11institutetext: University of Toronto, Toronto, Canada 22institutetext: The
Hospital for Sick Children, Toronto, Canada 33institutetext: Vector Institute,
Toronto, Canada

# 3D Reasoning for Unsupervised Anomaly Detection in Pediatric WbMRI

Alex Chang 11 2 2 3 3 Vinith Suriyakumar 11 2 233 Abhishek Moturu 1122 3 3
James Tu 33 Nipaporn Tewattanarat 22 Sayali Joshi 22 Andrea Doria 22 Anna
Goldenberg 11 2 2 3 3

###### Abstract

Modern deep unsupervised learning methods have shown great promise for
detecting diseases across a variety of medical imaging modalities. While
previous generative modeling approaches successfully perform anomaly detection
by learning the distribution of healthy 2D image slices, they process such
slices independently and ignore the fact that they are correlated, all being
sampled from a 3D volume. We show that incorporating the 3D context and
processing whole-body MRI volumes is beneficial to distinguishing anomalies
from their benign counterparts. In our work, we introduce a multi-channel
sliding window generative model to perform lesion detection in whole-body MRI
(wbMRI). Our experiments demonstrate that our proposed method significantly
outperforms processing individual images in isolation and our ablations
clearly show the importance of 3D reasoning. Moreover, our work also shows
that it is beneficial to include additional patient-specific features to
further improve anomaly detection in pediatric scans.

###### Keywords:

Cancer Detection Anomaly Detection Generative Model

##  1 Introduction

Deep learning has shown promise in detecting cancerous lesions in breast [22],
prostate [16] and lungs [15]. In earlier problem settings, there were too few
training examples as well as insufficient number of disease labels preventing
supervised methods from achieving good performance. Recently, unsupervised
deep learning methods have started to show promise in these challenging
settings by reformulating the problem as anomaly detection, using only the
disease-free images during training and diseased images for testing [18]. The
use of these methods is still relatively unexplored in pediatric cancer
detection setting. In pediatric cancer screening, especially in diseases of
cancer predisposition, which are intrinsically rare, whole-body MRI (wbMRI) is
an essential part of the screening protocol [8]. The goal is to detect lesions
as early as possible, which is a difficult task for radiologists because of
the small size of nodules and professional inexperience working with this
modality. Unsupervised deep learning for anomaly detection is yet to be
applied in this setting.

Existing techniques use variants of generative adversarial networks (GANs) [7]
and variational autoencoders (VAEs) [10] to model the distribution of non-
cancerous patients and doing out-of-distribution detection to highlight
anomalous cases, i.e. those with cancer [18]. Most studies applying these
techniques to MRIs have focused on single body parts such as the brain [12].
These settings are less difficult than the wbMRI setting because all of the
images are well registered and slices look similar throughout the volume.
wbMRIs present challenges such as poor registration between patient volumes,
large heterogeneity between the slices in the MRI volume, and lower signal-to-
noise ratio. These issues are exacerbated in the pediatric setting because
patients at various stages of growth and development look differently on this
and other imaging modalities. Due to these problems, traditional registration
is not applicable to the pediatric setting. We overcome these issues by
building on unsupervised methods to leverage 3D information, and patient
specific features.

We propose using a conditional VAE [19] with a sliding window approach to
overcome these challenges for cancer detection in pediatric wbMRI (Figure 1).
While prior work has previously dealt with 3D inputs by either using
computationally expensive 3D convolutions or by sampling individual slices as
2D inputs, our model accepts consecutive slices as multi-channel inputs and is
conditioned on the slice and window coordinates to facilitate 3D reasoning.
Moreover, we condition the model on patient features to disentangle the
learned latent space to more accurately model variations in different regions
of the body such as the brain, chest and abdomen. The contributions of our
study are four-fold:

  1. 1.

We show that processing multiple consecutive slices of 3D volumes
significantly improves unsupervised anomaly detection.

  2. 2.

We learn the distribution of large healthy body regions using a sliding window
approach by informing the model of the window location.

  3. 3.

We show that conditioning on patient meta-features can lead to substantial
improvements in anomaly detection.

  4. 4.

We are the first to perform anomaly detection for pediatric wbMRIs which are
more challenging than other settings due to difficulties in registration and
images involving multiple body regions.

Overall, we demonstrate multiple techniques towards more accurate anomaly
detection for cancer patients and we propose a method capable of processing
challenging whole-body MRI scans.

##  2 Related Work

###  2.1 Whole-Body MRIs in Pediatric Cancer Screening Protocols

Whole-body magnetic resonance imaging (wbMRI) is an essential part of well-
established cancer screening protocols [21]. These protocols have shown
improvements in early detection of cancer for both adult [2] and pediatric [8]
patients.

###  2.2 Deep Learning for Cancer Detection

Both supervised and unsupervised deep learning approaches have demonstrated
potential for improving cancer detection in a number of settings. Supervised
learning methods such as convolutional neural networks have been used to
improve cancer detection across imaging modalities such as MRIs [9], X-rays
[3], and CTs [14]. Different cancers where this has been demonstrated include:
breast cancer [22], prostate cancer [13], brain cancer [12], and lung cancer
[15]. Specifically, for whole-body MRIs, supervised deep learning has shown
promise in screening adult patients [11]. To the best of our knowledge, these
methods have not been applied to pediatric wbMRI cancer screening, where due
to smaller sample sizes and potentially even higher class imbalance these
methods are not likely to succeed.

Unsupervised deep learning techniques for cancer detection have focused on
formulating the task as anomaly detection. This approach has shown superior
performance to supervised deep learning techniques when the proportion of
disease labels is low [6]. Unsupervised deep learning has also been used as a
feature extractor to improve cancer detection. One example of this is in
improving breast cancer detection [23]. Initial work has shown the promise of
generating single pediatric wbMRI slices and anomaly detection [6] but no work
to our knowledge takes the full complexity into account successfully.

###  2.3 Unsupervised Anomaly Detection with Generative Models

Variations of VAEs, GANs, and combinations of the two have been proposed as
solutions to anomaly detection. In cancer detection, these methods aim to
model the distribution of patients without cancer and then perform out-of-
distribution detection to highlight image regions with cancer. First, AnoGAN
has demonstrated potential in highlighting anomalous regions on optical CT
scans [18]. GANomaly[1] and f-AnoGAN[17] improve this framework by removing
the costly optimization steps required for testing AnoGAN. However, these
methods are restricted to detection in low-resolution patches. AnoVAEGAN
performed cancer segmentation in high-resolution brain MRI using a VAE-GAN
hybrid with spatial latent space [4]. Although they performed cancer detection
in volumetric brain MRI, an important drawback from their method is that they
process each slice as an independent sample, resulting in 2D modeling which do
not capture 3D context; only a postprocessing step is applied afterwards with
the combined slices. Finally, Viana et al [20] propose using a model based on
f-AnoGAN which successfully handles volumetric brain MRI but requires
computationally expensive 3D convolution operations.

Figure 1: Framework overview. The blue bounding box is determined by multi-
scale template matching and determines the width of the chest. The red box
then defines the region of interest. Square windows (yellow) within the region
of interest are inputted to the spatial VAE.

##  3 Method

###  3.1 Conditional $\beta$-Variational Auto-Encoder

Architecture: We utilize a VAE to learn the distribution of cancer-free images
such that the model fails to reconstruct anomalous regions (i.e. the cancerous
part) of the test images. Note that while we build on [4], we simplified this
model by omitting the adversary as it has been demonstrated to only marginally
improve performance. We input conditional variables to the encoder and decoder
networks by using a linear layer to predict a scaling and bias factor for the
batch normalization feature maps in each residual block. Unlike BigGAN, we do
not include the latent samples in this linear layer due to the large
dimensionality of the spatial latent space.

Training For image $\mathbf{x}\in\mathbb{R}^{c\times 256\times 256}$ where $c$
is the number of channels/slices, and conditional variable $y=($age, sex,
weight$,w_{z},w_{y})$ with slice height $w_{z}$ and window height $w_{y}$, the
VAE parametrizes latent distribution $q_{\phi}(\mathbf{z}|\mathbf{x},y)$ from
which latent vector $\mathbf{\hat{z}}\sim q_{\phi}(\mathbf{z}|\mathbf{x},y)$
is sampled via the reparametrization trick. Reconstruction
$\mathbf{\hat{x}}=D_{\theta}(\mathbf{\hat{z}},y)$ is then generated with
decoder $D_{\theta}$. We minimize the conditional $\beta$-VAE loss function

$||\mathbf{x}-\mathbf{\hat{x}}||^{2}+\beta
D_{KL}(q_{\phi}(\mathbf{z}|\mathbf{x},y)||\mathcal{N}(0,\mathbf{I}))$

Intuitively, $\beta$ hyperparameter represents the strength on the KL
Divergence regularization term which encourages the proposal distribution
$q_{\phi}(\mathbf{z}|\mathbf{x},y)$ to match a standard Normal prior. This
constraint ensures that the latent space maintains a degree of smoothness and
prevents the model from memorizing the training set as in the case of
autoencoders.

Table 1:  Processing volumes of wbMRI scans as multiple slices instead of individual images (1 slice) achieves significantly better results. Also, it is crucial to provide the coordinates of the windows ($w_{y},w_{z}$ ) in our sliding window approach.  Dataset |  Input |  AUPRC |  AUROC |  DICE
---|---|---|---|---
Chest w/ simulated nodules |  1 slice |  0.221 |  0.912 |  0.251
1 slice + $w_{z}$ |  0.283 |  0.941 |  0.298
Upper body w/ real nodules |  1 slice |  0.033 |  0.845 |  0.053
1 slice + $w_{z}$ |  0.124 |  0.844 |  0.141
1 slice + $w_{y}$ \+ $w_{z}$ |  0.129 |  0.910 |  0.159
3 slices + $w_{y}$ \+ $w_{z}$ |  0.141 |  0.901 |  0.161
5 slices + $w_{y}$ \+ $w_{z}$ |  0.174 |  0.891 |  0.189
7 slices + $w_{y}$ \+ $w_{z}$ |  0.169 |  0.869 |  0.178
9 slices + $w_{y}$ \+ $w_{z}$ |  0.179 |  0.894 |  0.191

Anomaly Detection To produce an anomaly mask that we report back to the user,
we reconstruct $\mathbf{x}$ with maximum likelihood $\mathbf{z}$ from
distribution $q_{\phi}(\mathbf{z}|\mathbf{x},y)$
($\mathbb{E}[q_{\phi}(\mathbf{z}|\mathbf{x},y)]$) and obtain the pixel-wise
squared-error loss mask. We then apply the post-processing steps by only
keeping loss values for which the reconstruction underestimates the pixel
value since lesion are brighter than normal tissues on STIR images and remove
small regions which are unlikely to represent lesions using a $3\times 5\times
5$ median filter. During training, we randomly sample a window with a random
height within the region of interest and during testing, we produce anomaly
masks for equally spaced vertical windows and average overlapping mask regions
on the same slice.

###  3.2 3D Context and Patient Features

In prior work, images were treated independently which failed to capture the
spatial correlation between slices when they are actually sampled from the
same 3D volume. Meanwhile, training a 3D convolutional model is
computationally expensive and requires a consistent shape for the input volume
which is difficult to achieve in pediatric population. We therefore leverage
information of neighbouring slices by using consecutive slices as a multi-
channel input. The residuals of the middle channel, which we observe to have
the lowest loss (and most accurate reconstruction), are then used to produce
the loss mask. We slide the window across slices and concatenate loss masks
before post-processing with the aforementioned method. Baur et al. demonstrate
that a spatial latent space is essential for accurate reconstructions of brain
MRI as it facilitates the comparison of similar structures [4]. Because we
employ a sliding window approach, we condition the model on the window’s
coordinates since image contents greatly depends on the location of the
window. We compute the slice coordinates using the slice spacing and thickness
and use the relative distance from the back of the patient. All conditioned
variables are then standardized.

##  4 Experiments

###  4.1 Experimental Setup

Dataset: We demonstrate the efficacy of our method on a dataset of coronal
STIR whole-body MRI provided by a children’s hospital containing 535 non-
cancerous images (22 to 45 slices each for a total of 18012 slices) and 27
annotated images with at least one lesion. Images are preprocessed with
histogram equalization, N4 bias field correction, and noise removal. Pediatric
wbMRI are more difficult to register due to the greater variation in body
positions. Because we also lack the annotations and atlas required for most
robust registration methods, we focus on modeling a region of interest which
includes the width of the chest from head to pelvis. We employ a pseudo-
registration method by first localizing the bounding box which corresponds to
each patient’s chest with multi-scale template matching. We then define the
region of interest as the chest width from top to middle pixel row of the
image (Figure 1). We proportionally resize each region of interest to a width
of 256 pixels. Because the height of the regions of interest differs between
images, we use a vertical sliding window approach to model one square
$256\times 256$ window at a time.

Table 2: Ablation on using various patient features (age, weight, sex). Here window coordinates are still provided and we process 5 MRI slices. Additional features improves performance substantially. However, $\beta$ needs to be tuned appropriately.  Slices(5) |  Age |  Weight |  Sex |  Tune $\beta$ |  AUPRC |  AUROC |  DICE
---|---|---|---|---|---|---|---
✓ |  |  |  |  |  0.174 |  0.891 |  0.189
✓ |  ✓ |  |  |  |  0.169 |  0.853 |  0.184
✓ |  |  ✓ |  |  |  0.135 |  0.822 |  0.133
✓ |  |  |  ✓ |  |  0.213 |  0.917 |  0.194
✓ |  ✓ |  ✓ |  ✓ |  |  0.171 |  0.894 |  0.196
✓ |  |  |  ✓ |  ✓ |  0.235 |  0.941 |  0.239

Metrics: We evaluate model performance with the Area Under the Precision-
Recall Curve (AUPRC), Area Under the Receiving-Operator Curve (AUROC), and
DICE coefficient. Both AUPRC and AUROC allow the evaluation of model
capabilities without deciding an operating point or decision threshold. We
note that the AUROC is sensitive towards the minority class in the context of
severe class imbalance. To select a threshold required to compute DICE, we use
the 99th percentile of non-zero loss values after post-processing.

Labeling: We conduct two sets of anomaly detection experiments, one on images
with synthetically generated tumors and one on real tumors labelled by a
radiologist. We generate the synthetic tumors by adding a Gaussian blob with
peak intensity from 0.6 to 3 consecutive slices of each test volume.

Implementation details: All models are trained with a 0.001 learning rate, 16
batch size, and Adam optimizer for 150 epochs and the spatial latent dimension
is maintained at $32\times 16\times 16$. We also anneal $\beta$ during
training to linearly increase during the first 20 epochs. We modify the
encoder and decoder networks from a series of ReLU activated convolutional
layers to a sequence of residual blocks which is used in the state-of-the-art
conditional GAN model (BigGAN [5]). All model training is done in PyTorch
v1.6.0 and required approximately 24 hours on 2 NVIDIA T4 GPUs.

Figure 2: Performance comparison for various settings of the regularization
hyperparameter, $\beta$, with 5 slice inputs and conditioning on patient sex.

###  4.2 Cancer Detection

3D Context: To answer whether the naive baseline approach benefits from
conditioning on the slice coordinate, we compare the methods using only crops
of the chest nodules in the test set (Table 1). We observe a 0.062 increase in
AUPRC performance and 0.047 increase in DICE. We then proceed to model the
larger region of interest using the sliding window approach. With the naive
independent window samples, we observe that the AUPRC and DICE scores are low
while the AUROC is relatively high compared to the value of a random
classifier. Upon only adding the slice location, the model considerably
improves (0.091 AUPRC and 0.088 DICE increases) while subsequently adding the
window height marginally improves AUPRC and DICE. Maintaining both window and
slice coordinates while increasing the number of slices/channels further
boosts performance up to 5 slices (0.017 AUPRC improvement over 1 slice) where
it starts to plateau. We therefore used 5 slices for the remaining experiments
to reduce runtime. This suggests that the 3D information from neighbouring
slices helps learn a more discriminative latent manifold.

Patient Features: Next, we experimented with the inclusion of patient’s age,
weight, and sex as these meta-features theoretically correlate with the
anatomy. In these experiments window coordinates are still provided and we
process 5 MRI slices to minimize excess computation. We observe that the model
benefits from the inclusion of the patient’s sex (0.061 AUPRC improvement)
while including age and weight worsen the performance. We hypothesize that
this is due to the fact that sex is a binary attribute which helps to
distinguish between real differences on the wbMRIs while substantial number of
training examples still exists for both groups (sexes). In contrast, the
continuous attributes of weight and age may be poorly represented for some
ranges, giving a greater chance of introducing biases with our relatively
small training set.

Figure 3: Stitched reconstructions (top row) and their resulting postprocessed
loss masks (bottom row) for the original image with ground-truth label in the
leftmost column. We observe a more accurate loss mask with the increase in
number of slices used for input. Moreover, we observe blurrier
reconstructions, capturing more global features with the increase of $\beta$.

Regularization: Table 2 demonstrates that $\beta=2$ and $\beta=3$ provide the
greatest AUPRC and DICE respectively, while too low or too high $\beta$ values
decrease the performance. Figure 3 shows that with a low $\beta$ value the
reconstructions are very close to the input but the model fails to learn the
healthy latent manifold which results in reconstructions of anomalies. As
$\beta$ increases, the cancerous image region ceases to be reconstructed, but
as $\beta$ gets too high the input’s details are lost which contributes to
more false positives.

##  5 Conclusion

We demonstrated applicability of unsupervised cancer detection from wbMRI in
pediatric population. Using a multi-channel sliding window approach, we
incorporated 3D information which overcomes several challenges specific to
this modality such as registration and high image dimensionality and show
improved performance over a 2D baseline. We show that conditioning on certain
meta-features which are indicative of window contents improves detection
performance and that tuning the regularization hyperparameter can further help
disentangle the latent space.

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[table]capposition=top

# Estimation of nuclear polarization via discrete measurement of NV center
spin evolution

Mateusz Kuniej  Institute of Theoretical Physics, Faculty of Fundamental
Problems of Technology, Wrocław University of Science and Technology, 50-370
Wrocław, Poland  Katarzyna Roszak  FZU - Institute of Physics of the Czech
Academy of Sciences, 182 00 Prague, Czech Republic

###### Abstract

We propose a method for the estimation of the initial polarization of spinful
nuclei of the ${}^{13}C$ isotope in diamond via a measurement of the evolution
of the coherence of an NV center spin qubit. Existing polarization measurement
methods are difficult to implement experimentally, because they require direct
interference in the environment of the qubit. Here, in order to obtain the
information, it is necessary to measure the qubit coherence at certain points
of time, which are unambiguously determined by the applied magnetic field. For
sufficiently high magnetic fields, the minimum value of the measured coherence
constitutes an upper bound on the product of the initial polarizations of each
environmental spin. The most significant advantage of the method, which allows
to infer initial values of nuclear polarizations without any direct access to
the environment, lies in its simplicity and the small amount of experimental
resources that it requires. We exemplify the operation of the scheme on a
realistic, randomly generated environment of eight nuclear spins, obtaining a
reasonably accurate estimation of the initial polarization.

##  I Introduction

Nitrogen-vacancy (NV) centers in diamond [1, 2, 3, 4] are attracting a lot of
attention for their applications in ultrasensitive magnetometry [5, 6, 7, 8,
9, 10, 11]. Since these applications rely on the effect that even very small
magnetic fields have on the evolution of the NV center spin qubit, they
require minimization of decoherence stemming from sources within the diamond
lattice where the NV center is embedded. The dominating decoherence mechanism
here is the hyperfine interaction with spinfull ${}^{13}C$ carbon atoms
present in the lattice [12, 13]. Since the ${}^{13}C$ isotope is rare (and
randomly distributed within the lattice), the NV center spin interacts with a
small number of environmental spins [14, 15], which leads to naturally
occurring long decoherence times [16, 17, 8]. Nevertheless, even slower
decoherence is necessary for effective ultrasensitive magnetometry, so
techniques such as dynamical decoupling (DD) [18, 19, 20, 21] and dynamical
nuclear polarization (DNP) [22, 23, 24, 25, 26, 27, 28, 29] have to be
employed.

Recently, several schemes for polarizing nuclear spins and its detection have
been developed. A very promising DNP method is based on optically pumped NV
centers where several experimental demonstrations have been reported [30, 22,
23, 31, 32]. The main disadvantage of those schemes is their sensitivity to
the magnetic field alignment along the NV crystal axis. The Polarization
readout by Polarization Inversion (PROPI) technique allowed to find new
polarization schemes that can be applied at arbitrary magnetic field strengths
and a wide range of magnetic field orientations [26].

One of the detection schemes reads out the magnetization of the nuclear spin
bath using PROPI, which provides a measure of polarization of the nuclear
spins surrounding the NV center and allows observation of hidden polarization
dynamics similar to Landau-Zener-Stückelberg oscillations [26]. More often
estimation of the value of the polarization is achieved using optically
detected magnetic resonance (ODMR) by integrating electron resonances [30] or
characterizing the ODMR peaks by transition probabilities in the hyperfine
basis states [33]. Another intriguing scheme exploits the Fourier-transformed
nuclear magnetic resonance spectrum in which Lorentzian peaks are proportional
to the nuclear polarization [31].

Here we propose a much simplified scheme for the detection of polarization of
the nuclear environment which requires only the measurement of the evolution
of the coherence of the NV center spin qubit, instead of measurements on the
nuclear spins directly. The qubit evolution in principle contains the full
information about the initial state of the environment. Yet, since the
interaction with each nuclear spin is different, due to the strong dependence
on the actual location of the spinful carbon isotope in the diamond lattice,
and the strength of the coupling between the qubit with each spin of the
environment is unknown, a straightforward readout of the initial polarization
from the qubit evolution is not feasible. Nevertheless, the dependence of
qubit coherence evolution on a frequency that is a direct function of the
applied homogeneous magnetic field allows us to determine time-instants for
which the function governing the effect of each nucleus on the qubit is
greatly simplified. Since the frequency is the same for each environmental
spin, a discrete measurement of qubit coherence, in the sense that only the
evolution at points of time which are governed by simplified functions is
probed, allows to obtain much greater information about the environment than
the full evolution does.

We find that this technique, which is limited to the measurement of qubit
coherence at points of time that are unambiguously determined by the applied
magnetic field, allows to find the upper bound on the product of initial
environmental polarizations. The drawbacks of the method include the need for
measurements up to relatively long times and the fact that the obtained value
is only an estimate of the actual polarization. On the other hand, the
straightforward nature of the technique allows for extracting information
about the environment without any measurements on it, and only accessing the
NV center spin qubit. As such, the method should be useful for preliminary
studies of the effectiveness of the methods used to polarize the spin
environment.

The article is organized as follows. In Sec. II we discuss the evolution of
the spin qubit in the presence of a nuclear environment. In Sec. III we show
how probing the qubit at discrete points of time can lead to a simplification
of the observed qubit evolution in such a way that allows to obtain more
information about the environment. We discuss two such regimes in Secs III.1
and III.2 and show exemplary results. Sec. IV concludes the paper.

##  II Nuclear environment and spin qubit

The diamond lattice consists mainly of ${}^{12}C$ spinless carbon atoms, but
there is about $1.1\%$ of the ${}^{13}C$ spinful carbon isotope. The latter
interact with the NV center spin via the hyperfine interaction resulting in
the dominant decoherence mechanism, so they constitute the spin environment
[34]. Hence, when describing the open-system dynamics of the NV center spin,
only a small number of environmental spins need to be considered.

The free Hamiltonian of the spin environment is given by

$\hat{H}_{E}=\gamma_{n}B_{z}\sum_{k=0}^{N-1}\hat{I}_{k},$ (1)

where the magnetic field $B_{z}$ is applied in the $z$ direction and
$\gamma_{n}=10.71$ MHz/T is the gyromagnetic ratio of the ${}^{13}C$ nuclei.
$\hat{\bm{I}}_{k}$ is the spin operator for the $k-$th nuclear spin. A term
describing the internuclear magnetic interactions can be omitted since they
are weak in comparison with the qubit-environment interaction and do not
affect any of the described processes in a substantial way [12].

To probe the initial polarization of the environment, we will employ a spin
qubit defined on two of the low energy NV center spin states, which constitute
an effective electronic spin triplet $S=1$ [35], with the spin levels
${|-1\rangle}$, ${|0\rangle}$, ${|1\rangle}$ being the the eigenvectors of the
$\hat{S}_{z}$ operator. The most widely employed qubit consists of the $m=0,1$
states, and in the qubit subspace the free qubit Hamiltonian is given by

$\hat{H}_{Q}=(\Delta-\gamma_{e}B_{z}){|1\rangle}\langle 1|.$ (2)

Here $\Delta=2.87$ GHz is the zero-field splitting and $\gamma_{e}=28.08$
MHz/T is the electron gyromagnetic ratio.

The hyperfine interaction [36] between this qubit and environment is described
by the Hamiltonian (the $m=0$ qubit state is decoupled from the environment)

$\hat{H}_{QE}={|1\rangle}\langle
1|\otimes\sum_{k=0}^{N-1}\sum_{j\in(x,y,z)}\\!\\!\\!\mathbb{A}_{k}^{z,j}\hat{I}_{k}^{j}.$
(3)

We have omitted the Fermi contact interaction which describes the non-zero
probability of finding the electron bound to the NV center on the location of
a given nucleus and is proportional to the electronic wavefunction density at
the location of a nucleus. Since the wavefunction of deep defects is strongly
localized, the Fermi contact term gives a measurable contribution to the
hyperfine coupling for nuclei that are at most $0.5$ nm away from the defect
[37]. The coupling constants in eq. (3) are given by

$\mathbb{A}_{k}^{z,j}=\frac{\mu_{0}}{4\pi}\frac{\gamma_{e}\gamma_{n}}{r_{k}^{3}}\left(1-3\frac{({\bf
r}_{k}\cdot\hat{{\bf j}})({\bf r}_{k}\cdot\hat{{\bf z}})}{r_{k}^{2}}\right),$
(4)

where $\mu_{0}$ is the magnetic permeability of the vacuum and the ${\bf
r}_{k}$ is a displacement vector between the NV center and the $k$-th spin.
$\hat{{\bf j}}=\hat{{\bf x}},\hat{{\bf y}},\hat{{\bf z}}$ are versors in three
distinct directions.

Since the qubit Hamiltonian (2) and the interaction term (3) commute, the
evolution of the qubit is limited to pure decoherence, a process during which
only the off-diagonal elements of the qubit density matrix (written in the
$\\{|0\rangle,|1\rangle\\}$ pointer basis of the qubit) evolve and decay. In
the following, we consider the initial state of the qubit always to be the
equal superposition state
${|\Psi\rangle}=\frac{1}{\sqrt{2}}({|0\rangle}+{|1\rangle})$.

The initial state of the spinful carbon nuclei ${}^{13}C$ after DNP is given
by

$\hat{R}(0)=\bigotimes_{k}\hat{\rho}_{k}(0),$ (5)

where we assumed that there are no initial correlations between the nuclei,
and $\hat{\rho}_{k}(0)$ denotes the density matrix of $k$-th nucleus (with
spin-$1/2$ nuclei) given by

$\hat{\rho}_{k}(0)=\frac{1}{2}(\mathds{1}+2p_{k}\hat{I}^{z}_{k}).$ (6)

Here $p_{k}\\!\in\\![-1,1]$ is the polarization of the $k$-th nucleus. Without
DNP the polarization $p_{k}$ is zero for all nuclear spins, and the initial
state of the environment is fully mixed.

Since there are no inter-nuclear correlations in the environmental initial
state and no terms in the Hamiltonian that can induce such correlations, the
evolution of the qubit coherence can be found analytically [38] and for $N$
nuclei is given by

$\rho_{01}(t)=\frac{1}{2}\left(\langle\hat{\sigma}_{x}\rangle-i\langle\hat{\sigma}_{y}\rangle\right)=\frac{1}{2}e^{-i(\Delta-\gamma_{e}B_{z})t}\prod_{k=0}^{N-1}L_{k}(t),$
(7)

where the phase oscillation is the result of the free qubit evolution. The
decoherence stemming from the interaction with the $k$-th nucleus is
quantified by

$\displaystyle L_{k}(t)$ $\displaystyle=$
$\displaystyle\left(a_{k}\sin\frac{\omega
t}{2}\sin\frac{\omega_{k}t}{2}+\cos\frac{\omega
t}{2}\cos\frac{\omega_{k}t}{2}\right)$
$\displaystyle+ip_{k}\left(a_{k}\cos\frac{\omega
t}{2}\sin\frac{\omega_{k}t}{2}-\sin\frac{\omega
t}{2}\cos\frac{\omega_{k}t}{2}\right).$

Here $\omega=\gamma_{n}B_{z}$ is a frequency which is determined by the
applied magnetic field and is both typically well-known and can be controlled.
All other parameters depend on the specific locations of the spinful carbon
isotopes in a given sample realization with respect to the qubit location. The
$k$-dependent frequency (which is also magnetic-field-dependent) is given by

$\omega_{k}=\sqrt{\left(\mathbb{A}_{k}^{z,x}\right)^{2}+\left(\mathbb{A}_{k}^{z,y}\right)^{2}+\left(\omega+\mathbb{A}_{k}^{z,z}\right)^{2}},$
(9)

and the amplitude is

$a_{k}=\frac{\omega+\mathbb{A}_{k}^{z,z}}{\omega_{k}}.$ (10)

These parameters vary depending on the specific system under study and are
usually not known.

##  III Discrete qubit evolution

It is evident from eq. (II), where the function $L_{k}(t)$ determines the
effect of a single environmental spin on the evolution of the qubit coherence
that the information about the initial polarization of spin $k$ given by
$p_{k}$ must manifest itself in the evolution. For an environment consisting
of a single spin, the imaginary part of the coherence (when the free evolution
of the qubit is not taken into account), $\langle\hat{\sigma}_{y}\rangle$, is
directly proportional to the magnitude of initial polarization, yet its
evolution also depends on quantities which are unknown and depend on the
location of the spinful nucleus in the diamond lattice. For environments that
contain more spins, the situation becomes more complex, as the qubit
decoherence (7) is proportional to a product of single-nuclear-spin functions
(II), so the distinction between real and imaginary components is lost while
the number of free parameters grows rapidly with the growing size of the
environment. Nevertheless, the information about initial environmental
polarization is there and it should be possible to extract it by studying
qubit decoherence curves.

In the following, we propose a straightforward way of extracting the upper
bound on the geometric mean value of initial nuclear polarizations. To this
end, we note that all of the terms in eq. (II) have a simple cosinusoidal or
sinusoidal dependence on the frequency $\omega$ determined fully by the
applied magnetic field. Since this frequency can be controlled and is known,
the measurement of the qubit decoherence function (7) can be confined to set
time-intervals that correspond to $\sin\frac{\omega t}{2}=0$ or
$\cos\frac{\omega t}{2}=0$. We will focus on both of these cases in our
further analysis.

###  III.1 $\cos\frac{\omega t^{\prime}}{2}=0$

$k$ |  $r_{k}$ (nm) |  $\mathbb{A}^{z,x}_{k}$ (1/$\mu$s) |  $\mathbb{A}^{z,y}_{k}$ (1/$\mu$s) |  $\mathbb{A}^{z,z}_{k}$ (1/$\mu$s)
---|---|---|---|---
$1$ | 2.16954 | -0.017784 | -0.007206 | -0.034589
$2$ | 2.33189 | 0.013514 | -0.003004 | -0.028353
$3$ | 1.03132 | 0.505446 | -0.135434 | 0.257915
$4$ | 2.45435 | -0.001277 | 0.004767 | -0.025882
$5$ | 2.55104 | 0.017444 | -0.028528 | 0.001358
$6$ | 1.54291 | -0.137190 | 0.036760 | 0.122291
$7$ | 1.03132 | -0.251613 | -0.251613 | -0.215682
$8$ | 2.35773 | -0.041094 | -0.011011 | 0.027203
Table 1: Coupling constants between the NV-center qubit and nuclear spins for
the randomly generated realization of the environment used in all figures. The
second column contains distances between the qubit and each nucleus. Figure 1:
Absolute value of the discrete qubit coherence evolution at times $t^{\prime}$
under the influence of an 8-nucleus environment and applied magnetic field of
$B_{z}=1$ T. Violet dots correspond to all $p_{k}=0.8$ and green squares to
$p_{k}=0.6$. The values of coherence corresponding to those polarizations
($8.39\times 10^{-2}$ for $p_{k}=0.8$ and $8.40\times 10^{-3}$ for
$p_{k}=0.6$) are marked by the dashed black lines.  Figure 2: A close-up of
the $p_{k}=0.8$ discrete qubit coherence evolution of Fig. 1 (violet dots).
The yellow curve shows the continuous line obtained from eq. (13), which
contains the information about the interaction and the initial polarization of
the environment, but is still vastly simplified with respect to the full qubit
decoherence curve.

Let us first study the situation when the value of the qubit coherence is
probed only at times $t^{\prime}=\frac{2\pi}{\omega}(n+\frac{1}{2})$, where
$n\in\mathbb{N}$. This situation corresponds to $\cos\frac{\omega
t^{\prime}}{2}=0$. This significantly simplifies the contribution of an
individual nuclear spin (II) yielding

$L_{k}(t^{\prime})=\pm a_{k}\sin\frac{\omega_{k}t^{\prime}}{2}\mp
ip_{k}\cos\frac{\omega_{k}t^{\prime}}{2}.$ (11)

It is relevant to note here that the amplitude $a_{k}$ given by eq. (10) is
real and $|a_{k}|\in[0,1]$, hence

$|L_{k}(t^{\prime})|=\left[(a_{k}^{2}-p_{k}^{2})\sin^{2}\frac{\omega_{k}t^{\prime}}{2}+p_{k}^{2}\right]^{\frac{1}{2}}.$
(12)

This means that $|L_{k}(t)|$ oscillates between $|a_{k}|$ and $|p_{k}|$ with
the frequency $\omega_{k}$, while whether $|a_{k}|$ or $|p_{k}|$ is the
maximum or the minimum depends on the actual values of the polarization of a
given nuclear spin and the location-dependent amplitude.

As a typical environment of an NV-center spin qubit involves more than one
spin, the bounds on the oscillation of

$|\rho_{01}(t^{\prime})|=\frac{1}{2}\prod_{k=0}^{N-1}|L_{k}(t^{\prime})|$ (13)

cannot be used to extract the initial polarization of the environment unless
one can ensure that the coefficients $|a_{k}|\approx 1$ for all $k$. This can
be achieved when [as seen in eq. (10)] the applied magnetic field dependent
frequency $\omega$ is large in comparison with the coupling constants of eq.
(4). We will comment on the actual magnitude of the necessary magnetic field
later, when we present results for a realistically modeled NV-center spin
qubit. For a sufficiently large magnetic field we have
$|L_{k}(t^{\prime})|\in\left[|p_{k}|,1\right]$ for all $k$, so the coherence
at the discrete time-instants $t^{\prime}$ (13) is bounded from below by the
product of the environmental spin polarizations,

$|\rho_{01}(t^{\prime})|\in\frac{1}{2}\left[\prod_{k=0}^{N-1}|p_{k}|,1\right].$
(14)

Hence, the minimum value of the coherence which can be measured constitutes an
upper bound on the product of the polarizations of the relevant environmental
spins. Note here that the time-dependence of $|\rho_{01}(t^{\prime})|$
involves a product of $N$ oscillatory functions, with different frequencies
dependent on the position of spin $k$. This means that obtaining a good
approximation of the product of environmental spin polarizations will require
measurement of the coherence (at discrete time instants) for long times. We
will again comment on the actual times necessary when presenting exemplary
evolutions and approximated polarizations below.

Figure 3: Prediction of average initial polarization as a function of time for
$p_{k}=0.8$ for different values of the magnetic field: $B_{z}=1$ T - violet
points; $B_{z}=0.75$ T - green stars; $B_{z}=5$ T - blue crosses; $B_{z}=0.25$
T - yellow squares. Points are marked on the plot only when the discrete qubit
evolution signified a new (lower) value than previously.

We demonstrate the operation of the scheme using an environment of $8$ spins
at randomly generated locations for a qubit initially in an equal
superposition state. The values of the coupling constants present in eq. (4)
(as well as their distance from the NV center) are given in Table 1. They were
calculated using the realistic model of the diamond crystal lattice with
randomly generated spinful ${}^{13}C$ atoms. Fig. 1 contains the discrete
evolution of the qubit coherence for all polarizations $p_{k}=p=0.8$ (violet
dots) and $p_{k}=p=0.6$ (green squares) at magnetic field $B_{z}=1$ T. All of
the initial polarization values were chosen to be equal only for convenience.
The method yields an upper bound on the product of nuclear polarizations, and
does not require them to be the same. Such a value of the magnetic field is
large for the given setup in the sense that all of the amplitudes $a_{k}\geq
0.997$. The values of the coherence which correspond to the actual product of
initial nuclear polarizations ($0.5p^{8}$ for this realization) are marked by
the horizontal dashed lines. Due to the large number of nuclear spins, even
the simplified function for the qubit decoherence, which is the outcome of the
specific choice of measurement times is quite complex. This can be clearly
seen on Fig. 2 which contains a close-up of the discrete evolution of qubit
coherence and the corresponding continuous curve given by eqs (12) and (13 for
$p=0.8$.

Fig. 3 contains a plot where the lowest predicted value of the average single-
nucleus polarization, $\overline{p}=\sqrt[N]{\prod_{k=0}^{N-1}p_{k}}$, is
marked as a function of the first time that this value was reached. Since in
the studied case we chose all $p_{k}=p$, we obviously have $\overline{p}=p$,
and the plot is made for $p=0.8$. The different types of points correspond to
different values of the magnetic field, with the lowest value being
$B_{z}=0.25$ T, which yields all amplitudes $a_{k}\geq 0.975$ for this
realization ensuring the capability of detection of the product of initial
nuclear polarization if the smallest $p_{k}<0.975$. As can be seen, the
minimum value obtained during the discrete measurements of the evolution
initially changes often, up to times of a few hundred $\mu$s, while the next
time period when the approximation is enhanced occurs at around $1600$ $\mu$s.
Further change occurs $\sim 10^{4}$ $\mu$s and is not included in the plot.

We find that the functions which are responsible for the discrete evolution of
the qubit are strongly and qualitatively influenced by the number of
environmental spins and the actual values of the coupling constants (the
distribution of the nuclear spins), while the initial polarization has a
mostly quantitative effect. If the magnetic field is high enough to guarantee
that the amplitudes $|a_{k}|$ are close to unity, then also its effect on the
the operation of the scheme is negligible as seen in Fig. 3.

###  III.2 $\sin\frac{\omega t^{\prime\prime}}{2}=0$

Predictably, the “opposite” situation when the qubit coherence is only probed
at times $t^{\prime\prime}=\frac{2\pi}{\omega}n$ when $\sin\frac{\omega
t^{\prime\prime}}{2}=0$ is also of interest for the approximation of initial
environmental polarization. In this situation, the contribution of the
individual nuclear spin is given by

$L_{k}(t^{\prime\prime})=\pm\cos\frac{\omega_{k}t^{\prime\prime}}{2}\pm
ip_{k}a_{k}\sin\frac{\omega_{k}t^{\prime\prime}}{2}.$ (15)

This yields the absolute value

$|L_{k}(t^{\prime\prime})|=\left[(1-a_{k}^{2}p_{k}^{2})\cos^{2}\frac{\omega_{k}t^{\prime\prime}}{2}+a_{k}^{2}p_{k}^{2}\right]^{\frac{1}{2}},$
(16)

with unambiguously defined upper and lower bounds
$|L_{k}(t^{\prime\prime})|\in[|a_{k}||p_{k}|,1]$. This means that the
approximation of the upper bound on $\prod_{k=0}^{N-1}|a_{k}||p_{k}|$ does not
require any additional assumptions (such as a high magnetic field required in
Subsection III.1), since

$|\rho_{01}(t^{\prime\prime})|\in\frac{1}{2}\left[\prod_{k=0}^{N-1}|a_{k}||p_{k}|,1\right].$
(17)

Nevertheless to obtain information about the initial polarizations not
weighted by the amplitudes $|a_{k}|$, knowledge of the product of amplitudes
$\prod_{k=0}^{N-1}|a_{k}|$ is required. This problem can be solved in two
ways. The first, analogous to the solution in Subsection III.1 requires a high
magnetic field to be applied, which results in
$\prod_{k=0}^{N-1}|a_{k}|\approx 1$ and consequently reduces the lower bound
on the coherence at time-instants $t^{\prime\prime}$ to
$\frac{1}{2}\prod_{k=0}^{N-1}|p_{k}|$. Results obtained in this regime are
very similar to the results of the previous subsection, so we do not include
additional plots. The relevant difference between this case and the results of
Sec. III.2 is the necessary value of the magnetic field. Previously the
magnetic field had to be high enough that the amplitude $|a_{k}|$ for each
individual nucleus would be greater than the polarization $p_{k}$ of this
nucleus, while here only the value of the product of all amplitudes is of
importance.

The other solution requires a measurement of $\prod_{k=0}^{N-1}a_{k}$, which
is possible using the method of Sec. III.1. Note that at times $t^{\prime}$,
if the environment has not been polarized, so all $p_{k}=0$, the qubit
coherence is bounded as

$|\rho_{01}(t^{\prime})|\in\frac{1}{2}\left[0,\prod_{k=0}^{N-1}|a_{k}|\right],$
(18)

regardless of the magnetic field, as long as the frequency $\omega$ does not
cancel out with any coupling constant $\mathbb{A}_{k}^{z,z}$ for some nuclear
spin (in which situation a small change of the applied field rectifies the
situation). Hence the maximum value of $|\rho_{01}(t^{\prime})|$ obtained
during the evolution of the coherence yields the lower bound on
$\prod_{k=0}^{N-1}|a_{k}|$ and can be used for the estimation of the product
of initial environmental polarizations from Eq. (17), as long as both
measurements are performed at the same magnetic field (since
$\prod_{k=0}^{N-1}|a_{k}|$ depends strongly on the magnetic field). This
second solution has the advantage that it does not require high magnetic
fields to ensure the accuracy of the estimation of the initial polarization,
but it does require a second measurement of qubit decoherence to be performed
for an unpolarized environment, and this measurement yields only a lower bound
on the actual value of the product of amplitudes.

##  IV Conclusion

We have proposed a method for the estimation of the product of initial
polarizations of the spinful nuclei of the ${}^{13}C$ carbon isotope in
diamond via the measurement of the coherence of an NV-center spin qubit which
evolves due to the interaction with such an environment. We have found, that
although the amount of unknown variables describing the interaction in
physical realizations and the complexity of the function that governs the
decoherence due to the presence of already a small number of carbon isotope
atoms, precludes a straightforward readout of the polarization, measurement of
the coherence only at some points of time allows to obtain much more
information. The points of time are chosen in such a way as to significantly
simplify the functions which are responsible for the influence of each
environmental spin on the qubit and they depend only on the applied magnetic
field. The minimum of the decoherence curves obtained in such a way are
directly proportional to the upper bound on the product of the initial
environmental polarizations (the proportionality factor is the initial qubit
coherence).

We exemplified the operation of the scheme on an environment of eight nuclear
spins distributed randomly in the vicinity of the qubit in the diamond crystal
lattice. Due to the nature of the short-range interaction between the qubit
and its environment, the number of nuclear spins giving a noticeable
contribution will not be significantly larger in the experimental setup. The
method yields an upper bound on the actual value of the product of
polarizations, and procures a rough estimate of the value at measurement times
$\sim 10^{2}$ $\mu$s, which is much improved for measurement times $\sim
10^{3}$ $\mu$s.

The biggest advantage of the proposed method is the simplicity of the required
measurements, which do not involve accessing the degrees of freedom of the
environment in any way. In fact, all of the information about the state of the
environment is gained due to information being transferred into the qubit
state during the joint system-environment evolution.

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