A New Mechanism of Open System Evolution and Its Entropy Using Unitary Transformations in Noncomposite Qudit Systems
Julio A. López-Saldívar Octavio Castaños Margarita A. Man'ko Vladimir I. Man'ko
quant-ph



Excited-state effects in nucleon structure on the lattice using hybrid interpolators
    Sergey N. Syritsyn
    December 30, 2023
====================================================================================




The evolution of an open system is usually associated with the interaction of
the system with an environment. A new method to study the open-type
system evolution of a qubit  state is established. This evolution is determined by a unitary transformation applied to 
the qutrit (three-level atom) state, which defines the qubit subsystems. This procedure can be used to obtain different qubit quantum channels
employing unitary transformations into the qutrit system. In particular, we
study the phase damping and spontaneous-emission quantum channels.
In addition, we mention a proposal for quasiunitary transforms of
qubits, in view of the unitary transform of the total qutrit system. The
experimental realization is also addressed. The probability
representation of the evolution and its information-entropic
characteristics are considered.




§ INTRODUCTION


The open system evolution of a qudit state is known to be the result
of interactions with an environment. Usually, the states of the
complete system are thought to evolve by a unitary transformation in the
Hilbert space Ĥ=Ĥ_q ⊗Ĥ_env, then the
density operator of the composite system leads us, using the partial
tracing procedure, to the density operator of the subsystem
ρ̂_q (qudit), and its evolution is induced by the unitary
evolution of the complete system. In this picture, the qubit state
dynamics needs the structure of the Hilbert space Ĥ
corresponding to the presence of two subsystems, qudit and
environment <cit.>. In this work, we suggest a new mechanism to study the
open system evolution, which does not demand the complete system to
have a subsystem.

We show that for any system without subsystems, there exist a unitary
evolution, which due to hidden correlations in the system,
evolves according to the Gorini–Kossakowski–Sudarshan–Lindblad
equation <cit.>. We demonstrate this picture using the example of a qutrit
(complete system without subsystems), where the open-like evolution
is available for their associated qubits.

In previous
works <cit.>, a new
method to define different qubit density matrices from a qudit
system was established. This procedure uses the occupation
probabilities and transition probability amplitudes for different
levels of a qudit system and groups them as if there exists two levels only. This is done by mapping the qudit density matrix to the closest higher
even-dimensional density matrix. The partial trace operation then is enacted on the
resulting matrix in order to obtain well-defined qubit density
matrices.

The obtained qubits have been used to define a new geometric
representation of the d-dimensional qudit states through d Bloch
vectors <cit.> associated with the generated qubits. Furthermore, it has been 
possible to describe quantum phenomena as the entanglement on a
two-qubit system in terms of standard
probabilities <cit.>.

The evolution of a qutrit density matrix can provide the quantum
channel, which maps the initial state ρ̂_a onto the
density matrix ρ̂_a^'. The proposed open-type evolution establishes a new mechanism, which will need a special state preparation and a specific unitary operation for the
qutrit system, as we will show later on. The experimental possibilities by which one can
realize this new mechanism are related to superconducting
circuit devices <cit.>.

Most quantum computing processes consider a set of pure qubit states,
which are transformed by unitary operators, also called gates, that
are used to implement different computing algorithms. In this
article, instead, we have density matrices (which might be describing a mixed state) of
larger qudit systems. The definition of a set of qubit states from a
qudit system is similar to the ideas established in <cit.>, where the
emulation of a spin system was obtained from qudit states, and in <cit.>, where the quantum logic of qubits was simplified by
the use of a higher dimensional Hilbert space; and in general, with all the
procedures that make use of larger Hilbert spaces. In this work, we
demonstrate that subsystems of qubits defined by larger systems can be used in
quantum information. A principal foundation of quantum computation
is the study of quantum channels. These channels are linked to
unitary transformations of the qubit density matrix. There exist several
channels that can describe the interaction between a quantum
system and its environment such as the bit-flip, depolarization,
spontaneous emission, phase, and amplitude damping channels. For this, the study of quantum channels has been of relevance in the
error correction theory of quantum computation <cit.>.

Here, we present different examples of quantum channels, which act
on the associated qubits to qudit states. These quantum channels
have the advantage of being represented as unitary transformations acting in
the qudit system, providing the possibility to study the qubits as
if they were interacting with an environment.

On the other hand, the study of the interaction of three-level
systems with electromagnetic fields has led to the discovery of
important phenomena, such as the presence of dark states <cit.>
together with black resonances <cit.> and electromagnetically-induced
transparency <cit.>. This is important to
our objectives as in some cases, the herein proposed qubit quantum channels can be obtained by a unitary transformation of dark states,
suggesting the possibility of checking our results experimentally.

The work is organized as follows: In Section <ref>, a review of the qubit density matrices that are associated with a qutrit state is given. Furthermore, the association of a unitary transform of the qutrit to the nonunitary transformations of the qubits is studied. In Section <ref>, the definitions of the qubit phase damping and spontaneous-emission quantum channels are reviewed. Later, the unitary transformations of a qutrit system are explicitly given, which yields the phase damping and spontaneous-emission channels on the associated qubits. A way to obtain a quasi-unitary transformation on the qubits is also explored. The change of entropy associated with the nonunitary evolution of the qubits is discussed in Section <ref>. Finally, some concluding remarks are given. 



§ NONUNITARY EVOLUTION FOR THE QUBIT DECOMPOSITION OF QUTRIT STATES

In a previous work <cit.>, we showed the existence of six
different qubit states associated with a general qutrit density matrix:

    ρ̂=( [ ρ_11 ρ_12 ρ_13; ρ_21 ρ_22 ρ_23; ρ_31 ρ_32 ρ_33 ])   .


To define these states, different maps of ρ̂ to a
4 × 4 density matrix, with one row and one column equal to zero
(in such a way that ensures an eigenvalue equal to zero), were used.
Then, the partial trace of the resulting 4 × 4 matrix was
performed as if it was describing a two-qubit system. The obtained qubit
partial density operators can be explicitly written as:

    ρ̂_1=( [ 1-ρ_33   ρ_13;   ρ_31   ρ_33 ]) ,   ρ̂_2=( [ 1-ρ_22   ρ_12;   ρ_21   ρ_22 ]) ,   ρ̂_3=( [   ρ_11   ρ_13;   ρ_31 1-ρ_11 ]) , 
    ρ̂_4=( [   ρ_22   ρ_23;   ρ_32 1-ρ_22 ]) ,   ρ̂_5=( [   ρ_11   ρ_12;   ρ_21 1-ρ_11 ]) ,   ρ̂_6=( [ 1-ρ_33   ρ_23;   ρ_32   ρ_33 ]) .


The qubit states can be characterized in different sets by their corresponding von Neumann entropy
S_k=- ρ_klnρ_k, with k=1,2,…,6. These qubits
correspond to the reduction of the three-level system to different
two-level systems by the summation of the population probabilities of
two levels into one.

When the qutrit state is transformed using a general three-dimensional unitary matrix
Û, i.e., , the qubits in Equation (<ref>) are transformed in
a nonunitary way. The transformed qubit density matrices can be written by
the following expressions:

    ρ̂_1^'   =   1/D([ D-M_3,1 N_1,3+M_2,1 N_2,3-M_1,1 N_3,3   M_3,3 N_1,3-M_2,3 N_2,3+M_1,3 N_3,3;   M_3,1 N_1,1-M_2,1 N_2,1+M_1,1 N_3,1   M_3,1 N_1,3-M_2,1 N_2,3+M_1,1 N_3,3 ]), 
    ρ̂_2^'   =   1/D( [ D+ M_3,2 N_1,2-M_2,2 N_2,2+M_1,2 N_3,2    M_3,3 N_1,2-M_2,3 N_2,2+M_1,3 N_3,2;   -M_3,2 N_1,1+M_2,2 N_2,1-M_1,2 N_3,1   -M_3,2 N_1,2+M_2,2 N_2,2-M_1,2 N_3,2 ]) , 
    ρ̂_3^'   =   1/D( [   M_3,3 N_1,1-M_2,3 N_2,1+M_1,3 N_3,1   M_3,3 N_1,3-M_2,3 N_2,3+M_1,3 N_3,3;   M_3,1 N_1,1-M_2,1 N_2,1+M_1,1 N_3,1 D-M_3,3 N_1,1+M_2,3 N_2,1-M_1,3 N_3,1 ]) , 
    ρ̂_4^'   =   1/D([  -M_3,2 N_1,2+M_2,2 N_2,2-M_1,2 N_3,2  -M_3,2 N_1,3+M_2,2 N_2,3-M_1,2 N_3,3;   M_3,1 N_1,2-M_2,1 N_2,2+M_1,1 N_3,2 D+M_3,2 N_1,2-M_2,2 N_2,2+M_1,2 N_3,2 ]), 
    ρ̂_5^'   =   1/D([   M_3,3 N_1,1-M_2,3 N_2,1+M_1,3 N_3,1   M_3,3 N_1,2-M_2,3 N_2,2+M_1,3 N_3,2;  -M_3,2 N_1,1+M_2,2 N_2,1-M_1,2 N_3,1 D-M_3,3 N_1,1+M_2,3 N_2,1-M_1,3 N_3,1 ]) , 
    ρ̂_6^'   =   1/D([ D-M_3,1 N_1,3+M_2,1 N_2,3-M_1,1 N_3,3  -M_3,2 N_1,3+M_2,2 N_2,3-M_1,2 N_3,3;   M_3,1 N_1,2-M_2,1 N_2,2+M_1,1 N_3,2   M_3,1 N_1,3-M_2,1 N_2,3+M_1,1 N_3,3 ]) ,

where N_jk=(ρ̂Û)_jk, D is the determinant of Û, and M_jk are the
components of the minors of matrix Û, i.e., its elements
are the determinants after eliminating the (4-j)^th row and
(4-k)^th column of Û. The transformed states are
characterized into different sets by their corresponding transformed entropies
S'_k=- ρ'_klnρ'_k. We emphasize that the
resulting qubit density matrices are associated, in general, with a nonunitary
evolution of the original qubits. This fact establishes a new
mechanism to obtain the open-like system evolution in a noncomposite
qutrit system. Additionally, this procedure can be extended to any qudit system,
in view of the general definition of the qubit density matrices obtained from a
qudit system <cit.>.

In <cit.>, we discussed that a two-qubit density matrix
with one of its rows and columns equal to zero describes separable
states, if one of the off-diagonal terms is equal to zero, for example, the state:

    ρ̂=(
    [ ρ_11 ρ_12 ρ_13    0; ρ_21 ρ_22 ρ_23    0; ρ_31 ρ_32 ρ_33    0;    0    0    0    0 ])

is separable iff ρ_23=0. To show this, one can consider the previous density matrix to be in the standard two-qubit representation | 00 ⟩, | 01 ⟩, | 10 ⟩, and | 11 ⟩. It can be seen that the partial transpose operation <cit.> implies the change ρ_12↔ρ_21, and for this reason, the eigenvalues of ρ̂ with ρ_23=0 are equal to the eigenvalues of its partial transpose. As the partial transpose is a nonnegative operator, then the system is separable. The separability
implies the invariance of the partial density matrices under local
unitary transformations. As this two-qubit density matrix has a pair of
row-column with a diagonal term equal to zero, the correspondence with a qutrit density
matrix can be made. On the other hand, the correspondence between
two-qubit local unitary transformations and qutrit unitary transformations can
be made in the same way, e.g., by eliminating one row and one column
of the two-qubit local transformation. This procedure allows us to define
different unitary transformations that almost leave the qubits
in Expression (<ref>) invariant.



§ PHASE DAMPING AND SPONTANEOUS-EMISSION CHANNELS.

It is known that the interaction of a qubit system with an
environment leads to several physical phenomena such as dissipation
and decoherence in the qubit subsystem; an example of these
interactions is the phase damping channel. In this channel, the evolution of the qubit plus environment (|⋯⟩_q
|⋯⟩_e) is given by a unitary transformation T̂, which acts
differently if the qubit is in the ground or excited state,
according to the following rules: T̂(| 0 ⟩_q | 0
⟩_e)=√(1-p)| 0 ⟩_q | 0
⟩_e+√(p)| 0 ⟩_q | 1 ⟩_e and
T̂(| 1 ⟩_q | 0 ⟩_e)=√(1-p)| 0
⟩_q | 0 ⟩_e+√(p)| 0 ⟩_q | 2
⟩_e with p being a probability, i.e., the environment
subsystem goes to a superposition of the states (| 0
⟩_e, | 1 ⟩_e), or to (| 0 ⟩_e, |
2⟩_e), if the environment is in | 0 ⟩_e, or | 1 ⟩_e, respectively <cit.>. This two-qubit unitary transformations
result in a nonunitary change when the partial trace over the
environment subsystem is taken:

    ( [ 1-ρ_22   ρ_12; ρ_12^*   ρ_22 ]) →( [      1-ρ_22   ρ_12(1-p); ρ_12^*(1-p)        ρ_22 ])   .


When the map is applied a very large number of times (→∞), it is straightforward that the initial state tends to the completely decoherent
state:

    ( [ 1-ρ_22   ρ_12; ρ_12^*   ρ_22 ]) →( [ 1-ρ_22      0;      0   ρ_22 ])   ,

with an exponential convergence.

The other example is the spontaneous-emission (also called the
amplitude-damping) quantum channel. In this channel, the dynamics of
the qubit system plus the environment is determined by a unitary
transform T̂, which only acts if the qubit system is in the
excited state | 1 ⟩_q, according to the following rules:
T̂(| 0⟩_q | 0 ⟩_e)=| 0⟩_q
| 0 ⟩_e and T̂(| 1⟩_q | 0
⟩_e)=√(1-p)| 1⟩_q | 0
⟩_e+√(p)| 0⟩_q | 1 ⟩_e, where p is
the probability <cit.>. This channel then defines a nonunitary evolution
over the qubit subsystem, which transforms the qubit density matrix
as follows:

    ( [ 1-ρ_22   ρ_12; ρ_12^*   ρ_22 ]) →( [  1-(1-p)ρ_22   ρ_12√(1-p); ρ_12^*√(1-p)    (1-p)ρ_22 ])   .


If this channel is applied a very large number of times (→∞), the density matrix converges to a ground state, i.e.,

    ( [ 1-ρ_22   ρ_12; ρ_12^*   ρ_22 ]) →( [ 1 0; 0 0 ])   .


In addition to these examples, there exists another type of quantum
channel defined in the theory of interaction between a
quantum system and an environment, which can be considered <cit.>.

It is possible to demonstrate that phase damping and spontaneous-emission quantum channels for qubits
ρ̂_1, …, ρ̂_6 in Equation (<ref>) can be
obtained by the use of particular unitary transformations of a qutrit
state ρ̂. To justify this, we assumed a two-qubit quantum
system where one of the levels cannot be populated, i.e., the
4 × 4 density matrix has an eigenvalue equal to zero, e.g.,


    ρ̂=(
    [ ρ_11 ρ_12 ρ_13    0; ρ_21 ρ_22    0    0; ρ_31    0 ρ_33    0;    0    0    0    0 ]) ;

it is clear that this density matrix is separable since
ρ_23=ρ_32^*=0. The partial density matrices can be
operated locally by unitary transformations of the form û_1
⊗û_2. When only one of the qubits is operated, i.e.,
when the unitary matrix corresponds to a controlled
operation <cit.>: û_1=Î or û_2=Î.
If û_2=Î, then the unitary transformation only operates
over the second qubit,

    û=([ u_11 u_12    0    0; u_21 u_22    0    0;    0    0 u_11 u_12;    0    0 u_21 u_22 ]) .


By means of this type of unitary matrix, one can define an operation
in the qutrit system that approximately only affects ρ̂_2.
This is done by ignoring the fourth row and the fourth column of
(<ref>); the resulting qutrit state is then operated by the
unitary matrix resulting from the elimination of the fourth row and
the fourth column of Equation (<ref>). For the operator to be still
unitary, the (3,3) entry must be replaced by one. Following these
and other analogous arguments, we study the application of the
unitary transforms:

    Û_1=( [ u_11 u_12    0; u_21 u_22    0;    0    0    1 ]) ,   Û_2=( [ u_11    0 u_12;    0    1    0; u_21    0 u_22 ]) ,   Û_3=( [    1    0    0;    0 u_11 u_12;    0 u_21 u_22 ])

on the qutrit density matrices:

    σ̂_1=([ ρ_11 ρ_12 ρ_13; ρ_21 ρ_22    0; ρ_31    0 ρ_33 ]) ,   σ̂_2=([ ρ_11    0 ρ_13;    0 ρ_22 ρ_23; ρ_31 ρ_32 ρ_33 ]) ,   σ̂_3=([ ρ_11 ρ_12    0; ρ_21 ρ_22 ρ_23;    0 ρ_23 ρ_33 ])   .


 The unitary transformations in Equation (<ref>) can be enacted on any of the density matrices in Equation (<ref>), which define a nonunitary transformation of the qubits defined in Equation (<ref>). These qubit transformations are found by the substitution of Equations (<ref>) and (<ref>) into Equation (<ref>), e.g., the unitary transformation Û_1^†σ̂_1 Û_1 results in the following transformations of the qubits:

    [                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 ρ̂_1^'=([       1-ρ_33 ρ_13  u_11^*;   ρ_31  u_11         ρ_33 ]) ,;                                                                                                                                                                                                                                                                                                                                     ρ̂_2^'=([ 1 - u_12^* (σ̂_1 Û_1)_12 - u_22^* (σ̂_1 Û_1)_22     u_11^* (σ̂_1 Û_1)_12 + u_21^* (σ̂_1 Û_1)_22;     u_12^* (σ̂_1 Û_1)_11 + u_22^* (σ̂_1 Û_1)_21     u_12^* (σ̂_1 Û_1)_12 + u_22^* (σ̂_1 Û_1)_22 ]) ,;                                                                                                                                                                                                                                                                                                                                   ρ̂_3^' = ([     u_11^* (σ̂_1 Û_1)_11 + u_21^* (σ̂_1 Û_1)_21                                    ρ_13  u_11^*;                                      ρ_31  u_11 1 - u_11^* (σ̂_1 Û_1)_11 - u_21^* (σ̂_1 Û_1)_21 ]) ,;                                                                                                                                                                                                                                                                                                                                        ρ̂_4^' = ( [   u_12^* (σ̂_1 Û_1)_12 + u_22^* (σ̂_1 Û_1)_22                                   ρ_13 u_12^*;                                     ρ_31 u_12 1-u_12^* (σ̂_1 Û_1)_12 - u_22^* (σ̂_1 Û_1)_22 ])   ,;                                                                                                                                                                                                                                                                           ρ̂_5^' =( [                     u_11^* (σ̂_1Û_1)_11 + u_21^* (σ̂_1Û_1)_21                     u_11^* (σ̂_1Û_1)_12 + u_21^* (σ̂_1Û_1)_22; u_12^* (σ̂_1Û_1)_11 (r11 u11 + r12 u21) + u_22^* (σ̂_1Û_1)_21                   1-u_11^* (σ̂_1Û_1)_11 - u_21^* (σ̂_1Û_1)_21 ]) ,;                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 ρ̂_6^' =([       1-ρ_33 ρ_13  u_12^*;   ρ_31  u_12         ρ_33 ]). ]


From these results, one can notice that the transformed qubits ρ̂_1^' and ρ̂_6^' correspond to the phase damping channel of ρ̂_1 with different damping parameters. Furthermore, the qubit states ρ̂_2^', ρ̂_5^' can be seen as quasi-unitary transformations of the initial states ρ̂_2, ρ̂_5, respectively. In a similar way, one can obtain all the possible unitary transformations of the density matrices in Equation (<ref>). These transformations lead to the identification of two types of quantum channels: the phase damping and a quasi-unitary operation described below.


The unitary transformation over the density matrices σ̂_1,
σ̂_2, and σ̂_3 results in a change over their associated
qubits ρ̂_1, …, ρ̂_6, to ρ̂_1^', …, ρ̂_6^', which denote the
qubits after the transformation. We have found the following interesting expressions:

    Û_1^†σ̂_̂1̂Û_1 ⇒ρ̂_1^' =( [      1-ρ_33 u_11^* ρ_13;    u_11ρ_31        ρ_33 ]),   ρ̂_6^' =( [      1-ρ_33 u_12^* ρ_13;    u_12ρ_31        ρ_33 ]); 
    Û_2^†σ̂_̂1̂Û_2 ⇒ρ̂_2^' =( [      1-ρ_22 u_11^* ρ_12;    u_11ρ_21        ρ_22 ]),   ρ̂_4^' =( [      1-ρ_33 u_12^* ρ_12;    u_12ρ_21        ρ_33 ]); 
    Û_2^†σ̂_̂2̂Û_2 ⇒ρ̂_2^' =( [      1-ρ_22 u_21^* ρ_32;    u_21ρ_23        ρ_22 ]),   ρ̂_4^' =( [        ρ_22    u_22ρ_23; u_22^* ρ_32      1-ρ_22 ]); 
    Û_3^†σ̂_̂2̂Û_3 ⇒ρ̂_3^' =( [        ρ_11    u_22ρ_13; u_22^* ρ_31      1-ρ_11 ]),   ρ̂_5^' =( [        ρ_11    u_21ρ_13; u_21^* ρ_31      1-ρ_11 ]); 
    Û_1^†σ̂_̂3̂Û_1 ⇒ρ̂_1^' =( [      1-ρ_33 u_21^* ρ_23;    u_21ρ_32        ρ_33 ]),   ρ̂_6^' =( [      1-ρ_33 u_22^* ρ_23;    u_22ρ_32        ρ_33 ]); 
    Û_3^†σ̂_̂3̂Û_3 ⇒ρ̂_3^' =( [        ρ_11    u_12ρ_12; u_12^* ρ_21      1-ρ_11 ]),   ρ̂_5^' =( [        ρ_11    u_11ρ_12; u_11^* ρ_21      1-ρ_11 ]).


In most of the cases, the resulting qubits ρ̂^'_j
correspond to the phase damping quantum channel of ρ̂_j, as can be seen in Expression (<ref>).
In this channel, the probability amplitudes given by the original
off-diagonal terms of the qubits are multiplied by a number. The damping parameters are associated with different entries of the unitary transformation u_jk, which in general are complex numbers. When the unitary transformation correspond to a real matrix, then the expression for the standard phase damping map is obtained. As you can see in Equation (<ref>), in some 
cases, the unitary transformations leads to the quantum channel of another
qubit, e.g., after the application of Û_1 to
σ̂_1, the qubit ρ̂_6^' is the phase
damping channel of ρ̂_1. Furthermore, in some other cases, the obtained density matrices correspond to transformations similar to the
phase damping channel of matrices outside the ones in
Equation (<ref>), e.g., ρ̂_4^' after the application of Û_2 to σ̂_1.
Although these matrices seem unrelated, they have the same form as
the phase damping channel. In the case of Û being
a rotation matrix with a time-dependent angle θ=ω t, the original qubit states can be recovered at the time t=2 π l
/ω, l=0,1,2,…. 

The unitary transformations (Û_1, Û_2, Û_3)
previously described can also lead to quasi-unitary transformations of
the qubits. In particular, for the unitary transformation Û_1^†σ̂_1 Û_1, one gets the quasi-unitary transformations: 


    ρ̂_2^'   =   𝒰̂^†ρ̂_2  𝒰̂+ρ_33( [    | u_12|^2 -u_11^* u_12; -u_11 u_12^*   -| u_12|^2 ])  , 
    ρ̂_5^'   =   𝒰̂^†ρ̂_5  𝒰̂+ρ_33([   -| u_21|^2 -u_21^* u_22; -u_21 u_22^*    | u_21|^2 ])   ,

 with 𝒰̂=( [ u_11 u_12; u_21 u_22 ]) being a two-dimensional unitary transformation. For the other qubits, one can also define quasi-unitary transformations as follows:


  (a) From the qutrit unitary transformation Û_1^†σ̂_3 Û_1,

    ρ̂_2^'   =   𝒰̂^†ρ̂_2  𝒰̂+ρ_33([  -| u_12|^2 u_11^* u_12; u_11 u_12^*   | u_12|^2 ])   , 
    ρ̂_5^'   =   𝒰̂^†ρ̂_5  𝒰̂+ρ_33([   | u_21|^2 u_21^* u_22; u_21 u_22^*  -| u_21|^2 ])   ,

 
  (b) For the transformation Û_2^†σ̂_1 Û_2,

    ρ̂_1^'   =   𝒰̂^†ρ̂_1  𝒰̂+ρ_22( [    | u_12|^2 -u_11^* u_12; -u_11 u_12^*   -| u_12|^2 ])  , 
    ρ̂_3^'   =   𝒰̂^†ρ̂_3  𝒰̂+ρ_22( [   -| u_21|^2 -u_21^* u_22; -u_21 u_22^*    | u_21|^2 ])  .


  (c) For the transformation Û_2^†σ̂_2 Û_2,

    ρ̂_1^'   =   𝒰̂^†ρ̂_1  𝒰̂+ρ_22( [    | u_12|^2 -u_11^* u_12; -u_11 u_12^*   -| u_12|^2 ])  , 
    ρ̂_3^'   =   𝒰̂^†ρ̂_3  𝒰̂+ρ_22( [   -| u_21|^2 -u_21^* u_22; -u_21 u_22^*    | u_21|^2 ])  ,


  (d) From Û_3^†σ̂_2 Û_3,

    ρ̂_4^'   =   𝒰̂^†ρ̂_4 𝒰̂+ρ_11( [   -| u_21|^2 -u_12^* u_22; -u_21 u_22^*    | u_21|^2 ])   , 
    ρ̂_6^'   =   𝒰̂^†ρ̂_6  𝒰̂+ρ_11( [  -| u_12|^2 u_11^* u_12; u_11 u_12^*   | u_12|^2 ])   .


  (e) Finally, for Û_3^†σ̂_3 Û_3,

    ρ̂_4^'   =   𝒰̂^†ρ̂_4 𝒰̂+ρ_11( [   -| u_21|^2 -u_12^* u_22; -u_21 u_22^*    | u_21|^2 ])   , 
    ρ̂_6^'   =   𝒰̂^†ρ̂_6  𝒰̂+ρ_11( [  -| u_12|^2 u_11^* u_12; u_11 u_12^*   | u_12|^2 ])   ,



For all the cases, 𝒰̂ is a two-dimensional unitary transformation.

As in the phase-damping case, one can think of a rotation matrix
with a time-dependent angle θ= ω t as the unitary
operation, i.e.,

    𝒰̂=([  cos (ω t) -sin (ω t);  sin (ω t)  cos (ω t) ])   ,

 which, in the case where t≈ 0, results in the following
 transformations:

    ρ̂_j^'= 𝒰̂^†ρ̂_j  𝒰̂- ρ_kk ω  t  σ̂_x+𝒪(t^2)   ,

where σ̂_x is the Pauli matrix and ρ_kk is a
diagonal component of ρ̂, which depends on j. Its value
is k=2 for j=1,3, k=3 for j=2,5, and k=1 for j=4,6. It
is necessary to point out that, for ρ̂_5^' associated
with Û_1^†σ̂_3Û_1, we need to replace
ρ_33 with -ρ_33 in Equation (<ref>).

In the case where the density matrices correspond to states, where
one of the accessible levels is not occupied, i.e.,

    σ̂_4=( [ ρ_11 ρ_12    0; ρ_21 ρ_22    0;    0    0    0 ]),   σ̂_5=( [ ρ_11    0 ρ_13;    0    0    0; ρ_31    0 ρ_33 ]),   σ̂_6=([    0    0    0;    0 ρ_22 ρ_23;    0 ρ_32 ρ_33 ]),

we obtain the expressions:

    Û^†_2 σ̂_4 Û_2 ⇒ρ̂^'_5=( [   ρ_11| u_11|^2     ρ_12 u^*_11;       ρ_21 u_11 1-ρ_11| u_11|^2 ]),   ρ̂^'_6 = ( [ 1-ρ_11| u_12|^2       ρ_21 u_12;     ρ_12 u^*_12   ρ_11| u_12|^2 ]), 
    Û_3^†σ̂_4 Û_3 ⇒ρ̂^'_1 = ( [ 1-ρ_22| u_12|^2       ρ_12 u_12;     ρ_21 u^*_12   ρ_22| u_12|^2 ]),   ρ̂^'_2=([ 1-ρ_22| u_11|^2       ρ_12 u_11;     ρ_21 u^*_11   ρ_22| u_11|^2 ]), 
    Û^†_1 σ̂_5 Û_1 ⇒ρ̂^'_3= ( [   ρ_11| u_11|^2     ρ_13 u^*_11;       ρ_31 u_11 1-ρ_11| u_11|^2 ]),   ρ̂^'_4= ( [   ρ_11| u_12|^2     ρ_13 u^*_12;       ρ_31 u_12 1-ρ_11| u_12|^2 ]), 
    Û^†_3 σ̂_5 Û_3 ⇒ρ̂^'_1= ( [ 1-ρ_33| u_22|^2       ρ_13 u_22;     ρ_31 u^*_22   ρ_33| u_22|^2 ]),   ρ̂^'_2= ( [ 1-ρ_33| u_21|^2       ρ_13 u_21;     ρ_31 u^*_21   ρ_33| u_21|^2 ]), 
    Û^†_1 σ̂_6 Û_1 ⇒ρ̂^'_3= ( [   ρ_22| u_21|^2     ρ_23 u^*_21;       ρ_32 u_21 1-ρ_22| u_21|^2 ]),   ρ̂^'_4= ( [   ρ_22| u_22|^2     ρ_23 u^*_22;       ρ_32 u_22 1-ρ_22| u_22|^2 ]), 
    Û^†_2 σ̂_6 Û_2 ⇒ρ̂^'_5= ( [   ρ_33| u_21|^2     ρ_32 u^*_21;       ρ_23 u_21 1-ρ_33| u_21|^2 ]),   ρ̂^'_6= ( [ 1-ρ_33| u_22|^2       ρ_23 u_22;     ρ_32 u^*_22   ρ_33| u_22|^2 ])   .


These transformations in many of the cases can represent the
spontaneous-emission quantum channel. As in the other examples
studied above, when the unitary matrices are rotated by angle
θ=ω t, the original qubit systems can be recovered at
times t=2 π l/ω; l=0,1,2,…. It is important to
mention that the states represented by Equation (<ref>) correspond
to three-level systems, where one of the levels is a dark state, and then only two
of the levels can be populated, which have been experimentally
obtained <cit.>. These kinds of systems have been of
relevance as they can be created by two-photon processes in a
three-level system <cit.> or by the adiabatic variation of the
Rabi frequencies associated with the transitions between the three
states <cit.>. For example, to obtain the state
σ̂_4, one can think of an atomic Λ-type
three-level system (| 1 ⟩, | 2 ⟩, | 3
⟩), which interacts with an environment <cit.>; see
Figure <ref>. The Hamiltonian associated with this system can be written in the form:

    Ĥ=(
    [  ω_1    0 ω_13;    0  ω_2 ω_23; ω_13 ω_23    0 ])   ,

where ω_1,2 are the energies of the states | 1⟩, | 2
⟩, respectively. By considering the energy of the ground state | 3 ⟩ equal to zero, ω_13 and ω_23 are the transition energies. Taking
the zero energy in the ground state | 3 ⟩, we can make
the replacements ω_13→ω_1   e^-i
ω_1 t and ω_23→ω_2 e^-i ω_2 t.
The time evolution of the density matrix can be obtained by the
expression:

    d/dtρ̂= i [ ρ̂, Ĥ ]+ ρ̂^'  ,

where the matrix ρ̂^' is given by the interaction of
the original density matrix with the environment:

    ρ̂^'= (
    [  γ_31ρ_33  -γ^'ρ_12 -γ_1 ρ_13;  -γ^'ρ_21  γ_32ρ_33 -γ_2 ρ_23; -γ_1 ρ_31 -γ_2 ρ_32    -γρ_33 ]),

where the parameters γ_31, γ_32, and γ are
the spontaneous-emission rates, which must satisfy
, and the relaxation terms for the
coherence components are named γ_1 and γ_2, which also
satisfy γ^'=γ_1+γ_2. The resulting
differential equations (<ref>) can be reduced by considering that the
variation of the parameters ρ_13, ρ_23, and ρ_33 over time is smaller compared to the spontaneous
emission and decoherence terms γ_31 and γ_32; this
is called the adiabatic hypothesis. Under this hypothesis, it is
possible to obtain a state with ρ_13=ρ_23=ρ_33=0,
as the solution of the evolution of the density matrix
σ̂_4 discussed above. 

Another way to obtain these types
of systems is the case where the environmental interaction is neglected, i.e., ρ̂^'=0 in Equation (<ref>). The corresponding Schrödinger equation is , with |ψ⟩= a_1 (t) e^-i ω_1 t|
1 ⟩+ a_2 (t) e^-i ω_2 t| 2 ⟩ + a_3 (t)
| 3 ⟩, which in view of the initial conditions
,
a_2(0)=-ω_1/√(ω_1^2+ω_2^2), a_3(0)=0
leads to the solution:

    a_1(t)=ω_2/√(ω_1^2+ω_2^2)  ,   
    a_2(t)=-ω_1/√(ω_1^2+ω_2^2)  ;   
    a_3 (t)=0   ,

so the level | 3 ⟩ is never populated. 

The density
matrices σ̂_5 and σ̂_6 can be obtained by means of
analogous procedures applied to the V and Ξ
configurations of the three-level system depicted in
Figure <ref>.







It is also important to mention that the unitary transformations defined
by the matrices Û_1, Û_2, and Û_3 in
Equation (<ref>) can be generated experimentally by different
proposed mechanisms, such as sliding mode control <cit.>,
adiabatic passage <cit.>, and the robust control
scheme <cit.>. We want to emphasize that the resulting quasi-unitary evolutions and the different quantum channels obtained in our work can have applications in quantum computing and quantum information theories. We think so because the quasi-unitary operations discussed here could be used as approximations to the standard quantum gates, and furthermore, the obtained quantum channels could also be used in the quantum correction algorithms found in the literature.



§ PROBABILITY REPRESENTATION OF THE QUBIT-STATE EVOLUTION

In the quantum tomographic approach of qubit states <cit.>, the states are identified with tomographic probability distributions. In the case of the minimal number of probability parameters, the density
matrix of the qubit (spin-1/2) state reads <cit.>:

    ρ̂=( [                 p_3 p_1-1/2-i (p_2-1/2); p_1-1/2+i (p_2-1/2)               1-p_3 ])   ,   ∑_j=1^3 (p_j-1/4)^2 ≤1/4  ,

where 0≤ p_k,≤1 with k=1,2,3 are the probabilities to obtain the
value +1/2 in the x, y, z axis, respectively. Thus, any
qubit state can be identified through the probabilities p_1,
p_2, and p_3, i.e., given the density operator, one can get the set ρ̂↔ p_1, p_2, p_3 and vice versa. In the case of qubits (<ref>) associated with the qutrit
state, the evolution of the probabilities after the unitary
operation of the qutrit is determined by Equation (<ref>). For example, we have a probabilistic representation corresponding to
ρ̂_5^' in the first formula of Equation (<ref>), i.e.,

    p_3 → p_3 | u_11|^2 ,    p_1-1/2-i(p_2-1/2) → (p_1-1/2-i(p_2-1/2))u_11^*   .


The change of probabilities can be characterized by the evolution of
the Tsallis and Shannon entropies. For example, in (<ref>), the
unitary matrix parameter u_11 determines the evolution of the
Shannon entropy related to a coin probability distribution
(p_3,1-p_3) (assume that we have two nonideal classical
coins I and II in such a game as coin flipping, coin tossing, or heads (up, ⊕) or tails (down, ⊖), which is the
practice of throwing a coin in the air and checking which side is
showing when it lands, in order to choose between two alternatives
P_k or (1-P_k); k=1,2). 
This evolution is of the form:

    S(Û)=-p_3 | u_11|^2 ln( p_3 | u_11|^2 )-(1-p_3 | u_11|^2) ln( 1-p_3 | u_11|^2 )   .


This entropy, as a function of the unitary evolution applied to the
qutrit state, characterizes some aspects of the open dynamics of
qubits. We point out that, as for p_3, there exist other classical
entropic characteristics associated with the evolution of p_1 and
p_2 given by Equation (<ref>).



§ CONCLUDING REMARKS

A new mechanism to study the open system evolution of a
noncomposite qudit system was established. As an example of
the general procedure, we considered a qutrit system. Associated with the qutrit system, one can
define different qubit density matrices, which evolve in an
open-like way when a unitary transformation is enacted on the qutrit.

The application of the resulting transformations for the qubits within the qutrit was also discussed. The quasi-unitary transformations obtained here might be used as an approximation to quantum gates, whereas the quantum channels could be employed in quantum correction protocols.


Different types of quantum channels can be observed using the qubit
decomposition of a qutrit system. In particular, the phase damping
and the spontaneous-emission channels were obtained using a unitary
transformation acting on specific qutrit density matrices. The phase
damping channel was obtained when a unitary transformation of the density
matrix with one off-diagonal term equal to zero was performed. A spontaneous-emission channel can be observed by unitary transformations
acting over a dark state, i.e., a three-level state where one of the
levels cannot be populated.

In addition to these channels, quasi-unitary transformations of the
qubit states can be defined. This was also done by the application of
a unitary matrix to the generic qutrit state.

The entropy evolution of the tomographic-probability distributions
determined by the system of qubits was discussed.

We can extend our analysis to other qudit systems without subsystems
since, o an arbitrary spin-j density matrix and the spin unitary
evolution, one can associate the smaller spin j'<j evolution.

The possible experimental implementation of the procedure was also
addressed, given that there exist several proposed ways to generate
the unitary transformations such as by sliding mode
control <cit.>, adiabatic passage <cit.>, or the robust control scheme <cit.>.








§ ACKNOWLEDMENTS

This work was partially supported by DGAPA-UNAM (under Project IN101619). 



999

weiss
Weiss, U. Quantum Dissipative Systems; World Scientific: Singapore, 1993.

kossa
Kossakowski, A. On quantum statistical mechanics of non-Hamiltonian systems. Rep. Math. Phys. 1972, 3, 247–274.

ingarden
Ingarden, R.S.; Kossakowski, A. On the connection of nonequilibrium information thermodynamics with non-Hamiltonian quantum mechanics of open systems. Ann. Phys. 1975, 89, 451–485.

gorini 
Gorini, V.; Kossakowski, A.; Sudarshan, E.C.G. Completely positive dynamical semigroups of N-level systems. J. Math. Phys. 1976, 17, 821–825.

lindblad 
Lindblad, G. On the generators of quantum dynamical semigroups. Commun. Math. Phys. 1976, 48, 119–130.

chernegaA
Chernega, V.N.; Man'ko, O.V.; Man'ko, V.I. Triangle Geometry of the Qubit State in the Probability Representation Expressed in Terms of the Triada of Malevich’s Squares. J. Russ. Laser Res. 2017, 38, 141–149.

chernegaB
Chernega, V.N.; Man'ko, O.V.; Man'ko, V.I. Probability Representation of Quantum Observables and Quantum States. J. Russ. Laser Res. 2017, 38, 324–333.

chernegaC
Chernega, V.N.; Man'ko, O.V.; Man'ko, V.I. Triangle Geometry for Qutrit States in the Probability Representation. J. Russ. Laser Res. 2017, 38, 416–425.

entropy18
López-Saldívar, J.A.; Castaños, O.; Nahmad-Achar, E.; López-Peña , R.; Man'ko, V.I.; Man'ko, M.A. Geometry and Entanglement of Two-Qubit States in
the Quantum Probabilistic Representation. Entropy 2018, 20, 630.

QIP
López-Saldívar, J.A.; Castaños, O.; Man'ko, M.A.; Man'ko, V.I. Qubit representation of qudit states: Correlations and state reconstruction. Quantum Inf. Process. 2019, 18, 210.

wallraff
Devoret, M.H.; Wallraff, A.; Martinis, J.M. Superconducting Qubits: A Short Review. arXiv 2004, arXiv:cond-mat/0411174v1.

superc
Devoret, M.H.; Schoelkopf, R.J. Superconducting Circuits for Quantum Information: An Outlook. Science 2013, 339, 1169–1174.

neeley
Neeley, M.; Ansmann, M.; Bialczak, R.C.; Hofheinz, M.; Lucero, E.; O'Connell, A.D.; Sank, D.; Wang, H.; Wenner, J.; Cleland, A.N.; et al. Emulation of a quantum spin with a
superconducting phase qudit. Science 2009, 325, 722–725.

lanyon
Lanyon, B.P.; Barbieri, M.; Almeida, M.P.; Jennewein, T.; Ralph, T.C.; Resch, K.J.; Pryde, G.J.; O’Brien, J.L.; Gilchrist, A.; White, A.G. Simplifying quantum logic using
higher-dimensional Hilbert spaces. Nat. Phys. 2009, 5, 134–140.

chuang
Chuang, I.L.; Nielsen, M.A. Quantum Computation and Quantum Information; Cambridge University Press: Cambridge, UK, 2000.


terhal
Terhal, B.M. Quantum Error Correction for Quantum Memories. Rev. Mod. Phys. 2015, 87, 307, doi:10.1103/RevModPhys.87.307.


dark1976
Arimondo, E.; Orriols, G. Nonabsorbing atomic coherences by coherent two-photon transitions in a three-level optical pumping. Lett. Nuovo C. 1976, 17, 333–338.
reso1987
Dalibard, J.; Reynaud, S.; Cohen-Tannoudji, C. La cascade radiative de l'atome habillé. In Interaction of Radiation with Matter; A Volume in Honour of Adriano Gozzini; Scuola Normale Superiore: Pisa, Italy, 1987.

eit1990
Harris, S.E.; Field, J.E.; Imamoglu, A. Nonlinear Optical Processes Using Electromagnetically Induced Transparency. Phys. Rev. Lett. 1990, 64, 1107, doi:10.1103/PhysRevLett.64.1107.

eit1993
Harris, S.E. Electromagnetically induced transparency with matched pulses. Phys. Rev. Lett. 1993, 70, 552, doi:10.1103/PhysRevLett.70.552.

eit2005
Fleischhauer, M.; Imamoglu, A.; Marangos, J.P. Electromagnetically induced transparency: Optics in coherent media. Rev. Mod. Phys. 2005, 77, 633.


horo
Horodecki, M.; Horodecki, P.; Horodecki, R. Separability of mixed states: Necessary and sufficient conditions. Phys. Lett. A 1996, 223, 1–8, doi:10.1016/S0375-9601(96)00706-2.

caruso
Caruso, F.; Giovannetti, V.; Lupo, C.; Mancini, S. Quantum channels and memory effects. Rev. Mod. Phys. 2014, 86, 1203.



exper1976
Alzetta, G.; Gozzini, A.; Moi, L.; Orriols, G. An Experimental Method for the Observation of R.F. Transitions and Laser Beat Resonances in Oriented Na Vapour. Il Nuovo C. B 1976, 36, 5–20.

brewer
Brewer, R.G.; Hahn, E.L. Coherent two-photon processes: Transient and steady-state cases. Phys. Rev. A 1975, 11, 1641.

zanon
Zanon-Willette, T.; de Clercq, E.; Arimondo, E. Ultrahigh-resolution spectroscopy with atomic or molecular dark resonances: Exact steady-state line shapes and asymptotic profiles in the adiabatic pulsed regime. Phys. Rev. A 2011, 84, 062502, doi:10.1103/PhysRevA.84.062502.

SMC09
Dong, D.; Petersen, I.R. Sliding mode control of quantum systems. New J. Phys. 2009, 11, 105033.

AD09
Chen, J.-M.; Liang, L.-M.; Li, C.-Z.; Deng, Z.-J. Arbitrary state controlled-unitary gate between two remote atomic qubits via adiabatic passage. Opt. Commun. 2009, 282, 4020–4024.

AD11
Guérin, S.; Hakobyan, V.; Jauslin, H.R. Optimal adiabatic passage by shaped pulses: Efficiency and robustness. Phys. Rev. A 2011, 84, 013423.

AD11-1
Torosov, B.T.; Guérin, S.; Vitanov, N.V. High-Fidelity Adiabatic Passage by Composite Sequences of Chirped Pulses. Phys. Rev. Lett. 2011, 106, 233001.

RC14
Zhang, J.; Greenman, L.; Deng, X.; Whaley, K.B. Robust Control Pulses Design for Electron Shuttling in Solid-State Devices. IEEE Trans. Control Syst. Technol. 2014, 22, 2354–2359.

RC17
Wu, C.; Qi, B.; Chen, C.; Dong, D. Robust Learning Control Design for Quantum Unitary Transformations. IEEE Trans. Cybern. 2017, 47, 4405–4417.

dod
Dodonov, V.V.; Man'ko, V.I. Positive distribution description for spin states. Phys. Lett. A 1997, 229, 335–339.

olga
Man'ko, V.I.; Man'ko, O.V. Spin state tomography. J. Exp. Theor. Phys. 1997, 85, 430–434.



