel
and $tW$ channel processes.) As shown in the right panel, the comparison of
the D0 s- and t-channel cross sections constrains models of new physics.

Figure 5:  (left) The combined CDF and D0 measurement of the forward-backward
asymmetry for $t\overline{t}$ production as a function of the mass of the
$t\overline{t}$ system. ”Forward” (”backward”) corresponds to the top quark
(antitop) being emitted in the hemisphere containing the outgoing beam
protons. (right) The combined CDF and D0 measurement of the posterior
probability density for the CKM parameter $|V_{tb}|^{2}$ .

Figure 6:  (left) The D0 top quark mass measurements using the Monte Carlo
template method. The connection of this mass to theoretically well-defined
masses is not precisely known. (right) A preliminary measurement [7] of the
top quark pole mass, $m_{\rm pole}=169.1\pm 2.5$ GeV, based on the comparison
of theoretical and measured differential $t\overline{t}$ cross sections as a
function of $p_{T}^{t}$ and $M(t\overline{t})$.

Although the kinematic range of the differential cross sections and most of
the search results suffer in comparison with the recent LHC measurments, many
of the measurements of the properties are competitive with the LHC, and some
measurements such as the $t\overline{t}$ forward-backward asymmetry and the
measurement of the s-channel single top quark cross section have been unique
Tevatron contributions.

In an enterprise as complex as building a modern collider detector, operating
it with maximum efficiency, managing the flood of data that it produces, and
generating physics results on scores of topics, it is impossible to single out
one individual for recognition. Thus the award of the 2019 EPS prize to the
collaboration is most appropriate. It is also apt that the prize citation
mentions not only the discovery of the top quark but also the detailed
measurements of its properties. Those measurements involved collaboration
members working over the full time span of the experiment and those
achievements are as important as the original discovery. Over the lifetime of
the D0 collaboration some 1500 physicists contributed to its success, and it
is appropriate that all join in expressing their appreciation for the prize
award.

Acknowledgements The outstanding performance of the Tevatron was essential for
the success of the D0 program. We greatly appreciate the outstanding work by
the staff of the Fermilab Accelerator Division and appreciate the support of
the D0 physics program by the Fermilab Computing and Particle Physics
divisions. We thank the staffs at collaborating institutions, and acknowledge
support from the DOE and NSF (USA); CEA and CNRS/IN2P3 (France); MON, NRC KI,
and RFBR (Russia); CNPq and FAPERJ (Brazil); DAE and DST (India); Colciencias
(Colombia); CONACyT (Mexico); NRF (Korea); FOM (The Netherlands); STFC and The
Royal Society (UK); MSMT (Czech Republic); BMBF and DFG (Germany); SFI
(Ireland); Swedish Research Council (Sweden); CAS and CNSF (China); and MESU
(Ukraine).

## References

  * [1] S. Abachi et al. ”The D0 Detector”, Nucl. Instrum Methods in Phys. Res. Sect. A338, 185 (1994).
  * [2] S. Abachi et al. ”Search for the Top Quark in $p\overline{p}$ Collisions at $\sqrt{s}=1.8$ TeV”, Phys. Rev. Lett. 72, 2138 (1994).
  * [3] F. Abe et al. ”Observation of Top Quark Production in $p\overline{p}$ Collisions with the Collider Detector at Fermilab”, Phys. Rev. Lett. 74, 2626 (1995); S. Abachi et al. ”Observation of the Top Quark”, Phys. Rev. Lett. 74, 2632 (1995).
  * [4] V.M. Abazov et al. ”The Upgraded D0 Detector”, Nucl. Instrum. Methods in Phys. Res. Sect. A 565, 463 (2006).
  * [5] The full list of D0 Run 2 publications can be found at https://www-d0.fnal.gov/d0_publications/d0_pubs_list_bytopic.html
  * [6] Eduard Boos et al. ”The top quark (20 years after the discovery)”, Physics Uspekhi 58 (12), 1133 (2015); arXiv 1509.03325 [hep-ex] (2015).
  * [7] https://www-d0.fnal.gov/Run2Physics/WWW/results/top.htm


# Special features of the Weyl-Heisenberg Bell basis imply unusual
entanglement structure of Bell-diagonal states

Christopher Popp  University of Vienna, Faculty of Physics, Währingerstrasse
17, 1090 Vienna, Austria<EMAIL_ADDRESS>Beatrix C. Hiesmayr
University of Vienna, Faculty of Physics, Währingerstrasse 17, 1090 Vienna,
Austria<EMAIL_ADDRESS>
(September 2023)

###### Abstract

Bell states are of crucial importance for entanglement based methods in
quantum information science. Typically, a standard construction of a complete
orthonormal Bell-basis by Weyl-Heisenberg operators is considered. We show
that the group structure of these operators has strong implication on error
correction schemes and on the entanglement structure within Bell-diagonal
states. In particular, it implies an equivalence between a Pauli channel and a
twirl channel. Interestingly, other complete orthonormal Bell-bases do break
the equivalence and lead to a completely different entanglement structure, for
instance in the share of PPT-entangled states. In detail, we find that the
standard Bell basis has the highest observed share on PPT-states and PPT-
entangled states compared to other Bell bases. In summary, our findings show
that the standard Bell basis construction exploits a very special structure
with strong implications to quantum information theoretic protocols if a
deviation is considered.

##  1 Introduction

Leveraging quantum phenomena in technology offers new resources that enable
methods with better performance than classically limited methods in the field
of information theory and related applications like communication,
computation, simulation, metrology or cryptography [1, 2, 3, 4, 5].

Entanglement is one of the main resources for novel methods for quantum
information processing like super-dense coding [6] or teleportation [7], as
well as in other fields like e.g. medical sciences [8, 9, 10, 11, 12]. Despite
its relevance for our quantum-theoretic understanding of nature [13, 14, 15]
and for practical applications in quantum technology, this phenomenon is far
from being well understood. Two entangled qubits are the simplest system to
observe entanglement. The qubit Bell states [16] are a set of maximally
entangled states, which form a basis of this bipartite Hilbert space. A
bipartite pure state is called “maximally entangled”, if the reduced state for
each subsystem is the maximally mixed state, so all information about the
state lies in the correlations between the subsystems. Most applications
leveraging entanglement as a resource use these special states. The amount of
entanglement two parties possess if they share a maximally entangled qubit
Bell state is named “ebit” [17]. Recently, multi-level quantum systems called
“qudits” (with the “$d$” indicating the general dimension of the system) have
been shown to offer potential advantages for applications [18, 19]. The notion
of Bell states can be generalized to the qudit systems [20], in which case two
parties holding a maximally entangled bipartite qudit state are said to
possess one “edit”.

A frequently used standard construction for a basis of bipartite Bell states
of arbitrary dimension is based on a set of operators called “Weyl-Heisenberg”
operators, which can be seen as generalization of the two-dimensional Pauli
matrices to other dimensions [7]. Applying these operators locally to the
standard maximally entangled state, a basis of Bell states can be generated
[21]. Due to their properties, this specific “standard” Bell basis is often
used in applications, but as we will show in this contribution, other Bell
bases exist, which differ in some relevant properties.

For real applications, decoherence due to interaction with the environment
generally disturbs a pure state and the resource in form of an edit is
destroyed or cannot be used effectively. For this reason, quantum error
correction [22, 23] and entanglement purification/distillation [24] are
required. While error correction transforms a certain disturbed state back to
some logical pure state, entanglement purification processes several weakly
entangled or disturbed states to produce fewer but stronger or maximally
entangled states. Both concepts have been generalized to qudit systems. The
Weyl-Heisenberg operators thereby play a crucial role in the form of defining
a “nice” error basis [25, 26, 27] or to use the corresponding Bell states as
target states [28, 29, 30]. One phenomenon, which does not appear for
bipartite qubits but for dimension of the subsystems $d\geq 3$ is bound
entanglement [31, 32, 33, 34, 35, 36]. Bound entangled states are entangled
states, but they cannot be used for entanglement purification. It is known
that all entangled states with positive partial transposition (PPT) are bound
entangled, while the existence of bound entangled states with negative partial
transposition (NPT) is still an open problem. In general, the separability
problem to decide whether a given PPT quantum state is entangled or separable
is an NP-hard problem [37, 38].

Bipartite quantum states that are diagonal in the standard Bell basis are
called “Bell-diagonal”. Bell-diagonal states arise naturally in noise
environments affecting Bell states or in several applications like
entanglement distillation or quantum key distribution [3, 39]. If they are
constructed via the Weyl-Heisenberg operators, special algebraic and geometric
properties of the states can be leveraged for some insight concerning this
complex structure of entanglement [21, 36, 40]. Recently, the authors combined
analytical and numerical methods to investigate the systems of bipartite
qutrits ($d=3$) and ququarts ($d=4$) in detail [41, 42, 43, 44]. As a result,
$95\%$/$77\%$ of all PPT (standard) Bell-diagonal qutrits/ququarts can be
classified as PPT-entangled or separable. Moreover, it was shown that the
group structure in the set of the standard Weyl-Heisenberg-constructed Bell
states is highly relevant for the entanglement structure.

In this work, we demonstrate properties of the standard Bell basis constructed
via the Weyl-Heisenberg operators and show that this basis has special
properties among a set of generalized Bell bases with significant implications
for the corresponding systems of Bell-diagonal states. The paper is organized
as follows: First, we define a set of operators called “Weyl-Twirl” operators,
for which all elements of the standard Bell basis are eigenstates and
demonstrate an equivalence between the map to the set of Bell-diagonal states
and the randomized application of those operators. Then, we present some
applications of this “Weyl-Twirl” with relevance for the separability problem
of Bell-diagonal states and a simple error correction scheme for Bell states.
Second, by generalizing the construction based on the Weyl-Heisenberg
operators, we present a family of Bell bases, which contains the standard Bell
basis, but are generally not unitarily equivalent. We show that, even though
the generalized Bell bases also consist of maximally entangled orthonormal
Bell states, several characteristics and properties of the standard Bell basis
relevant for applications do not exist in the generalized case. These
differences significantly affect the entanglement structure of Bell-diagonal
states and the detection of PPT-entangled states. Finally, we conclude with a
summary of our findings.

##  2 A Stabilizing Group for Maximally Entangled States

Here we define the Weyl-Heisenberg operators, introduce the channels and show
their equivalency. Then we discuss applications in quantum information theory.

###  2.1 Weyl-Heisenberg and Weyl-Twirl Operators

Let $\mathcal{H}=\mathcal{H}_{d}\otimes\mathcal{H}_{d}$ be the Hilbert space
of two qudits of dimension $d$. In this bipartite system, we consider
maximally entangled states, also called “Bell states”. An orthonormal basis of
$d^{2}$ Bell states spanning the Hilbert space can be defined by applying
certain local operators on one of the subsystems to a seed state, e.g. the
maximally entangled state
$\ket{\Omega_{00}}:=\frac{1}{\sqrt{d}}\sum_{i=0}^{d-1}\ket{ii}$. Applying the
Weyl-Heisenberg operators [7]

$\displaystyle
W_{k,l}:=\sum_{j=0}^{d-1}w^{jk}\ket{j}\bra{j+l},~{}~{}k,l=0,...,d-1$ (1)

with $w:=e^{\frac{2\pi i}{d}}$ to the (w.l.o.g.) first subsystem of
$\ket{\Omega_{00}}$ defines the “standard” Bell basis of bipartite qudits:

$\displaystyle\ket{\Omega_{k,l}}:=W_{k,l}\otimes\mathbb{1}_{d}\ket{\Omega_{00}},~{}~{}k,l=0,...,d-1$
(2)

In eq.(1) and in the following, addition and subtraction are always to be
understood$\mod d$. The indices $k$ and $l$ in $W_{k,l}$ relate to phase and
shift operations, respectively. To highlight this property, $W_{k,l}$ can also
be written as $W_{k,l}=Z(k)X(l)$, where $Z(k)=\sum_{j}w^{j\cdot
k}\outerproduct{j}{j}$ is the phase operator and
$X(l)=\sum_{j}\outerproduct{j-l}{j}$ denotes the shift operator.
The Weyl-Heisenberg operators obey the following algebraic relations:

$\displaystyle
W_{k_{1},l_{1}}W_{k_{2},l_{2}}=w^{l_{1}k_{2}}~{}W_{k_{1}+k_{2},l_{1}+l_{2}}$
(3) $\displaystyle W_{k,l}^{\dagger}=w^{kl}~{}W_{-k,-l}=W_{k,l}^{-1}$
$\displaystyle W_{k,l}^{\ast}=W_{-k,l}$ $\displaystyle
W_{k,l}^{T}=w^{-kl}W_{k,-l}.$

Here, $(\dagger),(\ast)$ and $(T)$ denote the adjoint, complex conjugation and
transposition with respect to the computational basis, respectively. These
relations imply a linear structure for the Bell states, which was recently
shown [42] to be highly relevant for the geometric properties of the set of
separable, PPT entangled and NPT entangled states that are diagonal in the
basis defined in eq.(2).
Consider now the set of $d^{2}$ unitary operators, which we call “Weyl-Twirl
operators”,

$\displaystyle T_{i,j}:=W_{i,j}\otimes W_{i,j}^{\ast},~{}~{}i,j=0,...,d-1\;.$
(4)

Applying the Weyl relations (3), one observes that this set of unitary
operators $\\{T_{i,j}|i,j=0,...,d-1\\}$ forms an abelian group under
multiplication with neutral element
$T_{0,0}=\mathbb{1}_{d}\otimes\mathbb{1}_{d}$:

$\displaystyle
T_{i_{1},j_{1}}T_{i_{2},j_{2}}=T_{i_{1}+i_{2},j_{1}+j_{2}}=T_{i_{2},j_{2}}T_{i_{1},j_{1}}$
(5) $\displaystyle T_{i,j}^{-1}=T_{-i,-j}\;.$

Using the well–known property of the maximally entangled state
$\mathbb{1}_{d}\otimes
M\ket{\Omega_{0,0}}=M^{T}\otimes\mathbb{1}_{d}\ket{\Omega_{0,0}}$ for any
matrix $M$ together with the Weyl relations (3), we can calculate the action
of $T_{i,j}$ on the Bell state $\ket{\Omega_{k,l}}$:

$\displaystyle T_{i,j}\ket{\Omega_{k,l}}$ $\displaystyle=(W_{i,j}\otimes
W_{i,j}^{\ast})(W_{k,l}\otimes\mathbb{1}_{d})\ket{\Omega_{0,0}}$ (6)
$\displaystyle=W_{i,j}W_{k,l}\otimes W_{-i,j}\ket{\Omega_{00}}$
$\displaystyle=W_{i,j}W_{k,l}W_{-i,j}^{T}\otimes\mathbb{1}_{d}\ket{\Omega_{00}}$
$\displaystyle=w^{jk-il}W_{k,l}\otimes\mathbb{1}_{d}\ket{\Omega_{00}}$
$\displaystyle=w^{jk-il}\ket{\Omega_{k,l}}$

Each Bell state $\ket{\Omega_{k,l}}$ is thus an eigenvector for each
$T_{i,j}$, with the phase of the eigenvalue being determined by the
corresponding indices $(k,l)$ and $(i,j)$.

###  2.2 Pauli Projectors and the Weyl-Twirl Channel

The properties of quantum channels, i.e. completely positive maps from one
quantum state to another, lie at the heart of the dynamics of state
propagation (see e.g. [17] for an overview). For instance, it is of interest
to characterize channels which transform entangled states to separable states,
so called entanglement-breaking channels (for a recent study see Ref. [45]).
Here we consider bipartite “Pauli-type” channels, i.e. channels whose Kraus
operators are of the form $P_{k,l}$, that transform general states to Bell-
diagonal ones.

Let us define the Bell projectors by
$P_{k,l}:=\ket{\Omega_{k,l}}\bra{\Omega_{k,l}}$. The set $\mathcal{M}_{d}$ of
Bell-diagonal states is given as mixtures of the projectors $\\{P_{k,l}\\}$
with mixing probabilities $\\{c_{k,l}\\}$:

$\displaystyle\mathcal{M}_{d}:=\\{\rho=\sum_{k,l=0}^{d-1}c_{k,l}\,P_{k,l}~{}|~{}\sum_{k,l=0}^{d-1}c_{k,l}=1,c_{k,l}\geq
0\\}$ (7)

This object forms a mathematical simplex and is also known as the magic
simplex [40, 46, 21], where “magic” refers to the “magic” Bell basis for
bipartite qubits introduced by Wootters and Hill [47]. Let
$\rho=\sum_{k,l=0}^{d-1}\sum_{n,m=0}^{d-1}\rho_{k,l,m,n}\ket{\Omega_{k,l}}\bra{\Omega_{m,n}}\in\mathcal{H}$
be a bipartite state, represented in the Bell basis. We define the Pauli
channel $\mathcal{P}:\mathcal{H}\rightarrow\mathcal{M}_{d}$ as a map from the
total Hilbert space $\mathcal{H}$ to the set of Bell-diagonal states as
follows:

$\displaystyle\mathcal{P}(\rho)$
$\displaystyle:=\sum_{k,l=0}^{d-1}P_{k,l}\;\rho\;P_{k,l}=\sum_{k,l=0}^{d-1}\bra{\Omega_{k,l}}\rho\ket{\Omega_{k,l}}\,P_{k,l}$
(8)
$\displaystyle=\sum_{k,l=0}^{d-1}\rho_{k,l,k,l}\,P_{k,l}:=\sum_{k,l=0}^{d-1}c_{k,l}\,P_{k,l}$

Using the operators $T_{i,j}$ (4), we can define another channel, which we
name “Weyl-Twirl” channel:

$\displaystyle\mathcal{T}(\rho):=\frac{1}{d^{2}}\;\sum_{i,j=0}^{d-1}T_{i,j}\;\rho\;T_{i,j}^{\dagger}=\frac{1}{d^{2}}\;\sum_{i,j=0}^{d-1}W_{i,j}\otimes
W_{i,j}^{\ast}\;\rho\;(W_{i,j}\otimes W_{i,j}^{\ast})^{\dagger}$ (9)

We will show now that these two channels are equivalent, i.e.

######  Theorem 1.

$\mathcal{P}(\rho)\equiv\mathcal{T}(\rho)~{}~{}\forall\rho\in\mathcal{H}$ (10)

###### Proof.

Consider the action of $\mathcal{T}$ on a basis state
$\ket{\Omega_{k,l}}\bra{\Omega_{m,n}}$:

$\displaystyle\mathcal{T}(\ket{\Omega_{k,l}}\bra{\Omega_{m,n}})$
$\displaystyle=\frac{1}{d^{2}}\sum_{i,j}^{d-1}w^{j(k-m)-i(l-n)}\;\ket{\Omega_{k,l}}\bra{\Omega_{m,n}}$
$\displaystyle=\delta_{k,m}\delta_{l,n}\;\ket{\Omega_{k,l}}\bra{\Omega_{m,n}}$

Here, the first equality follows from (6) and the second equality from the
identity $\sum_{j=0}^{d-1}w^{jx}=d\delta_{x,0}$. Let
$\rho=\sum_{k,l,m,n=0}^{d-1}\rho_{k,l,m,n}\;\ket{\Omega_{k,l}}\bra{\Omega_{m,n}}\in\mathcal{H}$.
The equation above implies that only diagonal elements remain, i.e.:

$\displaystyle\mathcal{T}(\rho)=\sum_{k,l=0}^{d-1}\rho_{k,l,k,l}\ket{\Omega_{k,l}}\bra{\Omega_{k,l}}=\sum_{k,l=0}^{d-1}\rho_{k,l,k,l}\;P_{k,l}=\mathcal{P}(\rho)$

∎

Calling the application of the channel $\mathcal{T}$ “Weyl-Twirl” is motivated
by the fact that eq.(9) represents the random application of (bi-local)
operators $W_{i,j}\otimes W_{i,j}^{\ast}$. Operations of the form
$\rho\rightarrow\int(U\otimes U^{\ast})\rho(U\otimes U^{\ast})^{\dagger}dU$
with $U$ being a local unitary operator and $dU$ being the according Haar
measure are generally named “Twirl”. These operations leave certain diagonal
elements invariant, while eliminating off-diagonal elements. Operations of
this kind transform a state to Bell-diagonal form and are often relevant for
certain applications, e.g. entanglement purification schemes [24]. The channel
equivalence of the Pauli channel (eq.8) and the finite “Weyl-twirl” (eq.9))
shows that the operators $T_{i,j}$ have the special property that they leave
all Bell-diagonal elements invariant under random application, while
eliminating any off diagonal elements.

###  2.3  Applications of Weyl-Twirl operators and channel

The properties of the Weyl-Twirl operators $T_{i,j}$ and the channel
equivalence (eq.10) imply several properties that are relevant for
applications related to maximally entangled states. In the following, we will
briefly mention two examples.

####  2.3.1 The separability problem

A bipartite quantum state $\rho\in\mathcal{H}_{1}\otimes\mathcal{H}_{2}$ is
separable ($\rho\in SEP$), if it can be written as convex combination of pure
product states, that is $\rho=\sum_{i}q_{i}\rho_{1}^{i}\otimes\rho_{2}^{i}$
with $\rho_{1/2}^{i}\in\mathcal{H}_{1/2}$, $q_{i}>0$ and $\sum_{i}q_{i}=1$.
Due to the existence of PPT-entangled states, the decision problem, whether a
given bipartite state $\rho$ is separable ($\rho\in SEP$) or entangled
($\rho\notin SEP$), is generally a NP-hard problem [37, 38] called
“separability problem”. While all bipartite states with negative partial
transposition (NPT) are known to be entangled according to the Peres-
Horodecki/PPT-criterion [48], for subsystems of dimension $d\geq 3$, also
entangled states with non-negative partial transposition (PPT) exist [31]. No
general and efficient criterion is known that detects all PPT-entangled states
nor a general method to construct those. Consequently, given a bipartite PPT
state, it is generally unknown whether the state is entangled or not. However,
for the set of Bell-diagonal states $\mathcal{M}_{d}$ (7) in dimensions $d=3$
and $d=4$ the problem has efficiently be solved. In detail, for $d=3$ ($4$),
$95\%$ ($77\%$) of all PPT states have been classified as entangled or
separable, using a combination of analytical criteria and numerical methods
[41, 42]. Thus, for those dimension and the standard Bell basis the structure
of PPT-entangled states is known.

As demonstrated in referenced works, the combination of two numerical methods
are especially effective to distinguish separable and PPT-entangled Bell-
diagonal states:

  1. 1.

The construction of separable states $\rho_{s}\in\mathcal{M}_{d}\cap SEP$,
close to the border of the convex set of separable states, providing an inner
approximating polytope of $\mathcal{M}_{d}\cap SEP$

  2. 2.

The construction of optimal entanglement witnesses (see definition below),
providing an outer approximation of $\mathcal{M}_{d}\cap SEP$

Both methods rely on an efficient parameterization of separable Bell-diagonal
states $\rho_{s}\in\mathcal{M}_{d}\cap SEP$ that can be used to optimize
corresponding target quantities. Given an efficient parameterization of
unitaries and pure states [49, 50], a parameterization of separable and Bell-
diagonal states follows from the following corollaries of Theorem 1 :

######  Corollary 1.

Any separable state $\rho_{s}$ remains separable under application of the
Pauli channel $\mathcal{P}$, that is:

$\rho_{s}\in SEP\Rightarrow\mathcal{P}(\rho_{s})\in\mathcal{M}_{d}\cap SEP$

###### Proof.

Due to linearity, it suffices to show the corollary for a pure product state
$\rho_{s}=\ket{\psi_{1}}\bra{\psi_{1}}\otimes\ket{\psi_{2}}\bra{\psi_{2}}$. By
Theorem 1, we have:

$\displaystyle\mathcal{P}(\rho_{s})$ $\displaystyle=\mathcal{T}(\rho_{s})$
$\displaystyle=\frac{1}{d^{2}}\sum_{i,j=0}^{d-1}W_{i,j}\otimes
W_{i,j}^{\ast}\;\rho_{s}\;(W_{i,j}\otimes W_{i,j}^{\ast})^{\dagger}$
$\displaystyle=\frac{1}{d^{2}}\sum_{i,j=0}^{d-1}W_{i,j}\ket{\psi_{1}}\bra{\psi_{1}}W_{i,j}^{\dagger}\otimes
W_{i,j}^{\ast}\ket{\psi_{2}}\bra{\psi_{2}}(W_{i,j}^{\ast})^{\dagger}$

$W_{i,j}$ is unitary, so
$W_{i,j}\ket{\psi_{1}}\bra{\psi_{1}}W_{i,j}^{\dagger}$ and
$W_{i,j}^{\ast}\ket{\psi_{2}}\bra{\psi_{2}}(W_{i,j}^{\ast})^{\dagger}$ are
pure states. Therefore, the above expression represents an equal mixture of
pure product states with mixing probability $\frac{1}{d^{2}}$ and thus is a
separable mixed state. By definition, we also have
$\mathcal{P}(\rho_{s})\in\mathcal{M}_{d}$. ∎

The second corollary states that optimal and Bell-diagonal entanglement
witnesses are also optimal for $\mathcal{M}_{d}$. An entanglement witness $K$
is a Hermitian operator, for which the expectation $\Tr(K\rho_{s})$ is non-
negative for all separable states $\rho_{s}$, while at least one state
$\rho_{e}$ exists with $\Tr(K\rho_{e})<0$. In this case $K$ is said to detect
the entangled state $\rho_{e}$. $K$ is said to be optimal, if there exists a
separable state $\rho_{0}$, such that $\Tr(K\rho_{0})=0$. It has been shown
[40] that Bell-diagonal entanglement witnesses can detect all entangled states
in $\mathcal{M}_{d}$.

######  Corollary 2.

If a Bell-diagonal entanglement witness $K$ is optimal, then it is also
optimal for $\mathcal{M}_{d}$, i.e.
$\exists\tilde{\rho_{0}}\in\mathcal{M}_{d}\cap SEP$ s.th.
$\Tr(K\tilde{\rho_{0}})=0$.

###### Proof.

Let
$\rho_{0}=\sum_{k,l,m,n=0}^{d-1}\rho_{0(k,l,m,n)}\ket{\Omega_{k,l}}\bra{\Omega_{m,n}}\in
SEP$ be a state so that $\Tr(K\rho_{0})=0$. Define
$\tilde{\rho_{0}}:=\mathcal{P}(\rho_{0})$. By Corollary 1,
$\tilde{\rho_{0}}\in\mathcal{M}_{d}\cap SEP$. $K$ is of Bell-diagonal form and
Hermitian, so
$K=\sum_{k,l=0}^{d-1}\kappa_{k,l}P_{k,l},~{}~{}\kappa_{k,l}\in\mathbb{R}$.
This implies:

$0=\Tr(K\rho_{0})=\sum_{k,l=0}^{d-1}\kappa_{k,l}\rho_{0(k,l,k,l)}=\Tr(K\tilde{\rho_{0}})$

∎

Using the corollaries, a parameterization of $SEP$ together with convex
optimization methods can then be applied to $\mathcal{M}_{d}\cap SEP$, by
either optimizing for the convex set of $SEP$ and mapping the result to
$\mathcal{M}_{d}$ (eq.(8)), or by directly considering the action of
$\mathcal{P}$ for the parameterization.

####  2.3.2 Error correction for a maximally entangled qudit

The Weyl-Twirl operators $T_{i,j}$ (eq.4) allow for a simple error
identification and correction scheme for the process of sharing a maximally
entangled state without access to the state itself. Assume that in some
information processing task, the initial state $\ket{\Omega_{0,0}}$ is
transformed to $\ket{\Omega_{k,l}}$ with probability $p_{k,l}$ so that the
error channel state $\mathcal{E}(P_{0,0})$ is represented as

$\displaystyle\mathcal{E}(P_{0,0})=\sum_{k,l=0}^{d-1}p_{k,l}P_{k,l}\;.$ (11)

Another possibility that leads to the form of eq.(11) is a lost or unknown
measurement outcome in the Bell basis. The task is now to identify in which
Bell states $\ket{\Omega_{k,l}}$ the system is without measuring or disturbing
it and optionally to transform it back to the initial state
$\ket{\Omega_{0,0}}$. The idea is to use eq.(6) to store the state dependent
phase $\Phi=jk-il$ in an ancilla qudit. After the phase has been identified,
the state is known and can be transformed back to the initial state.
Consider a system to be in the initial state $\ket{\psi_{0}}$, where the Bell
state $\ket{\Omega_{k,l}}$ is unknown and the ancilla qudit in a fixed state,
i.e.

$\displaystyle\ket{\psi_{0}}=\ket{0}\otimes\ket{\Omega_{k,l}}\;.$

Suppose further that the following operations are available:

  * •

Generalized Hadamard/Fourier gate $F$ acting as
$F\ket{j}=\frac{1}{\sqrt{d}}\sum_{k=0}^{d-1}w^{kj}\ket{k}$

  * •

Controlled Weyl-Twirl gate $CT_{i,j}$ acting as
$CT_{i,j}\;\ket{m}\otimes\ket{n}=\ket{m}\otimes T_{i,j}^{m}\;\ket{n}$

Consider the application of the Fourier gate to the ancilla, followed by the
controlled Weyl-Twirl gate controlled by the ancilla with the unknown Bell
state as target, followed by the adjoint Fourier gate on the ancilla qudit.
These operations act as follows:

$\displaystyle\ket{\psi_{0}}$
$\displaystyle\rightarrow(F^{\dagger}\otimes\mathbb{1})CT_{i,j}(F\otimes\mathbb{1})(\ket{0}\otimes\ket{\Omega_{k,l}})$
$\displaystyle=(F^{\dagger}\otimes\mathbb{1})CT_{i,j}(\frac{1}{\sqrt{d}}\sum_{k}\ket{k}\otimes\ket{\Omega_{k,l}})$
$\displaystyle=(F^{\dagger}\otimes\mathbb{1})(\frac{1}{\sqrt{d}}\sum_{k}\ket{k}\otimes
T_{i,j}^{k}\ket{\Omega_{k,l}})$
$\displaystyle=(F^{\dagger}\otimes\mathbb{1})(\frac{1}{\sqrt{d}}\sum_{k}\ket{k}\otimes
w^{k\Phi}\ket{\Omega_{k,l}})$
$\displaystyle=(F^{\dagger}\frac{1}{\sqrt{d}}\sum_{k}w^{k\Phi}\ket{k})\otimes\ket{\Omega_{k,l}}$
$\displaystyle=F^{\dagger}F\ket{\Phi}\otimes\ket{\Omega_{k,l}}=\ket{\Phi}\otimes\ket{\Omega_{k,l}}$

Measurement of the ancilla qudit yields the phase $\ket{\Phi}=\ket{jk-il}$,
depending on the applied Weyl-Twirl operator through $i$ and $j$ and the
unknown Bell state indexed by $k$ and $l$. Due to the stabilizing property of
the operators $T_{i,j}$, the Bell state is not disturbed and the operation can
be repeatedly applied or run in parallel with additional ancilla qudits. The
applied Weyl-Twirl operators can be chosen according to the expected error. If
only phase or only shift errors appear, a single measurement with suitably
chosen $(i,j)$ can identify the error. If both shift and phase errors need to
be identified, then two measurements of $\Phi$ obtained with different $(i,j)$
for $T_{i,j}$ are required to identify the phase/shift error via $k$/$l$. The
correction operation to recover $\ket{\Omega_{0,0}}$ is then
$W_{k,l}^{\dagger}\otimes\mathbb{1}:\ket{\Omega_{k,l}}\rightarrow\ket{\Omega_{0,0}}$.

##  3 Generalized Bell-diagonal systems

Now we investigate cases, in which we still have a complete orthonormal Bell
basis, but not via the standard construction introduced in the last section.
In particular, this leads in general to a breaking of the channel equivalence
$\mathcal{P}\equiv\mathcal{T}$.

###  3.1 Generalized bases of Bell states

Bell states are characterized by the fact that the reduced state for any of
its subsystems is maximally mixed. Those states are called “locally maximally
mixed” and they imply that all information about the state is in the
correlations between the subsystems. The standard Bell basis defined in (2) is
of course an example of those states. However, there are bases that share this
property, but are not unitarily equivalent to the standard basis[40].
For $k,l=0,...,(d-1)$ consider the $d^{2}$ states

$\displaystyle\ket{\phi^{\alpha}_{k,l}}:=\frac{1}{\sqrt{d}}\sum_{s=0}^{d-1}w^{k(s-l)}\alpha_{s}\;\ket{s-l}\otimes\ket{s}$
(12)

with $\alpha_{s}=e^{i\phi_{s}}$ being a phase factor. For a suitable choice of
phase vectors
$\alpha{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}}=\\{\alpha_{s}\\}_{s=0}^{d-1}$,
these states form an orthonormal set of locally maximally mixed basis states.
Consider the corresponding projection operators

$\displaystyle\ket{\phi^{\alpha}_{k,l}}\bra{\phi^{\alpha}_{k,l}}=\frac{1}{d}\sum_{s,t=0}^{d-1}w^{k(s-t)}\alpha_{s}\alpha_{t}^{\ast}\;\ket{s-l}\bra{t-l}\otimes\ket{s}\bra{t},$

for which any partial trace of the first or second subsystem yields the
maximally mixed state:

$\displaystyle\tr_{1/2}(\ket{\phi^{\alpha}_{k,l}}\bra{\phi^{\alpha}_{k,l}})=\frac{1}{d}\sum_{s=0}^{d-1}\alpha_{s}\alpha_{s}^{\ast}\ket{s}\bra{s}=\frac{1}{d}\sum_{s=0}^{d-1}\ket{s}\bra{s}$
(13)

Furthermore, the orthonormality condition for $\ket{\phi^{\beta}_{m,n}}$ and
$\ket{\phi^{\alpha}_{k,l}}$ reads:

$\displaystyle\delta_{m,k}\delta_{n,l}=\bra{\phi^{\beta}_{m,n}}\ket{\phi^{\alpha}_{k,l}}=\frac{1}{d}\sum_{s,t}^{d-1}\alpha_{s}\beta_{t}^{\ast}\;w^{k(s-l)-m(t-n)}\;\bra{t-n}\ket{s-l}\bra{t}\ket{s}=\delta_{n,l}\frac{1}{d}\sum_{s}w^{s(k-m)}w^{mn-
kl}\alpha_{s}\beta_{s}^{\ast}$ (14)

This shows that for different shift indices ($n,l$), the Bell states (12) are
orthogonal, independent of the phases ($\alpha,\beta$). For equal shift
indices, we now require $\alpha_{s}=\beta_{s}$, which implies:

$\displaystyle\bra{\phi^{\beta}_{m,l}}\ket{\phi^{\alpha}_{k,l}}=\frac{1}{d}w^{l(m-k)}\sum_{s}w^{s(k-m)}\alpha_{s}\alpha_{s}^{\ast}=\delta_{m,k}$
(15)

Thus, in order to meet the requirements for orthonormality and being locally
maximally mixed, we define the matrix
$(\alpha_{s,t})_{s,t=0,...,d-1},~{}~{}|\alpha_{s,t}|=1~{}\forall s,t$ and use
it to define the “generalized” Bell basis
$\\{\ket{\Phi_{k,l}^{\alpha}}|k,l=0,...,d-1\\}$ with

$\displaystyle\ket{\Phi^{\alpha}_{k,l}}:=\frac{1}{\sqrt{d}}\sum_{s}^{d-1}w^{k(s-l)}\alpha_{s,l}\;\ket{s-l}\otimes\ket{s}$
(16)

In accordance with eq.(1) and eq.(2), we define unitary operators that map the
maximally entangled state $\ket{\Omega_{0,0}}$ to the generalized Bell basis
states:

$\displaystyle
V^{\alpha}_{k,l}:=\sum_{j}w^{jk}\alpha_{j+l,l}\;\ket{j}\bra{j+l},~{}~{}k,l=0,...,d-1$
(17)
$\displaystyle\ket{\Phi^{\alpha}_{k,l}}=V^{\alpha}_{k,l}\otimes\mathbb{1}\;\ket{\Omega_{0,0}}$
$\displaystyle
P_{k,l}^{\alpha}:=\ket{\Phi^{\alpha}_{k,l}}\bra{\Phi^{\alpha}_{k,l}}\;.$

Note that standard Weyl-Heisenberg operators $W_{k,l}$ and Bell basis states
$\ket{\Omega_{k,l}}$ are a special case of eq.(16), namely for
$\alpha_{s,t}=1~{}\forall s,t$. In this sense we can talk about a
generalization of the standard Weyl-Heisenberg operators and related Bell
states.

###  3.2 Properties of the generalized Bell bases

The construction of generalized Bell basis introduced in the previous section
(eq.(17)) looks rather similar to the standard case (eqs.(1), (2)) and
generates also a set of orthonormal, maximally entangled and locally maximally
mixed states. However, these Bell bases show significant differences with
implications for the entanglement properties of diagonal states. We first
state those differences and then numerically demonstrate their effect on the
entanglement structure of corresponding systems of Bell-diagonal states.
The following differences to the standard construction hold for general
transformation matrices $\alpha$ and can be checked by simple calculation or
counter examples:

  1. 1.

The linear group structure does generally not exist:
$V^{\alpha}_{k_{1},l_{1}}V^{\alpha}_{k_{2},l_{2}}\not\propto
V^{\alpha}_{k_{1}+k_{2},l_{1}+l_{2}}$

  2. 2.

The stabilizing property does generally not hold: $V^{\alpha}_{i,j}\otimes
V_{i,j}^{\alpha\ast}\ket{\Phi_{k,l}}\neq w^{\phi}\ket{\Phi_{k,l}}$

  3. 3.

Let the simplex [40]

$\displaystyle\mathcal{M}^{\alpha}_{d}:=\\{\rho=\sum_{k,l=0}^{d-1}c_{k,l}\;P^{\alpha}_{k,l}~{}|~{}\sum_{k,l=0}^{d-1}c_{k,l}=1,c_{k,l}\geq
0\\}$ (18)

be the set of states that are diagonal in the generalized Bell basis. The
generalized Pauli channel

$\mathcal{P}^{\alpha}:\mathcal{H}\rightarrow\mathcal{M}^{\alpha}_{d},~{}~{}\mathcal{P}^{\alpha}(\rho):=\sum_{k,l=0}^{d-1}P_{k,l}^{\alpha}\;\rho\;P_{k,l}^{\alpha}$

and the generalized Weyl-Twirl channel

$\mathcal{T}^{\alpha}(\rho):=\frac{1}{d^{2}}\sum_{i,j=0}^{d-1}V^{\alpha}_{i,j}\otimes
V^{\alpha\ast}_{i,j}\;\rho\;(V^{\alpha}_{i,j}\otimes
V^{\alpha\ast}_{i,j})^{\dagger}$

are generally not identical:
$\mathcal{P}^{\alpha}\not\equiv\mathcal{T}^{\alpha}$

  4. 4.

The generalized Pauli channel does generally not conserve separability, so
Corollary 1 does not hold, i.e.: $\rho_{s}\in
SEP\not\Rightarrow\mathcal{P}^{\alpha}(\rho_{s})\in\mathcal{M}^{\alpha}_{d}\cap
SEP$

  5. 5.

Due to the loss of the linear structure, also entanglement conserving
symmetries of $\mathcal{M}_{d}$ (see Ref. [42] and the references therein) are
lost.

###  3.3 Explicit counter examples for bipartite qutrits

Figure 1: These pictures show a $2$-dimensional slice through the magic
simplex for the state family,
$\rho(\alpha,\beta)=(1-\frac{\alpha}{5}-\frac{\beta}{4}-\frac{1}{\sqrt{3}})\frac{1}{9}\mathbb{1}_{9}+\frac{\alpha}{5}P_{0,0}+\frac{\beta}{8}(P_{0,1}+P_{0,2})+\frac{1}{3\sqrt{3}}(P_{1,0}+P_{1,1}+P_{1,2})$
which was investigated for the experimental proof of the existence of bound
entanglement [36]. Each colored point represents one state parameterized by
$(\alpha,\beta)$. If the color is green, then the corresponding state is NPT
entangled, else PPT. The pink and purple regions show states where the
realignment criterion (E2) and the witnesses used in the experiment [36]
detect PPT-entangled states, respectively. Figure (a) shows the result for
states constructed with the standard basis choice, all others for non-standard
bases. Figure (b) shows the result for arbitrary but small phases. All other
figures show the result for random phases in $[0,2\pi]$. We observe that the
PPT-region (blue and pink/purple) changes dramatically and with that the
possibility to find PPT-entangled states (pink/purple).

In Fig. 1 we present a visualization of the drastic change in the separability
structure of a state family within $\mathcal{M}_{d}^{\alpha}$ for dimension
$d=3$ for non-standard Bell basis choices compared to the standard Bell basis
choice. The state family consists of mixtures of Bell projectors, which were
experimentally investigated in Ref. [36]. This family is constructed to show
the greatest violation for a specific entanglement witness based on mutually
unbiased bases [51], which can in principle be realized in experiments. Fig. 1
(a) shows the result of the standard basis choice, including the region
detected by the entanglement witness. In Fig. 1 (b) the phases defining a
generalized Bell basis (eq.(17)) were randomly chosen but close to zero. One
observes that the PPT-region shrinks as well as the region detected by the
realignment criterion E2 (defined in the Appendix, named E2 according to Ref.
[41]) and the entanglement witness. The other figures (c)-(f) are based on
random choices of the phases for defining the generalized Bell projectors and
show how drastically the PPT region and the region detected by the realignment
criterion (E2) changes. The Weyl-Heisenberg structure is thus very important
for the separability structure within $\mathcal{M}_{3}^{\alpha}$.

###  3.4 General entanglement structure changes for bipartite qutrits

In the last section we have presented particular examples, but we can also
come up with general results for $d=3$ by exploiting and adapting the tools
developed in Refs. [41, 42]. A combination of analytical and numerical methods
[43] exploiting the Heisenberg-Weyl structure were used to analyze the
entanglement structure of Bell-diagonal qudits for $d=3$ and $d=4$, leading to
an efficient solution of the separability problem in this particular case. As
a consequence of the stated differences above, most of the tools cannot be
applied for mixtures of the generalized Bell basis states (16). However, some
methods are still applicable that allow us to compare the standard basis with
the generalized Bell bases for dimension $d=3$.
In order to compare the system $\mathcal{M}_{3}$ to
$\mathcal{M}^{\alpha}_{3}$, we use the same uniformly distributed sample set
of mixing probabilities $c_{k,l}$ to construct $10,000$ states in
$\mathcal{M}_{3}$ and in $\mathcal{M}^{\alpha}_{3}$ for various $\alpha$. In
particular, we generate $1,000$ matrices $\alpha$ with uniformly distributed
elements. For each $\alpha$, we define the Bell basis projectors
$P_{k,l}^{\alpha}$ and construct $10,000$ diagonal states according to (18).
We then analyze these states for PPT and entanglement detection by the
realignment criterion [52] (named E2 in Ref. [41]) and the quasipure
concurrence criterion [53] (named E3 in Ref. [41]) (see Appendix for
definitions), which do not depend on the entanglement class preserving
symmetries of the standard simplex $\mathcal{M}_{3}$, and are the most
successful analytical criteria among those investigated in Ref. [41, 42].

| Std. Basis | Min | Max | Mean
---|---|---|---|---
rPPT | $60.0\%$ | $49.0\%$ | $59.0\%$ | $51.6\%$
E2/PPT | $10.4\%$ | $2.1\%$ | $9.6\%$ | $3.6\%$
E3/PPT | $2.7\%$ | $0.6\%$ | $2.3\%$ | $1.0\%$
(E2&E3)/PPT | $1.8\%$ | $0.1\%$ | $1.4\%$ | $0.4\%$
Table 1: Minimum, maximum and mean statistics for 1,000 sample systems
$\mathcal{M}_{3}^{\alpha}$ and the reference values for the standard system
$\mathcal{M}_{3}$. Relative volume of PPT states (rPPT) and the share of PPT-
entangled states among them for the realignment criterion (E2/PPT), the
quasipure approximation criterion (E3/PPT) and the combined criterion
((E2&E3)/PPT).

| rPPT
---|---
E2/PPT | $0.99$
E3/PPT | $0.95$
(E2&E3)/PPT | $0.96$
Table 2: Correlation coefficient for the relative volume of PPT states (rPPT)
and the share of PPT-entangled states among the PPT states as detected by the
realignment criterion (E2/PPT), the quasipure concurrence criterion (E3/PPT)
and the combined criterion ((E2&E3)/PPT).

Table 2 shows the share of PPT states among all analyzed states (rPPT), the
share of PPT-entangled states detected by the realignment criterion among all
PPT states (E2/PPT), and the corresponding share for the quasipure concurrence
criterion (E3/PPT), as well as the share of entangled states that were
detected by both criteria simultaneously (E2&E3)/PPT. It shows that for the
1,000 realizations of $\mathcal{M}_{3}^{\alpha}$ the relative volume of PPT
states is between $49\%$ and $59\%$. The mean of $51.6\%$ indicates most of
the systems are closer to the minimum value than to the maximum. Indeed, each
of more than $85\%$ of the systems have less than $54\%$ PPT states.
Curiously, the PPT share for the standard system $\mathcal{M}_{3}$ is higher
than for any analyzed $\mathcal{M}_{3}^{\alpha}$. For values of all $\alpha$
being sufficiently close to $1$, i.e. for Bell basis being close to the
standard Bell basis, the observed quantities are arbitrarily close, but never
higher then the reference values for the standard system. The same statements
hold for the detection capabilities of E2, E3 and (E2&E3), which are always
lower than the value for the standard system and are typically closer to the
minimal observed value.
Table 2 presents strong positive correlations between the share of PPT states
in the systems $\mathcal{M}_{3}^{\alpha}$ and the relative detection
capabilities of the criteria E2, E3 and simultaneous detection (E2&E3).
Interestingly, the more PPT states are present, the larger the relative volume
of PPT-entangled states is, which are detected by those criteria. Moreover,
the relative number of simultaneously detected states is then also high, in
general. The strong correlations imply an almost linear dependence between
those quantities.
Based on these observations, one can assume that the more states with positive
partial transposition are present in a Bell-diagonal system, the larger the
share of PPT-entangled states, detected by the two effective entanglement
criteria E2 and E3, is. Additionally, a higher share of PPT states seems to
imply a higher share of PPT-entangled states that can be detected by both of
the criteria simultaneously. While in the standard system only $33\%$ of the
states detected by E3 are not also detected by E2, there are Bell-diagonal
systems with less PPT states, in which $83\%$ of the entangled states detected
by E3 are not detected by E2. Curiously, the standard system shows extreme
values for the relative volume of PPT states and the number of entangled
states among them.

##  4 Summary and outlook

In this work, we investigated and compared properties of bipartite maximally
entangled and locally maximally mixed Bell states, known as Bell-diagonal
states and showed that the special features of the Weyl-Heisenberg Bell basis
imply special features in the entanglement structure of Bell-diagonal states.

The frequently used “standard” construction of a $d^{2}$ dimensional Bell
basis of the joint Hilbert space $\mathcal{H}_{d}\otimes\mathcal{H}_{d}$ via
the Weyl-Heisenberg operators is presented and properties of this special
basis were derived that are strongly related to the separability problem and
other applications, for example, error corrections or channel equivalences. In
particular, utilizing the Weyl relations of the Weyl-Heisenberg operators
$W_{k,l}$, we showed that the “Weyl-Twirl” operators $W_{i,j}\otimes
W_{i,j}^{\ast}$ are diagonalized by all elements of the standard Bell basis.
We then leveraged this stabilizing property to show the equivalence of the
“Pauli” channel, $\mathcal{P}$, which projects onto Bell-diagonal states in
the standard Bell basis, and the “Weyl-Twirl” channel, $\mathcal{T}$, which
represents the randomized application of the Weyl-Twirl operators.

One implication of this channel equivalence, $\mathcal{P}\equiv\mathcal{T}$,
is that the separability is conserved under the channels and that optimal
Bell-diagonal entanglement witnesses remain optimal for the set of Bell-
diagonal states. We then demonstrated several applications of the Weyl-Twirl
operators. On the one hand, they allow the efficient parameterization of
separable Bell-diagonal states, relevant for polytope approximations of this
convex set and the construction of optimal Bell-diagonal entanglement
witnesses. These applications, enabled by the special structure of
investigated Bell states, allow the effective entanglement classification and
detection of PPT-entanglement for Bell-diagonal states. On the other hand, a
simple error detection and correction scheme for a maximally entangled qudit
was presented.

Highlighting the implication of the channel equivalence,
$\mathcal{T}\equiv\mathcal{P}$, we investigated systems of Bell-diagonal
states that are constructed from generalized Bell basis states. Those
generalized Bell states form an orthonormal basis and are locally maximally
mixed, similar to the standard basis based on Heisenberg-Weyl operators.
However, the channel equivalence $\mathcal{T}\equiv\mathcal{P}$ and other
properties are lost, which has strong implications on the entanglement
structure within the family of Bell-diagonal states. We summarize the most
relevant objects, their properties and relations in Fig.2 below. Thereby, we
compare the results related to the standard Bell basis to the generalized Bell
bases.

In detail, we analyzed this for bipartite qutrits. We derived the relative
volume of PPT states among Bell-diagonal states. For the standard basis, the
relative volume is $60\%$ while it was shown to drop to $49\%$ for some
generalized Bell bases. Interestingly, we found that the share of Bell-
diagonal PPT states is generally lower for the generalized Bell bases compared
to the standard Bell basis. Moreover, the relative detection rate for PPT-
entangled states of two, in the standard system highly effective, entanglement
criteria strongly correlates to the share of PPT states. The more states with
positive partial transposition exist for a system of Bell-diagonal states, the
higher is the share of detected PPT-entangled states among them for those two
criteria. Consequently, for the system based on the standard Bell basis, which
has the highest PPT share, also the highest relative amount of PPT-
entanglement is observed. Furthermore, we visualized the dramatic change of
the volume of PPT states in Fig. 1 for a family of states investigated
previously in experiment.

These results indicate that, among all in this way generalized Bell bases,
Bell-diagonal states related to the standard construction have some very
special properties. These properties have relevant implications on the
entanglement structure and on practical applications using those Bell states.
Numerical results suggest that either the two entanglement criteria are less
effective for Bell-diagonal states if the special properties of the standard
system are not given or that the structure and amount of PPT entanglement
depend on those properties.

In summary, our findings are the starting point to analyze different quantum
information theoretic protocols with regards to their dependence on the
underlying structure and properties of used Bell states.

Weyl operators: $W_{k,l}$ Weyl relations: Yes Bell basis: $\ket{\Omega_{k,l}}$
Weyl-Twirl: $T_{i,j}=W_{i,j}\otimes W_{i,j}^{\star}$
$T_{i,j}\ket{\Omega_{k,l}}=w^{jk-il}\ket{\Omega_{k,l}}$ Diagonal states:
$\mathcal{M}_{d}$ Channels: $\mathcal{P}\equiv\mathcal{T}$ Relative amount PPT
in $\mathcal{M}_{3}$: $60\%$ Detected PPT-entanglement: $11.3\%$ gen. Weyl
operators: $V^{\alpha}_{k,l}$ Weyl relations: No gen. Bell basis:
$\ket{\Phi^{\alpha}_{k,l}}$ Diagonal states: $\mathcal{M}^{\alpha}_{d}$ gen.
Weyl-Twirl: $T^{\alpha}_{i,j}=V^{\alpha}_{i,j}\otimes V^{\alpha\star}_{i,j}$
$T^{\alpha}_{i,j}\ket{\Phi^{\alpha}_{k,l}}\neq
w^{\phi}\ket{\Phi^{\alpha}_{k,l}}$ Channels:
$\mathcal{P^{\alpha}}\not\equiv\mathcal{T^{\alpha}}$ Relative amount PPT in
$\mathcal{M}_{3}^{\alpha}$: $\leq 60\%$ Detected PPT-entanglement: $\leq
11.3\%$ Figure 2: Comparison of relevant objects and their relation for the
standard and generalized Weyl-construction. Differences include the lack of
linear structure (Weyl relations) of generalized Weyl operators, the lack of
the stabilizing property of the Weyl-Twirl operators and the general
inequivalence of Pauli channel and Weyl-Twirl channel. In the generalized
case, the amount of PPT states and the relative amount of detected PPT-
entanglement is lower compared to the standard system.

## Data availability statement

All analyzed datasets were generated during the current study and are
available from the corresponding author on reasonable request.

## Code availability statement

The software used to generate the reported results is published as repository
and registered open source package “BellDiagonalQudits.jl” [43] available at
https://github.com/kungfugo/BellDiagonalQudits.jl.

## References

  * [1] J.. Cirac, A.. Ekert, S.. Huelga and C. Macchiavello  “Distributed quantum computation over noisy channels”  In _Phys. Rev. A_ 59 American Physical Society, 1999, pp. 4249–4254  DOI: 10.1103/PhysRevA.59.4249
  * [2] I.. Georgescu, S. Ashhab and Franco Nori  “Quantum simulation”  In _Rev. Mod. Phys._ 86 American Physical Society, 2014, pp. 153–185  DOI: 10.1103/RevModPhys.86.153
  * [3] Artur K. Ekert  “Quantum cryptography based on Bell’s theorem”  In _Phys. Rev. Lett._ 67 American Physical Society, 1991, pp. 661–663  DOI: 10.1103/PhysRevLett.67.661
  * [4] Yu-Bo Sheng, Lan Zhou and Gui-Lu Long  “One-step quantum secure direct communication”  In _Science Bulletin_ 67.4, 2022, pp. 367–374  DOI: 10.1016/j.scib.2021.11.002
  * [5] Fu-Guo Deng, Gui Lu Long and Xiao-Shu Liu  “Two-step quantum direct communication protocol using the Einstein-Podolsky-Rosen pair block”  In _Phys. Rev. A_ 68 American Physical Society, 2003, pp. 042317  DOI: 10.1103/PhysRevA.68.042317
  * [6] Charles H. Bennett and Stephen J. Wiesner  “Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states”  In _Phys. Rev. Lett._ 69 American Physical Society, 1992, pp. 2881–2884  DOI: 10.1103/PhysRevLett.69.2881
  * [7] Charles H. Bennett et al.  “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels”  In _Phys. Rev. Lett._ 70 American Physical Society, 1993, pp. 1895–1899  DOI: 10.1103/PhysRevLett.70.1895
  * [8] P. Moskal et al.  “Feasibility studies of the polarization of photons beyond the optical wavelength regime with the J-PET detector”  In _Eur. Phys. J. C_ 78.11, 2018, pp. 970  DOI: 10.1140/epjc/s10052-018-6461-1
  * [9] B.. Hiesmayr and Pawel Moskal  “Genuine Multipartite Entanglement in the 3-Photon Decay of Positronium”  In _Scientific Reports_ 7, 2017, pp. 15349  DOI: 10.1038/s41598-017-15356-y
  * [10] Beatrix.. Hiesmayr and Pawel Moskal  “Witnessing Entanglement In Compton Scattering Processes Via Mutually Unbiased Bases”  In _Scientific Reports_ 9, 2019, pp. 8166  DOI: 10.1038/s41598-019-44570-z
  * [11] W. Krzemien, E. Pérez del Río and K. Kacprzak  “Feasibility of Ortho-positronium Lifetime Studies with the J-PET Detector in Context of Mirror Matter Models”  In _Acta Phys. Pol. B_ 51.1, 2020, pp. 165  DOI: 10.5506/APhysPolB.51.165
  * [12] R.Y. Shopa et al.  “Optimisation of the event-based TOF filtered back-projection for online imaging in total-body J-PET”  In _Medical Image Analysis_ 73, 2021, pp. 102199  DOI: 10.1016/j.media.2021.102199
  * [13] A. Einstein, B. Podolsky and N. Rosen  “Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?”  In _Phys. Rev._ 47 American Physical Society, 1935, pp. 777–780  URL: https://link.aps.org/doi/10.1103/PhysRev.47.777
  * [14] J.. Bell  “On the Einstein Podolsky Rosen paradox”  In _Physics Physique Fizika_ 1 American Physical Society, 1964, pp. 195–200  DOI: 10.1103/PhysicsPhysiqueFizika.1.195
  * [15] B. Hensen et al.  “Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres”  In _Nature_ 526, 2015, pp. 682–684  DOI: 10.1038/nature15759
  * [16] Samuel L. Braunstein, A. Mann and M. Revzen  “Maximal violation of Bell inequalities for mixed states”  In _Phys. Rev. Lett._ 68 American Physical Society, 1992, pp. 3259–3261  URL: https://link.aps.org/doi/10.1103/PhysRevLett.68.3259
  * [17] Mark M. Wilde  “Quantum Information Theory”  Cambridge University Press, 2017  DOI: 10.1017/9781316809976
  * [18] Daniele Cozzolino, Beatrice Da Lio, Davide Bacco and Leif Katsuo Oxenløwe  “High-Dimensional Quantum Communication: Benefits, Progress, and Future Challenges”  In _Advanced Quantum Technologies_ 2.12, 2019, pp. 1900038  DOI: https://doi.org/10.1002/qute.201900038
  * [19] Yuchen Wang, Zixuan Hu, Barry C. Sanders and Sabre Kais  “Qudits and High-Dimensional Quantum Computing”  In _Frontiers in Physics_ 8, 2020, pp. 479  DOI: 10.3389/fphy.2020.589504
  * [20] Denis Sych and Gerd Leuchs  “A complete basis of generalized Bell states”  In _New Journal of Physics_ 11.1 IOP Publishing, 2009, pp. 013006  URL: https://doi.org/10.1088/1367-2630/11/1/013006
  * [21] Bernhard Baumgartner, Beatrix C. Hiesmayr and Heide Narnhofer  “A special simplex in the state space for entangled qudits”  In _J. Phys. A: Math. Theor._ 40, 2007, pp. 7919  DOI: https://doi.org/10.1088/1751-8113/40/28/S03
  * [22] Charles H. Bennett, David P. DiVincenzo, John A. Smolin and William K. Wootters  “Mixed-state entanglement and quantum error correction”  In _Phys. Rev. A_ 54 American Physical Society, 1996, pp. 3824–3851  DOI: 10.1103/PhysRevA.54.3824
  * [23] A.. Calderbank, E.. Rains, P.. Shor and N… Sloane  “Quantum Error Correction and Orthogonal Geometry”  In _Phys. Rev. Lett._ 78 American Physical Society, 1997, pp. 405–408  DOI: 10.1103/PhysRevLett.78.405
  * [24] Charles H. Bennett et al.  “Purification of Noisy Entanglement and Faithful Teleportation via Noisy Channels”  In _Phys. Rev. Lett._ 76 American Physical Society, 1996, pp. 722–725  DOI: 10.1103/PhysRevLett.76.722
  * [25] A. Ashikhmin and E. Knill  “Nonbinary quantum stabilizer codes”  In _IEEE Transactions on Information Theory_ 47.7, 2001, pp. 3065–3072  DOI: 10.1109/18.959288
  * [26] E Knill  “Non-binary Unitary Error Bases and Quantum Codes”  In _arXiv:quant-ph/9608048_ , 1996  URL: https://arxiv.org/abs/quant-ph/9608048
  * [27] Priya J. Nadakarni and Shayan Srinivasa Garani  “Non-binary entanglement-assisted stabilizer codes”  In _Quantum Inf Process_ 20.8, 2021, pp. 258  URL: https://doi.org/10.1007/s11128-021-03174-1
  * [28] Michał Horodecki and Paweł Horodecki  “Reduction criterion of separability and limits for a class of distillation protocols”  In _Phys. Rev. A_ 59 American Physical Society, 1999, pp. 4206–4216  DOI: 10.1103/PhysRevA.59.4206
  * [29] Gernot Alber, Aldo Delgado, Nicolas Gisin and Igor Jex  “Efficient bipartite quantum state purification in arbitrary dimensional Hilbert spaces”  In _Journal of Physics A: Mathematical and General_ 34.42, 2001, pp. 8821  DOI: 10.1088/0305-4470/34/42/307
  * [30] Karl Gerd H. Vollbrecht and Michael M. Wolf  “Efficient distillation beyond qubits”  In _Phys. Rev. A_ 67 American Physical Society, 2003, pp. 012303  DOI: 10.1103/PhysRevA.67.012303
  * [31] Michał Horodecki, Paweł Horodecki and Ryszard Horodecki  “Mixed-State Entanglement and Distillation: Is there a “Bound” Entanglement in Nature?”  In _Phys. Rev. Lett._ 80 American Physical Society, 1998, pp. 5239–5242  DOI: 10.1103/PhysRevLett.80.5239
  * [32] Pratapaditya Bej and Saronath Halder  “Unextendible product bases, bound entangled states, and the range criterion”  In _Physics Letters A_ 386, 2021, pp. 126992  DOI: https://doi.org/10.1016/j.physleta.2020.126992
  * [33] Joshua Lockhart, Otfried Gühne and Simone Severini  “Entanglement properties of quantum grid states”  In _Phys. Rev. A_ 97 American Physical Society, 2018, pp. 062340  DOI: 10.1103/PhysRevA.97.062340
  * [34] Dagmar Bruß and Asher Peres  “Construction of quantum states with bound entanglement”  In _Phys. Rev. A_ 61 American Physical Society, 2000, pp. 030301  DOI: 10.1103/PhysRevA.61.030301
  * [35] Paul B. Slater  “Jagged Islands of Bound Entanglement and Witness-Parameterized Probabilities.”  arXiv, 2019  URL: https://doi.org/10.48550/arXiv.1905.09228
  * [36] Beatrix C Hiesmayr and Wolfgang Löffler  “Complementarity reveals bound entanglement of two twisted photons”  In _New J. Phys_ 15, 2013, pp. 083036  DOI: https://doi.org/10.1088/1367-2630/15/8/083036
  * [37] Leonid Gurvits  “Classical Deterministic Complexity of Edmonds’ Problem and Quantum Entanglement”  In _Proceedings of the Thirty-Fifth Annual ACM Symposium on Theory of Computing_ , STOC ’03  San Diego, CA, USA: Association for Computing Machinery, 2003, pp. 10–19  DOI: 10.1145/780542.780545
  * [38] Sevag Gharibian  “Strong NP-Hardness of the Quantum Separability Problem”  In _Quantum Inf. and Comp._ 10, 2008  DOI: 10.26421/QIC10.3-4-11
  * [39] Dagomir Kaszlikowski et al.  “Quantum cryptography based on qutrit Bell inequalities”  In _Phys. Rev. A_ 67 American Physical Society, 2003, pp. 012310  DOI: 10.1103/PhysRevA.67.012310
  * [40] Bernhard Baumgartner, Beatrix C. Hiesmayr and Heide Narnhofer  “State space for two qutrits has a phase space structure in its core”  In _Phys. Rev. A_ 74, 2006, pp. 032327  DOI: 10.1103/PhysRevA.74.032327
  * [41] Christopher Popp and Beatrix C Hiesmayr  “Almost complete solution for the NP-hard separability problem of Bell diagonal qutrits”  In _Scientific Reports_ 12, 2022, pp. 12472  URL: https://doi.org/10.1038/s41598-022-16225-z
  * [42] Christopher Popp and Beatrix C Hiesmayr  “Comparing bound entanglement of bell diagonal pairs of qutrits and ququarts”  In _Scientific Reports_ 13, 2023, pp. 2037  URL: https://doi.org/10.1038/s41598-023-29211-w
  * [43] Christopher Popp  “BellDiagonalQudits: A package for entanglement analyses of mixed maximally entangled qudits”  In _Journal of Open Source Software_ 8.81 The Open Journal, 2023, pp. 4924  DOI: 10.21105/joss.04924
  * [44] Beatrix C Hiesmayr  “Free versus bound entanglement, a NP-hard problem tackled by machine learning”  In _Scientific Reports_ 11, 2021, pp. 19739  DOI: 10.1038/s41598-021-98523-67
  * [45] Fattah Sakuldee and Łukasz Rudnicki  “Bounds on the breaking time for entanglement-breaking channels”  In _Phys. Rev. A_ 107 American Physical Society, 2023, pp. 022430  DOI: 10.1103/PhysRevA.107.022430
  * [46] B. Baumgartner, Beatrix C. Hiesmayr and H. Narnhofer  “The geometry of bipartite qutrits including bound entanglement”  In _Physics Letters A_ 372.13, 2008, pp. 2190–2195  DOI: https://doi.org/10.1016/j.physleta.2007.11.028
  * [47] Sam A. Hill and William K. Wootters  “Entanglement of a Pair of Quantum Bits”  In _Phys. Rev. Lett._ 78 American Physical Society, 1997, pp. 5022–5025  DOI: 10.1103/PhysRevLett.78.5022
  * [48] Asher Peres  “Separability Criterion for Density Matrices”  In _Phys. Rev. Lett._ 77 American Physical Society, 1996, pp. 1413–1415  DOI: 10.1103/PhysRevLett.77.1413
  * [49] Christoph Spengler, Marcus Huber and Beatrix C. Hiesmayr  “Composite parameterization and Haar measure for all unitary and special unitary groups”  In _J. Math. Phys._ 53, 2012, pp. 013501  DOI: https://doi.org/10.1063/1.3672064
  * [50] Christoph Spengler, Marcus Huber and Beatrix C. Hiesmayr  “A composite parameterization of unitary groups, density matrices and subspaces”  In _J. Phys. A: Math. Theor_ 43, 2010, pp. 385306  DOI: https://doi.org/10.1088/1751-8113/43/38/385306
  * [51] B C Hiesmayr et al.  “Detecting entanglement can be more effective with inequivalent mutually unbiased bases”  In _New Journal of Physics_ 23.9 IOP Publishing, 2021, pp. 093018  URL: https://doi.org/10.1088/1367-2630/ac20ea
  * [52] Kai Chen and Ling-An Wu  “A matrix realignment method for recognizing entanglement”  In _Quantum Inf. and Comp._ 3, 2002  DOI: 10.26421/QIC3.3-1
  * [53] Joonwoo Bae et al.  “Detection and typicality of bound entangled states”  In _Phys. Rev. A_ 80 American Physical Society, 2009, pp. 022317  DOI: 10.1103/PhysRevA.80.022317
  * [54] William K. Wootters  “Entanglement of Formation of an Arbitrary State of Two Qubits”  In _Phys. Rev. Lett._ 80 American Physical Society, 1998, pp. 2245–2248  DOI: 10.1103/PhysRevLett.80.2245

## Acknowledgments

B.C.H. and C.P. acknowledge gratefully that this research was funded in whole,
or in part, by the Austrian Science Fund (FWF) project P36102-N. For the
purpose of open access, the author has applied a CC BY public copyright
licence to any Author Accepted Manuscript version arising from this
submission.

## Author contributions statement

C.P. carried out the analytical and numerical analyses, developed the new
methods and implemented the software.
B.C.H. revised the analyses and added results and proposed improvements.
C.P and B.C.H. both edited the manuscript.

## Competing interests

All authors declare no financial or non-financial competing interests.

## Additional information

Correspondence and requests for materials should be addressed to C.P..

## Appendix

### Applied entanglement criteria

Peres-Horodecki/PPT criterion
The partial transpose $\Gamma$ acts on the computational basis states of a
bipartite state as
$(\outerproduct{i}{j}\otimes\outerproduct{k}{l})^{\Gamma}:=\outerproduct{i}{j}\otimes\outerproduct{l}{k}$.
If the partial transpose for a density matrix of a bipartite state has at
least one negative eigenvalue (in which case it is said to be “NPT”), it is
entangled [48]. For $d=2$ it detects all entangled states, but for $d\geq 3$
it is only sufficient due to the existence of PPT-entangled states.

E2: Realignment criterion
The realignment operation $R$ is defined for the computational basis states as
$(\outerproduct{i}{j}\otimes\outerproduct{k}{l})_{R}:=\outerproduct{i}{k}\otimes\outerproduct{j}{l}$.
If the sum of singular values of the density matrix of a realigned state
$\sigma_{R}$ are larger than $1$, then $\sigma$ is entangled [52].

E3: Quasipure concurrence criterion
The quasipure approximation [53] $C_{qp}$ provides an efficiently computable
lower bound on the concurrence [54] $C$. Let $i$ and $j$ ($k$ and $l$)
enumerate the computational basis vectors of the first (second) partition of
the Hilbert space. Define
$A:=4\sum_{i<j,k<l}\ket{ikjl}-\ket{jkil}-\ket{iljk}+\ket{jlik}\times h.c.$ and
let a state be represented in its spectral composition
$\rho=\sum_{i}\mu_{i}\ket{\Psi_{i}}\bra{\Psi_{i}}$. With a dominant
eigenvector $\ket{\Psi_{0}}$, define $\ket{\xi}\propto
A\ket{\Psi_{0}}\otimes\ket{\Psi_{0}}$ and

$T_{i,j}:=\sqrt{\mu_{i}\mu_{j}}\bra{\Psi_{i}}\otimes\bra{\Psi_{j}}\ket{\xi}.$

The quasipure concurrence is defined by the singular values $S_{i}$ of the
matrix $T_{i,j}$

$C_{qp}(\rho):=\max(0,S_{0}-\sum_{i>0}S_{i})\leq C(\rho)$

and detects entanglement for positive values.
For the standard system $\mathcal{M}_{d}$, the approximation takes an explicit
form: Here, the singular values $S_{k,l}$ are explicitly given by

$\displaystyle
S_{k,l}=\sqrt{\frac{d}{2(d-1)}c_{k,l}[(1-\frac{2}{d})c_{n,m}\delta_{k,n}\delta_{l,m}+\frac{1}{d^{2}}c_{(2n-k)mod~{}d,(2m-l)mod~{}d}]}$

where $(n,m)$ is a multi-index of the coordinate of the largest value
$\\{c_{k,l}\\}$ [53].