Clarifications needed on why certain arguments related to quantum maps dubbed as false As I was learning more about the evolution of open quantum systems, I came across this question. Reading through the answers, I found this paper by A. Shaji and E.C.G. Sudarshan.
The mathematical arguments provided in the article clearly imply that NCP (not completely positive) maps are just as good as CP (completely positive) maps for certain cases. An example is provided to illustrate that if the reduced dynamics of a system is not positive on some of its states, then that must be treated as an indication of entanglement between the system and its environment. To restore physical consistency, a compatibility domain is defined in which the action of the NCP map is positive on all the states within the domain. The line of reasoning questions the justification we put forward in restricting the reduced dynamics of open quantum systems to exclusively CP maps.
Ron Maimon comments this at the end of that answer (and the comment had a few upvotes too):

Shaji and Sudarshan's paper is completely bogus.

Of course, NCP maps fail to preserve linearity as stated in this answer and hence might not be of much use. Still, I fail to understand why is the paper entirely bogus. Could someone please clarify what important points am I missing related to the article?
 A: Introduction
With some assumptions and approximations, the time-evolution of an open quantum system can often be adequately modelled using completely positive (CP) maps, as reviewed in [3], [4], and [5], specifically using CP trace-preserving (CPTP) maps, which are also called quantum channels. (The definitions are given below.) For example, page 1 in [6] says:


In quantum information theory, noisy evolution is usually modelled by completely positive, trace preserving (CPT) maps.


One reason for focusing on CP maps is convenience: the general properties of CP maps have been thoroughly characterized mathematically. For example, any such map may be expressed using a Kraus representation. Page 1 in reference [7] mentions a more fundamental reason for the widespread use of CP maps:

Given a factorized initial system-environment state and a unitary interaction between system and environment the reduced time evolution has to be CP...

But what if the initial system-environment state is not factorized? Should we then use a non-CP map to represent the system's dynamics, as suggested in [1]? We need to be careful here, because "not using a CP map" is different than "using a map that is non-CP." The issue is with using a map at all, whether or not it is CP. This answer explains why.

Positive maps and the partial trace
Consider a Hilbert space ${\cal H}={\cal H}_S\otimes{\cal H}_R$ where $S$ is the system of interest and $R$ is a "reservoir" (aka environment). An operator $A$ on ${\cal H}$ is called positive if it satisfies
$$
\langle\psi|A|\psi\rangle\geq 0
\tag{1}
$$
for all $|\psi\rangle\in{\cal H}$. A better name might be "non-negative," but I'll stick with traditional language and call it "positive" even though the right-hand side of (1) can be zero. Let $\rho$ be any density matrix (unit-trace positive operator) on ${\cal H}$, and let $U$ be any unitary transformation on ${\cal H}$, so that 
$$
U\rho U^\dagger
\tag{2}
$$ 
is another density matrix on ${\cal H}$. Let 
$$
\rho_S\equiv\text{Trace}_R(\rho)
\tag{3}
$$ 
denote the partial trace of $\rho$ over the reservoir $R$, so that $\rho_S$ is the reduced density matrix of the system $S$. The reduced density matrix $\rho_S$ is a positive operator on ${\cal H}_S$, which can be proven using the definition of the partial trace, using the fact that $\rho$ is a positive operator on ${\cal H}$. Here's the proof: for any $|\phi\rangle\in{\cal H}_S$, we have
\begin{align}
\langle\phi|\rho_S|\phi\rangle
  &= \langle\phi|\text{Trace}_R(\rho)|\phi\rangle \\
  &= \sum_k\langle\phi, k|\rho|\phi,k\rangle
\tag{4}
\end{align}
with 
$$
|\phi,k\rangle \equiv |\phi\rangle\otimes |k\rangle
\tag{5}
$$
and where the vectors $|k\rangle$ are an orthonormal basis for ${\cal H}_R$. Now observe that if $\rho$ is positive, then every term in the sum (4) is non-negative. This completes the proof that $\rho_S$ is positive in ${\cal H}_S$. The proof that $\rho_S$ has unit trace (if $\rho$ does) also follows easily from (4).

Illustration
If we start with a valid density matrix $\rho$ on the combined system, then the partial trace yields a valid reduced density matrix $\rho_S$. To illustrate what can go wrong if we're not careful, consider this two-qubit example from section 3 of [1]:
$$
\rho=\frac{1}{4}\left(
\left[\matrix{1&0\cr 0&1}\right]
\otimes
\left[\matrix{1&0\cr 0&1}\right]
+
\left[\matrix{0&1\cr 1&0}\right]
\otimes
\left[\matrix{0&0\cr 0&-2}\right]
\right),
\tag{6}
$$
which the paper [1] writes as ${\cal R}=\frac{1}{4}(1-\sigma_1+\sigma_1\tau_3)$ where $\sigma_k$ and $\tau_k$ are Pauli matrices acting on the first and second qubits in a two-qubit Hilbert space. The corresponding reduced matrix $\rho_S$ of the first qubit is not positive. There is no contradiction here, because the example (6) is not a valid density matrix on the two-qubit Hilbert space. In particular, it is not postive, because
$$
\langle\psi|\rho|\psi\rangle < 0
\hskip2cm
\text{when }
|\psi\rangle\propto \left[\matrix{1\cr 1}\right]
\otimes
\left[\matrix{0\cr 1}\right].
\tag{7}
$$
To get a valid reduced density matrix $\rho_S$, we should start with a valid density matrix $\rho$ on the combined system. 

The dynamics of an open quantum system
The focus here is on properties of the "map"
$$
\rho_S\to \rho_S'
\tag{8}
$$
defined by
$$
\rho_S\equiv\text{Trace}_R(\rho)
\hskip2cm
\rho_S'\equiv\text{Trace}_R(U\rho U^\dagger)
\tag{9}
$$
for some fixed unitary matrix $U$ on ${\cal H}$. If $\rho$ is a valid density matrix, then so are $\rho_S$ and $\rho_S'$. This was proven above. 
Now, suppose that the total state $\rho$ has the form
$$
\rho=\rho_S\otimes\rho_R.
\tag{10}
$$
If the factor $\rho_R$ is fixed, then equations (8)-(9) define a legitimate map from density matrices on ${\cal H}_S$ to density matrices on ${\cal H}_S$. In fact, in this case, the map (8) is completely positive (theorem 3.3 in [2]). This means that the associated map 
$$
\rho_S\otimes I\to\rho_S'\otimes I
$$ 
is positive whenver $I$ is the identity operator on some auxiliary Hilbert space ${\cal H}_W$ (sometimes called the witness). 
On the other hand, if $\rho$ does not have the form (10), then it is not clear to me that the form of $\rho$ can be naturally restricted in a way that makes (8) a map in the strict sense. If $\rho$ is not restricted, then for a generic $U$, we may be able to vary $\rho$ in a way that varies $\rho_S'$ without varying $\rho_S$. This is an obstacle to thinking of (8) as a map, in the strict sense of the word "map." To illustrate this, consider a two-qubit Hilbert space ${\cal H}$ with $\rho=\rho_S\otimes\rho_R$ and let $U$ be the unitary transformation that swaps the two factors. Then $\rho_S'=\rho_R$ even though $\rho_S$ is independent of $\rho_R$. So if $\rho_R$ is not fixed, then (8) is not a map at all, much less a CP map.
The transformation (8) still represents partial information about a unitary map on the larger Hilbert space (equations (9)); but (8) itself is not a map in the strict sense, because the same input $\rho_S$ can yield different outputs $\rho_S'$, even if the unitary transformation $U$ is fixed. The key points here are that if the dynamics is defined by taking a partial trace over part of a closed system with an unspecified initial state, then:


*

*In general, the dynamics of an open quantum system is not given by a CP map.

*In cases where we can't use a CP map, we typically can't use a map at all, CP or otherwise.
Again, though, with some assumptions and approximations, the time-evolution of $\rho_S$ can often be adequately modelled using CP maps, as reviewed in [3], [4], and [5]. We just need to remember the limitations of those assumptions and approximations.

References
[1] Shaji and Sudarshan (2005), "Who's afraid of not completely positive maps?" Physics Letters A 341: 48-54, https://web2.ph.utexas.edu/~gsudama/pub/2005_003.pdf
[2] Wood (2009), "Non-Completely Positive Maps: Properties and Applications," https://arxiv.org/abs/0911.3199
[3] Chruściński and Pascazio (2007), "A Brief History of the GKLS Equation," https://arxiv.org/abs/1710.05993
[4] Breuer (2008), "Non-Markovian quantum dynamics and the method of correlated projection superoperators," https://arxiv.org/abs/0707.0172
[5] Hall et al (2010), "Canonical form of master equations and characterization of non-Markovianity," https://arxiv.org/abs/1009.0845
[6] Cubitt et al (2010), "The Complexity of Relating Quantum Channels to Master Equations," https://pdfs.semanticscholar.org/c0db/f877764741361f2dcde2f440d5b29c5cea3e.pdf
[7] Vacchini (2016), "Generalized master equations leading to completely positive dynamics," https://arxiv.org/abs/1611.01150
