• Markovianity"
• Quantum
• Positivity
• Choi

Quantum non Markovianity is an open eld of quantum information that
displays an increasing interest in last years. In particular, my work faced
up one of the most general open problem regarding open quantum systems,
which can be summarized by the following statement:
Given a general quantum master equation, then non-Markovian, the necessary
and su cient conditions such that the resulting dynamics is completely
positive are unknown. In particular, I have approached to this problem for the
particular dynamics described by a Generalized Amplitude Damping Channel.
But what does we means when we talk about quantum master equation? and
in particular quantum non-Markovianity? what is the complete positivity?
So, in order to present the particular problem I have dealt with, a panoramic
about open quantum systems from Quantum Information point of view is
necessary, which makes the concepts introduced more clear.
The problem
To work with open quantum system a statistical approach to quantum mechanics
is required, then a system's description in terms of density matrices.
In my own case, I can limit onto two level open systems, the qubits.
One of the rst disadvantage encountered working with open quantum system
is the lost of the unitarity of the the time evolution, so of its description
in term of a Von Newman equation, used for closed systems.
So, instead of look at our system open system, let's consider it as a subsystem
of a bigger closed composite system SE: our system's evolution can be
obtained tracing away the degree of freedom of the ancillary system from
the known unitary evolution of the whole system. But this approach is not
enough to obtain a quantum master equation yet: it does not assures that
the evolved state remains a proper state of our system for each instant of
1
time. But which properties must be veri ed? The already mentioned complete
positivity is one of these.
To more easily nd an answer, a complementary approach to open quantum
system dynamics in terms of quantum dynamical map must be introduced.
A quantum dynamical map is a superoperator acting on the convex space
of density matrices, giving in output the evolved state t ( 0 ) = (t), so
giving a discrete in time approach to open quantum system's dynamics. In
this formalism it is easier to formalize the necessary requirement onto the
evolved state to remains a physical, that correspond to require to the dynamical
maps to be linear, trace preserving, positive and complete positive.
The last property assures the positivity preserving of the map's action for
each possible extension of the subsystem to a bigger composite system SA,
formally: ( S
I
A )( SA ) 0, for all ancillary system A, where SA is the
state of the joint system and
I
A is the identical superoperator.
Coming back to quantum master equations, performing the Markovian
assumption they assume the so called Lindblad form and the just given conditions
directly follow. What about quantum non Markovianity?
The best way to de ne a non-Markovian process is "every process that is not
Markovian", or more formally "each process whose master equation can't assume
a Lindblad form". But the lost of the Lindbladian form can cause the
lost of the requirements done onto the evolved state to preserve their physical
interpretation, in particular the general conditions to restore the complete
positivity are unknown.
A possible approach
A rst approach to this problem consists in the research of such conditions
on particular dynamics. In particular I have looked for such necessary and
su cient conditions for a perturbed generalized amplitude damping channel
for one qubit, which describes phenomena of lost of energy for a two levels
quantum system due to the coupling with a thermal bath at xed temperature.
The main problems that obstruct this already simpli ed approach are the not
existence of an analytic solution of the master equation and the verify of the
complete positivity, since it requires to check the positivity of the extended
maps onto all the possible external systems. I have bypassed rst studying
the perturbed process, also if this approach does not display a clear physical
interpretation. Instead, I have completely dealt with the second one using
an alternative formulation of complete positivity of M. Choi, which transfers
the requirement onto the positivity of a 4 4 matrices, the so called Choi
matrix, which depends on the parameters of the master equation through
the solution of the Bloch di erential equations.
The results found justify the choice of a perturbative approach to the particular
problem I have approached and con rm the statement that the lost
of Lindbladian form of the master equation makes it not necessarily able to
describe a physical process.