# Infrared Magneto-Polaritons in MoTe2 Mono- and Bilayers

Bo Han,1 Jamie M. Fitzgerald,2 Lukas Lackner,1 Roberto Rosati,2 Martin
Esmann,1 Falk Eilenberger,3,4,5 Takashi Taniguchi,6 Kenji Watanabe,7 Marcin
Syperek,8 Ermin Malic,2 and Christian Schneider1, Corresponding author:
<EMAIL_ADDRESS>1Institut für Physik, Fakultät V, Carl
von Ossietzky Universität Oldenburg, 26129 Oldenburg, Germany
2Department of Physics, Philipps-Universität Marburg, Renthof 7, D-35032
Marburg, Germany
3Fraunhofer-Institute for Applied Optics and Precision Engineering, 07745
Jena, Germany
4Institute of Applied Physics, Abbe Center of Photonics, Friedrich Schiller
Universität Jena, 07745 Jena, Germany
5Max Planck School of Photonics, 07745 Jena, Germany
6International Center for Materials Nanoarchitectonics, National Institute for
Materials Science, Tsukuba 305-0044, Japan
7Research Center for Functional Materials, National Institute for Materials
Science, Tsukuba 305-0044, Japan
8Department of Experimental Physics, Faculty of Fundamental Problems of
Technology, Wrocław University of Science and Technology, Wyb.Wyspiańskiego
27, 50-370 Wrocław, Poland

###### Abstract

MoTe2 monolayers and bilayers are unique within the family of van-der-Waals
materials since they pave the way towards atomically thin infrared light-
matter quantum interfaces, potentially reaching the important
telecommunication windows. Here, we report emergent exciton-polaritons based
on MoTe2 monolayer and bilayer in a low-temperature open micro-cavity in a
joint experiment-theory study. Our experiments clearly evidence both the
enhanced oscillator strength and enhanced luminescence of MoTe2 bilayers,
signified by a 38 % increase of the Rabi-splitting and a strongly enhanced
relaxation of polaritons to low-energy states. The latter is distinct from
polaritons in MoTe2 monolayers, which feature a bottleneck-like relaxation
inhibition. Both the polaritonic spin-valley locking in monolayers and the
spin-layer locking in bilayers are revealed via the Zeeman effect, which we
map and control via the light-matter composition of our polaritonic
resonances.

††preprint: arXiv:

##  I Introduction

Two-dimensional semiconductors of transition metal dichalcogenides (TMDCs) are
an excellent research platform for solid-state cavity quantum electrodynamics
due to their strong light-matter interactions and intriguing spin-valley
locking [1, 2, 3]. Their optical properties hinge on very robust valley
excitons with binding energies of hundreds of meV [4, 5, 6]. Composite quasi-
particles such as exciton-polaritons in TMDC-microcavity systems can inherit
physical properties from both the cavity modes and the optically active
material, including the ultralight effective mass [7], magnetic responses [8,
9], and non-linearities due to exchange-correlation [10, 11], dipolar
interaction [12], phase space filling [13] and moiré confinement [14].
Therefore, they are remarkable systems to explore collective phenomena such as
bosonic condensation [15, 16], coherent light emission[17, 18, 19, 20],
polariton blockade [21, 22] and correlated magnetism [23].

Among the TMDC family members, MoTe2 features a unique band structure with
both the monolayer (ML) and bilayer (BL) exhibiting a direct band gap [24, 25,
26, 27, 28, 29]. The ground-state excitonic resonances lie in the near
infrared spectral range $\sim$1.1 $\mu$m [30, 29], making MoTe2 a good
candidate for optoelectronic [31] and quantum optical applications [32] in the
optical telecommunication window [33]. The extraordinary electronic properties
[34, 35] also render MoTe2 a promising material for studies of integer and
fractional (anomalous) (spin) quantum Hall effects [36, 37, 38, 39]. However,
the strong light-matter coupling regime of MoTe2 has not received notable
attention so far compared to other TMDC materials..

In this work, we demonstrate the first experimental measurements on the strong
coupling of excitons in MoTe2 ML and BL with discretized cavity modes in a
low-temperature open optical micro-cavity. Our study reveals that the BL
features an increase in Rabi splitting and coupling strength by 38 % compared
to the ML, while maintaining otherwise identical conditions. Interestingly, we
observe a bottleneck-like inhibition of relaxation in ML structures,
effectively enhancing the luminescence from the upper polariton branch (UPB),
while, in stark contrast, the majority of population relaxes to the lower
polariton branch (LPB) in the BL. Our experiments are complemented by a many-
particle theory revealing the microscopic mechanisms behind this qualitatively
different behaviour in MoTe2 ML and BL. Using systematic magneto-optics
measurements, we explore the polaritonic Zeeman effect and reveal g-factors
which are controlled by the cavity-exciton detuning, manifesting the valley
and layer degree of freedom in the ML and BL, respectively. We also discover
an unconventional enhancement of the coupling strength of both the ML and BL
in the magnetic field.

##  II Strong Coupling and quasi-particle relaxation

The MoTe2 ML and BL are encapsulated [40] with thin hexagonal boron nitride
flakes and deposited on a distributed Bragg reflector (DBR) with a stop band
centered at 1050 nm (see Fig. S1(a,b,e) of Supplementary Material [41]). The
samples are loaded in a closed-cycle low-temperature cryostat (3.5 K) equipped
with a superconducting magnet. First, we perform the photoluminescence (PL)
and differential reflectivity (DR) measurements on the encapsulated MoTe2
flakes by using a linearly polarized 765 nm continuous wave laser and a
tungsten-halogen lamp, respectively. The indiscernible Stokes shift between
the PL and DR proves the high quality of our samples (see Fig. S1(f-i) [41]).
More experimental details are summarized in Section III of [41].

Figure 1:  (a) Schematic of the open cavity embedded with MoTe2 ML and BL in
magnetic field. The valley Zeeman effect for layers with inversion symmetry is
sketched on the right panel. (b) DR as a function of cavity detuning with a
cavity loaded with a ML (P1), featuring exciton energy $E_{X}^{ML}$=1.1719 eV
and a Rabi-splitting of $V_{ML}^{0T}$=13 meV. (c) DR in BL as a function of
cavity detuning, featuring $E_{X}^{BL}$=1.1513 eV and $V_{BL}^{0T}$=18 meV.
The colorbar is identical for (b) and (c).

For strong coupling measurements, we use a top mirror that is a gold-coated
silica mesa with a pre-manufactured hemispheric lens structure of 6 $\mu$m
diameter and 300 nm depth (see Fig. S1(c,d) [41]). A schematic of the cavity
setting is shown in Fig. 1(a). The lens and bottom DBR form discretized Tamm-
plasmon lens modes (Q-factors $\sim$400) that can be readily measured in PL or
DR. The focused excitation spots are comparable to the lens diameter. Our
previous works have demonstrated the capability of discretized Tamm modes in
tuning the emitters’ spontaneous emission rates in the weak coupling regime
[42, 43]. DC voltage applied on the actuator can change the cavity length with
sub-nm positioning resolution. The PL and DR measurements are thus performed
with a fine detuning step size of 0.1 meV. The detuning $\Delta=E_{C}-E_{X}$
is defined as the energetic difference between the cavity and exciton modes.
Both measured ML and BL samples are deliberately tuned to couple with the 8th
longitudinal lens modes, corresponding to a cavity length of approximately
4.28 $\mu$m at $\Delta=0$.

In cavity-length dependent DR measurements in Fig. 1(b,c), we observe clear
anti-crossing features. We apply coupled oscillators theory to calculate the
energy of both polariton branches as a function of the cavity-exciton detuning
via $E_{UPB(LPB)}$=$(E_{X}+E_{C}\pm\sqrt{V^{2}+\Delta^{2}})/2$. The fitting of
polariton resonances in ML and BL are superimposed on the experimental DR
results in Fig. 1(b,c), as well as in the PL measurements in Fig. 2(a,b). This
approach yields a coupling strength of $V_{ML}^{0T}$=13 meV for the MoTe2 ML
(position 1) and $V_{BL}^{0T}$=18 meV for the BL. The superscript 0 T
indicates the absence of an externally magnetic field. We encounter a slight
variation of the coupling strength for different sample positions of the ML on
the order of the polariton linewidth (see Fig. S5-S6 for measurements on
different positions: P2 and P3 [41]). Indeed, the observed increase of the
Rabi-splitting by a factor of 1.38 for the BL is in agreement with the Tavis-
Cummings model predicting a scaling with the square root of the number of
oscillators [44, 45, 46].

A striking difference between ML and BL polaritons becomes obvious when
comparing the PL intensity between UPB and LPB for the two cases. Figure 2(a)
depicts the PL from the polaritonic modes for the ML sample (P1), whereas the
BL case is plotted in Fig. 2(b) with identical excitation conditions. For BL-
polaritons, the UPB emission is much weaker than the emission from the LPB,
which is indicative of efficient relaxation of population to the energetically
lower state. In stark contrast, for ML-polaritons, the UPB is generally more
intense than the LPB (see Fig. S5(a) and Fig. S6(a) for other sample positions
[41]). We plot the emission intensity ratio between the UPB and LPB in Fig.
2(e) to quantify the relaxation strength for all detunings: in the BL case,
the intensity ratio is generally smaller than 0.4, which is indicative for
enhanced PL from the LPB, as a consequence of efficient relaxation. However,
in the case of the ML, the ratio exceeds unity even for very red detuning, and
increases dramatically towards the zero and blue detuning regimes.

Figure 2:  PL spectra of exciton-polaritons in (a) ML (P1) and (b) BL as a
function of detuning. The ML polaritons feature a significant PL emission from
the UPB, whereas BL polaritons exhibit significant particle relaxation into
the LPB. The coupled oscillator models for DR results as in Fig. 1 are also
superimposed on the PL anti-crossings with excellent consistency. Theoretical
description of the PL from (c) MoTe2 ML- and (d) BL-polaritons, based on the
material-realistic parameters from the experiments. Both absolute intensities
are normalized to the BL emission, sharing the same colorbar of (d). (e)
Experimentally extracted PL intensity ratio between the UPB and LPB in MoTe2
ML and BL as a function of the detuning. (f) Schematics of the polariton
relaxation regime. The detuning is set to -17.5 meV to have the LPB (K=0)
lying 21 meV below the BL exciton, matching the energy of the homopolar A1
optical phonons. The coupling strength and exciton energy at K=0 are
experimental values. The opening of relaxation to LPB (K=0) is marked by the
blue dashed arrow. Figure 3:  Polariton Zeeman effect in spectral and
polarization-resolved PL measurements at +9 T. The color-coded Zeeman plots in
(a) ML (P3) and (b) BL are defined by subtracting the $\sigma-$ polarized
spectrum from the $\sigma+$ polarized one at the same cavity detuning. (c) ML
(P3) and (d) BL representative PL spectra of polariton Zeeman splits at
$\Delta=$0 and $\pm$5 meV, corresponding to the dashed line marks in (a) and
(b). The Rabi splittings are $V_{ML}^{+9T}$= 17.5 meV for the ML, and
$V_{BL}^{+9T}$=19.5 meV for the BL. Experimental (dots) and theoretical
(dashed lines) polariton g-factors in (e) ML (P3) and (f) BL. The error bars
are derived by summing up the energetic error bars of the fitted polariton
peaks in Fig. S5(c,e) [41].

We model the polariton population and relaxation in our systems using a
material-realistic many-particle theory [41]. Exciton-polaritons are modelled
within a Wannier-Hopfield framework [47], using the experimentally extracted
light-matter coupling parameters as well as a bare cavity linewidth of $1.5$
meV. Polariton-phonon scattering is described via the deformation potential
and second-order Born-Markov approximation [48, 49]. In particular, we
explicitly account for the A1 homopolar phonon with an energy of $21$ meV,
which strongly interacts with excitons in MoTe2 [30, 50]. The values for the
deformation potentials are extracted from experimentally measured temperature-
dependent linewidths of the 1s exciton in both ML and BL MoTe2 (see Fig. S7
[41]). We find an increased exciton-phonon coupling for the BL, in agreement
with previous studies [26]. The frequency-dependent PL at steady-state
conditions is given by the polariton Elliot Formula [48]

$I(\omega)=\sum_{\textit{n=\text{LPB,UPB}}}\frac{\gamma_{n}\Gamma_{n}}{(E_{n}-\hbar\omega)^{2}+(\gamma_{n}+\Gamma_{n})^{2}}N_{n}^{0}$
(1)

where $\gamma$ and $\Gamma$ are the polariton radiative and phonon-induced
decay rates, respectively, $E_{n}$ is the polariton energy, and $N_{n}^{0}$ is
the Boltzmann distribution of the quasi-particles. We fix the effective
temperature of the exciton gas to $60$ K to consider the realistic case of
imperfect thermalization. The results of the model are plotted for the case of
ML- (Fig. 2(c)) and BL-polaritons (Fig. 2d)). Strikingly, the dramatic
increase of the lower polariton PL for the BL is well captured in our
microscopic description, which verifies the impact of the A1 phonon mode on
the relaxation dynamics. The more efficient polariton relaxation observed for
the BL is attributed to the combined effect of the enhanced exciton-phonon
coupling and larger Rabi splitting. The BL polariton has a larger excitonic
character at the negative detunings when the scattering channel from the
exciton reservoir opens up (see Fig. 2(f)), i.e.,
$E_{\text{LPB}}+E_{\text{A${}_{1}$}}\approx E_{\text{X}}$, where E${}_{A_{1}}$
is the A1 phonon energy. This, together with the larger exciton-phonon
scattering in the BL, leads to a significantly larger occupation of the BL-LPB
at all detunings.

##  III Polaritonic Magneto-optics

Polaritons are light-matter hybrid quasi-particles whose magnetic responses
stem from their excitonic nature. The polaritonic g-factor thus needs to be
rescaled by the excitonic Hopfield coefficient
$|X_{k}|^{2}$=$(\sqrt{\Delta_{k}^{2}+V^{2}}+\Delta_{k})/2\sqrt{\Delta_{k}^{2}+V^{2}}$,
a parameter characterizing the excitonic proportion in the polariton. However,
the polaritonic Zeeman phenomena in TMDCs have not been systematically studied
with different light-matter compositions.

Due to time-reversal symmetry, the spin, orbital and valley indices in the ML
take opposite signs between adjacent K valleys, leading to a locking of spin
and valley degrees of freedom in bands with strong spin-orbit coupling. In
contrast, the BL retains the inversion symmetry, which leads to the intriguing
phenomenon of spin-layer locking [51, 52, 53]. Both phenomena reveal
distinctive spectral signatures in polarization spectroscopy in the presence
of external magnetic fields, as sketched in Fig. 1(a). In order to extract the
exciton out-of-plane g-factor, magneto-DR is firstly performed between $\pm$9
T in a Faraday geometry in the absence of the top mirror, ensuring weak
coupling conditions. We utilize linearly polarized excitation, and measure
counter-circularly polarized signals ($\sigma\pm$). The results are shown in
Fig. S4 [41]. The out-of-plane excitonic g-factors ($g_{x}$) are calculated by
fitting the Zeeman splitting of the valley excitons by using
$\Delta$E=$g_{x}\mu_{B}B$, where $\mu_{B}$ is the Bohr magneton and $B$ is the
magnetic flux density. The values of $g^{ML}_{X}$=-4.6 and $g^{BL}_{X}$=-4.1
are in excellent agreement with previous reports [54, 55, 56].

We notice that the modified sample position (P3), which is chosen for the
magneto-optical study of the ML-polaritons features a slightly larger coupling
strength of 16 meV measured at zero-field (see Fig. S3(a) and Fig. S5(a,b)
[41]), which we attribute to locally varying dielectric screening and charging
effects in the ML. Polaritonic Zeeman effects in both the MoTe2 ML and BL are
then extracted from magneto-PL measurements (Fig. 3). We utilize the same
polarization configuration for the polaritonic magneto-optics measurements.
The polaritonic Zeeman splitting as a function of the detuning is then
straightforwardly calculated via the energetic differences between $\sigma+$
and $\sigma-$ resonances at +9 T.

We first plot the spectrally resolved detuning-dependent polarization patterns
for ML and BL polaritons in Fig. 3(a,b), which are obtained by subtracting the
$\sigma-$ polarized spectrum from the $\sigma+$ polarized one at the same
cavity detuning. The same effects are also revealed in the color-coded
magneto-DR difference spectra that are compiled in Fig. S5(d,f) [41]. We can
immediately see the detuning dependent dichroism, which becomes especially
prominent in the excitonic regime of the polariton branches, for example, at
$\Delta=\pm 20$ meV for the LPB and UPB, respectively. Magneto-PL and -DR
difference spectra on another sample position (P2) of the ML, which display
the same polaritonic Zeeman effects, are presented in Fig. S6 [41].

The PL spectra at detunings of 0 and $\pm$5 meV are also shown in Fig. 3(c)
for the ML case, and Fig. 3(d) for the BL case. The Zeeman splitting becomes
very apparent in this representation, and acquires a similar magnitude for the
ML and BL polaritons for comparable detunings. From the detuning dependent
Zeeman effect, we can model the UPB and LPB g-factors as a function of
detuning as:

$g_{UPB(LPB)}=\frac{g_{x}}{2}\pm\frac{\sqrt{V^{2}+\Delta_{1}^{2}}}{2\mu_{B}B}\mp\frac{\sqrt{V^{2}+\Delta_{2}^{2}}}{2\mu_{B}B}$
(2)

where $\Delta_{1}=E_{C}-E_{X}^{\sigma^{+}}$ and
$\Delta_{2}=E_{C}-E_{X}^{\sigma^{-}}$ are the actual cavity detunings with
respect to each excitonic Zeeman split, and
$\Delta=(\Delta_{1}+\Delta_{2})/2$. The zero-detunings of $\Delta_{1}$ and
$\Delta_{2}$ are marked by double-sided arrows in Fig. S5(c,e) [41] that
present the Lorentzian fitting results of the polariton Zeeman splits in ML
and BL at +9 T. The experimentally extracted polaritonic g-factors of each
branch in the ML (Fig. 3(e)) and BL (Fig. 3(f)) are in excellent agreement
with our theoretically derived values from Eq. (2). The degeneracy of the
valley as well as the layer-locked polaritons in the ML and BL is thus lifted,
while the strong coupling effect is verified as a viable tool to tune the
resulting optical dichroism.

It is furthermore worth noting that, comparing the coupling strength at +9 T
(Fig. 3(c,d)) to the scenario without the magnetic field (Fig. S3(b,h) [41]),
the Rabi splittings are enhanced by 1.5 meV for both the ML and BL,
corresponding to an effective enhancement of the coupling strength by
approximately 8-9 %. This behavior occurs consistently for various sample
positions (see Fig. S6(e,f) [41]). The coupling strength enhancement by
magnetic field was previously reported for polaritons in III-V semiconductor
quantum wells embedded in monolithic cavities [57], where the magnetic
compression of exciton wavefunction increases the exciton oscillator strength
and consequently enhances the vacuum-field Rabi splitting [58]. Our encounter
is thus very unconventional, since the exciton Bohr radius ($a_{B}\sim$1-2 nm)
is significantly smaller than the estimated magnetic length
$L_{B}$=$\sqrt{\hbar/(e{\cdotp}B)}\approx$ 8.6 nm.

##  IV Conclusions

We report the first optical microscopy measurements on the emergence of
exciton-polaritons in MoTe2 monolayer and bilayer. In contrast to the bilayer
that exhibits an efficient population relaxation to the lower polariton
branch, the relaxation in the monolayer features a pronounced bottleneck
phenomenon, which has been modeled using a microscopic many-particle theory
describing the scattering of exciton-polaritons with A1 optical phonons. We
have also verified the strong couplings via magneto-optics measurements, where
the polariton valley and layer degeneracies are lifted in monolayer and
bilayer, respectively. We can thus extract the polaritonic g-factors as a
function of the cavity-exciton detuning. Our work paves the way for further
research involving cavity-mediated phenomena in MoTe2-based van-der-Waals
heterostructures, including the study of correlated phenomena, Telluride-based
dipolaritons, and polariton lasers operated at telecommunication wavelengths.

###### Acknowledgements.

The project is funded by Deutsche Forschungsgemeinschaft (DFG) Lantern project
(funding numbers: Schn1376 11.1). C.S. acknowledges DFG within the initiative
for major equipment (Project INST184-220). B.H. acknowledges Alexander von
Humboldt-Stiftung for the fellowship grant. M.E. acknowledges funding from the
Carl von Ossietzky Universität Oldenburg through a Carl von Ossietzky Young
Researchers’ Fellowship. F.E. acknowledges support by DFG SFB 1375 (NOA) and
BMBF FKZs 16K1SQ087K and 13XP5053A. J.F, R.R and E.M acknowledge funding from
DFG via SFB 1083 and the regular project 524612380. K.W. and T.T. acknowledge
support from the JSPS KAKENHI (grant nos. 21H05233 and 23H02052) and World
Premier International Research Center Initiative (WPI), MEXT, Japan. M.S.
acknowledges funding from the project No. 2019/35/B/ST5/04308 financed by the
Polish National Science Center (NCN).

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LAQC for X states]Local available quantum correlations of X states: The asymmetric case

David M. Bellorin R.
Departamento de Física, Universidad Simón Bolívar, AP 89000, Caracas 1080, Venezuela.

Hermann L. Albrecht Q.
[Corresponding Author<EMAIL_ADDRESS>Departamento de Física, Universidad Simón Bolívar, AP 89000, Caracas 1080, Venezuela.

Douglas F. Mundarain

Departamento de Física, Universidad Católica del Norte, Casilla, 1280 Antofagasta, Chile.

Local available quantum correlations (LAQC), as defined by Mundarain et al., are analyzed for non-symmetric 2-qubit X  states, that is, X-states that are not invariant under the exchange of subsystems and therefore have local Bloch vectors whose norms are different. A simple analytic expression for their LAQC quantifier is obtained. As an example, we analyze the local application of the amplitude damping channel for Werner states and general canonical X states. Although this local quantum channel can create quantum discord in some cases, no such outcome is possible for LAQC, which hints toward their monotonicity under LOCC operations. This work, along with our previous result for so-called symmetric and anti-symmetric X states, gives a complete characterization and analytical expressions for the LAQC quantifier for the whole family of canonical 2-qubit X states.

§ INTRODUCTION

Quantum information theory has been one of the most active physics research fields in the past couple of decades. A key ingredient for the advantages rendered by quantum mechanics within information theory is correlations. From the early days of the quantum revolution until the 2000s, only entanglement [Horodecki et al., 2009] was the known quantum correlation. Since the development in 2001 of quantum discord [Ollivier and Zurek, 2001, Henderson and Vedral, 2001], there has been an effort to study and develop new types of quantum correlations that can be applied in quantum information processing and communications [Modi et al., 2012].

Within these studies, the role of local measurements is crucial for developing new criteria for quanticity. A quantum correlation has to quantify the ability of a local observer to infer the results of another one from his own. The abovementioned quantum discord, for instance, is defined by comparing the quantum mutual information of the original state with a corresponding post-measurement state in the absence of readout. That is,

\begin{equation}\label{eq:QD-Def}
D_A(\rho_{AB})\equiv \min_{\qty{\Pi_i^A}}\qty{I(\rho_{AB})-I[(\Pi^A\otimes\mathbbm{1}_2)\rho_{AB}]},
\end{equation}

\begin{equation}\label{eq:InfoMutua}
I(\rho_{AB})\equiv S\qty(\rho_A)+S\qty(\rho_B)-S\qty(\rho_{AB}),
\end{equation}

with $\rho_A$ and $\rho_B$ the reduced operators, i.e. marginals.

In the above definition (<ref>), the local measurement is performed on subsystem A, and the corresponding post-measurement states is usually referred to as classical-quantum (or A-classical) state,

\begin{equation}\label{eq:Estados_Cl-Q}
\rho^{cq}_{AB}=\sum_i\, p_i\,\dyad{i}\otimes\rho_B^i= \sum_i\, p_i\,\Pi^{(A)}_i\otimes\rho_B^i,
\end{equation}

and all such 2-qubit states constitute a set, denoted by $\Omega_o$.

Analogously, quantum discord can also be established for a local measurement on subsystem B:

\begin{equation}\label{eq:QD-Def-B}
D_B(\rho_{AB})\equiv \min_{\{\Pi_i^B\}}\left\{I(\rho_{AB})-I[(\mathbbm{1}_2\otimes\Pi^B)\rho_{AB}]\right\},
\end{equation}

which leads to defining the set $\Omega_o'$ for quantum-classical (B-classical) states as in (<ref>). There have been other quantum correlations defined using the set $\Omega_o$ (or  $\Omega_o'$) [Daki ćć et al., 2010, Paula et al., 2013, Spehner and Orszag, 2013] called Geometric Discords. Since the comparison involves local measurements on only one of the subsystems, they are in general not symmetric, $D_A(\rho_{AB})\neq{}D_B(\rho_{AB})$.

Trying to overcome this asymmetry, another set of quantum correlations and quantifiers was developed [Luo, 2008, Wu et al., 2009, Girolami et al., 2011, Wu et al., 2015] by considering post-measurement states in the absence of readout whose measurements were performed locally on both subsystems:

\begin{equation}\label{eq:Estados-Clasicos}
\rho_{AB}^c = \sum_{i,j} p_{ij} \Pi^{(A)}_i\otimes\Pi^{(B)}_j.
\end{equation}

Such states are called classical states. A special subset is defined when $p_{ij}$ can be factorized so that

\begin{equation}
\rho_{AB}^{\Pi}=\sum_{i,j} p_{i} \Pi^{(A)}_i\otimes{}p_{j}\Pi^{(B)}_j=\rho^{(A)}\otimes\rho^{(B)}.
\end{equation}

Such states are referred to as product or uncorrelated states, and the sets of classical and product 2-qubit states are labeled as $\Omega_c$ and $\Omega_{p}$, respectively.

In [Mundarain and de Guevara, 2015], Mundarain and Ladrón de Guevara developed the so-called Local Available Quantum Correlations (LAQC). It is  a symmetric quantum correlation slightly different from the one presented by Wu et al. in [Wu et al., 2015], defined in terms of mutual information of local bipartite measurements on the complementary basis of a previously determined optimal computational one.

This work is focused on an important family of 2-qubit states, the so-called X states [Yu and Eberly, 2007]

\begin{eqnarray}\label{eq:estadosX}
\rho^X =\mqty(
\rho_{11} & 0 & 0 & \rho_{14} \\
0 & \rho_{22} & \rho_{23} & 0 \\
0 & \rho_{32} & \rho_{33}& 0 \\
\rho_{41}& 0 & 0 & \rho_{44}).
\end{eqnarray}

These seven-parameter 2-qubit states have been extensively studied and used in QIT [Quesada et al., 2012]. One of the main reasons for this is that any arbitrary 2-qubit state $\rho_{AB}$ can be mapped to a $\rho^X$ while preserving its main characteristics, e.g. quantum correlations [Mendonça et al., 2014, Hedemann, 2018].

Moreover, the calculation of several quantum correlation quantifiers is simpler for X states. For instance, concurrence was introduced by Wootters [Wootters, 1998] as a bona fide entanglement measure defined as

\begin{equation}\label{eq:DefConcurrencia}
\mathcal{C}\qty(\rho_{AB}) \equiv \max\qty{0,\lambda_1-\lambda_2-\lambda_3-\lambda_4},
\end{equation}

where $\qty{\lambda_i}$ are the decreasing ordered eigenvalues of

\begin{equation}
\mathbf{R}=\sqrt{\sqrt{\rho_{AB}}\,\tilde{\rho_{AB}}\sqrt{\rho_{AB}}},
\end{equation}

with $\tilde{\rho_{AB}} = (\sigma_y\otimes\sigma_y)\rho_{AB}^*(\sigma_y\otimes\sigma_y)$, $\rho_{AB}^*$ the complex conjugate of $\rho_{AB}$, and $\sigma_y$ the corresponding Pauli matrix (<ref>). A direct calculation shows that this entanglement measure takes a much simpler expression for X states [Yu and Eberly, 2007]:

\begin{equation}\label{eq:Concurrencia_X}
\mathcal{C}_X = \frac{1}{2}\,\max \left\{0,\mathcal{C}_1,\mathcal{C}_2\right\}
\end{equation}

\begin{equation*}
\mathcal{C}_1\equiv 2\left(|\rho_{14}|-\sqrt{\rho_{22}\,\rho_{33}}\right)\qc \mathcal{C}_2\equiv 2\left(|\rho_{23}|-\sqrt{\rho_{11}\,\rho_{44}}\right).
\end{equation*}

Regarding quantum discord, a closed analytical expression for a generic 2-qubit state cannot be obtained [Girolami and Adesso, 2011]. For X states, although so far there is no such result, several approximations have been proposed [Ali et al., 2010, Quesada et al., 2012, Lu et al., 2011, Vinjanampathy and Rau, 2012, Li et al., 2011, Huang, 2013, Namkung et al., 2015, Maldonado-Trapp et al., 2015, Rau, 2018].

Moreover, since X states exhibit a $\mathfrak{su}(2)\times\mathfrak{su}(2)\times\mathfrak{u}(1)$ invariance symmetry [Rau, 2009], the definition of an X state can be generalized beyond (<ref>). Also, this algebraic characterization allows for defining general local and global quantum channels that map an X-state into another one.

In a previous paper [28], we determined the LAQC's quantifier for those canonical X states with $\rho^{X}_{22}=\rho^{X}_{33}$ or $\rho^{X}_{11}=\rho^{X}_{44}$, which we labeled as symmetric and anti-symmetric X-states, respectively. Such states have equal-normed Bloch vectors and define an equivalence class that exhibits a symmetry under subsystems exchange $\vb{A}\leftrightarrow\vb{B}$. Our goal in this paper is to focus on the complementary equivalence class. That is those X states whose local Bloch vectors have different magnitude. Since such states no longer exhibit the abovementioned symmetry, we label this equivalence class as asymmetric X states.

We have structured this paper as follows. We start reviewing the procedure for determining the local available quantum correlations quantifier [Mundarain and de Guevara, 2015]. Next, we study its calculation for canonical X states whose local Bloch vectors have a different norm. Then, we present some examples and discuss the effect of local quantum channels on the LAQC’s quantifier. Finally, we summarize the conclusions from our results.

§ LOCAL AVAILABLE QUANTUM CORRELATIONS OF 2-QUBITS

Any density operator $\rho$ can always be written in terms of different local bases. For bipartite qubit systems, given two general bases, $\qty{\ket{k,m}}$ and $\qty{\ket{\mu,\nu}}$, it's density operator $\rho_{AB}$ can be written in either basis as

\begin{equation}\label{eq:rho_Bases}
\rho_{AB}=\sum_{klmn}\rho_{kl}^{mn}\,\dyad{k,m}{l,n}=\sum_{\mu\nu\eta\gamma}R_{\mu\eta}^{\nu\gamma}\,\dyad{\mu,\nu}{ \eta,\gamma}
\end{equation}

where $k, l, m, n,\mu,\nu,\eta,\gamma \in \{0,1\}$. Both bases are equivalent under a local unitary transformations

\begin{equation}\label{eq:BasesTransform}
\ket{\mu,\nu}=\qty(\mathbbm{U}_{\vb{A}}\otimes{U}_{\vb{B}})\ket{k,m}\qc\mathbbm{U}_A,\mathbbm{U}_B\in{U}(2).
\end{equation}

Any such basis can be used as a computational one, whose elements are the eigenvectors of $\sigma_{\vu{u}}\equiv\va*{\sigma}\cdot\vu{u}$, with $\va*{\sigma}$ the vector whose local components are the Pauli matrices

\begin{equation}\label{eq:MatricesPauli}
\sigma_1=\mqty(\pmat{1})\qc\sigma_2=\mqty(\pmat{2})\qc\sigma_3=\mqty(\pmat{3}),
\end{equation}

and $\vu{u}\in\mathbbm{E}^3$ is a generic unitary vector, whose components can be written in terms of the parametrization of the respective $U(2)$ transformation.

For classical states, there exists a local basis for which the density operator $\rho_{AB}^c$ (<ref>) is diagonal. One can define a particular classical state $X_\rho\in\Omega_c$ related to $\rho_{AB}$ as the one induced by a measurement $\mathbbm{M}_o$ that minimizes

\begin{equation}\label{eq:S(rho||X)}
\end{equation}

where $S\qty(\rho||\chi)=-\mathrm{Tr}(\rho\mathrm{log}_2\chi)-S(\rho)$ is the relative entropy and

\begin{align}
\chi_\rho =&\sum_{\mu\nu}R_{\mu\nu}\,\dyad{\mu,\nu)},\\
R_{\mu\nu} =& \expval{\rho_{AB}}{\mu,\nu)}\label{eq:Coeff_Rij}.
\end{align}

The minimization in (<ref>) is equivalent to determining the coefficients $\qty\big{R_{ij}^{opt}}$ when $\chi_\rho=X_\rho$. These coefficients are associated with a new basis, labeled as the optimal computational basis $\qty\big{\ket{i,j}^{opt}}$, and the local available quantum correlations (LAQC) are defined in terms of it.

To determine $\qty\big{\ket{i,j}^{opt}}$, we define a general orthonormal basis for each subsystem:

\begin{equation}\label{eq:BaseOrtonormalGen}
\begin{aligned}
\ket{\mu_0^{(n)}} &= \cos\frac{\theta_n}{2}\ket{0^{(n)}} +\sin\frac{\theta_n}{2}\exp^{i\phi_n}\ket{1^{(n)}},\\
\ket{\mu_1^{(n)}} &= -\sin\frac{\theta_n}{2}\ket{0^{(n)}} +\cos\frac{\theta_n}{2}\exp^{i\phi_n}\ket{1^{(n)}},
\end{aligned}
\end{equation}

where $n=1$ denotes subsystem A and $n=2$, subsystem B. Such basis is the result of applying

\begin{equation}\label{eq:U(2)-Gen}
\mathbbm{U}_n = \mqty(\cos\frac{\theta_n}{2} & -\sin\frac{\theta_n}{2}  \\
\sin\frac{\theta_n}{2}\exp^{i\phi_n} & \cos\frac{\theta_n}{2}\exp^{i\phi_n})\;\;\in\;U(2)
\end{equation}

to the original computational basis of each subsystem. The classical correlations quantifier is given by  [Mundarain and de Guevara, 2015, Modi et al., 2010]

\begin{equation}\label{eq:CorrClasicas}
\mathcal{C}(\rho) = I(X_\rho).
\end{equation}

The mutual information (<ref>) may be written as

\begin{equation}\label{eq:Info_Mutua-Prob}
\begin{split}
I\qty(\rho_{AB}) =& \sum_{i,j} P_{\theta,\phi}(i_A,j_B)\\
& \qq{}\times\log_2\qty[\frac{P_{\theta,\phi}(i_A,j_B)}{P_{\theta_{A},\phi_{A}}(i_A)P_{\theta_B,\phi_B}(j_B)}],
\end{split}
\end{equation}

\begin{equation}\label{eq:P_theta_phi}
P_{\theta,\phi}(i_A,j_B) = \expval{\rho_{AB}}{\mu_i^{(1)},\mu_j^{(2)}}
\end{equation}

are the probability distributions corresponding to $\rho_{AB}$ and

\begin{align}
P_{\theta_{A},\phi_{A}}(i_A) &= \expval{\rho_A}{\mu_i^{(1)}},\\
P_{\theta_{B},\phi_{B}}(j_B) &= \expval{\rho_B}{\mu_i^{(2)}},
\end{align}

are the ones corresponding to its reduced operators $\rho_A$ and $\rho_B$. The minimization of the relative entropy (<ref>) required to define $X_{\rho}$ yields a minima for the classical correlations quantifier defined in (<ref>). It is straightforward to realize that these probability distributions are directly related to the $\qty\big{R_{ij}^{opt}}$ coefficients when $\qty{\ket{\mu_i^{(1)},\mu_j^{(2)}}}$ is the optimal computational basis.

With the optimal computational basis determined, the state $\rho_{AB}$ is rewritten and the complementary basis defined as

\begin{eqnarray}\label{eq:u0-u1}
\ket{u_0^{(m)}}&=&\frac{1}{\sqrt{2}}\qty(\ket{0^{(m)}}_{opt}+\exp^{i\Phi_m}\ket{1^{(m)}}_{opt}),\nonumber\\
\ket{u_1^{(m)}}&=&\frac{1}{\sqrt{2}}\qty(\ket{0^{(m)}}_{opt}-\exp^{i\Phi_m}\ket{1^{(m)}}_{opt}).
\end{eqnarray}

This basis results from applying the transformation (<ref>) with $\theta_n=\frac{\pi}{2}$ and a new angle $\phi_n\rightarrow\Phi_n$ via (<ref>). The corresponding probability distributions $P(i_A,j_B,\Phi)$ and marginal probability distributions $P^{(n)}(i_n,\Phi)$ analogue to (<ref>) and (<ref>) are determined. The maximization of $I\qty(\Phi_1,\Phi_2)$ (<ref>) corresponds to the LAQC quantifier:

\begin{equation}\label{eq:LAQC-quant}
\mathcal{L}(\rho_{AB}) \equiv \max_{\qty{\Phi_1,\Phi_2}} I\qty(\Phi_1,\Phi_2).
\end{equation}

§ LAQC OF ASYMMETRIC X STATES

We start by introducing what are often called canonical X states, which are X states whose density matrix has only real entries. For such states, we have that

\begin{equation}
\rho_{14}=\rho_{41} \qq{and} \rho_{23}=\rho_{32}.
\end{equation}

Therefore, we can rewrite (<ref>) as

\begin{eqnarray}\label{eq:Estados_X-abcdzw}
\rho^X =\mqty(
a & 0 & 0 & w \\
0 & b & z & 0 \\
0 & z & c& 0 \\
w& 0 & 0 & d),
\end{eqnarray}

where $a,b,c,w,z\,\in\,\mathbbm{R}$ are the five independent real parameters, and

\begin{align}
\end{align}

are the constraints on these parameters so $\rho^X$ is a well-behaved density matrix, i.e. has $\Tr\qty(\rho^X)=1$ and is semi-positive definite.

By introducing

\begin{equation}
\sigma_0 =\mathbbm{1}_2 = \mqty(\imat{2}),
\end{equation}

a Hermitian base for the space of linear operators acting on the qubit Hilbert space is obtained, which allows to write any qubit state as

\begin{equation}\label{eq:Bloch-Qubit}
\rho=\sum_{i}c_{i}\dyad{i}=\frac{1}{2}\qty(\mathbbm{1}_2+\va*{\tau}\cdot\va*{\sigma}),
\end{equation}

where $\va*{\tau}$ is the Bloch vector of $\rho$. This representation can be readily extended to 2-qubit systems, often referred to as the Fano form or Fano-Bloch representation [Fano, 1983],

\begin{equation}\label{eq:2qubits-Bloch}
\begin{split}
\rho = \frac{1}{4}\Bigg[ \mathbbm{1}_4 + \qty(\va{x}\cdot\va*{\sigma})\otimes\mathbbm{1}_2& + \mathbbm{1}_2\otimes \qty(\va{y}\cdot\va*{\sigma})\\ &+\va*{\sigma}\cdot\mathbbm{T}\cdot\va*{\sigma})\Bigg],
\end{split}
\end{equation}

where $\va{x}$ and $\va{y}$ are the local Bloch vectors of subsystems A and B, respectively, and $\mathbbm{T}$ is the correlations tensor. The components of the local Bloch vectors $x_i$ and $y_i$ as well as the ones of the correlations tensor $T_{ij}$ are given by

\begin{align}
\end{align}

For canonical X states (<ref>), we have that

\begin{equation}\label{eq:estadosX-Bloch}
\begin{split}
\rho^X = \frac{1}{4}\Bigg( \mathbbm{1}_4 + x_3\,\sigma_3\otimes\mathbbm{1}_2& + \mathbbm{1}_2\otimes y_3\,\sigma_3\\ &+\sum_{n=1}^3 T_{n}\sigma_n\otimes\sigma_n\Bigg).
\end{split}
\end{equation}

These Bloch representation is an alternative parametrization to (<ref>), with $x_3$, $y_3$, and $T_n$ related to the previous $a,b,c,d,z,$ and $w$ by

\begin{equation}\label{eq:Bloch_Xstates-abcdwz}
\begin{aligned}
\end{aligned}\qq{}
\begin{aligned}
\end{aligned}
\end{equation}

The family of 2-qubit X-states can be divided in two categories, depending on whether their local Bloch vectors have equal norm or not. The symmetrical and anti-symmetrical category were already analyzed in [28]. We will focus our attention on X states with  $\abs{x_3}\neq{}\abs{y_3}$, which is equivalent to requiring that $a\neq{}d$ and $b\neq{}c$. We label this set as asymmetric X states, $\rho^{X_{ns}}$. Since such states are not invariant under subsystem exchange $\vb{A}\leftrightarrow\vb{B}$, all four parameters $\theta_1$, $\theta_2$, $\phi_1$, and $\phi_2$ in (<ref>) have to be taken into account. A direct computation of the $R_{ij}$ coefficients (<ref>) for the general computational basis (<ref>) for $\rho^X$ (<ref>) leads to

\begin{align}
%        \\
%    &+\frac{1}{2}\qty[w\cos\qty(\phi_1+\phi_2)+z\cos\qty(\phi_1-\phi_2)]\sin\theta_1\sin\theta_2\nonumber
%        \\
%    &+\frac{1}{2}(c-d)\qty(1+\cos\theta_2)+d\nonumber
%        \\
=\,& \frac{1}{4}\Big[1+x_3\cos\theta_1 +y_3\cos\theta_2+T_3\cos\theta_1\cos\theta_2\nonumber
\\
\\
\\
%        \\
%   &-\frac{1}{2}\qty[w\cos\qty(\phi_1+\phi_2)+z\cos\qty(\phi_1-\phi_2)]\sin\theta_1\sin\theta_2\nonumber
%        \\
%    &-\frac{1}{2}(c-d)\qty(1+\cos\theta_2)+c\nonumber
%        \\
=\,& \frac{1}{4}\Big[1+x_3\cos\theta_1 -y_3\cos\theta_2-T_3\cos\theta_1\cos\theta_2 \nonumber
\\
\\
\\
%        \\
%    &-\frac{1}{2}\qty[w\cos\qty(\phi_1+\phi_2)+z\cos\qty(\phi_1-\phi_2)]\sin\theta_1\sin\theta_2\nonumber
%        \\
%    &-\frac{1}{2}(c-d)\qty(1+\cos\theta_2)+b\nonumber
%        \\
=\,& \frac{1}{4}\Big[1-x_3\cos\theta_1 +y_3\cos\theta_2-T_3\cos\theta_1\cos\theta_2 \nonumber
\\
\\
\\
%        \\
%    &+\frac{1}{2}\qty[w\cos\qty(\phi_1+\phi_2)+z\cos\qty(\phi_1-\phi_2)]\sin\theta_1\sin\theta_2\nonumber
%    \\
%    &-\frac{1}{2}(a-b)\qty(1+\cos\theta_2)+a\nonumber
%        \\
=\,& \frac{1}{4}\Big[1-x_3\cos\theta_1 -y_3\cos\theta_2+T_3\cos\theta_1\cos\theta_2\nonumber
\\
\\
\end{align}

For the reduced matrices we have that

\begin{align}
\\
\end{align}

It is easily verified that for

\begin{eqnarray}\label{eq:Theta-OptimalCompBasis}
\theta_i =(2k+1)\frac{\pi}{2} \qq{and} \theta_j=\tilde{k}\pi,
\end{eqnarray}

with $k=0,1$, $\tilde{k}=0,1,2$, and $i,j=1,2$,

we have that

\begin{equation}
\frac{R_{ij}}{R_{i}^{(\vb{A})}R_{j}^{(\vb{B})}}=1.
\end{equation}

Therefore, by choosing $\theta_1$ and $\theta_2$ so that (<ref>) is satisfied, with $\phi_1$ and $\phi_2$ arbitrary, we obtain that the classical correlations quantifier $\mathcal{C}\qty(\rho^X)$ (<ref>) is

\begin{equation}\label{eq:CorrClass-Xstates-Asymmetric}
\mathcal{C}\qty(\rho^{X_{ns}}) = 0.
\end{equation}

Since such angles correspond to possible optimal computational bases, we have to rewrite $\rho^{X_{ns}}$ (<ref>) in terms of them and then determine the corresponding probability distributions $P_{\theta_1,\theta_2}(i_A,j_B,\phi_1,\phi_2,\Phi_1,\Phi_2)$. There are two distinct possibilities for $\theta_1$ and $\theta_2$ for the computational basis,

\begin{equation}\label{eq:thetas-cases}
\begin{cases}
\theta_1&=0\qc \theta_2=\frac{\pi}{2},\\
\theta_1&=\frac{\pi}{2}\qc \theta_2=0.
\end{cases}
\end{equation}

All $\theta_i =(2k+1)\frac{\pi}{2}$ and $\theta_j=\hat{k}\pi$ lead to the same expressions of $P(i_A,j_B,\Phi_1,\Phi_2)$ for one of the above cases.

For $\theta_1=\tilde{k}\pi$ and $\theta_2 =(2k+1)\frac{\pi}{2}$ $(k=0,1, \tilde{k}=0,1,2)$, we have that

\begin{align}
P_{\qty(0,\frac{\pi}{2})}(0,0,\Psi_1,\Psi_2) &= \frac{1}{4}\big(1+y_3\cos\Psi_2\nonumber\\
&\qq{ }+T_2\sin\Psi_1\sin\Psi_2\big),
\\
P_{\qty(0,\frac{\pi}{2})}(0,1,\Psi_1,\Psi_2) &= \frac{1}{4}\big(1-y_3\cos\Psi_2\nonumber\\
&\qq{ }-T_2\sin\Psi_1\sin\Psi_2\big),
\\
P_{\qty(0,\frac{\pi}{2})}(1,0,\Psi_1,\Psi_2) &= \frac{1}{4}\big(1+y_3\cos\Psi_2\nonumber\\
&\qq{ }-T_2\sin\Psi_1\sin\Psi_2\big),
\\
P_{\qty(0,\frac{\pi}{2})}(1,1,\Psi_1,\Psi_2) &= \frac{1}{4}\big(1-y_3\cos\Psi_2\nonumber\\
&\qq{ }+T_2\sin\Psi_1\sin\Psi_2\big),
\end{align}

\begin{eqnarray}
\Psi_i\equiv\Phi_i-\phi_1\;\;\; (i=1,2)\qc \Psi_i\in\qty[-2\pi,2\pi].
\end{eqnarray}

The corresponding marginal probabilities are given by

\begin{align}
P_{\qty(0,\frac{\pi}{2})}^{(\vb{A})}(0,\Psi_1,\Psi_2) &=\frac{1}{2} = P_{\qty(0,\frac{\pi}{2})}^{(\vb{A})}(1,\Psi_1,\Psi_2),\\
P_{\qty(0,\frac{\pi}{2})}^{(\vb{B})}(0,\Psi_1,\Psi_2) &=\frac{1}{2}\qty(1+y_3\cos\Psi_2),\\ P_{\qty(0,\frac{\pi}{2})}^{(\vb{B})}(1,\Psi_1,\Psi_2)&=\frac{1}{2}\qty(1-y_3\cos\Psi_2).
\end{align}

Similar expressions are obtained when $\theta_1 =(2k+1)\frac{\pi}{2}$ and $\theta_2=\tilde{k}\pi$ $(k=0,1)$, with $x_3$ instead of the previous $y_3$.

\begin{align}
P_{\qty(\frac{\pi}{2},0)}(0,0,\Psi_1,\Psi_2) &= \frac{1}{4}\big(1+x_3\cos\Psi_1\nonumber\\
&\qq{ }+T_2\sin\Psi_1\sin\Psi_2\big),
\\
P_{\qty(\frac{\pi}{2},0)}(0,1,\Psi_1,\Psi_2) &= \frac{1}{4}\big(1+x_3\cos\Psi_1\nonumber\\
&\qq{ }-T_2\sin\Psi_1\sin\Psi_2\big),
\\
P_{\qty(\frac{\pi}{2},0)}(1,0,\Psi_1,\Psi_2) &= \frac{1}{4}\big(1-x_3\cos\Psi_1\nonumber\\
&\qq{ }-T_2\sin\Psi_1\sin\Psi_2\big),
\\
P_{\qty(\frac{\pi}{2},0)}(1,1,\Psi_1,\Psi_2) &= \frac{1}{4}\big(1-x_3\cos\Psi_1\nonumber\\
&\qq{ }+T_2\sin\Psi_1\sin\Psi_2\big),
\end{align}

and the corresponding marginal probabilities

\begin{align}
P_{\qty(\frac{\pi}{2},0)}^{(\vb{A})}(0,\Psi_1,\Psi_2) &=\frac{1}{2}\qty(1+x_3\cos\Psi_1),\\ P_{\qty(\frac{\pi}{2},0)}^{(\vb{A})}(1,\Psi_1,\Psi_2)&= \frac{1}{2}\qty(1-x_3\cos\Psi_1),\\
P_{\qty(\frac{\pi}{2},0)}^{(\vb{B})}(0,\Psi_1,\Psi_2) &=\frac{1}{2}= P_{\qty(\frac{\pi}{2},0)}^{(\vb{B})}(1,\Psi_1,\Psi_2).
\end{align}

The maximization required in (<ref>) is readily achieved for $\Psi_i=\pm(2k+1)\frac{\pi}{2}$, with $i=1,2$, and $k=0,1$. Therefore, we  have a single expression for the LAQC quantifier of asymmetric X states:

\begin{equation}\label{eq:LAQC-Asymm-Xstates}
\begin{aligned}
\mathcal{L}\qty(\rho^{X_{ns}}) =\,& \frac{1+T_2}{2}\log_2\qty(1+T_2)\\
&\qq{}+ \frac{1-T_2}{2}\log_2\qty(1-T_2).
\end{aligned}
\end{equation}

Notice that this expression corresponds to the function $g_2$ (<ref>) defined in [28].

§ LAQC AND LOCAL QUANTUM CHANNELS: AMPLITUDE DAMPING

The amplitude damping (AD) channel [Nielsen and Chuang, 2010] describes the process of energy dissipation (e. g. spontaneous emission) into the environment. The Kraus operators [Kraus, 1971] for this quantum operation on one qubit are given by

\begin{equation}\label{eq:Kraus_AD}
\vb{E}^{(AD)}_0 = \mqty(1 & 0\\ 0 & \sqrt{1-p\,})\,\qc\quad \vb{E}^{(AD)}_1 = \mqty(0 & \sqrt{p\,}\\ 0 & 0),
\end{equation}

where $p$ is the channel's parameter, which can be thought of as the probability for the transition $\ket{1} \rightarrow \ket{0}$.

We are interested in applying the AD channel to a bipartite qubit system. Within the operator-sum representation, the extension is readily achieved via

\begin{equation}\label{eq:Kraus_Interaccion}
\rho\longrightarrow \rho'=\sum_{i,j} \left(\vb{K}^{(A)}_i\otimes\vb{K}^{(B)}_j\right)\rho\left(\vb{K}^{(A)}_i\otimes\vb{K}^{(B)}_j\right)^\dagger
\end{equation}

where $\vb{K}^{(I)}_i$ is the $i$-th Kraus operator of a given quantum channel acting the $I$ subsystem. Since we are interested in a local quantum operation, without loss of generality we focus in what follows on a quantum channel acting only on subsystem B. Therefore, we have that

\begin{equation}\label{eq:Kraus_AD-2qubit}
\rho'=\sum_{i} \left(\mathbbm{1}\otimes\vb{E}_i\right)\rho\left(\mathbbm{1}\otimes\vb{E}_i\right)^\dagger.
\end{equation}

In [Rau, 2009], Rau established the algebraic characterization of 2-qubit X states in terms of the $\mathfrak{su}(2)\times{}\mathfrak{su}(2)\times{}\mathfrak{u}(1)$ subalgebra of $\mathfrak{su}(4)$. Operators belonging to it map a given X state into another one, and the above-given Kraus operators of the amplitude damping channel (<ref>) belong to such subalgebra.

§.§ Werner states

Werner states, $\rho_w$, can be written as:

\begin{equation}\label{eq:rhoWerner}
\rho_w= z\dyad{\Psi^-} +\frac{1-z}{4}\,\mathbbm{1}_4,
\end{equation}

where $z\in[0,1]$ and $\ket{\Psi^-}=\frac{1}{\sqrt{2}}\qty(\ket{01}-\ket{10})$ is the singlet, one of the four maximally entangled 2-qubit states known as Bell states. In previous references [Albrecht Q. et al., 2018, 28], we analyzed the behavior of LAQC for such states under Markovian decoherence, considering global Depolarizing, Phase Damping, and Amplitude Damping channels. In each case we considered the same type of quantum operation acting locally on both subsystems with the same interaction parameter.

The state $\rho_w^{(AD)}$ resulting from applying the above Kraus operators via (<ref>) is an asymmetric X state with non-null Bloch parameters given by

\begin{equation}\label{eq:WernerAD-Bloch}
\rho_w^{(AD)}:\;\;\begin{cases}
y_3 = p,\\
T_1 = T_2 = -\sqrt{1-p\,}\,z, \\
T_3 = -\qty(1-p)\,z.
\end{cases}
\end{equation}

From eq. (<ref>), it is direct to verify that

\begin{align}
\mathcal{L}\qty(\rho_w^{(AD)}) =\,&\frac{1+\sqrt{1-p\,}\,z}{2}\log_2\qty[1+\sqrt{1-p\,}\,z]\\ &\,+\frac{1-\sqrt{1-p\,}\,z}{2}\log_2\qty[1-\sqrt{1-p\,}\,z].\nonumber
\end{align}

Using (<ref>), the concurrence is given by

\begin{equation}
\mathcal{C}_{w^{AD}} = \max\qty{0,C_1},
\end{equation}

\begin{align*}
&\;\;\;- \sqrt{(1-p)(1-z)\qty\big[1+p-z(1-p)]\,}\bigg),
\end{align*}

and its QD (<ref>), computed using the algorithm proposed by Quesada et al. [Quesada et al., 2012], is given by

\begin{equation}
\begin{aligned}
D_B\qty(\rho_w^{(AD)}) =\,&\frac{3(1-z)}{4}\log_2\qty[\frac{1-z}{4}]\\
&\;-\frac{1-p}{2}\log_2\qty[1-p]\\ &\;-\frac{1+p}{2}\log_2\qty[1+p] +N_1,
\end{aligned}
\end{equation}

\begin{align*}
N_1 =\,& 1 -\frac{1}{2}\log_2\qty[(p-1) z^2+1]\\ &\qq{ }+\frac{z\sqrt{1-p\,}}{\ln(2)}\, \mathrm{arctanh}\qty(z\sqrt{1-p\,}).
\end{align*}

LAQC (above), QD (middle), and Concurrence (below) for a Werner state under local amplitude damping.

In Figure <ref> we present the graphical behavior of LAQC, QD, and concurrence. As was the case with other quantum channels, in particular global AD, Werner states again exhibits `Entanglement Sudden Death' [Yu and Eberly, 2009]. For LAQC and QD, these quantum correlations only vanish asymptotically.

As was observed in other cases, QD is larger than LAQC, as can be seen in Figure <ref>, where we show the surface

\begin{equation}\label{eq:S(z,p)}
\mathcal{S}(z,p)=D_A\qty(\rho_w^{(AD)}) -\mathcal{L}\qty(\rho_w^{(AD)}).
\end{equation}

We have that $\mathcal{S}(z,p)\geq0$ for all $z\in[0,1]$ and $p\in[0,1]$.

Surface $\mathcal{S}(z,p)$ (<ref>) resulting from the difference between QD and LAQC for a Werner state under local amplitude damping.

§.§ General canonical X states

In [Hu et al., 2012], Hu et al. derived the necessary and sufficient conditions for the local creation of quantum correlations. The amplitude damping channel is among the quantum operations that satisfies those conditions, and it has been established that it can create quantum discord [Abad et al., 2012, Xu, 2012, Gwóźdź and Jakóbczyk, 2013]. Whether LAQC can be locally created or not is still an open question and our present results allows us to address this questions for the amplitude damping channel.

The state $\rho'$ resulting from applying this quantum operation to a general canonical X state with Bloch parameters $\qty{x_3,y_3,T_1,T_2,T_3}$ via (<ref>) has the following non-null Bloch parameters:

\begin{equation}\label{eq:Xstate_AD-Bloch}
\begin{aligned}
&x_3' = x_3, &T_1'&=\sqrt{1-p\,} \,T_1,\\
&y_3'= (1-p)y_3 +p\qc&T_2'&=\sqrt{1-p\,} \,T_2,\\
&&T_3'&=(1-p)T_3 +px_3.
\end{aligned}
\end{equation}

Since the LAQC quantifier (<ref>) of $\rho'$ is a function of $T_2'$ only, then it is straightforward to verify

\begin{equation}\label{eq:rhoX_AD-inequality}
\mathcal{L}\qty(\rho^{X})\geq\mathcal{L}\qty\big(\rho') \qq{for all} p\in[0,1],
\end{equation}

whether $\rho^{X}$ is symmetric, antisymetric, or asymmetric.

If the initial state is asymmetric, the above inequality is trivial to verify. On the other hand, if $\rho^{X}$ is symmetric or anti-symmetric, we have to resort to the LAQC quantifier derived in [28]:

\begin{equation}\label{eq:LAQC-EstadosX-s_as}
\mathcal{L}\qty(\rho^{X_{\smqty{s\\ as}}}) = \max\qty(g_j,g_{\pm})
\end{equation}

\begin{align}
g_j\equiv\,&  \frac{1+T_j}{2}\log_2\qty(1+T_j) + \frac{1-T_j}{2}\log_2\qty(1-T_j),\label{eq:gi(Ti)}\\
g_{\pm}\equiv\,& \frac{1\pm{}T_3 + 2 x_3}{4}\log_2\qty[\frac{1\pm{}T_3 + 2 x_3}{(1+x_3)^2}]\nonumber\\
&\;\;+\frac{1\pm{}T_3 - 2 x_3}{4}\log_2\qty[\frac{1\pm{}T_3 - 2 x_3}{(1-x_3)^2}]\nonumber\\
&\;\;+ \frac{1-T_3}{2}\log_2\qty(\frac{1-T_3}{1-x_3^2})\label{eq:g+},
\end{align}

for $j=1,2$, and with $g_{+}$ for the symmetric case and $g_{-}$ for the anti-symmetric one. From eq. (<ref>), we have that

\begin{equation}
\mathcal{L}\qty(\rho^{X_{\smqty{s\\ as}}})\geq{}g_2\geq{}g_2',
\end{equation}

where $g_2'$ refers to the function (<ref>) with $T_2'$ (<ref>). Therefore, it is straightforward to realize that eq. (<ref>) holds. Thus, the action of a local amplitude damping channel cannot create LAQC.

§ CONCLUSIONS

We studied local available quantum correlations (LAQC) [Mundarain and de Guevara, 2015] for 2-qubit asymmetric X states. In doing so, we obtained an exact and unique analytical expression of its quantifier. Alongside our previous results for symmetric and anti-symmetric X states [28], the present results complete the study of the LAQC for the canonical 2-qubit X states family.

We included the analysis of the action of a local amplitude damping channel. It has been established that such a channel can create quantum discord, but this is not the case for LAQC. This impossibility hints toward its monotonicity under LOCC operations.

§ ACKNOWLEDGMENTS

This work was partially funded by the 2020 BrainGain Venezuela grant awarded to H. Albrecht by the Physics without Frontiers program of the ICTP. The authors are thankful to Gloria Buendia for her comments and discussions. Albrecht and Bellorin would also like to thank the support given by the research group GID-30, Teoría de Campos y Óptica Cuántica, at the Universidad Simón Bolívar, Venezuela.

§ QUANTUM DISCORD OF 2-QUBIT X STATES

In [Lu et al., 2011] Lu et al. introduced the so-called maximal-correlation-direction measurements (MCDM) which were later used by Quesada et al. [Quesada et al., 2012] to develop his algorithm to approximate the quantum discord of X states as

\begin{equation}
D_B\qty(\rho^X) \approx S\qty(\rho_B) - S\qty(\rho^X) + \min\qty{N_1, N_2},
\end{equation}

where $N_1$ (<ref>) and $N_2$ (<ref>) are the optimization functions derived using the MCDM. In this appendix we briefly summarize their procedure and how to determine these functions.

The starting point is to introduce a generic computational basis as the one given in (<ref>) for subsystem $\vb{B}$ and defining the corresponding projectors

\begin{equation}
\Pi^{(2)}_i \equiv \dyad{\mu_i^{(2)}}\qc \Pi^{(\vb{AB})}_i=\mathbbm{1}^{(1)}_2\otimes\Pi^{(2)}_i.
\end{equation}

The state $\rho^{X}$ is then rewritten as

\begin{equation}
\begin{aligned}
\rho^X_{A|\Pi^B_i} &= \frac{1}{p_i}\, \Pi^{(\vb{AB})}_i\rho^X \Pi^{(\vb{AB})}_i,\\
p_i &= \Tr[\Pi^{(\vb{AB})}_i\rho^X \Pi^{(\vb{AB})}_i],
\end{aligned}
\end{equation}

and then the conditional entropy

\begin{equation}\label{CondEntr}
S\qty(A|\qty{\Pi^B_i}) = \sum_i p_i S\qty(\rho^X_{A|\Pi^B_i})
\end{equation}

is determined. As is done in [Liao et al., 2015], the critical points of this equation that are independent of the density matrix's elements are studied. With them, the angles $\theta_2$ and $\phi_2$ corresponding to the minima are determined and the following optimization functions are found:

\begin{align}
N_1 =& 1 -\frac{1+\sqrt{x_3^2+T_1^2\,}}{2} \log_2 \qty(1+\sqrt{x_3^2+T_1^2\,})\nonumber\\
&-\frac{1-\sqrt{x_3^2+T_1^2\,}}{2}\log_2 \qty(1-\sqrt{x_3^2+T_1^2\,}),\label{eq:N1-QD-Quesada}\\
N_2 =\,& - a \log_2 \qty[\frac{a}{a+c}] - b \log_2 \qty[\frac{b}{b+d}] \nonumber\\
&\;\;- c \log_2 \qty[\frac{c}{a+c}] - d \log_2 \qty[\frac{d}{b+d}],\label{eq:N2-QD-Quesada}
\end{align}

where $a,b,c$, and $d$ (<ref>) are given in terms of $x_3$, $y_3$, and $T_3$ by

\begin{align}
a & = \frac{1}{4}\qty(1+x_3+y_3+T_3),\\
b& = \frac{1}{4}\qty(1+x_3-y_3-T_3),\\
c& = \frac{1}{4}\qty(1-x_3+y_3-T_3),\\
d& = \frac{1}{4}\qty(1-x_3-y_3+T_3).
\end{align}

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