# Measure of invertible dynamical maps under convex combinations of
noninvertible dynamical maps

Vinayak Jagadish<EMAIL_ADDRESS>Institute of Theoretical Physics,
Jagiellonian University, ul. Łojasiewicza 11, 30-348 Kraków, Poland  R.
Srikanth  Theoretical Sciences Division, Poornaprajna Institute of Scientific
Research (PPISR), Bidalur post, Devanahalli, Bengaluru 562164, India
Francesco Petruccione  School for Data Science and Computational Thinking,
Stellenbosch University, Stellenbosch 7600, South Africa  National Institute
for Theoretical and Computational Sciences (NITheCS), Stellenbosch 7600, South
Africa  Quantum Research Group, School of Chemistry and Physics, University of
KwaZulu-Natal, Durban 4001, South Africa

###### Abstract

We study the convex combinations of the $(d+1)$ generalized Pauli dynamical
maps in a Hilbert space of dimension $d$. For certain choices of the
decoherence function, the maps are noninvertible and they remain under convex
combinations as well. For the case of dynamical maps characterized by the
decoherence function $(1-e^{-ct})/n$ with the decoherence parameter $n$ and
decay factor $c$, we evaluate the fraction of invertible maps obtained upon
mixing, which is found to increase superexponentially with dimension $d$.

##  I Introduction

For a quantum system undergoing an open evolution petruccione , the time
evolution of the system of interest is described by a quantum dynamical map
which is a time-continuous family of completely positive (CP) and trace-
preserving linear time-dependent maps, $\\{\Phi(t):t\geq
0,\Phi(0)=\mathbbm{1}\\}$, acting on the bounded operators of the Hilbert
space of the system of interest sudarshan_stochastic_1961 ; Quanta77 . The
properties of the dynamical map is related to the time-local generator $L(t)$
gorini_completely_1976  in the time-local master equation,
$\dot{\Phi}(t)=L(t)\Phi(t)$, with

$\begin{split}L(t)[\rho]=&-\imath[H(t),\rho]\\\
&+\sum_{i}\gamma_{i}(t)\left(G_{i}(t)\rho
G_{i}(t)^{\dagger}-\frac{1}{2}\\{G_{i}(t)^{\dagger}G_{i}(t),\rho\\}\right).\end{split}$
(1)

Here $H(t)$ is the effective Hamiltonian, $G_{i}(t)$’s are the noise
operators, and $\gamma_{i}(t)$’s the decoherence rates, describing the
continuous-in-time description of the dynamics of the system of interest. The
dynamical map is divisible if

$\Phi(t_{f},t_{i})=K(t_{f},t)\Phi(t,t_{i}),\quad\forall t_{f}\geq t\geq
t_{i}\geq 0.$ (2)

The map is CP divisible if for all $t$’s, the propagator $K(t_{f},t)$ is CP.
If the decay rates $\gamma_{i}(t)$ are positive at all times, the map is CP
divisible. Otherwise, the map is CP indivisible. According to the CP
divisibility criterion rivas_entanglement_2010  of quantum Markovianity, a
quantum dynamical map is Markovian if it is CP divisible at all instants of
time. Any deviation from CP divisibility is, therefore, an indicator of non-
Markovianity. This is also equivalent to temporary negativity of the decay
rates in the time-local master equation hall2010 . If the decay rates are
constants, the generator corresponds to a Markovian semigroup. Such maps are
time homogenous, i.e.,

$\Phi(t_{f},t_{i})=\Phi(t_{f}-t_{i},0):=\Phi(t_{f}-t_{i}).$ (3)

Convex combinations of quantum dynamical maps have been studied by various
authors. An example to create a non-Markovian evolution by mixing two
Markovian semigroups was reported in Ref. chruscinski_non-markovianity_2015 .
A similar idea appeared in an earlier paper wolf_assessing_2008  prior to the
advent of the divisibility and distinguishability criteria for quantum non-
Markovianity. In Ref. breuer_mixing-induced_2018 , the counterintuitive
behavior was explained in terms of information flow between the system and the
environment. An example of a convex combination of two non-Markovian dynamical
maps leading to a Markovian evolution was discussed in Ref. wudarski2016 . It
was shown that an eternally non-Markovian evolution (with one decay rate being
negative at all times, $t>0$) results from a convex combination of Pauli
semigroups megier_eternal_2017 . More generally, eternally non-Markovian
evolution obtained by the convex combination of semigroups of generalized
Pauli dynamical maps has been studied in Ref. siudzinska_jpa_2020 . Recently,
the geometry of the region of non-Markovianity obtained by convex combinations
of Pauli semigroups and time-dependent Markovian Pauli dynamical maps was
addressed jagadish_convex_2020 ; jagadish_nonqds_2020 . These results showed
that the sets of CP divisible and CP indivisible Pauli dynamical maps are
nonconvex. A measure of non-Markovianity was introduced desantis2020  based on
the minimal amount of extra Markovian noise that needs to be added via convex
mixing in order to make the dynamical map Markovian.

In Eq. (2), if for a particular instant of time $t=t^{*}$, the map
$\Phi(t^{*},t_{i})$ is noninvertible, then there are multiple initial states
that are mapped to the same output state. In this case, the propagator
$K(t_{f},t^{*})=\Phi(t_{f},t_{i})\Phi(t^{*}t_{i})^{-1}$ is undefined. Such
points $t^{*}$ are referred to as singular points hou_singularity_2012  of the
map. The presence of singularities in the map is reflected in the
corresponding time-local master equation where one or more of the decay rates
could go to infinity. Despite the fact that the dynamical map could be
momentarily noninvertible, the singularities are usually not pathological, in
the sense that the density operator of the system of interest is well defined
jagadish_measure2_2019 ; jagadish_initial_2021 . Recently, convex combinations
of dynamical maps with singularities have caught attention
utagi_singularities_2021 ; siudzinska_markovian_2021 ;
jagadish2022noninvertibility ; siudzinska_jpa_2022 . For generalized Pauli
dynamical maps, it was shown that noninvertible maps cannot be produced by
mixing invertible maps. The surprising result that convex combinations of
noninvertible qudit Pauli maps can produce a semigroup was reported in Ref.
siudzinska_markovian_2021 . Subsequently, it was also shown that
noninvertibility of the generalized Pauli input maps is necessary for this
jagadish2022noninvertibility . This prompts the question of whether the
noninvertibility is sufficient. The answer is no: Some mixtures produce
noninvertible maps. A natural followup to this line of inquiry would be to
quantify the fraction of the mixtures of noninvertible maps that produce
(non)invertible maps.

The physical motivation is ultimately about fighting noise in practical
quantum information processing, especially in the context of near-term quantum
devices. In principle, noise corresponding to a noninvertible map cannot be
corrected. An experimentalist would want to avoid parameter regimes that would
generate such noise. In a futuristic quantum computer that is fully fault
tolerant and operating within the required noise thresholds, this is less of a
problem. However, as we enter the noisy intermediate scale quantum era
preskill_quantum_2018 , where quantum devices are expected to lack the depth
required to implement full quantum error correction, avoidance of
noninvertibility can be a matter of practical concern.

The paper is organized as follows. We discuss the case of mixing noninvertible
Pauli dynamical maps in the following Section. In Sec. III, we extend our
analysis to the case of generalized Pauli dynamical maps. Finally, we discuss
the results and conclude in Sec. IV.

##  II Convex Combination of noninvertible Pauli dynamical maps

###  II.1 Obtaining invertible maps and measure of the corresponding set

Consider the three Pauli dynamical maps,

$\Phi_{i}(t)[\rho]=[1-p(t)]\rho+p(t)\sigma_{i}\rho\sigma_{i},i=1,2,3$ (4)

with $p(t)$ being the decoherence function and $\sigma_{i}$ the Pauli
matrices. The convex combination of the three Pauli dynamical maps Eq. (4),
each mixed in proportions of $x_{i}$ is, therefore,

$\tilde{\Phi}(t)=\sum_{i=1}^{3}x_{i}\Phi_{i}(t),\quad(x_{i}>0,\sum_{i}x_{i}=1).$
(5)

We will call the three $\Phi_{i}(t)$’s input maps and $\tilde{\Phi}(t)$ the
output map. Let each of the maps be characterized by the same decoherence
function,

$p(t)=\frac{1-e^{-ct}}{n},\quad n\geq 1,c>0.$ (6)

From the corresponding time-local master equation for each of the input maps
$\Phi_{i}$,

$L_{i}(t)[\rho]=\gamma(t)(\sigma_{i}\rho\sigma_{i}-\rho),$ (7)

with the decay rate

$\gamma(t)=\frac{c}{(n-2)e^{ct}+2},$ (8)

it can be seen that for $n\geq 2$, the decay rates never become infinite at
finite times $t$, and therefore, the input maps are all invertible. The output
map $\tilde{\Phi}$ [Eq. (5 ] satisfies the eigenvalue relation,

$\tilde{\Phi}(t)[\sigma_{i}]=\lambda_{i}(t)\sigma_{i},$ (9)

with

$\lambda_{i}(t)=1-2(1-x_{i})p(t),$ (10)

with $p(t)$ given by Eq. (6). The eigenvalues $\lambda_{i}$ becomes singular
at

$t^{*}=\frac{1}{c}\ln[\frac{2(1-x_{i})}{2(1-x_{i})-n}\Big{]}.$ (11)

From Eq. (11), the requirement for invertibility is

$x_{i}\geq 1-\frac{n}{2}.$ (12)

For invertible inputs, $n\geq 2$, the right-hand side above is negative and,
thus, any mixing coefficient $x_{i}$ satisfies the lower bound in Eq. (12).
This bound monotonically increases as $n$ decreases in Eq. (12). If
$n<\frac{4}{3}$, then $x_{i}<\frac{1}{3}$, a constraint that cannot be
satisfied by all three maps (since in that case we would have
$\sum_{i}x_{i}<1$ ). We now have the following result.

When the input map is invertible ($n\geq 2$), the output map is necessarily
invertible. For noninvertible inputs, if $n<\frac{4}{3}$, the output map is
necessarily noninvertible. For input maps that are of the intermediate
noninvertible range ($2>n>\frac{4}{3}$ ), the output can be invertible or not,
depending on the mixing coefficients. In the intermediate range of
noninvertible inputs $n\in(\frac{4}{3},2)$, a fraction of the output maps will
be invertible. In the case of $n=4/3$, for the output map to be invertible,
Eq. (12) implies that all $x_{i}\geq\frac{1}{3}$. Therefore, the output map is
noninvertible for all values of mixing coefficients $x_{i}$’s except at the
point of equal mixing where all $x_{i}$’s are 1/3 each. From Eq. (10), one can
see that the eigenvalues of the output map, $\lambda_{i}(t)=e^{-ct}$ for the
case of equal mixing, which is indicative of a semigroup. For the intermediate
case, i.e., $n\in(4/3,2)$, let us evaluate the measure $\Delta_{\rm invert}$
of invertible maps over all possible mixtures.

To this end, we note that $x_{1}$ can take values in the range
$[1-\frac{n}{2},1-2\times(1-\frac{n}{2})]=[1-\frac{n}{2},n-1]$, where the
lower bound is determined by Eq. (12), and the upper bound arises due to
allocating this minimal mass of probability to the other two variables. The
variable $x_{2}$ then is allowed to take values in the range $1-\frac{n}{2}$
and $\frac{n}{2}-x_{1}$. We have

$\displaystyle\Delta_{\rm invert}$ $\displaystyle=$
$\displaystyle\frac{1}{\mu}\int_{1-\frac{n}{2}}^{\frac{n}{2}-x_{1}}dx_{2}\int_{1-\frac{n}{2}}^{n-1}dx_{1}$
(13) $\displaystyle=$ $\displaystyle\frac{(4-3n)^{2}}{8},$

where the normalization factor
$\mu=\int_{0}^{1-x_{1}}dx_{2}\int_{0}^{1}dx_{1}=\frac{1}{2}$. Equation (13)
shows that the measure of invertible maps falls monotonically through the
range of $[1,0]$ as $n$ is varied in the intermediate range $(\frac{4}{3},2)$.
The above arguments can be readily extended to the case of qudits as performed
in Sec. III.

###  II.2 More general decoherence functions

Going beyond the form of the decoherence function as in Eq. (6), one can
consider other forms of noninvertible maps. Obviously, there is a great deal
of freedom here, and one might wish to start with simple and natural forms. We
consider the case where each of the noninvertible maps is characterized by the
same decoherence function,

$p(t)=\frac{1-\cos(\omega t)}{2},$ (14)

in place of Eq. (6). The corresponding time-local generator can be evaluated
to be

$L_{i}(t)[\rho]=\frac{\omega}{2}\tan(\omega
t)(\sigma_{i}\rho\sigma_{i}-\rho).$

The decay rate $\frac{\omega}{2}\tan(\omega t)$ is indicative of the
noninvertibility of the input maps. The output map $\tilde{\Phi}(t)$ Eq. (5)
satisfies the eigenvalue relation given by Eq. (9), i.e.,
$\tilde{\Phi}(t)[\sigma_{i}]=\lambda_{i}(t)\sigma_{i},$ with

$\lambda_{i}(t)=x_{i}+(1-x_{i})\cos(\omega t).$ (15)

The eigenvalues $\lambda_{i}(t)$ become singular at

$t^{*}=\frac{1}{\omega}\Big{[}\cos^{-1}\Big{(}\frac{x_{i}}{x_{i}-1}\Big{)}\Big{]}.$
(16)

From this, it can be seen that for the output map to be invertible (i.e.,
there exists no finite time $t$ at which the eigenvalues can be zero), all
three mixing coefficients $x_{i},$ should be greater than 0.5, which is
impossible. This entails that the resultant map is noninvertible for all
choices of convex combinations. A similar behavior is seen for the choice of
$p(t)=\sin(\omega t)$. We note that this generalizes the example in Ref.
siudzinska_markovian_2021 , where the shift of the singularity in the input
maps under convex combination is noted.

Suppose the choice of the decoherence function $p(t)$ in Eq. (10) is such that
$p(t)$ monotonically increases up to a finite time $t^{\\#}$, such that
$p(t^{\\#})=\frac{1}{2}$, and it remains constant thereafter. For example:

$p(t)=f(t)[1-\Theta(t-t^{\\#})]+\frac{1}{2}\Theta(t-t^{\\#}),$ (17)

where $f(t)$ is any monotonically increasing function such that
$f(0)=0,f(t^{\\#})=\frac{1}{2}$. Thus, each input map is maximally dephasing
and thereby noninvertible (all “azimuthal” points on the Bloch ball being
mapped to a single point on the respective axis). Quite generally, Eq. (10)
implies that the potential singular point in the output map is the solution
$t^{*}$ for the equation

$p(t^{*})=\frac{1}{2(1-x_{i})}.$ (18)

But this is necessarily greater than $\frac{1}{2}$, considering that all
$x_{i}$’s are nonzero. It follows that this defines a set of noninvertible
input maps that produces an invertible output.

The above examples in the subsection make the following pattern evident: The
decoherence function $p(t)$ can be such that it just makes the input maps
noninvertible, but does not increase far enough to make the output map
noninvertible; or it can go all the way to 1, making noninvertibility
inevitable. We, therefore, understand that the choice of exponential function
as in Eq. (6) is a good candidate to get invertible or noninvertible maps by
tuning the parameter $n$ to enable an intermediate behavior. Similar arguments
also apply to the case of generalized Pauli maps, discussed below.

##  III Convex combination of noninvertible Generalized Pauli dynamical maps

Based on the concept of mutually unbiased bases (MUBs), Nathanson and Ruskai
introduced ruskai2007  a generalization of the Pauli dynamical maps to the
case of maps on qudits. A set of ($d+1$)-orthonormal bases
$\\{|\xi_{i}^{(\alpha)}\rangle,i=0,\ldots,d-1\\}$ in $\mathbb{C}^{d}$ such
that $\langle\xi_{i}^{(\alpha)}|\xi_{j}^{(\beta)}\rangle|^{2}=1/d$ whenever
$\alpha\neq\beta$, is a MUB. If the dimension of the Hilbert space $d$ is an
integer power of a prime number, then $d+1$ MUBs exist. Defining the rank-1
projectors
$P_{j}^{(\alpha)}=|\xi_{j}^{(\alpha)}\rangle\langle\xi_{j}^{(\alpha)}|$, one
introduces the ($d+1$)-unitary operators,

$U_{\alpha}=\sum_{j=0}^{d-1}\omega^{j}P_{j}^{(\alpha)},\qquad\omega=e^{2\pi\imath/d}.$
(19)

We consider the convex combination of $d+1$-generalized Pauli dynamical maps
($i=1,...d+1)$ chruscinski2016pra ,

$\Phi_{i}(t)[\rho]=[1-p(t)]\rho+p(t)\sum_{k=1}^{d-1}U_{i}^{k}\rho
U_{i}^{k\thinspace\dagger},$ (20)

each mixed in proportions of $x_{i}$ as

$\tilde{\Phi}(t)=\sum_{i=1}^{d+1}x_{i}\Phi_{i}(t),\quad(x_{i}>0,\sum_{i}x_{i}=1).$
(21)

The map $\tilde{\Phi}(t)$ satisfies the eigenvalue relation,

$\tilde{\Phi}(t)[U_{i}^{k}]=\lambda_{i}(t)U_{i}^{k},k=1,...,d-1,$ (22)

with

$\lambda_{i}(t)=1-\frac{d}{d-1}(1-x_{i})p(t).$ (23)

For the choice of $p(t)$ as in Eq. (6), the eigenvalues become singular at

$t^{*}=\frac{1}{c}\ln[\frac{d(1-x_{i})}{d(1-x_{i})-n(d-1)}\Big{]}.$ (24)

For the cases of Pauli dynamical maps and generalized Pauli maps, it was shown
that when the input map is invertible, the resultant map is necessarily
invertible siudzinska_markovian_2021 ; utagi_singularities_2021 . Invertible
input maps correspond to $n\geq\frac{d}{d-1}$. From Eq. (24), for
noninvertible inputs, if $n<\frac{d^{2}}{d^{2}-1}$, the resultant map is
necessarily noninvertible. We have the following result.

######  Theorem 1.

For generalized Pauli input maps that are of the intermediate noninvertible
range ($\frac{d}{d-1}>n\geq\frac{d^{2}}{d^{2}-1}$), the fraction of output
maps that are invertible is given by
$\left[\frac{(d^{2}(n-1)-n)}{d}\right]^{d}$.

###### Proof.

From Eq. (24) it follows that the requirement for invertibility is

$x_{i}\geq 1-\frac{n(d-1)}{d}\equiv g(d,n).$ (25)

The input map is invertible for $n\geq\frac{d}{d-1}$, and is a semigroup for
$n=\frac{d}{d-1}$. For invertible inputs $n\geq\frac{d}{d-1}$, the right-hand
side above is negative and, thus, any mixing coefficient $x_{i}$ satisfies the
lower bound in Eq. (25). This bound monotonically increases as $n$ decreases
in Eq. (25). If $n<\frac{d^{2}}{d^{2}-1}$, then $x_{i}<\frac{1}{d+1}$, a
constraint that cannot be satisfied by all three maps (since in that case we
would have $\sum_{i}x_{i}<1$).

In the intermediate range of noninvertible inputs
$n\in(\frac{d^{2}}{d^{2}-1},\frac{d}{d-1})$, a fraction of the output maps
will be invertible. To evaluate this, first we note that $x_{1}$ (any given
$x_{j}$) must range from the minimum of $g(d,n)$ in view of Eq. (25), with, at
least, this mass of probability being assigned to the remaining $d$ maps.
Therefore, the allowed maximum is bounded by $1-dg(d,n)$. Accordingly,

$x_{1}\in[g(d,n),1-dg(d,n)]=[g(d,n),(d-1)(n-1)].$ (26)

Similarly, $x_{2}$ (or, any but $x_{1}$) must range from the minimum of
$g(d,n)$ in view of Eq. (25) with, at least, this mass of probability being
assigned to the remaining $(d-1)$ maps less the mass $x_{1}$ already assigned
to the first map #1. Therefore, the allowed range is given by

$x_{2}\in[g(d,n),1-(d-1)g(d,n)-x_{1}].$ (27)

Proceeding analogously, for a general mixing variable $x_{j}$, we have

$x_{j+1}\in[g(d,n),f(j)-X_{j}],$ (28)

where $f(j)$ and $X_{j}$ are as defined by

$\displaystyle f(j)$ $\displaystyle\equiv 1-(d-j)g(d,n),$ $\displaystyle
X_{j}$ $\displaystyle\equiv x_{j}+x_{j-1}+\cdots+x_{1}.$ (29)

Given dimension $d$ and parameter $n$, the measure of the invertible region in
the space of resultant maps is evaluated to be

$\begin{split}\Delta_{\rm
invert}(d,n)&=\frac{1}{\mu}\int_{g(d,n)}^{(d-1)(n-1)}dx_{1}\cdots\\\
&\int_{g(d,n)}^{f(j)-X_{j}}dx_{j+1}\cdots\int_{g(d,n)}^{f(d-1)-X_{d-1}}dx_{d}\\\
&=\left[\frac{d^{2}(n-1)-n}{d}\right]^{d}.\end{split}$ (30)

Here,

$\mu=\int_{0}^{1-X_{j}}dx_{d}\int_{0}^{1-X_{j-1}}dx_{d-1}\cdots\int_{0}^{1}dx_{1}=\frac{1}{d!}$
(31)

is the normalization constant. ∎

Equation (30) implies that for any given dimension $d$, the invertible
fraction $\Delta_{\rm invert}$ ranges from $\Delta_{\rm
invert}(d,\frac{d^{2}}{d^{2}-1})=0$ to $\Delta_{\rm
invert}(d,\frac{d}{d-1})=1$. For fixed $n$, and sufficiently large $d$, we
find that $\Delta_{\rm invert}$ approaches unity with a superexponential rate
$O(d^{d})$. However, to plot this, we remark that the intermediate region
$[\frac{d^{2}}{d^{2}-1},\frac{d}{d-1}]$ quadratically shrinks into a narrow
interval that converges towards unity. Therefore, to depict the dependence of
the invertible fraction on $d$, we must choose a range of $d$ values such that
a given $n$ value falls within their corresponding interval. An example is
given in Fig. 1, varying $d$ discretely over the prime powers from 7 to 32.
The intermediate regions of $n$ corresponding to these two values are,
respectively, [1.021, 1.167] and [1.001, 1.032]. Here we choose $n=1.03$,
which is contained in both intervals and, by interpolation, in the intervals
corresponding to all the intermediate values of $d$. (The intervals
corresponding to prime powers equal to and below $d=5$ have no overlap with
the intervals corresponding to prime powers equal to and above $d=32$, and,
hence, are excluded in this representation).

Figure 1: Logarithm plot of the measure of the invertible fraction as a
function of $d$ for $n$ := 1.03, which falls within the interval
$[\frac{d^{2}}{d^{2}-1},\frac{d}{d-1}]$ for $d$ taking prime powers between
and including 7 and 32. The plot depicts how the invertible fraction
$\Delta_{\rm invert}$ rises at a superexponential rate $O(d^{d})$ in the
considered parameter space.

##  IV Discussions and conclusions

A quantum resource theory (RT) formalizes the separation of a given set of
objects of interest (typically represented as states) into resource states and
free (nonresource) states. The idea is that there are free (or “easy”)
operations in the RT such that the set of free states is closed under the
action of free operations. For example, in a resource theory of quantum
entanglement, the entangled (respectively, separable) states are the resource
(respectively, free) states, while local operations and classical
communication (LOCC) constitute the free operations. It is well known that
separable states cannot become entangled under LOCC. A RT, therefore,
addresses the question of tasks which are (im)possible using free operations
chitambar2019quantum . In a convex RT, the free states form a convex set.

One could consider a quantum RT for noninvertibility, considering how this
feature can be converted into corresponding limitations on quantum information
processing. Consider, for example, a map $\mathcal{E}$ described by the noise
$\rho\rightarrow\sum_{j}K_{j}\rho K_{j}^{\dagger}$, where $K_{j}$’s are Kraus
operators, acting on one-half of the maximally entangled state
$\frac{1}{\sqrt{D}}\sum_{k=1}^{D}\ket{k,k}_{A,B}$ between systems $A$ and $B$.
The resulting mixed entangled state can act as a noisy quantum teleportation
map. Given an initial state $\rho$ of system $S$, this map teleports $\rho$
according to

$\displaystyle\rho$ $\displaystyle\rightarrow$
$\displaystyle\sum_{rjklm}\bra{l,l}_{SA}\rho_{S}\otimes
K_{r}\ket{j}_{A}\bra{k}K_{r}^{\dagger}\otimes\ket{j}_{B}\bra{k}m,m\rangle_{SA}$
(32) $\displaystyle=$ $\displaystyle\sum_{r}K_{r}^{\dagger}\rho
K_{r}\equiv\mathcal{E}_{T}(\rho),$

i.e., a map characterized by Kraus operators $\\{K_{j}\\}$ is mapped to one
characterized by the Kraus operators $\\{K_{j}^{\dagger}\\}$. It follows that
if the dynamical map described by $\mathcal{E}$ is noninvertible, then so is
the effective teleportation map $\mathcal{E}_{T}$. Thus, we find that quantum
noninvertibility as a negative resource, i.e., one that is convertible to a
limiting (rather than enabling) feature of noisy teleportation, an idea that
can be extended to certain other quantum information processing tasks.

In this framework, invertible maps would correspond to free states and
noninvertible maps to resource states. Free operations must be resource
nonincreasing. For this we consider, the composition of maps, i.e., a
sequential application. Invertible maps being bijective, the composition of
bijective maps is also a bijection. Since bijective maps have a unique
inverse, the composed map, therefore, is also invertible. Thus, the set of
invertible maps is closed under composition. Given such a RT, our results
imply that in the context of qudit Pauli dynamical maps, noninvertibility is a
convex resource. Mixing invertible maps can never result in noninvertibility.
On the other hand, a mixture of noninvertible maps satisfying the condition
$n\in[\frac{d}{d-1},\frac{d^{2}}{d^{2}-1}]$ produces a set of invertible maps
with a finite nonzero measure. The convexity of the invertible set may be
contrasted with the nonconvexity of the set of quantum Markovian (CP
divisible) dynamical maps. On that basis, it was argued that CP indivisibility
does not yield a convex resource theory jagadish_convex_2020 .

In this paper, we have studied the properties of dynamical maps that result
under mixing of generalized Pauli dynamical maps that are noninvertible. In
any given dimension $d$, (non)invertibility of the input maps is parametrized
through a parameter $n$ , which ensures invertibility when sufficiently large.
Analogously, for noninvertible dynamical maps with sufficiently small $n$, the
resultant dynamical map is necessarily noninvertible. We quantify the fraction
of invertible maps in the intermediate range of $n$. Specifically, we have
shown that the set of such invertible maps is of nonzero measure.

###### Acknowledgements.

V.J. acknowledges financial support by the Foundation for Polish Science
through the TEAM-NET Project (Contract No. POIR.04.04.00-00-17C1/18-00). R.S.
acknowledges the support of the Department of Science and Technology (DST),
India, Grant No.: MTR/2019/001516. The work of F.P. is based upon research
supported by the South African Research Chair Initiative of the Department of
Science and Innovation and National Research Foundation (NRF) (Grant No. UID:
64812).

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# Dual function spin-wave logic gates based on electric field control magnetic
anisotropy boundary

Kang Wang  Center for Spintronics and Quantum Systems, State Key Laboratory
for Mechanical Behavior of Materials, School of Materials Science and
Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi, 710049, China  Shaojie
Hu<EMAIL_ADDRESS>Center for Spintronics and Quantum Systems,
State Key Laboratory for Mechanical Behavior of Materials, School of Materials
Science and Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi, 710049,
China  Fupeng Gao  Center for Spintronics and Quantum Systems, State Key
Laboratory for Mechanical Behavior of Materials, School of Materials Science
and Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi, 710049, China
Miaoxin Wang  Center for Spintronics and Quantum Systems, State Key Laboratory
for Mechanical Behavior of Materials, School of Materials Science and
Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi, 710049, China  Dawei
Wang  School of Microelectronics $\&$ State Key Laboratory for Mechanical
Behavior of Materials, Xi’an Jiaotong University, Xi’an 710049, China

###### Abstract

Spin waves (SWs) have been considered a promising candidate for encoding
information with lower power consumption. Here, we propose the dual function
SW logic gates based on the electric field controlling the SW propagation in
the Fe film of $\rm Fe/BaTiO_{3}$ heterostructure with the motion of magnetic
anisotropy boundary (MAB). We show micromagnetic simulations to validate the
AND-OR and NAND-NOR logic gates. Our research may find a path for simplifying
integrated logic circuits using such dual function SW logic gates.

With the rapid development in the miniaturization of electronic devices, power
consumption has become one important issue because of the unbearable Joule
heating.Vogel (2007); Theis and Wong (2017) To overcome this issue, seeking
new charge-neutral information carriers has spurred great attention. One
representative charge-neutral information carrier is spin angular momentum,
the electron’s another degree of freedom. Spin waves (SWs), the collective
spin angular momentum excitations in magnetically ordered material and their
associated quanta, magnons, are considered the promising information carrier
for the next-generation lower power consumption, higher speed, and higher
density devices. The phaseKhitun and Wang (2005); Klingler _et al._ (2014,
2015); Fischer _et al._ (2017), amplitudeKostylev _et al._ (2005); Schneider
_et al._ (2008); Lee and Kim (2008); Jamali _et al._ (2013), and
polarizationLan _et al._ (2017); Yu _et al._ (2020) of SWs can be employed to
encode information. Generally, the nanostructured SW logic gates are designed
on the uniform magnetized ferromagnetic materials by using all kinds of
methods to control the propagation of SWs, including magnetic fieldJamali _et
al._ (2013), electric currentKostylev _et al._ (2005); Schneider _et al._
(2008); Lee and Kim (2008) and voltageWang _et al._ (2017); Rana and Otani
(2018).

The logic gates based on electric field or voltage can be easily programmable
and compatible with nanoscale microwave devicesRana and Otani (2019). Rana et
al.Rana and Otani (2018) have proposed SW XNOR and universal NAND logic gates
based on the voltage-controlled magnetic anisotropy (VCMA), which mainly
arises from the change of electronic occupation state near the interface
between the magnetic films and nonmagnetic heavy metals controlled by the
voltageDuan _et al._ (2008); Tsujikawa and Oda (2009); Rana and Otani (2019).
Magnetoelectric effectsSpaldin and Fiebig (2005); Taniyama (2015); Song _et
al._ (2017) provide another path for electric-field control of SWs in
multiferroic materialsBalinskiy _et al._ (2018); Sadovnikov _et al._ (2018).
The most representative multiferroic system is artificial multiferroic
materials. Just recently, electric field controlled spin wave propagation had
been experimentally demonstrated in artificial multiferroic
heterostructure.Qin _et al._ (2021) Magnetoelectric effects could help us to
design much more functional SW logic gates based on multiferroic structure.
Here, the dual function SW logic gates are proposed by controlling the motion
of magnetic anisotropy boundary (MAB) in epitaxial Fe film on ferroelectric
$\rm BaTiO_{3}$ substrate with altering in-plane and out-of-plane polarization
domains. The AND-OR gate and NAND-NOR gate are validated by micromagnetic
simulation.

Figure 1:  (a) Schematic of the simulation model. Fe film is grown on the $\rm
BaTiO_{3}$ with alternating in-plane polarization a-domain and out-of-plane
polarization c-domain. The origin of the coordinate system is set at the
center of the Fe film. (b) The image plot of the dispersion relation of the
SWs. The dash lines are theoretically calculated results. (c) A snapshot of
the $M_{x}$ component of the magnetization taken at _t_ = 4 ns, _y_ = $-1$ nm
with a fixed _f_ = 12.6 GHz.

We perform micromagnetic simulations by MuMax3Vansteenkiste _et al._ (2014) to
study the propagation of Damon-Eshbach (DE) SWsDamon and Eshbach (1961) in the
epitaxial Fe film on ferroelectric $\rm BaTiO_{3}$ substrate. Figure 1(a)
shows one unit structure in our simulation, including one 2 nm thick Fe layer,
100 nm thick $\rm BaTiO_{3}$, and the bottom electrode layer. The width and
length of the structure are 800 nm and 2000 nm, respectively. The magnetic Fe
layer includes uniaxial and cubic magnetic anisotropy regions via inverse
magnetostriction owning to the different strains between Fe film and two kinds
of domains in $\rm BaTiO_{3}$ layerLahtinen _et al._ (2012). The easy axis of
uniaxial magnetic anisotropy on the top of in-plane ferroelectric a-domain is
along the _y_ axis, while the easy axes of cubic anisotropy on the top of out-
of-plane ferroelectric c-domain are along the direction with 45° to the _x_
axis, as shown by the white double head arrows. The MAB is at the middle of
the Fe film (_x_ = 0, along _y_ axis) and is pinned on the ferroelectric
domain wall of the $\rm BaTiO_{3}$ substrate. The width of MAB in Fe film is
same to the ferroelectric domain wall in $\rm BaTiO_{3}$ (2–5 nmZhang _et al._
(1992); Hlinka and Márton (2006); Zhang and Goddard (2006)), which is far
smaller than the wavelength of SWs (several hundred nanometers). So, we
neglect the width of the MAB in simulation. The parameters of Fe film used in
simulation are the followingFranke _et al._ (2015): saturation magnetization
$M_{\mathrm{s}}$ = 1.7 $\times$ $10^{6}$ A/m, exchange stiffness constant
$A_{\mathrm{ex}}$ = 2.1 $\times$ $10^{-11}$ J/m, Gilbert damping constant
$\alpha$ = 0.01. The uniaxial and cubic magnetic anisotropy constants are the
experiment valuesQin _et al._ (2021) of $K_{\mathrm{u}}$ = 1.5 $\times$
$10^{4}$ $\rm{J/m^{3}}$ and $K_{\mathrm{c}}$ = 4.4 $\times$ $10^{4}$
$\rm{J/m^{3}}$. The cell size is 2 $\times$ 2 $\times$ 2 $\rm nm^{3}$, which
is smaller than the exchange length ($l_{\mathrm{ex}}\approx 3.4$ nm). In
order to avoid SWs’ reflection at the boundary, the damping is set to 1 at
both ends (width = 100 nm) of the Fe film. An external magnetic field
$\mu_{0}H_{\mathrm{ext}}$ = 100 mT is applied to avoid the formation of
magnetic domain wall and magnetize the Fe film along _y_ axis. To obtain the
dispersion relation of the DE SWs in uniaxial and cubic anisotropy regions, a
sinc based exciting field, $\bm{\mathrm{h}}(t)=h_{0}\mathrm{sinc}(2\pi
f_{\mathrm{c}}(t-t_{0}))\hat{e}_{z}$, with $\mu_{0}h_{0}$ = 10 mT,
$f_{\mathrm{c}}$ = 50 GHz, $t_{0}$ = 5 ns, is applied locally to a 2 $\times$
800 $\times$ 2 $\rm nm^{3}$ central section of the Fe film, which is the SW
source port (S) as indicated by the red region in Fig. 1(a). The dispersion
relationKumar _et al._ (2011); Venkat _et al._ (2013) of SWs can be obtained
by performing a two-dimensional Fourier transform on $m_{x}$
($m_{x}=M_{x}/M_{\mathrm{s}}$), as shown in Fig. 1(b). The curve of the left
branch is the dispersion relation of SWs propagating along the the negative
direction of _x_ axis in the uniaxial anisotropy region and the right branch
is the dispersion relation of SWs propagating along the positive direction of
_x_ axis in the cubic anisotropy region. The theory curves of SW dispersion
relations are calculated by adopting the Eq.4 and Eq.5. It is clear to show
that the SWs are only excited in the cubic anisotropy region in the frequency
range of 9.5 GHz <  _f_ < 14.5 GHz. To further clarify the property, we excite
SWs at the source port with a sinusoidal field
$\bm{\mathrm{h}}(t)=h_{0}\sin(2\pi ft)\hat{e}_{z}$ with $\mu_{0}h_{0}$ = 10 mT
and a fixed _f_ = 12.6 GHz. A snapshot of the $M_{x}$ component of the
magnetization taken at _t_ = 4 ns is presented in Fig. 1(c). SWs only
propagate in the region with cubic anisotropy.

An out-of-plane electric field controlling ferroelectric domain wall motion
can realize the movement of the MABFranke _et al._ (2015). When the electric
field is along the direction of the polarization in the c-domain, the c-domain
will expand while the a-domain shrinks by the lateral domain wall motion.
Suppose the electric field is inverse to the direction of the polarization in
the c-domain. In that case, the c-domain will shrink and the a-domain expands.
So, the electric field can drive the ferroelectric domain wall’s motion, thus
driving the concurrent motion of the MAB. The velocity of MAB, that is, the
velocity of ferroelectric domain wall driven by electric field pulse, can be
up to 1000 m/sStadler and Zachmanidis (1963); Boddu _et al._ (2017). Because
the wall velocity is still smaller than the speed of sound in $\rm BaTiO_{3}$
($v_{\mathrm{sound}}$ = 2000 m/sTagantsev _et al._ (2010)), the strain state
can be considered quasistatic during the motion of ferroelectric domain wall
and the strength and symmetry of the magnetic anisotropy are invariant during
the MAB back and forth. The velocity $v$ of the MAB is implemented by shifting
the MAB over one discretization cell ( $\delta x$ = 2 nm) during each time
window $\delta t=\delta x/v$. In the simulation, the velocity of MAB is 1000
m/s under the electric field 31.4 MV/m obtained by applying 3.14 V
perpendicular voltage, which is inferred from previous experimental
resultsStadler and Zachmanidis (1963).

Figure 2:  (a) The position variation of the MAB with voltage pulse. (b), (c)
and (d) are the schematics of model and the snapshots of $M_{x}$ taken at _y_
= $-1$ nm when MAB at _x_ = $-600$ nm, $-100$ nm and 400 nm (adopting the
coordinate system in Fig. 1(a)), respectively. The dashed lines mark the
position of MAB.

Figure 2(a) shows the motion of MAB driven by voltage pulse and Fig. 2(b)–(d)
shows the schematics of model and the snapshots of $M_{x}$ when MAB at _x_ =
$-600$ nm, $-100$ nm and 400 nm, respectively. The SWs with _f_ = 12.6 GHz are
excited from the source port. The position of MAB is driven to _x_ = $-600$ nm
by using the positive electric voltage (1.4 ns – 2ns) shown in Fig. 2(a). As
expected, the SWs are blocked in uniaxial anisotropy region shown by the
snapshot of SWs taken at _t_ = 5 ns in Fig. 2(b). After applying one negative
pulse voltage (7 ns – 7.5 ns), the MAB moves to the position _x_ = $-100$ nm.
And, the SWs only propagate in the cubic anisotropy region from the snapshot
of $M_{x}$ taken at _t_ = 10.5 ns, shown in Fig. 2(c). When another 0.5 ns
negative voltage pulse is applied (12.5 ns – 13 ns), the MAB moves to the
position _x_ = $400$ nm, and the SW source port is out-of cubic anisotropy
region. There are no SWs in Fe film. Thus, the transmission and block of SWs
in Fe film can be controlled by voltage through driving the motion of MAB.
Based on this unique property, we propose the design of dual function SW logic
gates.

Figure 3(a) is the schematic of the dual function AND-OR gate, which includes
two input ports (A and B), two output ports (C and D), and a source port of
SWs. Initially, the MAB is set at _x_ = 400 nm parallel to _y_ axis. The
position of input A and B ports are arbitrary on the top of the Fe film. In
order to make the two input ports independent, the B port includes a time
delay unit, which can delay the voltage pulse for 0.5 ns. The output ports C
and D take the _x_ component of magnetization at region (498 nm <  _x_ < 500
nm, $-2$ nm <  _y_ < 0 nm) and ($-500$ nm <  _x_ < $-498$ nm, $-2$ nm <  _y_ <
0 nm), respectively, as indicated by the black square regions. Magnetic tunnel
junctions can perform this function and convert SW signals to electrical
signals. An external magnetic field $\mu_{0}H_{\mathrm{ext}}$ = 100 mT is
applied to magnetize the Fe film along _y_ axis and a sinusoidal field
$\bm{\mathrm{h}}(t)=h_{0}\sin(2\pi ft)\hat{e}_{z}$ with $\mu_{0}h_{0}$ = 10 mT
and _f_ = 12.6 GHz is set at the source port. The bottom electrode is
grounding.

Figure 3:  (a) Schematic diagram of the AND-OR gate. Initially, the MAB is set
at _x_ = 400 nm. Input A and B are the input ports and input B includes a time
delay unit that can delay the input signal for 0.5 ns. C and D ports are
output ports. The bottom electrode is grounding. The simulation results of the
AND-OR gate with different inputs are shown in (b) – (e), respectively. The
amplitudes of SWs in 11 ns – 11.5 ns are the logic operation results and the
time domain is marked by the red dash lines. Finishing one logic operation
needs 4.5 ns. (f) The truth table of the AND-OR gate.

The simulation results of the AND-OR gate with different inputs are shown in
Fig. 3(b)–(e). The simulation result of A = 0 and B = 0 is shown in Fig. 3(b).
Obviously, the MAB keeps still at position _x_ = 400 nm, because there is no
electric field across $\rm BaTiO_{3}$. The outputs of C and D are both small
amplitudes representing ’0’ due to the blocking of SWs shown in Fig. 2(d). The
simulation result of A = 1 and B = 0 is shown in Fig. 3(c). The input ’1’
signal includes one 0.5 ns positive voltage pulse and 0.5 ns negative voltage
pulse as shown in Fig. 3(c). The positive pulse is used to start logical
operation by driving the MAB moving 500 nm along the negative direction of _x_
axis in 0.5 ns, while the negative pulse is used to reset the position of MAB.
The related motion of MAB and output signals of the two ports are also shown
in Fig. 3(c). In 9 ns – 9.5 ns, the MAB is driven from _x_ = 400 nm to _x_ =
$-100$ nm. After oscillation in 9.5 ns – 11 ns, the amplitudes of SWs at C and
D represent C = 1 and D = 0 in the time range of 11 ns – 11.5 ns, marked by
the red dash lines. After that, the negative pulse drives the MAB back to _x_
= 400 nm during 11.5 ns – 12 ns. The outputs will be the same as the initial
and one whole logical operation is finished. Fig. 3(d) shows the simulation
result of A = 0 and B = 1. It’s very similar to the previous operation. The
only difference is the 0.5 ns delay of the voltage pulse in B port. But the
outputs of ports C and D are still ’1’ and ’0’, respectively. If A and B ports
are both ’1’, the driving time of MAB will be 1 ns due to the 0.5 ns delay
between A and B shown in Fig. 3(e). The MAB is driven to _x_ = $-600$ nm from
_x_ = 400 nm during the period of 9 ns – 10 ns. And the amplitudes of SWs for
ports C and D are over 1000 A/m at the time range of 11 ns – 11.5 ns. The
operation results of C and D are both ’1’. Then, the MAB is driven back to _x_
= 400 nm during 11.5 ns – 12.5 ns. It should be mentioned that the amplitude
of the SWs in C port at 11 ns – 11.5 ns is a bit larger than that in Fig. 3(c)
and (d), which mainly arises from the weaker interference between the
reflected SWs from the MAB and the initial SWs. In short summary, the logic
operation of the AND-OR gate can be divided into three stages, preparation
before output (9 ns – 11 ns), output (11 ns – 11.5 ns) and reset (11.5 ns –
13.5 ns). The truth table of the AND-OR gate is shown in Fig. 3(f). The input
A, B and output C compose an OR gate while the input A, B and output D compose
an AND gate.

Figure 4:  (a) Schematic diagram of the NAND-NOR gate. Initially, the MAB is
set at _x_ = $-600$ nm. The Fe film is grounding and the input voltage signals
are injected from bottom electrode. Other configurations and the input signals
are same to the AND-OR gate. The simulation results of the NAND-NOR gate with
different inputs are shown in (b) – (e), respectively. The amplitudes of SWs
in 11 ns – 11.5 ns are the logic operation results and the time domain is
marked by the red dash lines. Finishing one logic operation needs 4.5 ns. (f)
The truth table of the NAND-NOR gate.

In order to implement full Boolean logic operation, we also need the NAND and
NOR gates. Fig. 4(a) is the schematic of the dual function NAND-NOR gate.
Here, the initial MAB is set at _x_ = $-600$ nm. The main change of the NAND-
NOR gate is that the two input ports A and B are connected with the bottom
electrode and the Fe layer is grounding. The output ports and source port are
the same with the AND-OR gate. When A and B are both ’0’, the MAB keeps still
at _x_ = $-600$ nm. When only one of A and B is ’1’, the MAB is driven to _x_
= $-100$ nm for logic operation and then driven back to reset. When A and B
are both ’1’, the MAB is driven to _x_ = 400 nm for logic operation and then
driven back to reset. The simulation results of the NAND-NOR logic gate with
different inputs are shown in Fig. 4(b)–(e), respectively. The output result
of A = 0 and B = 0 is shown in Fig. 4(b). The MAB is still at _x_ = $-600$ nm
because of the zero-field force. The amplitudes of the SWs are over 1000 A/m,
both for C and D ports. This indicates that the output ports C and D are both
’1’. The simulation result of A = 1 and B = 0 is shown in Fig. 4(c). The MAB
is driven from _x_ = $-600$ nm to _x_ = $-100$ nm for positive pulse during 9
ns – 9.5 ns. Then, the amplitude of the SWs at C port is slightly reduced
before the negative reset pulse, which can be explained by the interference
between the reflected SWs from MAB and the initial SWs. For the D port, the
amplitude of the SWs is almost vanishing. So we can recognize the output C = 1
and D = 0 before the reset of the MAB. Fig. 4(d) shows the simulation result
for A = 0 and B = 1. The output is same as A = 1 and B = 0. The last
configuration of A = 1 and B = 1 is shown in Fig. 4(e). In 9 ns – 10 ns, the
MAB is driven to _x_ = 400 nm from _x_ = $-600$ nm. After oscillation in 10 ns
– 11 ns, the amplitudes of C and D in 11 ns – 11.5 ns are the operation result
representing C = 0 and D = 0. In 11.5 ns – 12.5 ns, the MAB is driven back to
_x_ = $-600$ nm. After oscillation in 12.5 ns – 13.5 ns, one logic operation
is finished. The logic operation of the NAND-NOR gate can also be divided into
three stages, preparation before output (9 ns – 11 ns), output (11 ns – 11.5
ns) and reset (11.5ns – 13.5 ns). One circle logic operation needs 4.5 ns. The
truth table of the NAND-NOR gate is listed in Fig. 4(f). The input A, B and
output C compose a NAND gate, while the input A, B and output D compose a NOR
gate.

In conclusion, we propose dual function SW logic gates based on the motion of
MAB controlled by electric field. AND-OR gate and NAND-NOR gate are validated
by micromagnetic simulation. Owning to the inherent dual function, the
proposed SW logic gates have potential of building low power consumption and
high density devices. Our results will also motivate further experimental
studies for the development of dual function SW devices.

###### Acknowledgements.

This work is partially supported by National Key Research Program of China
(Grant No. 2017YFA0206202), International Postdoctoral Exchange Fellowship
Program (20190083), Natural Science Foundation of Shaanxi Province
(2021JM-022).

##  Appendix A Dispersion relation of SWs

Kalinikos et alKalinikos and Slavin (1986) developed the theory of dispersion
relation of SWs taking into account both dipole-dipole and exchange
interactions. The general formula of dispersion relation of DE SWs is :

$f=\frac{\gamma\mu_{0}}{2\pi}\sqrt{(H_{i}+\frac{2A_{\mathrm{ex}}}{\mu_{0}M_{\mathrm{s}}}k^{2})(H_{i}+M_{\mathrm{s}}+\frac{2A_{\mathrm{ex}}}{\mu_{0}M_{\mathrm{s}}}k^{2})+M^{2}_{\mathrm{s}}F}$
(1)

where $\gamma$ is the gyromagnetic ratio, $\mu_{0}$ is vacuum permeability,
$H_{i}$ is the internal static magnetic field, $A_{\mathrm{ex}}$ is exchange
stiffness constant, $M_{\mathrm{s}}$ is saturation magnetization, $k$ is the
wave vector of SWs,
$F=\frac{1-\mathrm{e}^{-kd}}{kd}(1-\frac{1-\mathrm{e}^{-kd}}{kd})$, $d$ is the
thickness of the film. This formula doesn’t include magnetic anisotropy field.
So we need calculate magnetic anisotropy field and give the formula of
dispersion relation in uniaxial and cubic anisotropy region, respectively.

The dynamic unit magnetization can be written as:
$\mathbf{m}=\frac{\mathbf{M}}{M_{\mathrm{s}}}\approx\hat{e}_{y}+m_{x}\hat{e}_{x}+m_{z}\hat{e}_{z}$,
where $m_{x}$, $m_{z}$ $\ll$ 1\. The magnetic anisotropy fields in uniaxial
and cubic anisotropy regions can be calculated by equation (2) and (3) (ignore
high-order $m_{x}$ and $m_{z}$ terms for linear approximation).

$\mathbf{H}_{\mathrm{u}}=\frac{2K_{\mathrm{u}}}{\mu_{0}M_{\mathrm{s}}}(\mathbf{u}\cdot\mathbf{m})\mathbf{u}=\frac{2K_{\mathrm{u}}}{\mu_{0}M_{\mathrm{s}}}\hat{e}_{y}$
(2)

where $\mathbf{H}_{\mathrm{u}}$ is the uniaxial anisotropy field,
$K_{\mathrm{u}}$ is the uniaxial anisotropy constant, $\mathbf{u}$ is the unit
vector of the easy axis of uniaxial anisotropy, along _y_ axis.

$\displaystyle\mathbf{H}_{\mathrm{c}}=-\frac{2K_{\mathrm{c}}}{\mu_{0}M_{\mathrm{s}}}($
$\displaystyle((\mathbf{c_{2}}\cdot\mathbf{m})^{2}+(\mathbf{c_{3}}\cdot\mathbf{m})^{2})((\mathbf{c_{1}}\cdot\mathbf{m})\mathbf{c_{1}})\quad+$
(3)
$\displaystyle((\mathbf{c_{1}}\cdot\mathbf{m})^{2}+(\mathbf{c_{3}}\cdot\mathbf{m})^{2})((\mathbf{c_{2}}\cdot\mathbf{m})\mathbf{c_{2}})\quad+$
$\displaystyle((\mathbf{c_{1}}\cdot\mathbf{m})^{2}+(\mathbf{c_{2}}\cdot\mathbf{m})^{2})((\mathbf{c_{3}}\cdot\mathbf{m})\mathbf{c_{3}})\quad)$
$\displaystyle=\frac{K_{\mathrm{c}}}{\mu_{0}M_{\mathrm{s}}}m_{x}\hat{e}_{x}-\frac{K_{\mathrm{c}}}{\mu_{0}M_{\mathrm{s}}}\hat{e}_{y}-\frac{2K_{\mathrm{c}}}{\mu_{0}M_{\mathrm{s}}}m_{z}\hat{e}_{z}$

where $\mathbf{H}_{\mathrm{c}}$ is the cubic anisotropy field,
$K_{\mathrm{c}}$ is the cubic anisotropy constant, $\mathbf{c}_{1}$,
$\mathbf{c}_{2}$ and $\mathbf{c}_{3}$ are ($\sqrt{2}/2$, $\sqrt{2}/2$, 0),
($-\sqrt{2}/2$, $\sqrt{2}/2$, 0) and (0, 0, 1), respectively, the unit vector
of the easy axes of cubic anisotropy.

So, the formulas of dispersion relation of DE SWs in uniaxial and cubic
anisotropy region are

$f=\frac{\gamma\mu_{0}}{2\pi}\sqrt{(H_{\mathrm{ext}}+\frac{2K_{\mathrm{u}}}{\mu_{0}M_{\mathrm{s}}}+\frac{2A_{\mathrm{ex}}}{\mu_{0}M_{\mathrm{s}}}k^{2})(H_{\mathrm{ext}}+\frac{2K_{\mathrm{u}}}{\mu_{0}M_{\mathrm{s}}}+M_{\mathrm{s}}+\frac{2A_{\mathrm{ex}}}{\mu_{0}M_{\mathrm{s}}}k^{2})+M^{2}_{\mathrm{s}}F}$
(4)

and

$f=\frac{\gamma\mu_{0}}{2\pi}\sqrt{(H_{\mathrm{ext}}-\frac{2K_{\mathrm{c}}}{\mu_{0}M_{\mathrm{s}}}+\frac{2A_{\mathrm{ex}}}{\mu_{0}M_{\mathrm{s}}}k^{2})(H_{\mathrm{ext}}+\frac{K_{\mathrm{c}}}{\mu_{0}M_{\mathrm{s}}}+M_{\mathrm{s}}+\frac{2A_{\mathrm{ex}}}{\mu_{0}M_{\mathrm{s}}}k^{2})+M^{2}_{\mathrm{s}}F}$
(5)

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