# Boosting segmentation performance across datasets using histogram
specification with application to pelvic bone segmentation

###### Abstract

Accurate segmentation of the pelvic CTs is crucial for the clinical diagnosis
of pelvic bone diseases and for planning patient-specific hip surgeries. With
the emergence and advancements of deep learning for digital healthcare,
several methodologies have been proposed for such segmentation tasks. But in a
low data scenario, the lack of abundant data needed to train a Deep Neural
Network is a significant bottle-neck. In this work, we propose a methodology
based on modulation of image tonal distributions and deep learning to boost
the performance of networks trained on limited data. The strategy involves
pre-processing of test data through histogram specification. This simple yet
effective approach can be viewed as a style transfer methodology. The
segmentation task uses a U-Net configuration with an EfficientNet-B0 backbone,
optimized using an augmented BCE-IoU loss function. This configuration is
validated on a total of 284 images taken from two publicly available CT
datasets, TCIA (a cancer imaging archive) and the Visible Human Project. The
average performance measures for the Dice coefficient and Intersection over
Union are 95.7% and 91.9%, respectively, give strong evidence for the
effectiveness of the approach, which is highly competitive with state-of-the-
art methodologies.

Index Terms—  Pelvic bone segmentation, data pre-processing, histogram
specification, U-Net, fine-tuning.

##  1 Introduction

In recent years, due to the increase in the incidence of pelvic injuries from
traffic-related accidents [1], pelvic bone diseases within the aging
population, and sufficient access to computed tomography (CT) imaging,
automated pelvic bone segmentation in CT has gained considerable prominence.
The segmentation results assist physicians in the early detection of pelvic
injury and help expedite surgical planning and reduce the complications caused
by pelvic fractures [2]. In CT data, structures like the bone marrow and bone
surface appear as dark and bright regions due to their low and high densities
compared to the surrounding tissues. However, given the variations in image
quality between different CT datasets, distinguishing bone structures from the
image background becomes cumbersome and leads to erroneous segmentation
outputs. These issues indicate a need for a novel solution to develop a simple
yet effective methodology for the accurate segmentation of pelvic bones from
varying CT data.

Contribution of this paper: The key novelties of this work are as follows:

  1. 1.

introduction of an encoder-decoder network, trained on limited data, for high
accuracy segmentation of pelvic bones

  2. 2.

boosting model performance on unseen data by employing histogram specification

The exact details of the approach are deferred until Sec. 3.3. Fig. 1
illustrates the results of the proposed method.

$\begin{array}[]{cc}\includegraphics[width=99.58464pt,height=99.58464pt]{images1/3.png}&\includegraphics[width=99.58464pt,height=99.58464pt]{images1/4.png}\\\
\mbox{TCIA}&\mbox{(a1)}\\\
\includegraphics[width=99.58464pt,height=99.58464pt]{images1/t1.png}&\includegraphics[width=99.58464pt,height=99.58464pt]{images1/t2.png}\\\
\mbox{VHBD}&\mbox{(b1)}\\\ \end{array}$

Fig. 1: (a1) and (b1) – illustrate the segmentation outputs, for input images
from TCIA [3] and VHBD [4], respectively.

##  2 Prior Art

Fig. 2: Workflow of U-Net architecture with pre-trained backbone, detailing
pelvic bone segmentation.

Recent literature has seen many applications for the segmentation of the
pelvis from CT imaging data. Traditional methods such as thresholding and
region growth [5], deformable surface model [6], and others, have been
commonly used to perform bone segmentation. However, these approaches often
suffer from low accuracy due to varying image properties such as intensity,
contrast, and the inherent variations between the texture of the bone
structures (bone marrow and surface boundary) and the surrounding tissues. To
overcome these challenges, supervised methods such as statistical shape models
(SSM) and atlas-based deep learning (DL) methods have made significant
contributions to segmentation tasks. Wang et al. [7, 8] suggested using a
multi-atlas segmentation with joint label fusion for detecting regions on
interest from CT images. Yokota et al. [9] showcased a combination of
hierarchical and conditional SSMs for the automated segmentation of diseased
hips from CT data. Chu et al. [10] presented a multi-atlas based method for
accurately segmenting femur and pelvis. Zeng et al. [11] proposed a supervised
3D U-Net with multi-level supervision for segmenting femur in 3D MRI. Chen et
al. [12] showcased a 3D feature enhanced network for quickly segmenting femurs
from CT data. Chang et al. [13] proposed patch-based refinement on top of a
conditional random field model for fine segmentation of healthy and diseased
hips. Liu et al. [14] used 3D U-Nets in two-stages (trained on approximately
270K images) with a signed distance function for producing bone fragments from
image-stacks. In the following section, we discuss a new technique addressing
accurate segmentation of the pelvis from CT images of varying qualities.

##  3 Proposed Methodology

The efficacy of using Encoder-Decoder architectures for designing high
accuracy segmentation models for biomedical applications has been showcased in
recent literature [15, 11, 14]. We employ a similar architecture, with various
encoder modules for feature extraction and a decoder module for semantic
segmentation. The details of the encoder and decoder modules are explained in
the following.

###  3.1 Encoder Module

In simple terms, an encoder takes the input image and generates a high-
dimensional feature vector aggregated over multiple levels. We deploy a choice
of the following well-known architectures as the encoder module:

####  3.1.1 ResNet

Residual networks (ResNet) introduced residual mappings to solve the vanishing
gradient problem in deep neural networks [16]. ResNets are easy to optimize
and gain accuracy even with deeper models.

####  3.1.2 Inception V3

Inception Networks are computationally efficient architectures, both in terms
of the model parameters and their memory usage. Adapting the Inception network
for different applications while ensuring that changes do not impede its
computational efficiency is difficult. Inception V3 introduced various
strategies for optimizing network with ease of model adaptation capabilities
[17].

####  3.1.3 EfficientNet

Conventional methods make use of scaling to increase the accuracy of the
models. The models are scaled by increasing the depth/width of the network or
using higher resolution input images. EfficientNet results from a novel
scaling method that uses a compound coefficient to uniformly scale the network
across all dimensions [18].

###  3.2 Decoder Module

The decoder module is responsible for generating a semantic segmentation mask
using the aggregated high-dimensional features extracted by the encoder
module. We make use of the popular U-Net model specially designed for medical
imaging as the decoding module [15].

###  3.3 Histogram Specification

Histogram specification, or histogram matching, is a traditional image
processing technique [19] that matches the input image’s histogram to a
reference histogram. Histogram specification involves computing the cumulative
distribution function (CDF) of histograms from both the target and the
reference, following which a transformation function is obtained by mapping
each gray level $[0,255]$ from the target’s CDF (input) to the gray level in
the reference CDF. In this work, we construct the reference histogram by
averaging over histograms from every image in the training set. Using this
technique as a pre-processing step for the test data serves an important
purpose, as the distribution of the test data is converted to a similar form
seen by the network during training.

##  4 Experimental Validation

Table 1: An overview of the datasets used in this work. Dataset | Resolution | Train-set | Val-set | Test-set
---|---|---|---|---
# Images | (%) | (%) | (%)
TCIA [3] | 512 x 512 | 407 | 58 | 117
582 | (70%) | (10%) | (20%)
VHBD [4] | 512 x 512 | – | – | 167
167 | (100%)
VHBD-2 [4] | 512 x 512 | 116 | 17 | 34
167 | (70%) | (10%) | (20%)

###  4.1 Datasets

The input data preparation and label annotation were done using the tools from
Image-J software. A summary of TCIA–cancer imaging archive [3] and
VHBD–Visible human project [4] datasets, image resolution, the number of
images used in this study, and the respective data-splits for training-
validation-testing, are shown in Table 1.

###  4.2 Performance Measures

To quantify the quality of segmentation, we compute standard performance
measures for segmentation tasks commonly used in literature, specifically, the
mean Dice coefficient (mDice) and mean Intersection over Union (mIoU) [20,
21]. For a given segmentation output ($A$) and the ground truth ($B$), the
Dice coefficient is given by
$\text{Dice}=\frac{2\,|A\;\cap\;B|}{|A|\;+\;|B|}$, which can be interpreted as
a weighted average of the precision and recall, and
$\text{IoU}=\frac{|A\;\cap\;B|}{|A\;\cup\;B|}\text{,}$ (also known as Jaccard
index) is commonly used for comparing the similarity between sets ($A$) and
($B$), while penalizing their diversity.

###  4.3 Network Training

The implementations used were based on the documentation from [22]. The models
used [16, 17, 18] were pre-trained on the Imagenet [23] dataset to improve the
generalization capability on unseen data and achieve faster convergence. For
the base-model, we use ResNet-34 [16] as the encoder and a U-Net decoder. We
initialize the base-model with random weights (rnwt) and train without any
data-augmentation (noaug) on images from [3], using an Nvidia RTX 2070 GPU,
and an ADAM optimizer with a learning rate of 0.001, momentum of 0.9 and a
weight decay of 0.0001, for 40 epochs. We chose a 70% : 10% : 20% split of the
data (shown in the first row of Table 1), where the 70% was utilized for
training and the 10% of the data was utilized for validation. The remaining
20% for testing was completely unseen during training. About 50 passes of
random image batches, of size eight, from the training set, were used in each
epoch. The model was then validated on the 10% data to evaluate the
performance based on binary-cross-entropy loss (bce) and record the
corresponding weights. After training, the weights that gave the best
performance on the validation set were selected for the base-model, which was
then evaluated on the unseen test-sets, i.e., 20% of [3] and 100% of [4],
respectively, whose performance is showcased in the first row of Table 2.

Extending beyond the base-model, data augmentation (aug) was performed using
horizontal and vertical flips, affine transforms, image intensity modulation
and blurring, for increasing training data size and to help reduce over-
fitting. In addition, we try to find the best overall segmentation performance
and generalization capability to completely unseen data, through further
extension of the base-model with different configurations, using the
following:

  * •

encoder modules using ResNet-34 [16], Inception V3 [17] and EfficientNet-B0
[18], initialized with Imagenet weights (imwt) for transfer learning

  * •

re-configuration of input data, or not, to the pre-trained model’s format and
its pre-processing functions (ppr), for extraction of better features

  * •

loss functions like Dice loss (dice), IoU loss (iou) and combined bce-iou
loss, in place of bce loss, for propagating strong gradients for better
optimization and learning

$\begin{array}[]{ccc}\includegraphics[width=71.13188pt,height=71.13188pt]{images1/inp.png}&\includegraphics[width=71.13188pt,height=71.13188pt]{images1/dl1.png}&\includegraphics[width=71.13188pt,height=71.13188pt]{images1/dl2.png}\\\
\mbox{Input}&\mbox{(a)}&\mbox{(b)}\\\ \end{array}$

Fig. 3: Pelvic bone segmentation on TCIA data using: (a) Base U-Net with
random weight initialization for ResNet-34 encoder, with no data-augmentation,
optimized using BCE loss (least performing); and (b) fine-tuned U-Net with
Imagenet weight initialization for EfficientNet-B0 encoder, with data-
augmentation and input reconfiguration, optimized using combined BCE-IoU loss
(best performing), are overlaid onto the binary ground-truth; yellow - TP;
black - TN; green - FP; red - FN.

$\begin{array}[]{ccc}\includegraphics[width=71.13188pt,height=71.13188pt]{images1/a.png}&\includegraphics[width=71.13188pt,height=56.9055pt]{images1/b1.png}&\includegraphics[width=71.13188pt,height=71.13188pt]{images1/c.png}\\\
\mbox{TCIA}&\mbox{(a1)}&\mbox{(a2)}\\\
\includegraphics[width=71.13188pt,height=71.13188pt]{images1/d.png}&\includegraphics[width=71.13188pt,height=56.9055pt]{images1/e1.png}&\includegraphics[width=71.13188pt,height=71.13188pt]{images1/f.png}\\\
\mbox{VHBD (target)}&\mbox{(b1)}&\mbox{(b2)}\\\
\includegraphics[width=71.13188pt,height=71.13188pt]{images1/g.png}&\includegraphics[width=71.13188pt,height=56.9055pt]{images1/h1.png}&\includegraphics[width=71.13188pt,height=71.13188pt]{images1/i.png}\\\
\mbox{H-VHBD}&\mbox{(c1)}&\mbox{(c2)}\\\ \end{array}$

Fig. 4: Performance in segmentation with histogram specification: (a1-c1) show
the respective histograms of the input images; (a2-c2) show the pelvic bone
segmentations overlaid on the ground-truth; and (b2-c2) decisively show the
improvement in segmentation from matching target’s histogram to the reference.
yellow - TP; black - TN; green - FP; red - FN. Table 2: Performance comparison
of different U-Net configurations for pelvic bone segmentation on unseen data
from TCIA, VHBD, and H-VHBD, i.e., VHBD after histogram specification.

U-Net Configurations | TCIA | VHBD | H-VHBD
---|---|---|---
mIoU |  mDice |  mIoU |  mDice |  mIoU |  mDice
Res34-rnwt-noaug-bce |  0.788 $\pm$ 0.033 |  0.867 $\pm$ 0.025 |  0.131 $\pm$ 0.033 |  0.186 $\pm$ 0.037 |  0.774 $\pm$ 0.021 |  0.865 $\pm$ 0.015
Res34-imwt-aug-bce |  0.919 $\pm$ 0.007 |  0.957 $\pm$ 0.004 |  0.746 $\pm$ 0.028 |  0.840 $\pm$ 0.021 |  0.900 $\pm$ 0.004 |  0.947 $\pm$ 0.002
Res34-imwt-aug-dice |  0.925 $\pm$ 0.007 |  0.960 $\pm$ 0.004 |  0.523 $\pm$ 0.039 |  0.645 $\pm$ 0.038 |  0.907 $\pm$ 0.005 |  0.951 $\pm$ 0.002
Res34-imwt-aug-bce-iou |  0.922 $\pm$ 0.007 |  0.959 $\pm$ 0.004 |  0.790 $\pm$ 0.018 |  0.877 $\pm$ 0.012 |  0.906 $\pm$ 0.004 |  0.950 $\pm$ 0.002
IncepV3-imwt-aug-bce |  0.876$\pm$ 0.012 |  0.932 $\pm$ 0.007 |  0.663 $\pm$ 0.026 |  0.784 $\pm$ 0.019 |  0.906 $\pm$ 0.006 |  0.950 $\pm$ 0.003
IncepV3-ppr-imwt-aug-bce |  0.921 $\pm$ 0.011 |  0.957 $\pm$ 0.007 |  0.808 $\pm$ 0.014 |  0.890 $\pm$ 0.009 |  0.913 $\pm$ 0.005 |  0.954 $\pm$ 0.002
EffiB0-imwt-aug-bce |  0.923 $\pm$ 0.007 |  0.959 $\pm$ 0.004 |  0.835 $\pm$ 0.015 |  0.906 $\pm$ 0.010 |  0.901 $\pm$ 0.006 |  0.947 $\pm$ 0.003
EffiB0-ppr-imwt-aug-bce-iou |  0.924 $\pm$ 0.008 |  0.960 $\pm$ 0.004 |  0.836 $\pm$ 0.011 |  0.909 $\pm$ 0.006 |  0.914 $\pm$ 0.005 |  0.955 $\pm$ 0.002
ABLATION STUDY (♣) |  0.913 $\pm$ 0.007 |  0.954 $\pm$ 0.004 |  0.679 $\pm$ 0.047 |  0.801 $\pm$ 0.032 |  0.873 $\pm$ 0.013 |  0.931 $\pm$ 0.007

  * *

Encoder module \- Res34, IncepV3, EffiB0 are ResNet-34, Inception Net-V3,
EfficientNet-B0, respectively.

  * *

Encoder Weights \- rnwt and imwt are random weights and Imagenet weights,
respectively.

  * *

Augmentation \- aug and noaug means training with and without data-
augmentation, respectively.

  * *

Loss \- bce, dice, iou are the Binary Cross Entropy loss, Dice Loss and IoU
loss, respectively .

  * *

ppr\- configure input to the pre-trained backbone’s format.

  * *

Grey background\- indicates improvement due to histogram-specification based
pre-processing.

###  4.4 Results

The detailed comparisons of the different U-Net configurations’ segmentation
performance on test-sets with 95% confidence intervals are shown in Table 2.
The segmentation outputs from the least-performing (base-model) and best-
performing (fine-tuned U-Net with Imagenet weight initialization for
EfficientNet-B0 encoder [18], with data-augmentation and input re-
configuration, optimized using combined BCE-IoU loss) DL models are showcased
in Fig. 3 (a) &(b). The predicted outputs are overlaid onto the ground-truth
and color-coded (yellow - TP; black - TN; green - FP; red - FN) for
visualizing the quality of segmentation. The results shown in Fig. 4(b2) &(c2)
illustrate the desired effect on segmentation due to histogram specification.
The reduction in the number of pixels labeled as FPs & FNs, and improvement in
number of TPs from the overlays decisively show the significance of pre-
processing test-data, which clearly boosts the model’s segmentation
performance. Furthermore, the comparitive results tabulated in the last two
columns of Table 2 give strong evidence for the success of the proposed
methodology on all the specified model configurations.

On analyzing the data shown in Table 3, the proposed methodology’s overall
performance on the test-sets surpassed several state-of-the-art techniques
that were trained on similarly sized datasets, with the exception of Liu et
al. [14] who performed training on approximately 270,000 images. Since data
drives any model, the proposed methodology (trained only on 407 images) shows
room for further improvement in segmentation under the availability of larger
datasets.

Table 3: Overall performance comparison for pelvic bone segmentation with
state-of-the-art techniques.

Methodology | Dataset | mIoU | mDice
---|---|---|---
| (# Images) |  |
Yokota et al. [9] | Private (100) | — | 0.928
Chu et al. [10] | Private (318) | — | 0.941
Chang et al. [13] |  Private ($\sim$3420) | — | 0.949
Liu et al. [14] |  DS$\ddagger$($\sim$63K) | — | 0.984
Proposed method | TCIA,VHBD (284) | 0.919 | 0.957

  * (DS$\ddagger$) KITS19, CERVIX, ABDOMEN, MSD T10, COLONOG, CLINIC; Train:Test $\approx$ 270K: 63K; K = $10^{3}$

###  4.5 Ablation Study

Images from [3, 4], with the data splits shown in rows 1 and 3 of Table 1, are
used for training. The best-model was trained on the joint data whose test-
data performance is shown in Table 2 (♣). The results showed that training the
model on joint data degrades the performance on both datasets. The data
imbalance and the varying image tonal distributions play a significant role in
influencing the segmentation performance. And by using the proposed
methodology, the model overcomes data imbalance and generalizes well to unseen
datasets, which boosts its overall segmentation performance.

##  5 Conclusion

To sum up, in this work, we presented a novel methodology for the automated
segmentation of pelvic bones from axial CT images. We addressed the unmet need
for superior pelvic bone segmentation methodology for images with varying
properties by using histogram specification. This simple yet powerful approach
of pre-processing the test-data improved segmentation performance by a
significant margin, with the quantitative results confirming its validity.
Through our approach, the encoder-decoder configuration overcame a significant
hurdle of varying intensity distributions in CT images, which led to superior
segmentation quality. Moreover, after validating the results on publicly
available TCIA and VHBD datasets, the proposed methodology has been shown to
be highly competent with-respect-to existing state-of-the-art techniques.

Through this study, we saw that, although deep learning has pushed the limits
for image processing applications, traditional image processing techniques are
not necessarily obsolete and that combining the two approaches can lead to
superior performance in segmentation.

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# Distinguishability-based genuine nonlocality with genuine multipartite
entanglement

Zong-Xing Xiong1, Mao-Sheng Li2, Zhu-Jun Zheng2, Lvzhou Li1 1Institute of
Quantum Computing and Computer Theory, School of Computer and Engineering, Sun
Yat-Sen University, Guangzhou 510006, China
2Department of Mathematics, South China University of Technology, Guangzhou
510640, China

###### Abstract

A set of orthogonal multipartite quantum states is said to be
distinguishability-based genuinely nonlocal (also genuinely nonlocal, for
abbreviation) if the states are locally indistinguishable across any
bipartition of the subsystems. This form of multipartite nonlocality, although
more naturally arising than the recently popular “strong nonlocality” in the
context of local distinguishability, receives much less attention. In this
work, we study the distinguishability-based genuine nonlocality of a special
type of genuinely multipartite entangled states — the Greenberger-Horne-
Zeilinger (GHZ)-like states. We first show that any $5$ states of the three-
qubit GHZ basis are genuinely nonlocal, while any $4$ states of them are not.
Then for more general tripartite systems, we present a universal bound about
the cardinality for an arbitrary set of GHZ-like states to be genuinely
nonlocal. Although not necessary, entanglement is believed to raise difficulty
in state discrimination in many situations. In the literature, there has been
lots of studies in favor of this perspective, including the efforts seeking
for small nonlocal sets consisting of maximally entangled states in bipartite
systems. Here in the tripartite case, where GHZ-like states are studied, we
also find the existence of some small genuinely nonlocal sets: we show that
the cardinality can scale down to linear in the local dimension $d$. This
result not only substantiates the aforemention perspective in multipartite
scenario, but also suggests that there might exist substantial difference
between strong nonlocality and the normal distinguishability-based
multipartite nonlocality.

###### pacs:

03\. 67. Hk, 03. 65. Ud

Keywords: local distinguishability, quantum nonlocality, multipartite
entanglement, GHZ-like states

##  I Introduction

Quantum nonlocality is one of the most surprising properties in quantum
mechanics. The most well-known manifestation of nonlocality — Bell nonlocality
Bell1964 ; Clauser1969 ; Brunner2014 , which is revealed by violation of Bell
type inequilities Clauser1969 ; Freedman1972 ; Aspect1981 ; Aspect1982 ;
Hensen2015 ; Handsteiner2017 ; Rosenfeld2017 ; BIG Bell Test Collaboration ,
can only arise from entangled states. Apart from Bell nonlocality, there are
also other forms of nonlocality. Among them, the distinguishability-based
nonlocality, which concerns the problem about local discrimination of some set
of orthogonal quantum states has attracted much attention. It serves to
explore the fundamental question about locally accessing global information
and also the relation between entanglement and locality, which are of central
interest in quantum information theory. Unlike Bell nonlocality, however,
entanglement is not necessary for this kind of nonlocality. In Bennett99 ,
Bennett et al. presented the first example of orthogonal product states that
are indistinguishable when only local operations and classical communications
are allowed, which was known as “quantum nonlocality without entanglement”.
From then on, such nonlocality based on distinguishability has been studied
extensively (see Bennett99 ; Bennett1999 ; Walgate00 ; Walgate02 ; DiVicenzo03
; Horodecki03 ; Niset06 ; Ghosh01 ; Ghosh04 ; Fan04 ; Nathanson05 ;
Bandyopadhyay11 ; Tian2015 ; Tian15 ; Zhang2015 ; Li2019 ; LiMS2022 ; Yu12 ;
Nathanson13 ; Cosentino13 ; CosentinoR14 ; Li15 ; Yu15 ; Xu2016 ; Zhang2016 ;
Zhang2017 ; Halder2018 ; Li2018 ; Jiang2020  for an incomplete list).
Moreover, the local discrimination of quantum states has been practically
applied in a number of distributed quantum protocols such as quantum data
hiding Terhal2001 ; DiVicenzo2002 ; Eggeling2002 ; Matthews2009  and quantum
secret sharing Markham2008 ; Hillery1999 ; Schauer2010 ; Rahaman2015 ;
Wang2017 .

Although not necessary for distinguishability-based nonlocality, the existence
of entanglement is believed to raise difficulty in quantum state
discrimination in many occasions. Therefore, besides attention to the local
distinguishability of orthogonal product states, a lot of studies focus on the
local distinguishability of entangled states Ghosh01 ; Ghosh04 ; Fan04 ;
Tian2015 ; Zhang2015 ; Li2019 ; LiMS2022 ; Nathanson05 ; Bandyopadhyay11 ;
Yu12 ; Nathanson13 ; Cosentino13 ; CosentinoR14 ; Tian15 ; Li15 ; Yu15 . Among
them, the seeking of small sets of locally indistinguishable maximally
entangled states is the most attractive and much efforts have been paid to
study this problem Nathanson05 ; Bandyopadhyay11 ; Yu12 ; Nathanson13 ;
Cosentino13 ; CosentinoR14 ; Tian15 ; Li15 ; Yu15 .

Not long ago, Halder et al. Halder19  introduced the concept of local
irreduciblity, which is a stronger form of local indistinguishability. A set
of orthogonal multipartite quantum states is said to be locally irreducible if
one cannot eleminate one or more states from the whole set, on restriction
that only orthogonality-preserving local measurements are allowed. They
revealed the phenomenon of “strong nonlocality without entanglement”, which is
a nontrivial generation of Bennett’s nonlocality without entanglement
Bennett99 . Strong nonlocality of a set of orthogonal multipartite quantum
states refers to local irreducibility of these states through every
bipartition of the subsystems and the authors of Halder19  presented the first
examples of strong nonlocal sets of product states in
$\mathbb{C}^{3}\otimes\mathbb{C}^{3}\otimes\mathbb{C}^{3}$ and
$\mathbb{C}^{4}\otimes\mathbb{C}^{4}\otimes\mathbb{C}^{4}$. After that, much
attention have been paid to study the strong nonlocality of multipartite
quantum states Zhang19 ; Yuan20 ; Shi20 ; Wang21 ; Shi22 ; Shi2022 ;
Bandyopadhyay21 ; Li22 . In Shi20 , the authors constructed sets of strongly
nonlocal entangled states in tripartite system
$\mathbb{C}^{d}\otimes\mathbb{C}^{d}\otimes\mathbb{C}^{d}$ for $d\geq 3$, but
the entangled states therein are not genuinely entangled. Soon after, Wang et
al. successfully constructed strongly nonlocal sets with genuine multipartite
entanglement by relating the strong nonlocality of these special sets to the
connectivities of some graphs Wang21 . They constructed strongly nonlocal sets
of genuinely entangled states in tripartite system
$\mathbb{C}^{d}\otimes\mathbb{C}^{d}\otimes\mathbb{C}^{d}$ where $d\geq 3$,
with cardinality $d^{3}-(d-2)^{3}$ (resp. $d^{3}-(d-2)^{3}+2$) when $d$ is odd
(resp. even). In Rout19 , Rout et al. first discussed the concept of
distinguishability-based genuine nonlocality (there the authors named it
genuine nonlocality simply; we also adopt this abbreviation here and after
sometimes when no ambiguity occurs): a set of orthogonal multipartite quantum
states is said to be (distinguishability-based) genuinely nonlocal if the
states are locally indistinguishable across any bipartition of the subsystems.
Further in Li21 ; Rout21 , the authors constructed sets of multipartite
product states which are genuinely nonlocal. Namely, they found genuine
nonlocality without entanglement. This concept of multipartite nonlocality is
weaker but more natural than strong nonlocality in the context of local
distinguishability. It’s obvious that strong nonlocality would imply genuine
nonlocality Halder19 , yet in what extent the former is stronger than the
latter, little is known.

In this paper, we are to study the genuine nonlocality of a special type of
genuinely multipartite entangled states — the Greenberger-Horne-Zeilinger
(GHZ)-like states ghz1990 . More explicitly, the states discussed are on
tripartite systems with the same local dimension and they possess equal amount
of genuine entanglement. Similar to the bipartite cases, where
distinguishability of maximally entangled states are studied, one of the
motivations of our work is to reveal the enigmatic relation between
distinguishability-based nonlocality and quantum entanglement. We discover the
existence of genuinely nonlocal sets of GHZ-like states whose cardinality are
rather small: they can scale down to linear in the local dimension $d$.
Moreover, our results suggest that there might be a significant gap between
the strength of the two kinds of multipartite nonlocality: strong nonlocality
and genuine nonlocality.

The rest of this paper is organized as follows: Sec. II presents a summary of
the necessary background. In Sec. III, we discuss the genuine nonlocality of
the three-qubit GHZ basis. In Sec. IV, we study the genuine nonlocality of
GHZ-like states on tripartite system
$\mathbb{C}^{d}\otimes\mathbb{C}^{d}\otimes\mathbb{C}^{d}$ and give a
universal bound of the cardinality for any set to be genuinely nonlocal. Then,
we discuss a special type of GHZ-like states — the lattice-type GHZ-like
states in Sec. V. Taking the advantage of the struture of this type of
genuinely entangled states and also the method of semidefinite program, we’ve
been able to construct small genuinely nonlocal sets whose cardinality scale
linearly with respect to the dimension $d$. Finally, we draw a conclusion and
discuss some relating problems in Sec. VI.

##  II Preliminaries

Genuine multipartite entanglement: In bipartite system
$\mathcal{H}_{A}\otimes\mathcal{H}_{B}$, a pure state $|\Psi\rangle_{AB}$ is
called entangled if it cannot be written as the tensor product of two pure
states of the two subsystems, namely, it’s not of the form
$|\Psi\rangle_{AB}=|\alpha\rangle_{A}\otimes|\beta\rangle_{B}$. For pure state
$|\Psi\rangle_{A_{1}\cdots A_{n}}$ in multipartite system
$\mathcal{H}_{A_{1}}\otimes\cdots\otimes\mathcal{H}_{A_{n}}$, it is called
genuinely (multipartite) entangled if it is entangled for arbitrary
bipartition of the subsystems $\\{A_{1},\cdots,A_{n}\\}$. The most well known
genuinely (tripartite) entangled states are the GHZ state
$(|000\rangle+|111\rangle)/\sqrt{2}$ and the W state
$(|001\rangle+|010\rangle+|100\rangle)/\sqrt{3}$ in three-qubit system
$\mathbb{C}^{2}\otimes\mathbb{C}^{2}\otimes\mathbb{C}^{2}$. In fact, they are
the only two types of genuine tripartite entanglement in
$\mathbb{C}^{2}\otimes\mathbb{C}^{2}\otimes\mathbb{C}^{2}$ up to SLOCC
equivalence Dur2000 ; Acin2001 .

Greenberger-Horne-Zeilinger (GHZ)-like states ghz1990 : For tripartite system
$\mathbb{C}^{d_{A}}\otimes\mathbb{C}^{d_{B}}\otimes\mathbb{C}^{d_{C}}$, the
state

$\frac{e^{i\theta_{1}}|i_{1}j_{1}k_{1}\rangle+e^{i\theta_{2}}|i_{2}j_{2}k_{2}\rangle+\cdots+e^{i\theta_{w}}|i_{w}j_{w}k_{w}\rangle}{\sqrt{w}}$

(where $w\leq min\\{d_{A},d_{B},d_{C}\\}$ and $i_{m}\neq i_{n},j_{m}\neq
j_{n},k_{m}\neq k_{n}$ whenever $m\neq n$) can be proved to be genuinely
entangled. We call such form of states GHZ-like states with weight $w$. When
$d_{A}=d_{B}=d_{C}=w$, we call these states full-weighted GHZ-like states.
Similar to the role played by maximally entangled states in bipartite systems,
full-weighted GHZ-like states have been considered as generic resources in
quantum information processing tasks and are believed to have maximal amount
of multipartite entanglement Joo2003 ; Xie2021 ; Li2022 .

Three-qubit GHZ basis: In
$\mathbb{C}^{2}\otimes\mathbb{C}^{2}\otimes\mathbb{C}^{2}$, the $8$ orthogonal
states:

$\begin{split}|\psi_{0}\rangle=\frac{|000\rangle+|111\rangle}{\sqrt{2}},\
|\psi_{1}\rangle=\frac{|001\rangle+|110\rangle}{\sqrt{2}},\
|\psi_{2}\rangle=\frac{|010\rangle+|101\rangle}{\sqrt{2}},\
|\psi_{3}\rangle=\frac{|011\rangle+|100\rangle}{\sqrt{2}}\\\
|\psi_{7}\rangle=\frac{|000\rangle-|111\rangle}{\sqrt{2}},\
|\psi_{6}\rangle=\frac{|001\rangle-|110\rangle}{\sqrt{2}},\
|\psi_{5}\rangle=\frac{|010\rangle-|101\rangle}{\sqrt{2}},\
|\psi_{4}\rangle=\frac{|011\rangle-|100\rangle}{\sqrt{2}}\end{split}$ (1)

consist a set of basis for the three-qubit system. These states are all full-
weighted GHZ-like with equal amount of genuine entanglement. We call them the
three-qubit GHZ basis and they are important in a number of quantum
information processing scenarios Markham2008 ; Hillery1999 ; Schauer2010 ;
Rahaman2015 ; Wang2017 ; Karlsson1998 ; Shi2000 ; Joo2003 .

Local distinguishability: A set of orthogonal (pure) multipartite quantum
states, which is priorly known to several spatially separated parties, is said
to be locally distinguishable if the parties are able to tell exactly which
state they share through some protocol, provided only local measurements and
classical communications are allowed. Pratically, since the mathematical
structure of LOCC measurement is rather complicated, in literature it is
usually the separable measurement or the positive-partial-transpose (PPT)
measurement that is considered Bandyopadhyay13 ; Bandyopadhyay15 ; Duan09 ;
Yu14 ; Li17 .

Local irreducibility: A set of multipartite orthogonal quantum states is said
to be locally irreducible if it is not possible to locally eliminate one or
more states from the set while preserving orthogonality of the postmeasurement
states. Typical examples of locally irreducible sets include the two-qubit
Bell basis and the three-qubit GHZ basis Halder19 . While a locally
irreducible set must be locally indistinguishable, the opposite is generally
not true.

Distinguishability-based genuine nonlocality and strong nonlocality: A set of
orthogonal multipartite quantum states are called (distinguishability-based)
genuinely nonlocal if they are locally indistinguishable across every
bipartition of the subsystems. If the states are not locally reducible across
any bipartition, then they are called strongly nonlocal. Likewise, any
strongly nonlocal set is also genuinely nonlocal, while the converse is not
true. As we will show immediately in the next section, while the three-qubit
GHZ basis (1) failed to be strongly nonlocal Halder19 , they are genuinely
nonlocal beyond question.

##  III Genuine nonlocality of the three-qubit GHZ basis

In Halder19 , the authors discussed the strong nonlocality of the three-qubit
GHZ basis (1). They showed that these states are locally irreducible, while
they are not strongly nonlocal. Namely, when all the parties are spatially
separated, the three-qubit GHZ basis cannot be reduced locally, i.e., it is
impossible to locally eliminate one or more states from the whole set while
preserving orthogonality of the postmeasurement states (from this, the local
indistinguishability of the GHZ basis (1) can also be deduced). However, this
is no longer true if any two parties are allowed to join together. This can
easily be seen. For example, when $B,C$ join together, they can perform a two-
outcome joint measurement: $P_{1}^{(BC)}=|00\rangle\langle
00|+|11\rangle\langle 11|$, $P_{2}^{(BC)}=|01\rangle\langle
01|+|10\rangle\langle 10|$. The GHZ basis are then reduced into two sets:
$\mathcal{A}_{1}=\\{|000\rangle\pm|111\rangle,|011\rangle\pm|100\rangle\\}$
and
$\mathcal{A}_{2}=\\{|001\rangle\pm|110\rangle,|010\rangle\pm|101\rangle\\}$,
according to which of the two outcomes they get. The same holds when $A,B$ or
$C,A$ are joined.

Here, instead of discussing about the reducibility, we focus on the bipartite
distinguishability problem of the GHZ basis. Notice that in the last
paragraph, both of the reduced sets $\mathcal{A}_{1}$ and $\mathcal{A}_{2}$
are locally equivalent to the two-qubit Bell basis
$\\{|00\rangle\pm|11\rangle,|01\rangle\pm|10\rangle\\}$, which is known to be
locally indistinguishable (see FIG 1). Hence, in bipartition $A|BC$, the GHZ
basis (1) is not locally distinguishable. Neither in $B|CA$ or $C|AB$
likewise. This means that the set of three-qubit GHZ basis is genuinely
nonlocal. Then a natural question arises: what is the smallest subset of
states in (1) such that it retains the property of genuinely nonlocal? To
answer this, we have the following two propositions which show that any $5$
states of the GHZ basis are genuinely nonlocal while any $4$ of them are not:

######  Theorem 1.

Any $5$ states of the three-qubit GHZ basis (1) is locally indistinguishable
across any bipartition $A|BC$, $B|CA$ or $C|AB$. That is, they are genuinely
nonlocal.

The proof is rather simple, once we notice that the GHZ basis (1), no matter
by which bipartition, can be reduced into two sets, with each being locally
equivalent to the Bell basis
$\\{|00\rangle\pm|11\rangle,|01\rangle\pm|10\rangle\\}$. Since there are $5$
states to be distinguished, at least $3$ states of them come from the same
set. However, it’s well known that any $3$ states among the two-qubit Bell
basis are locally indistinguishable Ghosh01 . Hence, any $5$ states of the
three-qubit GHZ basis have the property of genuinely nonlocal. Actually, this
is a special case of the more general result which we will show immediately in
the next section (Corollary 1).

######  Theorem 2.

Any $4$ states of the three-qubit GHZ basis (1) can be locally distinguished
by at least one of the bipartitions $A|BC$, $B|CA$ or $C|AB$.

###### Proof.

This can be seen with the help of a schematic picture for the
distinguishability of the GHZ basis in different bipartitions (FIG 1).
Obviously, the $4$ states to be distinguished must be located on no less than
two vertices of the tetrahedron in FIG 1: If these $4$ states are located on
the four vertices in the tetrahedron, they can obviously be distinguished
across all the bipatitions. For example, in bipartition $A|BC$, the qubit
holders $BC$ can first perform local measurement
$\\{P_{1}^{(BC)}=|00\rangle\langle 00|+|11\rangle\langle 11|$,
$P_{2}^{(BC)}=|01\rangle\langle 01|+|10\rangle\langle 10|\\}$ to reduced the
states into two disjoint sets, with each being a $2$-ary subset of
$\\{\psi_{0},\psi_{7},\psi_{3},\psi_{4}\\}$ and
$\\{\psi_{1},\psi_{6},\psi_{2},\psi_{5}\\}$ respectively. Since any two states
are always locally distinguishable Walgate00 , these $4$ states can be locally
distinguished across $A|BC$. The same holds for $B|CA$ and $C|AB$; If the $4$
states to be distinguished are distributed on three vertices, they must be
located on a certain face of the tetrahedron. Without loss of generality,
suppose that these states are $\\{\psi_{0},\psi_{7},\psi_{1},\psi_{2}\\}$. In
this case, only in bipartition $A|BC$ can they be distinguished, for in $B|CA$
(resp. $C|AB$), the subset $\\{\psi_{0},\psi_{7},\psi_{2}\\}$ (resp.
$\\{\psi_{0},\psi_{7},\psi_{1}\\}$) is locally equivalent to three states of
the Bell basis, which are known to be locally indistinguishable; If the $4$
states to be distinguished are on two vertices, they must be located on a
certain edge. For example, for $\\{\psi_{0},\psi_{7},\psi_{3},\psi_{4}\\}$,
they are not locally distinguishable through $A|BC$ while they can be
distinguished through $B|CA$ and $C|AB$. As a result, any $4$ states of the
three-qubit GHZ basis (1) can be locally distinguished by at least one of the
three bipartitions, i.e., they are not genuinely nonlocal.

Figure 1:  Schematic picture for local distinguishability of the three-qubit
GHZ basis in all bipartitions: Each vertices of the tetrahedron represents one
pair of the GHZ basis (the pair with only phase difference). Four states on
one edge are locally equivalent to the two-qubit Bell basis
$\\{|00\rangle\pm|11\rangle$, $|01\rangle\pm|10\rangle\\}$, within bipartition
corresponding to the colour of that edge. As an example, in bipartition
$A|BC$, the four states of $\mathcal{A}_{1}$ are locally equivalent to the
four Bell basis. The same holds for $\mathcal{A}_{2}$. The two disjoint sets
are on opposite edge of the tetrahedron coloured blue, and states from
different sets can be told apart if $BC$ perform local measurement
$\\{P_{1}^{(BC)}=|00\rangle\langle 00|+|11\rangle\langle 11|$,
$P_{2}^{(BC)}=|01\rangle\langle 01|+|10\rangle\langle 10|\\}$.

##  IV More general cases — GHZ-like states on three-qudit systems

In Wang21 , the authors studied the local reducibility of tripartite genuinely
entangled states and constructed different sets of GHZ-like states with same
weight that are strongly nonlocal. For example, in
$\mathbb{C}^{3}\otimes\mathbb{C}^{3}\otimes\mathbb{C}^{3}$ they constructed a
strongly nonlocal set of $26$ GHZ-like states with weight $2$ :
$\mathcal{N}=\mathcal{N}_{1}\cup\mathcal{N}_{2}\cup\mathcal{N}_{3}\cup\mathcal{N}_{4}$,
where

$\begin{split}&\mathcal{N}_{1}=\\{|0\rangle|i\rangle|j\oplus
1\rangle\pm|2\rangle|i\oplus
1\rangle|j\rangle_{{}_{ABC}}|(i,j)\in\mathbb{Z}_{2}\times\mathbb{Z}_{2}\\},\\\
&\mathcal{N}_{2}=\\{|i\oplus
1\rangle|0\rangle|j\rangle\pm|i\rangle|2\rangle|j\oplus
1\rangle_{{}_{ABC}}|(i,j)\in\mathbb{Z}_{2}\times\mathbb{Z}_{2}\\},\\\
&\mathcal{N}_{3}=\\{|i\rangle|j\oplus 1\rangle|0\rangle\pm|i\oplus
1\rangle|j\rangle|2\rangle_{{}_{ABC}}|(i,j)\in\mathbb{Z}_{2}\times\mathbb{Z}_{2}\\},\\\
&\mathcal{N}_{4}=\\{|0\rangle|0\rangle|0\rangle\pm|2\rangle|2\rangle|2\rangle_{{}_{ABC}}\\}.\end{split}$
(2)

In $\mathbb{C}^{4}\otimes\mathbb{C}^{4}\otimes\mathbb{C}^{4}$, they also
presented a strongly nonlocal set of full-weighted GHZ-like basis (see Example
3 of Wang21 ).

It is immediate that these sets are also genuinely nonlocal. However, since
strong nonlocality is undoubtedly stronger than genuinely nonlocality, it’s
natural to ask whether some of these states can be excluded so that the
remaining states maintain the property of genuine nonlocality (similar to the
case of three-qubit GHZ basis)? Fortunately, we have the following theorem
that shows us the answer. The technique we use here is the PPT-
indistinguishability which was first originated from Yu12 .

######  Theorem 3.

Let $\mathcal{S}=\\{|\Phi_{1}\rangle,\cdots,|\Phi_{s}\rangle\\}$ be a set of
orthogonal GHZ-like states with the same weight $w$ in tripartite system
$\mathbb{C}^{d}\otimes\mathbb{C}^{d}\otimes\mathbb{C}^{d}$. If
$|\mathcal{S}|=s>d^{3}/w$, then $\mathcal{S}$ must be gennuinely nonlocal.

###### Proof.

Suppose that the states in $\mathcal{S}$ can be locally distinguished through
the bipatition $A|BC$, then there exist LOCC POVMs $P_{1},\cdots,P_{s}\geq 0$
satisfying $P_{1}+\cdots+P_{s}=I_{{}_{ABC}}$ and
$Tr(P_{k}\Phi_{l})=\delta_{kl}\ (\forall k,l)$. These POVMs are all seperable
ones across $A|BC$, for all that is required to distinguish $\mathcal{S}$ is a
series of local operators and classical communications through the
bipartition. What’s more, they have positive partial transposition, i.e.,
$P_{k}^{\mathrm{T}_{A}}\geq 0\ (\forall k)$.

Let $\\{|i\rangle_{{}_{A}}\\}_{i=1}^{d}$ and
$\\{|i\rangle_{{}_{BC}}\\}_{i=1}^{d^{2}}$ be some fix basis of subsystem $A$
and $BC$ respectively. Since all states $|\Phi_{k}\rangle_{(k=1,\cdots,s)}$
have weight $w$, they can be written as the form
$|\Phi_{k}\rangle=(U_{A}\otimes
U_{BC})\frac{1}{\sqrt{w}}\sum_{i=1}^{w}|i\rangle_{A}\otimes|i\rangle_{BC}$,
where $U_{A}$, $U_{BC}$ are local unitary operators that depend on $k$ and

$\Phi_{k}^{\mathrm{T}_{A}}=(U_{A}^{\mathrm{T}}\otimes
U_{BC})\frac{1}{w}\sum_{i,j=1}^{w}|ji\rangle\langle
ij|_{{}_{A|BC}}(\overline{U}_{A}\otimes U_{BC}^{{\dagger}}).$

Notice that the operator $\frac{1}{w}\sum_{i,j=1}^{w}|ji\rangle\langle
ij|_{{}_{A|BC}}$ has only eigenvalues $\frac{1}{w}$ and ${-\frac{1}{w}}$ on
its support, so

$\frac{1}{w}\sum_{i,j=1}^{w}|ji\rangle\langle
ij|_{{}_{A|BC}}\leq\frac{1}{w}I_{{}_{ABC}}$

and hence $\Phi_{k}^{\mathrm{T}_{A}}\leq\frac{1}{w}I_{{}_{ABC}}$. Now by the
positivity of $P_{k}^{\mathrm{T}_{A}}$ we have

$1=Tr(P_{k}\Phi_{k})=Tr(P_{k}^{\mathrm{T}_{A}}\Phi_{k}^{\mathrm{T}_{A}})\leq
Tr(P_{k}^{\mathrm{T}_{A}}I/w),$

so $Tr(P_{k})=Tr(P_{k}^{\mathrm{T}_{A}})\geq w$ for all $k$. On the other
hand, since $P_{1}+\cdots+P_{s}=I$, we have

$d^{3}=Tr(I_{{}_{ABC}})=\sum_{k=1}^{s}Tr(P_{k})\geq sw.$

Therefore, if $|\mathcal{S}|=s>d^{3}/w$, the states in $\mathcal{S}$ can not
be locally distinguished through $A|BC$. The arguements above obviously also
hold for the other two bipatitions $B|CA$ and $C|AB$, which completes the
proof.

This theorem gives a universal bound of the cardinality for a general GHZ-like
set to be genuinely nonlocal. As a result, by letting $w=2$, we find that any
$14$ states among set (LABEL:333) has the property of genuine nonlocality,
which is quite out of our intuition. Also, in
$\mathbb{C}^{4}\otimes\mathbb{C}^{4}\otimes\mathbb{C}^{4}$, any $17$ states of
the set given by Example 3 of Wang21  are genuinely nonlocal. It would be
natural to ask: can we further find such genuinely nonlocal sets with less
states? We should point out here that the answer would be yes but somewhat
trivial, if the considered GHZ-like states are not full-weighted. For example,
considering the following set:

$\begin{split}&|i_{0}\rangle|j_{0}\rangle|k_{0}\rangle\pm|i_{1}\rangle|j_{1}\rangle|k_{1}\rangle_{{}_{ABC}}\\\
&|i_{0}\rangle|j_{0}\rangle|k_{1}\rangle\pm|i_{1}\rangle|j_{1}\rangle|k_{0}\rangle_{{}_{ABC}}\\\
&|i_{0}\rangle|j_{1}\rangle|k_{0}\rangle\pm|i_{1}\rangle|j_{0}\rangle|k_{1}\rangle_{{}_{ABC}}\\\
&|i_{0}\rangle|j_{1}\rangle|k_{1}\rangle\pm|i_{1}\rangle|j_{0}\rangle|k_{0}\rangle_{{}_{ABC}}\
\ (i_{0}\neq i_{1},j_{0}\neq j_{1},k_{0}\neq k_{1})\end{split}$

in tripartite system
$\mathbb{C}^{d}\otimes\mathbb{C}^{d}\otimes\mathbb{C}^{d}(d>2)$, which is
locally equivalent to the three-qubit GHZ basis (1). By Theorem 1, any $5$
states of them are genuinely nonlocal. That is to say, even if the system
dimension is arbitrarily high, one can always find a genuinely nonlocal set of
GHZ-like states with cardinality down to $5$, if full-weighted is not
requested. As a consequence, we find it more meaningful to consider the
genuine nonlocality of full-weightd GHZ-like states. For this, we have the
following result that is a direct corollary of Theorem 3:

######  Corollary 1.

For tripartite system
$\mathbb{C}^{d}\otimes\mathbb{C}^{d}\otimes\mathbb{C}^{d}$, any set of
$s>d^{2}$ orthogonal full-weighted GHZ-like states is gennuinely nonlocal.

When $d=2$, namely, in the three-qubit case, this bound is tight universally,
as it is shown by Theorem 2. However, when $d>2$, this is no longer the fact.
In the next section, we are studying the genuine nonlocality of a special type
of GHZ-like states — the lattice-type GHZ-like states, which are full-weighted
in tripartite systems
$\mathbb{C}^{2^{t}}\otimes\mathbb{C}^{2^{t}}\otimes\mathbb{C}^{2^{t}}\ (t\geq
1)$. We will show that while in some cases, the bound given by Corollary 1 is
tight, for some others however, it is by no means effective. That is, there
exist genuinely nonlocal sets of lattice-type GHZ-like states of which the
cardinality are significantly smaller than $d^{2}$.

##  V Lattice-type GHZ-like states

In this section, we will discuss the genuine nonlocality of a particular type
of GHZ-like states — the lattice-type GHZ-like states, which are tensor
products of the three-qubit GHZ basis. Let
$\overrightarrow{v}=(v_{1},\cdots,v_{t})\in\\{0,1,\cdots,7\\}^{t}$ be a
$t$-dimensional vector $(t\geq 1)$. Then the $t$-level lattice-type GHZ-like
state with index $\overrightarrow{v}$ in tripartite system
$\mathbb{C}^{2^{t}}\otimes\mathbb{C}^{2^{t}}\otimes\mathbb{C}^{2^{t}}$ is
defined by
$|\Psi_{\overrightarrow{v}}\rangle=|\psi_{v_{1}}\rangle\otimes\cdots\otimes|\psi_{v_{t}}\rangle$.
One can easily check that these states consist a basis of
$\mathbb{C}^{2^{t}}\otimes\mathbb{C}^{2^{t}}\otimes\mathbb{C}^{2^{t}}$ and
they are full-weighted GHZ-like. The idea here is similar to that of
Cosentino13 , where lattice-type maximally entangled states on bipartite
systems were considered. For abbreviation, here and after we also call
lattice-type GHZ-like states “GHZ-lattices” sometimes. It turns out that when
the multipartite states to be distinguished are all GHZ-lattices, the special
structure of these genuinely entangled states enable us to discuss their
genuine nonlocality with much less efforts. First, we show that in some of the
cases where GHZ-lattices are distinguished, the bound given by Corollary 1 is
tight:

######  Proposition 1.

Suppose $0\leq a<b<c<d\leq 7$, then the following set of full-weighted GHZ-
like states

$\begin{split}\mathcal{L}^{16}=\\{&|\psi_{a}\rangle\otimes|\psi_{a}\rangle,|\psi_{a}\rangle\otimes|\psi_{b}\rangle,|\psi_{a}\rangle\otimes|\psi_{c}\rangle,|\psi_{a}\rangle\otimes|\psi_{d}\rangle,\\\
&|\psi_{b}\rangle\otimes|\psi_{a}\rangle,|\psi_{b}\rangle\otimes|\psi_{b}\rangle,|\psi_{b}\rangle\otimes|\psi_{c}\rangle,|\psi_{b}\rangle\otimes|\psi_{d}\rangle,\\\
&|\psi_{c}\rangle\otimes|\psi_{a}\rangle,|\psi_{c}\rangle\otimes|\psi_{b}\rangle,|\psi_{c}\rangle\otimes|\psi_{c}\rangle,|\psi_{c}\rangle\otimes|\psi_{d}\rangle,\\\
&|\psi_{d}\rangle\otimes|\psi_{a}\rangle,|\psi_{d}\rangle\otimes|\psi_{b}\rangle,|\psi_{d}\rangle\otimes|\psi_{c}\rangle,|\psi_{d}\rangle\otimes|\psi_{d}\rangle\\}\end{split}$

in $\mathbb{C}^{4}\otimes\mathbb{C}^{4}\otimes\mathbb{C}^{4}$ is locally
distinguishable through at least one of the bipartitions $A|BC$, $B|CA$ and
$C|AB$.

###### Proof.

Note that the parties $A,B,C$ are actually holding two shares of three-qubit
GHZ states, with each one coming from the set
$\\{|\psi_{a}\rangle,|\psi_{b}\rangle,|\psi_{c}\rangle,|\psi_{d}\rangle\\}$.
By the proof of Theorem 2, these states on
$\mathbb{C}^{2}\otimes\mathbb{C}^{2}\otimes\mathbb{C}^{2}$ can always be
locally distinguished by at least one of the bipartitions, say $A|BC$, without
loss of generality. Then, $A$ and $BC$ can repeat the procedure twice to find
out which two of the GHZ states they share and the conclusion comes out.

For $d=2^{t}$ similarly, when the parties $A,B,C$ hold $t$ shares of three-
qubit GHZ states from
$\\{|\psi_{a}\rangle,|\psi_{b}\rangle,|\psi_{c}\rangle,|\psi_{d}\rangle\\}$,
at least one of the bipartitions can explicitly tell what they are if they
repeat the same procedure $t$ times. That is to say:

######  Corollary 2.

In tripartite system
$\mathbb{C}^{d}\otimes\mathbb{C}^{d}\otimes\mathbb{C}^{d}$ where
$d=2^{t}(t\geq 1)$, there exist $d^{2}$ orthogonal full-weighted GHZ-like
states that can be locally distinguished by at least one of the bipartitions
$A|BC$, $B|CA$ and $C|AB$.

As what we’ve shown in Sec. III, in the case $t=1$ this bound given by
Corollary 1 is universally tight. It’s then natural and meaningful to consider
the case when $t>1$: do there exist some genuinely nonlocal sets with
cardinality smaller than $d^{2}$? In what follows, with the help of
semidefinite program, we will show a positive answer to this question by
constructing small genuinely nonlocal sets of lattice-type GHZ-like states,
whose cardinality scale down to linear in the local dimension $d$.

Since the LOCC distinguishability is diffcult to tackle, here instead we study
the PPT-distinguishability of the multipartite states. For bipartite system
$\mathcal{H}_{A}\otimes\mathcal{H}_{B}$, if a set of orthogonal states is
locally distinguishable, the states must be also PPT-distinguishable.
Furthermore, the orthogonal bipartite states
$\\{|\phi_{1}\rangle,\cdots,|\phi_{s}\rangle\\}$ (here and after , it’s also
denoted as $\\{\phi_{1},\cdots,\phi_{s}\\}$ where
$\phi_{i}=|\phi_{i}\rangle\langle\phi_{i}|$’s are the respective density
operators) are PPT-dinstinguishable if and only if the following semidefinite
program has optimal value $\alpha=1$ Cosentino13 :

$\begin{split}\alpha=&\max_{{}_{P_{1},\cdots,P_{s}}}\phantom{=}\frac{1}{s}\sum_{k=1}^{s}Tr(P_{k}\phi_{k})\\\
s.t.&\phantom{=}P_{1}+\cdots+P_{s}=I_{{}_{AB}}\\\
&\phantom{=}P_{1},\cdots,P_{s}\geq 0\\\
&\phantom{=}\mathrm{T}_{A}(P_{1}),\cdots,\mathrm{T}_{A}(P_{s})\geq
0.\end{split}$ (3)

Inspired by the bipartite case, we can make use of this semidefinte program to
construct genuinely nonlocal sets on multipartite systems. For a set of
orthogonal states $\mathcal{S}=\\{|\Phi_{1}\rangle,\cdots,|\Phi_{s}\rangle\\}$
on tripartite system
$\mathcal{H}_{A}\otimes\mathcal{H}_{B}\otimes\mathcal{H}_{C}$, a sufficient
condition for their genuine nonlocality is that across each bipartition these
states are PPT-indistinguishable. More explicitly, consider the semidefinite
programs

$\begin{split}\alpha_{X}=&\max_{{}_{P_{1}^{(X)},\cdots,P_{s}^{(X)}}}\phantom{=}\frac{1}{s}\sum_{k=1}^{s}Tr[P_{k}^{(X)}\Phi_{k}]\\\
s.t.&\phantom{=}P_{1}^{(X)}+\cdots+P_{s}^{(X)}=I_{{}_{ABC}}\\\
&\phantom{=}P_{1}^{(X)},\cdots,P_{s}^{(X)}\geq 0\\\
&\phantom{=}\mathrm{T}_{X}[P_{1}^{(X)}],\cdots,\mathrm{T}_{X}[P_{s}^{(X)}]\geq
0\end{split}$ (4)

where $X=A,B$ or $C$ represents that we are dealing with the bipartition
$A|BC$, $B|CA$ or $C|AB$ respectively. If for a specific set $\mathcal{S}$ we
have $\alpha_{X}<1$ for $\forall X\in\\{A,B,C\\}$, namely, PPT-
indistinguishable through all three bipartitions, then the set is genuinely
nonlocal. For convenience of calculation, we further consider the dual
problems:

$\begin{split}\beta_{X}=&\min_{{}_{Y^{(X)},Q_{1}^{(X)},\cdots,Q_{s}^{(X)}}}\phantom{=}\frac{1}{s}\
Tr[Y^{(X)}]\\\
s.t.&\phantom{=}Y^{(X)}-\Phi_{k}\geq\mathrm{T}_{X}[Q_{k}^{(X)}]\ \
(k=1,\cdots,s)\\\ &\phantom{=}Q_{1}^{(X)},\cdots,Q_{s}^{(X)}\geq 0\end{split}$
(5)

where $X\in\\{A,B,C\\}$. By the Slater’s condition, we know that strong
duality holds and so $\alpha_{X}=\beta_{X}\ (X\in\\{A,B,C\\})$. Therefore,
given a set $\mathcal{S}=\\{|\Phi_{1}\rangle,\cdots,|\Phi_{s}\rangle\\}$ of
orthogonal tripartite states, once we found that the minimum $\beta_{X}<1$ for
$\forall X\in\\{A,B,C\\}$, the genuine nonlocality of $\mathcal{S}$ is then
concluded.

To use these semidefinite programs to construct genuinely nonlocal sets of
GHZ-lattices, notice that for subsystems $X\in\\{A,B,C\\}$,
$\mathrm{T}_{X_{1}\cdots
X_{t}}(\Psi_{\overrightarrow{v}})=\mathrm{T}_{X_{1}}(\psi_{v_{1}})\otimes\cdots\otimes\mathrm{T}_{X_{t}}(\psi_{v_{t}})$
where $\psi_{v_{i}}=|\psi_{v_{i}}\rangle\langle\psi_{v_{i}}|$
($\Psi_{\overrightarrow{v}}=|\Psi_{\overrightarrow{v}}\rangle\langle\Psi_{\overrightarrow{v}}|$)
are the density operators of the three-qubit GHZ basis ($t$-level GHZ-
lattices). The subscripts “${}_{X_{1}\cdots X_{t}}$” indicate that each
subsystem holds $t$ qubits and we sometimes abbreviate it as “X” when there’s
no ambiguity. We should also note that the partial transpose operations
$\mathrm{T}_{X}$’s act linearly on
$B(\mathcal{H}_{A}\otimes\mathcal{H}_{B}\otimes\mathcal{H}_{C})$, the linear
space of all linear operators on Hilbert space
$\mathcal{H}_{A}\otimes\mathcal{H}_{B}\otimes\mathcal{H}_{C}$. Moreover, when
GHZ-lattices are considered, we have the following property about
$\mathrm{T}_{A},\mathrm{T}_{B}$ and $\mathrm{T}_{C}$ which is crucial to all
subsequent discussions about the genuine nonlocality of these states:

######  Lemma 1.

In the linear space
$B(\mathbb{C}^{2}\otimes\mathbb{C}^{2}\otimes\mathbb{C}^{2})$ of all linear
operators on $\mathbb{C}^{2}\otimes\mathbb{C}^{2}\otimes\mathbb{C}^{2}$, the
linear maps $\mathrm{T}_{A},\mathrm{T}_{C}$ and $\mathrm{T}_{C}$ act
invariantly on the subspace spanned by $\\{\psi_{0},\cdots,\psi_{7}\\}$.
Moreover, on this subspace they have the matrix representation:

$\begin{split}&\mathrm{T}_{A}\begin{bmatrix}\psi_{0}\\\ \psi_{7}\\\
\psi_{3}\\\ \psi_{4}\\\ \psi_{1}\\\ \psi_{6}\\\ \psi_{2}\\\
\psi_{5}\end{bmatrix}=\left[\begin{array}[]{cc}\begin{matrix}&&\frac{1}{2}&&\frac{1}{2}&&\frac{1}{2}&-&\frac{1}{2}\\\
&&\frac{1}{2}&&\frac{1}{2}&-&\frac{1}{2}&&\frac{1}{2}\\\
&&\frac{1}{2}&-&\frac{1}{2}&&\frac{1}{2}&&\frac{1}{2}\\\
&-&\frac{1}{2}&&\frac{1}{2}&&\frac{1}{2}&&\frac{1}{2}\\\ \end{matrix}&0\\\
0&\begin{matrix}&&\frac{1}{2}&&\frac{1}{2}&&\frac{1}{2}&-&\frac{1}{2}\\\
&&\frac{1}{2}&&\frac{1}{2}&-&\frac{1}{2}&&\frac{1}{2}\\\
&&\frac{1}{2}&-&\frac{1}{2}&&\frac{1}{2}&&\frac{1}{2}\\\
&-&\frac{1}{2}&&\frac{1}{2}&&\frac{1}{2}&&\frac{1}{2}\\\
\end{matrix}\end{array}\right]\begin{bmatrix}\psi_{0}\\\ \psi_{7}\\\
\psi_{3}\\\ \psi_{4}\\\ \psi_{1}\\\ \psi_{6}\\\ \psi_{2}\\\
\psi_{5}\end{bmatrix},\\\ \\\ &\mathrm{T}_{B}\begin{bmatrix}\psi_{0}\\\
\psi_{7}\\\ \psi_{2}\\\ \psi_{5}\\\ \psi_{3}\\\ \psi_{4}\\\ \psi_{1}\\\
\psi_{6}\end{bmatrix}=\left[\begin{array}[]{cc}\begin{matrix}&&\frac{1}{2}&&\frac{1}{2}&&\frac{1}{2}&-&\frac{1}{2}\\\
&&\frac{1}{2}&&\frac{1}{2}&-&\frac{1}{2}&&\frac{1}{2}\\\
&&\frac{1}{2}&-&\frac{1}{2}&&\frac{1}{2}&&\frac{1}{2}\\\
&-&\frac{1}{2}&&\frac{1}{2}&&\frac{1}{2}&&\frac{1}{2}\\\ \end{matrix}&0\\\
0&\begin{matrix}&&\frac{1}{2}&&\frac{1}{2}&&\frac{1}{2}&-&\frac{1}{2}\\\
&&\frac{1}{2}&&\frac{1}{2}&-&\frac{1}{2}&&\frac{1}{2}\\\
&&\frac{1}{2}&-&\frac{1}{2}&&\frac{1}{2}&&\frac{1}{2}\\\
&-&\frac{1}{2}&&\frac{1}{2}&&\frac{1}{2}&&\frac{1}{2}\\\
\end{matrix}\end{array}\right]\begin{bmatrix}\psi_{0}\\\ \psi_{7}\\\
\psi_{2}\\\ \psi_{5}\\\ \psi_{3}\\\ \psi_{4}\\\ \psi_{1}\\\
\psi_{6}\end{bmatrix},\\\ \\\ &\mathrm{T}_{C}\begin{bmatrix}\psi_{0}\\\
\psi_{7}\\\ \psi_{1}\\\ \psi_{6}\\\ \psi_{2}\\\ \psi_{5}\\\ \psi_{3}\\\
\psi_{4}\end{bmatrix}=\left[\begin{array}[]{cc}\begin{matrix}&&\frac{1}{2}&&\frac{1}{2}&&\frac{1}{2}&-&\frac{1}{2}\\\
&&\frac{1}{2}&&\frac{1}{2}&-&\frac{1}{2}&&\frac{1}{2}\\\
&&\frac{1}{2}&-&\frac{1}{2}&&\frac{1}{2}&&\frac{1}{2}\\\
&-&\frac{1}{2}&&\frac{1}{2}&&\frac{1}{2}&&\frac{1}{2}\\\ \end{matrix}&0\\\
0&\begin{matrix}&&\frac{1}{2}&&\frac{1}{2}&&\frac{1}{2}&-&\frac{1}{2}\\\
&&\frac{1}{2}&&\frac{1}{2}&-&\frac{1}{2}&&\frac{1}{2}\\\
&&\frac{1}{2}&-&\frac{1}{2}&&\frac{1}{2}&&\frac{1}{2}\\\
&-&\frac{1}{2}&&\frac{1}{2}&&\frac{1}{2}&&\frac{1}{2}\\\
\end{matrix}\end{array}\right]\begin{bmatrix}\psi_{0}\\\ \psi_{7}\\\
\psi_{1}\\\ \psi_{6}\\\ \psi_{2}\\\ \psi_{5}\\\ \psi_{3}\\\
\psi_{4}\end{bmatrix}.\end{split}$ (6)

###### Proof.

It is routine to check this result. For example, notice that
$\mathrm{T}_{A}(\psi_{0})=(|000\rangle\langle 000|+|111\rangle\langle
111|+|100\rangle\langle 011|+|011\rangle\langle 100|)/2$, and
$\psi_{4}=(|011\rangle\langle 011|+|100\rangle\langle 100|-|011\rangle\langle
100|-|100\rangle\langle 011|)/2$. Then by adding these two operators we
eliminate the cross terms:

$\mathrm{T}_{A}(\psi_{0})+\psi_{4}=\frac{1}{2}(|000\rangle\langle
000|+|111\rangle\langle 111|+|011\rangle\langle 011|+|100\rangle\langle
100|).$ (7)

Moreover, $\psi_{0}+\psi_{7}=|000\rangle\langle 000|+|111\rangle\langle 111|$
while $\psi_{3}+\psi_{4}=|011\rangle\langle 011|+|100\rangle\langle 100|$.
Therefore, we get
$\mathrm{T}_{A}(\psi_{0})=(\psi_{0}+\psi_{7}+\psi_{3}-\psi_{4})/2$ as it is
shown in (LABEL:transform). By acting $\mathrm{T}_{A}$ on (7), we get
$\psi_{0}+\mathrm{T}_{A}(\psi_{4})=(\psi_{0}+\psi_{7}+\psi_{3}+\psi_{4})/2$
and so $\mathrm{T}_{A}(\psi_{4})=(-\psi_{0}+\psi_{7}+\psi_{3}+\psi_{4})/2$.
Similarly,

$\mathrm{T}_{A}(\psi_{3})+\psi_{7}=\mathrm{T}_{A}(\psi_{7})+\psi_{3}=\frac{1}{2}(\psi_{0}+\psi_{7}+\psi_{3}+\psi_{4}),$
(8)

hence $\mathrm{T}_{A}(\psi_{7})=(\psi_{0}+\psi_{7}-\psi_{3}+\psi_{4})/2,\
\mathrm{T}_{A}(\psi_{3})=(\psi_{0}-\psi_{7}+\psi_{3}+\psi_{4})/2$ and so forth
for the remain.

Consider the subset
$\mathcal{S}^{5}=\\{\psi_{1},\psi_{2},\psi_{3},\psi_{4},\psi_{5}\\}$ of three-
qubit GHZ basis (1), namely $\Phi_{k}=\psi_{k}\ (k=1,\cdots,5)$. With the help
of (LABEL:transform), one can check that what we list in the following
constitutes a feasible solution to the dual problems (5) (namely, the
conditions $Y^{(X)}-\Phi_{k}\geq\mathrm{T}_{X}(Q_{k}^{(X)})$ are satisfied for
$1\leq k\leq 5$ and $X\in\\{A,B,C\\}$):

$\begin{split}&Y^{(A)}=\frac{1}{2}\psi_{1}+\frac{1}{2}\psi_{2}+\psi_{3}+\psi_{4}+\frac{1}{2}\psi_{5}+\frac{1}{2}\psi_{6},\\\
&Q_{1}^{(A)}=\psi_{5},Q_{2}^{(A)}=\psi_{6},Q_{3}^{(A)}=Q_{4}^{(A)}=0,Q_{5}^{(A)}=\psi_{1};\\\
\\\
&Y^{(B)}=\frac{1}{2}\psi_{1}+\psi_{2}+\frac{1}{2}\psi_{3}+\frac{1}{2}\psi_{4}+\psi_{5}+\frac{1}{2}\psi_{6},\\\
&Q_{1}^{(B)}=\psi_{4},Q_{2}^{(B)}=0,Q_{3}^{(B)}=\psi_{6},Q_{4}^{(B)}=\psi_{1},Q_{5}^{(B)}=0;\\\
\\\
&Y^{(C)}=\psi_{1}+\frac{1}{2}\psi_{2}+\frac{1}{2}\psi_{3}+\frac{1}{2}\psi_{4}+\frac{1}{2}\psi_{5},\\\
&Q_{1}^{(C)}=\psi_{2},Q_{2}^{(C)}=\psi_{4},Q_{3}^{(C)}=\psi_{5},Q_{4}^{(C)}=\psi_{2},Q_{5}^{(C)}=\psi_{3}.\end{split}$

Here for example, we check the inequality constraints above only for
bipartition $A|BC$ and the others are similar by routine calculation:

$\displaystyle\mathrm{T}_{A}(Q_{1}^{(A)})=-\frac{1}{2}\psi_{1}+\frac{1}{2}\psi_{6}+\frac{1}{2}\psi_{2}+\frac{1}{2}\psi_{5}\leq-\frac{1}{2}\psi_{1}+\frac{1}{2}\psi_{2}+\psi_{3}+\psi_{4}+\frac{1}{2}\psi_{5}+\frac{1}{2}\psi_{6}=Y^{(A)}-\Phi_{1};$
$\displaystyle\mathrm{T}_{A}(Q_{2}^{(A)})=\frac{1}{2}\psi_{1}+\frac{1}{2}\psi_{6}-\frac{1}{2}\psi_{2}+\frac{1}{2}\psi_{5}\leq\frac{1}{2}\psi_{1}-\frac{1}{2}\psi_{2}+\psi_{3}+\psi_{4}+\frac{1}{2}\psi_{5}+\frac{1}{2}\psi_{6}=Y^{(A)}-\Phi_{2};$
$\displaystyle\mathrm{T}_{A}(Q_{3}^{(A)})=0\leq\frac{1}{2}\psi_{1}+\frac{1}{2}\psi_{2}+\psi_{4}+\frac{1}{2}\psi_{5}+\frac{1}{2}\psi_{6}=Y^{(A)}-\Phi_{3};$
$\displaystyle\mathrm{T}_{A}(Q_{4}^{(A)})=0\leq\frac{1}{2}\psi_{1}+\frac{1}{2}\psi_{2}+\psi_{3}+\frac{1}{2}\psi_{5}+\frac{1}{2}\psi_{6}=Y^{(A)}-\Phi_{4};$
$\displaystyle\mathrm{T}_{A}(Q_{5}^{(A)})=\frac{1}{2}\psi_{1}+\frac{1}{2}\psi_{6}+\frac{1}{2}\psi_{2}-\frac{1}{2}\psi_{5}\leq\frac{1}{2}\psi_{1}+\frac{1}{2}\psi_{2}+\psi_{3}+\psi_{4}-\frac{1}{2}\psi_{5}+\frac{1}{2}\psi_{6}=Y^{(A)}-\Phi_{5}.$

The target values for the three bipartitions are $0.8,0.8,0.6$ respectively.
Therefore, the set $\mathcal{S}^{5}$ is genuinely nonlocal. This consists with
what we’ve found in Sec. III. Now with these, we give a simple construction of
a genuinely nonlocal set of GHZ-lattices in
$\mathbb{C}^{4}\otimes\mathbb{C}^{4}\otimes\mathbb{C}^{4}$, of which the
cardinality is considerably smaller than $16$ (which is the bound given by
Corollary 1):

######  Proposition 2.

In $\mathbb{C}^{4}\otimes\mathbb{C}^{4}\otimes\mathbb{C}^{4}$, the following
set of orthogonal GHZ-like states with weight $4$

$\begin{split}\mathcal{S}^{10}=\\{\psi_{1}\otimes\psi_{0},\psi_{2}\otimes\psi_{0},\psi_{3}\otimes\psi_{0},\psi_{4}\otimes\psi_{0},\psi_{5}\otimes\psi_{0},\\\
\psi_{1}\otimes\psi_{7},\psi_{2}\otimes\psi_{7},\psi_{3}\otimes\psi_{7},\psi_{4}\otimes\psi_{7},\psi_{5}\otimes\psi_{7}\\}\end{split}$

is genuinely nonlocal.

###### Proof.

Notice that by (LABEL:transform) we have
$\mathrm{T}_{X}(\psi_{0}+\psi_{7})=\psi_{0}+\psi_{7}$ for $\forall
X\in\\{A,B,C\\}$. From the above discussion, the constrains
$Y^{(X_{1})}-\psi_{k}\geq\mathrm{T}_{X_{1}}(Q_{k}^{(X_{1})})$ are satisfied
for $1\leq k\leq 5$ and $X_{1}\in\\{A_{1},B_{1},C_{1}\\}$, so we have

$\begin{split}&Y^{(X_{1})}\otimes(\psi_{0}+\psi_{7})-\psi_{k}\otimes(\psi_{0}+\psi_{7})\\\
\geq&\ \mathrm{T}_{X_{1}}(Q_{k}^{(X_{1})})\otimes(\psi_{0}+\psi_{7})\\\ =&\
\mathrm{T}_{X_{1}X_{2}}[Q_{k}^{(X_{1})}\otimes(\psi_{0}+\psi_{7})].\end{split}$

If we set $Y^{(X_{1}X_{2})}=Y^{(X_{1})}\otimes(\psi_{0}+\psi_{7})$ and
$Q_{k}^{(X_{1}X_{2})}=Q_{k+5}^{(X_{1}X_{2})}=Q_{k}^{(X_{1})}\otimes(\psi_{0}+\psi_{7})\
(k=1,\cdots,5)$, then

$Y^{(X_{1}X_{2})}-\Phi_{j}\geq\mathrm{T}_{X_{1}X_{2}}(Q_{j}^{(X_{1}X_{2})})\ \
(j=1,\cdots,10)$

also satisfy the inequality constrains of dual programs (5), where

$\Phi_{j}=\left\\{\begin{array}[]{ll}\psi_{j}\otimes\psi_{0},&1\leq j\leq 5\\\
\psi_{j-5}\otimes\psi_{7},&6\leq j\leq 10\end{array}\right.$

are just elements of $\mathcal{S}^{10}$. Since $Tr(Y^{(A)})=Tr(Y^{(B)})=8$,
$Tr(Y^{(C)})=6$, all the minimums $\beta_{X}\ (X\in\\{A,B,C\\})$ are smaller
than 1. This means that $\mathcal{S}^{10}=\\{\Phi_{1},\cdots,\Phi_{10}\\}$ is
genuinely nonlocal.

The readers may be curious of the above construction procedure about
$\mathcal{S}^{5}$ and $\mathcal{S}^{10}$, for the feasible solutions to
semidefinte programs (5) we’ve presented (the $Y^{(X)}$’s and the
$Q_{k}^{(X)}$’s) are all lattice-type. To explain why, we present the
following lemma which tells that when GHZ-lattices are considered, any
feasible solution to the programs (5) corresponds to a lattice-type one that
has same target values.

######  Lemma 2.

If the states $|\Psi_{1}\rangle,\cdots,|\Psi_{s}\rangle$ to be distinguished
are all GHZ-lattices, then the semidefinite programs (5) [and (4) likewise]
can be reduced to linear programs.

###### Proof.

We check here the dual problem case and the primal case is likewise. First,
note that the set of GHZ basis (1) is the common eigenbasis of the following
set of commutative product operators on
$\mathbb{C}^{2}\otimes\mathbb{C}^{2}\otimes\mathbb{C}^{2}$:

$C=\\{\mathrm{III},\mathrm{XXX},\mathrm{IZZ},\mathrm{ZIZ},\mathrm{ZZI},\mathrm{XYY},\mathrm{YXY},\mathrm{YYX}\\},$

where $\\{\mathrm{X},\mathrm{Y},\mathrm{Z}\\}$ are the one-qubit Pauli
operators. More explicitly, they have spetrum decomposition:

$\begin{array}[]{cl}\mathrm{III}&=\psi_{0}+\psi_{1}+\psi_{2}+\psi_{3}+\psi_{4}+\psi_{5}+\psi_{6}+\psi_{7},\\\
\mathrm{XXX}&=\psi_{0}+\psi_{1}+\psi_{2}+\psi_{3}-\psi_{4}-\psi_{5}-\psi_{6}-\psi_{7},\\\
\mathrm{IZZ}&=\psi_{0}+\psi_{7}+\psi_{3}+\psi_{4}-\psi_{1}-\psi_{6}-\psi_{2}-\psi_{5},\\\
\mathrm{ZIZ}&=\psi_{0}+\psi_{7}+\psi_{2}+\psi_{5}-\psi_{1}-\psi_{6}-\psi_{3}-\psi_{4},\\\
\mathrm{ZZI}&=\psi_{0}+\psi_{7}+\psi_{1}+\psi_{6}-\psi_{2}-\psi_{5}-\psi_{3}-\psi_{4},\\\
\mathrm{XYY}&=\psi_{1}+\psi_{2}+\psi_{4}+\psi_{7}-\psi_{6}-\psi_{5}-\psi_{3}-\psi_{0},\\\
\mathrm{YXY}&=\psi_{1}+\psi_{3}+\psi_{5}+\psi_{7}-\psi_{6}-\psi_{4}-\psi_{2}-\psi_{0},\\\
\mathrm{YYX}&=\psi_{2}+\psi_{3}+\psi_{6}+\psi_{7}-\psi_{5}-\psi_{4}-\psi_{1}-\psi_{0}.\end{array}$
(9)

Consider the quantum operation

$\Delta(\rho)=\frac{1}{|C|}\sum_{U_{i}\in C}U_{i}\rho U_{i}^{{\dagger}}$ (10)

that acts on system
$\mathbb{C}^{2}\otimes\mathbb{C}^{2}\otimes\mathbb{C}^{2}$. Substituting (9)
into (10), one can check that all the cross terms in expression eliminate and
hence $\Delta$ acts as the dephasing operation under the three-qubit GHZ
basis:

$\Delta(\rho)=\frac{1}{8}\sum_{i=0}^{7}|\psi_{i}\rangle\langle\psi_{i}|\rho|\psi_{i}\rangle\langle\psi_{i}|.$
(11)

It’s obvious that $\Delta$ is positive and trace-preserving. What’s more, it
commutes with the partial transpose operations $\mathrm{T}_{X}$’s
$(X\in\\{A,B,C\\})$. To see this, just write $\Delta$ in the form (10), and
check if each $\mathrm{T}_{X}$ commutes with each term $U_{i}\rho
U_{i}^{{\dagger}}$. This is routine, for example

$\begin{split}\mathrm{T}_{B}(\mathrm{YYX}\cdot\rho\cdot\mathrm{YYX})&=\mathrm{YY^{T}X}\cdot\mathrm{T}_{B}(\rho)\cdot\mathrm{YY^{T}X}\\\
&=(-\mathrm{YYX})\cdot\mathrm{T}_{B}(\rho)\cdot(-\mathrm{YYX})\\\
&=\mathrm{YYX}\cdot\mathrm{T}_{B}(\rho)\cdot\mathrm{YYX},\end{split}$ (12)

and so on. Hence, $\mathrm{T}_{X}[\Delta(\rho)]=\Delta[\mathrm{T}_{X}(\rho)]$
for $X\in\\{A,B,C\\}$. Now let $\Lambda=\Delta^{\otimes t}$, that is

$\Lambda(\zeta)=\frac{1}{8^{t}}\sum_{i_{1}=0}^{7}\cdots\sum_{i_{t}=0}^{7}\zeta_{i_{1}\cdots
i_{t}}\cdot|\psi_{i_{1}}\cdots\psi_{i_{t}}\rangle\langle\psi_{i_{1}}\cdots\psi_{i_{t}}|,$

where $\zeta$ is any state in tripartite system
$\mathbb{C}^{2^{t}}\otimes\mathbb{C}^{2^{t}}\otimes\mathbb{C}^{2^{t}}$ and
$\zeta_{i_{1}\cdots
i_{t}}=\langle\psi_{i_{1}}\cdots\psi_{i_{t}}|\zeta|\psi_{i_{1}}\cdots\psi_{i_{t}}\rangle$.
It also acts as the dephasing operation under basis
$\\{|\psi_{i_{1}}\rangle\otimes\cdots\otimes|\psi_{i_{t}}\rangle\\}_{i_{1},\cdots,i_{t}}$
and has the same property as $\Delta(\cdot)$. Since this operation is
invariant on the GHZ-lattices
$\Psi_{k}=\psi_{k_{1}}\otimes\cdots\otimes\psi_{k_{t}}$, by acting it on any
specific feasible solution of (5), one can achieve another feasible solution
consisting only of diagonal operators while the target values stay still. That
is to say, the semidefinite programs (5) reduce to linear programs

$\begin{split}\beta_{X}=&\min\frac{1}{s}\sum_{l_{1}=0}^{7}\cdots\sum_{l_{t}=0}^{7}(\overrightarrow{y^{{}_{(X)}}})_{l_{1}\cdots
l_{t}}\\\ s.t.\ \ &(\overrightarrow{y^{{}_{(X)}}})_{i_{1}\cdots
i_{t}}-\sum_{l_{1}\cdots l_{t}}(T^{{}_{(X)}})^{\otimes t}_{i_{1}\cdots
i_{t},l_{1}\cdots l_{t}}\cdot(\overrightarrow{q_{k}^{(X)}})_{l_{1}\cdots
l_{t}}\geq\delta_{i_{1}\cdots i_{t},k_{1}\cdots k_{t}}\\\
&(\overrightarrow{q_{k}^{(X)}})_{i_{1}\cdots i_{t}}\geq 0\\\
(i_{1},\cdots,&i_{t}\in\\{0,\cdots,7\\};\ k=1,\cdots,s;\
X\in\\{A,B,C\\})\end{split}$ (13)

where “$\delta$” is the Kronecker delta and we suppose that the GHZ-lattices
$|\Psi_{k}\rangle$ have index $k_{1},\cdots,k_{t}\ (k=1,\cdots,s)$;
$(\overrightarrow{y^{{}_{(X)}}})$’s, $(\overrightarrow{q_{k}^{{}_{(X)}}})$’s
are $8^{t}\times 1$ vectors representing the diagonal elements of $Y^{(X)}$’s
and $Q_{k}^{(X)}$’s $(1\leq k\leq s;\ X\in\\{A,B,C\\})$ respectively;
$T^{(X)}$’s $(X\in\\{A,B,C\\})$ are the tranformation matrices of partial
transpose $\mathrm{T}_{X}$ under $\\{\psi_{0},\cdots,\psi_{7}\\}$ as shown by
(LABEL:transform) in Lemma 1 (but with index in ascending order).

This result greatly reduces the complexity of the problem, for all operators
we need to consider now are just diagonal (lattice-type) ones. We should also
point out here that the feasible solutions we presented in the cases
$\mathcal{S}^{5}$ and $\mathcal{S}^{10}$ are actually optimal. Moreover, by
running programs (13) numerically on computer, we found the existence of
genuinely nonlocal subsets of $\mathcal{S}^{10}$ with cardinality down to 8!
In other words, the genuine nonlocality of $\mathcal{S}^{10}$ can actually be
deduced trivially from its subset (a set of states is of course
indistinguishable if its proper subset is already indistinguishable).

######  Theorem 4.

In $\mathbb{C}^{4}\otimes\mathbb{C}^{4}\otimes\mathbb{C}^{4}$, the following
$8$-ary subset of $\mathcal{S}^{10}$:

$\begin{split}\mathcal{S}^{8}=\\{\psi_{1}\otimes\psi_{0},\psi_{2}\otimes\psi_{0},\psi_{3}\otimes\psi_{0},\psi_{1}\otimes\psi_{7},\psi_{2}\otimes\psi_{7},\psi_{3}\otimes\psi_{7},\psi_{4}\otimes\psi_{7},\psi_{5}\otimes\psi_{7}\\}\end{split}$

still has the property of genuine nonlocality.

###### Proof.

We present here the explicit (optimal) solution to the dual problems (5) with
respect to all the bipartitions.

$A|BC$:

$\begin{split}Y^{(A_{1}A_{2})}=&\frac{1}{2}\psi_{1}\otimes\psi_{0}+\frac{1}{2}\psi_{1}\otimes\psi_{7}+\frac{1}{2}\psi_{2}\otimes\psi_{0}+\frac{1}{2}\psi_{2}\otimes\psi_{7}+\psi_{3}\otimes\psi_{0}+\psi_{3}\otimes\psi_{7}\\\
+&\psi_{4}\otimes\psi_{7}+\frac{1}{2}\psi_{5}\otimes\psi_{0}+\frac{1}{2}\psi_{5}\otimes\psi_{7}+\frac{1}{2}\psi_{6}\otimes\psi_{0}+\frac{1}{2}\psi_{6}\otimes\psi_{7},\end{split}$
$\begin{split}&Q_{1}^{(A_{1}A_{2})}=\psi_{5}\otimes\psi_{0}+\psi_{5}\otimes\psi_{7},\
Q_{2}^{(A_{1}A_{2})}=\psi_{6}\otimes\psi_{0}+\psi_{6}\otimes\psi_{7},\
Q_{3}^{(A_{1}A_{2})}=0,\\\
&Q_{4}^{(A_{1}A_{2})}=\psi_{5}\otimes\psi_{0}+\psi_{5}\otimes\psi_{7},Q_{5}^{(A_{1}A_{2})}=\psi_{6}\otimes\psi_{0}+\psi_{6}\otimes\psi_{7},Q_{6}^{(A_{1}A_{2})}=Q_{7}^{(A_{1}A_{2})}=0,\\\
&Q_{8}^{(A_{1}A_{2})}=\psi_{1}\otimes\psi_{0}+\psi_{1}\otimes\psi_{7};\end{split}$
(14)

$B|CA$:

$\begin{split}Y^{(B_{1}B_{2})}=&\frac{1}{2}\psi_{1}\otimes\psi_{0}+\frac{1}{2}\psi_{1}\otimes\psi_{7}+\psi_{2}\otimes\psi_{0}+\psi_{2}\otimes\psi_{7}+\frac{1}{2}\psi_{3}\otimes\psi_{0}+\frac{1}{2}\psi_{3}\otimes\psi_{7}\\\
+&\frac{1}{2}\psi_{4}\otimes\psi_{0}+\frac{1}{2}\psi_{4}\otimes\psi_{7}+\psi_{5}\otimes\psi_{7}+\frac{1}{2}\psi_{6}\otimes\psi_{0}+\frac{1}{2}\psi_{6}\otimes\psi_{7},\end{split}$
$\begin{split}&Q_{1}^{(B_{1}B_{2})}=\psi_{4}\otimes\psi_{0}+\psi_{4}\otimes\psi_{7},\
Q_{2}^{(B_{1}B_{2})}=0,\
Q_{3}^{(B_{1}B_{2})}=\psi_{6}\otimes\psi_{0}+\psi_{6}\otimes\psi_{7},\\\
&Q_{4}^{(B_{1}B_{2})}=\psi_{4}\otimes\psi_{0}+\psi_{4}\otimes\psi_{7},\
Q_{5}^{(B_{1}B_{2})}=0,\
Q_{6}^{(B_{1}B_{2})}=\psi_{6}\otimes\psi_{0}+\psi_{6}\otimes\psi_{7},\\\
&Q_{7}^{(B_{1}B_{2})}=\psi_{1}\otimes\psi_{0}+\psi_{1}\otimes\psi_{7},\ \
Q_{8}^{(B_{1}B_{2})}=0;\end{split}$ (15)

$C|AB$:

$\begin{split}Y^{(C_{1}C_{2})}=&\psi_{1}\otimes\psi_{0}+\psi_{1}\otimes\psi_{7}+\frac{1}{2}\psi_{2}\otimes\psi_{0}+\frac{1}{2}\psi_{2}\otimes\psi_{7}+\frac{1}{2}\psi_{3}\otimes\psi_{0}+\frac{1}{2}\psi_{3}\otimes\psi_{7}\\\
+&\frac{1}{2}\psi_{4}\otimes\psi_{0}+\frac{1}{2}\psi_{4}\otimes\psi_{7}+\frac{1}{2}\psi_{5}\otimes\psi_{0}+\frac{1}{2}\psi_{5}\otimes\psi_{7},\end{split}$
$\begin{split}&Q_{1}^{(C_{1}C_{2})}=0,\ \
Q_{2}^{(C_{1}C_{2})}=\psi_{4}\otimes\psi_{0}+\psi_{4}\otimes\psi_{7},\ \
Q_{3}^{(C_{1}C_{2})}=\psi_{5}\otimes\psi_{0}+\psi_{5}\otimes\psi_{7},\\\
&Q_{4}^{(C_{1}C_{2})}=\psi_{2}\otimes\psi_{0}+\psi_{2}\otimes\psi_{7},\
Q_{5}^{(C_{1}C_{2})}=\psi_{4}\otimes\psi_{0}+\psi_{4}\otimes\psi_{7},\
Q_{6}^{(C_{1}C_{2})}=\psi_{5}\otimes\psi_{0}+\psi_{5}\otimes\psi_{7},\\\
&Q_{7}^{(C_{1}C_{2})}=\psi_{2}\otimes\psi_{0}+\psi_{2}\otimes\psi_{7},\ \
Q_{8}^{(C_{1}C_{2})}=\psi_{3}\otimes\psi_{0}+\psi_{3}\otimes\psi_{7}.\\\
\end{split}$ (16)

The corresponding target values are $7/8,7/8,3/4$ respectively. The
feasibility of this solution can be checked routinely through
(LABEL:transform) by the readers. For example,

$\begin{split}Y^{(A_{1}A_{2})}-\psi_{1}\otimes\psi_{0}=&-\frac{1}{2}\psi_{1}\otimes\psi_{0}+\frac{1}{2}\psi_{1}\otimes\psi_{7}+\frac{1}{2}\psi_{2}\otimes\psi_{0}+\frac{1}{2}\psi_{2}\otimes\psi_{7}+\psi_{3}\otimes\psi_{0}+\psi_{3}\otimes\psi_{7}\\\
&\ \ \
+\psi_{4}\otimes\psi_{7}+\frac{1}{2}\psi_{5}\otimes\psi_{0}+\frac{1}{2}\psi_{5}\otimes\psi_{7}+\frac{1}{2}\psi_{6}\otimes\psi_{0}+\frac{1}{2}\psi_{6}\otimes\psi_{7}\\\
\geq&\
(-\frac{1}{2}\psi_{1}+\frac{1}{2}\psi_{6}+\frac{1}{2}\psi_{2}+\frac{1}{2}\psi_{5})\otimes(\psi_{0}+\psi_{7})\\\
=&\ \mathrm{T}_{A_{1}A_{2}}[Q_{1}^{(A_{1}A_{2})}]\end{split}$

and

$Y^{(C_{1}C_{2})}-\psi_{1}\otimes\psi_{0}\geq
0=\mathrm{T}_{C_{1}C_{2}}[Q_{1}^{(C_{1}C_{2})}]$

and so on. These together show that the set $\mathcal{S}^{8}$ is genuinely
nonlocal.

Similar to the proof of Proposition 2, one can inductively construct genuinly
nonlocal sets of lattice-type GHZ-like states in the cases $t\geq 3$. For
example, in $\mathbb{C}^{8}\otimes\mathbb{C}^{8}\otimes\mathbb{C}^{8}$ the set

$\begin{split}\mathcal{S}^{16}=\\{&\psi_{1}\otimes\psi_{0}\otimes\psi_{0},\psi_{2}\otimes\psi_{0}\otimes\psi_{0},\psi_{3}\otimes\psi_{0}\otimes\psi_{0},\psi_{1}\otimes\psi_{7}\otimes\psi_{0},\\\
&\psi_{2}\otimes\psi_{7}\otimes\psi_{0},\psi_{3}\otimes\psi_{7}\otimes\psi_{0},\psi_{4}\otimes\psi_{7}\otimes\psi_{0},\psi_{5}\otimes\psi_{7}\otimes\psi_{0},\\\
&\psi_{1}\otimes\psi_{0}\otimes\psi_{7},\psi_{2}\otimes\psi_{0}\otimes\psi_{7},\psi_{3}\otimes\psi_{0}\otimes\psi_{7},\psi_{1}\otimes\psi_{7}\otimes\psi_{7},\\\
&\psi_{2}\otimes\psi_{7}\otimes\psi_{7},\psi_{3}\otimes\psi_{7}\otimes\psi_{7},\psi_{4}\otimes\psi_{7}\otimes\psi_{7},\psi_{5}\otimes\psi_{7}\otimes\psi_{7}\\}\end{split}$

is genuinely nonlocal, and so on. To sum up:

######  Corollary 3.

In tripartite system
$\mathbb{C}^{d}\otimes\mathbb{C}^{d}\otimes\mathbb{C}^{d}$ where
$d=2^{t}(t>1)$, there exist genuinely nonlocal sets of full-weighted GHZ-like
states with cardinality no larger than $2d$.

Surely, some of the states therein might further be excluded as analogous to
what’s shown in Theorem 4, but what’s inspiring to us is that the size of such
genuinely nonlocal sets can scale down to linear in the local dimension $d$,
much smaller than the bound presented in Corollary 1!

##  VI Conclusion and discussion

In this paper, we have discussed the distinguishability-based genuine
nonlocality of a special type of genuinely multipartite entangled states — the
GHZ-like states. We first study the genuine nonlocality of the three-qubit GHZ
basis (1). Then using these results in the three-qubit case, we construct
genuinely nonlocal sets of “GHZ-lattices” in tripartite systems where the
local dimension are powers of $2$. Similar to the efforts seeking for small
nonlocal sets with maximally entangled states in bipartite systems, our result
in some sense substantiates the perspective that entanglement has the ability
to raise difficulty in local discrimination of quantum states. It turns out
that the size of these genuinely nonlocal sets with genuine multipartite
entanglement can scale down to $2d$ at least, where $d$ is the local
dimension.

On the other hand, the concept of genuine nonlocality, which concerns the
local distinguishability of multipartite quantum states through any possible
bipartition of the subsystems, is in our opinion more naturally arising than
the recently more popular “strong nonlocality”. However, little is known about
this natural form of distinguishability-based nonlocality. An interesting
question is to ask in what extent strong nonlocality is stronger than the
normal multipartite nonlocality? As far as we know presently, in the context
of strong nonlocality, no nonlocal set with such scale has been constructed
yet, no matter product states or genuinely entangled states were considered.
This comparison may hint that there might exist a significant gap between the
strength of strong nonlocality and the normal distinguishability-based
multipartite nonlocality.

There are also some questions left to be considered. For example, whether
these results can be generalized to the more general cases where the local
dimension $d$ are not powers of $2$? What is the situation when more parties
(say, four parties) are considered? Besides, notice that we’ve constructed
genuinely nonlocal sets mainly through PPT-indistinguishability in our work.
It’s then natural to ask whether there exist other ways to construct genunely
nonlocal sets, such that they have even smaller cardinality?

Acknowledgments

This work was supported by the Guangdong Basic and Applied Basic Research
Foundation under Grant No. 2020B1515020050, NSFC under Grant No. 12005092, Key
Research and Development Project of Guangdong Province under Grant No.
2020B0303300001 and the Guangdong Basic and Applied Basic Research Foundation
under Grant No. 2020B1515310016. The authors were also partly supported by the
Guangdong Basic and Applied Basic Research Foundation (Grant No.
2019A1515011703) and Laboratory Stabilization Support Fund (Grant No.
KY12400210010).

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