Sudarshan's dynamical maps

Question

This is a question about an equation in a paper by E.C.G. Sudarshan, P.M.Matthews and J. Rau. The authors introduce the concept of dynamical maps - objects that determine the time evolution of density matrices. When the system under study is hamiltonian the $B$-map is unitary and it satisfies the equation
begin{equation}tag{1}
sum_{r=1}^n B_{r,r^prime;r, s^prime} =  delta_{r^prime, s^prime}.
end{equation}
The paper uses summation convention but I choose to use the summation explicitly.
In section 3 they mention that using equation (1) one can prove that
begin{equation}tag{2}
B^2 - nB = 0.
end{equation}
I am unable to see how equation (2) follows from equation (1) even after using the fact that $B$ is hermitian. Can someone please help me obtain (2)?
A brief introduction to the $B$ matrix
Sudarshan's paper considers the transformation between density matrices at times $t_0$ and $t_1 > t_0$. One can write
begin{equation}tag{3}
rho_{r,s}(t_1) = A_{r,s;r^prime, s^prime}rho_{r^prime,s^prime}(t_0).
end{equation}
Summation convention is used here and the other equations following this one. The 'object' $A_{r,s;r^prime, s^prime}$ can be considered as an $n^2 times n^2$ matrix labelled by 'double indices' $(rs)$ and $(r^prime, s^prime)$. $n$ is the (finite) dimension of the Hilbert space on which the density matrix is defined. The $A$-matrices do not have properties convenient for further analysis. Therefore, the authors introduce the $B$-matrices, defining them as
begin{equation}tag{4}
B_{r,r^prime;s, s^prime} = A_{r,s;r^prime, s^prime}.
end{equation}
From the hermiticity of $rho$ one can conclude that
begin{equation}tag{5}
B_{r,r^prime;s, s^prime} = B_{s, s^prime; r,r^prime}^ast.
end{equation}
The positive definiteness of $rho$ gives,
begin{equation}tag{6}
z_{rr^prime}^ast B_{r,r^prime;s, s^prime}z_{ss^prime} ge 0.
end{equation}
Since the density matrices have a unit trace, we require
begin{equation}tag{7}
B_{r,r^prime;r, s^prime} =  delta_{r^prime, s^prime}.
end{equation}
This is same as equation (1) except that in the case of (1) I have used the summation sign.
An update 04-Jan-2021
The general form of the $B$-map that satisfies hermiticity condition of equation (5) and the trace condition of equation (7) is
begin{equation}tag{8}
B = begin{pmatrix}
B_{11;11} & B_{11;12} & B_{11;21} & B_{11;22} 
B_{11;12}^ast & B_{12;12} & B_{12;21} & B_{12;22} 
B_{11;21}^ast & B_{12;21}^ast & 1 - B_{11;11} & -B_{11;12} 
B_{11;22}^ast & B_{12;22}^ast & B_{11;12}^ast & 1 - B_{12;12}
end{pmatrix}
end{equation}
Unless we impose some more conditions we are unlikely to get the relation $B^2 = 2B$.

Amey Joshi · Accepted Answer

I am unable to prove $B^2 = nB$ using equations (5) and (7) in the question. However, the authors mention another way to get this result. When the density matrix evolves unitarily, the $B$-matrix can be written as
begin{equation}tag{9}
B_{rr^prime,ss^prime} = U_{rr}U^ast_{ss^prime}
end{equation}
so that
begin{equation}tag{10}
(B^2)_{rr^prime, ss^prime} = sum_{uv}B_{rr^prime, uv}B_{uv, ss^prime}
= sum_{uv}U_{rr}U^ast_{uv}U_{uv}U^ast_{ss^prime}.
end{equation}
We can simplify it as
begin{equation}tag{11}
(B^2)_{rr^prime, ss^prime} = U_{rr}U^ast_{ss^prime}sum_{uv}U^ast_{uv}
U_{uv} = U_{rr}U^ast_{ss^prime}sum_{uv}U_{uv}(U^dagger)_{vu}.
end{equation}
We can write the sum as
begin{equation}tag{12}
sum_{uv}U_{uv}(U^dagger)_{vu} = sum_uleft(sum_v U_{uv}(U^dagger)_{vu}
right) = sum_u delta_{uu} = n,
end{equation}
where we have used the fact that $U$ is unitary. From equation (9), (11) and (12) we get
begin{equation}tag{13}
(B^2)_{rr^prime, ss^prime} = nU_{rr}U^ast_{ss^prime} = nB_{rr,ss^prime},
end{equation}
where we used equation (9) to get the last term. We can write this
equation as
begin{equation}tag{14}
B^2 = nB.
end{equation}
I think equation (14) is valid only in the case of a unitary evolution and it cannot be proved using the trace condition alone.

Sudarshan's dynamical maps

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