# Measure of not-completely-positive qubit maps: the general case

Vinayak Jagadish  Quantum Research Group, School of Chemistry and Physics,
University of KwaZulu-Natal, Durban 4001, South Africa  National Institute for
Theoretical Physics (NITheP), KwaZulu-Natal, South Africa  R. Srikanth
Poornaprajna Institute of Scientific Research, Bangalore- 560 080, India
Francesco Petruccione  Quantum Research Group, School of Chemistry and
Physics, University of KwaZulu-Natal, Durban 4001, South Africa  National
Institute for Theoretical Physics (NITheP), KwaZulu-Natal, South Africa

###### Abstract

We show that the set of not-completely-positive (NCP) maps is unbounded,
unless further assumptions are made. This is done by first proposing a
reasonable definition of a valid NCP map, which is nontrivial because NCP maps
may lack a full positivity domain. The definition is motivated by specific
examples. We prove that for valid NCP maps, the eigenvalue spectrum of the
corresponding dynamical matrix is not bounded. Based on this, we argue that in
general the volume measure of qubit maps, including NCP maps, is not well
defined.

##  I Introduction

For a long time, the study of open quantum systems Breuer and Petruccione
(2002) remained confined to maps which are completely positive (CP). However,
phenomenological studies of spin relaxation showed that complete positivity
was not a necessary requirement Simmons and Park (1981); Raggio and Primas
(1982); Simmons and Park (1982). Based on the concept of linear assignment
maps, a better view on the issue of complete positivity was brought forward in
the debate between Pechukas and Alicki Pechukas (1994); Alicki (1995);
Pechukas (1995). With the understanding of the importance of initial
correlations Jordan  _et al._ (2004); Carteret  _et al._ (2008), non-Markovian
dynamics Breuer  _et al._ (2016); Li  _et al._ (2018) and advances in control
of quantum systems, not-completely-positive (NCP) maps are now studied ever
more actively Szarek  _et al._ (2008); Cuffaro and Myrvold (2013). It is also
now known that NCP maps arise naturally as intermediate maps of quantum non-
Markovian processes Rivas  _et al._ (2010).

Within the space of positive maps, the relative measure for CP and NCP maps
was addressed for Pauli unital channels in Jagadish  _et al._ (2019). The
present work revisits the question of measure of NCP maps, but by relaxing the
requirement that the map has a full positivity domain on the system of
interest, i.e., all states of the reduced system produce valid output states.
An example of this type is the map corresponding to the partial-trace
operation. But, more generally, we can allow maps, namely NCP maps, whose
positivity domain is restricted. An example of this type would be the
intermediate (NCP) map for the non-Markovian dephasing channel Shrikant  _et
al._ (2018), for which, at the singularity in the decoherence rate, only a set
of states of zero measure produces a valid output.

In Jagadish  _et al._ (2019), we showed that for CP maps, the eigenvalue
spectrum is always bounded. Here, our main result is that the NCP maps no
longer form a compact set, essentially because the eigenvalues of the
corresponding dynamical matrices are not bounded. Hence, for the set of qubit
maps, including NCP maps, the volume measure may not be well defined, in
general. For simplicity, we restrict ourselves to the case of unital maps. The
paper is organized as follows. In Sec. II, we discuss the preliminaries.
Section III sets the idea presented in the paper through three examples. The
main results are then presented and discussed in Sec. IV. We then conclude in
Sec. V.

##  II Preliminaries

###  II.1 Positivity domains and valid maps

Let $\mathcal{E}$ be a positive map acting on a qubit represented by the
density matrix

$\rho=\frac{1}{2}(\mathbbm{1}+a_{i}\sigma_{i})=\frac{1}{2}\left(\begin{array}[]{cc}1+a_{3}&a_{1}-\imath
a_{2}\\\ a_{1}+\imath a_{2}&1-a_{3}\end{array}\right),$ (1)

where the vector ${\bf a}=(a_{1}\,,\,a_{2}\,,\,a_{3})$, with

$a_{1}^{2}+a_{2}^{2}+a_{3}^{2}\leq 1,$ (2)

is called the Bloch vector. All one qubit states lie on or inside the “Bloch
ball,” which is the unit ball in the space $\mathbb{R}^{3}$ parametrized by
the axes $a_{1}$, $a_{2},$ and $a_{3}$.

The map $\mathcal{E}$ can be represented by a four dimensional Hermitian
matrix, usually referred to as the dynamical matrix Sudarshan  _et al._
(1961). The dynamical matrix is also called the Choi matrix Choi (1975). If
the dynamical matrix is positive, the map is completely positive (CP).
Otherwise, it is not-completely-positive (NCP). The trace of the dynamical
matrix acting on a qubit is 2. Positivity of a map addresses its action on
density matrices whereas complete positivity (CP) is a statement on the map
itself.

The important point to note is that the definition of a map makes sense only
for a valid set of states. For a CP map acting on a qubit, the domain is the
entire Bloch ball $\mathcal{B}$. But, for NCP maps, only certain states on the
Bloch ball act as a valid domain for their action.

######  Definition 1 (Positivity domain).

Given the map $\mathcal{E}$, the set
$\mathcal{P}_{\mathcal{E}}\equiv\\{p\in\mathcal{B}|\mathcal{E}(p)\in\mathcal{B}\\}$
constitutes the positivity domain of $\mathcal{E}$.

In words, $\mathcal{P}_{\mathcal{E}}$ is the subset of the Bloch ball
$\mathcal{B}$ whose image under map $\mathcal{E}$ falls within $\mathcal{B}$.

######  Definition 2 (Valid map).

A map $\mathcal{E}$ acting on a qubit is valid iff its associated positivity
domain is nonempty. i.e., $\mathcal{P}_{\mathcal{E}}\neq\emptyset$.

In view of this definition, a map is valid precisely if its positivity domain
is nonvanishing. This definition will be motivated in the examples discussed
later.

Note that the evolution of a qubit may contain a singularity at some instance,
meaning that the decoherence rate in the master equation at that point, and
hence the (instantaneous) intermediate map $\mathcal{E}^{\ast}$, diverges. The
map can still be valid if the positivity domain
$\mathcal{P}_{\mathcal{E}^{\ast}}$ is nonvanishing. That this is physically
well motivated will be clarified in examples discussed later.

###  II.2 Unital maps acting on a qubit

The general form of a dynamical matrix $\tilde{B}$ acting on a qubit which is
unital and trace preserving (TP) can be parametrized as follows:

$\tilde{B}=\left(\begin{array}[]{cccc}a&x&y&z\\\ x*&1-a&w&-y\\\
y*&w*&1-a&-x\\\ z*&-y*&-x*&a\\\ \end{array}\right).$ (3)

This is obtained by considering a four dimensional Hermitian matrix such that
$\mathrm{tr}_{1}\tilde{B}=\mathbbm{1}=\mathrm{tr}_{2}\tilde{B}$. This means
that the partial trace with respect to each of the two bipartite subsystems is
identity. Note that $a$ is real and $x,y,z,$ and $w$ are in general complex.
More general maps, allowing damping, can be considered, but the above suffices
to prove our main result.

##  III Motivating the concept of the validity of a map

Allowing NCP maps gives considerable freedom in what maps we can consider,
even if they are not CP. Our earlier definition of validity is the only
restriction we place on what we consider as valid maps. The trivial map that
maps every qubit state to the fixed state $\begin{pmatrix}2&0\\\
0&-1\end{pmatrix}$ is invalid (as this is not a bona fide density matrix),
whereas a map that takes all states to the maximally mixed state, is indeed
valid, according to our definition.

###  III.1 A numerical example

This numerically simulated example is to show that the eigenvalues of the
dynamical matrix representing a NCP map on a qubit can exceed 2 in absolute
values, unlike that for CPTP maps. Consider a NCP map, whose dynamical matrix
is given by

$B_{NCP}=\left(\begin{array}[]{cccc}0.20&0.95&0.70&0.10\\\
0.95&0.80&0.30&-0.70\\\ 0.70&0.30&0.80&-0.95\\\ 0.10&-0.70&-0.95&0.20\\\
\end{array}\right).$ (4)

Clearly, the map is unital and trace preserving. It has eigenvalues 2.32409,
-1.12409, 0.669258, 0.130742. As we can see, one eigenvalue is greater than 2,
which indicates that the eigenvalues are not bounded by 2, for general NCP
maps acting on a qubit. Its action on a general qubit density operator
produces an output with eigenvalues

$\displaystyle\lambda_{1}$ $\displaystyle\approx$ $\displaystyle
0.5\,-0.47\sqrt{4.2a_{1}^{2}-1.3a_{1}a_{3}+0.04a_{2}^{2}+2.6{a_{3}}^{2}},$
$\displaystyle\lambda_{2}$ $\displaystyle\approx$ $\displaystyle
0.5+0.47\sqrt{4.2{a_{1}}^{2}-1.3{a_{1}}{a_{3}}+0.04{a_{2}}^{2}+2.6{a_{3}}^{2}}.$

from which it follows that any choice of $a_{j}$ that satisfies Eq. (2) and
$4.2{a_{1}}^{2}-1.3{a_{1}}{a_{3}}+0.04{a_{2}}^{2}+2.6{a_{3}}^{2}\geq 0$ (for
example, $a_{3}=0.5,a_{1}=0.05,a_{2}=0.1$) is an element of the positivity
domain of the map. Since this is a non-vanishing set, it follows from our
definition that this constitutes a valid map. Note that in this case, the set
$\mathcal{P}_{\mathcal{E}}$ has a non zero measure. However, this is not
necessary. So long as the positivity domain is non-vanishing, the map is well
defined. This is shown to be the case in the next two examples, where the
positivity domain is of measure zero.

###  III.2 Two successive CNOT gates

Consider the action of two successive CNOT gates, $U=U_{CNOT}\circ U_{CNOT},$
on the initial product state
$\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)_{A}\otimes|\phi\rangle_{B}$ of two
qubits labeled $A$ and $B$.

Let $|\psi^{\prime}\rangle$ denote the total state of the systems $A$ and $B$
after the first $U_{CNOT}$ has been applied:

$\displaystyle|\psi^{\prime}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(|0\rangle_{A}\otimes|0\rangle_{B}+|1\rangle_{A}\otimes|1\rangle_{B});\thinspace\text{for}\thinspace|\phi\rangle=|0\rangle,$
(5) $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(|1\rangle_{A}\otimes|0\rangle_{B}+|0\rangle_{A}\otimes|1\rangle_{B});\thinspace\text{for}\thinspace|\phi\rangle=|1\rangle.$

Since $U=\mathbbm{1}$, the output state of $B$ is the same as its input, which
could be $|0\rangle_{B}$ or $|1\rangle_{B}$. Therefore, to discern the reduced
dynamics of $B$ between the application of the two CNOT operations, one would
need a map that takes the identity state as the input, and outputs either
$|0\rangle_{B}$ or $|1\rangle_{B}$.

Note that this is a one-to-many relation and hence non-linear behavior Schmid
_et al._ (2018) would invalidate it as a map in certain conventional
situations. However, this intermediate map corresponding to the application of
the second CNOT is indeed valid according to our definition. This is
physically well motivated since this intermediate map, only when supplemented
to the CP map representing the evolution up to the application of the first
CNOT, recreates the full identity map after the application of the second
CNOT.

The problem may be understood as an instance of a singularity in the
intermediate map, while the full map is well defined. Consider the control
qubit to be in the state $\cos\theta\ket{0}+\sin\theta\ket{1}$ and the system
to be the target qubit. Let us represent the map acting on the density matrix
expressed as a column vector, and call it the $A$ matrix following Sudarshan
_et al._ (1961); Jagadish and Petruccione (2018). Denote the $A$ matrix
corresponding to the first application of CNOT to be $A_{1}$, and that of the
second one to be $A_{2}$. Then we have $A_{1}\cdot A_{2}=\mathbbm{1}_{4}$,
from which it follows that

$A_{2}=(A_{1})^{-1},$ (6)

meaning that $A_{2}$ corresponds to the inverse of the first, which is seen to
be a NCP map as shown below.

The $A$ matrix for the first application of CNOT is

$A_{1}=\left(\begin{array}[]{cccc}\cos^{2}(\theta)&0&0&\sin^{2}(\theta)\\\
0&\cos^{2}(\theta)&\sin^{2}(\theta)&0\\\
0&\sin^{2}(\theta)&\cos^{2}(\theta)&0\\\
\sin^{2}(\theta)&0&0&\cos^{2}(\theta)\\\ \end{array}\right),$ (7)

which corresponds to a bit flip channel. In view of Eq. (6), it follows from
Eq. (7) that

$A_{2}=\sec(2\theta)\left(\begin{array}[]{cccc}\cos^{2}(\theta)&0&0&-\sin^{2}(\theta)\\\
0&\cos^{2}(\theta)&-\sin^{2}(\theta)&0\\\
0&-\sin^{2}(\theta)&\cos^{2}(\theta)&0\\\
-\sin^{2}(\theta)&0&0&\cos^{2}(\theta)\\\ \end{array}\right),$ (8)

from which it follows that the associated dynamical matrix is

$\mathfrak{B}_{2}=\sec(2\theta)\left(\begin{array}[]{cccc}\cos^{2}(\theta)&0&0&\cos^{2}(\theta)\\\
0&-\sin^{2}(\theta)&-\sin^{2}(\theta)&0\\\
0&-\sin^{2}(\theta)&-\sin^{2}(\theta)&0\\\
\cos^{2}(\theta)&0&0&\cos^{2}(\theta)\\\ \end{array}\right),$ (9)

whose non vanishing eigenvalues are seen to be
$-2\sin^{2}(\theta)\sec(2\theta)$ and $2\cos^{2}(\theta)\sec(2\theta)$.
Exactly one of the two eigenvalues is negative for any choice of $\theta\neq
0,n\pi$ ($n$ an integer), showing that $A_{2}$ is NCP for such choices. The
parameter $\theta=\frac{\pi}{4}+n\pi$ represents a singularity, where the
intermediate map represented by $A_{2}$ diverges, and corresponds to the
nonlinearity mentioned above, which is now seen as a manifestation of the
singularity rather than grounds to invalidate the map.

The operator sum representation for the map $\mathfrak{B}_{2}$ is

$\rho^{\prime}=\sqrt{1+\sec(2\theta)}\mathbbm{1}\rho\mathbbm{1}+\sqrt{1-\sec(2\theta)}\sigma_{x}\rho\sigma_{x}.$
(10)

Consider the fixed points of $\sigma_{x}$, i.e., states that are invariant
under the action of $\sigma_{x}$. They will also be invariant under the above
map, irrespective of $\theta$. Thus, when $\theta=\frac{\pi}{4}$, only these
states, being invariant, will produce a valid output, whereas any other states
will lead to divergent outputs. Thus, the set of these fixed points,
constitute the positivity domain.

In particular, these states have the form

$\rho^{X}_{p}\equiv
p\ket{+}\bra{+}+(1-p)\ket{-}\bra{-}=\begin{bmatrix}1&2p-1\\\
2p-1&1\end{bmatrix},$ (11)

where $0\leq p\leq 1$. Under the action of the above map, the state (11)
evolves to
$[1+\sec(2\theta)]\rho^{X}_{p}+[1-\sec(2\theta)]\rho^{X}_{p}=\rho^{X}_{p}$,
independently of $\theta$. At $\theta=\pi/4$, the function $\sec(2\theta)$ and
hence the map diverge, but the set $\Sigma^{X}_{\sigma}$ of all points of the
form $\rho^{X}_{p}$ is unaffected by the infinity and thus constitutes our
positivity domain. Notice that $\Sigma^{X}_{\sigma}$ represents a straight
line connecting two extreme points on the Bloch ball, and thus has measure
zero.

It is important to stress here that even when the (intermediate) map is
singular– i.e., the absolute values of the eigenvalues of the dynamical matrix
increase without bound to infinity (such that their sum is 2)–the set
$\mathcal{P}_{\mathcal{E}^{\ast}}$ is non vanishing, ensuring that the map is
well-defined. This is of course validated by the fact that the full map itself
corresponds to identity operation and is thus well defined. The singularity
thus gives an extreme instance of the unboundedness of eigenvalues of valid
dynamical maps.

###  III.3 Intermediate map with non-Markovian dephasing

Consider an example of a non-Markovian channel based on a familiar process,
namely quantum dephasing. This is described by the Lindblad equation,

$\frac{d\rho}{dq}=\lambda(q)[-\rho(q)+\sigma_{z}\rho(q)\sigma_{z}],$ (12)

where $\lambda$ is the decoherence rate and the parameter $q$ rises
monotonically from 0 to (asymptotically with time) $\frac{1}{2}$. For
Markovianity to hold in the sense of CP divisibility, $\lambda$ must always be
positive as a function of $q$ . Furthermore, for the noise to be described by
a quantum dynamical semigroup, $\lambda$ must be a positive constant.

In a non-Markovian dephasing channel such as quantum random telegraph noise
(RTN) Daffer  _et al._ (2004), the negativity of $\lambda$ arises from the
presence of terms with (co)sinusoidal dependence on time, but in the non-
Markovian dephasing model proposed in Shrikant  _et al._ (2018)

$\lambda(q)=\frac{\frac{1}{2}(\alpha_{-}+\alpha_{+})-q}{(q-\alpha_{+})(q-\alpha_{-})},$
(13)

where $\alpha_{-}$ is a parameter that lies in the range $[\frac{1}{2},1]$ and
$\alpha_{-}<\alpha_{+}$. Thus, the regions for which $q>\alpha_{-}$ lead to
non-Markovianity. Furthermore, $q=\alpha_{-}$ represents a singularity, which
we discuss below.

In Ref. Shrikant  _et al._ (2018), the authors choose

$\alpha_{\pm}=\frac{1}{2\nu}(\nu+1\pm\sqrt{\nu^{2}+1}),$

which corresponds to the Kraus operators $K_{I}=\sqrt{(1-\beta)}\mathbbm{1}$
and $K_{Z}=\sqrt{\beta}\sigma_{z}$, with $\beta=[1+\nu(1-q)]q$, where the
dimensionless quantity $\beta$ can turn negative for sufficiently large value
of parameter $\nu$. In this model, $0\leq\nu\leq 1$ and serves as a non-
Markovianity parameter.

The singularity mentioned above occurring at $q=\alpha_{-}$ can be physically
interpreted as follows. At this point, all initial states of the form

$\rho(0)\equiv\left(\begin{array}[]{cc}r&s\\\ s^{\ast}&1-r\end{array}\right),$

are mapped to

$\rho(\alpha_{-})\equiv\left(\begin{array}[]{cc}r&0\\\
0&1-r\end{array}\right),$

creating an instance of non invertibility. The family of states
$\rho(\alpha_{-})$ corresponds to a straight line in the Bloch ball from the
north to south pole, having measure zero. (For general consistency conditions
pertaining to non-invertible time evolution, see Andersson  _et al._ (2007);
Rivas  _et al._ (2014).)

However, as in the previous example, the singularity is not pathological, and
only momentary. This follows from the fact that the infinitesimal evolution
determined by Eq. (12),

$d\rho=\lambda(\alpha_{-})[-\rho(q)+\sigma_{z}\rho(q)\sigma_{z}]dq,$ (14)

is well-behaved when acting on (only) the elements of the family
$\rho(\alpha_{-})$. The intermediate map (14) thus gives us an example of a
map whose positivity domain is of zero measure.

We shall now state and prove our main result.

##  IV Results and Discussion

The following result on unboundedness, which is shown for the unital maps,
also can be extended to non-unital maps. But, by virtue of applying it to
unital qubit maps, it obviously implies the unboundedness of the general set
of maps (CP or NCP) on qubits.

######  Theorem 1.

The set of maps (including NCP maps) acting on a qubit is neither closed nor
bounded.

We represent maps in the space of eigenvalues of the dynamical matrix. By the
Choi-Jamiolkowski isomorphism Jamiołkowski (1972), the set of CP maps is
isomorphic to that of two-qubit states, and hence bounded. Thus, we shall be
concerned with the (un)boundedness of NCP maps. By definition, the positivity
domain corresponding to any valid NCP map acting on a qubit should be non
empty. Our proof proceeds by providing an example that evinces the
unboundedness of the eigenvalues of $\tilde{B}$ of Eq. (3). Specifically,
consider Eq. (9) which is in the form of Eq. (3) with
$a=z=\sec(2\theta)\cos^{2}(\theta),w=\sec(2\theta)\sin^{2}(\theta),$ and the
rest are all zero. Clearly, the eigenvalues $-2\sin^{2}(\theta)\sec(2\theta)$
and $2\cos^{2}(\theta)\sec(2\theta)$ diverge to $\infty$ at
$\theta=\frac{\pi}{4}$, providing an instance of the unboundedness of
eigenvalues of NCP maps. This unboundedness of eigenvalues arises only for NCP
maps, where the eigenvalues can take negative values. That the eigenvalues are
non bounded implies that the space of NCP maps is not closed.

For NCP maps that arise as intermediate maps of a CP map, even if they
diverge, one can find the positivity domain to be non vanishing, affirming the
validity of the NCP map. An instance was discussed in the context of example
of Sec. III.2. As another instance, consider the action of the map as in Eq.
(3) on Eq. (1). We assume all parameters are real, for simplicity. The output
density matrix has eigenvalues

$\lambda_{\pm}=\frac{1}{2}\Big{(}1\pm\sqrt{a_{1}^{2}\left((w+z)^{2}+4x^{2}\right)+4a_{1}a_{3}((2a-1)x+y(w+z))+a_{2}^{2}(w-z)^{2}+a_{3}^{2}\left((1-2a)^{2}+4y^{2}\right)}\Big{)}.$
(15)

In this case, setting the positivity condition on the two eigenvalues, one can
see that the point $(a_{1},a_{2},a_{3})=(0,0,0)$ corresponding to the
maximally mixed state serves as one point in the positivity domain of the NCP
map. In general, one can find sets of points $(a_{1},a_{2},a_{3})$ in the
Bloch ball such that the eigenvalues $\lambda_{\pm}$ in Eq. (15) are both
positive. This set will constitute the positivity domain of the map (for an
illustration, see below). $\blacksquare$

For the case of Eq. (4) in the example discussed in Sec. III.1, the positivity
domain is plotted in Fig. (1). That this domain is a subset of the Bloch ball
implies that the map in question is not positive, and thereby goes beyond the
restriction under which a simple measure of NCP maps was found in Ref.
Jagadish  _et al._ (2019).

Figure 1: (Color online) The positivity domain for the NCP map represented by
the dynamical matrix in Eq. (4). The outer ball (of unit radius) shown in
white is the Bloch ball.

This result for the unital map can be extended (albeit in more cumbersome way)
to the non unital case, also. However, the present example suffices for the
proof.

It follows from the unboundedness of NCP maps, that the set of qubit maps,
including NCP maps, do not form a compact set. Now, if the set of divergent
maps form a discrete set of singularities, then one might hope that the set of
maps obtained by excepting these points could be endowed with a bona fide
measure. However, it is easy to see that the divergent maps themselves form a
continuous family.

To see this, note that we can replace the CNOT in the double-CNOT example with
any other entangling operation, and we obtain a divergent map corresponding to
the intermediate map represented by the second application of the control
operation. Indeed, consider any control-$Q(\theta,\phi,\xi)$, where
$Q(\theta,\phi,\xi)$ is any unitary, i.e.,

$I_{2}\otimes I_{2}+I_{2}\otimes Q(\theta,\phi,\xi),$

and the angles $\theta,\phi,$ and $\xi$ may be varied continuously. This
strongly suggests that a well-defined measure of maps for this general case
does not seem to arise, unless further restrictions are made.

We enumerate a few such interesting cases:

  1. 1.

If we restrict to the case of Pauli channels, subject to the condition of
positivity (i.e., having full positivity domain), then the volume measure of
NCP maps is twice that of CP maps Jagadish  _et al._ (2019).

  2. 2.

A similar result also holds if the Pauli channels are rotated by a fixed
unitary $U$. This corresponds to the simplex whose four vertices are
$UPU^{\dagger}$, where $P$ is a Pauli matrix. This may be considered as a
rotated version of the Pauli tetrahedron $\mathcal{T}$, discussed in Jagadish
_et al._ (2019).

Finally, note that the double-CNOT, or equivalently a double-C-phase gate,
provides perhaps the simplest method to practically realize NCP maps, and even
a singularity, conveniently on a quantum computer in a way that is well within
the reach of present-day quantum technology. Indeed, the double-control
application of any two-qubit gate will be acceptable, based on the argument
given above.

##  V Conclusions

Positivity domains for NCP qubit maps, arising from initial correlations with
an environment, were first addressed in Jordan  _et al._ (2004), where it was
shown that they can be strict subsets of the Bloch ball. Here, we show that
arbitrary (even divergent) NCP maps, subject only to the trace-preserving
condition, can be physically valid in the sense of having a non-vanishing
(possibly zero-measure) positivity domain.

The physical significance of this result is that the volume measure of qubit
maps, including NCP maps, may in general not be well defined, even though the
dynamics is quite regular Chruściński and Kossakowski (2010). Specific
restrictions may be imposed in certain cases, as in Jagadish  _et al._ (2019),
to define a measure.

##  VI Acknowledgements

The work of V.J. and F.P. is based upon research supported by the South
African Research Chair Initiative of the Department of Science and Technology
and National Research Foundation. R.S. thanks the Defense Research and
Development Organization (DRDO), India for the support provided through the
Project No. ERIP/ER/991015511/M/01/1692.

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