1. Introduction

The study of quantum correlations is at the core of Quantum Information Theory (QIT).
Entanglement ^{[1]} had been considered to solely
encompass what Schrödinger himself esteemed to be “*the*
characteristic trait of quantum mechanics, the one that enforces its entire
departure from classical lines of thought” ^{[2]}. The development of Quantum Discord (QD) by Olliver and Zurek, and
independently by Henderson and Vedral ^{[3]}, in
2001 showed that there are quantum correlations that are not included within the
separability criteria of entanglement. Using Werner states as an example, both
articles show that there are states that are not entangled, *i.e.*
null concurrence ^{[4]}, and yet exhibit nonzero
QD. This has given a new impulse to a highly dynamical subfield of QIT, the study of
new quantifiers for quantum correlations.

Local measurements are the key ingredient to properly define correlations. They are
important because correlations must quantify the ability of one local observer to
infer the results of a second local observer from his own local results. The
aforementioned Quantum Discord ^{[3]}:

is based on comparing the quantum Mutual Information, defined for the original state

with a corresponding classical-quantum (or A-classical) state

which is a postmeasurement state in the absence of readout, where the measurement is
performed locally over the A subsystem of

Quantifiers of quantum correlations using either A-classical or B-classical states are called Discords and are, in general, not symmetrical.

Other quantifiers ^{[5]} are based on the
difference of a quantity (*e.g*. mutual information, relative
entropy, etc.) with respect to systems in which both subsystems have been locally
measured. These type of states are labeled as strictly classical

where

Quantifiers of this sort include Measurement-Induced Disturbance (MID), introduced by
Luo ^{[6]}, as well as its ameliorated form
(AMID), introduced by Wu *et al*. ^{[7]}.

General quantum correlations defined in terms of local bipartite measurements were
considered recently by Wu *et al*. in ^{[8]}, where they introduce and study non-symmetric quantum correlations
using the Holevo quantity ^{[9]} and, in a brief
final appendix, they define symmetric quantum correlations in terms of mutual
information. The LAQCs developed in ^{[10]}
focused on a slightly different version of those symmetric correlations, preserving
the requirement that any available ones must always be defined in terms of mutual
information of local bipartite measurements.

This work is focused on analytically calculating the LAQCs quantifier for the family of BD states, given by

where the coefficients *i.e*. has non-negative
eigenvalues) and ^{[11]} and phase damping ^{[12]}. We will also make use of the Bloch representation for
2-qubits, given by

where

The present article is structured as follows: in Sec. 2 we review the main results
obtained in ^{[10]} by defining our procedure
for calculating the local available quantum correlations quantifier. Section 3 is
dedicated to the explicit calculation of this quantifier for Bell diagonal (BD)
states. We start by performing the calculation for a highly symmetrical subset of BD
states, namely Werner states. These results are then generalized for the whole BD
states family. Section 4 is devoted to the subject of Markovian decoherence. We
start by presenting the Kraus operators formalism and proceed to analyze two
dissipative quantum channels, namely depolarizing ^{[11]} and phase damping ^{[12]},
acting on the set of Werner states and determining the dissipative dynamics of the
LAQCs quantifier by means of our previous result for BD states. Finally, Sec. 5 is
devoted to the summary.

2. Local available quantum correlations for 2-qubits

A density operator

where

Any such basis for the Hilbert space of qubits can be thought of as a new
computational basis, *i.e*. the basis of eigenvector of

Since strictly classical states are states which are diagonal in some local basis,
one can define

where

and

Whitout loss of generality, the search for

Therefore, analyzing the criteria for minimization of the aforementioned relative
entropy is related to the behavior of the coefficients

It is important to keep in mind that this process is equivalent to finding the
unitary vectors

In this context, Mundarain *et al*. define the classical correlations
quantifier as

where *et al*. ^{[13]}, the relative entropy of a generic state,
*e.g*. *i.e*.

where

where

Once the optimal angles

where ^{[8]}. In doing so, we are now able to determine
the local available quantum correlations, which are quantified in terms of the
maximal mutual information for measurements performed on

and by means of (17), we determine the mutual information

3. LAQCs for Bell Diagonal states

3.1 Werner States

As to better illustrate the calculation of the LAQCs quantifier, we start by
determining it for a highly symmetrical subset of BD states (7), namely Werner
states,

where ^{[3]}, these states
have non-vanishing quantum correlations, *i.e*. their quantum
discord is only null for

The density matrix for the Werner states, using the standard computational matrix, is written as:

By means of (14), the elements

Since we are using (17) to minimize (11), all that is needed are the optimal
angles

where

To determine the LAQCs quantifier for the Werner states, we need to define the
complementary basis. Since we can consistently measure the classical
correlations on the

where once again we have that

Therefore, we have that for Werner states, there is the same amount of classical correlations as there are locally available quantum correlations.

3.1.1. Comparing with other quantifiers

We briefly compare our result (26) for the LAQCs quantifier with other
quantum correlations quantifiers, such as quantum discord ^{[3]} and concurrence, a quantifier for
entanglement.

It is well known that concurrence^{[4]}, has a simple expression for Werner states, given by:

The expression for quantum discord for Werner states is derived from the
analytical one obtained by Luo in ^{[14]} for the more general case of BD states, given by

Using the fact that

Comparison of the LAQCs quantifier with concurrence and quantum discord is
shown graphically in Fig. 1. As
observed in an example presented in ^{[10]}, the quantifier for the LAQCs has values lower than the
ones for Quantum Discord. In the aforementioned case, the 2-qubit pure state *i.e.* equal to 1.

Nevertheless, this does not imply that both quantifiers will necessarily show
in general a similar qualitative behavior. As was also pointed out in ^{[10]}, for the family of mixed states

3.2. General case

We now proceed to the general case of BD states (7). Following the same procedure
as before, we determine the coefficients

Since all BD states have maximally mixed marginals, we can again make use of the
symmetry under exchange of subsystems A

From (31) it is straightforward to realize that

In this case, the minimization will depend on whether

As happened for Werner states, due to the symmetry of BD states, the density
matrix associated with (7) is invariant under the aforementioned unitary
transformations (13) for the previously chosen optimal computational basis.
Identifying

As previously done for the Werner states, the LAQCs quantifier is then calculated
in the basis (18), with

where we also have that

where once again we have that

4. Decoherence

Modeling the behavior of any real quantum system must take into account that it will
not be completely isolated. There will be a much larger system surrounding the
quantum one, called environment, which in general will have infinite degrees of
freedom. This interaction between quantum system and environment, albeit efforts to
minimize it, will induce a process of decoherence and relaxation. This in turn may
hinder the ability of the system to maintain quantum correlations, therefore
affecting its ability to perform certain tasks in quantum computing, among others.
The study of this process can be done, under the Markovian approximation, either by
using a master equation, *i.e*. the Lindblad equation ^{[15]}, also referred to as the
Lindblad-Kossakowski equation ^{[16]}, or a
quantum dynamical semigroup approach, *i.e*. Kraus operator ^{[17]} formalism. In what follows we will make
use of the later, with common interactions to both subsystems, *i.e.*
with the interaction parameter

Within this framework, we will study two dissipative quantum channels: Depolarizing
^{[11]} and Phase Damping Channel ^{[12]}.

4.1 Depolarizing Channel

This quantum operation represents the process of substituting an initial single
qubit state ^{[11]}. Its Kraus operators are given
by

Applying these operators on a Werner state (20) via (37), it is straightforward to verify that the resulting density operator has the following Bloch parameters:

which corresponds to a Werner state where the action of this noisy quantum
channel contracts the state parameter

Let us now compare this with other quantum correlations quantifiers. It is well
known that Werner states exhibit entanglement sudden death (ESD) ^{[18]}, as can easily be seen by using

For quantum discord ^{[3]}, by means of (29)
and using

The behavior of the LAQCs quantifier, concurrence and quantum discord for a Werner state under the action of a Depolarizing Channel is shown graphically in Fig. 2. It is worthy noticing that, since the resulting state of this quantum channel is still a Werner state, the qualitative behavior of both QD and LAQCs quantifiers is indeed similar as previously shown, maintaining the relation of the quantifier for QD being greater than the one for LAQCs.

4.2. Phase Damping Channel

One of the quantum channels analyzed by Werlang *et al*. ^{[19]} in order to show the robustness of
Quantum Disord to decoherence is the *Phase Damping Channel*
acting on a Werner state. This noisy channel describes the loss of quantum
information without loss of energy ^{[12]}.
The Kraus operators for this quantum channel are given by:

Applying these operators on a Werner state (20) via (37), the resulting density matrix has the following Bloch parameters:

which corresponds to a BD state (7) with

Even though the resulting quantum state is no longer a Werner state, since

Concurrence for (44) is given by:

and Quantum Discord is readily obtained from (28) and (44), yielding:

The behavior of the LAQCs quantifier, quantum discord and Concurrence for a Werner state under the action of a Phase Damping Channel is shown graphically in Fig. 3. As can be inferred from this graphics, the qualitative behavior of both QD and LAQCs is in this case also quite similar, maintaining the expected relation of QD being larger than LAQCs.

5. Conclusions

We have successfully evaluated the LAQCs quantifier for the family of BD states,
obtaining analytical formulas for it. To do so, we started with a much simpler case,
the subfamily of Werner states, as to better illustrate the procedure for
determining the LAQCs quantifier. For this subset of BD states, its behavior has
been graphically presented, comparing it with both concurrence ^{[4]} and quantum discord ^{[3},^{14]}. In this case QD and LAQCs
exhibit similar qualitative behavior and, as expected, the LAQCs quantifier is lower
in value than QD.

The dissipative dynamics of the 2-qubit LAQCs quantifier under Markovian decoherence
was studied for Werner states using the Kraus operators formalism in two cases:
Depolarizing channel ^{[11]} and Phase Damping
channel ^{[12]}. Analytical expressions were
obtained for both cases and presented graphically. As was previously reported for
Quantum Discord ^{[19]}, LAQCs also do not
exhibit the sudden-death behavior shown by entanglement, *i.e.*
concurrence.

It is important to notice that we are maintaining the usual notation for Concurrence
by using the letter