# Generation and robustness of non-local correlations induced by Heisenberg
XYZ and intrinsic decoherence models: $(x,y)$-spin-orbit interactions and
$x$\- magnetic field

F. Aljuaydi  Corresponding author<EMAIL_ADDRESS>S. N. Almutairi
Department of Mathematics, College of Science and Humanities, Prince Sattam
bin Abdulaziz University, Al Kharj 11942, Saudi Arabia  A.-B. A. Mohamed
Department of Mathematics, College of Science and Humanities, Prince Sattam
bin Abdulaziz University, Al Kharj 11942, Saudi Arabia  Department of
Mathematics, Faculty of Science, Assiut University, Assiut, Egypt

###### Abstract

In this work, the Milburn intrinsic decoherence model is used to investigate
the role of spin-spin Heisenberg-XYZ interaction supported by spin-orbit
Dzyaloshinsky–Moriya (DM) interactions of $x$ and $y$-directions together in
the non-local correlation (NLC) dynamics of Local quantum Fisher information
(LQFI), local quantum uncertainty (LQU), and Log-negativity’s entanglement.
The two-qubit-Heisenberg-XYZ (non-X)-states’ non-local correlation generations
are explored under the effects of the uniformity and the inhomogeneity of an
applied $x$-direction external inhomogeneous magnetic field (EIMF). Our
meticulous exploration of the obtained results shows that the spin-spin
Heisenberg XYZ and $x,y$-spin-orbit interactions have a high capability to
raise non-local correlations in the presence of a weak external magnetic
field. The raised non-local correlation can be improved by strengthening the
spin-spin and $x,y$-spin-orbit interactions and increasing the EIMF’s
inhomogeneity and uniformity. Non-local correlation oscillations’ amplitudes
and fluctuations are increased. The degradations of the NLCs’ generations in
the presence of intrinsic decoherence (NLCs’ robustness against intrinsic
decoherence) can be decreased by strengthening the spin-spin interactions.
They can be increased by increasing the intensities of $x,y$-spin-orbit
interactions as well as increasing the EIMF’s inhomogeneity and uniformity.

Keywords: non-local correlation; Heisenberg XYZ states; magnetic fields: spin-
orbit interaction

##  I Introduction

Among the numerous quantum systems proposed to implement quantum information
and computation [1, 2], superconducting circuits, trapped ions, and
semiconductor quantum dots are essential techniques for realizing quantum bits
(qubits). Based on electron spins trapped in quantum dots, a quantum computer
protocol has been initially proposed [3, 4, 5], the electron having a spin of
1/2 is the simplest natural qubit. Recently, quantum computation (as a single-
spin-qubit geometric gate) with electron spins (single-spin-qubit geometric
gate) has been realized in quantum dots [6, 7]. Due to tunneling the electrons
from one to the other, the spin-spin coupling and spin-orbit coupling of the
interaction between two qubits can be realized by considering a two-qubit
system represented by two coupled quantum dots’ two electrons. Therefore,
Heisenberg XYZ models describing spin-spin interactions are among of the
important proposed qubit systems. Two qubit Heisenberg XYZ models have been
realized in different systems, including bosonic atoms inside an optical
lattice [8], trapped ions [9] and superconductor systems [10], and linear
molecules [11]. Heisenberg XYZ models have been updated to include spin-orbit
interactions [12, 13] with the first order of SO coupling ( that is known by
Dzyaloshinsky–Moriya interactions [14] (realizing by an antisymmetric
superexchange La2CuO4 interaction [15]), the second order of SO coupling
(Kaplan-Shekhtman-Entin-Wohlman-Aharony interaction [16]). Spin-1/2 Heisenberg
XYZ models also have been updated to include dipole-dipole interaction [17],
and inhomogeneous external magnetic fields (IEMFs) [18, 19].
Exploring two-qubit information dynamics in different proposed qubit systems
to two-qubit resources, relating to different types of nonlocal correlations
(as entanglement, quantum discored, …), is one of the most required research
fields in implementing quantum information and computation [20]. Quantum
entanglement (QE) (realizing by quantifiers’ entropy [21], concurrence [22],
negativity, and log-negativity [23], …) is an important type of qubits’
nonlocal correlations (NLCs) [24, 25] and applications have an important role
in quantum information fields. Where QE has a wide range of applications in
implementing quantum computation, teleportation [26, 27], quantum optical
memory [28], and quantum key distribution [29]. After implementing quantum
discord as another type of qubits’ NLCs beyond entanglement [30], several
NLCs’ quantifiers have been introduced to address other NLCs [31, 32] by using
Wigner–Yanase (WY) skew information [33] and quantum Fisher information (QFI)
[34]. Where, WY-skew-information minimization (local quantum uncertainty [35]
LQU) and the WY-skew-information maximization (uncertainty-induced nonlocality
[36]) have been introduced to quantify other NLCs beyond entanglement. Also,
the minimization of QFI (local quantum Fisher information, LQFI) was used to
implementing other qubits’ NLCs [37, 38]. LQU has a direct connection to LQFI
[39, 40], establishing more two-qubit NLCs in several proposed qubit systems
[46, 47]: as hybrid-spin systems (under random noise [41] and intrinsic
decoherence [42]), two-coupled double quantum dots [43], the mixed-spin
Heisenberg [44], Heisenberg XXX system [45].
The information dynamics of the two-spin Heisenberg XYZ states have been
investigated, by using Milburn intrinsic decoherence model [48], of
entanglement teleportation based on the Heisenberg XYZ chain [49, 50], Fisher
of Heisenberg XXX states’LQFI beyond IEMF effects [51], quantum correlations
of concurrence and LUQ [52]. The previous works have focused on exploring the
time evolution of the two-spin Heisenberg-XYZ states’ NLCs with limited
conditions on the spin-spin and spin-orbit interactions, and the applied
magnetic fields, to ensure residing two-qubits X-states [53, 54, 55, 56, 57,
58, 59, 60, 61]. Therefore, by using the Milburn intrinsic decoherence and
Heisenberg XYZ models are used to investigate the non-local correlation
dynamics of LQFI, LQU, and log-negativity (LN) for general two-qubit-
Heisenberg-XYZ (non-X)-states, inducing by other specific conditions on the
spin-spin and spin-orbit interactions, as well as the applied magnetic fields.
The manuscript structure is prepared to include the Milburn intrinsic
decoherence equation including the Heisenberg XYZ model and its solution in
Sec. (II). But in Sec. (III), we introduce the definition of the NLCs’
quantifiers of LQFI, LQU, and LN. Sec. (IV) presents the outcomes of the
dependence of the NLCs’ quantifiers on the physical parameters. Our
conclusions are provided in Sec. (V).

##  II The Heisenberg spin model

Here, Milburn intrinsic decoherence model and Heisenberg XYZ model are used to
examine the embedded capabilities in spin-spin interaction and spin-orbit
interaction (that describes Dzyaloshinsky–Moriya (DM) $x,y$-interactions with
the first order of SO couplings $D_{x}$ and $D_{x}$) to generate essential two
SO-qubits’ nonlocal correlations (NCs) under the effects of the uniformity
$B_{m}$ and the inhomogeneity $b_{m}$ of applied external inhomogeneous
magnetic field (EIMF). For two spin-qubits (each $k$-qubit ($k=A,B$) is
described by upper $|1_{k}\rangle$ and lower $|1_{k}\rangle$ states), the
Hamiltonian of the system is written as

$\displaystyle\\!\\!\\!\\!\\!\hat{H}=\\!\\!\sum_{\alpha=x,y,z}J_{\alpha}\hat{\sigma}^{\alpha}_{A}\hat{\sigma}^{\alpha}_{B}+\sum_{k=A,B}\vec{B}_{k}.\vec{\sigma}_{k}+\vec{D}.(\vec{\sigma}_{A}\times\vec{\sigma}_{B}).$
(1)

$\vec{\sigma}_{k}=(\hat{\sigma}_{k}^{x},\hat{\sigma}_{k}^{y},\hat{\sigma}_{k}^{z})$
with $\hat{\sigma}_{k}^{x,y,z}$ represent the $k$-qubit Pauli matrices.
$\vec{B}_{k}=(B^{x}_{k},B^{y}_{k},B^{z}_{k})$ is the vector of the external
magnetic field applying on $k$-spin,
$\vec{B}_{k}.\vec{\sigma}_{k}=B^{x}_{k}\hat{\sigma}_{k}^{x}+B^{y}_{k}\hat{\sigma}_{k}^{y}+B^{z}_{k}\hat{\sigma}_{k}^{z}$.
In our work, we consider that the EIMF is applied only in the $x$-direction:
$\vec{B}_{k}=(B^{x}_{k},0,0)$, $B^{x}_{A}=B_{m}+b_{m}$, and
$B^{x}_{B}=B_{m}-b_{m}$. $\vec{D}=(D_{x},D_{y},D_{z})$ is the spin-orbit/DM
interaction vector. Therefore, we have
$\vec{D}.(\vec{\sigma}_{A}\times\vec{\sigma}_{B})=D_{x}\hat{C}_{x}+D_{y}\hat{C}_{y}+D_{z}\hat{C}_{z}$
with
$\hat{C}_{\alpha}=\hat{\sigma}_{A}^{\alpha+1}\hat{\sigma}_{B}^{\alpha+2}-\hat{\sigma}_{A}^{\alpha+2}\hat{\sigma}_{B}^{\alpha+1}(\alpha=x,y,z)$.
Here, we take only the $x,y$-spin-orbit interactions:
$\vec{D}=(D_{x},D_{y},0)$. The capacitive spin-spin and $x,y$-spin-orbit
interactions, under the effects of the EIMF characteristics, can be used to
build two-spin-qubit correlations. The considered Hamiltonian is written as

$\displaystyle\hat{H}$ $\displaystyle=$
$\displaystyle\sum_{i=x,y,z}J_{i}\hat{\sigma}^{i}_{A}\hat{\sigma}^{i}_{B}+\sum_{i=x,y}D_{i}\hat{C}_{i}$
(2)
$\displaystyle\qquad\quad+(B_{m}+b_{m})\hat{\sigma}^{x}_{A}+(B_{m}-b_{m})\hat{\sigma}^{x}_{B}.$

The time evilution of the two spin-qubits’ nonlocal correlations (NCs) will be
explored by using Milburn intrinsic decoherence model [48], which is given by

$\frac{d}{dt}\hat{M}(t)=-i[\hat{H},\hat{M}]-\frac{\gamma}{2}[\hat{H},[\hat{H},\hat{M}]],$
(3)

$\hat{M}(t)$ is the density matrix of the generated two-spin-qubits state.
$\gamma$ is the intrinsic spin-spin decoherence (ISSD) coupling.
Here, the two-spin-qubits eigenvalues $V_{k}$ ($k=1,2,3,4$) and the
eigenstates $|V_{k}\rangle$ of the Hamiltonian of Eq. (2) will be calculated,
numerically. And hence, in the two-spin-qubits basis:
$\\{|1_{A}1_{B}\rangle,|1_{A}0_{B}\rangle,|0_{A}1_{B}\rangle,|0_{A}0_{B}\rangle\\}$,
the two-spin-qubits state dynamics can obtained numerically by using the
solution of Eq. (3) giving by

$\hat{M}(t)=\\!\\!\sum^{4}_{m,n=1}\\!\\!U_{mn}(t)\,S_{mn}(t)\,\langle
V_{m}|\hat{M}(0)|V_{n}\rangle\,|V_{m}\rangle\langle V_{n}|.$ (4)

The unitary interaction $U_{mn}(t)$ and the ISSD coupling $S_{mn}(t)$ effects
are controlled by the following terms:

$\displaystyle U_{mn}(t)$ $\displaystyle=$ $\displaystyle
e^{-i(V_{m}-V_{n})t},$ $\displaystyle S_{mn}(t)$ $\displaystyle=$
$\displaystyle e^{-\frac{\gamma}{2}(V_{m}-V_{n})^{2}t}.$ (5)

The Eq.(4) is used to calculate and explore, numerically, the dynamics of the
nonlocal correlations residing within the two-spin-qubits states’ Heisenberg
XYZ model under the effects of the $x,y$-spin-orbit interactions (spin-orbit
interactions acting along the $x-$ and $y-$ directions) and an applied
external magnetic field applying along $x$-direction.

##  III Non-local correlation (NLC) quantifiers

Here, the two-spin-qubits’ NLCs will be measured by the following LQFI, LQU,
and logarithmic negativity (LN):

  * •

LQFI
LQFI can be used as a two-spin-Heisenberg-XYZ correlation quantifier beyond
entanglement, which is recently introduced as another correlation type. After
calculating the two-spin eigenvalues $\pi_{k}$ ($k=1,2,3,4$) and the two-spin
eigenstates $|\Pi_{k}\rangle$ of Eq. (4 having the representation matrix:
$M(t)=\sum_{m}\pi_{m}|\Pi_{m}\rangle\langle\Pi_{m}|$ with $\pi_{m}\geq 0$ and
$\sum_{m}\pi_{m}=1$, the LQFI is calculated by using the closed expression
[38, 34, 37] giving by

$F(t)=1-\pi_{R}^{\max},$

$\pi_{R}^{\max}$ represents the highest eigenvalue of the symmetric matrix
$R=[r_{ij}]$. Based on the Pauli spin-$\frac{1}{2}$ matrices
$\sigma^{i}\,(i=1,2,3)$ and the elements
$\xi_{mn}^{i}=\langle\Pi_{m}|I\otimes\sigma^{i}|\Pi_{n}\rangle$, the symmetric
matrix elements $r_{ij}$ are given by

$r_{ij}=\sum_{\pi_{m}+\pi_{n}\neq
0}\frac{2\pi_{m}\pi_{n}}{\pi_{m}+\pi_{n}}\xi_{mn}^{i}(\xi_{nm}^{j})^{\dagger}.$

For a two-spin-qubits maximally correlated state, the LQFI function has
$F(t)=1$. The case of $0<F(t)<1$ means that the states have partial LQFI’s
nonlocal correlation.

  * •

LQU
LQU of Wigner–Yanase (WY) skew information [33] is realized to use as an
another type of two spin-qubits’ nonlocal correlations [33, 35, 36]. For the
two spin-qubits’ density matrix $M(t)$ of Eq. (4), the LQU can be calculated
by [35]

$\displaystyle U(t)$ $\displaystyle=$ $\displaystyle
1-\lambda_{max}(\Lambda_{AB}),$ (6)

$\lambda_{max}$ designs the largest eigenvalue of the $3\text{x}3$-matrix
$\Lambda=[a_{ij}]$, which have the elements:

$\displaystyle a_{ij}=\text{T}r\mathbf{\big{\\{}}\sqrt{M(t)}(\sigma_{i}\otimes
I)\sqrt{M(t)}(\sigma_{j}\otimes I)\mathbf{\big{\\}}}.$

  * •

Logarithmic negativity (LN)
We employ the logarithmic negativity [23] to measure of the generated two-
spin-qubits entanglement. The LN expression is based on the negativity’s
definition $\mu_{t}$ [23] (which is defined as the absolute sum of the
matrix’s negative eigenvalues $(M(t))^{T}$ of the partial transposition of the
two spin qubits density matrix $M(t)$ of Eq. (4). The LN can be expressed as:

$\displaystyle N(t)$ $\displaystyle=$ $\displaystyle\log_{2}[1+2\mu_{t}],$ (7)

The $N(t)=0$ for a disentangled two-spin state, $N(t)=1$ for a maximally
entangled two-spin state, and $0\leq N(t)\leq 1$ for a partially entangled
two-spin state.

##  IV Two spin-Heisenberg-XYZ-qubits dynamics

Figure 1: The dynamics of the generated local-QFI, local-QU, and log-
negativity correlations due to the couplings
$(J_{\alpha},J_{y},J_{z})=(0.8,0.8,0.8)$ are shown under the effects of the
applied magnetic field $(B_{m},b_{m})=(0.3,0.5)$ and the $D_{x,y}$
interactions: $(D_{x},D_{y})=(0.0,0.0)$ in (a), $(D_{x},D_{y})=(0.5,0.0)$ in
(b), and $(D_{x},D_{y})=(0.5,0.5)$ in (c).

Figure 2: The LQFI (red solid curve), LQU (boule dash-dotted curve), and log-
negativity (green dashed curve) dynamics of Fig.1c are plotted for different
Heisenberg-XYZ couplings: $(J_{x},J_{y},J_{z})=(1,0.5,1.5)$ in (a) and
$(J_{x},J_{y},J_{z})=(5,1,1.5)$ in (b), and strong $x,y$-spin-orbit
interactions $D_{x}=D_{y}=2$ in (c).

Figure 3: The LQFI (red solid curve), LQU (boule dash-dotted curve), and log-
negativity (green dashed curve) dynamics of Fig.2a (for
$(J_{x},J_{y},J_{z})=(1,0.5,1.5)$, $(B_{m},b_{m})=(0.3,0.5)$, and
$D_{x}=D_{y}=0.5$) are plotted for different large magnetic-field
uniformities: $B_{m}=2$ in (a) and $B_{m}=10$ in (b).

Figure 4: The LQFI (red solid curve), LQU (boule dash-dotted curve), and log-
negativity (green dashed curve) dynamics of Fig.2a (for
$(J_{x},J_{y},J_{z})=(1,0.5,1.5)$, $(B_{m},b_{m})=(0.3,0.5)$, and
$D_{x}=D_{y}=0.5$) are plotted for different large magnetic-field
inhomogeneities: $b_{m}=2$ in (a) and $b_{m}=10$ in (b).

Figure 5: The LQFI (red solid curve), LQU (boule dash-dotted curve), and log-
negativity (green dashed curve) dynamics is shown in the presence of the ISSD
$\gamma=0.05$ and the magnetic field $(B_{m},b_{m})=(0.3,0.5)$ with the
couplings $J_{\alpha}=0.8$ for different couplings: $D_{k}=0(k=x,y)$ in (a),
$D_{k}=0.5$ in (b), and $D_{k}=2$ in (c).

Figure 6: The two spin-qubits correlation dynamics of the of Fig.5b and c is
shown but for strong spin-spin couplings $(J_{x},J_{y},J_{z})=(1,0.5,1.5)$.

Figure 7: The dynamics of the LQFI (red solid curve), LQU (boule dash-dotted
curve), and log-negativity (green dashed curve) of Fig.3a is shown but for
large EIMF’s uniformity $(B_{m},b_{m})=(2,0.5)$ in (a) and large EIMF’s
inhomogeneity $(B_{m},b_{m})=(0.3,2)$ in (b).

Here, we will explore the role of $J_{\alpha}$-spin-spin interactions
supported by $x,y$-spin-orbit interactions in the generation dynamics of the
two-qubit non-local correlations (of LQFI, local LQU, and LN’s entanglement
general Heisenberg-XYZ (non-X)-states in the presence of an $x-$direction
EIMF. To explore the generation of the two-spin-qubits non-local correlations,
we consider that the two spins are initially in their uncorrelated upper
states $|1_{A}\rangle\otimes|1_{B}\rangle$, which its density matrix has no
nonlocal correlations of the considered quantifiers. Our focus is on the
$J_{\alpha}$-spin-spin interaction effects ($D_{x}$ and $D_{y}$), and
inhomogeneous $x-$direction magnetic field parameters $(B_{m}$ and $b_{m})$ in
the presence of the intrinsic spin decoherence (ISSD) coupling.
As our first analysis, we display the dynamics of the two spin qubits nonlocal
correlations of the LQFI, LQU, and LN, generating due to the couplings
$(J_{x},J_{y},J_{z})=(0.8,0.8,0.8)$ supported by different intensities of
$x,y$-spin-orbit interactions in the presence of the inhomogeneous
$x-$direction magnetic field having weak uniformity and inhomogeneity
$(B_{m},b_{m})=(0.3,0.5)$. In the absence of the intrinsic spin-spin
decoherence $\gamma=0$ and the $x,y$-spin-orbit interactions
$(D_{x},D_{y})=(0.0,0.0)$, the Fig.1(a) illustrates that the two-spin-qubits
LQFI, LQU, and log-negativity grow to reach their maximum. The nonlocal
correlations of the LQFI, LQU, and log-negativity undergo slow quasi-regular
oscillations having the same frequencies and different amplitudes. LQFI and
LQU have the same behavior, i.e., the two-spin-qubits correlation is called
the ”Fisher-Wigner–Yanase nonlocal correlation”. The log-negativity amplitude
is always greater than those of the LQFI and LQU. Under these circumstances of
weak coupling regime of $J_{\alpha}=0.8$ and the applied inhomogeneous
$x-$direction magnetic field (weak uniformity and inhomogeneity), the initial
pure-uncorrelated two-spin state undergoes different time-dependent partially
correlated states, except particular time, it transforms maximally correlated
states. The two-spin states have maximal correlations of Fisher-Wigner–Yanase
nonlocal correlation ($F(t)=U(t)=1$) and log-negativity $N(t)=1$ at the same
time. At particular times, we observe that partially two-spin entangled states
have no LQFI or LQU correlation.
The effects of weak intensities of $x,y$-spin-orbit interactions are shown in
Fig.1(b). As is clear in this figure, the regularity and fluctuations of the
generated Fisher-Wigner–Yanase and log-negativity nonlocal correlations are
substantially more than previously presented in the absence of the $x,y$-spin-
orbit interaction. The weak $D_{x}$-spin-orbit interaction
$(D_{x},D_{y})=(0.5,0)$ dramatically improves the appearance of the intervals
of the maximal Fisher-Wigner–Yanase and log-negativity nonlocal correlations,
as well as the intervals in which two-spin entangled states have no LQFI or
LQU correlation. In Fig.1(c), we combined the $D_{x}-$ and $D_{y}-$spin-orbit
interactions $(D_{x},D_{y})=(0.5,0.5)$ into $x,y$-spin-orbit interactions. As
is clear in this figure, the NLC fluctuations between their partial and
maximal values are substantially fewer than previous results in Figs.1(a,b).
Furthermore, the NLC frequency has been reduced while the lower bounds of the
Fisher-Wigner–Yanase and log-negativity nonlocal correlations are shifted up.
This means that the combined $D_{x}-$ and $D_{y}-$spin-orbit interactions
$(D_{x},D_{y})=(0.5,0.5)$ improves the generated partial two-spin-qubits
Fisher-Wigner–Yanase and log-negativity correlations.
Fig.2 shows that the higher couplings of $J_{\alpha}$-spin-spin interactions
and $x,y$-spin-orbit interactions ($(J_{x},J_{y},J_{z})=(1,0.5,1.5)$ in (a)
and $(J_{x},J_{y},J_{z})=(5,1,1.5)$ in (b), and strong $D_{x,y}$-spin-orbit
interaction $D_{x}=D_{y}=2$ in (c)) have a high ability to enhancing the
arisen two-spin-qubits’ NLC of the Fisher-Wigner–Yanase and log-negativity. By
comparing the generated spin-spin NLCs showing in Figs.1(c) and 2(a), we find
that the relative strong couplings of $J_{\alpha}$-spin-spin interactions
$(J_{x},J_{y},J_{z})=(1,0.5,1.5)$ (which are supported by a weak
$D_{x,y}$-spin-orbit interactions $(D_{x},D_{y})=(0.5,0.5)$) increases the
amplitudes and frequencies of the Fisher-Wigner–Yanase and log-negativity’s
oscillations. Fig.2 shows that the higher $J_{\alpha}$-couplings lead to the
spin-spin NLCs’ oscillations have more regularity and fluctuations. The time
positions of the maximal Fisher-Wigner–Yanase and log-negativity correlations
are enhanced. Fig.2(c) is plotted to see the capability the increase of the
$x,y$-spin-orbit interactions $D_{x}=D_{y}=2$ supporting by weak spin-spin
interactions $J_{\alpha}=0.8)$ to enhance the generated spin-spin NLCs when
the external magnetic field applied with weak determinants
$(B_{m},b_{m})=(0.3,0.5)$. By comparing the qualitative dynamics of the
generated Fisher-Wigner–Yanase and log-negativity correlations shown in Fig.1c
($D_{x}=D_{y}=0.5$) with that shown in Fig.2c ($D_{x}=D_{y}=2$), we can deduce
that the $D_{x,y}$-spin-orbit interactions have a high role in enhancement the
generated Fisher-Wigner–Yanase and log-negativity correlations, their
amplitudes are increased and their oscillations have more fluctuations between
their extreme values. In addition, the strong $x,y$-spin-orbit interactions
potentially strength and speed the generation of the Fisher-Wigner–Yanase and
log-negativity correlations, due to the $J_{\alpha}$-spin-spin interactions.
Figures 3 show Fisher-Wigner–Yanase and log-negativity nonlocal correlation
dynamics of Fig.2a (where $(J_{x},J_{y},J_{z})=(1,0.5,1.5)$, $b_{m}=0.5$, and
$D_{x}=D_{y}=0.5$) are plotted for different uniformities of the applied EIMF.
Fig. 3a shows the generated Fisher-Wigner–Yanase and log-negativity
correlations, due to the $J_{\alpha}$-spin-spin and $x,y$-spin-orbit
interactions, previously presented after applying an external magnetic field
(having a small inhomogeneity $b_{m}=0.5$ and a large uniformity $B_{f}=2$).
In this case, the increase of the EIMF uniformity delays the growth of the
LQFI, LQU as well as log-negativity. It increases the two-spin state’s
fluctuations between different partially and maximally correlated states. The
generations of the Fisher-Wigner–Yanase and log-negativity correlations shown
in Fig.2a (with $B_{m}=0.5$) and Fig.3a (with $B_{m}=2$) with those shown in
Fig.3b (with $B_{m}=10$) confirm that the increase of the EIMF uniformity will
enhance the ability of the strong $J_{\alpha}$-spin-spin interactions
supported by weak $x,y$-spin-orbit interactions to create partially and
maximally correlated states with more stability; however, the generated spin-
spin NLCs are more sensitive to the EIMF uniformity.
In the forthcoming analysis of Fig.4, we keep the system with the same
parameters’ values of Fig.2a (where $(J_{x},J_{y},J_{z})=(1,0.5,1.5)$,
$B_{m}=0.3$, and $D_{x}=D_{y}=0.5$) and consider different magnetic-field
inhomogeneities: $b_{m}=2$ in (a) and $b_{m}=10$ in (b). In this case of
Fig.4a, we notice that the larger EIMF uniformities enhance the efficiency of
the generation of the Fisher-Wigner–Yanase and log-negativity correlations.
The EIMF uniformity increases the two-spin state’s fluctuations between
different partially and maximally correlated states. The time positions of the
maxima ($F(t)=U(t)=N(t)\approx 1$ and minima (zero-value)
($F(t)=U(t)=N(t)\approx 0$ of the generated Fisher-Wigner–Yanase and log-
negativity correlations are enhanced. In the case of
$(J_{x},J_{y},J_{z})=(1,0.5,1.5)$, the increase of the EIMF inhomogeneity have
a high role in enhancement the generated two-spin-qubits’ NLCs. Where NLCs
oscillations’ amplitudes and fluctuations are increased (see Fig.4b).
The next illustration of Figs.5-7 is obtained to show the nonlocal correlation
dynamics of the LQFI, LQU, and log-negativity in the presence of the non-zero
ISSD coupling $\gamma=0.05$. By comparing the results of Fig.1a ($\gamma=0.0$)
with these of Fig.5a ($\gamma=0.05$), we find that the LQFI, LQU, and log-
negativity shows a decaying oscillatory dynamical evolutions. The generations
of the Heisenberg-XYZ (non-X)-states’NLCs (due to $J_{\alpha}=0.8$ spin-spin
couplings the applied magnetic field $(B_{m},b_{m})=(0.3,0.5)$ without spin-
orbit interaction) are weaken and have different amplitudes (which are
decreased by increasing the ISSD coupling). After particular interval time,
with non-zero ISSD coupling, the LQFI and LQU present different nonlocal
correlations having different amplitudes with the same behaviors. Moreover,
the NLCs’ robustness (against the ISSD effect) of the LQFI and log-negativity
is more than that of the LQU.
As shown in Figs.5b and c, the increase of the intensities of $x,y$-spin-orbit
interactions ($D_{k}=0(k=x,y)$ in (a), $D_{k}=0.5$ in (b), and $D_{k}=2$ in
(c)) reduce the NLCs’ robustness (against the ISSD effect) of the LQFI, LQU
and log-negativity correlation, the NLCs’ amplitudes significantly decrease as
the $x,y$-spin-orbit interactions increase. Moreover, LQFI and LQU display
sudden changes at different times. The sudden-changes phenomenon has been
studied theoretically [62] and experimentally [63] (see Figs.5b and c). For
very strong $x,y$-spin-orbit interactions $D_{k}=2$ (see Fig.5c), we observe
that the two-spin-qubits log-negativity drops instantly to zero at a
particular time for a long time (sudden-death LN-entanglement phenomenon),
then the disentangled two-spin states have only different stable NLCs of the
LQFI and LQU. We can deduce that the NLCs’ decay resulting from ISSD can be
enhanced by increasing the intensities of $x,y$-spin-orbit interactions.
By comparing the Figs.5(b,c) and Figs.6(b,c), we find that, the strong spin-
spin couplings $(J_{x},J_{y},J_{z})=(1,0.5,1.5)$ reduce the ISSD effect and
improve the NLCs’ robustness (against the ISSD effect) of the LQFI, LQU, and
LN. For very strong $x,y$-spin-orbit interactions $D_{k}=2$ (see Fig.6b), the
sudden-death LN-entanglement phenomenon does not occur, except at the time
$t\approx 0.5\pi$ it occurs instantaneously. The generated two-spin states
have different stable partial NLCs of the LQFI, LQU, and LN. In this case, the
NLCs’ decay resulting from ISSD can be weakened by strengthening the spin-spin
interactions.
In the presence of the ISSD effect $\gamma=0.05$, Fig.7 shows the generated
NLCs of the Fig.3a (or it shows the degradation of NLCs of Fig.6a) after
strengthening the EIMF’s uniformity $(B_{m},b_{m})=(2,0.5)$ in (a) and EIMF’s
inhomogeneity $(B_{m},b_{m})=(0.3,2)$ in (b). From Fig.7a, we observe that the
large EIMF’s uniformity $B_{m}=2$ increases the NLCs’ decay resulting from
ISSD. The time intervals in which the disentangled two-spin states have only
different stable NLCs of the LQFI and LQU appeared. Moreover, the NLCs’
robustness (against the ISSD effect) of the LQFI and LN is reduced by
increasing the large EIMF’s uniformity. The outcomes of Fig.7b illustrate that
strengthening the EIMF’s inhomogeneity $b_{m}=2$ also increases the
degradation of the NLC functions. In this case of the parameters: $b_{m}=2$,
$(J_{x},J_{y},J_{z})=(1,0.5,1.5)$, and $D_{k}=0.5$, we observe that: (1) the
generated NLCs (LQFI, LQU and entanglement) of the Fig.3a degrade (due to and
ISSD effect) and reach their partial stable oscillatory behaviors, quickly,
comparing with the case where the small value of $b_{m}=0.5$ of Fig.6a. We
find that the ability of the EIMF’s inhomogeneity to enhance the ISSD effect
is small compared to that of the EIMF’s uniformity.

##  V Conclusion

In this investigation, the Milburn intrinsic decoherence model and Heisenberg
XYZ model are used to examine the embedded capabilities in spin-spin
interaction and spin-orbit interaction (that describes $x,y$-DM interactions)
to generate nonlocal correlations (realizing by LQFI, LQU, and LN) of general
two-spin-qubits (non-X)-states under the effects of the EIMF’s uniformity and
the inhomogeneity. In the presence and absence of the ISSD, the dependence of
the generated nonlocal correlations on the parameters, of spin-spin
interaction and spin-orbit interactions as well as of the EIMF’s uniformity
and the inhomogeneity, are explored. It is found that the spin-spin Heisenberg
XYZ and $x,y$-spin-orbit interactions have a high capability to raise non-
local correlations in the presence of a weak external magnetic field. The
spin-orbit interactions have a high role in the enhancement of the generated
two-spin-qubits Fisher-Wigner–Yanase and log-negativity correlations, their
oscillations’ amplitudes and fluctuations are increased. In the presence and
absence of the ISSD, the NLCs’ generations are weakened and have different
amplitudes, decreasing by increasing the ISSD coupling. The NLCs’ robustness
(against the ISSD effect) of the LQFI and log-negativity is more than that of
the LQU. The phenomenon of the sudden changes occurs during the LQU and LQFI
dynamics whereas the sudden death occurs during log-negativity-entanglement
dynamics. The NLCs’ decay resulting from ISSD can be enhanced by increasing
the intensities of $x,y$-spin-orbit interactions. Strengthening the spin-spin
interactions weakens the NLCs’ decay resulting from ISSD. The generated NLCs
degrade (due to an ISSD effect with a large IMF’s inhomogeneity) and reach
their partially stable oscillatory behaviors, quickly. The ability of the
IMF’s inhomogeneity to increase the ISSD effect is small compared to that of
the EIMF’s uniformity.

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# Perturbation analysis for Moore-Penrose inverse of closed operators on
Hilbert spaces

FAPENG DU
School of Mathematical and Physical Sciences, Xuzhou Institute of Technology
Xuzhou 221008, Jiangsu Province, P.R. China
E-mail<EMAIL_ADDRESS>YIFENG XUE
Department of mathematics, East China Normal University
Shanghai 200241, P.R. China

###### Abstract

In this paper, we investigate the perturbation for the Moore-Penrose inverse
of closed operators on Hilbert spaces. By virtue of a new inner product
defined on $H$, we give the expression of the Moore-Penrose inverse
$\bar{T}^{\dagger}$ and the upper bounds of $\|\bar{T}^{\dagger}\|$ and
$\|\bar{T}^{\dagger}-T^{\dagger}\|$. These results obtained in this paper
extend and improve many related results in this area.

2000 Mathematics Subject Classification: 15A09, 47A55

Key words: generalized inverse, Moore-Penrose inverse, stable perturbation,
closed operators

##  1 Introduction

An operator $\bar{T}=T+\delta T$ is called the stable perturbation of $T$ if
$R(\bar{T})\cap N(T^{+})=\\{0\\}$. This notation is introduced by Chen and the
second author in [2, 3]. Later it is generalized to the Banach algebra by the
second author in [15] and to Hilbert $C^{*}$–module by Xu, Wei and Gu in [17].
Using this notation the upper bounds for generalized inverse or Moore–Penrose
inverse of bounded linear operators are discussed(See all references). A
classical result about upper bounds is

$\|\bar{T}^{\dagger}\|\leq\frac{\|T^{\dagger}\|}{1-\|T^{\dagger}\|\|\delta
T\|},\quad\frac{\|\bar{T}^{\dagger}-T^{\dagger}\|}{\|T^{\dagger}\|}\leq\frac{1+\sqrt{5}}{2}\frac{\|T^{\dagger}\|}{1-\|T^{\dagger}\|\|\delta
T\|}.$

In recent years, the perturbation analysis for generalized inverses of closed
operators has been appeared. Some results similar to the perturbation analysis
of bounded linear operators are obtained when $\delta T$ is a T–bounded linear
operator(see [9],[10],[13]).

But there are some unsolved questions. What is the result of the perturbation
for closed operators $T\in C(X,Y)$ when $\delta T$ is a linear operators? What
is the expression of the Moore-Penrose inverse $(T+\delta T)^{\dagger}$ and
how to estimate the upper bounds of $\|\bar{T}^{\dagger}\|$ and
$\|\bar{T}^{\dagger}-T^{\dagger}\|$ when $X,\,Y$ are Hilbert spaces ? The
first question has been solved in [7]. Now we discuss the second question in
this paper.

Let $H,K$ be Hilbert spaces, $T\in C(H,K)$ defined on $D(T)$, $\delta T\in
L(H,K)$ be a linear operators. We introduce a new norm $\|\cdot\|_{T}$ on
$D(T)$ such that $(D(T),\|\cdot\|_{T})$ be a Hilbert spaces and give the
expression of $(T+\delta T)^{\dagger}$ and the upper bounds of
$\|\bar{T}^{\dagger}\|$ and $\|\bar{T}^{\dagger}-T^{\dagger}\|$ when $\delta
T$ is a bounded linear operators on $(D(T),\|\cdot\|_{T})$.

##  2 Preliminaries

Let $X,Y$ be Banach spaces, $L(X,Y),\,C(X,Y)$ and $B(X,Y)$ denote the set of
linear operators, densely-defined closed operators and bounded linear
operators from $X$ to $Y$, respectively. For an operator $T\in L(X,Y)$,
$D(T),\,R(T),\,\ker T$ denoted by the domain, the range and the null spaces of
$T$, respectively.

Let $V$ be a closed subspace of $X$. Recall that $V$ is complemented in $X$ if
there is a closed subspace $U$ in $X$ such that $V\cap U=\\{0\\}$ and $X=V+U$.
In this case, we set $X=V\dotplus U$ and $U=V^{c}$.

###### Definition 2.1

[7] Let $T\in C(X,Y)$. If there is $S\in C(Y,X)$ with $D(S)\supset R(T)$ and
$R(S)\subset D(T)$ such that

$TST=T\;\text{on}\ D(T),\quad STS=S\ \text{on}\;D(S),$

then $S$ is called a generalized inverse of $T$, which is also denoted by
$T^{+}$.

Clearly, $P=I-ST$ (resp. $Q=TS$) are idempotent operators on $D(T)$ (resp.
$D(S)$) with $R(P)=\ker T$ (resp. $R(Q)=R(T)$).

###### Proposition 2.1

Let $T\in C(X,Y)$. Then $T^{+}\in C(Y,X)$ exists if and only if

$X=\ker T\oplus\overline{R(T^{+})},\quad Y=\overline{R(T)}\oplus\ker T^{+}.$

In addition, $T^{+}$ is bounded if $R(T)$ closed.

Proof. $(\Rightarrow).$ If $T^{+}\in C(Y,X)$, then we have

$D(T)=R(S)+\ker T,\quad D(S)=R(T)+\ker S.$

So the assertion follows since $D(T)$ (rsep. $D(S)$) are densely in $X$(rsep.
$Y$).

$(\Leftarrow).$ See Proposition 2.2 in [7].

###### Lemma 2.1

[7] Let $T\in C(X,Y)$ such that $T^{+}$ exists. Let $\delta T\colon D(\delta
T)\rightarrow D(T^{+})$ be a linear operators. Assume that $I+\delta
TT^{+}\colon D(T^{+})\rightarrow D(T^{+})$ is bijective. Put $\bar{T}=T+\delta
T$ and $G=T^{+}(I+\delta TT^{+})^{-1}$. Then the following statements are
equivalent:

  1. $(1)$

$R(\bar{T})\cap\ker T^{+}=\\{0\\};$

  2. $(2)$

$\bar{T}G\bar{T}=\bar{T},\;G\bar{T}G=G$ and $R(\bar{T}^{+})=R(T^{+})$,
$\ker\bar{T}^{+}=\ker T^{+}$.

  3. $(3)$

$(I+\delta TT^{+})^{-1}\bar{T}$ maps $\ker T$ into $R(T);$

  4. $(4)$

$(I+\delta TT^{+})^{-1}R(\bar{T})=R(T);$

  5. $(5)$

$(I+T^{+}\delta T)^{-1}\ker T=\ker\bar{T}$.

Let $H$ and $K$ be Hilbert spaces. For $T\in C(H,K)$, let
$P_{\overline{R(T)}}$ (resp. $P_{\ker T}$) denote the orthogonal projection
from $K$ (resp. $H$) to $\overline{R(T)}$ (resp. $\ker T$).

###### Definition 2.2

Let $T\in C(H,K)$. Then there is a unique $S\in C(K,H)$ with
$D(S)=R(T)+R(T)^{\perp}$ and $R(S)=\ker T^{\perp}\cap D(T)$ such that

$\displaystyle TST$ $\displaystyle=T\ \text{on}\ D(T),\,\ $ $\displaystyle\,\
STS$ $\displaystyle=S\ \text{on}\ D(S),$ $\displaystyle TS$
$\displaystyle=P_{\overline{R(T)}}\ \text{on}\ D(S),\,\ $ $\displaystyle\,\
ST$ $\displaystyle=I-P_{\ker T}\ \text{on}\ D(T).$

The operator $S$ is called the Moore–Penrose inverse of $T$, denoted by
$T^{\dagger}$. Clearly, $\ker T^{\dagger}=R(T)^{\perp}$ and
$R(T^{\dagger})=\ker T^{\perp}\cap D(T)$. In addition, if $R(T)$ is closed,
then $S$ is bounded.

##  3 Perturbation analysis of M-P inverse on Hilbert spaces

In this section, we investigate the expression of M-P inverse
$\bar{T}^{\dagger}$ and the upper bound of $\|\bar{T}^{\dagger}\|$ and
$\|\bar{T}^{\dagger}-T^{\dagger}\|$.

$\forall x\in H$, let

$\|x\|_{G}=\|x\|+\|Tx\|,$

then we know $T$ is closed if and only if $(D(T),\|\cdot\|_{G})$ is a Banach
space([11, P191]). Clearly $T$ is a bounded linear operators on
$(D(T),\|\cdot\|_{G})$ since $\|Tx\|\leq\|x\|_{G}$.

Denote $(\cdot,\cdot)_{H}$ be a inner product on $H$. $\forall x,y\in D(T)$,
let

$(x,y)_{T}=(x,y)_{H}+(Tx,Ty)_{K}.$

It is easy to check that $(x,y)_{T}$ is a inner product on $D(T)$.

Let

$\|x\|^{2}_{T}=(x,x)_{T},$

then

$\|x\|^{2}_{T}=(x,x)_{T}=(x,x)_{H}+(Tx,Tx)_{K}=\|x\|^{2}+\|Tx\|^{2},$

that is,

$\|x\|_{T}=(\|x\|^{2}+\|Tx\|^{2})^{\frac{1}{2}}.$

Since

$\frac{\sqrt{2}}{2}\|x\|_{G}\leq\|x\|_{T}\leq\|x\|_{G},$

we know $\|\cdot\|_{G}$ equivalence to $\|\cdot\|_{T}$. So $T$ is closed if
and only if $(D(T),\|\cdot\|_{T})$ is a Hilbert space. For convenience, we
denote $(D(T),\|\cdot\|_{T})$ by $D_{T}$ in the context.

Consider a mapping as following:

$\displaystyle\tau:D(T)\subset H\rightarrow D_{T}$ $\displaystyle\tau
x=x,\quad\forall x\in D(T)$

Clearly, $\tau$ is defined on $D(T)$ and $R(\tau)=D_{T}$.

Let $x_{n}\subset D(T)$ and $x_{n}\xrightarrow{\|\cdot\|}x,\;\tau
x_{n}\xrightarrow{\|\cdot\|_{T}}y$, then

$0\leftarrow\|\tau x_{n}-y\|^{2}_{T}=\|x_{n}-y\|^{2}+\|T(x_{n}-y)\|^{2}.$

So $\|x_{n}-y\|\rightarrow 0$. This indicate $y=\tau x=x\in D(T)$. Hence,
$\tau\in C(H,D_{T})$.

Clearly,

$\displaystyle\tau^{\dagger}$ $\displaystyle=\rho\in B(D_{T},H);$
$\displaystyle\rho x$ $\displaystyle=x,\;x\in D_{T}.$

###### Lemma 3.1

[6] Let $A\in C(L,K),B\in C(H,L)$ with $R(A),R(B),R(AB)$ closed and
$R(B)\subseteq D(A)$. Assume that $AB\in C(H,K)$. Then

$\displaystyle(AB)^{\dagger}$
$\displaystyle=P_{\ker(AB)^{\perp}}(B^{\dagger}(A^{\dagger}ABB^{\dagger})^{\dagger}A^{\dagger})\times$
$\displaystyle\\{A(A^{\dagger}ABB^{\dagger})(A^{\dagger}ABB^{\dagger})^{\dagger}A^{\dagger}+(A^{\dagger})^{*}(A^{\dagger}ABB^{\dagger})(A^{\dagger}ABB^{\dagger})^{\dagger}A^{*}-I\\}^{-1}.$

###### Lemma 3.2

Let $T\in C(H,K)$, then $T^{+}\in B(K,H)$ if and only if $T^{+}\in
B(K,D_{T})$, and in this case

$\|T^{+}\|^{2}\leq\|T^{+}\|^{2}_{T}\leq\|T^{+}\|^{2}+\|TT^{+}\|^{2}.$

Proof.

If $T^{+}\in B(K,H)$, then $TT^{+}\in B(K)$. $\forall x\in K$,

$\|T^{+}x\|^{2}_{T}=\|T^{+}x\|^{2}+\|T(T^{+}x)\|^{2}\leq(\|T^{+}\|^{2}+\|TT^{+}\|^{2})\|x\|^{2}.$

Hence, $T^{+}\in B(K,D_{T})$ and
$\|T^{+}\|^{2}_{T}\leq\|T^{+}\|^{2}+\|TT^{+}\|^{2}$.

Conversely, if $T^{+}\in B(K,D_{T})$, then $\forall x\in K$,

$\|T^{+}x\|^{2}=\|T^{+}x\|^{2}_{T}-\|TT^{+}x\|^{2}\leq\|T^{+}x\|^{2}_{T}.$

Hence, $T^{+}\in B(K,H)$ and $\|T^{+}\|\leq\|T^{+}\|_{T}$.

From the above, we have

$\|T^{+}\|^{2}\leq\|T^{+}\|^{2}_{T}\leq\|T^{+}\|^{2}+\|TT^{+}\|^{2}.$

###### Lemma 3.3

Let $T\in C(H,K)$ with $R(T)$ closed. If $T$ has generalized inverse $T^{+}$,
then $T^{\dagger}\in B(K,H)$ and

$T^{\dagger}=-P_{\ker T^{\perp}}(I+P(I-P-P^{*})^{-1})T^{+}(I-Q-Q^{*})^{-1}.$

Proof. Since $R(T)$ closed, we have $T^{+}\in B(K,H)$. So $T^{+}\in
B(K,D_{T})$ by Lemma 3.2. Thus, $Q=TT^{+}\in B(K),\;P=I-T^{+}T\in B(D_{T})$
are idempotent operators. Now we consider the Moore-Penrose inverse
$T^{\dagger}$ of $T$ on $D_{T}$. From [4], we have $T^{\dagger}\in B(K,D_{T})$
and

$T^{\dagger}=-(I+P(I-P-P^{*})^{-1})T^{+}(I-Q-Q^{*})^{-1}.$

Since $T^{\dagger}\in B(K,D_{T})$, we have $T^{\dagger}\in B(K,H)$ by Lemma
3.2. Noting that $T\in C(H,K)$ is a compound operator by $T\in B(D_{T},K)$ and
$\tau\in C(H,D_{T})$. Therefore, by Lemma 3.1, we have

$T^{\dagger}=-P_{\ker T^{\perp}}(I+P(I-P-P^{*})^{-1})T^{+}(I-Q-Q^{*})^{-1}.$

###### Theorem 3.1

Let $T\in C(H,K)$ with $T^{\dagger}\in B(K,H)$, $\delta T\in B(D_{T},K)$ such
that $\bar{T}=T+\delta T$ closed, $D(T)\subseteq D(\delta T)$. If $I+\delta
TT^{\dagger}$ is invertible and $R(\bar{T})\cap N(T^{\dagger})=\\{0\\}$, then
$\bar{T}^{\dagger}\in B(K,H)$ and

$\bar{T}^{\dagger}=-P_{\ker\bar{T}^{\perp}}(I+\bar{P}(I-\bar{P}-\bar{P}^{*})^{-1})G(I-\bar{T}G-(\bar{T}G)^{*})^{-1},$

where $G=T^{\dagger}(I+\delta TT^{\dagger})^{-1},\;\bar{P}=I-G\bar{T}$.

Proof. $\forall x\in D(T)$, there is an $M$ such that $\|\delta Tx\|\leq
M\|x\|_{T}$ since $\delta T\in B(D_{T},K)$. Thus, $\forall y\in K$,

$\|\delta TT^{\dagger}y\|^{2}\leq\|\delta
T\|^{2}_{T}\|T^{\dagger}y\|^{2}_{T}\leq\|\delta
T\|^{2}_{T}(\|T^{\dagger}\|^{2}+1)\|y\|^{2}.$

Hence $G=T^{\dagger}(I+\delta TT^{\dagger})^{-1}\in B(K,H)$ be the generalized
inverse of $\bar{T}$ by Lemma 2.1. By Lemma 3.3, $\bar{T}^{\dagger}\in B(K,H)$
and

$\bar{T}^{\dagger}=-P_{\ker\bar{T}^{\perp}}(I+\bar{P}(I-\bar{P}-\bar{P}^{*})^{-1})G(I-\bar{T}G-(\bar{T}G)^{*})^{-1}.$

###### Remark 3.1

If $\delta T$ is T–bounded, i.e., there are constants $a,\,b>0$ such that

$\|\delta Tx\|\leq a\|x\|+b\|Tx\|,\quad\forall\,x\in D(T),$

then $\delta T\in B(D_{T},K)$. Indeed,

$\|\delta Tx\|^{2}\leq(a\|x\|+b\|Tx\|)^{2}\leq
2(\max(a,b))^{2}(\|x\|^{2}+\|Tx\|^{2})=2(\max(a,b))^{2}\|x\|^{2}_{T}.$

Let $M,N$ are two closed subspaces of $H$. Set

$\delta(M,N)=\sup\\{dist(\mu,N)|\|\mu\|=1,\mu\in M\\}.$

We call $\hat{\delta}(M,N)=\max\\{\delta(M,N),\delta(N,M)\\}$ the gap between
subspaces $M$ and $N$.

###### Proposition 3.1

[11]

(1)

$\delta(M,N)=0$ if and only if $M\subset N$

(2)

$\hat{\delta}(M,N)=0$ if and only if $M=N$

(3)

$\hat{\delta}(M,N)=\hat{\delta}(N,M)$

(4)

$0\leq\delta(M,N)\leq 1$, $0\leq\hat{\delta}(M,N)\leq 1$

(5)

$\hat{\delta}(M,N)=\|P-Q\|$, where $P,Q$ are orthogonal projection on $M,N$,
respectively.

For convenience, we set $\|\delta T\|_{T}=\underset{\|x\|_{T}\leq
1}{\sup}\dfrac{\|\delta Tx\|}{\|x\|_{T}}$ if $\delta T\in B(D_{T},K)$.

###### Lemma 3.4

Under the assumptions of Theorem 3.1, we have

  1. 1.

$\delta(R(T),R(\bar{T}))\leq\|\delta
T\|_{T}(\|T^{\dagger}\|^{2}+1)^{\frac{1}{2}}.$

  2. 2.

$\delta(\ker T,\ker(\bar{T}))\leq\|\bar{T}^{\dagger}\|\|\delta T\|_{T}.$

Proof. Noting that $\|\delta Tx\|\leq\|\delta T\|_{T}\|x\|_{T}$ and
$\bar{T}^{\dagger}\in B(K,H)$.

$(1).$ Let $u\in R(T)$ with $\|u\|=1$, then there is a $x\in D(T)$ such that
$u=Tx$.

$\displaystyle dist(u,R(\bar{T}))$
$\displaystyle\leq\|u-\bar{T}(T^{\dagger}Tx)\|=\|\delta TT^{\dagger}Tx\|$
$\displaystyle\leq\|\delta T\|_{T}\|T^{\dagger}Tx\|_{T}$
$\displaystyle\leq\|\delta T\|_{T}(\|T^{\dagger}\|^{2}+1)^{\frac{1}{2}}\|u\|.$

Hence, $\delta(R(T),R(\bar{T}))\leq\|\delta
T\|_{T}(\|T^{\dagger}\|^{2}+1)^{\frac{1}{2}}$.

$(2).$ Let $x\in\ker T$ with $\|x\|=1$, then $Tx=0$

$\displaystyle dist(x,\ker(\bar{T}))$
$\displaystyle\leq\|x-(I-\bar{T}^{\dagger}\bar{T})x\|=\|\bar{T}^{\dagger}\delta
Tx\|$ $\displaystyle\leq\|\bar{T}^{\dagger}\|\|\delta Tx\|$
$\displaystyle\leq\|\bar{T}^{\dagger}\|_{T}\|\delta T\|_{T}\|x\|.$

Hence, $\delta(\ker T,\ker(\bar{T}))\leq\|\bar{T}^{\dagger}\|\|\delta
T\|_{T}$.

###### Lemma 3.5

Let M and N be closed subspaces of H. Suppose that $M\cap
N^{\perp}=\\{0\\}$.Then

$\delta(M,N)=\|P_{M}-P_{N}\|.$

Proof. If $\delta(M,N)=1$, then $1=\hat{\delta}(M,N)=\|P_{M}-Q_{N}\|$, Thus
$\delta(M,N)=\|P_{M}-P_{N}\|$.
Assume that $\delta(M,N)=\delta<1$, then $\forall x\in M$,

$\|(I-P_{N})P_{M}x\|=dist(P_{M}x,N)\leq\|P_{M}x\|\delta(M,N)\leq\delta\|x\|.$

So by Lemma 3 of [14], we know $\delta(M,N)=\|P_{M}-P_{N}\|$.

###### Lemma 3.6

Under the assumptions of Theorem 3.1, we have

$\|TT^{\dagger}-\bar{T}\bar{T}^{\dagger}\|=\delta(R(T),R(\bar{T})).$

Proof. Since $T^{\dagger}\in B(K,H)$, we have
$\ker(T^{\dagger})=R(T)^{\perp}$. So
$\ker(T^{\dagger})=\ker(\bar{T}^{\dagger})$ implies
$R(T)\cap\ker(\bar{T}^{\dagger})=\\{0\\}$. By Lemma 3.5, we know

$\|TT^{\dagger}-\bar{T}\bar{T}^{\dagger}\|=\delta(R(T),R(\bar{T})).$

###### Theorem 3.2

Under the assumptions of Theorem 3.1, we have

(1)

$\|\bar{T}^{\dagger}\|\leq\|(1+\delta
TT^{\dagger})^{-1}\|(\|T^{\dagger}\|^{2}+1)^{\frac{1}{2}}$.

(2)

$\|\bar{T}^{\dagger}-T^{\dagger}\|\leq\dfrac{1+\sqrt{5}}{2}\|\bar{T}^{\dagger}\|\|\delta
T\|_{T}(\|T^{\dagger}\|^{2}+1)^{\frac{1}{2}}$.

Proof. $(1).$ Since $\delta T\in B(D_{T},K)$, we have

$\|\delta TT^{\dagger}x\|\leq\|\delta
T\|_{T}\|\|T^{\dagger}x\|_{T}\leq\|\delta
T\|_{T}(\|T^{\dagger}\|^{2}+1)^{\frac{1}{2}}\|x\|.$

Hence, $\delta TT^{\dagger}\in B(K)$. By Lemma 2.1 ,we have

$G=T^{+}(I+\delta TT^{+})^{-1}\in B(K,H),\;P=I-G\bar{T}\in
B(D_{T}),\;Q=\bar{T}G\in B(K)$

and $\|(I+P(I-P-P^{*})^{-1})x\|_{T}\leq\|x\|_{T},\;x\in D(T).$ So,

$\displaystyle\|\bar{T}^{\dagger}x\|^{2}$
$\displaystyle\leq\|\bar{T}^{\dagger}x\|^{2}_{T}=\|-(I+P(I-P-P^{*})^{-1})G(I-Q-Q^{*})^{-1}x\|^{2}_{T}$
$\displaystyle\leq\|G(I-Q-Q^{*})^{-1}x\|^{2}_{T}$
$\displaystyle\leq\|T^{\dagger}(1+\delta
TT^{\dagger})^{-1}(I-Q-Q^{*})^{-1}x\|^{2}+\|TT^{\dagger}(1+\delta
TT^{\dagger})^{-1}(I-Q-Q^{*})^{-1}x\|^{2}$
$\displaystyle\leq(\|T^{\dagger}\|^{2}+1)\|(1+\delta
TT^{\dagger})^{-1}\|^{2}\|x\|^{2}.$

$(2).$ Since $D(T)$ dense in $H$, we can extend $I-T^{\dagger}T$ to the whole
spaces $H$ such that $P_{\ker T}|_{D(T)}=I-T^{\dagger}T$ and $P_{\ker T}$ is
an orthogonal projection form $H$ onto $\ker T$. Similarly, we extend
$I-\bar{T}^{\dagger}\bar{T}$ to the whole spaces $H$ such that
$P_{\ker(\bar{T})}|_{D(\bar{T})}=I-\bar{T}^{\dagger}\bar{T}$ and
$P_{\ker(\bar{T})}$ is an orthogonal projection form $H$ onto $\ker(\bar{T})$.

Clearly, $\forall x\in D(T)$,
$(T^{\dagger}T-\bar{T}^{\dagger}\bar{T})x=(P_{\ker(\bar{T})}-P_{\ker T})x.$

Noting that $\ker T\cap\ker(\bar{T})^{\perp}=\\{0\\}$, we have $\|P_{\ker
T}-P_{\ker(\bar{T})}\|=\delta(\ker T,\ker(\bar{T}))$ by Lemma 3.5.

$\forall y\in K$ and $\|y\|=1$,

$\displaystyle\bar{T}^{\dagger}y-T^{\dagger}y$
$\displaystyle=-\bar{T}^{\dagger}\delta
TT^{\dagger}TT^{\dagger}y+\bar{T}^{\dagger}(\bar{T}\bar{T}^{\dagger}-TT^{\dagger})(I-TT^{\dagger})y$
$\displaystyle+(P_{\ker(\bar{T})}-P_{\ker T})T^{\dagger}y.$

Using the proof of Proposition 7 in [14], we have

$\|\bar{T}^{\dagger}y-T^{\dagger}y\|^{2}_{T}\leq\frac{3+\sqrt{5}}{2}\|\bar{T}^{\dagger}\|^{2}\|\delta
T\|^{2}_{T}(\|T^{\dagger}\|^{2}+1).$

Since
$\|\bar{T}^{\dagger}y-T^{\dagger}y\|\leq\|\bar{T}^{\dagger}y-T^{\dagger}y\|_{T}$,
we have

$\|\bar{T}^{\dagger}-T^{\dagger}\|\leq\frac{1+\sqrt{5}}{2}\|\bar{T}^{\dagger}\|\|\delta
T\|_{T}(\|T^{\dagger}\|^{2}+1)^{\frac{1}{2}}.$

##  4 Perturbation analysis for $Tx=b$ in Hilbert spaces

In this section, we consider the perturbation of the least square solution of
the following two equations

(1)

$Tx=b,$

(2)

$\bar{T}\bar{x}=\bar{b},$ $(\bar{b}=b+\delta b)$

As we know the solutions of

$\|Tx-b\|=\min_{z\in D(T)}\|Tz-b\|$

are $x=T^{\dagger}b+(I-T^{\dagger}T)z,\forall z\in D(T)$, denoted by $S(T,b)$,
i.e.

$S(T,b)=\\{x:x=T^{\dagger}b+(I-T^{\dagger}T)z,\forall z\in D(T)\\}$

Similarly,

$S(\bar{T},\bar{b})=\\{\bar{x}:\bar{x}=\bar{T}^{\dagger}\bar{b}+(I-\bar{T}^{\dagger}\bar{T})z,\forall
z\in D(\bar{T})\\}$

###### Theorem 4.1

Under the assumptions of Theorem 3.1, we have
$(1)$ For any solution $x=T^{\dagger}b+(I-T^{\dagger}T)z$ in $S(T,b)$, there
exist $\bar{x}\in S(\bar{T},\bar{b})$ such that

$\|\bar{x}-x\|\leq\|\bar{T}^{\dagger}\|\|(b-Tx)+(\delta b-\delta Tx)\|.$

$(2)$ For any solution
$\bar{x}=\bar{T}^{\dagger}\bar{b}+(I-\bar{T}^{\dagger}\bar{T})z$ in
$S(\bar{T},\bar{b})$, there exist $x\in S(T,b)$ such that

$\|\bar{x}-x\|\leq\|T^{\dagger}\|\|(\bar{b}-\bar{T}\bar{x})-(\delta b-\delta
T\bar{x})\|.$

Proof. $(1)$ Taking

$\bar{x}=\bar{T}^{\dagger}\bar{b}+(I-\bar{T}^{\dagger}\bar{T})(T^{\dagger}b+(I-T^{\dagger}T)z).$

Then

$\displaystyle\|\bar{x}-x\|$
$\displaystyle=\|\bar{T}^{\dagger}\bar{b}+(I-\bar{T}^{\dagger}\bar{T})(T^{\dagger}b+(I-T^{\dagger}T)z)-(T^{\dagger}b+(I-T^{\dagger}T)z)\|$
$\displaystyle=\|\bar{T}^{\dagger}\bar{b}-(\bar{T}^{\dagger}\bar{T})(T^{\dagger}b+(I-T^{\dagger}T)z)\|$
$\displaystyle=\|\bar{T}^{\dagger}\delta
b+(I-\bar{T}^{\dagger}\bar{T})T^{\dagger}b-\bar{T}^{\dagger}\bar{T}(I-T^{\dagger}T)z+(\bar{T}^{\dagger}-T^{\dagger})b\|$
$\displaystyle=\|\bar{T}^{\dagger}\delta
b+(I-\bar{T}^{\dagger}\bar{T})T^{\dagger}b-\bar{T}^{\dagger}\bar{T}(I-T^{\dagger}T)z$
$\displaystyle+(-\bar{T}^{\dagger}\delta
TT^{\dagger}b+\bar{T}^{\dagger}(I-TT^{\dagger})b-(I-\bar{T}^{\dagger}\bar{T})T^{\dagger}b)\|$
$\displaystyle=\|\bar{T}^{\dagger}(\bar{b}-\bar{T}x)\|$
$\displaystyle\leq\|\bar{T}^{\dagger}\|\|(b-Tx)+(\delta b-\delta Tx)\|.$

$(2)$ Taking

$x=T^{\dagger}b+(I-T^{\dagger}T)(\bar{T}^{\dagger}\bar{b}+(I-\bar{T}^{\dagger}\bar{T})z).$

By similarly computation to $(1)$, we have

$\|\bar{x}-x\|\leq\|T^{\dagger}\|\|(\bar{b}-\bar{T}\bar{x})-(\delta b-\delta
T\bar{x})\|.$

###### Theorem 4.2

Under the assumptions of Theorem 3.1, we have

$\displaystyle\|\bar{T}^{\dagger}\bar{b}-T^{\dagger}b\|$
$\displaystyle\leq\|(1+\delta
TT^{\dagger})^{-1}\|(\|T^{\dagger}\|^{2}+1)^{\frac{1}{2}}\|\delta b\|$
$\displaystyle+\frac{1+\sqrt{5}}{2}\|(1+\delta TT^{\dagger})^{-1}\|\|\delta
T\|_{T}(1+\|T^{\dagger}\|^{2})^{\frac{1}{2}}\|b\|.$

Proof. By Theorem 3.2, we have

$\displaystyle\|\bar{T}^{\dagger}\bar{b}-T^{\dagger}b\|$
$\displaystyle=\|\bar{T}^{\dagger}\delta b+(\bar{T}^{\dagger}-T^{\dagger})b\|$
$\displaystyle\leq\|\bar{T}^{\dagger}\delta
b\|+\|\bar{T}^{\dagger}-T^{\dagger}\|\|b\|$ $\displaystyle\leq\|(1+\delta
TT^{\dagger})^{-1}\|(\|T^{\dagger}\|^{2}+1)^{\frac{1}{2}}\|\delta b\|$
$\displaystyle+\frac{1+\sqrt{5}}{2}\|(1+\delta TT^{\dagger})^{-1}\|\|\delta
T\|_{T}(1+\|T^{\dagger}\|^{2})^{\frac{1}{2}}\|b\|.$

##  5 Conclusion

In this paper, we extend the perturbation analysis of Moore-Penrose inverse
for bounded linear operators to closed operators. By virtue of a new inner
product, we give the expression of the Moore-Penrose inverse
$\bar{T}^{\dagger}$ and the upper bounds of $\|\bar{T}^{\dagger}\|$ and
$\|\bar{T}^{\dagger}-T^{\dagger}\|$. As an application, we study the
perturbation of the least square solution. These results enrich and improve
the perturbation theory of Moore-Penrose inverse described in [16].

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