#  Data-driven criteria for quantum correlations

Mateusz Krawczyk  Institute of Theoretical Physics, Wrocław University of
Science and Technology, 50-370 Wrocław, Poland  Jarosław Pawłowski  Institute
of Theoretical Physics, Wrocław University of Science and Technology, 50-370
Wrocław, Poland  Maciej M. Maśka  Institute of Theoretical Physics, Wrocław
University of Science and Technology, 50-370 Wrocław, Poland  Katarzyna Roszak
Institute of Physics (FZU), Czech Academy of Sciences, Na Slovance 2, 182 00
Prague, Czech Republic

###### Abstract

We build a machine learning model to detect correlations in a three-qubit
system using a neural network trained in an unsupervised manner on randomly
generated states. The network is forced to recognize separable states, and
correlated states are detected as anomalies. Quite surprisingly, we find that
the proposed detector performs much better at distinguishing a weaker form of
quantum correlations, namely, the quantum discord, than entanglement. In fact,
it has a tendency to grossly overestimate the set of entangled states even at
the optimal threshold for entanglement detection, while it underestimates the
set of discordant states to a much lesser extent. In order to illustrate the
nature of states classified as quantum-correlated, we construct a diagram
containing various types of states – entangled, as well as separable, both
discordant and non-discordant. We find that the near-zero value of the
recognition loss reproduces the shape of the non-discordant separable states
with high accuracy, especially considering the non-trivial shape of this set
on the diagram. The network architecture is designed carefully: it preserves
separability, and its output is equivariant with respect to qubit
permutations. We show that the choice of architecture is important to get the
highest detection accuracy, much better than for a baseline model that just
utilizes a partial trace operation.

Quantum correlations play a pivotal role in near-future quantum technologies,
from quantum cryptography [1, 2], networks [3, 4], to computation [5, 6]. The
source of their usefulness is that they allow for the state of one subsystem
to be influenced by operations and measurements on the other, which yields
effects that cannot be reproduced by classical physics. A good example here is
teleportation [7, 8, 9], which allows for the transfer of an unknown quantum
state between distant locations by means of local operations and measurements,
and the transfer of classical information, due to a maximally entangled state
being shared by two parties.

In quantum information theory, there is a long-standing question about quantum
correlations in mixed states [10, 11]. For pure states, any correlations
between subsystems are quantum in their nature (entanglement), and the set of
separable (not entangled) states is equivalent to the set of product states.
In the case of mixed states, the fact that a density matrix cannot be written
in product form does not guarantee the existence of quantum correlations,
since this formalism allows for the description of classical (purely
statistical) correlations, i.e. $\hat{\rho}_{AB}=1/2\left(|00\rangle\langle
00|+|11\rangle\langle 11|\right)$, whereas a product state contains no
correlations whatsoever.

One definition of the set of states that contain no quantum correlations,
namely the set of mixed separable states, is given by [12, 13, 14]

$\hat{\rho}_{AB}^{sep}=\sum_{i}p_{i}|\psi_{i}\rangle_{AA}\langle\psi_{i}|\otimes|\phi_{i}\rangle_{BB}\langle\phi_{i}|,$
(1)

where $p_{i}$ are probabilities and the indices $A$ and $B$ label the
subsystems. It is relevant to note here that there is no limitation on states
$|\psi_{i}\rangle_{A}$ or $|\phi_{i}\rangle_{B}$, which do not have to form a
basis. By this definition, a bipartite state is separable if and only if it
can be created from a product state using only local operations and classical
communication. The quantification of entanglement (for mixed states, entangled
states are defined as not-separable) is classed as an NP-complete problem
[15], and it does not simplify even when a state only needs to be qualified as
separable or entangled.

A state with no quantum correlations can also be defined as a state for which
no information about one subsystem can be learned by measurements performed on
the other subsystem. Such states are said to contain zero discord [16, 17,
18], and are defined by Eq. (1) with the added constraints that the states
$|\psi_{i}\rangle_{A}$ must be part of an orthonormal basis, and the same for
states $|\phi_{i}\rangle_{B}$. Quantifying the amount of discord in a given
state is also NP-complete [19], but it is harder than quantifying entanglement
for small systems [20, 21, 22]. On the other hand, simply qualifying a state
as discordant or not is a straightforward task [20, 23]. The set of non-
discordant states is a subset of separable states. Currently, entanglement is
the accepted, standard metric for quantum correlations, yet it should be noted
that discord has one important advantage over entanglement, namely that it has
a much stronger relationship to measurable quantities, while the detection of
entanglement generally requires quantum tomography [24, 25].

In the following, we would like to go beyond the typical scheme of describing
the correlations in quantum states using analytical metrics and propose the
study of quantum correlations using machine learning (ML) models forcing them
to learn similarities between different classes of quantum states. We use
neural networks (NNs) which have proven their usefulness in many applications,
such as computer vision or signal processing [26, 27], but recently also in
quantum physics, e.g., in many-body theory [28, 29, 30, 31] or in entanglement
modelling [32, 33, 34]. However, in our approach, we carefully design the
model architecture to be separability-preserving and qubit permutations
equivariant, which is part of the general trend of building NN models that
respect symmetries of a given problem, for example, being rotation and
translation group-equivariant [35, 36, 37, 38].

We present results where the ML model is first trained to distinguish between
separable and entangled states (on both mixed and pure states) and then tested
on similar sets. These sets (both in the training and testing phases) apart
from entangled states, contain different variants of separable states, with
special emphasis lain on discordant separable states. While the ML model
trained only on pure states can only distinguish between product and non-
product mixed states, hence signifying correlations, but not being able to
discriminate between classical, quantum-classical (discord), and quantum-
quantum (entanglement) correlations, quite surprisingly, we find that the
performance of the separator trained also on mixed states is much better at
signifying quantum discord than entanglement. This corroborates the intuition
that a type of quantum correlations which are defined on the properties of
measurement outcomes, as opposed to correlations defined on specifics of state
preparation, should be more evident in a given state, even though the ML model
was not trained directly on non-discordant states (but on separable ones).

Since the goal of using the neural network is not only to classify the
discordance or entanglement of states, but also to produce a similarity
measure, we propose to train the model in the unsupervised anomaly detector
scheme [39]. Anomaly detection is a standard problem in ML where a model is
trained to reconstruct normal data and then fails when trying to reconstruct
abnormal data (anomaly). The reconstruction error, i.e. the model “loss
function” measures the similarity of a given unknown sample to the training
data, and if it is greater than some assumed threshold, then the sample is
treated as an anomaly. In our case, anomalies are states that are quantum-
correlated. The model is trained to recognize separable states of three
qubits. We study 3-qubit states to have a non-trivial yet manageable Hilbert
space, keeping in mind that the proposed model construction can be easily
extended to larger qubit registers.

To build the model, we use convolutional NNs (with a discrete convolution
operation [40, 41, 42]) as they are commonly applied to build robust
representations of correlations on data on regular grids such as matrices.
First, we need to organize the convolution layers, so that the resulting NN
can separate the input density matrix into single qubits matrices. The
constructed NN should preserve pure separable (product) states by definition
and also not change the separability of mixed states. A closer look at the
model scheme presented in Fig. 1 shows that the convolution layers are
organized in a way that is a generalization of the partial trace operation, in
the sense that the kernel simplified to identity matrix would calculate the
partial trace over the remaining qubits. Such an architecture implies qubit
permutation equivariance, meaning that permutation of qubits in the input
matrix gives the same permutation of the reconstructed state and preserves
separability of any reconstructed state.

Figure 1: The 3-qubit neural network separator model. The input density matrix
is convolved separately with three convolution layers, each with $4\times 4$
kernels that work in parallel. The stride and dilation parameters defining
these convolution layers are arranged in such a way that they operate on each
qubit subspace separately (details can be found in SM [43]). After the
convolution layers, shape-preserving fully-connected layers are applied
independently for each qubit matrix.

Additionally, to extend the model complexity we include four fully-connected
(FC) layers with nonlinear activation functions (multilayer perceptron) after
the convolutional ones. This extension has a physical justification that will
become clear later. Importantly, to preserve the separability property, these
layers are applied independently to each block representing the qubit matrix.
Moreover, to allow the neural network to generalize and reconstruct also
separable mixed states, more sets of convolutions, i.e. channels, have been
used, giving a separable density matrix that is a sum over $N_{K}$ channels.
Details can be found in the Supplemental Material (SM) [43]. The reconstructed
density matrix is given by

$\hat{\rho}_{ABC}=\frac{1}{N}\sum_{i=1}^{N_{K}}\hat{\rho}_{Ai}\otimes\hat{\rho}_{Bi}\otimes\hat{\rho}_{Ci},$
(2)

where we calculate the sum of the Kronecker product of output single qubit
matrices ($\hat{\rho}_{Ai}$, $\hat{\rho}_{Bi}$, $\hat{\rho}_{Ci}$) over
different $N_{k}$ channels, numbered by $i$. Note that the reconstructed
matrix is separable, but, in general, it does not fulfill the additional
criteria which would make it a non-discordant state from definition. Since a
perfect decoupling into single-qubit matrices is possible only for separable
states, the minimal distance between the original and reconstructed density
matrices, obtained by the training of the neural network, can be seen as a
measure of the separability of states.

Both the original and reconstructed density matrices have to be included in
the NN loss function, which describes how different the reconstructed state is
from the original state. In quantum information theory, one of the natural
choices for a difference between density matrices is the Bures distance [44],
but numerical experiments showed much better efficiency for training our NN
model with the well-established $L^{1}$ norm,

$\mathcal{L}(\rho_{ABC})=\frac{1}{d^{2}}\sum_{i,j}|\rho^{ij}_{ABC}-\hat{\rho}^{ij}_{ABC}|.$
(3)

Here $\rho^{ij}_{ABC}$ and $\hat{\rho}^{ij}_{ABC}$ denote the elements of the
original and reconstructed density matrices labeled by the indices $i$ and
$j$, and $d^{2}$ is the number of matrix elements. Using this loss function,
we train the NN model so that the total reconstruction loss, summed over all
training examples, is minimized. Note that to train the network, which we will
call a separator, we do not need the labeled data. Instead, we compare inputs
with outputs, which means that our training scheme is fully unsupervised.
These types of ML models are also called autoencoders (we comment on this in
more detail in SM [43]).

Importantly, by choice of the model architecture, the separator network is
suited to reconstruct separable states, hence, for these states we expect the
value of the loss function to be close to zero. In contrast, for entangled
states, the model is not able to reconstruct the input, and the difference
between the two matrices will result in a significantly higher value of the
loss $\mathcal{L}$. In order to distinguish between separable and entangled
states, we set the anomaly threshold $\tau$ in such a way that if
$\mathcal{L}(\rho_{ABC})$ is greater than $\tau$ for a given state
$\rho_{ABC}$, then this state is classified as entangled. Otherwise, it is
marked as separable.

The most critical aspect of training an ML model is having a well-balanced and
diversified data set. The modern approach in ML is to synthesize data with
already known properties (e.g. forming separate classes in case of classifier
training). To achieve this goal, we select different methods for generating
random density matrices. The main idea is to generate random states using the
quantum circuits approach [34], but in order to have a well-diversified data
set, we also use a technique based on sampling from the uniform Haar measure
[45] and an additional parameterized method designed by us (generation details
can be found in SM [43]).

We construct a training set composed of 530 000 pure and mixed separable
states including a class of non-discordant states and a validation set of 50
000 is constructed in the same way. Furthermore, we generate two test sets
involving both separable and entangled states. The first of those
($S_{\mathrm{pure}}$) is constructed from 15 000 random pure separable states
and 15 000 random pure entangled states. This data set is generated evenly
using both the quantum circuit approach and sampling from the Haar measure.
The second test set ($S_{\mathrm{mixed}}$) includes 65 000 mixed states,
constructed from pure states either by mixing different states with random
probabilities or by reducing a larger multi-qubit space to the desired few-
qubit size via the partial trace.

To classify states as separable during the generation process, we use the
negativity [46, 47], an entanglement measure based on the PPT-criterion of
entanglement [48, 49], which is one of the few measures that can be found
directly from the density matrix for systems larger than two qubits. It is
defined as the absolute sum of the negative eigenvalues of a bipartite density
matrix after partial transposition is performed on either of the subsystems.
The generation methods for mixed entangled states are specifically adjusted
not to produce bound entangled states [50], which cannot be detected by
negativity. To qualify discord, we use the criterion of Ref. [23], which is
very straightforward to check, especially if one of the subsystems is small
(relevant details are given in SM [43]). Incidentally, it is common that a
state is discordant only with respect to one part of a bipartition, so a state
has to contain no discord with respect to both subsystems to be classified as
a non-discordant state [18]. For both entanglement and discord, the lack of
correlations is checked three times: between qubit $i$ and the rest of the
3-qubit system, with $i=A,B,C$.

| test
---
set
$\langle\mathcal{L}\rangle$ for different subsets
| separable | non-discordant | discordant | entangled
$S_{\mathrm{pure}}$ | $5.9\text{\times}{10}^{-4}$ | $5.9\text{\times}{10}^{-4}$ | $0.042$ | $0.042$
$S_{\mathrm{mixed}}$ | $5.2\text{\times}{10}^{-3}$ | $1.0\text{\times}{10}^{-3}$ | $0.017$ | $0.021$
Table 1: Average separator loss for different subsets in pure
$S_{\mathrm{pure}}$ and mixed $S_{\mathrm{pure}}$ test sets quantifying
similarity between different types of states and separable ones.

In ML a single pass through the entire training set, during which we update
model parameters (typically using the gradient of a loss function in the
parameter space), is called an epoch. The separator model was trained for 20
epochs, in a way that the model parameters for the epoch with the lowest
averaged reconstruction loss $\langle\mathcal{L}\rangle$ on the validation set
were saved and used for evaluation (testing) purposes.

For the test set that contains only pure states, $S_{\mathrm{pure}}$, the
average loss $\langle\mathcal{L}\rangle$ for separable states is equal to
$5.9\text{\times}{10}^{-4}$, while in the case of entangled states, it is much
larger, giving $\langle\mathcal{L}\rangle=0.042$. Obviously, for the discord
the numbers are the same – see Table 1. Choosing the threshold value $\tau$ to
obtain the highest possible accuracy (balanced accuracy, $BA$, defined in SM
[43]) of the model predictions yields $\tau=$1.25\text{\times}{10}^{-3}$$ with
the corresponding classification accuracy of $$99.63$\%$ for the whole
$S_{\mathrm{pure}}$ set.

For the test set with mixed states, $S_{\mathrm{mixed}}$, in the case of non-
discordant states, the average loss $\langle\mathcal{L}\rangle$ is only about
$1.5$ times higher than for uncorrelated pure states, meaning that the
original and reconstructed states are the same to an equivalent extent, while
for mixed separable states it is $40$ times higher, signaling that, on
average, mixed separable states differ significantly more from the
reconstruction. Moreover, discordant and entangled states display roughly half
$\langle\mathcal{L}\rangle$ for mixed states compared to pure correlated
states, which is in agreement with the fact that mixed states are on average
less correlated than pure states. Overall, the difference between zero and
nonzero discord states (measured by reconstruction loss $\mathcal{L}$) is over
an order of magnitude larger than between separable and entangled states.

Let us now test the separator performance using the mixed test set,
$S_{\mathrm{mixed}}$, and compare its practical usefulness for either discord
or entanglement detection. We performed two independent tests. In the first,
we took discord labels and tested the ability to classify discord, and in the
second, we took separability labels for the same set $S_{\mathrm{mixed}}$ and
tested the ability to classify entanglement. Since for mixed states, a
discordant state can also be separable, one cannot simultaneously balance non-
discordant vs. discordant and separable vs. entangled states in a single mixed
test set, therefore we use $BA$ (see SM [43]) as the model performance
measure.

Figure 2: (a) Separator performance for detecting discordant (green) and
entangled (red) states, tested on $S_{\mathrm{mixed}}$. The results present:
(left) precision vs. recall curves, and (right) balanced accuracy, $BA$
depending on the threshold $\tau$ value. For comparison, partial trace-based
baseline model performance is also presented by blue and orange curves,
respectively. (b) Same as (a) but for separator model with removed FC layers –
cf. Fig. 1. (c) Same as in (a) but for the separator trained on the non-
product states only.

Fig. 2 contains results of separator performance for detecting discordant
(green curves) and entangled (red) states on the $S_{\mathrm{mixed}}$ set.
Fig. 2(a), right, shows that the model has much higher $BA$ when detecting
discord than entanglement in a wide range of thresholds, reaching $BA=93$% for
$\tau=$0.0023$$ in case of discord detection, while for entanglement the
maximum $BA=82$% is reached at $\tau=$0.0051$$. Moreover, when training the
model on the subspace of the original training domain where the product states
have been excluded, as presented in Fig. 2(c), we clearly get better
performance for discord in terms of the larger area under the precision-recall
curve (left). Performance comparisons for training separately on some other
classes of states can be found in SM [43].

The convolution part in the separator performs an operation that is a
generalization of the partial trace. When removing the FC layers that extend
the convolution layers, we observe that the model simply collapses to the
partial trace baseline (example kernels for both cases can be found in SM
[43]) and is analogous to what is obtained when training only on pure
separable states. Fig. 2(b) contains results for a separator with the FC
layers removed: comparing accuracy with a partial trace baseline (blue and
orange curves) we get practically the same accuracies.

We present the confusion matrices for the separator serving as a
separability/entanglement classifier in Table 2 and a non-
discordance/discordance classifier in Table 3. Overall, the separator tends to
grossly overestimate the set of entangled states as seen in Table 2, where for
both thresholds presented ($\tau=0.0023$ corresponding to the best discord
classification and $\tau=0.0051$ to the best entanglement classification) the
number of separable states wrongly labeled as entangled is of the same order
of magnitude as the number of correctly detected entangled states. The number
of erroneously classified separable states is comparatively small. On the
contrary, the set of discordant states tends to be underestimated, with a
discordant state being more likely to be labeled as not correlated than the
other way around. This occurs to a much lesser extent than the overestimation
of entanglement as seen in Table 3. The results here are more balanced, with
erroneous classifications averaging at a number an order of magnitude lower
than the correct predictions.

Labels Predictions | separable | entangled
---|---|---
separable $\hskip 11.99998pt\tau=0.0023$ | 28340 | 15257
entangled $\hskip 10.00002pt\tau=0.0023$ | 305 | 21098
separable $\hskip 11.99998pt\tau=0.0051$ | 30696 | 12901
entangled $\hskip 10.00002pt\tau=0.0051$ | 942 | 20461
Table 2: Confusion matrix for separable/entangled states classification at $\tau=0.0023$ and $\tau=0.0051$. Labels Predictions | non-discordant | discordant
---|---|---
non-discordant $\tau=0.0023$ | 25114 | 1192
discordant $\hskip 6.99997pt\tau=0.0023$ | 3531 | 35163
non-discordant $\tau=0.0051$ | 25822 | 484
discordant $\hskip 6.99997pt\tau=0.0051$ | 5816 | 32878
Table 3: Confusion matrix for non-discordant/discordant states classification
at $\tau=0.0023$ and $\tau=0.0051$.

In order to acquire a deeper understanding of how well the sets of non-
discordant and separable states are reconstructed by the ML classifier at
different thresholds, we analyze a family of 3-qubit states which are easy to
parameterize, and study a 2D diagram forming a map with regions differing in
the type of correlations present (or absent) as seen in the inset of Fig. 3.

The family of states is constructed as follows. We take a separable mixed
state of the form $\rho=p\rho_{1}+(1-p)\rho_{2}$, where probability $p$ is one
of the parameters, and the density matrices $\rho_{1}$ and $\rho_{2}$
correspond to 3-qubit pure states,
$|\Psi_{i}\rangle=\bigotimes_{k}|\psi_{i}\rangle$, with $k=A,B,C$ and

$\displaystyle|\psi_{1}\rangle$ $\displaystyle=$ $\displaystyle
a|0\rangle+\sqrt{1-a^{2}}|1\rangle,$ (4a) $\displaystyle|\psi_{2}\rangle$
$\displaystyle=$ $\displaystyle e^{-i\frac{\phi}{2}}\sqrt{1-a^{2}}|0\rangle-
ae^{i\frac{\phi}{2}}|1\rangle,$ (4b)

respectively. Coefficient $a$ and phase factor $\phi$ constitute two other
parameters that characterize the family of states. It now contains only
separable states. For $p=0,1$ the state is pure (and consequently of product
form), while it is mixed non-discordant for $\phi=0$ or $a=0,1$. To introduce
entanglement into the system we find the spectral decomposition of
$\rho=\sum_{i}\lambda_{i}|\varphi_{i}\rangle\langle\varphi_{i}|$ and increase
the largest eigenvalue, $\lambda_{0}$,

$\rho\rightarrow\frac{\sum_{i\neq
0}\lambda_{i}|\varphi_{i}\rangle\langle\varphi_{i}|+(\lambda_{0}+c)|\varphi_{0}\rangle\langle\varphi_{0}|}{\sum_{i}\lambda_{i}+c},$
(5)

making $c$ the final parameter which characterizes this family of states.

Figure 3: Reconstruction loss $\mathcal{L}$ for the already trained separator
model tested on the family of 3-qubit states with their
separability/discordance conditions known and parameterized on 2D map (see
inset for the map division into the classes).

This allows us to plot the loss function (3) for selected cross sections of
the Hilbert space and visualize how surfaces of equal $\mathcal{L}$ compare to
the sets of non-discordant and separable states. In Fig. 3 only the darkest
regions in the inset are entangled, all other regions are separable and are
further divided into product states, non-discordant mixed states, and
discordant separable states. The respective regions are obtained by careful
choice of parameters $p$, $a$, $\phi$, and $c$ along the horizontal and
vertical axes.

The main plot in Fig. 3 shows the loss $\mathcal{L}$ on this parameter space.
The specifics of how the different parameters change in each region are
described in detail in SM [43]. It is noticeable that, while the partial
trace-based model is capable only of distinguishing between product and non-
product states (we show the plot and discuss this in detail in SM [43]), the
neural network distinguishes all non-discordant states. The results presented
in Fig. 3 are in agreement with the general results of Fig. 2, and Tables 2
and 3, demonstrating why the discord detection results in false negatives
(overestimating the set of non-discordant states), while entanglement
detection results in so many false positives.

In conclusion, we designed and trained a neural network to distinguish between
separable and entangled states on a set of mixed and pure states. For optimum
threshold values, the network performs with $82\%$ balanced accuracy for mixed
states, as compared to over $99\%$ classification accuracy of the same network
for pure state entanglement. To understand this discrepancy outside of the
trivial acknowledgment that determining mixed state entanglement is a much
more complicated task than pure state entanglement, we have compared the
results obtained on the mixed test set with other classes of uncorrelated
states. We have found that the set of states labeled as uncorrelated by the
separator overlaps best with the set of non-discordant states. This is
confirmed by the $93\%$ balanced accuracy of detecting such states,
outperforming the detection of both separable and product states. This result
is highly surprising, because it shows that, by ML determination, random mixed
non-discordant states are much more similar to states from a random set of
pure separable states (as the lowest loss is observed for such states) than
random mixed separable states. We supplement plots of ML metrics and confusion
matrices with a map of numerically detected quantum correlations for cross
sections of the 3-qubit Hilbert space, which allows us to understand the
nature of the sets of states which are qualified as uncorrelated by ML. The
surface where the loss is near zero (no quantum correlations) reproduces the
set of non-discordant states extremely well, while the discordant separable
states are largely classified as quantum-correlated, illustrating why the
separator grossly overestimates entanglement while only marginally
underestimating discordance. The network architecture is designed carefully to
preserve separability (i.e. it does not introduce any entanglement when
reconstructing the output) and its output is equivariant with respect to qubit
permutations. We observe that the simplified version of the separator model,
which realizes just the generalization of a partial trace operation, simply
collapses to partial trace during the training. However, when extending it to
a neural model capable to learn other representations by adding fully-
connected layers working separately within each qubit subspace, it learns more
complex patterns and improves its performance over a partial trace baseline.

Acknowledgements. M.K. and J.P. acknowledge support from National Science
Centre (Poland) under Grant No. 2021/43/D/ST3/01989. M.M.M. acknowledges
support from the National Science Centre (Poland) under Grant No.
DEC-2018/29/B/ST3/01892.

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Supplemental Material for “Data-driven criteria for quantum correlations”

##  .1 Details of the separator neural network

In the following section, we analyze the structure of the neural network (NN)
model trained to reconstruct separable states and detect non-separable states
as an anomaly.

Figure S1: The separator autoencoder composed of trainable encoder neural
network (gray triangle) and analytical decoder that simply calculates the
Kronecker product.

It is easy to construct a many-qubit density matrix as a product of single-
qubit density matrices. However, the opposite operation is much more
complicated. Then, this task can be seen as an inverse problem. Here, we
propose to solve it with the help of a modified autoencoder-type NN,
schematically depicted in Fig. S1. The typical autoencoder is composed of two
connected networks: the encoder and the decoder. Usually, the output of the
encoder is narrower than its input, so this part compresses the data. Next,
the decoder decompresses it, and both parts of the autoencoder are trained to
have the output as close as possible to the input. Our idea is to use the
encoder to transform 3-qubit density matrices into three single-qubit matrices
in such a way that their product, calculated by the decoder, is as close to
the original density matrix as possible. The autoencoder will be implemented
as a NN, with only the encoder part trainable. The decoder simply calculates
the Kronecker product of the single-qubit density matrices, as described in
Eq. (2).

The backbone of the encoder is composed of convolution layers, which map the
input density matrix into single qubit subspaces. They are followed by fully
connected (FC) layers that extend the model and introduce non-linearity;
however, they do not introduce entanglement, since they are applied
independently for each of the reconstructed qubits.

Let us focus on a $d=3$ qubit system $\rho_{ABC}$, bearing in mind that the
proposed reconstruction can be easily extended to larger qubit registers. We
require a neural network model $\mathcal{N}$, which we call a separator, to
have the following properties:

$\displaystyle\mathcal{N}(\rho_{A}\otimes\rho_{B}\otimes\rho_{C})=c\,\rho_{A}\otimes\rho_{B}\otimes\rho_{C},$
(S1)
$\displaystyle\mathcal{N}\left(\sum_{i}p_{i}\,\rho_{Ai}\otimes\rho_{Bi}\otimes\rho_{Ci}\right)=\sum_{i}\hat{p}_{i}\,\hat{\rho}_{Ai}\otimes\hat{\rho}_{Bi}\otimes\hat{\rho}_{Ci},$
(S2)

with some multiplicative constant $c$. To do so one can adjust the convolution
layers $\mathcal{C}^{s,l}$, with kernel $\mathcal{K}$, step size (stride) $s$,
and dilation $l$:

$\mathcal{C}_{s,l}(\rho)^{i,j}=(\mathcal{K}\ast_{l}\rho)^{si,sj}=\sum_{k,q}\mathcal{K}^{k,q}\rho^{si+l(k),sj+l(q)},$
(S3)

$i,j=0,1$, and $k,q=0,...,2^{d-1}-1$, so that the kernels overlap with the
respective blocks of the density matrix and therefore convolving them against
the density matrix results with output matrices representing systems with a
reduced number of qubits. Dilation can be just a simple scaling of the sum
index: $l(k)=l\,k$, with rate $l$ meaning that the kernel only touches the
signal at every $l$-th entry, or more complex indexing.

Specifically, choosing $\mathcal{C}_{A}=\mathcal{C}_{4,1}$ with kernel
$\mathcal{K}$ of size $4\times 4$ allows the neural network to learn the
structure of repeating blocks $\rho_{B}\otimes\rho_{C}$ and therefore to
reduce the input density matrix to:

$\mathcal{C}_{4,1}(\rho_{A}\otimes\rho_{B}\otimes\rho_{C})=c\,\rho_{A},$

with $c$ being some multiplicative constant, as in Eq. (S1). To extract the
last one, we need $\mathcal{C}_{C}=\mathcal{C}_{1,2}$. To extract the second
qubit, more complex dilation is needed in $\mathcal{C}_{B}$. All kernels are
visualized in Fig. 1. The input density matrix, divided into 2 channels
(respectively for the real and imaginary parts), is convolved separately with
3 convolution layers, each with $4\times 4$ kernels. The first one is set up
using a stride equal to 4, as a result producing a matrix corresponding to the
first qubit subspace. In the second convolution, the kernel is divided into 4
blocks of size $2\times 2$, separated with dilations resulting in the second
qubit subspace. The third convolution consists of the kernel dilated with
extra space between every weight, therefore extracting the information about
the last qubit. Finally, after applying all the convolutions, one can combine
them and construct the output separable density matrix
$\mathcal{C}_{A}(\rho_{ABC})\otimes\mathcal{C}_{B}(\rho_{ABC})\otimes\mathcal{C}_{C}(\rho_{ABC})=\hat{\rho}_{A}\otimes\hat{\rho}_{B}\otimes\hat{\rho}_{C}$.

It is obvious that in this way the defined separator model $\mathcal{N}$
fulfils Eq. S1. However, to enable the neural network to generalize and
reconstruct separable mixed states, i.e., obey Eq. S2, a larger number of
(triples of) kernels is chosen (we use $N_{K}=24$ kernels in each type of
convolution layer). To simplify the implementation and keep neural network
values real, we doubled the number of kernels, that is, $2N_{K}$, so half of
the channels processed the real and the other half the imaginary part of the
density matrix. At this point, we also add four FC layers for each qubit
($A,B,C$) separately to extend the complexity of the model. Subsequently, we
calculate the sum of the Kronecker product of output single qubit matrices
($\hat{\rho}_{Ai}$, $\hat{\rho}_{Bi}$, $\hat{\rho}_{Ci}$) over different
kernels numbered by $i$ and arrive at the reconstructed density matrix as in
Eq. 2.

Both the original and reconstructed density matrices are included in the loss
function of the neural network defined in Eq. 3. Using this loss function, we
train the separator model so that the total reconstruction loss, summed over
all training examples, is minimized. Note that to train the network, we do not
need the data to be labeled. Instead, we simply compare inputs with outputs,
and therefore, our model scheme is fully unsupervised. The trainable weights
of the kernels are adjusted using the procedure called gradient descent, where
the gradients of the loss are calculated with respect to the weights.
Subsequently, all parameters of the model, e.g. each kernel weights
$\mathcal{K}_{ij}$ are updated as follows:

$\mathcal{K}^{ij}_{\mathrm{new}}=\mathcal{K}^{ij}_{\mathrm{old}}-\alpha\frac{\partial\mathcal{L}}{\partial\mathcal{K}^{ij}_{\mathrm{old}}},$
(S4)

where $\alpha$ is the learning rate. We proceed similarly for weights in FC
layers. This procedure is repeated until the reconstruction loss (Eq. 3)
reaches (we hope) a global minimum, which means that the model is already
trained. Importantly, by the choice of the model architecture, the separator
network is suited to reconstruct separable states, and for these states, we
expect the value of the loss function to be close to zero. In contrast, in the
case of an entangled state, the model will not be able to reconstruct the
input, and the output density matrix should differ from the input one,
resulting in a significantly higher value of the loss. Therefore, to
distinguish between separable and entangled states, we set the anomaly
threshold $\tau$ in such a way that if $\mathcal{L}(\rho_{ABC})$ is larger
than $\tau$ for a given state $\rho_{ABC}$, then this state is classified as
entangled and, otherwise, it is marked as separable. Interestingly, in
addition to classification purposes, the loss value $\mathcal{L}(\rho_{ABC})$
can also be interpreted as a similarity measure between the states used in the
training dataset and given tested state $\rho_{ABC}$.

Figure S2:  Same as in Fig. 2 but for separator trained only on some part of
the original training set: pure separable states (Pure), product states
(Prod), non-discordant states (ZD), separable states (Sep) and non-product
separable states (NPS). Results for separator without four fully-connected
layers (containing only convolutional ones) are marked as ”No FC4”.

In Fig. S2 we present the extended results for the separator model with and
without four FC layers trained on different subsets of the original training
set. As before, the neural network is evaluated on the $S_{\mathrm{mixed}}$
set. We start to train the model in the simplest possible scenario where
dataset consists of 190000 pure separable states. The model trained on such a
dataset collapses to the baseline trace model. It is expected behavior since
in this case the baseline model is a sufficient criterion for distinguishing
between entangled and separable states. The same situation occurs when we
extend our training set to the wider set of product states by adding 120000
mixed product states. Interestingly, in this case, the neural network reflects
partial trace operation even more precisely, which is revealed by a wider
range of thresholds producing overlapping accuracy. Including FC layers, and
thus introducing the nonlinearity, slightly limits this phenomenon, however,
since the baseline model is still a precise criterion of separability in the
case of product states, hence the neural network is not penalized for not
learning more complex operations. The situation changes when we fatherly
append 100000 non-product non-discordant states. Now, the baseline is no
longer reliable and the network needs to learn extra features in order to
correctly separate and reconstruct such states. Notably, the separator model
is capable of achieving this task only when extra FC layers are present.
Otherwise, it collapses to the baseline. This situation reoccurs for full
training set consisting of 530000 separable states. For the sake of
completeness, we additionally test our model after training it on the reduced
domain of non-product separable states. The data set in this case is
constructed separately from 360000 states, which are generated with the same
methodology as for the original training set. The main difference of the model
trained on such a subdomain is that even without any FC layers it does not
collapse to a partial trace operation, which may suggest that using only non-
product separable states for training can also be beneficial for the model
with FC layers.

Figure S3: Example weights of $4\times 4$ convolutional kernels for the
already trained 3-qubit separator model. The kernels reconstruct (a) real and
(b) imaginary parts of the first qubit (weights for two other qubits are
similar) for the plain separator model, with just convolutional layers.
Similarly, (c) and (d) show real and imaginary parts of kernels for separator
with additional fully-connected layers.

In Fig. S3 we present exemplary kernels learned for the separator model with
(right) and without FC layers (left) trained on the full data set (as in the
main text). Model with only convolutional layers collapses to partial trace
baseline which manifests itself in the fact that its kernels resemble identity
matrices (partial trace is equivalent to the convolutional kernel being equal
to the identity matrix).

In Fig. S4 we also present a reconstruction loss map analogous to the map in
Fig. 3 in the main text, but obtained using a partial trace as a baseline
replacement for the separator model. On this map, only product states are
clearly detected as not containing correlations, as is to be expected;
however, the region in the lower right square of the map, which represents
some space of mixed zero-discord states, is signified as containing fewer
correlations. Note that the other region of similar type (in the center of the
plot) is not correspondingly distinguished.

Figure S4: Reconstruction loss $\mathcal{L}$ for the partial trace-based model
tested on family of 3-qubit states with their separability/discordance
conditions known parameterized on 2D map.

##  .2 Classification metrics

In machine learning to evaluate classification or detection results, we
typically use various types of metrics that measure model performance.

The standard metrics [51] used in binary classification, also to tune the
model hyperparameters, e.g., classification threshold $\tau$, are precision
$PR=\frac{TP}{TP+FP}$ and recall $RC=\frac{TP}{TP+FN}$. Here $TP$, $TN$, $FP$,
and $FN$ denote respectively the number of true/false positives/negatives and
by negatives we mean non-discordant (or separable), while by positives –
discordant (or entangled) states.

The procedure called precision-recall tradeoff involves choosing a threshold
that maximizes the area under the precision-recall curve. However, due to the
fact that our dataset is imbalanced (we cannot have balanced
separable/entangled and non-discordant/discordant classes at the same time),
we use a metric typically defined for imbalanced data sets. Balanced accuracy
(BA) is defined as the average of recall obtained on each class [52] as:
$BA=\frac{1}{2}\left(\frac{TP}{TP+FN}+\frac{TN}{TN+FP}\right)$.

##  .3 Parameter changes on 3-qubit map

The way in which the parameter space of the family of states depicted on the
diagram map in Fig. 3. is defined is explained in the main text. Here, we
detail how the specific parameters are changed in the different regions of the
map to generate states corresponding to different classes in terms of quantum
and classical correlations.

Specifically, we start from point $A$, where the parameter $a=1/{\sqrt{2}}$
takes the maximal value, while all others are set to minimal values: $p=0$,
$\phi=0$ and $c=0$. Moving along the axis $AB$ we increase the parameter $p$
so that it takes the maximal value $1/2$ in point $B$. Similarly, coefficient
$a$ is decreased along the axis $BC$, so that in point $C$ it is equal to $0$.
Moving outside the square region $ABCD$ we fix parameters $a$ and $p$ to the
boundary values and start increasing the parameters $\phi$ and $c$. The phase
$\phi$ grows linearly along both horizontal and vertical axes in such a way
that inside the square $ABCD$ it is equal to $0$ and it is equal to $\pi/2$ at
the edges of the map. The parameter $c$ is increased along the counter-
diagonal in the top right square, taking the minimal value $0$ in point $B$
and maximal value $1$ in the top right corner of the map. The parameterization
is performed symmetrically on both sides of the diagonal of the map.
Therefore, the axis $AB$ in the upper right triangle corresponds to the axis
$AD$ in the lower left triangle. Similarly, $BC$ is reflected to $DC$. Keeping
that in mind, we note that parameter $c$ is additionally increased by $\Delta
c$ in the lower left triangle. This extra term grows linearly along the
counter-diagonal from $\Delta c=0$ in the center of square $ABCD$ to $\Delta
c=1$ in the bottom left corner of the map.

##  .4 Parameterized method of generating 3-qubit density matrices

In this section we describe the original parameterized method used to generate
random mixed states when building the training set. To generate such states,
we also used the quantum circuits approach and a technique based on sampling
from the uniform Haar measure – both are described in [34].

Let us start by defining a 3-qubit pure separable state,

$\begin{split}|\psi_{SEP}\rangle&=|\psi_{1}\rangle\otimes|\psi_{2}\rangle\otimes|\psi_{3}\rangle\\\
&=(a_{1}|0\rangle+b_{1}|1\rangle)\otimes(a_{2}|0\rangle+b_{2}|1\rangle)\otimes(a_{3}|0\rangle+b_{3}|1\rangle)\\\
&=a_{1}a_{2}a_{3}|000\rangle+a_{1}a_{2}b_{3}|001\rangle+a_{1}a_{3}b_{2}|010\rangle\\\
&+a_{1}b_{2}b_{3}|011\rangle+a_{2}a_{3}b_{1}|100\rangle+a_{2}b_{1}b_{3}|101\rangle\\\
&+a_{3}b_{1}b_{2}|110\rangle+b_{1}b_{2}b_{3}|111\rangle,\end{split}$ (S5)

where the complex coefficients of each state fulfil,

$|a_{i}|^{2}+|b_{i}|^{2}=1.$ (S6)

We then introduce entanglement into the system for each pair of qubits via a
phases $\phi_{ij}$,

$\begin{split}|\psi_{ENT}\rangle&=a_{1}a_{2}a_{3}|000\rangle+a_{1}a_{2}b_{3}|001\rangle+a_{1}a_{3}b_{2}|010\rangle\\\
&+a_{1}b_{2}b_{3}e^{i\phi_{23}}|011\rangle+a_{2}a_{3}b_{1}|100\rangle\\\
&+a_{2}b_{1}b_{3}e^{i\phi_{13}}|101\rangle+a_{3}b_{1}b_{2}e^{i\phi_{12}}|110\rangle\\\
&+b_{1}b_{2}b_{3}e^{i(\phi_{12}+\phi_{13}+\phi_{23})}|111\rangle.\end{split}$
(S7)

Finally, the decoherence coefficients $c_{i}$ are included for each qubit
separately in order to obtain mixed states, mimicking the operation of single-
qubit dephasing channels,

$\displaystyle\rho_{i}$
$\displaystyle=|\psi_{i}\rangle\langle\psi_{i}|=\left(\begin{matrix}|a_{i}|^{2}&a_{i}b_{i}^{*}\\\
a_{i}^{*}b_{i}&|b_{i}|^{2}\\\ \end{matrix}\right)$ (S8)
$\displaystyle\xrightarrow{\mathrm{decoherence}}\left(\begin{matrix}|a_{i}|^{2}&c_{i}a_{i}b_{i}^{*}\\\
c_{i}^{*}a_{i}^{*}b_{i}&|b_{i}|^{2}\\\ \end{matrix}\right)=\tilde{\rho}_{i}.$
(S9)

Combining all of the above, one arrives at the following:

$\rho=|\psi_{ENT}\rangle\langle\psi_{ENT}|\xrightarrow{\mathrm{decoherence}}\tilde{\rho},$
(S10)

where the density matrix $\tilde{\rho}$ is parameterized with independent
coefficients $a_{i}$ and $c_{i}$ as well as phases $\phi_{ij}$. In order to
obtain a diversified set of states, we propose to choose these parameters
uniformly at random over the interval $[0,1]$ for $a_{i}$ and $c_{i}$, and
over the interval $[0,2\pi]$ for phases $\phi_{i}$. Moreover, we randomize the
states by evolving them with local single qubit gates:

$U(\theta,\phi,\lambda)=\left(\begin{array}[]{cc}\cos\frac{\theta}{2}&-e^{i\lambda}\sin\frac{\theta}{2}\\\
e^{i\phi}\sin\frac{\theta}{2}&e^{i(\phi+\lambda)}\cos\frac{\theta}{2}\end{array}\right),$
(S11)

where $\theta$, $\phi$ and $\lambda$ are some arbitrary Euler’s angles. Hence,
sampling the angles randomly over the interval $[0,2\pi]$, independently for
separate qubits, one can write the final randomized density matrix as:

$\tilde{\rho}_{RAND}=(U_{1}\otimes U_{2}\otimes
U_{3})\tilde{\rho}(U_{1}\otimes U_{2}\otimes U_{3})^{\dagger}.$ (S12)

##  .5 Classifying states as non-discordant

To classify a state as discordant or not, we use an if-and-only-if criterion
for non-discordant states from Ref. [23]. The criterion states that a
bipartite state with subsystems of arbitrary dimensions $N$ and $M$ has zero
quantum discord with respect to the system of dimension $M$, if and only if
all blocks of the $(NM)\times(NM)$ density matrix, which are obtained by
partitioning the density matrix into $N^{2}$ square matrices of dimension $M$
are normal matrices and commute with each other.

The required partition is performed starting from the density matrix written
in the form

$\hat{\sigma}=\sum_{kq}\sum_{nm}P_{kq}^{nm}|k\rangle\langle
q|\otimes|n\rangle\langle m|,$ (S13)

where the indices and corresponding states $k,q$ correspond to the system of
dimension $N$, while the indices and states $n,m$ correspond to the subsystem
of dimension $M$. The partition into blocks requires leaving only the elements
which correspond to matrix elements $|k\rangle\langle q|$ in the subspace of
the system of dimension $N$,

$\hat{\sigma}_{kq}=\langle k|\hat{\sigma}|q\rangle.$ (S14)

Each choice of $k,q$ yields a different matrix.

A normal matrix is one that commutes with its Hermitian conjugate (the matrix
obtained through the partition does not need to be density matrices), thus
fulfilling

$[\hat{\sigma}_{kq},\hat{\sigma}^{\dagger}_{kq}]=0,$ (S15)

whereas commutation obviously requires that for all $k,q$ and
$k^{\prime},q^{\prime}$ one has

$[\hat{\sigma}_{kq},\hat{\sigma}_{k^{\prime}q^{\prime}}]=0.$ (S16)

Both criteria are fulfilled, if and only if the state has zero discord with
respect to the subsystem of dimension $M$.

It is relevant to keep in mind that a zero discord state must not have any
correlations of this type both with respect to the subsystem of size $M$ and
the one of size $N$, so in the case of our three qubit system and each of the
three qubit/2-qubits partitions, will require two checks for zero discord, one
with respect to the qubit, and the other with respec to the 2-qubit subsystem.
In total, this means that checking for the presence of discord in a given
3-qubit state is performed six times due to the different partitions and the
asymmetry of the discord.

##  .6 Numerical Methods

Data generation is performed with the usage of qiskit [53] library. All data
sets can be generated in $\sim 8$ hours. Neural network model (encoder and
decoder part) is implemented using Pytorch [54] library with an automatic
differentiation engine (autograd) used to track gradient of the model
parameters. The training of the single model on the GPU Nvidia RTX 3070 takes
$\sim 1$ day.