# A Lattice-Reduction Aided Vector Perturbation Precoder Relying on Quantum
Annealing

Samuel Winter, Yangyishi Zhang, , Gan Zheng, , Lajos Hanzo S. Winter and Y.
Zhang are with Research and Network Strategies, BT Group (Email:
{samuel.2.winter, yangyishi.zhang}@bt.com), G. Zheng is with University of
Warwick (Email: gan.zheng@warwick.ac.uk), and L. Hanzo is with University of
Southampton (Email: lh@ecs.soton.ac.uk).S. Winter and Y. Zhang have been
supported by Innovate UK project no. 10031626. L. Hanzo would like to
acknowledge the financial support of the EPSRC projects EP/W016605/1,
EP/X01228X/1 and EP/Y026721/1 as well as of the European Research Council’s
Advanced Fellow Grant QuantCom (Grant No. 789028). G. Zheng has been supported
by EP/X04047X/1.

###### Abstract

Quantum annealing (QA) is proposed for vector perturbation precoding (VPP) in
multiple input multiple output (MIMO) communications systems. The mathematical
framework of VPP is presented, outlining the problem formulation and the
benefits of lattice reduction algorithms. Lattice reduction aided quantum
vector perturbation (LRAQVP) is designed by harnessing physical quantum
hardware, and the optimization of hardware parameters is discussed. We observe
a 5dB gain over lattice reduction zero forcing precoding (LRZFP), which
behaves similarly to a quantum annealing algorithm operating without a lattice
reduction stage. The proposed algorithm is also shown to approach the
performance of a sphere encoder, which exhibits an exponentially escalating
complexity.

###### Index Terms:

Quantum annealing, downlink precoding, vector perturbation, MIMO

##  I Introduction

Multiple input multiple output (MIMO) systems form the backbone of wireless
communications links; they are capable of improving the links’ capacity beyond
that of single-antenna communications. Multiple antennas may also be harnessed
for supporting multiple users by exploiting the unique user-specific channel
impulse responses (CIRs), provided that they are accurately estimated. Then,
these coordinated antennas used at the base station (BS) can transmit and
receive multiple data streams, whilst compensating the inter-link
interferences. More explicitly, when considering the downlink (DL), the users
may estimate the DL CIRs and signal it back to the BS. By exploiting the CIR,
the BS then applies DL transmit precoding (TPC) to pre-compensate for the DL
CIRs about to be encountered.

Nonlinear TPC schemes have been shown to provide better performance than
linear schemes, but at the cost of exponentially escalating complexity [1, 2]
versus the number of antennas. In this treatise we conceive a vector
perturbation precoder (VPP), which transforms the original TPC design problem
of maximizing the signal-to-noise ratio (SNR) into the closest vector problem
(CVP). The CVP involves finding a lattice point that is closest to a given
target vector that does not lie on the lattice, in an $N_{r}$ dimensional
complex vector space. Further restrictions on the legitimate vector search
space are imposed due to the limited linear dynamic range of the radio
frequency (RF) chain of each antenna, and the limited symbol-duration
available to find a solution at the target symbol rate.

Noisy intermediate scale quantum (NISQ) devices are currently being assessed
in terms of their suitability for wireless systems. In this context, a quantum
search algorithm was developed for multiuser detection that can match the
performance of a classical counterpart [3]. Although, at the time of writing,
implementations using quantum hardware are limited by the available capability
of the NISQ devices, they hold the promise of reducing computational
complexity. The maximum likelihood (ML) detection problem has been solved for
small-scale systems using the quantum approximate optimization algorithm [4].
Indeed, the employment of quantum annealing (QA) in ML detection is already a
well studied problem [5, 6]. Tabi et al. [5] employ QA in massive MIMO
communications, and probe the limits of what is possible based on the
available hardware. It is also shown that for a quadrature amplitude
modulation (QAM) scheme of order 16, QA was able to match the performance of a
classical ML detector. As a further advance, De Luna et al. [6] demonstrate
the ability to parallelize computations for DWave processors, but at the time
of writing the number of qubits remains limited. The limitations imposed by
the number of bits per symbol are attributed to the resolution of the
annealing control. As for DL TPC, Kasi et al. present a novel preprocessing
technique that relaxes the hardware requirements, for a QA VPP solver, at a
modest performance erosion [7]. Their experiments show steps taken to mitigate
hardware constraints, however a strong dependence on the time allowed for the
QA shows a strong susceptibility to noise.

Against this background, a novel QA based framework is proposed for VPP, which
substantially improves the robustness against hardware limitations. The
algorithm, unlike the previous work, is designed with the physical mechanism
of QA in mind. Explicitly, we conceive a powerful preprocessing procedure for
the QA by harnessing the Lenstra-Lenstra-Lovasz (LLL) algorithm [8] for
reducing the search space of the vectors and adjusting the quantum mechanical
form of the problem. Additionally, we incorporate a noise resilient encoder
for reducing the impact of quantum hardware impairments on the system
performance. The performance of the proposed lattice reduction aided quantum
vector perturbation (LRAQVP) scheme is experimentally validated on real
quantum hardware as well as benchmarked against its classical lattice reduced
zero forcing (LRZFP) counterpart.

Notation: scalar quantities are represented by normal letters, vector
quantities by lowercase boldface letters, and matrices by uppercase boldface
letters. The Euclidean and Frobenius norms are represented by
$\lVert\cdot\rVert$ and $\lVert\cdot\rVert_{F}$ respectively; the floor
operator by $\lfloor\cdot\rfloor$. The (right pseudo) inverse of a matrix is
denoted by $\left(\cdot\right)^{+}$ and the matrix/vector transpose by
$\left(\cdot\right)^{\top}$. The functions $\mathfrak{Re}[\cdot]$ and
$\mathfrak{Im}[\cdot]$ extract the real and imaginary parts of its argument.

##  II System Model

A BS having $N_{t}$ transmit antennas serves $N_{r}$ users, each with a single
antenna, where $N_{r}\leq N_{t}$. The symbol vector
$\mathbf{u}\in\mathbb{C}^{N_{r}}$ contains the symbols transmitted to each
user. The elements of $\mathbf{u}$ are drawn from a uniform distribution of
complex square QAM symbols; this choice of modulation is however not a
requirement for VPP [9]. The average transmit power of a symbol is 1. We
define $\rho=\mathcal{E}_{s}/N_{s}$ as the average symbol energy
$\mathcal{E}_{s}$ over the noise variance per symbol $N_{s}$. The received
signal vector is

$\mathbf{y}=\sqrt{\frac{\rho}{\lVert\mathbf{x}\rVert^{2}}}\mathbf{H}\mathbf{x}+\mathbf{n},$
(1)

where $\mathbf{H}\in\mathbb{C}^{N_{r}\times N_{t}}$ represents the channel
state information, which is assumed to be perfectly known, and
$\mathbf{n}\in\mathbb{C}^{N_{r}}$ is the noise vector. The elements of
$\mathbf{H}$ and $\mathbf{n}$ are drawn from the complex Gaussian distribution
$\mathcal{CN}(0,1)$. Lastly, the precoded transmit vector of

$\mathbf{x}=\mathbf{G}(\mathbf{u}-\tau\mathbf{l})$ (2)

is dependent on the (pseudo)inverse of $\mathbf{H}^{+}=\mathbf{G}$ and on an
integer perturbation vector $\mathbf{l}\in\mathbb{G}^{N_{r}}$, where
$\mathbb{G}$ represents the set of Gaussian integers. The perturbation vector
is scaled to match the width of the QAM scheme by $\tau=2c_{max}+\Delta$,
where $\Delta$ represents the distance between two neighbouring constellation
points on an Argand plot of the modulation scheme, and $c_{max}$ is the
largest possible real part of a value in the modulation scheme.

To recover the symbols after transmission, we define
$\Theta_{\tau}(x)=\tau\lfloor x/\tau+\frac{1}{2}\rfloor$ and
$\Gamma_{\tau}(x)=x-\Theta_{\tau}(x)$ to represent the integer and fractional
parts of the $\mathbf{u}$ vector with respect to $\tau$. The functions act
element-wise on a vector. The recovered signal of

$\bar{\mathbf{u}}=\mathbf{u}+\Gamma_{\tau}\left(\sqrt{\frac{\lVert\mathbf{x}\rVert^{2}}{\rho}}\mathbf{n}\right)$
(3)

demonstrates the reduced impact of noise, when $\lVert\mathbf{x}\rVert^{2}$ is
minimized. This yields an instance of the closest vector problem

$\mathbf{l}_{\mathrm{opt}}=\underset{\mathbf{l}}{\operatorname{argmin}}\lVert\mathbf{G}(\mathbf{u}-\tau\mathbf{l})\rVert^{2},$
(4)

which is well known to be an NP hard problem.

##  III Quantum Annealing

The QA algorithm is an instance of adiabatic quantum computing that solves a
specific set of problems, namely quadratic unconstrained binary optimization
(QUBO) [10]. Inspired by the adiabatic theorem of quantum mechanics, QA finds
the lowest energy configuration of a programmable magnetic quantum spin
network that can be described by the Ising model. In QA, an appropriately
selected initial Hamiltonian $\mathcal{H}_{I}$ is slowly evolved into the
problem Hamiltonian $\mathcal{H}_{P}$ that maps the QUBO onto the spin states
$\lvert-1\rangle$ and $\lvert+1\rangle$.

The system commences its evolution from the ground state of

$\mathcal{H}_{I}=\sum_{i\in\mathsf{V}}\sigma_{x}^{i},$ (5)

where $\sigma_{x}^{i}$ is the Pauli spin $x$ operator acting on the $i$th
qubit; $\mathsf{V}$ is the set of qubits. This represents the application of a
transverse magnetic field. The final measurements of spin are then taken along
the $z$ axis, thus the qubits start in an equi-probable superposition of the
spin states. The problem is encoded into the magnetic biases $h_{i}$ and
couplings between the qubits $J_{ij}$ along the $z$ axis. This yields a
problem Hamiltonian of the form

$\mathcal{H}_{P}=\sum_{i\in\mathsf{V}}h_{i}\sigma_{z}^{i}+\sum_{\langle
i,j\rangle\in\mathsf{E}}J_{ij}\sigma_{z}^{i}\sigma_{z}^{j},$ (6)

where $\mathsf{E}$ is the set of physical couplings between qubits $i$ and
$j$. The adiabatic theorem states that a quantum system will remain in its
ground state, provided that it transitions to a different Hamiltonian slowly
enough. The full time-dependent Hamiltonian is formulated as

$\mathcal{H}=A(s)\mathcal{H}_{I}+B(s)\mathcal{H}_{P},$ (7)

where $s=t/T_{a}$ and the function $A(s)$ and $B(s)$ are subject to the
conditions that $A(0)=B(1)=1$ and $A(1)=B(0)=0$, while $T_{a}$ is the
annealing time which must be long enough for the adiabatic theorem to hold. If
$T_{a}$ is too short, the probability of remaining in the ground state is
reduced, making the computation less likely to give an optimal solution [10].

The spin variables may be transformed onto binary variables as follows
$s_{i}=2q_{i}-1$. Our QUBO may be represented as

$\mathbf{q}_{opt}=\underset{\mathbf{q\in\mathbb{B}^{M}}}{\operatorname{argmin}}\quad\mathbf{q}^{\top}\mathbf{Q}\mathbf{q},$
(8)

where the legitimate values of the scalar
$\mathbf{q}^{\top}\mathbf{Q}\mathbf{q}$ are related to the energies of the
quantum spin network, while $\mathbb{B}=\\{0,1\\}$, and $\mathbf{q}$ has a
length of $M$. Since we have $q_{i}^{2}=q_{i}$, the magnetic bias terms
($h_{i}$) are absorbed into the diagonal of $\mathbf{Q}$.

This work focuses on integer programming where the space of legitimate
solutions is infinite i.e. any integer vector. Legitimate solutions of a QUBO
take the form of any binary vector of a fixed dimension, this space is
unconstrained but finite. Thus, the integer solution space is itself
constrained to a finite number of vectors. This leads to the use of an
encoding where the integer vector

$\mathbf{l}=\mathbf{C}\mathbf{q}.$ (9)

The encoding matrix $\mathbf{C}\in\mathbb{Z}^{N\times M}$ is a block matrix
containing integers, where each element in $\mathbf{q}\in\mathbb{B}^{M}$
represents whether the integer in $\mathbf{C}$ is part of the sum that yields
an element in $\mathbf{l}\in\mathbb{Z}^{N}$. Each column of $\mathbf{C}$
contains only a single integer. Encodings may be degenerate, i.e. an integer
may not have a unique representation in binary form. The more integers that
are encoded, the higher the qubit requirement becomes, and the higher the
number of legitimate energy levels of the Ising model Hamiltonian becomes. The
energy range of the QA hardware is fixed, and the problem values are scaled to
fit into this range. Hence, for a higher number of energy levels, there is
also a lower expected energy gap between the ground state and the excited
states of the system, making the annealing more susceptible to noise as a
consequence of the adiabatic theorem [11]. To compensate for the reduced
energy level gaps, $T_{a}$ must be increased to ensure a higher probability of
measuring an optimal solution, however this comes at the cost of delicate
quantum state being perturbed by environmental noise. This decoherence of the
quantum state [12] and qubit manufacturing imperfections [13] are inherent
sources of error to QA.

Once the QUBO has been formulated, it may be represented by a graph structure
of logical variables. The physical hardware however already has a fixed
structure, so the problem graph must be embedded in the hardware graph as a
graph minor, where chains of physical qubits represent a single logical qubit.
The problem is then transformed to its Ising formulation $\mathcal{H}_{P}$,
and $\mathcal{H}$ evolves according to (7). The chains are positively coupled
($J_{ij}\geq 0$), ensuring that neighbouring spins remain aligned.

##  IV Algorithm Design

The problem formulated in (4) is calculated in terms of complex variables. To
encode the integers in the problem we rewrite the variables using the
isomorphism between $\mathbb{C}$ and $\mathbb{R}^{2}$,

$\mathbf{M}\rightarrow\begin{pmatrix}\mathfrak{Re}[\mathbf{M}]&-\mathfrak{Im}[\mathbf{M}]\\\
\mathfrak{Im}[\mathbf{M}]&\mathfrak{Re}[\mathbf{M}]\end{pmatrix}\qquad\mathbf{v}\rightarrow\begin{pmatrix}\mathfrak{Re}[\mathbf{v}]\\\
\mathfrak{Im}[\mathbf{v}]\end{pmatrix},$ (10)

where $\mathbf{M}$ and $\mathbf{v}$ are an arbitrary complex matrix and
vector. This is used to rewrite (4) in terms of only real variables, allowing
for the encoding of real integers, and the lattice reduction algorithm to act
as expected over a real valued lattice basis.

Due to the limited resource of NISQ devices, the problem formulation and
preprocessing are vital for generating useful results. Again, the LLL
algorithm [8] is employed for generating a short basis on which to solve the
CVP. The matrix $\mathbf{G}$ can be said to form a lattice basis having
columns that are the basis vectors. In the context of Figure 1, the original
basis is $\\{\mathbf{g}_{1},\mathbf{g}_{2}\\}$. The LLL algorithm factors the
original lattice $\mathbf{G}=\mathbf{F}\mathbf{Z}^{+}$ into the reduced
lattice $\mathbf{F}$ and the unimodular matrix $\mathbf{Z}$. Another property
of $\mathbf{F}$ is that of having a reduced Frobenius norm
$\lVert\mathbf{F}\rVert_{F}\leq\lVert\mathbf{G}\rVert_{F}$. LLL reduces the
lengths of the basis vectors, ensuring short vectors are typically represented
by small integers. In the context of Figure 1,
$\\{\mathbf{f}_{1},\mathbf{f}_{2}\\}$ is the reduced basis. To formulate the
CVP in the reduced basis, let

$\displaystyle\mathbf{u^{\prime}}$
$\displaystyle=\Gamma_{\tau}(\mathbf{Z}^{+}\mathbf{u})$ (11)
$\displaystyle\mathbf{l^{\prime}}$
$\displaystyle=\mathbf{Z}^{+}+\mathbf{l}\Theta_{\tau}(\mathbf{Z}^{+}\mathbf{u})$
(12)

and then substitute them into (2), which yields

$\mathbf{x}=\mathbf{F}(\mathbf{u^{\prime}}-\tau\mathbf{l^{\prime}}),$ (13)

noting that in the case of $\mathbf{l}=\mathbf{0}$ it is equivalent to the
LRZFP.

Figure 1: The impact of lattice reduction on limited search spaces in 2D. The
bases $\\{\mathbf{g}_{1},\mathbf{g}_{2}\\}$ and
$\\{\mathbf{f}_{1},\mathbf{f}_{2}\\}$ describe the same lattice. The
$\mathbf{u}$ vector in green is closest to the encircled lattice point. The
search points in the non-reduced basis do not include this closest point,
showing the necessity of lattice reduction in limited search space examples.

The elements of the integer perturbation vector are typically in the set
$\\{-1,0,1\\}$ [14]. This, along with the lattice reduction, justifies the
choice to only encode the integers in this set. To encode the integers, the
scheme laid out in [15] is used, but this is only valid for positive integers.
A translation is used for rectifying the mismatch between the two sets
allowing the search space to shift to $\\{0,1,2\\}$, where
$\mathbf{u^{\prime\prime}}=\mathbf{u^{\prime}}+\tau\mathbf{t}$ and
$\mathbf{l^{\prime\prime}}=\mathbf{l^{\prime}}+\mathbf{t}$ with
$\mathbf{t}=\begin{pmatrix}1&1&\ldots&1\end{pmatrix}^{\top}$. The problem can
then be transformed into a QUBO accordingly. We encode a single integer,

$l_{i}=\begin{pmatrix}1&1\end{pmatrix}\begin{pmatrix}q_{i0}\\\
q_{i1}\end{pmatrix},\vspace{-0.16cm}$ (14)

showing that the integer ‘0’ is represented in binary form as ‘00‘’, while ‘1’
has two representations - ‘01’ and ‘10’ - and ‘2’ is represented as ‘11’. The
redundancy in encoding reduces the number of energy levels in $\mathcal{H}$ by
introducing degeneracy, making the encoding more resilient to noise. The
smaller number of energy levels will be scaled to the annealer’s hardware
capabilities, leaving a larger energy gap between the ground and excited
states of $\mathcal{H}$ compared to that of a more dense encoding, hence
allowing for reduced annealing times.

To transform the problem into a QUBO, we substitute (9) and (13) into (4),
followed by expanding it, yielding

$\displaystyle\begin{split}\lVert\mathbf{x}\rVert^{2}={}&\tau^{2}\mathbf{q}^{\top}\mathbf{C}^{\top}\mathbf{F}^{\top}\mathbf{F}\mathbf{C}\mathbf{q}-2\tau\mathbf{u^{\prime\prime}}^{\top}\mathbf{F}\mathbf{C}\mathbf{q}\\\
&+\mathbf{u^{\prime\prime}}^{\top}\mathbf{F}^{\top}\mathbf{F}\mathbf{u^{\prime\prime}}\end{split}$
(15) $\displaystyle={}$
$\displaystyle\mathbf{q}^{\top}\mathbf{Q}\mathbf{q}+c,$ (16)

where we have
$\mathbf{Q}=\tau^{2}\mathbf{C}^{\top}\mathbf{F}^{\top}\mathbf{F}\mathbf{C}+\operatorname{diag}(-2\tau\mathbf{u^{\prime\prime}}^{\top}\mathbf{F}\mathbf{C})$
and
$c=\mathbf{u^{\prime\prime}}^{\top}\mathbf{F}^{\top}\mathbf{F}\mathbf{u^{\prime\prime}}$.
The $\operatorname{diag}(\cdot)$ function takes the row vector argument and
places its elements on the diagonal of a matrix.

The graph $\mathbf{Q}$ represents is typically complete for this problem. The
physical connections of qubits on the quantum annealer do not match up with
the structure, thus the QUBO must be embedded in the hardware graph, where
chains of physical qubits represent a single logical qubit. The ‘minor miner’
algorithm [16] generates a new graph that matches the hardware connections,
and an implementation can be found in the Ocean SDK [17]. The chains require a
positive coupling ($J_{chain}>0$) so that the spins remain aligned. To assign
the chain strength, the uniform toque compensation function in the Ocean SDK
is used that calculates,

$J_{chain}=p\cdot\sqrt{\sum_{\langle
i,j\rangle\in\mathsf{E}}\frac{J_{ij}^{2}}{N_{J}}}\cdot\sqrt{k_{avg}},$ (17)

where $k_{avg}$ is the average degree of a node in the problem graph, and
$N_{J}$ is the number of off diagonal elements of $\mathbf{Q}$. The chain
strength parameter $p$ may be varied to find the optimal performance of the
algorithm. At this stage, $\mathcal{H}_{P}$ has been formulated and may be
solved with the QA algorithm. The algorithm framework is summarized in Alg. 1

Input : $\mathbf{H}$, $\mathbf{u}$

Output : $\mathbf{l^{\prime\prime}}$

1 Invert $\mathbf{H}$ and apply the LLL algorithm [8] to generate $\mathbf{F}$
and $\mathbf{Z}$.

2 Generate the encoding matrix $\mathbf{C}$ and calculate the QUBO using (15).

3 Embed the QUBO in the hardware graph and calculate $J_{chain}$ according to
(17).

4 Apply the QA algorithm and invert the binary result back into an integer
vector according to (9).

Algorithm 1 QA-based LR-VP Precoding

##  V Results

The experimental parameters used in the section are summarized in Table I,
unless explicitly stated that they different. The number of transmitted
symbols applies only to the symbol error rate (SER) simulations.

TABLE I: Experimental parameters Parameter | Value
---|---
Encoding | 64 QAM
$N_{r}\times N_{t}$ | 10$\times$10
Number of transmitted symbols | 150 000
$T_{a}$ | 100 $\mu$s
$p$ | 1.5

Tuning $T_{a}$ and $p$ are important steps in order to ensure optimal
performance of QA. Furthermore, the dependence on these parameters gives an
insight into the robustness of the algorithm against noise and the limited
dynamic range of the hardware. In total 150 channels and signals were
generated at random and (4) was solved using the DWave Advantage quantum
annealer, with $T_{a}$ varying from 20 $\mu$s to 400 $\mu$s and $p$ fixed at
1.5. We observed that, without showing graphically due to page limit, the
choice of $T_{a}$ had no impact on the distributions of
$\lVert\mathbf{x}\rVert^{2}$, indicating that there is a robust production of
a large enough energy gap between the ground and first excited states of
$\mathcal{H}$. The lack of an increase in median values with increasing
$T_{a}$ shows that other sources of noise are not impacting the quality of the
results. For the rest of the experiments $T_{a}$ is fixed at 100 $\mu$s as a
conservative value, selected for reducing the likelihood of measuring
suboptimal solutions. The optimization of $p$ was carried out similarly with
$p$ varying between 0.6 and 1.8. The distributions of
$\lVert\mathbf{x}\rVert^{2}$ are shown in Fig. 2, where the optimal median
value lies at $p=1.5$. The large median and variance at low values of $p$ are
due to broken chains leading to suboptimal results being calculated however,
when $p\geq 1.2$ we see a compression in the distributions indicating robust
chains and reliable solutions. The increase of the median value at $p=1.8$ is
due to the limited dynamic range of the qubit couplers. All values in
$\mathcal{H}_{P}$ get scaled to fit the range of the hardware, thus the larger
value of $J_{chain}$ lead to a compression of gaps between the energy levels
of the problem leading to a greater susceptible to noise. The distributions of
$\lVert\mathbf{x}\rVert^{2}$ shown in Fig. 2 are displayed with the outliers
removed. The outliers are those that lie outside 1.5 times the interquartile
range (the height of the boxes in Fig. 2), above the upper quartile, and below
the lower quartile.

Figure 2: Distributions of $\lVert\mathbf{x}\rVert^{2}$, calculated via the
LRAQVP algorithm, with varying chain strength parameter. The reduced median
values show the impact of increased chain strength counteracting the noise,
ensuring better solution quality. The limited dynamic range of the quantum
annealer becomes the dominant factor in noise above the optimal value of $p$.
There are 150 samples in each distribution.

Fig. 3 shows performance in terms of the optimization objective, several
algorithms were compared over a sample size of 2000 random channels and
signals. The Hyper-sphere Approximation (HA) is calculated for each channel
according to Lemma 1 in [18]. The HA represents a lower bound of the SER for
the average of all the symbols under a sphere encoder of exponentially
escalating complexity. QVP was found to have a similar distribution to LRZFP,
but LRAQVP is approaching the HA in terms of its median performance. In all
the channels, a non-zero perturbation vector $\mathbf{l}$ was found by the
LRAQVP algorithm.

Figure 3: The distributions of $\lVert\mathbf{x}\rVert^{2}$ for Zero forcing,
the LRZFP protocol and the LRAQVP presented here. The HA lower bound represent
a spherical approximation to what the closest vector could be. The HA is a
lower bound on the median performance that is averaged over all symbol vectors
that could pass through a given channel.

SER simulations were carried out to compare the algorithms in a communications
system. We also compare the empirical results to the theoretical uniform input
limit (UIL). The prediction for LRZFP has the associated shaping loss
compensated for [19]. Despite being valid only for large $\rho$, it fits the
empirical data well. $T_{a}$ and $p$, were held constant at 100 $\mu$s and 1.5
for LRAQVP and QVP. Since LRAQVP is expected to be a more reliable algorithm,
the annealing time may be longer than it is strictly necessary. Lastly, the HA
prediction constitutes a lower bound on the expected SER of the sphere encoder
[18]. For LRAQVP and QVP, each perturbation vector was calculated using the
DWave Advantage. The end to end SER performance is much improved over that of
LRZFP, and approaches the HA. There is an approximately 5 dB gain over LRZFP.
The predictions and the results of the SER simulations are shown in Fig. 4.

Figure 4: The SER simulations for a 10x10 MIMO system. A 5 dB gain can be seen
from LRZFP to LRAQVP protocols. QVP can be seen to behave similarly to LRZFP,
however it scales with much higher computational complexity, and performs
worse in the high SNR regime. The uniform input limits (UILs) are theoretical
predictions for the zero forcing protocols used to verify the empirical
results. HA is a lower bound on the performance, that LRAQVP is approaching.

The complexities of the core algorithms discussed are displayed in Table II.
Explicitly, the complexity of QA is governed by the annealing time required
for the adiabatic theorem to hold [20]. During the annealing, there is only
one evaluation of $\lVert\mathbf{x}\rVert^{2}$, whilst for a classical
algorithm, $\lVert\mathbf{x}\rVert^{2}$ is iteratively evaluated. The number
of evaluations dictates its complexity. The upper bound on the number of
evaluations of $\lVert\mathbf{x}\rVert^{2}$, for a sphere encoder, is an
exponential function of the number of users  [18]. The complexity analysis
shows that there exists some $N_{r}$ at which it would be faster to use a
quantum annealer to find a solution to the CVP compared to a sphere encoder.
The total computation time for the QA section of LRAQVP, for a single channel
and signal vector, was 260 ms as 1000 anneals were performed to maximize the
likelihood of measuring the ground state. The 1000$\times$100 $\mu$s $=$ 100
ms of quantum annealing accounts for solving the problem and the remainder is
the time required for the chip to reset and for the temperature to reduce
again. The typical coherence times for 5G radio channels are on the order of
milliseconds [21], but they can be shorter depending on the mobility of a
user. Therefore, the impact of noise within QA must be much reduced so that
the number of anneals required is reduced to reach a practical timescale. A
solution may be to pair QA with channel stabilization techniques, increasing
the channel’s coherence time. For much larger antenna arrays and cell-free
communications, QA may be viewed as a door into nonlinear TPC on a massive
scale.

TABLE II: Time complexity in number of users. Algorithm | Complexity
---|---
ZFP | $\mathcal{O}(N_{r}^{2.373})$
LRZFP | $\mathcal{O}(N_{r}^{5})$
Sphere Encoder | $\mathcal{O}(e^{N_{r}})$
Quantum Annealing | $\mathcal{O}(e^{\sqrt{N_{r}}})$

##  VI Conclusions

The problem formulation and preprocessing in this LRAQVP framework has been
shown to give good performance for a quantum CVP solver in MIMO systems.
Defining the problem in terms of a short lattice basis allows searching over a
limited subset of the most relevant points on the lattice, with small integers
available for encoding on limited quantum hardware and paving the way for more
practical implementations relying on NISQ annealing hardware. The performance
of the LRAQVP framework has been experimentally validated on real QA hardware,
and it was also benchmarked against the classical LRZFP protocol. As all the
perturbation vectors were non-zero, this underlines the importance of an
effective CVP solver, even in conjugation with lattice reduction. The
complexity of classical algorithms demonstrates that QA-based protocols could
outperform them for larger problem sizes, while still being practically
feasible. However, optimal preprocessing of the problem demonstrates the
importance of lattice reduction in ensuring quantum advantage over classical
algorithms.

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# Quantum correlations and their significance in quantum teleportation under
$\cal{PT}$-symmetric operation

J. Ramya Parkavi1, R. Muthuganesan2, V. K. Chandrasekar1, and M. Lakshmanan3
1Department of Physics, Centre for Nonlinear Science & Engineering, School of
Electrical & Electronics Engineering, SASTRA Deemed University, Thanjavur -613
401, Tamilnadu, India
2Department of Physics, Faculty of Nuclear Sciences and Physical Engineering,
Czech Technical University in Prague, Br̆ehová 7, 115 19 Praha 1-Staré Mĕsto,
Czech Republic
3Centre for Nonlinear Dynamics, School of Physics, Bharathidasan University,
Tiruchirappalli 620 024, Tamilnadu, India.

###### Abstract

In this article, we exploit the different notions of quantumness measure to
understand the properties of the Heisenberg XY model with and without
$\cal{PT}$-symmetric operation. In the absence of $\cal{PT}$-symmetry, we
study the significance of different measures, namely entanglement and
measurement-induced nonlocality (MIN), in the detection of the quantumness of
the Heisenberg XY model. It is observed that the quantum correlations and
teleportation fidelity monotonically decreases with respect to temperature.
Furthermore, the intervention of $\cal{PT}$-symmetric operation enhances the
strengths of quantum correlation. In addition, we highlight the role of the
system’s parameters and $\cal{PT}$-symmetric operation on the teleportation of
a quantum state. Our results also emphasize that after the addition of
$\cal{PT}$-symmetric operation, the considered physical model remains a
versatile resource to achieve successful teleportation of the quantum state.

##  I Introduction

Characterizing quantum correlation in composite systems via local
measurements/operations is one of the fundamental areas of research in quantum
information theory. One of the most fundamental quantum correlation measures
is entanglement, which was introduced by Schr$\ddot{o}$dinger Schrodinger1935
due to the paradox demonstrated by Einstein, Podolsky, and Rosen (EPR)
Einstein1935 . It is argued that due to the entangled nature of systems, the
measurement on one of the subsystems will affect the outcome of measurements
performed in the other subsystem. This feature obviously contradicts local
realism, i.e., the quantum states of spatially separated non–interacting
particles are interdependent. The contradiction arising in the above argument
leads to the question of the completeness of quantum mechanics which was
rooted in their insistence on local realism. Thus, entanglement is viewed as
“spooky action” at a distance exhibited in the composite quantum systems by
virtue of the superposition principle. In an entangled composite quantum
system, the individual subsystems are strongly linked to each other even if
they are far apart and no longer interact with each other. The whole composite
quantum system can be well described as a definite pure state while for the
state of its subsystems, such a description is not possible.

Subsequently, assuming local realism of EPR, Bell attempted for the
completeness of quantum mechanics using the hidden variable theory Bell1964 ;
Bell1966 . One of the main manifestations of nonlocal attributes of many body
quantum states is entanglement which has no classical analog. In quantum
information theory, entanglement is used as a resource for different
applications: for instance, in quantum cryptography, or teleportation schemes,
or quantum computation, where it is used much like energy or information that
are considered as resources in classical processing. While the entanglement
between bipartite (two particles) states has been extensively investigated
over many decades, the entanglement of multipartite (many particles) states
are increasingly difficult for investigation. Even within the domain of
bipartite states, it is now fairly realized that entanglement is not the
complete manifestation of non-classical correlation between the particles.

In order to capture the complete spectrum of nonlocality, a new measure of
quantum correlation from the perspective of measurement called quantum discord
Ollivier2001  has been introduced. The operational meaning of this quantity is
the minimal loss of information of a state due to local quantum measurements.
Computation of discord involves complex optimization Girolami2011  and is also
shown to be a nondeterministic polynomial problem Huang2014 . For easy
computation, till date, various quantum correlations have been identified
based on the metric in state space Girolami2012 ; Dakic2010 ; Luo2011 ;
Spehner2017 ; Roga2016 . Measures based on the metric are experimentally
realizable Jin2012 ; Passante2012 ; Girolami2012 . One of the notable
correlation measures beyond entanglement is the measurement induced
nonlocality (MIN) Luo2011 . MIN characterizes the nonlocality from a
measurement perspective, which is different from the conventionally mentioned
quantum nonlocality related to entanglement or the violation of Bell’s
inequalities. This quantity can be considered as an indicator of the global
effect caused by locally invariant measurements. Up to date, several other
versions of MIN were suggested too HFan2015 ; MuthuPLA ; XiMIN ; HuMIN ;
HFanReview .

On the other side, non-Hermitian systems with parity-time ($\cal{PT}$)
symmetry proposed by Bender and Boettcher have attracted growing interest over
the past two decades. These systems support two distinct phases, namely
unbroken phase and broken phase. In these systems, the eigenvalues are real in
$\cal{PT}$ unbroken regime and the eigenvalues are complex in the
spontaneously broken $\cal{PT}$ regime ptbase1 ; ptbase2 ; ptbase3 . As a
result, the dynamics is drastically different in these two phases, and a
dynamical criticality occurs at the boundary between the two regimes called
exceptional points. The exploitation of quantum mechanical phenomena around
these exceptional points is a formidable task in the domain of quantum
information theory. Recently, their realization in different $\cal{PT}$\-
symmetric systems ijtp ; ourpaper ; bellPT  and then their applications were
demonstrated. For instance, the application of local non-Hermitian detuning on
entangled bipartite state significantly reduces the uncertainty of quantum
measurements entropynonherm . In particular, the information flow between a
$\cal{PT}$-symmetric non-Hermitian system and its environment has been studied
theoretically as well as experimentally and it has been shown that the
complete retrieval of information is possible in the unbroken
$\cal{PT}$-symmetric regimeinfflowtheory ; infflowexp . In Ref. rocatti , F.
Roccati et al. studied the dynamics of quantum correlations in a
$\cal{PT}$-symmetric gain-loss system which exhibits stationary quantum
correlations beyond entanglement. They also pointed out the usefulness of such
stationary quantum correlations in information encoding infencod , remote-
state preparation remote , quantum metrology and sensing quanmetrology , etc.
To an extent, the authors of highpt  experimentally demonstrated the quantum
information dynamics through entropy in a high dimensional
$\cal{PT}$-symmetric system with its exceptional point degeneracies.

Compared to the above systems, the Heisenberg spin chain model is noteworthy
which accounts for the enormous phenomena in magnetic systems which can be
implemented in the exploitation of quantum/magnetic phase transitions, quantum
correlations and teleportation. For example, teleportation which depends on
the entangled states was demonstrated with the effects of dipolar interaction
between two spin-$1/2$ particles and it was shown that even without the
violation of Bell’s inequality all the entangled states are useful for
teleportation dipolar . This is in contrast to the earlier studies which
showed that any two spin-1/2 state which violates the Bell-CHSH inequality
proposed by Clauser, Horne, Shimony and Holt is useful for teleportation
fidelbellreferee1 ; fidelbellreferee2 . The authors of Ref. DZ  also
highlighted that a minimal entanglement of the thermal state is needed to
realize the entanglement teleportation in a two-qubit Heisenberg chain model
with Dzyaloshinski-Moriya interaction.

However, to our knowledge, fewer research works related to the quantum
correlations and their significance in quantum teleportation have been
investigated, especially under the action of $\cal{PT}$-symmetric operation.
Two questions are putforth here: (i) How does the $\cal{PT}$-symmetric
operation affect the existence of quantum correlations in a Heisenberg XY spin
system? and (ii) Whether the $\cal{PT}$-symmetric nature enhances or
diminishes the teleportation process in the system? To answer these queries,
in our work, we mainly investigate the co-evolution of quantum correlations
and quantum teleportation of the Heisenberg XY spin model under the non-
unitary time evolution. We have measured the thermal as well as temporal
quantum correlations via four different measures, namely Bell nonlocality,
concurrence, measurement induced non-locality, and trace distance based
measurement induced nonlocality. In addition to these, we try to exploit the
teleportation process even in the absence of entanglement.

The article is structured as follows: First, we introduce the different
measures of quantum correlations in Sec. II. The considered Heisenberg XY
model is described in Sec. III. We display the results of thermal quantum
correlations and thermal teleportation for the considered model in Sec. IV.
The role of $\cal{PT}$-symmetric operation is explored in the model in Sec. V.
In Sec. V A, the quantum correlation measures and teleportation under
$\cal{PT}$-symmetric time evolution will be discussed. Finally, we summarize
the key findings of this paper in Sec. VI.

##  II Preliminaries

In this section, we introduce the quantum correlation quantifiers of bipartite
states. For this purpose, we consider a bipartite system represented by its
density matrix $\rho$ shared between the subsystems $a$ and $b$.

Entanglement: The amount of bipartite entanglement between the subsystems is
measured by the quantity concurrence and it is defined as concurform

$\displaystyle\mathcal{C}(\rho)=2~{}\text{max}~{}\left\\{0,~{}\lambda_{1}-\lambda_{2}-\lambda_{3}-\lambda_{4}\right\\},$
(1)

where $\lambda_{i}$’s are the square root of the eigenvalues of the matrix
$\Lambda=\sqrt{\rho}\tilde{\rho}\sqrt{\rho}$ and arranged in decreasing order.
Here
$\tilde{\rho}=(\sigma_{y}\otimes\sigma_{y})\rho^{*}(\sigma_{y}\otimes\sigma_{y})$
is the spin flipped density operator and the symbol $*$ denotes the usual
complex conjugate in the computational basis. The well-known fact is that
concurrence takes the minimum and maximum values from $0$ to $1$ which
correspond to the separable (unentangled) and maximally entangled states,
respectively.

Maximal Bell function:

In general, for any qubit-qubit system, Bell inequality is a well-known
technique to demonstrate the presence of nonlocality of a state. It is also
useful in the quantification of quantum correlation in physical systems. The
nonlocality of a quantum state can be detected by the violation of the Bell-
CHSH inequality, which is given by Clauser

$|\langle\mathcal{B}_{{\text{CHSH}}_{\rho}}\rangle|\leq 2,$ (2)

where
$|\langle\mathcal{B}_{{\text{CHSH}}_{\rho}}\rangle|=\text{Tr}(\rho~{}\mathcal{B}_{\text{CHSH}})$
and $\mathcal{B}_{\text{CHSH}}$ is the Bell operator associated with the
quantum CHSH inequality. To quantify the quantum correlation between the
subsystems via Bell inequality, the maximal Bell function is defined as
Horodecki ; Brunner

$\mathcal{B}_{\text{max}}=2\sqrt{M(\rho)},$ (3)

where $M(\rho)=\text{max}_{i<j}(R_{i}+R_{j})$ with $R_{i}(i=1,2,3)$ being the
eigenvalues of the correlation matrix $R^{t}R$. The matrix elements of $R$ are
defined as $r_{ij}=\text{Tr}(\rho(\sigma_{i}\otimes\sigma_{j}))$. The function
$\mathcal{B}(\rho)>2$ indicates the violation of Bell inequality and the
presence of nonlocal correlation of two-qubit quantum states.

Measurement-induced nonlocality: Recently, a new measure of quantum
correlation has been introduced from the perspective of measurements called
measurement-induced nonlocality (MIN). Originally, it was defined as the
maximal nonlocal effect due to locally invariant projective measurement via
Hilbert-Schmidt norm. Mathematically, the definition of MIN is given as
Luo2011

$\mathcal{N}_{2}\mathcal{(\rho)}=~{}^{\text{max}}_{\Pi^{a}}\|\rho-\Pi^{a}(\rho)\|^{2},$
(4)

where $\|\mathcal{O}\|^{2}=\text{Tr}(\mathcal{O}\mathcal{O}^{\dagger})$ is
Hilbert-Schmidt norm of operator $\mathcal{O}$ and the maximum is taken over
the locally invariant projective measurements on subsystem $a$ which does not
change the state. The post-measurement is defined as
$\Pi^{a}(\rho)=\sum_{k}(\Pi^{a}_{k}\otimes\mathds{1}^{b})\rho(\Pi^{a}_{k}\otimes\mathds{1}^{b})$,
with $\Pi^{a}=\\{\Pi^{a}_{k}\\}=\\{|k\rangle\langle k|\\}$ being the
projective measurements on the subsystem $a$, which do not change the marginal
state $\rho^{a}$ locally i.e., $\Pi^{a}(\rho^{a})=\rho^{a}$.

In fact the MIN has a closed formula for any $2\times n$ dimensional system
and is given as

$\mathcal{N}_{2}\mathcal{(\rho)}=\begin{cases}\text{Tr}(RR^{t})-\frac{1}{\|\textbf{x}\|^{2}}\textbf{x}^{t}RR^{t}\textbf{x}&\text{if}\quad\textbf{x}\neq
0,\\\ \text{Tr}(RR^{t})-R_{3}&\text{if}\quad\textbf{x}=0,\end{cases}$ (5)

where $R_{3}$ is the least eigenvalue of matrix $RR^{t}$, the superscript $t$
stands for the transpose of a matrix.

Figure 1: (a) Quantum correlations as a function of temperature $T$ for the
state of Heisenberg XY model in the absence of $\cal{PT}$-symmetric operation
for the values of $f=1$, $\gamma=0.05$, $B=1.5$, and $J=4.5$. (b) Quantum
correlations versus $J$ at $T=1$. The other parameters are the same as
considered in the figure (a). MIN, Trace MIN, Concurrence, and Bell inequality
are shown in magenta, green, blue, and red curves.

Trace distance-based MIN: The Hilbert-Schmidt norm based measures are not a
faithful measure of quantum correlations due to the local ancilla problem,
which is indicated by Piani Piani2012 . A natural way to circumvent this issue
is to redefine MIN in terms of contractive distance measure. Another alternate
form of MIN is based on trace distance ($l_{1}$ norm) HFan2015 , namely, trace
MIN (T-MIN) which resolves the local ancilla problem HFan2015 . The trace
distance-based MIN is defined as HFan2015

$\mathcal{N}_{1}\mathcal{(\rho)}:=~{}^{\text{max}}_{\Pi^{a}}\|\rho-\Pi^{a}(\rho)\|_{1},$
(6)

where $\|\mathcal{O}\|_{1}=\text{Tr}\sqrt{\mathcal{O}^{\dagger}\mathcal{O}}$
is the trace norm of operator $\mathcal{O}$. Here also, the maximum is taken
over all locally invariant projective measurements. For the considered system,
the closed formula of trace MIN $\mathcal{N}_{1}\mathcal{(\rho)}$ is given as

$\mathcal{N}_{1}\mathcal{(\rho)}=\begin{cases}\frac{\sqrt{\chi_{+}}~{}+~{}\sqrt{\chi_{-}}}{2\|\textbf{x}\|_{1}}&\text{if}\quad\textbf{x}\neq
0,\\\
\text{max}\\{|c_{1}|,|c_{2}|,|c_{3}|\\}&\text{if}\quad\textbf{x}=0,\end{cases}$
(7)

where $\chi_{\pm}~{}=~{}\alpha\pm 2\sqrt{\tilde{\beta}}~{}\|\textbf{x}\|_{1}$,
$\alpha=\|\textbf{c}\|^{2}_{1}~{}\|\textbf{x}\|^{2}_{1}-\sum_{i}c^{2}_{i}x^{2}_{i}$,
$\tilde{\beta}=\sum_{\langle ijk\rangle}x^{2}_{i}c^{2}_{j}c^{2}_{k}$, and
$c_{i}=Tr(\rho(\sigma_{i}\otimes\sigma_{i}))$. Here $|c_{i}|$’s are the
absolute values of $c_{i}$ and the summation runs over cyclic permutation of
$\\{1,2,3\\}$.

##  III The model

In general, thermal fluctuation degrades the quantum property of the system.
To study the role of thermal effects on quantum correlations and teleportation
technique, we consider a pair of spin-1/2 particles coupled by Heisenberg
interaction. The Hamiltonian of the XY Heisenberg spin model is given as

$\displaystyle\mathcal{H}=\frac{1}{2}\left[J((1+\gamma)\sigma^{x}_{1}\sigma^{x}_{2}+(1-\gamma)\sigma^{y}_{1}\sigma^{y}_{2})+B(\sigma^{z}_{1}+\sigma^{z}_{2})\right]$
(8)

where $J$ is exchange coupling between the spins, $\gamma$ represents the
anisotropy in $xy-$plane, $\sigma_{i}^{\delta}~{}(i=1,2;~{}\delta=x,y,z)$ are
the usual Pauli’s spin matrices and $B$ represents the external magnetic field
along the z direction.

Figure 2: (a) Fidelity of teleportation versus $T$ for different values of
$\gamma=-0.01$ (cyan curve),$-0.5$ (orange curve), $0.05$ (green curve), $1.0$
(blue curve) and for the magnetic field $B=1.5$ in the absence of non-unitary
time evolution (i.e. $\phi=0$, $t=0$). (b) Thermal teleportation as a function
of temperature $T$ and the magnetic field $B$ is plotted in color code. Other
parameters are $f=1$, $\gamma=0.05$, and $J=4.5$.

In the standard two-qubit computational basis
$\\{|00\rangle,|01\rangle,|10\rangle,|11\rangle\\}$, the eigenvalues and the
corresponding eigenvectors of the Hamiltonian are computed as

$\displaystyle E_{1,2}=\pm
J,~{}~{}~{}|E_{1,2}\rangle=\frac{1}{\sqrt{2}}\left(|01\rangle\pm|10\rangle\right),$
(9) $\displaystyle
E_{3,4}=\pm\sqrt{\eta},~{}~{}~{}|E_{3,4}\rangle=N_{\pm}\left[(B\pm\sqrt{\eta})|00\rangle+J\gamma|11\rangle\right],$
(10)

where $\eta=B^{2}+(J\gamma)^{2}$ and
$N_{\pm}=[(B\pm\sqrt{\eta})^{2}+(J\gamma)^{2}]^{-1/2}$ are the normalization
constants. At thermal equilibrium, the thermal density operator is defined as

$\displaystyle\rho(T)=\frac{1}{\mathcal{Z}}\exp{\left(-\beta\mathcal{H}\right)}=\frac{1}{\mathcal{Z}}\sum_{i=1}^{4}\varepsilon_{i}|E_{i}\rangle\langle
E_{i}|$ (11)

where ${\mathcal{Z}}=\text{Tr}(\exp\left(-\beta\mathcal{H}\right))$ being the
canonical ensemble partition function of the system,
$\varepsilon_{i}/\mathcal{Z}$ are the eigenvalues of $\rho(T)$ and the inverse
thermodynamic temperature $\beta=1/(k_{B}T)$ wherein $k_{B}$ is the
Boltzmann’s constant which is considered as unity in the following for
simplicity. The thermal density matrix $\rho(T)$ can be written as

$\displaystyle\rho(T)=\frac{1}{\mathcal{Z}}\begin{pmatrix}\mu_{-}&0&0&\nu\\\
0&\kappa&\omega&0\\\ 0&\omega&\kappa&0\\\ \nu&0&0&\mu_{+}\end{pmatrix}.$ (12)

where the matrix elements are
$\mu_{\pm}=\text{cosh}(\beta\sqrt{\eta})\pm\frac{B}{\sqrt{\eta}}\text{sinh}(\beta\sqrt{\eta})$,
$\kappa=\text{cosh}(J\beta)$, $\omega=-\text{sinh}(J\beta)$,
$\nu=-\frac{J\gamma}{\sqrt{\eta}}\text{sinh}(\beta\sqrt{\eta})$ and the
partition function
${\mathcal{Z}}=2(\text{cosh}(\beta\sqrt{\eta})+\text{cosh}(J\beta))$.

##  IV Results and Discussions

###  IV.1 Thermal Quantum Correlations and Teleportation

In the following, we study the behaviors of thermal quantum correlation of the
Heisenberg spin model in terms of entanglement, Bell nonlocality, and
measurement-induced nonlocality (MIN). Fig. 1(a) shows the above-mentioned
four correlation measures as a function of the equilibrium temperature $T$. At
$T=0$, all the correlation measures are maximum and the state is maximally
entangled, which is understood from the spectra of the Hamiltonian. As the
temperature increases, all the correlation measures decrease gradually which
is mainly due to the influence of thermal fluctuations in the system. At a
sufficiently high temperature $(T=5)$, $\mathcal{C}(\rho)$ only vanishes, and
the other measures (MIN) have nonzero values. To be specific, we observe that
at temperature $T\approx 2$, the entanglement $\mathcal{C}(\rho)$ is non-zero,
and $\mathcal{B}(\rho)$ is less than $2$ which indicates that the quantum
state has no nonclassical correlation in terms of the Bell nonlocality.
Similarly, at $T\approx 5$, the entanglement is zero. On the other hand, both
the MIN and trace MIN decrease monotonically with temperature and show more
robustness against thermal agitation compared to other measures.

In order to understand the role of coupling strength on the correlation
between the spins, we have plotted the different correlation measures as a
function of $J$ in Fig. 1(b). From this, we observe that the entanglement is
zero in the ferromagnetic phase and nonzero in the anti-ferromagnetic phase.
Meanwhile, the thermal state has zero entanglements in the region of $0\leq
J\leq 0.9$ and for all the negative values of J. The maximal Bell function
having a value less than 2 also ensures the absence of nonlocal attributes in
the ferromagnetic phase. Nevertheless, MINs capture nonzero correlation
between the spins even in the ferromagnetic region ($J>0$), implying that the
measures may signal phase transition between the ferro- and antiferro-magnetic
phases. Similar results were observed in the antiferro-magnetic phases. On
further increasing the value of $J>0.9$, one can reach the maximum value of
entanglement and Bell’s inequality which implies that the states are entangled
in the anti-ferromagnetic phase. In addition, we note that the anisotropy
parameter $\gamma$ also strengthens the quantum correlations.

Next, we study the teleportation of any arbitrary state
$\rho_{in}=|\Upsilon\rangle\langle\Upsilon|$ in the Heisenberg XY spin system
through Pauli’s channels.

Figure 3:  Quantum correlations with respect to time $t$ for the state of
Heisenberg XY model in the presence of $\cal{PT}$-symmetric operation for the
values of $T=1$, $f=1$, $\gamma=0.05$, $B=1.5$, and $J=4.5$. (a) MIN and (b)
TMIN, (c) Bell inequality, and (d) Concurrence versus time. Solid line, dash-
dotted line, and dashed line are plotted for the values of
$\phi=\frac{\pi}{3}$, $\phi=\frac{\pi}{4}$, and $\phi=\frac{\pi}{6}$,
respectively.

To characterize the efficiency of the teleportation, we employ the fidelity
fidelformula1 ; fidelformula2 ; fidelformula3  between $\rho_{out}$ and
$\rho_{in}$ defined by

$\mathcal{F}(\rho)=\left\\{\text{Tr}\left(\sqrt{(\rho_{in})^{1/2}\rho_{out}(\rho_{in})^{1/2}}\right)\right\\}^{2}$
(13)

The output state is then given by

$\rho_{out}=\sum_{mn}p_{mn}(\sigma_{m}\otimes\sigma_{n})\rho_{in}(\sigma_{m}\otimes\sigma_{n})$
(14)

where $\sigma_{m,n}(m,n=0,1,2,3)$ signify the unit matrix and the components
of the spin vectors $\vec{\sigma}$, respectively,
$p_{mn}=tr[\mathcal{E}^{m}\rho(T)]tr[\mathcal{E}^{n}\rho(T)]$, and
$\sum_{mn}p_{mn}=1$. Here $\mathcal{E}^{0}=|\Psi^{-}\rangle\langle\Psi^{-}|$,
$\mathcal{E}^{1}=|\Phi^{-}\rangle\langle\Phi^{-}|$,
$\mathcal{E}^{2}=|\Phi^{+}\rangle\langle\Phi^{+}|$,
$\mathcal{E}^{3}=|\Psi^{+}\rangle\langle\Psi^{+}|$, in which
$|\Psi^{\pm}\rangle=(1/\sqrt{2})(|01\rangle\pm|10\rangle)$,
$|\Phi^{\pm}\rangle=(1/\sqrt{2})(|00\rangle\pm|11\rangle)$.

In order to evaluate the above, we consider a pure and maximally entangled
state $|\Upsilon\rangle=(|00\rangle+|11\rangle)/\sqrt{2}$ as an input for this
process. The fidelity of teleportation for the above considered initial state
and output state is displayed in Fig. 2 as a function of temperature and
magnetic field. The fidelity of thermal teleportation is maximum at near
$T=0$. As the temperature increases, the fidelity decreases monotonically and
reaches a non-zero steady state value even at high temperature as shown in
Fig. 2(a). The functional behaviors of the teleportation fidelity are exactly
the same for the values $\gamma=-0.01$, and $\gamma=0.05$ and are different
for the values $\gamma=-0.5$, and $\gamma=1.0$. More specifically, the maximum
value of teleportation fidelity decreases well for the value of $\gamma=1.0$
as compared to the other parametric values of anisotropy. The most interesting
fact that we observed here is that entanglement is the ultimate resource for
teleportation. Further, Guo-Feng Zhang has observed that in the Heisenberg
spin model teleportation is not possible where the entanglement is zero DZ .
Interestingly, we found that the fidelity is non-zero even in the absence of
entanglement. The next question arises, how does the maximum fidelity of
thermal teleportation gets affected when the entangled states are nonlocal in
the presence of external magnetic field $B$? To elucidate this, the fidelity
of thermal teleportation is presented in Fig. 2(b) as a function of $B$ and
$T$. In Fig. 2(b) we have red colour regions where the fidelity is maximum and
dark blue colour regions where the fidelity is minimum. For any value of $B$,
one can achieve thermal teleportation through the considered channels with
nonzero fidelity.

Figure 4:  Quantum correlations with respect to time $t$ for the state of
Heisenberg XY model in the presence of $\cal{PT}$-symmetric operation for
$\phi=\frac{\pi}{3}$. (a) at $T=4$ and (b) at $T=6$. MIN, Trace MIN,
Concurrence, and Bell inequality are shown in Magenta, green, Blue, and Red
curves. Other parametric values are the same as considered in Fig. 3.

##  V Role of $\cal{PT}$-Symmetry

In the previous section, we have observed that the thermal fluctuation
degrades the quantum correlation and efficiency of the teleportation protocol.
To minimize the thermal effects on measures, we evolve the thermal state
$\rho(T)$ under non-unitary operation. To investigate the dynamical properties
of different quantum correlation measures under non-unitary evolution, we
consider the $\cal{PT}$-symmetric non-Hermitian Hamiltonian as

$\displaystyle H=\begin{pmatrix}if\sin\phi&f\\\ f&-if\sin\phi\\\
\end{pmatrix}$ (15)

where $f$ is a real number with -$\frac{\pi}{2}\leq\phi\leq\frac{\pi}{2}$.
Note that, $\phi=\frac{\pi}{2}$ is the $\cal{PT}$-symmetry breaking point.
When $\phi=0$, the Hamiltonian $H$ is Hermitian and the $\cal{PT}$-symmetric
operation turns into local unitary operation. Based on the non-Hermitian
quantum theory, when -$\frac{\pi}{2}<\phi<\frac{\pi}{2}$, the time-evolution
operator $U(t)=e^{-iHt}$ has the following form

$\displaystyle
U(t)=\frac{1}{\cos\phi}\begin{pmatrix}\cos(\psi-\phi)&-i\sin\psi\\\
-i\sin\psi&\cos(\psi+\phi)\\\ \end{pmatrix}$ (16)

with $\psi=ft\cos\phi$. Here, the above considered non-Hermitian system acts
as the local environment for the subsystem $a$, and the global state $\rho(T)$
evolves under the nonunitary evolution.

Using the time evolution operator given in Eq. (16), the final state of the
system can be obtained by

$\rho^{{}^{\prime}}(t)=\frac{(U(t)\otimes\mathds{1})\rho(T)(U(t)^{\dagger}\otimes\mathds{1})}{Tr[(U(t)\otimes\mathds{1})\rho(T)(U(t)^{\dagger}\otimes\mathds{1})]}$
(17)

where $\mathds{1}$ is an identity matrix acting on the subsystem $b$ and
$\rho(T)$ is as given in Eq. (12).

Figure 5: (a) Teleportation fidelity $\cal{F}(\rho^{\prime})$ versus time $t$
for the state of Heisenberg XY model under $\cal{PT}$-symmetric operations for
various values of $\phi$, $\phi=\frac{\pi}{3}$ (dash-dotted line),
$\phi=\frac{\pi}{4}$ (solid line), and $\phi=\frac{\pi}{6}$ (dashed line). (b)
$\cal{F}(\rho^{\prime})$ versus $t$ for different values of $\gamma$,
$\gamma=-0.01$ (cyan curve),$-0.5$ (orange curve), $0.05$ (green curve), $1.0$
(blue curve). By fixing the temperature at $T=1$ and other values as
$\phi=\frac{\pi}{6}$, $f=1$, $B=1.5$, and $J=4.5$, we have plotted the
figures.

The elements of the matrix given in Eq. (17) are

$\displaystyle\rho^{{}^{\prime}}_{11}$ $\displaystyle=$
$\displaystyle\sec(\phi)\left(\mu_{-}\cos^{2}(\phi-\psi)+\kappa\sin^{2}(\psi)\right),$
$\displaystyle\rho^{{}^{\prime}}_{13}$ $\displaystyle=$ $\displaystyle
i\sec(\phi)\sin(\psi)(\mu_{-}\cos(\phi-\psi)-k\cos(\phi+\psi)),$
$\displaystyle\rho^{{}^{\prime}}_{33}$ $\displaystyle=$
$\displaystyle\sec(\phi)\left(\mu_{-}\sin^{2}(\psi)+\kappa\cos^{2}(\phi+\psi)\right),$
$\displaystyle\rho^{{}^{\prime}}_{22}$ $\displaystyle=$
$\displaystyle\sec(\phi)\left(\kappa\cos^{2}(\phi-\psi)+\mu_{+}\sin^{2}(\psi)\right),$
$\displaystyle\rho^{{}^{\prime}}_{24}$ $\displaystyle=$ $\displaystyle
i\sec(\phi)\sin(\psi)(\kappa\cos(\phi-\psi)-\mu_{+}\cos(\phi+\psi)),$
$\displaystyle\rho^{{}^{\prime}}_{44}$ $\displaystyle=$
$\displaystyle\sec(\phi)\left(\kappa\sin^{2}(\psi)+\mu_{+}\cos^{2}(\phi+\psi)\right),$
$\displaystyle\rho^{{}^{\prime}}_{12}$ $\displaystyle=$
$\displaystyle-i\sin^{2}(\psi)(\omega-\nu)(\tan(\phi)+\cot(\psi)),$
$\displaystyle\rho^{{}^{\prime}}_{14}$ $\displaystyle=$
$\displaystyle\sec(\phi)\left(\omega\sin^{2}(\psi)+\nu\cos(\phi-\psi)\cos(\phi+\psi)\right),$
$\displaystyle\rho^{{}^{\prime}}_{23}$ $\displaystyle=$
$\displaystyle\sec(\phi)\left(\omega\cos(\phi-\psi)\cos(\phi+\psi)+\nu\sin^{2}(\psi)\right),$
$\displaystyle\rho^{{}^{\prime}}_{34}$ $\displaystyle=$ $\displaystyle
i\sec(\phi)\sin(\psi)(\omega-\nu)\cos(\phi+\psi),$ (18)

with a common denominator factor
$M_{1}=\sec(\phi)(\mu_{-}+2\kappa+\mu_{+})-\tan(\phi)((\mu_{-}+\kappa)\sin(\phi-2\psi)+(\kappa+\mu_{+})\sin(\phi+2\psi)$
for all the matrix elements given above in Eq. (18). We also note that the
above matrix elements satisfy the relation
$\rho^{{}^{\prime}}_{ij}=\rho^{{}^{\prime}*}_{ji}$.

###  V.1 Correlations and teleportation under $\cal{PT}$-symmetric operation

To protect the resources from the thermal effects, we employ the
$\cal{PT}$-symmetry as a tool. The quantum correlation quantifiers such as
MIN, T-MIN, Bell nonlocality and entanglement (quantified by concurrence) for
the considered model after the $\cal{PT}$-symmetric operation are portrayed in
Figs. 3(a)-3(d) for different values of $\phi$ with the same parametric values
($J$, $B$ and $\gamma$) as considered in Fig. 1(a) and 2(a). As seen in Fig.
3, at $t=0$, the correlation measures are maximum. For a fixed temperature
$T=1$, we observe that the evolution of correlation measures under the
$\cal{PT}$-symmetric operation with respect to time $t$ are in periodic
evolution. The frequency of oscillations for all the functions increases
monotonically for different strengths of the non-Hermiticity parameter
$\phi=\frac{\pi}{3},\frac{\pi}{4},\text{and}~{}\frac{\pi}{6}$. However, the
maximum value of the amplitude is the same for all the values of $\phi$.

Further, the violation of Bell’s inequality during the time evolution now
indicates the nonlocality of the considered system. Note that the Bell’s
inequality ceases to violate with respect to temperature before the inclusion
of $\cal{PT}$-symmetric operation. More interestingly, the value of
$\mathcal{C}(\rho^{\prime})$ corresponding to the temperature which is frozen
to zero as shown in the dark blue curves in Fig. 1(a) in the absence of
$\cal{PT}$-symmetric operation now turns to oscillate periodically under the
non-unitary time evolution as depicted in Fig. 3(d). From the results, we
observe that the change of correlation measures is periodic with time $t$ and
it persists for a long enough time. The recovery of Bell’s nonlocality and
concurrence is possible for a certain range of time and thus we get the
maximum value of Bell nonlocality and concurrence by choosing the suitable
value of $t$ under $\cal{PT}$-symmetric operation.

A similar analysis is also done for higher temperatures such as $T=4$ and
$T=6$ and the correlation measures are plotted in Fig. 4. We observe that the
quantum correlation measures exhibit similar dynamics. While comparing the
correlation measures in the absence of $\cal{PT}$-symmetric operation, the
measures are now greatly enhanced due to the non-unitary evolution. Here
again, the entanglement and Bell nonlocality fail to quantify the quantum
correlation at high temperatures. On the other hand, the $\cal{PT}$-symmetry
operation strengthens the quantum correlation measures. It is worth mentioning
that the noise causes sudden death of entanglement pra ; suddendeath . On the
other hand, the dynamics of entanglement is protected under non-unitary
($\cal{PT}$-symmetric) operation. Notably, similar results were observed by
performing a local non-Hermitian on two qubit entangled system enhanceprotect
.

Next, we investigate the role of $\cal{PT}$-symmetry in the teleportation
technique. Fig. 5(a) shows the time evolution of fidelity of teleportation
$\mathcal{F}(\rho^{\prime})$ for different strengths of non-Hermiticity. The
fidelity of teleportation, which shows decaying dynamics in the absence of
$\cal{PT}$-symmetry now changes into periodic oscillations. The maximum value
of fidelity is the same as before and after the operation of
$\cal{PT}$-symmetry. However, as the degree of non-Hermiticy decreases, the
frequency of oscillations increases with the same value of maximum amplitude.
Fig. 5(b) is plotted for $\mathcal{F}(\rho^{\prime})$ with different values of
$\gamma$. The maximum value of fidelity increases with the lower value of
$\gamma$ mainly, say $\gamma=-0.01$, as shown in the cyan colour curve in Fig.
5(b). The oscillation pattern is the same in terms of amplitude and frequency
as observed for the value of $\gamma=0.05$, as shown in green colour curves in
Fig. 5(b). Conversely, the maximum fidelity decreases with the values of
$\gamma=-0.5$ and $\gamma=1.0$ as shown in orange and blue colour curves in
Fig. 5(b). In Fig. 6, the fidelity under $\cal{PT}$-symmetric operation is
observed for higher temperatures $T=4$ and $T=6$, which exhibits the same
behaviour as shown in the case of low temperature $T=1$. From this
investigation, we have found that performing non-unitary time evolution on
thermally entangled states is useful for the revival of the teleportation
process with maximum fidelity at certain values of $t$.

Figure 6: Teleportation fidelity $\cal{F(\rho^{\prime})}$ for the state of
Heisenberg XY model under $\cal{PT}$-symmetric operations with respect to time
$t$ for various values of $\phi$, $\phi=\frac{\pi}{3}$ (dash-dotted line),
$\phi=\frac{\pi}{4}$ (solid line), and $\phi=\frac{\pi}{6}$ (dashed line). (a)
for $T=4$, and (b) for $T=6$. The figures are plotted for $f=1$, $B=1.5$,
$\gamma=0.05$, and $J=4.5$.

##  VI Conclusions

In this paper, we have examined the quantum correlation measures and
teleportation techniques in the Heisenberg spin model with and without
$\cal{PT}$-symmetry. We have demonstrated the ability of different criteria to
capture the nonlocal attributes of the Heisenberg spin model. We captured the
entire spectrum of nonlocal aspects of the system in terms of measurement-
induced nonlocality even in the absence of entanglement. This result also
suggests that the absence of entanglement does not indicate the absence of
nonlocal properties of the quantum system. Moreover, we have implemented the
teleportation protocol in the spin system successfully even in the absence of
entanglement.

We have also studied the behavior of quantum correlation and quantum
teleportation under non-unitary evolution. In the light of our results, we
show that $\cal{PT}$-symmetry greatly enhances the quantum correlations and
efficiency of teleportation techniques which have potential applications in
quantum communication. Further, our exploitation highlights that efficient
information processing based on the measurement-induced nonlocality offer more
resistance to non-unitary time evolution and are completely distinguishable
from that of entanglement.

## Acknowledgements

J.R.P. thanks the Department of Science and Technology, Government of India,
for providing an INSPIRE Fellowship No. DST/INSPIRE Fellowship/2017/IF170539.
R.M. is grateful for the CTU Global Postdoc Fellowship Program and the
financial support from MŠMT RVO 14000. The work of V.K.C. forms part of the
research projects sponsored by SERB-DST-MATRICS Grant No. MTR/2018/000676 and
CSIR Project Grant No. 03(1444)/18/EMR-II. J.R.P., and V.K.C. also wish to
thank DST, New Delhi for computational facilities under the DST-FIST programme
(SR/FST/PS- 1/2020/135) to the Department of Physics. The work of M.L. is
supported by a DST-SERB National Science Chair position (NSC/2020/000029).

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