#  Discord-type quantum correlations in axially symmetric spin-(1/2, $S$)
systems

Saeed Haddadi<EMAIL_ADDRESS>Faculty of Physics, Semnan University,
P.O.Box 35195-363, Semnan, Iran  M. A. Yurischev<EMAIL_ADDRESS>Federal
Research Center of Problems of Chemical Physics and Medicinal Chemistry,
Russian Academy of Sciences, Chernogolovka 142432, Moscow Region, Russia

###### Abstract

A mixed spin-$(1/2,S)$ system with arbitrary spin $S$ and interactions
satisfying the U(1) axial symmetry is considered. Compact explicit forms of
the local quantum uncertainty (LQU) and local quantum Fisher information
(LQFI) are obtained directly through the entries and eigenvalues of the
density matrix. Using these forms, we perform a comparative study of discord-
type quantum correlations LQU and LQFI for the system at thermal equilibrium.
An increase in quantum correlations with increasing spin length $S$ is
discovered. Moreover, we find that as the system cools, quantum correlations
can undergo one or more abrupt transitions while the temperature changes
smoothly.

Quantum Fisher information, Wigner-Yanase skew information, Qubit-qudit
system, Axial U(1) symmetry, Sudden transitions

###### pacs:

03.65.Aa, 03.65.Ud, 03.67.-a, 75.10.Jm

##  I Introduction

Quantum information technologies are rapidly entering all aspects of human
life. The current state of quantum computing is called the Noisy Intermediate-
Scale Quantum (NISQ) era [1, 2, 3]. Today’s quantum computers are limited to
hundreds of qubits and have relatively short coherence times and significant
errors [2, 4]. In general, they are far from universal, fault-tolerant quantum
computers, which are believed to make it possible to effectively simulate any
processes occurring in Nature.

Nevertheless, existing quantum computers can already solve important real-
world problems. In search of the quantum advantage of these devices, NISQ
algorithms have been proposed for applications in various disciplines,
including physics, quantum chemistry, materials science, machine learning,
etc. The goal of such algorithms is to exploit the limited available resources
to perform classically complex tasks.

At the heart of quantum computing lie extraordinary correlations between
subsystems of a quantum-mechanical system. Until the 21st century, the quantum
entanglement was considered to be the only ingredient of quantum properties
leading to such correlations. It manifests itself in the Einstein-Podolsky-
Rosen gedanken (thought) experiment, Bell’s inequality test, quantum
cryptography, superdense coding, teleportation, and so on [5, 6] (see also
review articles [7, 8, 9, 10]).

Quantum entanglement was quantified in 1996, first for pure states [11, 12],
and then for mixed states [13]. According to the accepted definition, the
entanglement of a bipartite pure state is the von Neumann entropy of either of
the two subsystems. The entanglement (of formation) of a bipartite mixed state
is defined as the minimum entanglement of an ensemble over all ensembles
realizing the mixed state; using this definition, Wootters derived an exact
formula for general two-qubit systems [14]. Another important measure of the
quantum entanglement was proposed in 2002 by Vidal and Werner [15]. This
measure, called negativity, originated from the Peres-Horodecki PPT-criterion
[16, 17]. Moreover, many measures have been introduced so far to quantify the
entanglement of bipartite and multipartite systems, each of which is efficient
for specific quantum states [18, 19].

At one time it was believed that quantum entanglement was the only resource of
quantum speedup in quantum computation. However, in 1998, Knill and Laflamme
showed, using the model of deterministic quantum computation with one pure
qubit (DQC1) [20], that the computation can achieve an exponential improvement
in efficiency over classical computers even without containing much
entanglement. In 2000-2001, $\rm\dot{Z}$urek et al. developed the concept of
quantum discord – “a measure of the quantumness of correlations” [21, 22].
Simultaneously and independently, Vedral et al. [23, 24] proposed a measure
for the purely classical correlation, which, after subtracting it from the
total correlation, led to the same amount of quantum correlation as the
discord. Then, Datta et al. [25, 26] calculated discord in the Knill-Laflamme
DQC1 model and proved that it scales with the quantum efficiency, while
entanglement remains vanishingly small throughout the computation. This
attracted a lot of attention to the new measure of quantum correlation [27,
28, 29, 30, 31, 32].

Quantum discord and entanglement are the same for the pure quantum states.
However, discord can exist in separable mixed states, i.e., when quantum
entanglement is identically equal to zero. The set of separable states
possesses a nonzero volume in the whole Hilbert space of a system [33] (it is
a necessary condition for the arising of entanglement sudden death (ESD)
effect [34]), whereas the set of states with zero discord, vise versa, is
negligibly small [35]. This circumstance alone sharply distinguishes discord
from entanglement. Moreover, numerous theoretical and experimental
investigations of different quantum systems have clearly shown that while the
quantum entanglement and discord measure the same thing – the quantum
correlation – but as a matter of fact, discrepancies in quantitative and even
qualitative behavior are large [36, 37, 38]. Discord and entanglement behave
differently even for the simplest mixed states, namely the Werner and Bell-
diagonal ones (see, e.g., [39]). This has led many to talk about entanglement
and discord as different types of quantum correlations.

The subsequent proposals with more and more new measures of quantum
correlations [40, 41] caused a dilemma: should each measure be attributed to
its own correlation, or should it be argued that there is only one quantum
correlation, but the methods of describing it are not adequate? The physicists
community now prefers to talk about entanglement and discord-like quantum
correlations [42]. As is customary for brevity, we will also refer to the
various measures of quantum correlation as “quantum correlations”.
Nevertheless, the quantum correlation is one, but now there are only different
measures for it, which are still imperfect. We would like to have such
measures of quantum correlations that would be useful for estimating the
values of the speedup of quantum computing, the efficiency of quantum heat
engines, etc. Let the measures be different, but the results must be the same.
As, for example, there are various formulations of quantum mechanics which
differ dramatically in mathematical and conceptual overview, yet each one
makes identical predictions for all experimental results [43]. The existing
difficulties only increase interest in quantum correlations because “who
knows, studying the nature of correlations in the world around us might even
help us catch a glimpse of the theory that comes to supersede quantum physics”
[29].

The concept of quantum discord is based on local measurements and optimization
that were applied to the mutual information. This idea was then used for other
physical or information quantities, and to date, a large number of discord-
like measures of quantum correlation have been introduced, such as quantum
deficit, geometric quantum discord, measurement induced disturbance and so on
[29, 40, 41]. The local quantum uncertainty (LQU) and local quantum Fisher
information (LQFI), based respectively on the Wigner-Yanase and quantum Fisher
information, belong to this type of measures [44, 45].

We consider here two-site spin models. They naturally arise after tracing out
all degrees of freedom from a many-body system except for two spins. On the
other hand, there are heterodinuclear complexes with pronounced dimer magnetic
structures. For instance, the crystalline compound
[Ni(dpt)(H2O)Cu(pba)]$\cdot$2H2O, where dpt=bis-(3-aminopropyl)amine and
pba=1,3-propylenebis(oxamato), is regarded as the $S=1/2$ and 1 ferrimagnetic
dimer [46]. The copper(II)-chromium(III) complex [Cu(oxpn)Cr(5-Cl-
phen)2](NO3)3 with N,N′-bis(3-aminopropyl)oxamide and 5-chloro-1,
10-phenanthroline consists of (1/2, 3/2) clusters with ferromagnitic spin-
exchange couplings [47]. A hybrid system $(1/2,S)$ with $S=1$ is realized in
deuterium atoms in which hyperfine splitting is caused by the interaction of
nuclear and electron magnetic moments [48, 49, 50]. Besides, spin-orbit
coupling of electron spin with its orbital angular momentum leads to the
$(1/2,S)$ system in which “spin” $S$ plays a role of the orbital angular
momentum (here a qubit-qudit state is realized on one particle). There are
also specific compounds containing a stable isotope, such as 169Tm, with
nuclear spin 1/2 and isotopes 17O or 73Ge, whose nuclear spins are 5/2 and
9/2, respectively. Another example is the hydroxyl radicals $\cdot$OH, which
are very stable in the form of a rarefied gas.

Nonclassical correlations in various special cases of the mixed spin-$(1/2,S)$
model were considered in a number of works. For example, the quantum
entanglement, quantified by negativity, was investigated in Refs. [51, 52, 53,
54, 55]. Recently, two studies have been carried out on quantum correlations
in hybrid qubit-qutrit systems using negativity, as well as LQU and LQFI [56,
57]. In the paper [58], we provided exact formulas of LQU and LQFI for general
qubit-qubit X states. Extending this approach to arbitrary spin $S$, we
evaluate here the exact analytical expressions for LQU and LQFI in the case of
$(1/2,S)$ axisymmetric states and examine the behavior of thermal quantum
correlations for different choices of Hamiltonian parameters.

The present paper is structured as follows. In the next section, we establish
the general form of the axially symmetric Hamiltonian and density matrix.
Section III is devoted to the diagonalization of the density matrix. In
Section IV, we derive formulas suitable for actually calculating the LQU and
LQFI correlations. The spin-(1/2,$S$) system at thermal equilibrium is
considered in Sect. V. Section VI includes the numerical results and
discussion. Finally, concluding remarks close this paper.

##  II Hamiltonian and density matrix

We consider a U(1)-invariant spin system composed of two particles with spins
1/2 and $S\ (=1/2,\,1,\,3/2,\ldots)$. This means that the Hamiltonian of the
system commutes with the $z$-component of total spin, given by

${\cal S}_{z}=s_{z}\otimes I_{2S+1}+I_{2}\otimes S_{z}={\rm
diag}\,[1/2+S,1/2+S-1,\ldots,1/2-S,-1/2+S,\ldots,-1/2-S+1,-1/2-S],$ (1)

where $I_{2}$ and $I_{2S+1}$ are the identity operators of the second and
($2S+1$)-th orders, respectively. The matrix (1) of $2(2S+1)$-th order is
diagonal, and let its eigenvalues be numbered from 0 to $4S+1$. Except for the
extreme eigenvalues $\pm(1/2+S)$, the remaining $4S$ internal eigenvalues of
the matrix (1) are two-fold degenerate: the $k$-th and $(2S+k)$-th ones are
equal to each other ($k=1,\ldots,2S$).

The most general matrix which commutes with the $z$-component of total spin
(1) can be written as

${\cal A}=\left(\begin{array}[]{cccccccccc}a_{00}&&&&&&&&&\\\
&a_{11}&0&\ldots&0&a_{1,2S+1}&0&\ldots&0&\\\
&0&a_{22}&\ldots&0&0&a_{2,2S+2}&\ldots&0&\\\
&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots&\ddots&\vdots&\\\
&0&0&\ldots&a_{2S,2S}&0&0&\ldots&a_{2S,4S}&\\\
&a_{2S+1,1}&0&\ldots&0&a_{2S+1,2S+1}&0&\ldots&0&\\\
&0&a_{2S+2,2}&\ldots&0&0&a_{2S+2,2S+2}&\ldots&0&\\\
&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots&\ddots&\vdots&\\\
&0&0&\ldots&a_{4S,2S}&0&0&\ldots&a_{4S,4S}&\\\
&&&&&&&&&a_{4S+1,4S+1}\end{array}\right).\ $ (2)

This matrix has a block-diagonal structure $(1\times 1)\oplus(4S\times
4S)\oplus(1\times 1)$, where the interior $4S\times 4S$ subblock is sparse,
and its nonzero entries are located only on the main diagonal and two sub-
diagonals. Remarkably, the set of matrices of such a type are algebraically
closed: their sums and products preserve the same form. Below, for brevity,
such matrices generated by the axial symmetry will be called $\cal A$
matrices.

If the matrix (2) is Hermitian, as in the case of Hamiltonians or density
matrices, it may have $2(4S+1)$ real independent parameters. Expansion of the
matrix ${\cal H}$ into the spin-1/2 and spin-$S$ matrices and their products
and powers allows us to clarify the structure and physical meaning of
interactions in the system (see below).

For a given Hamiltonian $\cal H$ in the $\cal A$ form, the density matrix
$\rho$ can be obtained from solving some master equations. Let the initial
state has the $\cal A$ structure and the dynamics of the system is governed,
for example, by the quantum Liouville equation. Then, the $\cal A$ form of the
density matrix will be preserved throughout the entire evolution process due
to the above-mentioned property of the algebraic closedness of $\cal A$
matrices. The $\cal A$-form of the density matrix will also be realized for
systems in thermal equilibrium, since $\rho$ and $\cal H$ are related by
functional dependence. In these and similar cases, the density matrix under
examination can be written as

$\rho=\left(\begin{array}[]{clccccccrc}p_{0}&&&&&&&&&\\\
&a_{1}&0&\ldots&0&u_{1}&0&\ldots&0&\\\ &0&a_{2}&\ldots&0&0&u_{2}&\ldots&0&\\\
&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots&\ddots&\vdots&\\\
&0&0&\ldots&a_{2S}&0&0&\ldots&u_{2S}&\\\
&u_{1}^{*}&0&\ldots&0&a_{2S+1}&0&\ldots&0&\\\
&0&u_{2}^{*}&\ldots&0&0&a_{2S+2}&\ldots&0&\\\
&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots&\ddots&\vdots&\\\
&0&0&\ldots&u_{2S}^{*}&0&0&\ldots&a_{4S}&\\\
&&&&&&&&&p_{4S+1}\end{array}\right).\ $ (3)

Below, we restrict ourselves to these $\cal A$ quantum states.

##  III  Diagonal representation of the density matrix

###  III.1  Diagonalization of the matrix $\rho$

Let us turn to the diagonalization of matrix (3). Since the density matrix
$\rho$ commutes with the diagonal matrix ${\cal S}_{z}$, it has nonzero matrix
elements only between states with the same eigenvalue of ${\cal S}_{z}$. Here,
${\cal S}_{z}$ plays a role of “constant or integral of motion”; in this
connection see, e.g., Chapter 1 in two books [59, 60]. As mentioned above,
${\cal S}_{z}$ has two non-degenerate and $4S$ doubly degenerate eigenvalues.
Therefore, the density matrix (3) actually consists of two one-by-one and $2S$
two-by-two subblocks.111 This will be obvious if we reorder the vectors of the
original basis so that the $k$th and $(2S+k)$th vectors become neighbors, then
the density matrix (3) takes an explicit quasi-diagonal form:
$\hat{\rho}=[p_{0},\left(\begin{array}[]{cc}a_{1}&u_{1}\\\
u_{1}^{*}&a_{2S+1}\\\
\end{array}\right),\ldots,\left(\begin{array}[]{cc}a_{2S}&u_{2S}\\\
u_{2S}^{*}&a_{4S}\\\ \end{array}\right),p_{4S+1}].$ This form is implemented
for the system $(S,1/2)$.  The first and $(4S+1)$th “subblocks” have sizes
$1\times 1$, i.e. they are ready eigenvalues. The remaining $2S$ subblocks are
$2\times 2$-dimensional, and their diagonalization is easily accomplished by
directly calculating the eigenvectors of the corresponding subblocks (see, for
example, [61]). Eigenvectors are constructed using the $k$th and $(2S+k)$th
pair of vectors of the original basis. Indeed, a suitable fragment extracted
from the inner $4S\times 4S$ block of the density matrix (3) reads

$\left(\begin{array}[]{ccc}a_{k}&\ldots&u_{k}\\\ \vdots&\ddots&\vdots\\\
u_{k}^{*}&\ldots&a_{2S+k}\\\ \end{array}\right),\qquad k=1,\ldots,2S$ (4)

and then the corresponding eigenvectors are given as

$|k\rangle=\left(\begin{array}[]{c}{\tilde{q}}_{k}\\\ \vdots\\\
{\tilde{u}}_{k}^{*}\\\
\end{array}\right),\qquad|2S+k\rangle=\left(\begin{array}[]{c}{\tilde{u}}_{k}\\\
\vdots\\\ -{\tilde{q}}_{k}\\\ \end{array}\right),\qquad$ (5)

where

${\tilde{q}_{k}}=q_{k}/\sqrt{q_{k}^{2}+|u_{k}|^{2}},\qquad{\tilde{u}_{k}}=u_{k}/\sqrt{q_{k}^{2}+|u_{k}|^{2}},\qquad
q_{k}=\frac{1}{2}\Big{(}a_{k}-a_{2S+k}+\sqrt{(a_{k}-a_{2S+k})^{2}+4|u_{k}|^{2}}\Big{)}.$
(6)

This makes it possible to immediately write down the desired transformation,
which diagonalizes the density matrix (3):

$R=\left(\begin{array}[]{clccccccrc}1&&&&&&&&&\\\
&{\tilde{q}}_{1}&0&\ldots&0&{\tilde{u}}_{1}&0&\ldots&0&\\\
&0&{\tilde{q}}_{2}&\ldots&0&0&{\tilde{u}}_{2}&\ldots&0&\\\
&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots&\ddots&\vdots&\\\
&0&0&\ldots&{\tilde{q}}_{2S}&0&0&\ldots&{\tilde{u}}_{2S}&\\\
&{\tilde{u}}_{1}^{*}&0&\ldots&0&-{\tilde{q}}_{1}&0&\ldots&0&\\\
&0&{\tilde{u}}_{2}^{*}&\ldots&0&0&-{\tilde{q}}_{2}&\ldots&0&\\\
&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots&\ddots&\vdots&\\\
&0&0&\ldots&{\tilde{u}}_{2S}^{*}&0&0&\ldots&-{\tilde{q}}_{2S}&\\\
&&&&&&&&&1\end{array}\right),$ (7)

where ${\tilde{q}}_{k}$ and ${\tilde{u}}_{k}$ ($k=1,\ldots,2S$) are defined by
Eq. (6). Note that $R$ is the $\cal A$ matrix. Moreover, it is clear that
$R^{\dagger}=R$ and $R^{\dagger}R=I$, so this diagonalizing transformation is
both Hermitian and unitary.

Using Eqs. (3) and (7) we find

$\rho^{\prime}\equiv R\rho
R=[p_{0},p_{1},\ldots,p_{2S},p_{2S+1},\ldots,p_{4S},p_{4S+1}],$ (8)

where

$p_{k,2S+k}=\frac{1}{2}\Big{(}a_{k}+a_{2S+k}\pm\sqrt{(a_{k}-a_{2S+k})^{2}+4|u_{k}|^{2}}\Big{)}$
(9)

are the eigenvalues of the density matrix (3). From here and Eq. (6) it
follows that

$q_{k}=p_{k}-a_{2S+k}$ (10)

and

${\tilde{q}}_{k}^{2}=\frac{1}{2}\Big{(}1+\frac{a_{k}-a_{2S+k}}{p_{k}-p_{2S+k}}\Big{)},\qquad{\tilde{u}}_{k}^{2}=\frac{1}{2}\Big{(}1-\frac{a_{k}-a_{2S+k}}{p_{k}-p_{2S+k}}\Big{)},$
(11)

where as before $k=1,\ldots,2S$.

###  III.2  Spin matrices $\sigma_{\mu}\otimes I_{2S+1}$ in
$\rho^{\prime}$-representation

Let us transform the local Pauli matrices $\sigma_{\mu}\otimes I_{2S+1}$
($\mu=x,y,z$) of the particle $A$ into a diagonal representation of the
density operator. Using Eq. (7) we get

$R(\sigma_{x}\otimes
I_{2S+1})R=\left(\begin{array}[]{cccccccccccc}&{\tilde{u}}_{1}^{*}&&&&&-{\tilde{q}}_{1}&&&&&\\\
{\tilde{u}}_{1}&0&{\tilde{q}}_{1}{\tilde{u}}_{2}^{*}&\ldots&0&0&0&-{\tilde{q}}_{1}{\tilde{q}}_{2}&\ldots&0&0&\\\
&{\tilde{q}}_{1}{\tilde{u}}_{2}&0&\ldots&0&0&{\tilde{u}}_{1}{\tilde{u}}_{2}&0&\ldots&0&0&\\\
&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\\\
&0&0&\ldots&0&{\tilde{q}}_{2S-1}{\tilde{u}}_{2S}^{*}&0&0&\ldots&0&-{\tilde{q}}_{2S-1}{\tilde{q}}_{2S}&\\\
&0&0&\ldots&{\tilde{q}}_{2S-1}{\tilde{u}}_{2S}&0&0&0&\ldots&{\tilde{u}}_{2S-1}{\tilde{u}}_{2S}&0&{\tilde{q}}_{2S}\\\
-{\tilde{q}}_{1}&0&{\tilde{u}}_{1}^{*}{\tilde{u}}_{2}^{*}&\ldots&0&0&0&-{\tilde{q}}_{2}{\tilde{u}}_{1}^{*}&\ldots&0&0&\\\
&-{\tilde{q}}_{1}{\tilde{q}}_{2}&0&\ldots&0&0&-{\tilde{q}}_{2}{\tilde{u}}_{1}&0&\ldots&0&0&\\\
&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\\\
&0&0&\ldots&0&{\tilde{u}}_{2S-1}^{*}{\tilde{u}}_{2S}^{*}&0&0&\ldots&0&-{\tilde{q}}_{2S}{\tilde{u}}_{2S-1}^{*}&\\\
&0&0&\ldots&-{\tilde{q}}_{2S-1}{\tilde{q}}_{2S}&0&0&0&\ldots&-{\tilde{q}}_{2S}{\tilde{u}}_{2S-1}&0&{\tilde{u}}_{2S}^{*}\\\
&&&&&{\tilde{q}}_{2S}&&&&&{\tilde{u}}_{2S}&\end{array}\right),$ (12)

$R(\sigma_{y}\otimes
I_{2S+1})R\\!=\\!\\!\left(\begin{array}[]{cccccccccccc}&-i{\tilde{u}}_{1}^{*}&&&&&i{\tilde{q}}_{1}&&&&&\\\
i{\tilde{u}}_{1}&0&-i{\tilde{q}}_{1}{\tilde{u}}_{2}^{*}&\ldots&0&0&0&i{\tilde{q}}_{1}{\tilde{q}}_{2}&\ldots&0&0&\\\
&i{\tilde{q}}_{1}{\tilde{u}}_{2}&0&\ldots&0&0&i{\tilde{u}}_{1}{\tilde{u}}_{2}&0&\ldots&0&0&\\\
&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\\\
&0&0&\ldots&0&-i{\tilde{q}}_{2S-1}{\tilde{u}}_{2S}^{*}&0&0&\ldots&0&i{\tilde{q}}_{2S-1}{\tilde{q}}_{2S}&\\\
&0&0&\ldots&i{\tilde{q}}_{2S-1}{\tilde{u}}_{2S}&0&0&0&\ldots&i{\tilde{u}}_{2S-1}{\tilde{u}}_{2S}&0&-i{\tilde{q}}_{2S}\\\
-i{\tilde{q}}_{1}&0&-i{\tilde{u}}_{1}^{*}{\tilde{u}}_{2}^{*}&\ldots&0&0&0&i{\tilde{q}}_{2}{\tilde{u}}_{1}^{*}&\ldots&0&0&\\\
&-i{\tilde{q}}_{1}{\tilde{q}}_{2}&0&\ldots&0&0&-i{\tilde{q}}_{2}{\tilde{u}}_{1}&0&\ldots&0&0&\\\
&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\\\
&0&0&\ldots&0&-i{\tilde{u}}_{2S-1}^{*}{\tilde{u}}_{2S}^{*}&0&0&\ldots&0&i{\tilde{q}}_{2S}{\tilde{u}}_{2S-1}^{*}&\\\
&0&0&\ldots&-i{\tilde{q}}_{2S-1}{\tilde{q}}_{2S}&0&0&0&\ldots&-i{\tilde{q}}_{2S}{\tilde{u}}_{2S-1}&0&-i{\tilde{u}}_{2S}^{*}\\\
&&&&&i{\tilde{q}}_{2S}&&&&&i{\tilde{u}}_{2S}&\end{array}\right)$ (13)

and

$R(\sigma_{z}\otimes I_{2S+1})R=\left(\begin{array}[]{clccccccrc}1&&&&&&&&&\\\
&{\tilde{q}}_{1}^{2}-|{\tilde{u}}_{1}|^{2}&0&\ldots&0&2{\tilde{q}}_{1}{\tilde{u}}_{1}&0&\ldots&0&\\\
&0&{\tilde{q}}_{2}^{2}-|{\tilde{u}}_{2}|^{2}&\ldots&0&0&2{\tilde{q}}_{2}{\tilde{u}}_{2}&\ldots&0&\\\
&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots&\ddots&\vdots&\\\
&0&0&\ldots&{\tilde{q}}_{2S}^{2}-|{\tilde{u}}_{2S}|^{2}&0&0&\ldots&2{\tilde{q}}_{2S}{\tilde{u}}_{2S}&\\\
&2{\tilde{q}}_{1}{\tilde{u}}_{1}^{*}&0&\ldots&0&-{\tilde{q}}_{1}^{2}+|{\tilde{u}}_{1}|^{2}&0&\ldots&0&\\\
&0&2{\tilde{q}}_{2}{\tilde{u}}_{2}^{*}&\ldots&0&0&-{\tilde{q}}_{2}^{2}+|{\tilde{u}}_{2}|^{2}&\ldots&0&\\\
&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots&\ddots&\vdots&\\\
&0&0&\ldots&2{\tilde{q}}_{2S}{\tilde{u}}_{2S}^{*}&0&0&\ldots&-{\tilde{q}}_{2S}^{2}+|{\tilde{u}}_{2S}|^{2}&\\\
&&&&&&&&&-1\end{array}\right).$ (14)

Unlike the first two matrices, the latter has the $\cal A$ form. The entries
$\langle m|\sigma_{\mu}\otimes I_{2S+1}|n\rangle$ of the matrices (12) – (14)
will be needed in the next section.

##  IV  Quantum correlations

###  IV.1  LQU

The total uncertainty of an observable $H$ in the quantum-mechanical state
$\rho$ is usually expressed by the variance

${\rm Var}(\rho,H)=\langle H^{2}\rangle_{\rho}-\langle H\rangle_{\rho}^{2}.$
(15)

On the other hand, the quantum contribution to the total statistical error may
be reliable quantify via the Wigner-Yanase skew information [62, 63, 64]

${\cal I}(\rho,H)=-\frac{1}{2}{\rm Tr}[\sqrt{\rho},H]^{2},$ (16)

where $[.,.]$ denotes the commutator. Notice that the skew information (16) is
not grater than the variance (15), namely

${\cal I}(\rho,H)\leq{\rm Var}(\rho,H),$ (17)

where the equality is achieved for pure states when classical ignorance does
not occur [44]. This makes it possible to introduce a discord-type measure
$\cal U$ (also called LQU) of quantum correlations in any bipartite system
$AB$ as follows [44]

${\cal U}(\rho)=\min_{H_{A}}{\cal I}(\rho,H_{A}),$ (18)

in which the minimum is taken over all local observables $H_{A}$ on the
subsystem $A$. It is worth mentioning that the LQU is a genuine quantifier of
quantum correlations, and it has been shown that the LQU meets all the
physical conditions of a measure of quantum correlations.

Importantly, the authors [44] were able to perform optimization for qubit-
qudit systems and presented the measure (18) in the simple form

${\cal U}=1-\lambda_{max}^{(W)},$ (19)

where $\lambda_{max}^{(W)}$ denotes the maximum eigenvalue of the $3\times 3$
symmetric matrix $W$ whose entries are

$W_{\mu\nu}={\rm Tr}\\{\rho^{1/2}(\sigma_{\mu}\otimes{\rm
I}_{2S+1})\rho^{1/2}(\sigma_{\nu}\otimes{\rm I}_{2S+1})\\}$ (20)

with $\mu,\nu=x,y,z$ and $\sigma_{x,y,z}$ are the Pauli matrices as before.
This opens up the possibility of real calculating LQU.

Using Eqs. (12)–(14), we find that the matrix $W$ is diagonal, $W_{yy}=W_{xx}$
and

$\displaystyle\frac{1}{2}W_{xx}={\tilde{q}}_{1}^{2}\sqrt{p_{0}p_{2S+1}}+{\tilde{q}}_{2S}^{2}\sqrt{p_{2S}p_{4S+1}}+|{\tilde{u}}_{1}|^{2}\sqrt{p_{0}p_{1}}+|{\tilde{u}}_{2S}|^{2}\sqrt{p_{4S}p_{4S+1}}$
$\displaystyle+\sum_{k=1}^{2S-1}\big{[}{\tilde{q}}_{k}^{2}{\tilde{q}}_{k+1}^{2}\sqrt{p_{k}p_{2S+1+k}}+{\tilde{q}}_{k}^{2}|{\tilde{u}}_{k+1}|^{2}\sqrt{p_{k}p_{k+1}}+|{\tilde{u}}_{k}|^{2}{\tilde{q}}_{k+1}^{2}\sqrt{p_{2S+k}p_{2S+1+k}}+|{\tilde{u}}_{k}|^{2}|{\tilde{u}}_{k+1}|^{2}\sqrt{p_{k+1}p_{2S+k}}\big{]}$
(21)

and

$W_{zz}=p_{0}+p_{4S+1}+\sum_{k=1}^{2S}\big{[}(p_{k}+p_{2S+k})({\tilde{q}}_{k}^{2}-|{\tilde{u}}_{k}|^{2})^{2}+8{\tilde{q}}_{k}^{2}|{\tilde{u}}_{k}|^{2}\sqrt{p_{k}p_{2S+k}}\big{]}.$
(22)

Therefore, LQU would be given by

${\cal U}=\min{\\{{\cal U}_{0},{\cal U}_{1}\\}},$ (23)

where two branches (sub-functions) are ${\cal U}_{0}=1-W_{zz}$ and ${\cal
U}_{1}=1-W_{xx}$.

Substituting expressions (11) into Eqs. (IV.1) and (22), we arrive at the
final formulas for the branches

${\cal
U}_{0}=\sum_{k=1}^{2S}\Bigg{[}(\sqrt{p_{k}}-\sqrt{p_{2S+k}})^{2}-\frac{(a_{k}-a_{2S+k})^{2}}{(\sqrt{p_{k}}+\sqrt{p_{2S+k}})^{2}}\Bigg{]}$
(24)

and

${\cal
U}_{1}=1-2\Bigg{[}\frac{a_{2S+1}+\sqrt{p_{1}p_{2S+1}}}{\sqrt{p_{1}}+\sqrt{p_{2S+1}}}\sqrt{p_{0}}+\frac{a_{2S}+\sqrt{p_{2S}p_{4S}}}{\sqrt{p_{2S}}+\sqrt{p_{4S}}}\sqrt{p_{4S+1}}+\sum_{k=1}^{2S-1}\frac{(a_{k}+\sqrt{p_{k}p_{2S+k}})(a_{2S+1+k}+\sqrt{p_{k+1}p_{2S+1+k}})}{(\sqrt{p_{k}}+\sqrt{p_{2S+k}})(\sqrt{p_{k+1}}+\sqrt{p_{2S+1+k}})}\Bigg{]}.$
(25)

It is remarkable that both quantities ${\cal U}_{0}$ and ${\cal U}_{1}$ are
expressed directly only through the matrix elements of the density matrix (3)
and its eigenvalues, therefore, the same is true for the measure ${\cal U}$.

###  IV.2  LQFI

Quantum Fisher information (QFI) is the cornerstone of quantum estimation
theory [65, 66, 67, 68, 69]. Suppose there is unitary evolution of the quantum
state $\varrho=e^{iH\epsilon}\rho e^{-iH\epsilon}$ with some observable $H$,
and the parameter $\epsilon$ wanted to be estimated. The QFI can be defined as
follows

$F(\varrho,H)=\frac{1}{4}{\rm Tr}(\varrho L_{\epsilon}^{2}),$ (26)

where $L_{\epsilon}$ is the symmetric logarithmic derivative (SLD) operator
satisfying the Lyapunov equation (known from control theory)

$\frac{\partial\varrho}{\partial\epsilon}=\frac{1}{2}(\varrho
L_{\epsilon}+L_{\epsilon}\varrho).$ (27)

Then the accuracy of the estimate $\epsilon$ is limited by the quantum Rao-
Cram$\rm{\acute{e}}$r inequality [66]

$\Delta\epsilon\geq\frac{1}{\sqrt{NF(\varrho,H)}},$ (28)

where $N$ is the number of measurements.

Another application of QFI is to use it as a quantifier of quantum
correlations. Girolami et al. [45] showed that the quantity

${\cal F}(\rho)=\min_{H_{A}}F(\rho,H_{A}),$ (29)

where $H_{A}$ (as in the case of LQU) is a local observable, again satisfies
all the necessary criteria to be defined as a measure of quantum correlations.
Note that here the (local) QFI is minimized, unlike the case of the lower
bound (28). The measure $\cal F$ is called interferometric power [45] or LQFI
[70, 71]. Moreover, if the subsystem $A$ is a qubit, then minimization in (29)
can be done and LQFI is given as [45]

${\cal F}=1-\lambda_{max}^{(M)},$ (30)

where $\lambda_{max}^{(M)}$ is the largest eigenvalue of the real symmetric
$3\times 3$ matrix $M$ with entries [72]

$M_{\mu\nu}=\sum_{{m,n};\ {p_{m}+p_{n}\neq
0}}\frac{2p_{m}p_{n}}{p_{m}+p_{n}}\langle m|\sigma_{\mu}\otimes{\rm
I}_{2S+1}|n\rangle\langle n|\sigma_{\nu}\otimes{\rm I}_{2S+1}|m\rangle.$ (31)

From a programming point of view, the right side of this equation represents
two nested loops.

Again using Eqs. (12)–(14), we find that the matrix $M$ is diagonal,
$M_{yy}=M_{xx}$, and

$\displaystyle\frac{1}{4}M_{xx}$ $\displaystyle=$
$\displaystyle\frac{p_{0}p_{2S+1}}{p_{0}+p_{2S+1}}{\tilde{q}}_{1}^{2}+\frac{p_{2S}p_{4S+1}}{p_{2S}+p_{4S+1}}{\tilde{q}}_{2S}^{2}+\frac{p_{0}p_{1}}{p_{0}+p_{1}}|{\tilde{u}}_{1}|^{2}+\frac{p_{4S}p_{4S+1}}{p_{4S}+p_{4S+1}}|{\tilde{u}}_{2S}|^{2}+\sum_{k=1}^{2S-1}\Big{[}\frac{p_{k}p_{2S+1+k}}{p_{k}+p_{2S+1+k}}{\tilde{q}}_{k}^{2}{\tilde{q}}_{k+1}^{2}$
(32) $\displaystyle+$
$\displaystyle\frac{p_{k}p_{k+1}}{p_{k}+p_{k+1}}{\tilde{q}}_{k}^{2}|{\tilde{u}}_{k+1}|^{2}+\frac{p_{2S+k}p_{2S+1+k}}{p_{2S+k}+p_{2S+1+k}}|{\tilde{u}}_{k}|^{2}{\tilde{q}}_{k+1}^{2}+\frac{p_{k+1}p_{2S+k}}{p_{k+1}+p_{2S+k}}|{\tilde{u}}_{k}|^{2}|{\tilde{u}}_{k+1}|^{2}\Big{]}$

and

$M_{zz}=p_{0}+p_{4S+1}+\sum_{k=1}^{2S}\big{[}(p_{k}+p_{2S+k})({\tilde{q}}_{k}^{2}-|{\tilde{u}}_{k}|^{2})^{2}+16\frac{p_{k}p_{2S+k}}{p_{k}+p_{2S+k}}{\tilde{q}}_{k}^{2}|{\tilde{u}}_{k}|^{2}\big{]}.$
(33)

It is interesting to draw our attention to the analogy in the structures, on
the one hand, between Eqs. (IV.1) and (32) and, on the other hand, between
Eqs. (22) and (33).

As with LQU, this makes it possible to actually calculate LQFI:

${\cal F}=\min{\\{{\cal F}_{0},{\cal F}_{1}\\}},$ (34)

where the branches (sub-functions) are equal to ${\cal F}_{0}=1-M_{zz}$ and
${\cal F}_{1}=1-M_{xx}$.

Substituting (11) into Eqs. (32) and (33), we come to a nice formula for the
${\cal F}_{0}$ branch

${\cal F}_{0}=4\sum_{k=1}^{2S}\frac{|u_{k}|^{2}}{a_{k}+a_{2S+k}}.$ (35)

Similarly, the final formula for the second branch ${\cal F}_{1}$ can be
written as

$\displaystyle{\cal
F}_{1}=1-4\Bigg{\\{}\frac{p_{0}(a_{2S+1}p_{0}+p_{1}p_{2S+1})}{(p_{0}+p_{1})(p_{0}+p_{2S+1})}+\frac{p_{4S+1}(a_{2S}p_{4S+1}+p_{2S}p_{4S})}{(p_{2S}+p_{4S+1})(p_{4S}+p_{4S+1})}+\frac{1}{2}\sum_{k=1}^{2S-1}\Bigg{[}\frac{(a_{k}p_{k+1}+p_{k}p_{2S+k})p_{k+1}}{(p_{k}+p_{k+1})(p_{k+1}+p_{2S+k})}$
$\displaystyle+\frac{a_{k}\\{p_{2S+1+k}[p_{2S+k}(p_{k}+p_{k+1})+p_{k}p_{k+1}]+p_{k}p_{k+1}p_{2S+k}\\}+p_{k}p_{2S+k}(p_{k}p_{2S+k}-p_{k+1}p_{2S+1+k})}{(p_{k}+p_{k+1})(p_{k+1}+p_{2S+k})(p_{2S+k}+p_{2S+1+k})(p_{2S+1+k}+p_{k})}(a_{2S+1+k}-a_{k+1})$
$\displaystyle+\frac{(a_{k}p_{2S+1+k}+p_{k}p_{2S+k})p_{2S+1+k}}{(p_{k}+p_{2S+1+k})(p_{2S+k}+p_{2S+1+k})}\Bigg{]}\Bigg{\\}}.$
(36)

The resulting formulas (24), (25), (35) and (IV.2) for the branches of quantum
correlations are surprisingly simple, compact and very comfortable to perform
quick calculations on a computer.

##  V  System at thermal equilibrium

Let’s now turn to the consideration of a spin system in the state of thermal
equilibrium with a reservoir at the temperature $T$.

###  V.1  Spin Hamiltonian and Gibbs density matrix

We take the Hamiltonian in the Hermitian $\cal A$ form

${\cal H}=\left(\begin{array}[]{clccccccrc}E_{0}&&&&&&&&&\\\
&h_{1}&0&\ldots&0&g_{1}&0&\ldots&0&\\\ &0&h_{2}&\ldots&0&0&g_{2}&\ldots&0&\\\
&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots&\ddots&\vdots&\\\
&0&0&\ldots&h_{2S}&0&0&\ldots&g_{2S}&\\\
&g_{1}^{*}&0&\ldots&0&h_{2S+1}&0&\ldots&0&\\\
&0&g_{2}^{*}&\ldots&0&0&h_{2S+2}&\ldots&0&\\\
&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots&\ddots&\vdots&\\\
&0&0&\ldots&g_{2S}^{*}&0&0&\ldots&h_{4S}&\\\
&&&&&&&&&E_{4S+1}\end{array}\right).\ $ (37)

The energy levels of the system are obviously $E_{0}$, $E_{4S+1}$ and $2S$
additional pairs

$E_{k,2S+k}=\frac{1}{2}\big{(}h_{k}+h_{2S+k}\pm R_{k}\big{)},$ (38)

where

$R_{k}=\sqrt{(h_{k}-h_{2S+k})^{2}+4|g_{k}|^{2}}$ (39)

and $k=1,\ldots,2S$.

The partition function $Z=\sum_{n}\exp(-E_{n}/T)$ of the system is given by

$Z=e^{-E_{0}/T}+e^{-E_{4S+1}/T}+2\sum_{k=1}^{2S}\cosh{\frac{R_{k}}{2T}}e^{-(h_{k}+h_{2S+k})/2T}.$
(40)

Statistical weights, i.e. density matrix eigenvalues (9), would be obtained as

$p_{0}=\frac{1}{Z}e^{-E_{0}/T},\qquad p_{k}=\frac{1}{Z}e^{-E_{k}/T},\qquad
p_{2S+k}=\frac{1}{Z}e^{-E_{2S+k}/T},\qquad
p_{4S+1}=\frac{1}{Z}e^{-E_{4S+1}/T}.$ (41)

The Gibbs density matrix is equal to

$\rho=\frac{1}{Z}\exp(-{\cal H}/T).$ (42)

Since $\cal H$ has the $\cal A$ form, this functional relationship also leads
to the density matrix of the same structure, Eq. (3). If the dimension of the
matrix $\cal H$ is finite, $\rho$ can be found manually or obtained on a
digital machine by direct symbolic (analytical) calculations. After this, the
results can be extended by induction to systems with arbitrary spin $S$. This
allows us to get all non-zero matrix elements of the inner block of the
density matrix (3) through the parameters of Hamiltonian:

$\displaystyle a_{k,2S+k}$ $\displaystyle=$
$\displaystyle\frac{1}{Z}\Big{(}\cosh{\frac{R_{k}}{2T}}\pm\frac{h_{2S+k}-h_{k}}{R_{k}}\sinh{\frac{R_{k}}{2T}}\Big{)}e^{-(h_{k}+h_{2S+k})/2T},$
(43) $\displaystyle
u_{k}=-\frac{2g_{k}}{ZR_{k}}\sinh{\frac{R_{k}}{2T}}e^{-(h_{k}+h_{2S+k})/2T}$

with $k=1,\ldots,2S$. In turn, these expressions open up an alternative
opportunity to find the eigenvalues of the Gibbs density matrix using Eq. (9).

###  V.2  Hamiltonian and interaction constants

Expanding the Hermitian matrix (37) into generalized Gell-Mann matrices and
then using the connections of these matrices with the spin ones, we obtain an
expression for the Hamiltonian with $2(4S+1)$ terms of the spin matrices,
their powers and products. Instead of this formal approach, one can also
construct the same form of the Hamiltonian by adding linear independent spin
terms with axial symmetry. In both cases, we come to the representation of the
Hamiltonian (37) in the form

${\cal
H}=B_{1}s_{z}+B_{2}S_{z}+J(s_{x}S_{x}+s_{y}S_{y})+J_{z}s_{z}S_{z}+D_{z}(s_{x}S_{y}-s_{y}S_{x})+K_{1}S_{z}^{2}+K_{2}s_{z}S_{z}^{2}+\ldots.$
(44)

We use a notation for the expanding coefficients which is suitable for the
magnetic compounds, and will below restrict ourselves to seven parameters,
namely, $B_{1}$ and $B_{2}$ are the $z$-components of an external magnetic
field applied to spins 1/2 and $S$, respectively; $J$ and $J_{z}$ are the
exchange Heisenberg constants; $K_{1}$ is the uniaxial one-ion anisotropy;
$K_{2}$ is the uniaxial two-ion anisotropy; $D_{z}$ is the $z$-component of
Dzyaloshinskii vector. It is interesting to note that the antisymmetric
Dzyaloshinskii-Moriya (DM) interaction satisfies axial symmetry, while the
symmetric combination $s_{x}S_{y}+s_{y}S_{x}$, which corresponds to the so-
called Kaplan–Shekhtman–Entin-Wohlman–Aharony (KSEA) interaction, does not
satisfy. Finally, if necessary, the Hamiltonian (44) can be extended by adding
new axially invariant terms.

In Eq. (44), $s_{\alpha}=\sigma_{\alpha}/2$, where $\sigma_{\alpha}$
($\alpha=x,y,z$) are again the Pauli matrices, and (setting $\hbar=1$)

$S_{z}={\rm diag}\,[S,S-1,\ldots,-S],$ (45)
$S_{x}=\left(\begin{array}[]{ccccccc}&b_{S}&&&&&\\\ b_{S}&&b_{S-1}&&&&\\\
&b_{S-1}&&b_{S-2}&&&\\\ &&&&&&\\\ &&&&&b_{-S+1}\\\
&&&&b_{-S+1}&&\end{array}\right),$ (46)
$S_{y}=\left(\begin{array}[]{ccccccc}&-ib_{S}&&&&&\\\ ib_{S}&&-ib_{S-1}&&&&\\\
&ib_{S-1}&&-ib_{S-2}&&&\\\ &&&&&&\\\ &&&&&-ib_{-S+1}\\\
&&&&ib_{-S+1}&&\end{array}\right)$ (47)

with $b_{m}=\sqrt{(S+m)(S-m+1)}/2$.

Inserting the Pauli and spin matrices (45) - (47) into Eq. (44) and then
comparing the resulting open form of (44) with the matrix (37), we get
expressions for the quantities $E_{0,4S+1}$, $h_{k,k+2S}$ and $g_{k}$
($k=1,\dots,2S$) through the input parameters of the model, for each desired
spin value $S$. For example, if $S=1$, then

$E_{0,5}=K_{1}+\frac{J_{z}+K_{2}}{2}\pm\Big{(}\frac{B_{1}}{2}+B_{2}\Big{)},\quad
h_{1,4}=\pm\frac{B_{1}}{2},\quad
h_{2,3}=-\frac{J_{z}}{2}+K_{1}\pm\Big{(}\frac{B_{1}+K_{2}}{2}-B_{2}\Big{)},\quad
g_{1}=g_{2}=\frac{J+iD_{z}}{\sqrt{2}}.$ (48)

Now, one can get statistical weights (41) and entries (43), that opens up the
possibility of calculating branches (24), (25), (35), (IV.2) and, finally,
quantum correlations $\cal U$ and $\cal F$ through the interaction constants
of the Hamiltonian (44).

##  VI  Behavior of quantum correlations

Let us move on to consider the properties of spin-$(1/2,S)$ systems with a
Hamiltonian (44). We will limit ourselves to seven specified interaction
parameters because they are most important for magnetic materials.

###  VI.1  High-temperature wings of quantum correlations

Begin our analysis with the behavior of quantum correlations at high
temperatures $T\to\infty$. Using symbolic calculations on a digital computer,
we extract high-temperature asymptotics from the above general formulas. The
results are as follows.

Spin $S=1$:

${\cal F}_{0}=2\,{\cal
U}_{0}=\frac{J^{2}+D_{z}^{2}}{3T^{2}}+O(1/T^{3}),\quad{\cal F}_{1}=2\,{\cal
U}_{1}=\frac{3B_{1}^{2}+2K_{2}^{2}+2(J^{2}+J_{z}^{2}+D_{z}^{2}+2B_{1}K_{2})}{12T^{2}}+O(1/T^{3}).$
(49)

Spin $S=3/2$:

${\cal F}_{0}=2\,{\cal
U}_{0}=\frac{5(J^{2}+D_{z}^{2})}{8T^{2}}+O(1/T^{3}),\quad{\cal F}_{1}=2\,{\cal
U}_{1}=\frac{16B_{1}^{2}+41K_{2}^{2}+20(J^{2}+J_{z}^{2}+D_{z}^{2}+2B_{1}K_{2})}{64T^{2}}+O(1/T^{3}).$
(50)

Spin $S=2$:

${\cal F}_{0}=2\,{\cal
U}_{0}=\frac{J^{2}+D_{z}^{2}}{T^{2}}+O(1/T^{3}),\quad{\cal F}_{1}=2\,{\cal
U}_{1}=\frac{5B_{1}^{2}+34K_{2}^{2}+10(J^{2}+J_{z}^{2}+D_{z}^{2}+2B_{1}K_{2})}{20T^{2}}+O(1/T^{3}).$
(51)

Spin $S=5/2$:

${\cal F}_{0}=2\,{\cal
U}_{0}=\frac{35(J^{2}+D_{z}^{2})}{24T^{2}}+O(1/T^{3}),\quad{\cal
F}_{1}=2\,{\cal
U}_{1}=\frac{48B_{1}^{2}+707K_{2}^{2}+140(J^{2}+J_{z}^{2}+D_{z}^{2}+2B_{1}K_{2})}{192T^{2}}+O(1/T^{3}).$
(52)

These expressions show that the quantum correlations asymptotically fall to
zero according to the power law $T^{-2}$. The field $B_{2}$ and strength of
uniaxial single-ion anisotropy $K_{1}$ are absent here. The amplitudes of
these high-temperature wings (coefficients in the $1/T^{2}$ law) demonstrate
that the DM coupling $D_{z}$ and the Heisenberg interactions $J$ and $J_{z}$
enhance the quantum correlations, but the strength of these correlations
decreases when the sign of the $B_{1}K_{2}$ product is negative. Note that the
main high-$T$ terms ${\cal F}_{0,1}$ are twice as large as the corresponding
terms ${\cal U}_{0,1}$.

###  VI.2  $S$ as a quantum correlation enhancer

($\rm\bf a$)($\rm\bf b$)

Figure 1:  LQFI versus reduced temperature for the isotropic Heisenberg
systems $(1/2,S)$ with $S=1$ (blue), $S=3/2$ (green), $S=2$ (orange) and
$S=5/2$ (red). ($\rm\bf a$) $J>0$, antiferromagnetic interactions; ($\rm\bf
b$) $J<0$, ferromagnetic interactions. Here, ${\cal
H}=J{\vec{s}}\cdot{\vec{S}}$.

($\rm\bf a$)($\rm\bf b$)

Figure 2:  LQFI versus reduced temperature for the isotropic Heisenberg systems $(1/2,S)$ with $S=1$ (blue), $S=3/2$ (green), $S=2$ (orange) and $S=5/2$ (red). ($\rm\bf a$) $J>0$, antiferromagnetic interactions; ($\rm\bf b$) $J<0$, ferromagnetic interactions. Here, ${\cal H}=\frac{J}{S}{\vec{s}}\cdot{\vec{S}}$.  Table 1: Values of ground-state quantum correlations LQU and LQFI in the spin-$(1/2,S)$ XXX dimers with both antiferromagnetic ($J>0$) and ferromagnetic ($J<0$) couplings.  $S$ |  | $J>0$ |  | $J<0$
---|---|---|---|---
1/2 |  | 1 |  | 1/3=0.333…
1 |  | 8/9=0.888… |  | 4/9=0.444…
3/2 |  | 5/6=0.833… |  | 1/2=0.5
2 |  | 4/5=0.8 |  | 8/15=0.533…
5/2 |  | 7/9=0.777… |  | 5/9=0.555…

Now let us explore the behavior of quantum correlations throughout temperature
region. We start with the simplest case of the completely isotropic Heisenberg
XXX model setting $J_{z}=J$ and $B_{1}=B_{2}=K_{1}=K_{2}=D_{z}=0$, i.e.

${\cal H}=J{\bf s}\cdot{\bf S}.$ (53)

Calculations show that here ${\cal F}_{1}={\cal F}_{0}={\cal F}$ and ${\cal
U}_{1}={\cal U}_{0}={\cal U}$. Using Eq. (35), we find LQFI for different
values of spin $S$. So, when $S=1$, the quantum correlation $\cal F$ is equal
to

${\cal
F}^{(1)}(T)=\frac{16}{9}\frac{\sinh{(3J/4T)\tanh{(3J/4T)}}}{2+\exp{(3J/2T)}}e^{3J/4T}.$
(54)

In analogous manner, we obtain

${\cal
F}^{(3/2)}(T)=\frac{5\sinh^{2}{(J/T)}}{4+4\cosh{(2J/T)}-\sinh{(2J/T)}},$ (55)
${\cal
F}^{(2)}(T)=\frac{32\sinh^{2}{(5J/4T)}}{5[5+5\cosh{(5J/2T)}-\sinh{(5J/2T)}]},$
(56) ${\cal
F}^{(5/2)}(T)=\frac{35}{9}\frac{\cosh{(3J/T)}-1}{6+6\cosh{(3J/T)}-\sinh{(3J/T)}}$
(57)

for $S=3/2,\,2$ and 5/2, respectively. Similar formulas would be obtained for
higher values of spin $S$.

Figure 1 shows the behavior of the quantum correlation $\cal F$ as a function
of the reduced temperature $T/J$ (or $T/|J|$) for both antiferromagnetic and
ferromagnetic systems with $S=1,\dots,5/2$. Notice, ${\cal F}$ and ${\cal U}$
coincide at $T=0$. As seen from Fig. 1, the quantum correlations have finite
values at zero temperature. With increasing the temperature, the correlations
first maintain a quasi-stationary value and then monotonically tend to zero.

In the case of antiferromagnetic interaction, the ground-state quantum
correlations decrease with increasing spin $S$ (see Fig. 1a and Table 1). A
similar picture is observed for the entanglement [53, 73]. This was
interpreted as follows [53]. At $S\to\infty$, the system becomes more and more
classical and therefore the quantum correlations gradually weaken and
completely disappear in the limit of infinitely large spin $S$.

This interpretation is intuitively attractive, but unfortunately, it fails in
the case of ferromagnets. Really, as can be seen from Fig. 1b and Table 1
(column $J<0$), quantum correlations at zero temperature surprisingly increase
as spin $S$ increases. This is due to the peculiarities of the ground state.

Moreover, from Fig. 1 it is clearly seen that the high-$T$ wings of
correlations rise as $S$ grows, for both antiferromagnetic and ferromagnetic
systems. This also follows from Eqs. (49) - (52), since $T^{2}{\cal
F}/J^{2}=1/3,\,5/8,\,1$ and 35/24 for $S=1,\,3/2,\,2$ and 5/2, respectively.
Thus, an increase in quantum correlations is observed with increasing spin $S$
in the region of high temperatures.

It is interesting to draw a parallel with the behavior of quantum entanglement
in the XXX model (53). Using negativity [15] as a measure of entanglement, the
authors [53] found that the threshold temperature is equal to

$T_{\rm th}=\frac{2S+1}{2\ln{(2S+2)}}.$ (58)

Above this temperature, quantum entanglement vanishes. Note that quantum
entanglement is present only in the case of antiferromagnetic interactions
($J>0$) and is completely absent for ferromagnets ($J<0$). From (58), a
striking result follows [53]

$\lim_{S\to\infty}T_{\rm th}=\infty.$ (59)

In other words, the threshold temperature may be arbitrarily high when $S$ is
sufficiently large.

However, this conclusion is controversial. Indeed, before passing to the limit
$S\to\infty$, it is necessary to renormalize the spin operators:
$S_{\alpha}\to S_{\alpha}/\sqrt{S(S+1)}$ ($\alpha=x,y,z$) [74, 75]. Because of
this renormalization, the Hamiltonian (53) takes the form

${\cal H}=\frac{J}{\sqrt{S(S+1)}}\,{\bf s}\cdot{\bf S}\simeq\frac{J}{S}\,{\bf
s}\cdot{\bf S}$ (60)

and the threshold temperature (58) should be replaced by

$T_{\rm th}=\frac{2S+1}{2S\ln{(2S+2)}}.$ (61)

From here we come to another result, namely,

$\lim_{S\to\infty}T_{\rm th}=0.$ (62)

So, increasing the spin length $S$ reduces the threshold temperature and turns
it to zero in the limit of an infinitely large spin $S$.

The renormalized Hamiltonian (60) changes the behavior of quantum
correlations. The new curves are shown in Fig. 2 for $S>1/2$, i.e. for
relatively large spins. The main difference is occurred in the high-
temperature region. Now an increase in the spin $S$ here leads to a decrease
in quantum correlations. This is entirely consistent with intuitive
expectations.

It is important to emphasize the following. The low-temperature limit of
quantum correlations does not depend on the value of the interaction constant
in the Hamiltonian (depends only on the sign of $J$). This is clearly visible
from Figs. 1 and 2. Therefore, the values presented in Table 1 are universal.

###  VI.3  Sadden change phenomena

Quantum correlations LQU and LQFI are determined by Eqs. (23) and (34),
respectively. In general, these functions are piecewise-defined. The
minimization conditions included in their definitions can lead to an
interesting phenomenon, namely, the behavior of quantum correlations can
undergo sharp changes as the parameters of the system change smoothly. In the
mathematical literature, such phenomena are called catastrophes [76, 77].

Consider the system, in which $B_{1}=K_{2}=D_{z}=0$. As follows from Eqs. (49)
- (52), 0-wings are below 1-wings if $|J_{z}|>1$, and, conversely, above if
$|J_{z}|<1$.

In the absence of a second external field ($B_{2}=0$), pairs of branches
${\cal U}_{0}(T)$ and ${\cal U}_{1}(T)$, as well as ${\cal F}_{0}(T)$ and
${\cal F}_{1}(T)$ do not intersect. For $|J_{z}|>1$, the quantum correlations
${\cal U}(T)={\cal U}_{0}(T)$ and ${\cal F}(T)={\cal F}_{0}(T)$. If
$|J_{z}|<1$, then the correlations ${\cal U}(T)$ and ${\cal F}(T)$ are
completely determined by the branches ${\cal U}_{1}(T)$ and ${\cal F}_{1}(T)$,
respectively. In both cases, the quantum correlations are smooth functions.

Figure 3:  Branches of quantum correlations LQFI and LQU depending on the
temperature $T$ for the model with $B_{1}=0$, $B_{2}=-0.8$, $J=1$,
$J_{z}=0.3$, $K_{1}=0$, $K_{2}=0$ and $D_{z}=0$ at the spin lengths
$S=1,\,3/2,\,2$ and 5/2.

Figure 4:  Quantum correlations $\cal F$ and $\cal U$ as functions of
temperature $T$ for the model with $B_{1}=0$, $B_{2}=-0.8$, $J=1$,
$J_{z}=0.3$, $K_{1}=0$, $K_{2}=0$, and $D_{z}=0$ at different values of spin
length $S$.

The situation begins to change dramatically at $B_{2}\neq 0$. Figure 3 shows
the temperature dependencies of branches for different spin values $S$. The
intersections of branches are clearly observed here. The ${\cal U}_{0}(T)$
curves intersects the ${\cal U}_{1}(T)$ ones at the points with temperatures
0.215, 0.262, 0.292 and 0.323 for $S=1$, 3/2, 2 and 5/2, respectively. These
temperatures increase with increasing $S$. Similarly for the $\cal
F$-branches.

On the correlation curves, the above intersection points are visible as
fractures (sharp bends), see Fig. 4. They are points of non-analyticity. The
functions ${\cal U}(T)$ and ${\cal F}(T)$ are continuous, but their first
derivatives experience discontinuities of the first kind and therefore these
functions belong to the differentiability class $C^{0}$.

Experimental detection of singular points and study of the behavior of quantum
correlations near them carry valuable information about the systems under
examination. This information can make it possible to draw conclusions about
interaction constants and understand the role of certain bonds in processes
occurring in various substances.

Figure 5:  Branches of quantum correlations depending on temperature for the
model with $B_{1}=-0.65$, $B_{2}=0.9$, $J=-1.3$, $J_{z}=-1$, $K_{1}=0.4$,
$K_{2}=-1.02$ and $D_{z}=-0.3$ at spin lengths $S=1$ and 3/2.

As follows from Figs. 3 and 4, the above systems contain one sudden
transition. It is remarkable that a series of such transitions is possible.
Indeed, let $B_{1}=-0.65$, $B_{2}=0.9$, $J=-1.3$, $J_{z}=-1$, $K_{1}=0.4$,
$K_{2}=-1.02$ and $D_{z}=-0.3$. As can be seen from Fig. 5, the $\cal
U$-branches (blue curves) intersect twice and hence the behavior of LQU
suddenly changes at two different temperatures. It is interesting to note that
the second correlation, LQFI, preserves continuity and smoothness. This
indicates a peculiar hybrid behavior of the non-classical correlation in the
given system.

Figure 6:  Branches of quantum correlations depending on temperature for the
model with $B_{1}=-0.65$, $B_{2}=0.9$, $J=-1.3$, $J_{z}=-1$, $K_{1}=0.4$,
$K_{2}=-0.6$ and $D_{z}=-0.3$ at spin lengths $S=1$ and 3/2. Insets show the
vicinities of intersection points at higher temperatures.

Finally, let’s consider a model with slightly different parameters, namely,
$B_{1}$, $B_{2}$, $J$, $J_{z}$, $K_{1}$ and $D_{z}$ remain the same, but
$K_{2}$ is replaced by -0.6. Look at Fig. 6. Here, each pair of 0- and
1-branches intersects at two points. As a result, when the temperature changes
from zero to infinity, the quantum correlations $\cal U$ and $\cal F$ undergo
two sudden transitions. Figure 6 clearly shows the intersections of the curves
at low temperatures. The $\cal U$ branches (blue curves) intersect at
temperatures 0.164 and 0.06 for $S$=1 and 3/2, respectively. Similarly for
$\cal F$-branches (orange curves): they intersect at temperatures 0.375 and
0.13 for spin lengths $S$=1 and 3/2, respectively. Unfortunately, at higher
temperatures the intersections become less visible, so their surroundings are
shown in the insets. From these insets it is clear that at high temperatures
the behavior of quantum correlations is determined by 0-branches. The same
conclusion follows from the high-temperature decompositions (49) - (50).

##  VII  Summary and outlook

In this paper, we have derived analytical formulas (24), (25), (35) and (IV.2)
for the branches of quantum correlations LQU and LQFI. The obtained formulas
are surprisingly compact and allow one to calculate the local quantum
uncertainty and local quantum Fisher information for any spin-(1/2,$S$)
systems whose states commute with the $z$-component of the total spin.

The derived formulas were then applied to the systems in thermal equilibrium,
for which the behavior of quantum correlations has been studied in detail.
Among findings, we note the following. It is natural to expect that an
increase in the spin length, $S\to\infty$, should destroy quantumness of the
system. Our calculations confirm this observation, which is known for
antiferromagnets at ground states. However, contrary to intuitive
expectations, we found enhancement of quantum correlations at zero temperature
for systems with ferromagnetic couplings.

We also carefully studied the phenomena of sudden transitions where smooth
variations in temperature lead to sharp changes in the behavior of quantum
correlations. Remarkably, both single such transitions and series of them have
been revealed.

The obtained formulas can be used to study other properties of quantum
correlations, such as their dynamical behavior.

##

Acknowledgments: S. H. was supported by Semnan University under Contract No.
21270. M.A.Y. was supported in part by a state task, the state registration
number of the Russian Federation is $\\#$124013000760-0.

## CRediT authorship contribution statement

Saeed Haddadi: Methodology, Investigation. M. A. Yurischev: Writing - original
draft, Conceptualization.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.

## Data availability

No data was used for the research described in the article.

## ORCID iDs

Saeed Haddadi https://orcid.org/0000-0002-1596-0763

M. A. Yurischev https://orcid.org/0000-0003-1719-3884

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# Simplifying Semantic Annotations of SMCalFlow

###### Abstract

SMCalFlow [Semantic Machines et al., 2020] is a large corpus of semantically
detailed annotations of task-oriented natural dialogues. The annotations use a
dataflow approach, in which the annotations are programs which represent user
requests. Despite the availability, size and richness of this annotated
corpus, it has seen only very limited use in dialogue systems research work,
at least in part due to the difficulty in understanding and using the
annotations. To address these difficulties, this paper suggests a
simplification of the SMCalFlow annotations, as well as releases code needed
to inspect the execution of the annotated dataflow programs, which should
allow researchers of dialogue systems an easy entry point to experiment with
various dataflow based implementations and annotations.

languageresourceLanguage Resources

Simplifying Semantic Annotations of SMCalFlow

Joram Meron
---
Telepathy Labs GmbH
36 Militärstrasse, Zurich, Switzerland
<EMAIL_ADDRESS>

Abstract content

## 1\. Introduction

As in many other natural language processing tasks, dialogue systems have
achieved impressive advances due to the use of machine learning techniques.
These techniques typically require large amounts of high quality annotated
data in order to ensure that the resulting models will be able to generalize
correctly to unseen input.

Since the models used by dialogue systems need to also learn the effect of
previous turns in the dialogue context (as opposed to models which operate on
isolated sentences), even larger amounts of training data are needed.

Training data for dialogue systems typically includes the natural language
utterances of the user (”request”) and agent (”answer”), as well as some
structured data representing the state of the dialogue after the turn
(including any additional actions affected by the agent). While the user input
can be collected from naive users (e.g. using crowd sourcing platforms), the
agent response (both natural language and structured data) need skilled
annotators which have been trained specifically for the task.

Due to these difficulties, the number , and size, of available datasets for
training dialogue systems has been very limited - a few hundreds or thousands
of dialogues only, limiting the type of models which can be used. MultiWOZ
[Budzianowski et al., 2018], with 10K dialogues and 70K turns was until
recently the largest available set, and is being widely used in many research
works.

More recently, SMCalFlow [Semantic Machines et al., 2020] was released,
comprising of more than 40K dialogues (totalling more than 155K turns) of
natural (non-scripted) task-oriented user-agent interactions in several
domains (calendar events, weather, places and people), with semantically rich
annotation.

The Dialogues were collected via a Wizard-of-Oz process. At each turn, a
crowdworker acting as the user was presented with a dialogue as context and
asked to append a new utterance. An annotator acting as the agent labelled the
utterance, and then selected a natural-language response from a set of
candidates produced by the language generation model. Annotators were provided
with detailed guidelines containing example annotations and information about
available library functions.

Despite the size of this dataset, and the high level of detail given by the
annotations, it was not adopted by the dialogue systems research community.
The assumption in this paper is that this is the result, at least in part, of
the difficulty in understanding and using this dataset by the research
community. This difficulty is due to two factors: 1) The annotation scheme is
complex, and lacks sufficient documentation to explain it, and 2) tools to
inspect and verify that the annotations are correct were not released.

This paper111Published in the Proceedings of the 18th Joint ACL - ISO Workshop
on Interoperable Semantic Annotation within LREC2022 addresses these
difficulties by 1) suggesting a simplified annotation scheme, which, hopefully
is easier to understand, and 2) releasing the necessary code to inspect the
annotation results. It is hoped that with these contributions, the research
community will be encourage to explore and exploit the potential of this rich
dataset.

## 2\. Dataflow Dialogues

SMCalFlow uses dataflow (DF) computational graphs, composed of a rich set of
both general and application specific functions (see figures 1 and 4), to
represent the user requests as rich compositional (hierarchical) expressions.
These computational graphs can be executed, which results in manipulating the
computational graphs, generating an answer (possibly an error message), and
optionally producing some side effects through API’s to external services
(e.g. updating the user’s calendar appointments on an external database).

The prominent features of this paradigm are:

  * •

The dialogue history is represented as a set of graphs, where each
computational graph typically represents one user turn.

  * •

It has a refer operation to search over the current and previous computational
graphs (as well as external resources) which allows easy look-up and re-use of
graph nodes which occurred previously in the dialogue.

  * •

It has a revise operation which allows modification and reuse of previous
computations

  * •

It has an exception mechanism which allows convenient interaction with the
user (e.g. asking for missing information, and resuming the computation once
the information is supplied).

These features correspond to essential phenomena in natural conversations
(referring to previous turns, modifying previous requests, reacting to wrong
information, etc.), which allows the system to effectively handle these kinds
of user requests.

## 3\. Simplifying SMCalFlow

In this work, a simplified annotation is presented, with the motivation to
reduce the effort on the annotator/reader, without increasing the learning
effort for the machine translation models used to convert the users’ natural
language requests to the target annotation format.

As described below, this simplification requires some additional logic to be
implemented in the execution engine, as well as in the individual functions,
but this additional logic is typically trivial.

The starting point of this work is SMCalFlow, with its original annotation
style. Because of its size, and the limited resources available in this work,
manual modification of individual annotations were not feasible. Instead, the
modifications had to be done fully automatically, using a programmatic
solution to do the conversion. The consequences of this decision are:

  * •

The new annotations are still tied to the original ones, so some of the design
decisions made by the original annotators are difficult to change (as opposed
to the case where the new annotation would start from scratch).

  * •

Specifically, any mistakes or anomalies in the original annotations are
carried over to the simplified annotations.

  * •

An automatic conversion mechanism had to be created and configured to convert
the annotations correctly.

While DF is not inherently complicated, finding a good design is a challenging
task. A novel aspect of this challenge is the need for the design to function
correctly within the DF paradigm (e.g. use the refer and revise operators).
Indeed, one of the motivations of this work is the hope that the community can
suggest interesting new designs, which can serve as templates for further
applications.

### 3.1. Simplification Mechanism

The simplification was performed by implementing a set of tree transformation
rules, which convert specified sub-trees of the original expressions into
simplified sub-trees. The transformation code is part of the release, and can
be used to replicate the work reported here.

The simplification is applied to the whole dataset, resulting in a simplified
dataset, which can then be fed into the exact same machine translation
training and evaluation pipeline used in the original paper.

For convenience, the simplified format uses Python style expressions (as
opposed to the Lisp style S-expressions in the original dataset), as this
format is generally more familiar (the released system itself is written in
Python).

### 3.2. Simplification Approach

The design principles for the simplifications were:

  1. 1.

Retain only necessary information

  2. 2.

Avoid explicit logical steps

  3. 3.

Move logic from the annotation to the implementation of the individual
functions

  4. 4.

Group and reuse repeating sequences of functions

  5. 5.

Relax strict type constraints

  6. 6.

Reduce unnecessary compositions

Practically this means: Try to omit any information which can be
deterministically inferred - keep only information which can not be inferred.
Specifically, logical steps which can be inferred from context, are moved from
the annotation into the implementation of the functions. For example:

  * •

Explicit type casts which are clear from the context can be omitted.

  * •

When needed information is missing in the user input, but can be inferred from
the computation, the simplified annotation should leave the inference of the
missing information to the function implementation.

  * •

The simplified annotation tries to avoid fragments of the original annotation
which serve only ”formal” purposes, and instead tries to style the annotation
to be closer to a more natural/comprehensible description of the user requests
(and in general be closer to the surface form of the user request, as can be
seen in the examples).

Below are examples of original vs. simplified annotations.

#### 3.2.1. Example 1

The user’s request is:

"Delete the meeting with John’s supervisor tomorrow".

Figures 1 and 2 show the original and simplified annotations for this request.
Figure 3 show the annotations as computational graphs.

This example illustrates a few of the simplification ideas:

  * •

Computational steps which always appear together are bundled into one step:
’DeletePreflightEventWrapper’ and ’DeleteCommitEventWrapper’ correspond to two
sub steps of the act of deleting an event. Here, they are simplify by
combining them into one step ’DeleteEvent’.

  * •

Relax strict type constraints in the annotation. In the original annotation,
’DeletePreflightEventWrapper’ can accept only an integer input (representing
the unique id of the event to be deleted). In the simplified version, the
implementation of the ’DeletePreflightEventWrapper’ can handle additional
types of input, by calling the necessary type conversion, i.e.: if the input
is an ’Event’ type, then extract its ’Event.id’ value, and if the input is a
set of events, then additionally invoke a call to the ’singleton’ function.

  * •

Avoid explicit logical steps. In the original annotation, the process of
searching for a person is an explicit part of the annotation (see the input to
the ’FindManager’ function). In the simplified annotation, this logic is added
to the implementation of ’FindManager’ , so the annotation can be simply
’FindManager(John).

  * •

Avoid unnecessary compositions and annotations which serve only ”formal”
purpose. In the original annotation, ’RecipientWithNameLike’ implements a
compositional pattern, where one of the inputs is an empty constraint, which
is dropped in the simplified annotation (in this case, the whole surrounding
block is also removed).

Figure 1:  Example 1 - original annotation Figure 2:  Example 1 - simplified
annotation Figure 3:  Example 1 - original and simplified annotations shown as
graphs

#### 3.2.2. Example 2

The user’s request is:

"I want John, Emily, John’s supervisor and Bob to attend".

Figures 4 and 5 show the original and simplified annotations for this request.

This example illustrates some simplification ideas:

  * •

Simplification of the assignment construct, avoiding unnecessary assignments
(which are used only once) - in this example, variable x1 is used only once,
so it is substituted directly into the main expression.

  * •

Reducing the use of compositions, in favour of flatter expressions. In this
example, instead of chaining constraints using ’andConstraint’, the simplified
annotation uses a flat ’AND’ construct.

Figure 4:  Example 2 - original annotation Figure 5:  Example 2 - simplified
annotation Figure 6:  The result of transforming and evaluating the simplified
annotation for the request ”Delete the meeting with John’s supervisor
tomorrow”. Nodes from the simplified annotation are shown in gray. Yellow
nodes indicate nodes added automatically by expansion logic, green nodes
indicate information extracted from external DB (the result of searching for
John and his manager, and for the event matching the requested constraints).
Blue dashed lines indicate execution results (some nodes are omitted to reduce
clutter).

### 3.3. Executing Simplified Annotations

At execution time, an additional step transforms the simplified annotation to
a fully executable expression. This is done, again, by implementing tree
transformation rules (for each function), which can add deterministically
inferable missing information/steps (e.g. casting input to the right type, or
performing other conversions/functions based on input type).

This step could be viewed, in principle, as the inverse of the dataset
simplification step, but in practice the run-time transformation of the
simplified annotation is often quite different from the original annotation,
due to different design decisions and function implementations.

Figure 6 shows the result of transformation and execution of the simplified
annotation for example dialogue 1 above. The transformed graph is clearly
different from the original annotation’s graph.

### 3.4. Simplification Results

Since the original code to execute SMCalFlow was not released (and
documentation not supplied), it is impossible to verify that the suggested
simplifications implement/execute the exact same logic (in fact this was one
of the motivations for this paper). It can only be left to the readers to
inspect the simplified annotations and the code and draw their own
conclusions.

Qualitative evaluation confirmed correct execution of a sample of expressions,
but further work is needed to obtain more significant quantitative evaluation.

|  Program Length
---|---
Original Annotation |  (11, 37, 58)
Simplified Annotation |  (2, 11, 20)
Table 1:  Program length of the two annotation styles. Length is measured as
number of seq2seq target tokens, when translating user request to annotation.
Showing (.25, .50, .75) quantiles over the entire dataset.

Table 1 shows the results of a comparison of the annotation lengths of the
original and simplified annotations, confirming that the simplification does
make the annotation significantly shorter.

The example annotations shown above should show that the simplified
annotations are not just significantly shorter, but are also significantly
simpler to understand, which should reduce annotation efforts when creating
new training data.

Table 2 shows that the simplification did not reduce (and maybe increased) the
accuracy of machine translation of natural language user requests to dataflow
expressions. Refinement of the simplification rules may result in further
improvements.

| 1k | 3k | 10k | 33k
---|---|---|---|---
Original | 30.2±3.6 | 41.8±7.9 | 55.7±7.0 | 72.8
Simplified | 35.9±4.2 | 47.7±7.0 | 62.1±1.9 | 73.8
Table 2:  Translation accuracy (exact match) as function of training data
size, showing average and std. (in percent) over 7 randomly selected samples
per size.

## 4\. Further Work

The work presented in this paper is still in progress, trying to improve the
simplified annotation format and the automatic simplification.

Accordingly, the implementation of the executable functions will continue to
evolve, to be able to correctly execute modified annotation formats.

While the automatic simplification covers all of the dataset, the
implementation of functions has concentrated mostly (but not exclusively) on
turns dealing with the calendar domain (which is the most complex domain in
this dataset).

Further ideas and work on the simplified annotation definition (and
transformation process) from the community are encouraged. With the released
code, researchers should be able to experiment with new ideas and share them
with the community.

Additional areas of interest may include:

  * •

Evaluation: in addition to the exact-match metric for translation accuracy,
other metrics can be used, such as comparison of execution results, graph
structure similarity, etc.

  * •

Using the graph structure: the graph structure (at different points of the
execution) can be used by prediction models.

  * •

Different design patterns which are beneficial to specific parts of the
system. For example, the execution of a computation graph could emit various
types of information which would then be useful for subsequent prediction
models.

## 5\. Conclusion

A simplification of the SMCalFlow annotations has been presented. Some
simplification principles have been suggested, and an automatic conversion
tool has been implemented. Examples have been given to show that the
simplified annotations are significantly shorter, as well as easier to
understand, than the original annotations.

The code for reproducing this work
222https://github.com/telepathylabsai/OpenDF allows to run annotation
simplification as well as executing these annotations to inspect and verify
they satisfy user requests, and should lower the barrier of entry into
Dataflow dialogue design for interested researchers, allowing them to
experiment with new ideas.

## 6\. Bibliographical References

## References

  * Budzianowski et al., 2018 Budzianowski, P., Wen, T.-H., Tseng, B.-H., Casanueva, I., Ultes, S., Ramadan, O., and Gašić, M.  (2018).  MultiWOZ - a large-scale multi-domain Wizard-of-Oz dataset for task-oriented dialogue modelling.  In Proceedings of the 2018 Conference on Empirical Methods in Natural Language Processing, Brussels, Belgium.
  * Semantic Machines et al., 2020 Semantic Machines et al.  (2020).  Task-oriented dialogue as dataflow synthesis.  Transactions of the Association for Computational Linguistics, 8:556–571.