# Construction of propagators for divisible dynamical maps

Ujan Chakraborty  Department of Physical Sciences, Indian Institute of Science
Education and Research Kolkata, Mohanpur, Nadia - 741246, West Bengal, India
Dariusz Chruściński  Institute of Physics, Faculty of Physics, Astronomy and
Informatics, Nicolaus Copernicus University, Grudziadzka 5/7, 87-100 Toruń,
Poland

###### Abstract

Divisible dynamical maps play an important role in characterizing Markovianity
on the level of quantum evolution. Divisible maps provide important
generalization of Markovian semigroups. Usually one analyzes either completely
positive or just positive divisibility meaning that the corresponding
propagators are defined in terms of completely positive or positive maps,
respectively. For maps which are invertible at any moment of time the very
existence of propagator is already guaranteed and hence the only issue is
(complete) positivity and trace-preservation. However, for maps which are not
invertible the problem is much more involved since even the existence of a
propagator is not guaranteed. In this paper we propose a simple method to
construct propagators of dynamical maps using the concept of generalized
inverse. We analyze both time-continuous and time-discrete maps. Since the
generalized inverse is not uniquely defined the same applies for the
corresponding propagator. In simple examples of qubit evolution we analyze it
turns out that additional requirement of complete positivity possibly makes
the propagator unique.

##  I Introduction

The evolution of open quantum systems Open1 ; Open2  attracts a lot attention
due to rapidly developing modern fields of science and quantum technologies
like quantum communication or quantum computation QIT . Any quantum evolution
is represented by a quantum dynamical map, that is, a continuous family of
linear maps $\\{\Lambda_{t}\\}_{t\geq 0}$

$\Lambda_{t}:{{\rm L}}(\mathcal{H})\to{{\rm L}}(\mathcal{H}),\ \ (t\geq 0),$
(1)

where ${{\rm L}}(\mathcal{H})$ denotes the space of linear operators acting on
the system’s Hilbert space $\mathcal{H}$. One requires that for any $t\geq 0$
the map $\Lambda_{t}$ is completely positive and trace-preserving (CPTP)
Paulsen ; Stormer ; Wolf . Moreover, it satisfies the standard initial
condition $\Lambda_{t=0}={\rm id}$, where ${\rm id}$ denotes the identity map
in ${{\rm L}}(\mathcal{H})$. Hence, for any $t\geq 0$ the map $\Lambda_{t}$
represents a quantum channel, a basic object of quantum information QIT ; Wolf
.

Very often one also considers a different scenario: time-discrete dynamical
map $\\{\Lambda_{n}\\}_{n\geq 0}$, where $n=0,1,\ldots$, and
$\Lambda_{n}:\mathrm{L}(\mathcal{H}_{0})\to\mathrm{L}(\mathcal{H}_{n})$, where
$\mathcal{H}_{n}$ is a Hilbert space at ‘time’ $n$. In principle for different
$n$ the corresponding Hilbert space may be different. Such scenario is
important to deal with many processes like encodings, decodings, quantum
measurement, and many others. In this paper we consider both time-continuous
and time-discrete scenarios.

In the time-continuous case it well known that under appropriate Markovian
approximation the evolution of an open quantum system can be described by
quantum dynamical semigroup satisfying the following master equation

$\dot{\Lambda}_{t}=\mathcal{L}\Lambda_{t}\ ,\ \ \Lambda_{0}={\rm id},$ (2)

where $\mathcal{L}:{{\rm L}}(\mathcal{H})\to{{\rm L}}(\mathcal{H})$ has the
celebrated Gorini-Kossakowski-Lindblad-Sudarshan (GKLS) structure GKS ; L

$\mathcal{L}(\rho)=-i[H,\rho]+\sum_{\alpha}\gamma_{\alpha}\left(L_{\alpha}\rho
L_{\alpha}^{\dagger}-\frac{1}{2}\\{L^{\dagger}_{\alpha}L_{\alpha},\rho\\}\right),$
(3)

with Hermitian $H^{\dagger}=H\in{{\rm L}}(\mathcal{H})$, arbitrary noise
(Lindblad) operators $L_{\alpha}\in{{\rm L}}(\mathcal{H})$, and
dissipation/decoherence rates $\gamma_{\alpha}\geq 0$ (we keep $\hbar=1$).
Markovian approximation leading to (2) assumes weak coupling between system
and its environment and separation of characteristic time scales. Such
approximation works well in quantum optical systems where the coupling between
atoms/molecules and electromagnetic field is weak. Recently, the notion of
non-Markovian quantum evolution received considerable attention (see review
papers NM1 ; NM2 ; NM3 ; NM4 ; Piilo ). There is no unique universal approach
to deal with quantum (non)Markovianity. In this paper we concentrate on two
very popular approaches based on divisibility of dynamical maps RHP  and so
called information flow BLP . The very concept of divisibility of quantum
channels was initiated in Wolf-1 ; Wolf-2  and recently reviewed in Mario .
According to RHP  the evolution is Markovian iff there exist a family of CPTP
propagators $V_{t,s}$ $(t\geq s)$ such that $\Lambda_{t}=V_{t,s}\Lambda_{s}$
(see Section II for more details). One calls such evolution CP-divisible.
Requiring that $V_{t,s}$ defines a positive (and not necessarily completely
positive) map one calls $\\{\Lambda_{t}\\}_{t\geq 0}$ P-divisible (for the
intricate relation between P- and CP-divisibility cf. Fabio ). Actually, one
may introduce a whole hierarchy of so called $k$-divisibility, where $k$ runs
from $k=1$ (P-divisibility) up to $d={\rm dim}\,\mathcal{H}$ (CP-divisibility)
Sabrina . In a recent paper Farrukh  so called KS-divisibility was analyzed
based on so called Kadison-Schwarz property of the propagator $V_{t,s}$ which
is stronger than P- but weaker than CP-divisibility. An interesting relation
of divisibility of quantum dynamical maps and collision models was studied in
Sergey-2017 . Moreover $k$-divisibility was linked to discrimination of
quantum channels Junu-DC . For recent review of various properties of quantum
open systems related to divisibility of dynamical maps see recent review Chem
. On the other hand authors of BLP  proposed the following approach: they call
quantum evolution represented by $\\{\Lambda_{t}\\}_{t\geq 0}$ Markovian if
for any pair of states $\rho_{1}$ and $\rho_{2}$ so called information flow

$\frac{d}{dt}\|\Lambda_{t}(\rho_{1}-\rho_{2})\|_{1}\leq 0,$ (4)

where $\|X\|_{1}$ denotes the trace norm. Since
$\|\Lambda_{t}(\rho_{1}-\rho_{2})\|_{1}$ corresponds to distinguishability of
time evolved states at time $t$ inequality (4) states that distinguishability
monotonically decreases in time. It is interpreted as a flow of information
from the system to the environment. Violation of (4) is therefore interpreted
as an information back-flow and provides clear evidence of memory effects and
hence non-Markovianity NM1 ; NM2 ; NM3 ; NM4 . These two approaches are not
independent Angel . Actually, CP-divisibility implies (4). However, violation
of (4) is immediately recognized as a clear sign of non-Markovianity. The
approach based on information flow is very popular in the literature.
Intriguing connections of divisibility, information back-flow and quantum
correlations were reported recently in Janek ; Acin-cor .We stress, however,
that there are other approaches to quantum non-Markovianity (see detailed
review in NM4  and the recent papers Modi-1 ; Modi-2  relating
(non)Markovianity and quantum stochastic processes).

In this paper we analyze the problem of construction of propagators $V_{t,s}$
(or $V_{j,i}$ in the time-discrete case). We propose a very simple approach
based on the concept of generalized inverse which reduces to the standard
inverse if the map is invertible. Generalized inverse and hence the
corresponding propagator is not uniquely defined. We provide several simple
examples of qubit evolution for which the non-uniqueness of the propagator is
removed when one requires that the propagator is completely positive and
trace-preserving.

##  II Divisible dynamical maps

###  II.1 Time-continuous maps

Consider a dynamical map $\left\\{\Lambda_{t}\right\\}_{t\geq 0}$ acting on
${{\rm L}}(\mathcal{H})$. We assume that the family of maps
$\left\\{\Lambda_{t}\right\\}_{t\geq 0}$ is differentiable (w.r.t. $t$), and
${\rm dim}\,\mathcal{H}=d<\infty$.

###### Definition 1

A quantum dynamical map $\left\\{\Lambda_{t}\right\\}_{t\geq 0}$ is called
divisible if for any $t\geq 0$

$\Lambda_{t}=V_{t,s}\Lambda_{s},$ (5)

and $V_{t,s}:{{\rm L}}(\mathcal{H})\to{{\rm L}}(\mathcal{H})$.
$\\{\Lambda_{t}\\}_{t\geq 0}$ is called CP-divisible iff $V_{t,s}$ is CPTP,
and P-divisible iff $V_{t,s}$ is PTP.

Clearly, any invertible map $\Lambda_{t}$ is always divisible since the
propagator $V_{t,s}$ can be uniquely defined by
$V_{t,s}=\Lambda_{t}\Lambda_{s}^{-1}$. Moreover, the propagator $V_{t,s}$
satisfies local composition law

$V_{t,u}V_{u,s}=V_{t,s},$ (6)

for $t\geq u\geq s$. In this case one proves the following

######  Theorem 1 (Angel )

Let us assume that $\\{\Lambda_{t}\\}_{t\geq 0}$ is an invertible dynamical
map. Then $\Lambda_{t}$ is P-divisible iff

$\frac{d}{dt}\|\Lambda_{t}(X)\|_{1}\leq 0,$ (7)

for any Hermitian $X\in\mathrm{L}(\mathcal{H})$. It is CP-divisible iff

$\frac{d}{dt}\|({\rm id}_{d}\otimes\Lambda_{t})(X)\|_{1}\leq 0,$ (8)

for any Hermitian ${X}\in\mathrm{L}(\mathcal{H}\otimes\mathcal{H})$.

Note that if $X$ is traceless then $X=a(\rho_{1}-\rho_{2})$, where
$a\in\mathbb{R}$, and hence (7) recovers BLP criterion (4). Interestingly, one
proves

######  Theorem 2 (BOGNA )

Let us assume that $\\{\Lambda_{t}\\}$ is an invertible dynamical map. Then
$\\{\Lambda_{t}\\}$ is CP-divisible iff

$\frac{d}{dt}\|[{\rm
id}_{d+1}\otimes\Lambda_{t}]({\rho}_{1}-{\rho}_{2})\|_{1}\leq 0,$ (9)

for any pair of density operators ${\rho}_{1}$, ${\rho}_{2}$ in
$\mathcal{B}(\mathcal{H}^{\prime}\otimes\mathcal{H})$ with
$\dim(\mathcal{H}^{\prime})-1=\dim(\mathcal{H})=d$.

The essence of (9) is that one enlarges the dimension of the ancilla
$d\rightarrow d+1$, but uses only traceless operators $X=a(\rho_{1}-\rho_{2})$
like in the original approach to the information flow BLP .

For non-invertible maps the divisibility is not guarantied. Actually, one
proves

######  Proposition 1 (PRL-2018 )

The dynamical map $\\{\Lambda_{t}\\}_{t\geq 0}$ is divisible if and only if

${\rm Ker}(\Lambda_{s})\subseteq{\rm Ker}(\Lambda_{t}),$ (10)

for any $t>s$.

One has the following generalization of Theorem 1

######  Theorem 3 (PRL-2018 )

If the dynamical map $\\{\Lambda_{t}\\}_{t\geq 0}$ satisfies

$\frac{d}{dt}\|({\rm id}_{d}\otimes\Lambda_{t})(X)\|_{1}\leq 0,$ (11)

for any Hermitian ${X}\in\mathrm{L}(\mathcal{H}\otimes\mathcal{H})$, then
there exists completely positive propagator $V_{t,s}:{{\rm
L}}(\mathcal{H})\to{{\rm L}}(\mathcal{H})$ which is trace preserving on the
image of $\Lambda_{s}$.

Moreover, in the qubit case ($d=2$) one has the following

######  Theorem 4 (CC19 )

The qubit dynamical map $\\{\Lambda_{t}\\}_{t\geq 0}$ is CP-divisible iff it
satisfies

$\frac{d}{dt}\|({\rm id}_{2}\otimes\Lambda_{t})(X)\|_{1}\leq 0,$ (12)

for any Hermitian ${X}\in\mathrm{L}(\mathcal{H}\otimes\mathcal{H})$.

###  II.2 Time-discrete maps

Consider now the family $\\{\Lambda_{n}\\}_{n\geq 0}$ of CPTP maps

$\Lambda_{n}:\mathrm{L}(H_{S})\to\mathrm{L}(\mathcal{H}_{n})$ (13)

where $\mathcal{H}_{S},\mathcal{H}_{1},\ldots$ are finite dimensional Hilbert
spaces of dimensions $d_{S},d_{1},d_{2},\ldots$, respectively. Moreover, we
assume that $\Lambda_{0}:\mathrm{L}(H_{S})\to\mathrm{L}(H_{S})$ is an identity
map. We call time discrete dynamical map $\\{\Lambda_{n}\\}_{n\geq 0}$ CP-
divisible iff there exists a family of CPTP propagators

$V_{j,i}:\mathrm{L}(\mathcal{H}_{i})\to\mathrm{L}(\mathcal{H}_{j}),\ \ \
(j>i)$ (14)

such that

$\Lambda_{j}=V_{j,i}\Lambda_{i}\ .$ (15)

Consider now an ensemble of states $\rho^{x}$ prepared with probability
$p_{x}$: $\mathcal{E}=\\{p_{x},\rho^{x}\\}_{x}$. To distinguish between these
states one defines guessing probability

$\mathrm{P}_{\rm guess}(\mathcal{E})=\max\sum_{x\in\mathcal{X}}p_{x}{\rm
Tr}(P^{x}\rho^{x}),$ (16)

where the maximum is over all POVMs $\\{P^{x}\\}_{x}$ defined on the Hilbert
space $\mathcal{H}$. Buscemi and Datta datta  introduced the following
interesting concept enabling one to compare quantum channels

###### Definition 2

Time discrete dynamical map $\\{\Lambda_{n}\\}_{n\geq 0}$ is information
decreasing iff for any $j>i$ one has

$\mathrm{P}_{\rm guess}(\mathcal{E}_{j})\leq\mathrm{P}_{\rm
guess}(\mathcal{E}_{i}),$ (17)

where $\mathcal{E}_{n}=\\{p_{x},\Lambda_{n}(\rho^{x})\\}_{x}$.
$\\{\Lambda_{n}\\}_{n\geq 0}$ is completely information decreasing iff ${\rm
id}_{S}\otimes\Lambda_{n}$ is information decreasing (${\rm id}_{S}$ stands
for the identity map on $\mathrm{L}(\mathcal{H}_{S})$).

######  Theorem 5 (datta )

Time discrete dynamical map $\\{\Lambda_{n}\\}_{n\geq 0}$ is CP-divisible iff
it is completely information decreasing.

Note, that if $\mathcal{E}$ consists of two members
$\mathcal{E}=\\{p_{1},\rho_{1};p_{2},\rho_{2}\\}$, then HEL  (see also JUNU
for the review)

$\mathrm{P}_{\rm
guess}(\mathcal{E})=\frac{1}{2}\|p_{1}\rho_{1}-p_{2}\rho_{2}\|_{1},$ (18)

and hence the monotonicity condition (17)

$\|[{\rm id}_{S}\otimes\Lambda_{j}](X)\|_{1}\leq\|[{\rm
id}_{S}\otimes\Lambda_{i}](X)\|_{1},$ (19)

defines a necessary condition for CP-divisibility.

###### Proposition 2

If ${\rm Im}(\Lambda_{n})=\mathrm{L}(\mathcal{H}_{n})$ for all $n>0$, then
$\\{\Lambda_{n}\\}_{n\geq 0}$ is CP-divisible iff (19) is satisfied for all
$j>i$.

Proof: We show that (19) implies CP-divisibility. Note, that
$\\{\Lambda_{n}\\}_{n\geq 0}$ is divisible, that is, there exists the family
$V_{j,i}$ satisfying (14). We can define

$V_{j,i}:=\Lambda_{j}\Lambda_{i}^{-{1_{R}}},$ (20)

where $\Lambda_{i}^{-1_{R}}$ denotes a (linear) right inverse of
$\Lambda_{i}$, that is, one has
$\Lambda_{i}\Lambda_{i}^{-{1_{R}}}={\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}_{\mathcal{H}_{i}}$,
but in general $\Lambda_{i}^{-{1_{R}}}\Lambda_{i}\neq{\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}_{\mathcal{H}_{S}}$. A right inverse exists, because ${\rm
Im}(\Lambda_{i})=\mathrm{L}(\mathcal{H}_{i})$. A left inverse does not exist
unless $\rm{Ker}(\Lambda_{i})=\\{0\\}$, and we do not require it to exist here
either.

We see that $V_{j,i}$ defined thus is a valid propagator: For all
$X\in\rm{L}(\mathcal{H}_{S})$

$\Lambda_{i}^{-1_{R}}\Lambda_{i}(X)=X+\delta$ (21)

for some $\delta\in\rm{Ker}(\Lambda_{i})$, and hence

$V_{j,i}\Lambda_{i}(X)=(\Lambda_{j}\Lambda_{i}^{-1_{R}})\Lambda_{i}(X)=\Lambda_{j}(X+\delta)=\Lambda_{j}(X)$
(22)

as $\rm{Ker}(\Lambda_{i})\subseteq\rm{Ker}(\Lambda_{j})$

Now, for any Hermitian $X\in\mathrm{L}(\mathcal{H}_{S}\otimes\mathcal{H}_{S})$
one has

$\|[{\rm id}_{S}\otimes\Lambda_{j}](X)\|_{1}=\|[{\rm id}_{S}\otimes
V_{j,i}\Lambda_{i}](X)\|_{1}\leq\|[{\rm id}_{S}\otimes\Lambda_{i}](X)\|_{1},$
(23)

which means that

$\|[{\rm id}_{S}\otimes V_{j,i}](Y)\|_{1}\leq\|Y\|_{1},$ (24)

for any $Y=[{\rm id}_{S}\otimes\Lambda_{i}](X)$. Now, since ${\rm
Im}(\Lambda_{i})=\mathrm{L}(\mathcal{H}_{i})$, $V_{j,i}$ is trace-preserving
and ${\rm id}_{S}\otimes V_{j,i}$ defines a contraction w.r.t. trace norm, and
thus one finds that $V_{j,i}$ is CPTP (note that ${\rm
dim}\,\mathcal{H}_{S}\geq{\rm dim}\,\mathcal{H}_{n}$ for $n>0$). $\Box$

Clearly, $\Lambda_{i}^{-{1_{R}}}$ is not uniquely defined but it is trace-
preserving. Note however that

###### Proposition 3

The corresponding propagator $V_{j,i}$ (20) is uniquely defined, independent
of the choice of right inverse $\Lambda_{i}^{-1_{R}}$.

Proof: Suppose that there are two propagators $V_{j,i}$ and $\tilde{V}_{j,i}$
defined via

$V_{j,i}:=\Lambda_{j}\Lambda_{i}^{-{1_{R}}}\ ,\ \ \
\tilde{V}_{j,i}:=\Lambda_{j}\tilde{\Lambda_{i}}^{-{1_{R}}},$

where $\Lambda_{i}^{-{1_{R}}}$ and $\tilde{\Lambda_{i}}^{-{1_{R}}}$ are two
different right inverses of $\Lambda_{i}$. Now, let

$X=\Lambda_{i}^{-{1_{R}}}(Y)\ ,\ \ \
\tilde{X}=\tilde{\Lambda_{i}}^{-{1_{R}}}(Y).$

Note, that $\Lambda_{i}\Lambda_{i}^{-{1_{R}}}(Y)=Y$ and
$\Lambda_{i}\tilde{\Lambda_{i}}^{-{1_{R}}}(Y)=Y$, and hence

$\Lambda_{i}(X)=Y=\Lambda_{i}(\tilde{X}),$

which shows that $X-\tilde{X}\in{\rm Ker}(\Lambda_{i})$. Finally, divisibility
requires that ${\rm Ker}(\Lambda_{j})\supseteq{\rm Ker}(\Lambda_{i})$ and
hence $\Lambda_{j}(X)=\Lambda_{j}(\tilde{X})$ which implies
$V_{j,i}(Y)=\tilde{V}_{j,i}(Y)$. $\Box$

###### Corollary 1

In particular, if $\mathcal{H}_{S}=\mathcal{H}_{n}$ for $n>0$, and
$\Lambda_{n}$ is invertible, then the map $\\{\Lambda_{n}\\}_{n\geq 0}$ is CP-
divisible iff (19) holds for all Hermitian
$X\in\mathrm{L}(\mathcal{H}_{S}\otimes\mathcal{H}_{S})$.

##  III Construction of propagators

For invertible maps the propagator $V_{t,s}$ is uniquely defined via
$V_{t,s}=\Lambda_{t}\Lambda_{s}^{-1}$. Suppose now that the dynamical map
$\\{\Lambda_{t}\\}_{t\geq 0}$ is divisible but not necessarily invertible.
There is a natural construction of a propagator via the following formula

$V_{t,s}=\Lambda_{t}\Lambda_{s}^{-},$ (25)

where $\Lambda_{s}^{-}:{{\rm L}}(\mathcal{H})\to{{\rm L}}(\mathcal{H})$
denotes a generalized inverse of $\Lambda_{s}$ Kumar ; GI ; Horn ; Yanai  (cf.
Appendix for more details). A generalized inverse $\Lambda_{s}^{-}$ coincides
with the inverse $\Lambda_{s}^{-1}$ for invertible maps and it is defined by
the following property

$\Lambda_{s}\Lambda_{s}^{-}\Lambda_{s}=\Lambda_{s}.$ (26)

Clearly, $\Lambda_{s}^{-}$ is not uniquely defined. Suppose, that the image of
$\Lambda_{s}$ is a proper subspace of ${{\rm L}}(\mathcal{H})$, and let
$\mathcal{C}_{s}$ be a complementary subspace such that for any $s$ one has

${{\rm L}}(\mathcal{H})={\rm Im}(\Lambda_{s})\oplus\mathcal{C}_{s}.$ (27)

It should be stressed that $\mathcal{C}_{s}$ is not uniquely defined.
Actually, any linear subspace $\mathcal{C}_{s}$ such that ${\rm
dim}(\mathcal{C}_{s})+{\rm dim}({\rm Im}(\Lambda_{s}))=d^{2}$, and
$\mathcal{C}_{s}\cap{\rm Im}(\Lambda_{s})={\\{0\\}}$ does the job. Now, for
any $Y\in{{\rm L}}(\mathcal{H})$ we have the unique decomposition

$Y=Y_{1}\oplus Y_{0}\in{\rm Im}(\Lambda_{s})\oplus\mathcal{C}_{s},$

where, by the last statement, we mean that $Y_{1}\in{\rm Im}(\Lambda_{s})$ and
$Y_{0}\in\mathcal{C}_{s}$. The following linear map is a generalized inverse
$\Lambda_{s}^{-}$:

$\Lambda_{s}^{-}(Y):=X_{1}+B_{s}(Y_{0}),$ (28)

where $\Lambda_{s}(X_{1})=Y_{1}$ and $B_{s}:\mathcal{C}_{s}\to{\rm
Ker}(\Lambda_{s})$ (cf. Kumar ; GI  and the Appendix) is an arbitrary linear
map. One can therefore define

$V_{t,s}(Y)=\Lambda_{t}\Lambda_{s}^{-}(Y)=\Lambda_{t}(X_{1}+B_{s}(Y_{0}))=\Lambda_{t}(X_{1}),$
(29)

due to $B_{s}(Y_{0})\in{\rm Ker}(\Lambda_{s})\subseteq{\rm Ker}(\Lambda_{t})$.
Hence such a propagator is characterized by the family of subspaces
$\\{\mathcal{C}_{t}\\}_{t\geq 0}$ transversal to images of
$\\{\Lambda_{t}\\}_{t\geq 0}$. Such propagators all agree on
$\rm{Im}(\Lambda_{s})$, due to the kernel non-decreasing property.

###### Proposition 4

$V_{t,s}=\Lambda_{t}\Lambda_{s}^{-}$ satisfies composition law (6).

Proof: One has

$V_{t,u}V_{u,s}=V_{t,u}(\Lambda_{u}\Lambda_{s}^{-})=(V_{t,u}\Lambda_{u})\Lambda_{s}^{-}=\Lambda_{t}\Lambda_{s}^{-}=V_{t,s},$
(30)

which ends the proof. $\Box$

###### Proposition 5

$V_{t,s}=\Lambda_{t}\Lambda_{s}^{-}$ (29) is trace-preserving iff the subspace
$\mathcal{C}_{s}$ contains only traceless operators.

Proof: One has

${\rm Tr}\,V_{t,s}(Y)={\rm Tr}\,\Lambda_{t}(X_{1})={\rm
Tr}\,\Lambda_{s}(X_{1}),$

due to trace-preservation of the maps $\Lambda_{t}$ and $\Lambda_{s}$. Now,
using $Y=\Lambda_{s}(X_{1})\oplus Y_{0}$, one finds ${\rm
Tr}\,\Lambda_{s}(X_{1})={\rm Tr}\,Y-{\rm Tr}\,Y_{0}={\rm Tr}\,Y$, if and only
if ${\rm Tr}\,Y_{0}=0$. $\Box$

Note, that $V_{t,s}=\Lambda_{t}\Lambda_{s}^{-}$ satisfies

${\rm Im}(V_{t,s})={\rm Im}(\Lambda_{t})\subseteq{{\rm L}}(\mathcal{H}),$ (31)

for any $t\geq s$.

###### Proposition 6

If the propagator $V_{t,s}$ satisfy (31), then
$V_{t,s}=\Lambda_{t}\Lambda_{s}^{-}$ for some generalized inverse
$\Lambda_{s}^{-}$.

Proof: Note, that the most general propagator satisfying (31) reads as follows

$V_{t,s}(Y)=\Lambda_{t}(X_{1})+\Phi_{t,s}(Y_{0}),$ (32)

where $X_{1}$ satisfies $Y_{1}=\Lambda_{s}(X_{1})\in{\rm Im}(\Lambda_{t})$,
and $\Phi_{t,s}:\mathcal{C}_{s}\to{\rm Im}(\Lambda_{t})$ is some linear map.
One has therefore $V_{t,s}(Y)=\Lambda_{t}(X_{1}+\delta)$, where
$\Lambda_{t}(\delta)=\Phi_{t,s}(Y_{0})\in{\rm Im}(\Lambda_{t})$. Hence,
$V_{t,s}(Y)=\Lambda_{t}(X_{1}^{\prime})$, where $X_{1}^{\prime}=X_{1}+\delta$
corresponds to another decomposition $Y=Y_{1}^{\prime}\oplus Y_{0}^{\prime}$,
w.r.t. the new family of transversal subspaces
$\\{\mathcal{C}^{\prime}_{t}\\}_{t\geq 0}$, with
$Y_{1}^{\prime}=\Lambda_{s}(X_{1}^{\prime})$ $\Box$

Clearly, the same techniques applies for the time-discrete dynamical maps
$\\{\Lambda_{n}\\}_{n\geq 0}$. One defines

$V_{j,i}:=\Lambda_{j}\Lambda_{i}^{-},$ (33)

which reduces to $\Lambda_{j}\Lambda_{i}^{-1_{R}}$ iff ${\rm
Im}(\Lambda_{i})=\mathrm{L}(\mathcal{H}_{i})$.

###### Example 1

Consider the following dynamical map

$\Lambda_{t}(\rho)=[1-f(t)]\rho+f(t)\omega_{t}{\rm Tr}\,\rho$ (34)

where $\omega_{t}$ is a time-dependent density operator, and
$f:\mathbb{R}_{+}\to[0,1]$ with the initial condition $f(0)=0$. Now, the map
is invertible iff $f(t)<1$ for all $t$. Suppose, that $f(t_{*})=1$ for some
$t_{*}>0$. Then the map is divisible iff $f(t)=1$ for all $t\geq t_{*}$.
Hence, for $t\geq t_{*}$ the map $\Lambda_{t}(\rho)=\omega_{t}{\rm Tr}\,\rho$
projects any density operator into $\omega_{t}$ and clearly is not invertible.
Now, the propagator is defined by fixing the family of subspaces
$\mathcal{C}_{s}$ transversal to the image of $\Lambda_{s}$

$V_{t,s}(Y)=\Lambda_{t}(X_{1}),$ (35)

where $Y=\Lambda_{s}(X_{1})\oplus Y_{0}$. We consider $t\geq s\geq t_{*}>0$.
There are two natural choices of $\mathcal{C}_{s}$. Taking
$\mathcal{C}_{t}={\rm Ker}(\Lambda_{t})=\\{X\,|\,{\rm Tr}X=0\\}$ one has

$Y=\omega_{s}{\rm Tr}Y\oplus[Y-\omega_{s}{\rm Tr}Y]=Y_{1}\oplus
Y_{0}=\Lambda_{s}(X_{1})\oplus Y_{0},$ (36)

and hence

$V_{t,s}(Y)=\Lambda_{t}(X_{1})=\omega_{t}{\rm Tr}Y\ .$ (37)

Clearly, one can choose

$\Lambda_{s}^{-}(Y)=\omega_{s}\,{\rm Tr}\,Y,$ (38)

for $s\geq t_{*}$. Interestingly, $\Lambda_{s}^{-}=\Lambda_{s}$ and hence in
this case $\Lambda_{s}^{-}$ is reflexive (satisfies (115)). Note, however,
that neither (122) nor (123) is satisfied. Propagator (37) is completely
positive and trace preserving. Moreover, it does satisfy the composition law.
Indeed, one has

$V_{t,s}V_{s,u}(Y)=V_{t,s}[\omega_{s}{\rm Tr}Y]=\omega_{t}{\rm
Tr}Y=V_{t,u}(Y).$

A second natural choice is to take $\mathcal{C}_{t}={\rm
Ker}(\Lambda^{*}_{t})=\\{X\,|\,{\rm Tr}(\omega_{t}X)=0\\}$, where
$\Lambda_{t}^{*}:{{\rm L}}(\mathcal{H})\to{{\rm L}}(\mathcal{H})$ denotes the
dual (that is, a Heisenberg picture of the dynamical map) defined via

$(\Lambda_{t}(X),Y)=(X,\Lambda_{t}^{*}(Y)),$ (39)

for all $X,Y\in{{\rm L}}(\mathcal{H})$, and $(X,Y):={\rm Tr}(X^{\dagger}Y)$
stands for the Hilbert-Schmidt inner product. One has

$Y=\omega_{s}\frac{{\rm Tr}(Y\omega_{s})}{{\rm
Tr}(\omega_{s}^{2})}\oplus\left[Y-\omega_{s}\frac{{\rm Tr}(Y\omega_{s})}{{\rm
Tr}(\omega_{s}^{2})}\right]=Y_{1}\oplus Y_{0}=\Lambda_{s}(X_{1})\oplus Y_{0},$
(40)

and hence

$V_{t,s}(Y)=\omega_{t}\frac{{\rm Tr}(Y\omega_{s})}{{\rm
Tr}(\omega_{s}^{2})}=\omega_{t}\frac{(Y,\omega_{s})}{(\omega_{s},\omega_{s})}.$
(41)

One finds

$\Lambda_{s}^{-}(\rho)=\omega_{s}\frac{{\rm Tr}(\omega_{s}\rho)}{{\rm
Tr}(\omega_{s}\omega_{s})}.$ (42)

Note, that formula (42) implies $\Lambda_{s}\Lambda_{s}^{-}=\Lambda_{s}^{-}$,
and hence

$\Lambda_{s}^{-}\Lambda_{s}\Lambda_{s}^{-}=\Lambda_{s}^{-}\Lambda_{s}^{-}=\Lambda_{s}^{-},$

which shows that (42) is also reflexive. However, neither (122) nor (123) is
satisfied.

The second propagator is completely positive but not trace-preserving due to
the fact that ${\rm Ker}(\Lambda^{*}_{t})$ does not contain traceless
operators only. However, it does satisfy the composition law. Indeed, one has

$V_{t,s}V_{s,u}(Y)=V_{t,s}(\omega_{s})\frac{(Y,\omega_{u})}{(\omega_{u},\omega_{u})}=\omega_{t}\frac{(\omega_{s},\omega_{s})}{(\omega_{s},\omega_{s})}\frac{(Y,\omega_{u})}{(\omega_{u},\omega_{u})}=V_{t,u}(Y).$

Clearly both propagators agree on the image of $\Lambda_{s}$, that is, for
$Y=\alpha\,\omega_{s}$ one has

$\frac{(Y,\omega_{s})}{(\omega_{s},\omega_{s})}=\omega_{t}{\rm Tr}Y.$

##  IV Propagator from spectral properties of dynamical maps

Generalized inverse $\Lambda_{s}^{-}$ is highly non unique. There is, however,
a natural way to define a generalized inverse (and hence the propagator) using
spectral properties of the dynamical map. In this section we analyze both
diagonalizable and non-diagonalizable cases.

###  IV.1 Diagonalizable dynamical maps

Let us assume that dynamical map $\\{\Lambda_{t}\\}$ is diagonalizable, that
is, for any $t\geq 0$ one has

${\Lambda}_{t}F_{\alpha}(t)=\lambda_{\alpha}(t)F_{\alpha}(t)\ ,\ \ \
\Lambda^{*}_{t}G_{\alpha}(t)=\overline{\lambda_{\alpha}(t)}G_{\alpha}(t)\ ,$
(43)

for some $F_{\alpha}(t),G_{\alpha}(t)\in L(H)$, $\alpha=0,1,\ldots,d^{2}-1$.
One has

$(G_{\alpha}(t),F_{\beta}(t))=\delta_{\alpha\beta}.$ (44)

One has for the spectral resolution

${\Lambda}_{t}=\sum_{\alpha=0}^{d^{2}-1}\lambda_{\alpha}(t)\mathcal{P}_{\alpha}(t),$
(45)

where

$\mathcal{P}_{\alpha}(t)(\rho):=F_{\alpha}(t)(G_{\alpha}(t),\rho),$ (46)

are projectors (not necessarily Hermitian) satisfying

$\mathcal{P}_{\alpha}(t)\mathcal{P}_{\beta}(t)=\delta_{\alpha\beta}\mathcal{P}_{\alpha}(t).$
(47)

Note, that trace-preservation condition implies $\lambda_{0}(t)=1$ and
$G_{0}(t)={\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}$. Suppose that the eigenvalues satisfy
$\lambda_{\alpha}(s)=0$ for $\alpha>n$ and are non-zero for $\alpha\leq n$.
Divisibility implies that $\lambda_{\alpha}(t)=0$ for $\alpha>n$ for all
$t>s$. Let us define generalised inverse via

${\Lambda}^{-}_{s}=\sum_{\alpha=0}^{n}\lambda^{-1}_{\alpha}(s)\mathcal{P}_{\alpha}(s).$
(48)

The corresponding propagator reads

${\Lambda}_{t}{\Lambda}^{-}_{s}(\rho)=\sum_{\alpha,\beta=0}^{n}\frac{\lambda_{\alpha}(t)}{\lambda_{\beta}(s)}\,C_{\alpha\beta}(t,s)\,F_{\alpha}(t)\,(G_{\beta}(s),\rho),$
(49)

where the ‘correlator’ $C_{\alpha\beta}(t,s)$ reads

$C_{\alpha\beta}(t,s)=(G_{\alpha}(t),F_{\beta}(s)).$ (50)

###### Proposition 7

Generalized inverse defined via (48) is reflexive, that is,
$\Lambda_{t}^{-}\Lambda_{t}\Lambda_{t}^{-}=\Lambda_{t}^{-}$. Moreover, both
projectors

${\Lambda}_{t}{\Lambda}_{t}^{-}={\Lambda}_{t}^{-}{\Lambda}_{t}=\sum_{\alpha=0}^{n}\mathcal{P}_{\alpha}(t)\,$

project to the image of ${\Lambda}_{t}$.

###### Corollary 2

If $\Lambda_{t}$ is normal, that is, $F_{\alpha}(t)=G_{\alpha}(t)$, then
generalized inverse defined via (48) is Moore-Penrose generalized inverse.

The corresponding propagator $\Lambda_{t}\Lambda_{s}^{-}$ is evidently trace-
preserving. Moreover, in this case the family of subspaces $\mathcal{C}_{t}$
satisfy $\mathcal{C}_{t}={\rm Ker}(\Lambda_{t})$.

###### Example 2

Let us observe that dynamical map in Example 1 is diagonalizable. Indeed, one
finds

$\Lambda_{t}(\omega_{t})=\omega_{t},\ \ \
\Lambda_{t}(F_{\alpha})=[1-f(t)]F_{\alpha},$ (51)

where $F_{\alpha}$ is traceless for $\alpha=1,\ldots,d^{2}-1$. The dual map
reads

$\Lambda_{t}^{*}(X)=[1-f(t)]X+f(t){\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}{\rm
Tr}(\omega_{t}X),$ (52)

and hence

$\Lambda^{*}_{t}({\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm 1\mskip-5.0mul}})={\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}},\ \ \
\Lambda_{t}(G_{\alpha})=[1-f(t)]G_{\alpha},$ (53)

where $G_{\alpha}$ satisfy ${\rm Tr}(G_{\alpha}\omega_{t})=0$. Hence the
spectral resolution reads as follows

$\Lambda_{t}(\rho)=\omega_{t}{\rm
Tr}\,\rho+[1-f(t)]\sum_{\alpha=1}^{d^{2}-1}F_{\alpha}(G_{\alpha},\rho).$ (54)

Now, for $s\geq t_{*}$, it reduces to

$\Lambda_{s}(\rho)=\omega_{s}{\rm Tr}\,\rho,$ (55)

that is, there is only one non-vanishing eigenvalue $\lambda_{0}(t)=1$. Hence,
the formula (48) leads to

$\Lambda^{-}_{s}(\rho)=\Lambda_{s}(\rho)=\omega_{s}{\rm Tr}\,\rho,$ (56)

which reproduces (37).

###  IV.2 Non-diagonalizable dynamical maps

Consider now a general case corresponding to the following spectral Jordan
decomposition Kato

${\Lambda}_{t}=\sum_{\alpha=0}^{d^{2}-1}(\lambda_{\alpha}(t)\mathcal{P}_{\alpha}(t)+\mathcal{N}_{\alpha}(t)),$
(57)

where $\mathcal{N}_{\alpha}(t)$ are nilpotent maps satisfying

$\mathcal{P}_{\alpha}(t)\mathcal{N}_{\beta}(t)=\mathcal{N}_{\beta}(t)\mathcal{P}_{\alpha}(t)=\delta_{\alpha\beta}\mathcal{N}_{\alpha}(t),$
(58)

and $\mathcal{P}^{n_{\alpha}}_{\alpha}(t)=0$, with $1\leq n_{\alpha}\leq{\rm
rank}\,\mathcal{P}_{\alpha}(t)$. Suppose that for $t\geq t_{*}$ the
eigenvalues satisfy $\lambda_{\alpha}(t)=0$ for $\alpha>n$. One has therefore

${\Lambda}_{t}=\sum_{\alpha=0}^{n}(\lambda_{\alpha}(t)\mathcal{P}_{\alpha}(t)+\mathcal{N}_{\alpha}(t))+\sum_{\alpha=n+1}^{d^{2}-1}\mathcal{N}_{\alpha}(t).$
(59)

Clearly nilpotent maps are represented by Jordan blocks.

###### Proposition 8

Let $J_{k}(\lambda)$ be a Jordan block of size $k$. One finds

$J_{k}(0)J^{T}_{k}(0)J_{k}(0)=J_{k}(0),$ (60)

and

$J^{T}_{k}(0)J_{k}(0)J^{T}_{k}(0)=J^{T}_{k}(0),$ (61)

and hence $J^{T}_{k}(0)$ is a reflexive generalized inverse of $J_{k}(0)$.

##  V Bloch representation and Propagators

For a qubit system one often uses well known Bloch representation

$\rho=\frac{1}{2}({\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}+\sum_{k=1}^{3}r_{k}\sigma_{k}),$ (62)

where $\mathbf{r}=(r_{1},r_{2},r_{3})^{T}$ is the Bloch vector corresponding
to $\rho$. Now, for a qubit map $\Phi$ one defines a real $4\times 4$ matrix

$T_{\alpha\beta}=\frac{1}{2}{\rm Tr}(\sigma_{\alpha}\Phi(\sigma_{\beta})),$
(63)

with $\sigma_{0}={\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}$. $T_{\alpha\beta}$ has the following
structure

$T_{\alpha\beta}=\left(\begin{array}[]{c|c}1&0\\\
\hline\cr\mathbf{x}&\Delta\end{array}\right),$ (64)

where $\mathbf{x}\in\mathbb{R}^{3}$ and $\Delta$ is a $3\times 3$ real matrix
and the map $\rho\to\Phi(\rho)$ in the Bloch representation is realized via
the following affine transformation

$\mathbf{r}\to{\Delta}\mathbf{r}+\mathbf{x}.$ (65)

This representation may be generalized for arbitrary dimension $d$: let
$\tau_{\alpha}$ ($\alpha=0,1,\ldots,d^{2}-1$) be Hermitian orthonormal basis
in ${{\rm L}}(\mathcal{H})$ such that $\tau_{0}={\mathchoice{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}/\sqrt{d}$. Any CPTP map $\Phi:{{\rm L}}(\mathcal{H})\to{{\rm
L}}(\mathcal{H})$ gives rise to real $d^{2}\times d^{2}$ matrix

$T_{\alpha\beta}={\rm Tr}(\tau_{\alpha}\Phi(\tau_{\beta})).$ (66)

Consider now the dynamical map $\\{\Lambda_{t}\\}$. One finds
$T_{\alpha\beta}(t)={\rm Tr}(\tau_{\alpha}\Lambda_{t}(\tau_{\beta}))$

$T_{\alpha\beta}(t)=\left(\begin{array}[]{c|c}1&0\\\
\hline\cr\mathbf{x}_{t}&\Delta_{t}\end{array}\right),$ (67)

where $\mathbf{x}_{t}\in\mathbb{R}^{d^{2}-1}$, and $\Delta_{t}$ is a real
$(d^{2}-1)\times(d^{2}-1)$ matrix. In general a generalised inverse $T^{-}(t)$
is not trace-preserving. There is, however, a natural class of trace-
preserving generalised inverses corresponding to the following matrix
representation

$T^{-}(t)=\left(\begin{array}[]{c|c}1&0\\\
\hline\cr\mathbf{y}_{t}&\Gamma_{t}\end{array}\right)$ (68)

where, the defining condition $T(t)T^{-}(t)T(t)=T(t)$ implies that
$\Gamma_{t}=\Delta^{-}_{t}$, and

$\Delta_{t}(\mathbf{y}_{t}+\Delta_{t}^{-}\mathbf{x}_{t})=0,$

that is, $\mathbf{y}_{t}+\Delta^{-}_{t}\mathbf{x}_{t}\in{\rm
Ker}(\Delta_{t})$. One finds for the matrix representation of propagator

$T(t,s):=T(t)T^{-}(s)=\left(\begin{array}[]{c|c}1&0\\\
\hline\cr\mathbf{x}_{t}+\Delta_{t}\mathbf{y}_{s}&\Delta_{t}\Delta_{s}^{-}\end{array}\right).$
(69)

This propagator is by construction trace-preserving.

##  VI Propagators for Qubit Dynamical Maps

In the qubit case any quantum channel $\Lambda:M_{2}\to M_{2}$ can be
represented as follows RSW02 ; QIT ; Wolf

$\Lambda=\mathcal{U}\,\Phi\,\mathcal{V}^{*},$ (70)

where $\mathcal{U}$ and $\mathcal{V}$ are unitary channels, and $\Phi$ has the
following Bloch representation $T_{\mathcal{\alpha\beta}}=\frac{1}{2}{\rm
Tr}(\sigma_{\alpha}\Phi(\sigma_{\beta}))$

$T=\begin{pmatrix}1&0&0&0\\\ x_{1}&\lambda_{1}&0&0\\\ x_{2}&0&\lambda_{2}&0\\\
x_{3}&0&0&\lambda_{3}\end{pmatrix},$ (71)

where $\lambda_{i}$’s are (up to a sign) the singular values of $\Delta$
defined in (67). Now, since $\mathcal{U}$ and $\mathcal{V}$ are invertible,
one has for the generalized inverse of $\Lambda$

$\Lambda^{-}=\mathcal{V}\Phi^{-}\mathcal{U}^{*},$ (72)

that is, the generalized inverse of $\Lambda$ is completely determined by that
of $\Phi$. In CC19  the following theorem was proved

###### Theorem 6

There is no CPTP projector $\mathcal{P}:M_{2}\to M_{2}$ projecting $M_{2}$ to
the 3-dimensional subspace of $M_{2}$.

Here we provide an independent proof based on the very concept of generalized
inverse. Note, that any projector $\mathcal{P}$ such that ${\rm
Im}(\mathcal{P})=\Sigma$ may be realized via

$\mathcal{P}=\Lambda\Lambda^{-},$ (73)

where $\Lambda$ is a map satisfying ${\rm Im}(\mathcal{P})=\Sigma$ (cf.
Appendix). Using representation (70) one has

$\mathcal{P}=\Lambda\Lambda^{-}=\mathcal{U}\,\Phi\,\Phi^{-}\,\mathcal{U}^{*}.$
(74)

Since the image of $\Lambda$ is 3-dimensional let as assume that in the
formula (71) one has $\lambda_{3}=0$ and let us look for the general inverse
represented by the general formula (68)

$T^{-}=\begin{pmatrix}1&0&0&0\\\ y_{1}&\alpha_{11}&\alpha_{12}&\alpha_{13}\\\
y_{2}&\alpha_{21}&\alpha_{22}&\alpha_{23}\\\
y_{3}&\alpha_{31}&\alpha_{32}&\alpha_{33}\end{pmatrix}.$ (75)

Using defining property $TT^{-}T=T$ one finds

$\alpha_{11}=\frac{1}{\lambda}_{1}\ ,\ \ \alpha_{22}=\frac{1}{\lambda}_{2}\ ,\
\ \alpha_{12}=\alpha_{21}=0,$

together with

$y_{1}=-\frac{x_{1}+\lambda_{1}\alpha_{13}x_{3}}{\lambda_{1}}\ ,\ \ \
y_{2}=-\frac{x_{2}+\lambda_{2}\alpha_{23}x_{3}}{\lambda_{2}}.$ (76)

The remaining parameters are completely free. Hence the Bloch representation
of the corresponding projector $TT^{-}$ reads

$TT^{-}=\begin{pmatrix}1&0&0&0\\\ -\beta_{1}x_{3}&1&0&\beta_{1}\\\
-\beta_{2}x_{3}&0&1&\beta_{2}\\\ x_{3}&0&0&0\end{pmatrix},$ (77)

with $\beta_{1}=\lambda_{1}\alpha_{13}$ and
$\beta_{2}=\lambda_{2}\alpha_{23}$. One easily finds the corresponding map
$\Phi\Phi^{-}$:

$\Phi\Phi^{-}({\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm 1\mskip-5.0mul}})={\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}-x_{3}(\beta_{1}\sigma_{1}+\beta_{2}\sigma_{2}-\sigma_{3})\ ,\
\ \ \Phi\Phi^{-}(\sigma_{3})=\beta_{1}\sigma_{1}+\beta_{2}\sigma_{2},$ (78)

and $\Phi\Phi^{-}(\sigma_{1})=\sigma_{1}$,
$\Phi\Phi^{-}(\sigma_{2})=\sigma_{2}$. To check complete positivity one has to
analyze the spectrum of the corresponding Choi matrix

$C=\sum_{i,j=1}^{2}E_{ij}\otimes\Phi\Phi^{-}(E_{ij}),$ (79)

with $E_{ij}:=|i\rangle\langle j|$. Using

$E_{11}=\frac{1}{2}({\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}+\sigma_{3})\ ,\ \
E_{22}=\frac{1}{2}({\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}-\sigma_{3}),\ \
E_{12}=\frac{1}{2}(\sigma_{1}+i\sigma_{2}),$

and $E_{21}=E_{12}^{\dagger}$, one finds

$C=\frac{1}{2}\left(\begin{array}[]{cc|cc}1+x_{3}&(1-x_{3})(\beta_{1}-i\beta_{2})&0&2\\\
(1-x_{3})(\beta_{1}+i\beta_{2})&1-x_{3}&0&0\\\ \hline\cr
0&0&1+x_{3}&-(1+x_{3})(\beta_{1}-i\beta_{2})\\\
2&0&-(1+x_{3})(\beta_{1}+i\beta_{2})&1-x_{3}\end{array}\right).$ (80)

Now, observe that $2\times 2$ submatrix

$\begin{pmatrix}1+x_{3}&2\\\ 2&1-x_{3}\end{pmatrix}$

is not positive, and hence the projector $\Phi\Phi^{-}$ is not completely
positive.

Consider now the structure of projectors onto 2-dimensional subspaces of
$M_{2}$, that is, let us assume that in the formula (71) one has
$\lambda_{2}=\lambda_{3}=0$. Now, using again the defining property
$TT^{-}T=T$ one finds

$\alpha_{11}=\frac{1}{\lambda}_{1}\ ,\ \
y_{1}=-\frac{x_{1}+\lambda_{1}\alpha_{12}x_{2}+\lambda_{1}\alpha_{13}x_{3}}{\lambda_{1}}.$

The remaining parameters are completely free. Hence the Bloch representation
of the corresponding projector $TT^{-}$ reads

$TT^{-}=\begin{pmatrix}1&0&0&0\\\
-(\gamma_{2}x_{2}+\gamma_{3}x_{3})&1&\gamma_{2}&\gamma_{3}\\\ x_{2}&0&0&0\\\
x_{3}&0&0&0\end{pmatrix},$ (81)

with $\gamma_{2}=\lambda_{1}\alpha_{12}$ and
$\gamma_{3}=\lambda_{1}\alpha_{13}$. One easily finds the corresponding map
$\Phi\Phi^{-}$:

$\Phi\Phi^{-}({\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm 1\mskip-5.0mul}})={\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm
1\mskip-5.0mul}}-(\gamma_{2}x_{2}+\gamma_{3}x_{3})\sigma_{1}+x_{2}\sigma_{2}+x_{3}\sigma_{3},$

together with

$\Phi\Phi^{-}(\sigma_{1})=\sigma_{1}\ ,\ \
\Phi\Phi^{-}(\sigma_{2})=\gamma_{2}\sigma_{1}\ ,\ \
\Phi\Phi^{-}(\sigma_{3})=\gamma_{3}\sigma_{1},$

and hence the corresponding Choi matrix reads

$C=\frac{1}{2}\left(\begin{array}[]{cc|cc}1+x_{3}&\gamma_{3}(1-x_{3})-x_{2}(i+\gamma_{2})&0&1+i\gamma_{2}\\\
\gamma_{3}(1-x_{3})+ix_{2}(i-\gamma_{2})&1-x_{3}&1+i\gamma_{2}&0\\\ \hline\cr
0&1-i\gamma_{2}&1+x_{3}&-\gamma_{3}(1+x_{3})-x_{2}(i+\gamma_{2})\\\
1-i\gamma_{2}&0&-\gamma_{3}(1+x_{3})+x_{2}(i-\gamma_{2})&1-x_{3}\\\
\end{array}\right).$ (82)

###### Lemma 1

The Choi matrix (82) is positive semidefinite iff $x_{2}=x_{3}=0$ and
$\gamma_{1}=\gamma_{2}=0$.

Indeed, let us observe that the following $2\times 2$ submatrix

$\begin{pmatrix}1+x_{3}&1+i\gamma_{2}\\\ 1-i\gamma_{2}&1-x_{3}\end{pmatrix}$

is positive iff $1-x_{3}^{2}\geq 1+\gamma_{2}^{2}$ which implies
$x_{3}=\gamma_{2}=0$. Hence (82) reduces to

$C=\frac{1}{2}\left(\begin{array}[]{cc|cc}1&\gamma_{3}-ix_{2}&0&1\\\
\gamma_{3}+ix_{2}&1&1&0\\\ \hline\cr 0&1&1&-\gamma_{3}-ix_{2}\\\
1&0&-\gamma_{3}+ix_{2}&1\\\ \end{array}\right).$ (83)

The eigenvalues of (83) read as follows

$1-\sqrt{\gamma_{3}^{2}+x_{2}^{2}+1},\ 1-\sqrt{\gamma_{3}^{2}+x_{2}^{2}+1},\
1+\sqrt{\gamma_{3}^{2}+x_{2}^{2}+1},\ 1+\sqrt{\gamma_{3}^{2}+x_{2}^{2}+1}$

and hence it is evident that they are all non-negative iff
$x_{2}=\gamma_{3}=0$. $\Box$

This way we proved that $TT^{-}$ defines Bloch representation of CPTP
projector onto 2-dimensional subspace in $M_{2}$ if and only if

$TT^{-}=\begin{pmatrix}1&0&0&0\\\ 0&1&0&0\\\ 0&0&0&0\\\ 0&0&0&0\end{pmatrix}.$
(84)

Note, that the corresponding Kraus representation of $\Phi\Phi^{-}$ reads
$\Phi\Phi^{-}(X)=\frac{1}{2}(X+\sigma_{1}X\sigma_{1})$ and hence that of
$\Lambda\Lambda^{-}$ reads as follows

$\Lambda\Lambda^{-}(X)=\frac{1}{2}(X+U\sigma_{1}U^{\dagger}XU\sigma_{1}U^{\dagger}),$
(85)

where we used $\mathcal{U}(X)=UXU^{\dagger}$. Interestingly, this implies that
any CPTP projector onto 2-dim. subspace is unital.

Finally, if the image is 1-dimensional then the corresponding projector reads

$TT^{-}=\begin{pmatrix}1&0&0&0\\\ x_{1}&0&0&0\\\ x_{2}&0&0&0\\\
x_{3}&0&0&0\end{pmatrix},$ (86)

where the Bloch vector $\mathbf{x}=(x_{1},x_{2},x_{3})$ satisfies
$|\mathbf{x}|\leq 1$.

##  VII Examples

In this Section we illustrate the construction of qubit propagators for
divisible dynamical maps using Bloch representation.

###### Example 3

A well known example of a commutative diagonalizable qubit dynamical map is
generated by the following generator (it was already analyzed in PRL-2018 )

$\mathcal{L}_{t}=\gamma_{1}(t)\mathcal{L}_{1}+\gamma_{2}(t)\mathcal{L}_{2}+\gamma_{3}(t)\mathcal{L}_{3},$
(87)

where $\mathcal{L}_{k}(\rho)=\frac{1}{2}(\sigma_{k}\rho\sigma_{k}-\rho)$. The
corresponding dynamical map reads

$\Lambda_{t}(\rho)=\sum_{\alpha=0}^{3}p_{\alpha}(t)\sigma_{\alpha}\rho\sigma_{\alpha},$
(88)

with $\sigma_{0}={\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}$, and has the following representation

$\Lambda_{t}=\begin{pmatrix}1&0&0&0\\\ 0&\lambda_{1}(t)&0&0\\\
0&0&\lambda_{2}(t)&0\\\ 0&0&0&\lambda_{3}(t)\end{pmatrix},$ (89)

with

$\lambda_{i}(t)=\exp(-\Gamma_{j}(t)-\Gamma_{k}(t)),$

where $\\{i,j,k\\}$ is a permutation of $\\{1,2,3\\}$, and
$\Gamma_{k}(t)=\int_{0}^{t}\gamma_{k}(\tau)d\tau$. The map $\Lambda_{t}$ is
invertible if all $\Gamma_{k}(t)$ are finite for finite times. Now, if for
example one has $\Gamma_{1}(t_{*})=\infty$, then
$\lambda_{2}(t_{*})=\lambda_{3}(t_{*}){=0}$ which means that the image of
$\Lambda_{t_{*}}$ is $2$-dimensional and of course it is orthogonal to the
$2$-dimensional kernel. Now, for any $s>t_{*}$ (assuming that the image of
$\Lambda_{s}$ is 2-dimensional)

$\Lambda_{t}\Lambda_{s}^{-}=\begin{pmatrix}1&0&0&0\\\
0&\lambda_{1}(t)/\lambda_{1}(s)&0&0\\\ 0&0&0&0\\\ 0&0&0&0\end{pmatrix}.$ (90)

Moreover, $V_{t,t}$ defines CPTP projector.

###### Example 4

Consider the qubit dynamical map given by:

$\Lambda_{t}(X)=[1-f(t)]X+f(t)\omega{\rm Tr}(X)$ (91)

where $\omega$ is a density matrix, and $f:\mathbb{R}_{\geq 0}\to[0,1]$ is a
monotonic function with $f(0)=0$ and $f(t)=1$ for all $t\geq t_{*}$. It is
direct to see that, $\\{\Lambda_{t}\\}_{t\geq 0}$ is CP-divisible. In the
Pauli basis, with $X=(x_{0}\ x_{1}\ x_{2}\ x_{3})^{T}$, and $\omega=(1\
\omega_{1}\ \omega_{2}\ \omega_{3})^{T}$ the map can be written as:

$\Lambda_{t}=\begin{pmatrix}1&0&0&0\\\ f(t)\omega_{1}&(1-f(t))&0&0\\\
f(t)\omega_{2}&0&(1-f(t))&0\\\ f(t)\omega_{3}&0&0&(1-f(t))\end{pmatrix}$ (92)

For $s<t_{*}$, the map $\Lambda_{s}$ is invertible, and

$\Lambda_{s}^{-1}=\begin{pmatrix}1&0&0&0\\\
-\frac{f(s)\omega_{1}}{1-f(s)}&\frac{1}{1-f(s)}&0&0\\\
-\frac{f(s)\omega_{2}}{1-f(s)}&0&\frac{1}{1-f(s)}&0\\\
-\frac{f(s)\omega_{3}}{1-f(s)}&0&0&\frac{1}{1-f(s)}\end{pmatrix}$ (93)

and hence, we have the unique propagator

$V_{t,s}=\Lambda_{t}\Lambda_{s}^{-1}=\begin{pmatrix}1&0&0&0\\\
\omega_{1}\frac{f(t)-f(s)}{1-f(s)}&\frac{1-f(t)}{1-f(s)}&0&0\\\
\omega_{2}\frac{f(t)-f(s)}{1-f(s)}&0&\frac{1-f(t)}{1-f(s)}&0\\\
\omega_{3}\frac{f(t)-f(s)}{1-f(s)}&0&0&\frac{1-f(t)}{1-f(s)}\end{pmatrix}$
(94)

Evidently, this is a channel for $0\leq s\leq t_{*}$, and in particular, for
$0\leq s\leq t_{*}\leq t$, it is a CPTP projection operator onto the 1
dimensional subspace spanned by $\omega$. Now, for $0\leq t_{*}\leq s\leq t$,
we have

$\Lambda_{s}=\Lambda_{t}=\begin{pmatrix}1&0&0&0\\\ \omega_{1}&0&0&0\\\
\omega_{2}&0&0&0\\\ \omega_{3}&0&0&0\end{pmatrix}.$ (95)

A direct computation shows that the following matrix is a generalised inverse
of $\Lambda_{s}$ giving rise to the same propagator $V_{t,s}$ through
$\Lambda_{t}\Lambda_{s}^{-}$:

$\Lambda_{s}^{-}=\begin{pmatrix}1&0&0&0\\\
\alpha_{10}&\alpha_{11}&\alpha_{12}&\alpha_{13}\\\
\alpha_{20}&\alpha_{21}&\alpha_{22}&\alpha_{23}\\\
\alpha_{30}&\alpha_{31}&\alpha_{32}&\alpha_{33}\end{pmatrix}$ (96)

where all the $\alpha_{ij}$’s are completely arbitrary.

Now, we give another example, of a non-diagonalizable, non CP-divisible (but
still divisible) qubit dynamical map:

###### Example 5

We consider the following Bloch representation of a qubit channel

$\Psi=\begin{pmatrix}1&0&0&0\\\ 0&0&0&1\\\ 0&0&0&0\\\ 0&0&0&0\end{pmatrix}\ .$
(97)

One sees that such a matrix is non-diagonalizable, has rank 2, singular values
$1,1,0,0,$ eigenvalues $1,0,0,0$, and Kraus rank $2$ with the following Kraus
operators

$K_{1}=\frac{1}{\sqrt{2}}\begin{pmatrix}0&0\\\ -1&1\end{pmatrix},\
K_{2}=\frac{1}{\sqrt{2}}\begin{pmatrix}1&1\\\ 0&0\end{pmatrix}\ .$ (98)

Now, we define the dynamical map as follows

$\Lambda_{t}(X)=[1-f(t)]X+f(t)\Psi(X)$ (99)

where $f(t)\in[0,1]$. Suppose, again that $f(t)=1$ for $t\geq t_{*}$. Clearly,
the map is divisible. Now, for $s<t_{*}$ one finds for the propagator

$V_{t,s}=\Lambda_{t}\Lambda_{s}^{-1}=\begin{pmatrix}1&0&0&0\\\
0&\frac{1-f(t)}{1-f(s)}&0&\frac{f(t)-f(s)}{(1-f(s))^{2}}\\\
0&0&\frac{1-f(t)}{1-f(s)}&0\\\ 0&0&0&\frac{1-f(t)}{1-f(s)}.\end{pmatrix}$
(100)

Simple analysis of the corresponding Choi matrix shows that in general
$V_{t,s}$ is not completely positive and hence $\Lambda_{t}$ is not CP-
divisible. Now, for $t_{*}\leq s$ one has $\Lambda_{t}=\Lambda_{s}=\Psi$, and
the most general (and trace-preserving) generalised inverse reads

$\Lambda_{s}^{-}=\begin{pmatrix}1&0&0&0\\\
\alpha_{10}&\alpha_{11}&\alpha_{12}&\alpha_{13}\\\
\alpha_{20}&\alpha_{21}&\alpha_{22}&\alpha_{23}\\\
0&1&\alpha_{32}&\alpha_{33}\end{pmatrix}$ (101)

where all the $\alpha_{ij}$’s are completely arbitrary, and the corresponding
propagator reads:

$V_{t,s}=\Lambda_{t}\Lambda_{s}^{-}=\begin{pmatrix}1&0&0&0\\\
0&1&\alpha_{32}&\alpha_{33}\\\ 0&0&0&0\\\ 0&0&0&0\end{pmatrix}.$ (102)

One finds for the Choi matrix

$C=\left(\begin{array}[]{cccc}1&\alpha_{33}&0&1+i\alpha_{32}\\\
\alpha_{33}&1&1+i\alpha_{32}&0\\\ 0&1-i\alpha_{32}&1&-\alpha_{33}\\\
1-i\alpha_{32}&0&-\alpha_{33}&1\\\ \end{array}\right),$ (103)

with the corresponding eigenvalues
$1\pm\sqrt{\alpha_{32}^{2}+\alpha_{33}^{2}+1}$. Clearly, $C$ is positive
definite if and only if $\alpha_{32}=\alpha_{33}=0$. Hence, the requirement of
complete positivity makes $V_{t,s}$ unique. We stress in this example the
image is not complimentary to the kernel, and hence one cannot make the choice
$\mathcal{C}_{s}=\rm{Ker}({\Lambda_{s}})$.

###### Example 6

Consider now a phase covariant evolution governed by the following time-local
master equation

$\dot{\Lambda}_{t}=\mathcal{L}_{t}\Lambda_{t}\ ,\ \ \Lambda_{0}={\rm id},$
(104)

with the following time-local generator Maniscalco-NJP ; Sergey ; OSID

$\mathcal{L}_{t}=\gamma_{+}(t)\mathcal{L}_{+}+\gamma_{-}(t)\mathcal{L}_{-}+\gamma_{3}(t)\mathcal{L}_{3},$
(105)

where

$\displaystyle\mathcal{L}_{+}(\rho)$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\sigma_{+}\rho\sigma_{-}-\frac{1}{2}\\{\sigma_{-}\sigma_{+},\rho\\}),$
$\displaystyle\mathcal{L}_{-}(\rho)$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\sigma_{-}\rho\sigma_{+}-\frac{1}{2}\\{\sigma_{+}\sigma_{-},\rho\\}),$
$\displaystyle\mathcal{L}_{3}(\rho)$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\sigma_{z}\rho\sigma_{z}-\rho),$

with $\sigma_{\pm}=(\sigma_{x}\pm i\sigma_{y})/2$. In general it defines a
non-commutative family of maps, that is,
$\mathcal{L}_{t}\mathcal{L}_{s}\neq\mathcal{L}_{s}\mathcal{L}_{t}$. The
corresponding dynamical map
$\Lambda_{t}=\mathcal{T}e^{\int_{0}^{t}\mathcal{L}_{\tau}d\tau}$ is given by

$\rho=\left(\begin{array}[]{c c}1-p&\alpha\\\ \alpha^{*}&p\end{array}\right)\
\to\ \rho_{t}=\left(\begin{array}[]{c c}1-p(t)&\alpha(t)\\\
\alpha(t)^{*}&p(t)\end{array}\right),$ (106)

where

$p(t)=e^{-\Gamma(t)}[G(t)+p]\ ,\ \
\alpha(t)=\alpha\,e^{-(\Gamma(t)/2+{\Gamma_{3}}(t))},$

with

$\Gamma_{3}(t)=\int_{0}^{t}\gamma_{3}(\tau)d\tau\ ,\ \
\Gamma(t)=\frac{1}{2}\int_{0}^{t}(\gamma_{+}(\tau)+\gamma_{-}(\tau))d\tau\ ,\
\ G(t)=\frac{1}{2}\int_{0}^{t}e^{\Gamma(\tau)}\gamma_{-}(\tau)d\tau\ .$

One finds the corresponding Bloch representation

$\Lambda_{t}=\begin{pmatrix}1&0&0&0\\\
0&e^{-(\Gamma(t)/2+{\Gamma_{3}}(t))}&0&0\\\
0&0&e^{-(\Gamma(t)/2+{\Gamma_{3}}(t))}&0\\\
1-e^{-\Gamma(t)}(2G(t)+1)&0&0&e^{-\Gamma(t)}\end{pmatrix},$ (107)

which already has the Ruskai representation RSW02 . Now, if
$\Gamma(t)\neq+\infty\neq\Gamma_{3}(t)$, the map is invertible, and hence
divisible. For divisibility while non-invertible, we need the following: rank
first decreases to $2$ from $4$ at $t=t_{1}$, and $\Gamma_{3}(t)=+\infty$ for
$t\geq t_{1}$. Hence, for $s\geq t_{1}$, we have

$\Lambda_{s}=\begin{pmatrix}1&0&0&0\\\ 0&0&0&0\\\ 0&0&0&0\\\
1-e^{-\Gamma(s)}(2G(s)+1)&0&0&e^{-\Gamma(s)}\end{pmatrix}.$ (108)

The most general trace-preserving generalized inverse of this map can be
written as follows

$\Lambda_{s}^{-}=\begin{pmatrix}1&0&0&0\\\
y_{1}&\alpha_{11}&\alpha_{12}&\alpha_{13}\\\
y_{2}&\alpha_{21}&\alpha_{22}&\alpha_{23}\\\
-e^{\Gamma(s)}+2G(s)+1&\alpha_{31}&\alpha_{32}&e^{\Gamma(s)}\end{pmatrix}$
(109)

where all the $y_{i}$’s and $\alpha_{ij}$’s are arbitrary. Hence, the
corresponding propagator, for $t\geq s\geq t_{1}>0$ can be written as:

$V_{t,s}=\Lambda_{t}\Lambda_{s}^{-}=\begin{pmatrix}1&0&0&0\\\ 0&0&0&0\\\
0&0&0&0\\\
\xi(t,s)&\alpha_{31}e^{-\Gamma(t)}&\alpha_{32}e^{-\Gamma(t)}&e^{-(\Gamma(t)-\Gamma(s))}\end{pmatrix}\
,$ (110)

where

$\xi(t,s)=(1-e^{-\Gamma(t)}(2G(t)+1))-e^{-(\Gamma(t)-\Gamma(s))}(1-e^{-\Gamma(s)}(2G(s)+1))\
.$

One finds for the projection

$V_{s,s}=\Lambda_{s}\Lambda_{s}^{-}=\begin{pmatrix}1&0&0&0\\\ 0&0&0&0\\\
0&0&0&0\\\
0&\alpha_{31}e^{-\Gamma(s)}&\alpha_{32}e^{-\Gamma(s)}&1\end{pmatrix}$ (111)

The Choi matrix of this map reveals that it is CP iff
$\alpha_{31}=\alpha_{32}=0$, which is, when it is just the depolarizing
channel. If the rank decreases once more at $t=t_{2}\geq t_{1}$, that is, if
$\Gamma(t)=+\infty$ for all $t\geq t_{2}$, then for $t\geq t_{2}$

$\Lambda_{t}=\begin{pmatrix}1&0&0&0\\\ 0&0&0&0\\\ 0&0&0&0\\\
1&0&0&0\end{pmatrix},$ (112)

which is a CPTP projector onto the space spanned by vacuum state
$|0\rangle\langle 0|$.

##  VIII Conclusion

CP-divisibility is a recent topic of growing interest in the field of open
quantum systems. Though extensive characterizations of necessary and (some)
sufficient conditions for CP-divisibility do exist in the literature Angel ;
BOGNA ; PRL-2018 ; CC19 ; datta , so far, to the best of our knowledge, before
us no work has been done in the direction of explicit construction of
propagators for divisible but non-invertible dynamical maps. We have
successfully proved that if a propagator has range coinciding with the image
of the map itself, we can always obtain it through a suitable generalized
inverse. For all our examples, this is possible through a reflexive
generalized inverse even. Several (non-unique) generalized inverses give the
same propagator. Moreover, for such propagators, for all our examples, we see
that the CPTP propagator is unique, though the map is non-invertible, which is
an interesting observation perhaps not predictable through earlier works in
this direction. It turns out that the generalized inverse is a very powerful
tool for attacking such problems. For example, it was a well-established fact
CC19  that there is no qubit channel which is also a projection onto a
3-dimensional subspace, but it was relatively cumbersome to prove. Through the
generalized inverse, we have in effect exhaustively characterized all trace-
preserving projections for qubit channels, CP and otherwise. And now, with a
concrete structure for the propagator at hand, we hope to make some concrete
statement about CP-divisibility or P-divisibility in general, perhaps even
solve the long-standing problem of whether CP-divisibility is equivalent to
monotonicity of the trace norm beyond qubit dynamics. Future work is intended
along these directions. Also, it should be mentioned that in our work, the
generalized inverses considered are not ”dynamical”, that is, they have not
been interpreted to represent some physical process. They are not necessarily
CP, even when the corresponding propagators are CPTP. It is perhaps possible
to view generalized inverses too as ”time reversals” in some sense, possibly
along the lines of TimeRev . Perhaps this could be done through Kraus
operators, as done in TimeRev . If possible, this would give a strong
”physical” flavour to our work on obtaining propagators through generalized
inverses. Possible connections between their work and ours, are under
consideration as well.

##  Appendix A Generalized inverse – classification

For invertible maps the propagator is uniquely defined as follows
$V_{t,s}=\Lambda_{t}\Lambda_{s}^{-1}$. The same applies for the propagator
(20) in the time-discrete case. For maps which are not invertible (or ${\rm
Im}(\Lambda_{j})$ is a proper subspace of $\mathrm{L}(\mathcal{H}_{j})$) this
prescription is no longer true. However, there is a natural generalization of
these constructions in terms of generalized inverse of the map $\Lambda_{t}$.
In this section we briefly recall basic information about generalized inverse
for linear operators Kumar ; GI ; Horn ; Yanai .

Consider a linear operator $A:\mathcal{H}_{1}\to\mathcal{H}_{2}$ and denote by
$A^{\dagger}$ a dual operator $A^{\dagger}:\mathcal{H}_{2}\to\mathcal{H}_{1}$
defined by

$\langle y|Ax\rangle_{2}=\langle A^{\dagger}y|x\rangle_{1},$ (113)

for all $x\in\mathcal{H}_{1}$ and $y\in\mathcal{H}_{2}$
($\langle\,\cdot\,|\,\cdot\,\rangle_{k}$ denotes an inner product in
$\mathcal{H}_{k}$). One defines a generalized inverse GI ; Horn  of $A$ to be
a linear operator $A^{-}:\mathcal{H}_{2}\to\mathcal{H}_{1}$ satisfying

$AA^{-}A=A.$ (114)

Clearly, $A^{-}$ is arbitrarily defined outside the image of A.

###### Definition 3

A generalized inverse $A^{-}$ is called reflexive if

$A^{-}AA^{-}=A^{-}.$ (115)

Actually, the additional property (115) is equivalent to ${\rm
Rank}(A^{-})={\rm Rank}(A)$ (in general one has ${\rm Rank}(A^{-})\geq{\rm
Rank}(A)$).

Consider now a singular value decomposition (SVD) of the complex $d_{2}\times
d_{1}$ matrix $A$

$A=U\Sigma V^{\dagger},$ (116)

where $U\in U(d_{2})$ and $V\in U(d_{1})$ (where $U(d)$ denotes the set of all
unitary $d\times d$ matrices), and $\Sigma$ is a $d_{2}\times d_{1}$ matrix
displaying the following block structure

$\Sigma=\left(\begin{array}[]{cc}D&0\\\ 0&0\end{array}\right),$ (117)

and $D$ is a diagonal $r\times r$ matrix with positive (diagonal) entries
($r\leq\min\\{d_{1},d_{2}\\}$). Any generalized inverse of $A$ has the
following form

$A^{-}=V\Sigma^{-}U^{\dagger},$ (118)

with

$\Sigma^{-}=\left(\begin{array}[]{cc}D^{-1}&X\\\ Y&Z\end{array}\right),$ (119)

and $X,Y,Z$ are completely arbitrary matrices with appropriate dimensions.

###### Proposition 9

Generalized inverse $A^{-}$

  1. 1.

satisfies (115) iff $Z=YDX$,

  2. 2.

satisfies (122) iff $Y=0$,

  3. 3.

satisfies (123) iff $X=0$,

  4. 4.

is Moore-Penrose generalized inverse iff $X=0$, $Y=0$ and $Z=0$.

##  Appendix B Generalized inverses and Projections

It turns out GI  that given any generalised inverse $A^{-}$ of $A$, there
exist transversal subspaces $V$ and $W$ of $\rm{Ker}(A)$ and $\rm{Im}(A)$
respectively, that is, $\mathcal{H}_{1}={\rm Ker}(A)\oplus V$ and
$\mathcal{H}_{2}=W\oplus{\rm Im}(A)$, with the induced decomposition
$x=x_{0}\oplus x_{1}\in\mathcal{H}_{1}$ and $y=y_{0}\oplus
y_{1}\in\mathcal{H}_{2}$, and a map $B:W\to\rm{Ker}(A)$, such that
$A^{-}(y)=By_{0}+x_{1}$. One finds therefore that

$AA^{-}=\Pi_{{\rm Im}(A)},$ (120)

defines a projector onto ${\rm Im}(A)$ along $W$, and

$A^{-}A=\Pi_{V},$ (121)

defines a projector onto $V$ along $\rm{Ker(A)}$.

Different choices of transversal subspaces $V$ and $W$ give us different
generalized inverses. Now, there are two special choices of subspaces $V$ and
$W$, that is, when $V\perp{\rm Ker}(A)$ and $W\perp{\rm Im}(A)$.

###### Proposition 10

One has

  * •

$V\perp{\rm Ker}(A)$ iff $V={\rm Im}(A^{\dagger})$,

  * •

$W\perp{\rm Im}(A)$ iff $W={\rm Ker}(A^{\dagger})$.

In this case $A^{-}A$ and $AA^{-}$ are orthogonal projectors.

###### Proposition 11

$V={\rm Im}(A^{\dagger})$ iff

$(A^{-}A)^{\dagger}=A^{-}A.$ (122)

Moreover, $W={\rm Ker}(A^{\dagger})$ iff

$(AA^{-})^{\dagger}=AA^{-}.$ (123)

Actually, there is a unique generalized inverse satisfying (115), (122) and
(123) and it is called Moore-Penrose generalized inverse.

###### Proposition 12

Let $V$ be a finite dimensional vector space, $P$ be any projection onto some
subspace $S\subset V$, and $A$ be any linear map in $\mathrm{L}(V)$ with
$\rm{Im}(A)=S$. Then, there exists a generalized inverse $A^{-}$ of $A$ such
that $AA^{-}=P.$

Proof: Let ${\rm dim}(V)=n$, and ${\rm dim}(S)=r<n$. To begin with let $S_{0}$
be the span of the first $r$ of the $n$ vectors in the canonical basis. Let
$P_{0}$ be an arbitrary projection onto this subspace. As a matrix resolved in
the canonical basis $P_{0}$ can be represented as follows

$P_{0}=\left(\begin{array}[]{cc}{\mathchoice{\rm 1\mskip-4.0mul}{\rm
1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}_{r}&X\\\
0&0\end{array}\right)\ ,$

where $X$ is an arbitrary $r\times(n-r)$ matrix. If $A=U\Sigma V^{\dagger}$
stands for SVD for $A$, then $S_{0}$ can be transformed to $S$ via $U$, and
$P=UP_{0}U^{\dagger}$. Now, we choose $P_{0}=U^{\dagger}PU$. This fixes the
matrix $X$. Now, $\Sigma=\left(\begin{array}[]{cc}D&0\\\
0&0\end{array}\right)$, where $D$ is an invertible diagonal matrix of
dimension $r\times r$. A valid generalized inverse of $\Sigma$ is,
$\Sigma^{-}=\left(\begin{array}[]{cc}D^{-1}&D^{-1}X\\\ 0&0\end{array}\right)$,
and we immediately observe that

$P_{0}=\Sigma\Sigma^{-}.$ (124)

Hence, we have $P=AA^{-}$, where $A^{-}=V\Sigma^{-}U^{\dagger}$ is a
generalized inverse of $A$.

## Acknowledgements

DC was supported by the National Science Centre project 2018/30/A/ST2/00837.
UC would like to thank the TAPS programme (2019) of the Faculty of Physics,
Astronomy and Informatics of the Nicolaus Copernicus University in Toruń for
his stay in Poland in summer 2019.

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