11institutetext: Vanderbilt University, Nashville TN 37215, USA
22institutetext: Vanderbilt University Medical Center, Nashville TN 37215, USA
33institutetext: NVIDIA Corporation, Santa Clara and Bethesda, USA

# A Unified Single-stage Learning Model for Estimating Fiber Orientation
Distribution Functions on Heterogeneous Multi-shell Diffusion-weighted MRI

Tianyuan Yao 11 Nancy Newlin 11 Praitayini Kanakaraj 11 Vishwesh Nath 33 Leon
Y Cai 11 Karthik Ramadass 11 Kurt Schilling 22 Bennett A. Landman 11 Yuankai
Huo 11

###### Abstract

Diffusion-weighted (DW) MRI measures the direction and scale of the local
diffusion process in every voxel through its spectrum in q-space, typically
acquired in one or more shells. Recent developments in micro-structure imaging
and multi-tissue decomposition have sparked renewed attention to the radial
b-value dependence of the signal. Applications in tissue classification and
micro-architecture estimation, therefore, require a signal representation that
extends over the radial as well as angular domain. Multiple approaches have
been proposed that can model the non-linear relationship between the DW-MRI
signal and biological microstructure. In the past few years, many deep
learning-based methods have been developed towards faster inference speed and
higher inter-scan consistency compared with traditional model-based methods
(e.g., multi-shell multi-tissue constrained spherical deconvolution). However,
a multi-stage learning strategy is typically required since the learning
process relied on various middle representations, such as simple harmonic
oscillator reconstruction (SHORE) representation. In this work, we present a
unified dynamic network with a single-stage spherical convolutional neural
network, which allows efficient fiber orientation distribution function (fODF)
estimation through heterogeneous multi-shell diffusion MRI sequences. We study
the Human Connectome Project (HCP) young adults with test-retest scans. From
the experimental results, the proposed single-stage method outperforms prior
multi-stage approaches in repeated fODF estimation with shell dropoff and
single-shell DW-MRI sequences.

###### Keywords:

DW-MRImulti-shell Deep learning.

##  1 Introduction

Diffusion-weighted magnetic resonance imaging (DW-MRI) is essential for the
non-invasive reconstruction of the microstructure of the human in vivo brain
[3, 24, 11]. Substantial efforts have shown that other advanced approaches can
recover more elaborate reconstruction of the microstructure [15, 18, 8] and
these methods are collectively referred to as high angular resolution
diffusion imaging (HARDI). HARDI methods have been broadly proposed in two
categories of single-shell acquisitions and multi-shell acquisitions (i.e.,
using multiple diffusivity values). A majority of single-shell HARDI methods
utilize spherical harmonics (SH) based modeling as in q-ball imaging (QBI)
[22], constrained spherical deconvolution (CSD) [21], and many others.
However, SH-based modeling cannot directly leverage additional information
provided by multi-shell acquisitions as the SH transformation does not allow
for a representation of the radial complexity that is introduced by the
diffusivity value. SH has been combined with other bases to represent multi-
shell data, e.g., solid harmonics [9], simple harmonic oscillator
reconstruction (SHORE) [6], and spherical polar Fourier imaging [5].

Deep learning (DL) has revolutionized many different domains in medical
imaging, and DW-MRI parameter estimation is no different. Lots of DW-MRI
methods have been developed that utilize the powerful data-driven capabilities
of deep learning, yielding improved accuracy over conventional fitting when
the acquisition scheme has a limited number of measurements. However, most
methods are only focused on the translation of single-shell data to DW-MRI
parameters and in contrast, the multi-shell methods get neglected due to the
complexity associated with multi-shell data. Moreover, the SHORE-based DL
methods typically used a multi-stage design. For instance, the algorithm must
first optimize a specific optimal SHORE representation and then optimize the
fiber orientation distribution function (fODF) estimation. Such methods are
prone to overfitting, lower inference time, and complicated parameter tuning.

As shown in Fig 1, in this paper, we propose a single-stage dynamic network
with both the q-space and radial space signal based on a spherical
convolutional neural network. We evaluated the resultant representation by
targeting it to multi-shell multi-tissue CSD (MSMT-CSD). Both fiber
orientation estimation and recovery of tissue volume fraction are evaluated.
The contribution of this paper is three-fold:

$\bullet$ We proposed a unified dynamic network with the single-stage
spherical convolutional neural network that can recover/predict
microstructural measures.

$\bullet$ The proposed method is universally applicable to perform learning-
based fODF estimation using a single deep model for various combinations of
multiple shells.

$\bullet$ The proposed method achieved an overall superior performance
compared with model-based and data-driven benchmarks.

Figure 1: Utilizing multi-shell DW-MRI signals in deep learning usually
requires independent models trained for each specific shell configuration as
conventional SH-based modeling cannot directly leverage additional information
(radial space) provided by multi-shell acquisitions. In our study, the dynamic
head aims to improve the network expressiveness by learning and adaptively
adjusting the first convolution layer for different shell configurations.

##  2 Related work

###  2.1 Multi-Shell Multi-Tissue Constrained Spherical Deconvolution

Spherical deconvolution [2] is a set of methods to reconstruct the local fODF
from DW-MRI data. They have become a popular choice for recovering the fiber
orientation due to their ability to resolve fiber crossings with small inter-
fiber angles in datasets acquired within a clinically feasible scan time. SD
methods are based on the assumption that the acquired diffusion signal in each
voxel can be modeled as a spherical convolution between the fODF and the fiber
response function (FRF), which describes the common signal profile from the
white matter (WM) bundles contained in the voxel. The term ”constrained” in
CSD [21] refers to constraints that are placed on the solution. The goal of
these constraints is to ensure that the estimated fiber orientations are
biologically plausible and to avoid overfitting noise in the data.

Multi-Shell Multi-Tissue Constrained Spherical Deconvolution (MSMT-CSD) [15]
is a technique developed to overcome the limitations of traditional single-
shell diffusion MRI methods, which are unable to resolve the complex fiber
orientations of multiple tissue types in the brain. MSMT-CSD is able to
separate the contribution of different tissue types (such as gray matter,
white matter, and cerebrospinal fluid) to the diffusion signal by modeling the
diffusion signal as a combination of multiple shells with different b-values.
This modeling-based method has been a conventional method for multi-tissue
micro-architecture estimation.

###  2.2 Learning-based estimation

Recently, machine learning (ML) and deep learning (DL) techniques have
demonstrated their remarkable abilities in neuroimaging. Such approaches have
been applied to the task of microstructure estimation [17], aiming to directly
learn the mapping between input DW-MRI scans and output fiber tractography
[19, 4] while maintaining the necessary biological characteristics and
reproducibility for clinical translation. Such studies have illustrated that
DL is a promising tool that uses nonlinear transforms to extract features from
high-dimensional data. Data-driven approaches can be useful in validating the
hypothesis of the existence of untapped information because they generalize
toward the ground truth.

###  2.3 Model based representation

The SHORE basis function has been shown to capture the representation of
multi-shell DW-MRI with minimal representation error and ensure the same when
modeling single-shell DW-MRI. SHORE [6] provides a more detailed
characterization of the diffusion process by modeling the water diffusion
profile as a combination of multiple frequencies, each representing a
different spatial scale of motion. This allows for a more accurate and
detailed representation of the underlying tissue microstructure. In our study,
we applied the SHORE basis function as the baseline representation for multi-
shell DW-MRI.

##  3 Methods

###  3.1 Preliminaries

Traditional deep learning frameworks are not generalizable to new acquisition
schemes. This complicates the application of a DL model to data acquired from
multiple sites. Our model aims to train a DL framework that can be adapted to
an arbitrary number of available multi-shell DW-MRI sequences. To serve this
motivation, we employ a dynamic head (DH) design to handle the multi-shell
problem on the three most common diffusivity values: 1000, 2000, and 3000
$s/mm^{2}$. Additionally, to tackle the problem of a varying number of
gradient directions on each shell (diffusivity value), we leverage the
spherical CNNs with the traditional ’modeling then feeding to FCN’ strategy.
In this study, we employ the fODF estimation as our chosen task to perform
assessments on different methods.

###  3.2 Dynamic head design

In our model, a single set of model parameters $\theta$ is learned to handle
all permutations of a different number of shells with the most commonly known
b-values. Note that with $K$ shells there are $2^{K}\shortminus 1$
configurations. To improve the network expressiveness, we devise a dynamic
head $D$ to adaptively generate model parameters conditioned on the
availability of input shells. We use a binary code $m\in\mathbb{R}^{K}$ where
0/1 represent the absence/presence of each shell. To mitigate the large input
variation caused by artificially zero-ed channels, we use the dynamic head to
generate the parameters for the first convolutional layer.

###  3.3 Spherical Convolution

To extract features from DW-MRI signals, the first and most common deep
learning network architecture applied to dMRI is the FCN [1, 16],
Conventionally these have been implemented following:

$\begin{split}t=F_{FCN}(s;\theta_{FCN})\\\ \end{split}$ (1)

where $F_{FCN}$ is a fully-connected network with trainable parameters
$\theta_{FCN}$. The dMRI signals serve as the input for the network, and it
does not consider the acquisition information, making the network unaware of
the acquisition scheme. This lack of knowledge poses an issue when
incorporating new data acquired at a different location with a distinct
acquisition scheme. The accuracy of estimation from a new set of DW-MRIs
depends on the consistency of the acquisition scheme with the training set.
Additionally, the FCN’s design does not account for rotational equivariance,
which could result in requiring a varied range of tissue microstructure
orientations in the training dataset for accurate estimation independent of
fiber orientation.

Theoretically, Spherical CNNs offer an advantage over FCNs regarding both the
robustness of the gradient scheme and the distribution of training data [12,
20]. The Spherical CNN’s architecture differs from FCNs, but not in the
conventional sense. Instead of convolution across multiple voxels, Spherical
CNNs perform convolution over the spherical image space. Hence, like FCNs,
they are voxelwise networks. At each voxel, the spherical image is created
from the dMRI signals and their corresponding gradient scheme. This
architecture can naturally address the limitations of FCNs in two ways.
Firstly, Spherical CNNs are aware of the gradient scheme in their input, which
is not the case with FCNs, as demonstrated in the equation below:

$\begin{split}t=F_{S-CNN}(s,G;\theta_{S-CNN})\\\ \end{split}$ (2)

where $F_{S-CNN}$ is a Spherical CNNs with trainable parameters
$\theta_{S-CNN}$. In Spherical CNNs, the gradient scheme is explicitly taken
into account in the input, enabling them to better handle variations in the
gradient scheme that can occur between different acquisition protocols or
imaging sites. Additionally, Spherical CNNs can better handle the distribution
of training data, as they are naturally suited to working with data on a
sphere. Overall, these benefits of spherical convolution can lead to improved
accuracy and robustness in the analysis of diffusion MRI signals.

Suppose $L_{t}$ is a loss function designed for the downstream task. Let $x$
be the input of the framework. During training, the shared network $F_{S-CNN}$
is trained using the input data and there are $2^{K}\shortminus 1$ unique
shell configurations, represented by:

$\tilde{x}^{k}=\delta^{k}x^{k},k\in\\{1,...,K\\},$ (3)

By targeting different diffusion properties $y$ and with the dynamic head
setting. The learning objective can be expressed as:

$\begin{split}\operatorname*{\arg\\!\min}_{\theta_{s}}L_{t}(y,F(\tilde{x}_{i}|\theta_{S-CNN}))\end{split}$
(4)

##  4 Experiments

###  4.1 Data and Implementation Details

We have chosen DW-MRI from the Human Connectome Project - Young Adult (HCP-ya)
dataset [23, 11], 45 subjects with the scan-rescan acquisition were used (a
total of 90 images). The acquisitions had b-values of 1000, 2000, 3000
$s/mm^{2}$ with 90 gradient directions on each shell. A T1 volume of the same
subject was used for WM segmentation using SLANT[13]. All HCP-ya DW-MRI was
distortion corrected with top-up and eddy[14]. 30 subjects are used as
training data while 10 subjects were used as evaluation and 5 subjects as
testing data.

We performed shell extraction on all the data. Every subject has seven
different shell configurations which are the permutations of all three
b-values $\\{$$\\{1K,2K,3K\\}$, $\\{1K,2K\\}$, $\\{2K,3K\\}$, $\\{1K,3K\\}$,
$\\{1K\\}$, $\\{2K\\}$, $\\{3K\\}$$\\}$. Ground-truth fODF maps were computed
from MSMT-CSD using the DIPY library with the default settings [10]. $8^{th}$
order SH were chosen for data representation with the ’tournier07’ basis [21].
The white matter fODF and the volume fraction which refers to the proportion
of the volume of the voxel that is occupied by each tissue type, are combined
together as the targeted sequence. The $6^{th}$ radial order SHORE basis is
employed as a baseline representation, the SHORE scaling factor $\zeta$
defined in units of $mm^{-2}$ as $\zeta=1/8\pi^{2}\tau MD$ is calculated based
on the mean diffusivity (MD) obtained from the data.

###  4.2 Experimental setting

We first trained separate models for each shell configuration. The models
consist of four fully connected layers with ReLU activation function. The
number of neurons per layer is 400, 48, 200, and 48. The input is the $1\times
50$ vector of the shore basis signal ODF, and the output is the combination
$1\times 45$ vectors of the SH basis WM fODF and the $1\times 3$ vector of
tissues fraction. The models are then tested on the different shell
configurations. By simply feeding all the shell configuration data (all
labeled with reconstructed fODF from data with all shells) to the FCN as a
baseline ’unified’ deep learning model.

We assess the impact of dynamic head strategy by evaluating the performance of
the unified models against independent models trained for each specific shell
configuration. Furthermore, the generalizability of the different
representations with dynamic head designs was assessed. For the spherical
convolution, we used an architecture known as the hybrid spherical CNN as
described in [7]. After the rotational invariant features are extracted. They
are concatenated and fed into fully connected layers(the same hidden size as
above) which perform the final estimation.

###  4.3 Evaluation metric

To evaluate the predictions from the proposed methods, we calculated the mean
squared error of the volume fractions with ground truth sequences. Then we
compute the angular correlation coefficient (ACC, Eq. 5) between the predicted
fODF and the ground truth fODF over the white matter region. ACC is a
generalized measure for all fiber population scenarios. It assesses the
correlation of all directions over a spherical harmonic expansion. In brief,
it provides the estimate of how closely a pair of fODFs are related on a scale
of -1 to 1, where 1 is the best measure. Here ‘u’ and ‘v’ represent sets of SH
coefficients.

$\begin{split}ACC=\frac{\sum_{k=1}^{L}\sum_{m=-k}^{k}(u_{km})(v^{*}_{km})}{[\sum_{k=1}^{L}\sum_{m=-k}^{k}|u_{km}|^{2}]^{0.5}\cdot[\sum_{k=1}^{L}\sum_{m=-k}^{k}|v_{km}|^{2}]^{0.5}}\end{split}$
(5)

##  5 Experimental Results

We compared the performances of the unified models against independent models
trained for each specific shell configuration. A qualitative result of fODF
predictions and GT are shown in Fig 2. As shown in Table 1, the independent
models that are trained are thus more likely to outperform others in their own
shell configuration and these models can be considered as the upper bounds for
each shell configuration. With the dynamic head settings, the unified model
with spherical convolution outperforms the other models in the single shell
configuration. Additionally, the ACC is a sensitive generalized metric, the
performances need further evaluation. We assessed how good our prediction was
by evaluating the scan/rescan consistency and volume fraction prediction 2.

Figure 2: This is a visualization of the fODF prediction and the correlation with the GT in different views. The background of the zoom-in patches shows the ACC spatial map with the GT signals. Table 1: Performances of the unified models against independent models in different shell configurations Model |  $C_{1}$ |  $C_{2}$ |  $C_{3}$ |  $C_{1,2}$ |  $C_{2,3}$ |  $C_{1,3}$ |  $C_{1,2,3}$ |  Ave.
---|---|---|---|---|---|---|---|---
$M_{1}$ |  0.808 |  0.725 |  0.732 |  0.752 |  0.734 |  0.751 |  0.788 |  0.756
$M_{2}$ |  0.762 |  0.815 |  0.756 |  0.744 |  0.749 |  0.745 |  0.774 |  0.764
$M_{3}$ |  0.757 |  0.724 |  0.814 |  0.734 |  0.753 |  0.760 |  0.779 |  0.760
$M_{1,2}$ |  0.745 |  0.732 |  0.743 |  0.831 |  0.788 |  0.778 |  0.789 |  0.772
$M_{2,3}$ |  0.734 |  0.744 |  0.738 |  0.802 |  0.825 |  0.786 |  0.786 |  0.774
$M_{1,3}$ |  0.737 |  0.745 |  0.745 |  0.785 |  0.793 |  0.832 |  0.784 |  0.774
$M_{1,2,3}$ |  0.752 |  0.734 |  0.742 |  0.762 |  0.756 |  0.772 |  0.853 |  0.767
All Data Feeding |  0.789 |  0.793 |  0.794 |  0.801 |  0.799 |  0.803 |  0.814 |  0.799
DH w. SHORE [6] |  0.782 |  0.788 |  0.784 |  0.823 |  0.817 |  0.825 |  0.843 |  0.809
DH w. SH |  0.805 |  0.809 |  0.814 |  0.818 |  0.812 |  0.812 |  0.832 |  0.815
DH w. SC (Ours) |  0.816 |  0.82 |  0.816 |  0.827 |  0.828 |  0.824 |  0.837 |  0.824
$M_{i}$, where i $\in[1,2,3]$ indicates the model is only trained on that shell configuration. $C_{i}$ indicates the testing data in that shell configuration. The best and second best performances are denoted by the red mark and blue mark. The average metrics of ACC are listed in the last column.  Table 2: FODF prediction assessment Model |  Shell configuration |  Tissue proportion prediction |  Scan-rescan consistency
---|---|---|---
Single model |  1K |  8.45E-04 |  0.862
|  2K |  7.92E-04 |  0.865
|  3K |  8.63E-04 |  0.857
|  1K, 2K |  7.32E-04 |  0.856
|  2K, 3K |  7.49E-04 |  0.86
|  1K, 3K |  8.02E-04 |  0.862
|  1K, 2K, 3K |  6.38E-04 |  0.865
DH w. SC |  1K |  7.27E-04 |  0.855
|  2K |  7.12E-04 |  0.86
|  3K |  7.35E-04 |  0.858
|  1K, 2K |  7.01E-04 |  0.858
|  2K, 3K |  6.79E-04 |  0.86
|  1K, 3K |  6.82E-04 |  0.864
|  1K, 2K, 3K |  5.92E-04 |  0.861
Silver standard : MSMT-CSD [15] |  |  0.856
Reconstruction result from msmt-CSD are applied as silver standard in the
evaluation. Wilcoxon signed-rank test is applied as statistical assessment for
scan-rescan consistency evaluation. It has significant difference ($p<0.001$)
compared with WM fODF. The MSE is reported for evaluation of VF predictions.
The ACC between scan/rescan DW-MRI over WM regions is reported.

##  6 Conclusion

In this paper, we propose a single-stage dynamic network with both the q-space
and radial space signal based on a spherical convolutional neural network.
Integrating dynamic head and spherical convolution removes the need to retrain
a new network for a known diffusivity value of DW-MRI. Besides, adjusting the
last multi-layer regression network to different targets, this plug-and-play
design of our method is potentially applicable to a wider range of diffusion
properties in neuroimaging.

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# Discriminating an Arbitrary Number of Pure Quantum States by the Combined
$\mathcal{CPT}$ and Hermitian Measurements

Yaroslav Balytskyi<EMAIL_ADDRESS>Department of Physics and Energy
Science, University of Colorado at Colorado Springs, Colorado Springs, CO
80933, USA  Sang-Yoon Chang<EMAIL_ADDRESS>Department of Computer Science,
University of Colorado at Colorado Springs, Colorado Springs, CO 80933, USA
Anatoliy Pinchuk<EMAIL_ADDRESS>Department of Physics and Energy Science,
University of Colorado at Colorado Springs, Colorado Springs, CO 80933, USA
Manohar Raavi<EMAIL_ADDRESS>Department of Computer Science, University of
Colorado at Colorado Springs, Colorado Springs, CO 80933, USA

###### Abstract

If the system is known to be in one of two non-orthogonal quantum states,
$|\psi_{1}\rangle$ or $|\psi_{2}\rangle$, $\mathcal{PT}$-symmetric quantum
mechanics can discriminate them, in principle, by a single measurement.

We extend this approach by combining $\mathcal{PT}$-symmetric and Hermitian
measurements and show that it’s possible to distinguish an arbitrary number of
pure quantum states by an appropriate choice of the parameters of
$\mathcal{PT}$-symmetric Hamiltonian.

Quantum state discrimination; $\mathcal{PT}$-symmetry; Unstructured database
search;

Quantum state discrimination is an important problem which has many
applications in quantum computing and quantum cryptography; see a review
article, Ref.Rev . In quantum computation it is known that an unstructured
database search can be mapped to the problem of distinguishing exponentially
close quantum states, Ref.Exp . In turn, it is related to the hash functions
widely used in cryptography, Ref.Hash .

Suppose, the system may be in one of two possible states, $|\psi_{1}\rangle$
or $|\psi_{2}\rangle$ and the task is to find in which of them the system is.
A simple classical example: The experimenter is told that the coin is either
fair or biased. The task is to determine by tossing the coin which option is
true.

Quantum state discrimination consists of finding an optimal observable and
strategy of measurements, Ref.Rev . In an conventional quantum mechanics, it’s
impossible to find the state of the system by a single measurement. However,
it is not possible in general to repeat the measurement since it changes the
state of the system. Therefore, to know the state of the system with a high
confidence level, one has to prepare a large number of identical samples.

However, $\mathcal{PT}$-symmetric quantum mechanics, RefsNH1 ; NH2 ; NH3 ,
offers new opportunities. If the $\mathcal{PT}$ symmetry of a Hamiltonian is
not broken, then its eigenvalues are real and Hamiltonian determines an inner
product giving an extra degree of freedom in comparison with a conventional
quantum mechanics.

While it’s an open question on how to properly treat the boundary between
Hermitian and non-Hermitian regimes, Refs.(Doubt1, ; Doubt2, ; Doubt3, ), it’s
worth considering a potential implications of an additional degree of freedom.

In particular, it was shown that in a $\mathcal{PT}$-symmetric quantum
mechanics it’s possible in principle to discriminate between 2 states just by
a single measurement, Ref.PTDis . On practice, discriminating two non-
orthogonal states in a closed system with one measurement can only be done
with probability less then one, otherwise it would be a violation of
unitarity. Therefore, even in the ideal noiseless case, sometimes we may have
to apply the measurement more then once.

Two alternative solutions were proposed:

  * •

Solution 1: Finding a $\mathcal{PT}$-symmetric Hamiltonian whose inner product
interprets $|\psi_{1}\rangle$ and $|\psi_{2}\rangle$ as being orthogonal under
the $\mathcal{CPT}$ scalar product.

  * •

Solution 2: Finding a $\mathcal{PT}$-symmetric Hamiltonian under which the
states $|\psi_{1}\rangle$ and $|\psi_{2}\rangle$ evolve into orthogonal states
under Hermitian scalar product.

We propose to combine Solution 1 and Solution 2 to extend this approach to be
able to discriminate more than two states.

Start with 3 states:

$|\psi_{1}\rangle=\begin{pmatrix}\cos\left(\frac{\pi-2\epsilon}{4}\right)\\\
-i\sin\left(\frac{\pi-2\epsilon}{4}\right)\end{pmatrix};\\\
|\psi_{2}\rangle=\begin{pmatrix}\cos\left(\frac{\pi+2\epsilon}{4}\right)\\\
-i\sin\left(\frac{\pi+2\epsilon}{4}\right)\end{pmatrix};\\\
|\psi_{3}\rangle=\begin{pmatrix}\cos\left(\frac{\pi-2\epsilon}{4}+\gamma\right)\\\
-ie^{i\delta}\sin\left(\frac{\pi-2\epsilon}{4}+\gamma\right)\end{pmatrix};$
(1)

and prepare two identical samples for measurements,
$\left(Sample_{1},Sample_{2}\right)$.

Consider a general $\mathcal{PT}$-symmetric Hamiltonian and an associated
$\mathcal{C}$ operator which defines $\mathcal{CPT}$ scalar product:

$H=H^{\mathcal{PT}}=\begin{pmatrix}re^{i\beta};&s;\\\
s;&re^{-i\beta};\end{pmatrix};\\\
\mathcal{C}=\frac{1}{\cos\left(\alpha\right)}\begin{pmatrix}i\sin\left(\alpha\right);&1;\\\
1;&-i\sin\left(\alpha\right);\end{pmatrix};$ (2)

where $\sin\left(\alpha\right)=\frac{r}{s}\sin\left(\beta\right)$. By setting
$\sin\left(\alpha\right)=\cos\left(\epsilon\right)$ we make $|\psi_{1}\rangle$
and $|\psi_{2}\rangle$ orthogonal in a sense of the $\mathcal{CPT}$ scalar
product. Note that $|\psi_{1}\rangle$ and $|\psi_{2}\rangle$ remain orthogonal
by $\mathcal{CPT}$ over time since $\left[\mathcal{C},\mathcal{PT}\right]=0$
and $\left[\mathcal{C},H\right]=0$ but Hermitian scalar product changes over
time since $H^{\dagger}\neq H$.

The $\mathcal{CPT}$ projection operators are:

$(|\psi_{1}\rangle\langle\psi_{1}|)_{\mathcal{CPT}}=\frac{1}{2\sin\left(\epsilon\right)}\begin{pmatrix}1+\sin\left(\epsilon\right);&-i\cos\left(\epsilon\right);\\\
-i\cos\epsilon;&-1+\sin\left(\epsilon\right);\end{pmatrix};$ (3)
$(|\psi_{2}\rangle\langle\psi_{2}|)_{\mathcal{CPT}}=\frac{1}{2\sin\left(\epsilon\right)}\begin{pmatrix}-1+\sin\left(\epsilon\right);&i\cos\left(\epsilon\right);\\\
i\cos\epsilon;&1+\sin\left(\epsilon\right);\end{pmatrix};$ (4)

Then, make $|\psi_{2}\rangle$ and $|\psi_{3}\rangle$ orthogonal in a sense of
Hermitian scalar product by the Hamiltonian evolution:

$\langle\psi_{2}|e^{iH^{\dagger}t}e^{-iHt}|\psi_{3}\rangle_{Hermitian}=0$ (5)

$e^{iH^{\dagger}t}e^{-iHt}=\frac{1}{\cos^{2}\left(\alpha\right)}\begin{pmatrix}\cos^{2}\left(\omega
t-\alpha\right)+\sin^{2}\left(\omega t\right);&-2i\sin^{2}\left(\omega
t\right)\sin\left(\alpha\right);\\\
2i\sin^{2}\left(\omega\right)\sin\left(\alpha\right);&\cos^{2}\left(\omega
t+\alpha\right)+\sin^{2}\left(\omega t\right);\end{pmatrix}$ (6)

The Hermitian scalar product $(\langle\psi_{2}|\psi_{3}\rangle)_{Hermitian}=0$
becomes zero after the time $\tau$:

$\tan\left(\omega\tau\right)=\frac{\sin\left(\epsilon\right)\left(\cos\left(\epsilon\right)\pm\sqrt{2\cos\left(\epsilon\right)\tan\left(\frac{\pi+2\epsilon}{4}\right)-1}\right)}{2\cos\left(\epsilon\right)\tan\left(\frac{\pi+2\epsilon}{4}\right)-\cos^{2}\left(\epsilon\right)-1}$
(7)

where positive or negative sign depends on the geometry of states and is
chosen in such a way that $\tau>0$.

Now, when $(\langle\psi_{1}|\psi_{2}\rangle)_{\mathcal{CPT}}=0$ and
$(\langle\psi_{2}|\psi_{3}\rangle)_{Hermitian}=0$, Fig.[1], make two
measurements:

Figure 1: $(\langle\psi_{1}|\psi_{2}\rangle)_{\mathcal{CPT}}=0$ and
$(\langle\psi_{2}|\psi_{3}\rangle)_{Hermitian}=0$ after the time $\tau$.

  * •

Apply the $\mathcal{CPT}$ projection operator on the first sample,

$(|\psi_{1}\rangle\langle\psi_{1}|)_{\mathcal{CPT}}|Sample_{1}\rangle=Measurement_{1}$
(8)

  * •

Apply the Hermitian projection operator on the second sample,

$(|\psi_{2}\rangle\langle\psi_{2}|)_{Hermitian}|Sample_{2}\rangle=Measurement_{2}$
(9)

We may get the following results:

  * •

If $Measurement_{1}=0$, the system is the state $|\psi_{2}\rangle$ and the
problem is solved (since
$(\langle\psi_{1}|\psi_{2}\rangle)_{\mathcal{CPT}}=0$)

  * •

If $Measurement_{1}\neq 0$, the system may be in the states $|\psi_{1}\rangle$
or $|\psi_{3}\rangle$ and we need to consider the second measurement.

  * •

If $Measurement_{1}\neq 0$ and $Measurement_{2}=0$, the system is in the state
$|\psi_{3}\rangle$ (since $(\langle\psi_{2}|\psi_{3}\rangle)_{Hermitian}=0$)

  * •

If $Measurement_{1}\neq 0$ and $Measurement_{2}\neq 0$, the system is in the
state $|\psi_{1}\rangle$ (since it is not in the states $|\psi_{2}\rangle$ and
$|\psi_{3}\rangle$ which we assumed to be pure states)

As we shown, the same Hamiltonian makes the corresponding $\mathcal{CPT}$ and
Hermitian products zero. Another possible solution would be to apply two
different $\mathcal{PT}$-symmetric Hamiltonians on $Sample_{1}$ and
$Sample_{2}$ to make $(\langle\psi_{1}|\psi_{2}\rangle)_{\mathcal{CPT}}=0$ and
$(\langle\psi_{2}|\psi_{3}\rangle)_{\mathcal{CPT}}=0$ and then apply two
$\mathcal{CPT}$ projections instead of the combined $\mathcal{CPT}$ and
Hermitian measurements.

This scheme can be extended for distinguishing an arbitrary number of pure
quantum states. Suppose, we have an arbitrary number of states

$|\psi_{1}\rangle,|\psi_{2}\rangle,\cdots,|\psi_{N}\rangle$ (10)

Prepare $N-1$ samples for measurements and consider three states from the
list, $i,j,k\in[1,N]$. Then do the following:

  * •

Make $(\langle\psi_{i}|\psi_{j}\rangle)_{\mathcal{CPT}}=0$ and
$(\langle\psi_{j}|\psi_{k}\rangle)_{Hermitian}=0$ (or
$(\langle\psi_{i}|\psi_{j}\rangle)_{\mathcal{CPT}}=0$ and
$(\langle\psi_{j}|\psi_{k}\rangle)_{\mathcal{CPT}}=0$)

  * •

Apply the $\mathcal{CPT}$ projection operator on the first sample,

$(|\psi_{i}\rangle\langle\psi_{i}|)_{\mathcal{CPT}}|Sample_{1}\rangle=Measurement_{1}$
(11)

  * •

Apply the Hermitian (or $\mathcal{CPT}$) projection operator on the second
sample,

$(|\psi_{j}\rangle\langle\psi_{j}|)_{Hermitian/\mathcal{CPT}}|Sample_{2}\rangle=Measurement_{2}$
(12)

We find a state of the system if one of the measurements is zero, or exclude
two possibilities from our list. By applying this procedure several times on
the remaining states from our list, we exhaust all possibilities and find the
state of the system assuming it’s in the pure state.

The question on whether this scheme could be extended to mixed states we refer
to our future research.

##  I Acknowledgments

Y.B. was supported by the UCCS Graduate Research Award.

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