pnasmathematics Kim Many sub-Neptune exoplanets are believed to have thick
atmospheres dominated by hydrogen, surrounding the interiors with mainly
silicates and oxides, assuming no reaction between them. From high-pressure
experiments on molten magnesium oxide in a dense hydrogen liquid, we found
that the reaction between hydrogen and the oxide melt results in the formation
of hydride and water under the pressure-temperature conditions expected for
the atmosphere-interior boundary of sub-Neptunes. At higher pressures, we
found enhanced chemical mixing between magnesium and hydrogen. The experiments
suggest that sub-Neptune exoplanets likely have a mineralogy different from
that of rocky planets, and the chemical reaction between the atmosphere and
interior can impact the size of sub-Neptunes and the compositions of their
atmospheres. T.K. performed the experiments, analyzed data, and wrote the
manuscript. S.-H.S conceptualised and supervised the project, acquired
funding, designed and performed experiments, analyzed data, and wrote the
manuscript. S.C., Y.-J.R., S.Y., and V.B.P. provided resources for the sample
analysis and supervised the experiments. All authors reviewed the manuscript.
The authors declare no competing interests. *To whom correspondence should be
addressed. E-mail<EMAIL_ADDRESS>and<EMAIL_ADDRESS>
# Stability of Hydrides in Sub-Neptune Exoplanets with Thick Hydrogen-Rich
Atmospheres

Taehyun Kim  School of Earth and Space Exploration, Arizona State University,
Tempe, AZ 85287, USA.  Corresponding authors  Xuehui Wei  School of Earth and
Space Exploration, Arizona State University, Tempe, AZ 85287, USA.  Stella
Chariton  Center for Advanced Radiation Sources, University of Chicago,
Argonne, IL 60439, USA.  Vitali B. Prakapenka  Center for Advanced Radiation
Sources, University of Chicago, Argonne, IL 60439, USA.  Young-Jay Ryu  Center
for Advanced Radiation Sources, University of Chicago, Argonne, IL 60439, USA.
Shize Yang  Eyring Materials Center, Arizona State University, Tempe, AZ
85287, USA.  Sang-Heon Shim  School of Earth and Space Exploration, Arizona
State University, Tempe, AZ 85287, USA.  Corresponding authors

###### Abstract

Many sub-Neptune exoplanets have been believed to be composed of a thick
hydrogen-dominated atmosphere and a high-temperature heavier-element-dominant
core. From an assumption that there is no chemical reaction between hydrogen
and silicates/metals at the atmosphere-interior boundary, the cores of sub-
Neptunes have been modeled with molten silicates and metals (magma) in
previous studies. In large sub-Neptunes, pressure at the atmosphere-magma
boundary can reach tens of gigapascals where hydrogen is a dense liquid. A
recent experiment showed that hydrogen can induce the reduction of Fe2+ in
(Mg,Fe)O to Fe0 metal at the pressure-temperature conditions relevant to the
atmosphere-interior boundary. However, it is unclear if Mg, one of the
abundant heavy elements in the planetary interiors, remains oxidized or can be
reduced by H. Our experiments in the laser-heated diamond-anvil cell found
that heating of MgO + Fe to 3500–4900 K (close to or above their melting
temperatures) in a H medium leads to the formation of Mg2FeH6 and H2O at 8–13
GPa. At 26–29 GPa, the behavior of the system changes, and Mg–H in an H fluid
and H2O were detected with separate FeHx. The observations indicate the
dissociation of the Mg–O bond by H and subsequent production of hydride and
water. Therefore, the atmosphere-magma interaction can lead to a fundamentally
different mineralogy for sub-Neptune exoplanets compared with rocky planets.
The change in the chemical reaction at the higher pressures can also affect
the size demographics (i.e., "radius cliff") and the atmosphere chemistry of
sub-Neptune exoplanets.

###### keywords:

Hydride $|$ Sub-Neptune exoplanets $|$ Hydrogen $|$ Magma

###### doi:

www.pnas.org/cgi/doi/10.1073/pnas.2309786120

This manuscript was compiled on

Thick hydrogen-rich atmosphere is likely common in early rocky planets and
among large low-density planets. In the early stage of planet formation, the
accretion of planetesimals forms a core comprised of Mg-rich silicates/oxides
and Fe-rich metal alloys (1). As the core becomes sufficiently large
($>$0.5–0.75 $M(E)$, Earth mass unit), it can accrete the nebular gas (mostly
hydrogen), and therefore, planets can develop hydrogen-rich primary
atmospheres (2, 3). Earth and Venus in our solar system and the larger
versions of them, such as super-Earth exoplanets ($\leq$1.7–1.8 $R(E)$, Earth
radius unit), have sufficient core masses to retain a hydrogen-rich atmosphere
at least early in their geological history (4). In such a setting, chemical
processes at the boundary between the atmosphere and magma ocean can impact
the properties and dynamics of the planets.

The exoplanet surveys for the past 2–3 decades have accumulated mass-radius
data for a large number of exoplanets. Such data have revealed previously
unknown features in the demographics of the exoplanets, providing a new
opportunity to advance our knowledge on their origin and evolution. One of the
significant discoveries from the surveys so far is that planets with sizes
between Earth and Neptune are common in our galaxy. Among these planets, sub-
Neptunes at 1.8–3 $R(E)$ have lower densities relative to rocky planets,
suggesting a significant mass of atmosphere for sub-Neptunes. Sub-Neptunes
likely have sufficient core masses to retain hydrogen-rich atmospheres (4).
Therefore, models involving the rocky/metallic core with the hydrogen-rich
atmosphere have been widely accepted for sub-Neptunes (5). In those planets,
particularly larger ones, the atmosphere is likely thick enough that the
pressure reaches tens of GPa at the boundary between the atmosphere and the
interior. A thick atmosphere acts as a thermal blanket for the core to
maintain a molten state (i.e., magma) for billions of years (6, 7, 8).
Therefore, the physical and chemical interaction between dense hydrogen and
molten silicates/oxides at high pressure-temperature ($P$-$T$) is the key to
understand the interiors and atmospheres of sub-Neptunes.

Until recently, however, most models for these planets assumed that dense
hydrogen and magma do not react with each other. Some recent models have
considered the ingassing of hydrogen in magma in the form of H2 (9) and redox
reactions involving Fe and Si (10) based on low pressure data. Recent models
for sub-Neptunes have shown that the reactions between the hydrogen-rich
atmosphere and magma can produce water and alter the composition of the
atmosphere and interior (10, 11). For large low-density planets, the endogenic
production in the interior can be an important source for water (11). It was
proposed that such endogenic production may be important for Earth’s water as
well (12). It is also possible that potential hydrogen-magma interaction may
impact the exoplanet demographics. For example, the occurrence of sub-Neptunes
show a precipitous drop at $3R(E)$, i.e., known as the radius cliff (13). It
was proposed that high pressure expected under the thick atmosphere of a
${\sim}3R(E)$ sub-Neptune may promote efficient physical ingassing of hydrogen
to the magma, preventing further growth of the planet beyond the radius cliff
(14).

Horn et al. (11) showed experimentally that hydrogen can reduce Fe2+ in
(Mg,Fe)O to Fe metal when the material is partially molten up to 4000 K at
20–40 GPa. The reduced Fe metal subsequently reacts with H and forms Fe-H
alloy. The reaction releases O atoms to an H medium, which then react with H
to form H2O. Therefore, the reaction allows for H ingassing as an alloy
component in Fe metal (H0) and hydroxyl ((OH)- or H2O) dissolved in magma.
Shinozaki et al. (15) showed the dissolution of SiO2 in an H2 liquid at 2–3
GPa and 1500–1700 K, forming Si–H. The reaction also produces H2O. Despite
these important developments, the stability of MgO, one of the dominant
components of planetary silicates and oxides, in an H-rich environment is
still not well known for the range of $P$-$T$ conditions expected for the
atmosphere-interior boundary of sub-Neptunes (0.1–30 GPa and $\geq$2000 K)
(16, 8). Therefore, the persistence of MgO after heating documented in the
experiments could be because of the kinetics of the solid-liquid reaction
instead of the stability of MgO. In this study, we report experiments on MgO +
Fe metal mixtures in a pure hydrogen medium at 3500–4900 K and 8–29 GPa,
conditions close to the melting of MgO (17) (Fig. 1). We will discuss
implications of the experimental results for sub-Neptune exoplanets.

## Methods

Thin foils of an MgO + Fe metal mixture were loaded in diamond-anvil cells
(DACs). The foils were propped by MgO grains, as spacers, and pure H2 gas was
loaded in the DACs (Fig. S1 and Text 1). Experiments were conducted at
beamline 13-IDD of GeoSoilEnviroCARS (GSECARS) at the Advanced Photon Source
(APS). A double-sided laser-heating setup was combined with X-ray diffraction
for measurements at high $P$-$T$ (18). Because of the diamond embrittlement by
H, we conducted pulsed laser heating (19, 20, 21) (Text 2) to reach
temperatures close to the melting of MgO. While the pulse heating method
enables us to reach temperatures above 2000 K for the H-rich sample, the
heating duration is 0.1–0.2 second, which is not sufficient for measuring
X-ray diffraction patterns with low noise. Furthermore, our heating targets
melting. When a crystalline sample undergoes melting, its sharp and intense
diffraction lines disappear, and broad diffraction features from melt appear.
In a DAC, such broad diffraction features are difficult to separate from the
strong background caused by Compton scattering from diamond anvils.
Furthermore, melting in an H fluid can result in the removal of materials at
the heating center (therefore a lack of diffraction at the heating center
(22); also found in our experiments as shown in Fig. S2a). Therefore, we focus
on analyzing the diffraction patterns measured before and after laser heating
at high pressures in this study.

Thermal radiation spectra were measured for the both sides of the sample for
temperature measurements. Pressures at high temperatures were calculated from
the measured MgO volumes combined with its equation of state (23). Diffraction
measurements (a beamsize of $3{\times}2\,\mu$m2) were performed at the center
of the heated area as well as its adjacent areas. Diffraction images were
measured for 0.1–0.2 second at high temperatures and 30 seconds after heating
at 300 K. Raman spectroscopy was conducted at GSECARS (24) and ASU using a
532-nm laser as an excitation source (Text 3). The samples were recovered and
prepared in a focused ion beam (FIB) instrument (Helios 5 UX, Thermo Fisher
Scientific Inc.) at ASU. The recovered samples were analyzed in scanning
transmission electron microscopy (ARM-200F, JEOL co., at ASU) for chemical
composition and imaging. A detailed description of our methods can be found in
SI Appendix.

## Results

Figure 1: Pressure and temperature conditions (circles) of the experimental
runs (Table S1). Black and white circles represent the presence and absence of
Mg2FeH6, respectively. A color gradient in the background indicates a possible
change in the chemical behavior of the system at 15–25 GPa. The arrow at the
top shows the estimated range for the pressure at the atmosphere-magma
boundary $P(\mathrm{AMB})$ in sub-Neptunes with masses of $9.2{-}13.6M(E)$ for
a radius of $3R(E)$ and 2% hydrogen (see SI Appendix). $R(E)$: Earth radius.
$M(E)$: Earth mass. The MgO melting curve is from (17). H-2023: a data point
for MgO + Fe in H from (11).  Figure 2: X-ray diffraction patterns measured
after the melting of MgO + Fe in H2 at (a) 4770 K and 12.5 GPa, and (b) 4510 K
and 28.5 GPa. The colored circles above the peak indicate the peak positions
of the observed phases in the legend. If peaks are either weak or not
observed, we used open circles. The Le Bail analysis of pattern (a) is
presented in Fig. S6. The wavelength of the X-ray beam is 0.3344 Å. SEM
analysis of the recovered sample for (a) is presented in Fig. S2.

At 8–13 GPa after heating, XRD patterns showed the peaks from Mg2FeH6,
Mg(OH)2, face-centered cubic (fcc) FeHx, and MgO (Figs 2a, S3, and S4a). As
shown in previous high-pressure studies, hydrogen alloys with liquid Fe metal
at high $P$-$T$ and form FeHx, $\mathrm{Fe+0.5}x\mathrm{H_{2}\rightarrow
FeH}_{x}$, which is also confirmed here (25, 26, 21) (in our experiments,
$x=1.0{-}1.2$; Fig. S5). Cubic Mg2FeH6 perovskite phase appeared after an MgO
+ Fe mixture was heated to temperatures above 3500 K, which is close to or
above the melting temperatures of MgO (Figs 2a, S3, and S4a). Three intense
peaks of Mg2FeH6 were clearly identified in diffraction images (Fig. S3). In
the recovered samples, we often found a hole at the heating center (Fig. S2a).
As shown in a previous study (22), convecting H fluid can remove some melt
from the hottest spot and in case the hole becomes sufficiently large, it can
be observed in the LHDAC samples with hydrogen. Therefore, the observation of
the holes in some samples can indicate melting. Horn et al. (11) studied the
same system but only at pressures above 20 GPa. Shinozaki et al. (27) found
MgO does not decompose in H2 at 1500 K and 15 GPa. Therefore, the formation of
Mg2FeH6 from MgO + Fe + H2, which implies the dissociation of the Mg–O bond,
likely requires heating to sufficiently high temperatures at pressures lower
than 15 GPa (Fig. 1):

$\mathrm{2MgO+Fe+5H_{2}\rightarrow Mg_{2}FeH_{6}+2H_{2}O}.$ (1)

Reaching to the melting temperature of MgO ($\sim$4000–5500 K) in an H medium
at high pressure is extremely challenging (11). In this study, the
experimental setup was optimized for achieving temperature as close to the
melting of MgO in the presence of H2, although this heating method makes it
challenging to measure diffraction patterns at simultaneously high $P$-$T$
(Text 2). Previous studies have shown that the breakdown of the Fe–O bond (28,
11) and the Si–O bond (15) in the presence of hydrogen. However, to our
knowledge, none of the previous studies have reported the breakdown of the
Mg–O bond and the formation of Mg,Fe-hydride at the $P$-$T$ conditions
relevant to the atmosphere-magma boundary of sub-Neptunes.

Because of the Mg–O bond break, some O atoms can be released from MgO and
react with either Fe or H in the system. The XRD patterns do not show any
peaks of FeO and Fe2O3 (Figs 2a, S3, and S4a). Under an H-rich environment,
iron oxides (FeO and Fe2O3) can be easily reduced to Fe metal by H2 at
2000–3000 K and 25 GPa (11), which explains no observation of iron oxides in
our experiments. The diffraction peaks of H2O are difficult to identify
because of the peak overlap with fcc FeHx expected at this pressure range. In
addition, H2O peaks have much weaker diffraction intensity. Related to H2O,
however, Mg(OH)2 peaks were observed (Figs 2a, S3, and S4a). Mg(OH)2 can form
through a reaction between MgO and H2O during temperature quench at lower
temperatures: MgO + H2O (from Eq. 1) $\rightarrow$ Mg(OH)2. Therefore, its
appearance indicates the formation of H2O as predicted in Eq. 1. In the laser-
heated area, the O–H vibrational modes of H2O-ice VII were detected in Raman
spectroscopy at high pressure and 300 K after temperature quench (Figs 3a and
3b). Raman spectroscopy also detected the O–H vibration from Mg(OH)2.

Figure 3: Raman spectra measured at (a) 7.0 GPa for the sample heated at 12.5
GPa and 4770 K (#31-005), and (b) 11.5 GPa and 3820 K (#31-003), and (c) 22.0
GPa for the sample heated at 26.0 GPa and 4870 K (#26-009) (Table S1). SEM
data for the sample in (c) is presented in Fig. S12. H2O: H2O-VII and Br:
Mg(OH)2.

At 26–29 GPa, XRD patterns showed the formation of fcc FeHx and Mg(OH)2,
whereas Mg2FeH6 peaks were not found (Figs 2b, S4b, and S4c). As discussed
above, the appearance of Mg(OH)2 requires H2O formation and therefore the
release of O from MgO through the breakage of some Mg–O bond (Eq. 1). Similar
to the case of Fe2+ in (Mg,Fe)O (11), it can be hypothesized that O is
released by Mg2+ reduction to Mg0 metal. It is also possible that Mg–O bond
breaks down, but Mg remains as Mg2+. Diffraction lines of Mg metal or its
hydride forms were not observed at high pressures after heating, thus Mg2+ was
unlikely reduced to Mg metal. If Mg2+ dissolves in H2 liquid during melting,
it would not create any diffraction peaks. Related to the system we study
here, a recent study showed that MgO can dissolve into H2O fluid at $\sim$30
GPa and above 1500 K (29). Similarly, Mg2+ and H may form an Mg–H bond. In our
experiments, when the sample was decompressed to $\sim$1 GPa, several
diffraction spots appeared in XRD images. Those spots were detected at the
same two theta angle at different azimuthal angles (Fig. S7). The two-theta
angle does not match with any diffraction peaks expected for Mg(OH)2, MgO, Fe
metal, and fcc FeHx. The observed peak is possibly from the crystallization of
a component dissolved in H2 liquid at high pressures during decompression.
However, no more clear diffraction features were observed and therefore we
could not identify the phase with XRD alone.

After heating at 26–29 GPa, an intense O–H vibration from Mg(OH)2 and H2O was
observed at 2800–3700 cm-1 as well as the lattice modes of Mg(OH)2 (30) and
H2O (31) at 200–1300 cm-1 (Figs 3c, S8, and S9). Upon decompression, the peak
shifts to lower wavenumbers but remains in the range (Fig. S10). The Mg–H
stretching vibration exists between 900 and 1300 cm-1 at 1 bar (32) (Figs S9
and S10). At 1 GPa, some additional Raman peaks appeared. The mode frequencies
of these peaks agree well with those expected for the lattice vibrational
modes of $\gamma$-MgH2 (33) (Figs S9 and S10b), although some level of
intensity contribution from weak H2O Raman modes cannot be ruled out (31).
From these observations, we interpret that some Mg2+ may dissolve into H2
fluid (i.e., Mg–H) at pressures above 25 GPa and $\sim$3500 K:

$\mathrm{MgO+2H_{2}\rightarrow MgH_{2}+H_{2}O}.$ (2)

The dissolved Mg–H precipitates with decompression because of its lowering
solubility in H2 at lower pressures.

In our experiments, a complete conversion of MgO to MgH2 was not observed. The
total heating duration was 0.1–0.2 second and it could not be extended any
further because of the possible mechanical failure of diamond anvils by H
(Text 2). The heating duration may not be sufficient for the completion of the
reaction (Figs 2, S3, and S4). It is also possible that H2O, produced from the
reaction, makes the condition more oxidizing than the initial redox condition
from a pure H2 medium. Therefore, the reaction (Eq. 2) may not proceed any
further.

Figure 4: STEM (a and b) and SEM (c and d) images of the heated areas in the
samples recovered from run #31-003 (11.5 GPa and 3820 K) and #26-009 (26 GPa
and 4870 K), respectively (Table S1). (a) An FIB section shows some of the
phases found in XRD. The small bright grains are bcc Fe metal (labeled as Fe)
converted from fcc FeHx during decompression (34). The diffraction pattern for
the converted Fe metal can be found in Fig. S7. The Fe grains in the left side
are not heated. (b) Bubble-shaped structures are found along the edge of MgO
grans (a white box in (a)). An intensity line profile between point A and
point B is shown in the inset. (c) A cross section of the laser-heated area in
the sample foil was made in FIB to reveal the structures. The center of the
heated area contains more MgO and the surrounding area contains more Fe metal.
Some Fe metal may migrate during melting along the radial thermal gradients.
(d) A magnified image (a white box in (c)) reveals the bubble-shaped
structures (highlighted by yellow arrows) within the heated area, similar to
the ones observed in (b) but in larger sizes (note the scale difference).

The recovered samples were analyzed in STEM (scanning transmission electron
microscopy) and SEM (scanning electron microscopy). The samples are fractured
and mechanically weak because of the O release, melting in a fluid H medium,
and hydrogen transition from solid/liquid to gas during decompression (Figs
4a, 4c, S2, S11, and S12). Therefore, it was difficult to recover all the
samples studied at the synchrotron. For the sample recovered from 11.5 GPa,
bubble-shaped structures were observed near the edge of MgO grains (Fig. 4b).
The size of the structure ranges between 50 and 100 nm. Inside the bubble-
shaped area, intensity was found to be as low as that found at the vacuum
(Fig. 4b). The intensity in an STEM annular dark-field (ADF) image is
proportional to $Z^{2-x}$, where $Z$ is the atomic numbers and $x$ is the
empirical parameters (35). Therefore, the bubble-like structures are either
empty or filled with low $Z$ elements, most likely H2 in this experiment.

In a sample recovered from 26 GPa, similar bubble-shaped structures are
observed with larger sizes, ranging between 100 and 300 nm (Figs 4c and 4d).
It is possible that such bubble-shaped structures were formed from the
mechanical capturing of H2 or H2O (formed by the reactions) fluid in the melt
during temperature quench. Alternatively, such structures can form from the
separation of two components that were once miscible at high temperature. In
this case, because of the much lower mutual solubility at lower temperatures,
the two components can separate into different phases during temperature
quenching, resulting in the formation of the bubble-shaped structure. In the
case of the albite–H2O system, they become miscible with each other at 0.73
GPa and 1673 K (36). During temperature quench, because of the low solubility
of H2O in albite at lower temperatures, the exsolved H2O can form bubble-
shaped structures in the albite crystal. Therefore, if the same process occurs
for the MgO–H system studied here, H once mixed with MgO melt may form the
bubble-shaped structures during temperature quench. At 25 GPa, the size of the
bubble-shaped structure is larger and Mg–H mode is observed in H2, indicating
that the mutual solubility during heating is much greater at the higher
pressures (Figs 4d and S12b).

Horn et al. (11) conducted experiments on (Mg,Fe)O in H at similar $P$-$T$
range. They reported the reduction of Fe2+ and the formation of Mg(OH)2, but
did not report the Mg–H mixing. However, they did not conduct TEM, SEM, and
Raman spectroscopy measurements which are critical for detecting the Mg–H
mixing in our study.

## Implications

In recent studies, possible chemical reactions between hydrogen and
silicates/metals have been drawn interest for their potential impacts on
planetary processes (11, 12, 21). A high-pressure study (11) showed that H can
induce the reduction of Fe2+ in (Mg,Fe)O to Fe metal. Because of the electron
$(e^{-})$ transfer from H0 to Fe2+ (the first two reactions in Eq. 3 below), O
atoms released from (Mg,Fe)O (the first reaction in Eq. 3) react with H+,
resulting in the formation of H2O (the third reaction in Eq. 3, which can be
obtained by combining first two reactions).

$\begin{gathered}\mathrm{Fe^{2+}O^{2-}}+2e^{-}\rightarrow\mathrm{Fe^{0}+O^{2-}}\\\
\mathrm{H^{0}_{2}}\rightarrow\mathrm{2H^{+}}+2e^{-}\\\
\mathrm{Fe^{2+}O^{2-}\leavevmode\nobreak\
(magma)+H^{0}_{2}\leavevmode\nobreak\
(atm)}\rightarrow\mathrm{Fe^{0}\leavevmode\nobreak\
(metal)+H^{+}_{2}O^{2-}\leavevmode\nobreak\ (atm\leavevmode\nobreak\
and\leavevmode\nobreak\ magma)}.\end{gathered}$ (3)

The endogenically produced water can partition into both magma and atmosphere
(“atm” in Eq. 3).

We found that MgO reacts with H at higher temperatures, breaking Mg–O bonds
and forming Mg–H bonds (hydride, H-), leading to the formation of Mg2FeH6 at
pressures below 13 GPa and the possible dissolution of Mg in H fluid (i.e.,
Mg–H) at pressures above 24 GPa. Our Raman measurements showed the formation
of H2O (and therefore proton, H+). From these observations, the following
chemical reactions with electron transfer can be inferred (the fourth reaction
can be obtained by combining first three reactions; note that we are talking
about a general case here for the Mg-related components which can be applied
to both Eqs 1 and 2):

$\begin{gathered}\mathrm{Mg^{2+}O^{2-}}\rightarrow\mathrm{Mg^{2+}}+\mathrm{O^{2-}}\\\
\mathrm{H^{0}_{2}}+2e^{-}\rightarrow\mathrm{2H^{-}}\\\
\mathrm{H^{0}_{2}}\rightarrow\mathrm{2H^{+}}+2e^{-}\\\
\mathrm{Mg^{2+}O^{2-}\leavevmode\nobreak\
(magma)+2H^{0}_{2}\leavevmode\nobreak\
(atm)}\rightarrow\mathrm{Mg^{2+}H^{-}_{2}+H^{+}_{2}O^{2-}(atm\leavevmode\nobreak\
and\leavevmode\nobreak\ magma)}.\end{gathered}$ (4)

The key difference from the Fe case (11) include: (1) Mg remains oxidized (the
first reaction in Eq. 4), (2) Mg forms bonding directly with H- (hydride) (the
second reaction in Eq. 4), and (3) electron transfer occurs between hydrogen
atoms to form protons which react with O to form water (the third reaction in
Eq. 4). As Mg is one of the dominant heavy elements in the planetary
interiors, the chemical process we identified here can add a significant
amount of H2O to the total endogenic H2O in addition to the H2O from the Fe2+
reduction found in (11). An important difference between Fe and Mg in the
reaction with H in a planetary setting is that after the reduction Fe metal
would sink toward the center (because of its high density), and therefore be
separated from the atmosphere and magma. However, Mg, by forming hydride
bonding, would likely remain in atmosphere and/or magma, possibly contributing
to radial compositional gradients at the shallower depths in the planets
together with endogenic H2O.

Figure 5: An exoplanet size demographics histogram up to $R/R(E)=6$ ($R$:
radius, $R(E)$: Earth’s radius; bottom) shown together with the implications
of our experiments on the radius cliff (top). Schematic diagrams for the
structures of sub-Neptunes with $P(\mathrm{AMB})<15$ GPa and
$P(\mathrm{AMB})>25$ GPa are shown (AMB: atmosphere-magma boundary). In the
histogram, three horizontal bars show the estimated radii of sub-Neptunes (2
wt% H2 atmosphere with an Earth-like interior composition) for different
pressures (in GPa) at the atmosphere-magma boundary (see Text 5 for method).

At the $P$-$T$ conditions (8–13 GPa and 3500–4800 K) relevant for the
interface between the atmosphere and the magma ocean in H-rich sub-Neptunes,
we observed the formation of Mg2FeH6 from the reaction between MgO + Fe and H.
For example, for GJ1214b ($6.55M(E)$ and $2.68R(E)$), 8 GPa and 3000 K were
estimated for the atmosphere-magma boundary (37). Mg2FeH6, the material formed
under the planetary conditions in our experiments below 13 GPa, has been
extensively studied in condensed matter physics and materials science because
of its importance for H storage applications (38). Our experiments show that
Mg2FeH6 can exist in a H-rich planet, i.e., the atmosphere-magma boundary
(AMB) of sub-Neptune exoplanets, potentially important for the H storage in a
natural setting as well (Fig. 5).

Spectroscopy studies have shown that the valence states of Mg and [FeH6] in
Mg2FeH6 are $+2$ and $-4$, respectively (39, 40). Therefore, reduction to
metal does not occur for Mg under H2, as we found no evidence for Mg metal or
alloy in the experiment. However, our experiments show that by breaking the
Mg–O bond MgO + H reaction can still produce H2O (Eqs 1 and 4). The valence
state of Fe in the [FeH6]4- complex has been debated: Fe2+ (39) or Fe0 (40). A
computational study (41) showed that the bonding between Fe and H is partially
covalent. Because of Mg2+ in Mg2FeH6 and the higher electronegativity of H
than Fe, [FeH6]4- should have some degree of ionic bonding, therefore a
negative charge for H. This means hydride bonding should exist regardless of
whether the valence state of Fe is 2+ or 0. Therefore, our experiments reveal
an intriguing possibility of co-ingassing of H- (in Mg2FeH6 hydride) and H+
(in H2O) in a planetary setting.

Our measurements combined with the existing data (42, 38) found that Mg2FeH6
has a substantially lower density than MgSiO3 and MgO, which are
representative silicate/oxide for the rocky part of planets (Fig. S13).
Mg2FeH6 has a perovskite-type structure, similar to the crystal structure of
MgSiO3 above 24 GPa (orthorhombic perovskite structure; bridgmanite). However,
Mg2FeH6 has a vacancy-ordered double perovskite structure (43). Bridgmanite,
with a normal perovskite structure, has no vacant site between the SiO6
octahedra which are connected with each other by the corner sharing. However,
in the crystal structure of Mg2FeH6 (space group: $Fm\bar{3}m$, #225), the
FeH6 octahedra, positioned at the lattice points of face-centered cubic, are
isolated with each other because of the ordered vacancies (40). Therefore, the
crystal structure of Mg2FeH6 and its high H content can explain its low
density. Given as much as 15–30% lower density than MgSiO3 between 1 bar and
20 GPa (we compare to MgSiO3 enstatite, a phase stable at the pressure range),
Mg2FeH6 likely remains less dense than silicate even at the high temperatures
of planetary interiors. Therefore, Mg2FeH6 can be buoyant when it crystallizes
in the magma ocean or in the solidified silicate rocky layer. Such buoyant
Mg2FeH6 at the interface between the atmosphere and the interior can be an
important agent for maintaining very reducing conditions for the underlying
interior. It is also feasible that other hydrides may form at the interface
(15). If dense hydrides can be formed, and ingassing of hydride (H-) could be
extended to much greater depths combined with the convective flow of the
interior. In (11), from the higher melting temperature of MgO, it was
postulated that MgO may crystallize first from the magma ocean beneath a thick
H-rich atmosphere and form a crust. They further hypothesized that the MgO-
rich crust could regulate the interaction between H-rich atmosphere and the
magma ocean. According to our experiments, Mg,Fe hydrides may form at the
interface and therefore such an MgO-rich crust unlikely forms.

Mg2FeH6 does not form at pressures above 25 GPa. Instead, the Raman data,
together with the electron microscopy data, indicate that Mg may dissolve into
a H fluid at this pressure range. In other words, MgO (possibly in a liquid
form) may become (at least partially) miscible with a H2 liquid.

According to the Kepler data, planet occurrence drops sharply at ${\sim}3R(E)$
dividing sub-Neptune exoplanets and Neptune-size exoplanets (i.e., radius
cliff; Fig. 5) (13, 44, 14). Kite et al. (14) proposed that a sudden increase
in the H solubility in magma by pressure effects may ingas much more H at the
radius cliff, rather than leave much hydrogen for building the atmosphere.
Because of the significant ingassing of H and much smaller molar volume of H
expected for the ingassed states compared with pure H2, the atmosphere does
not increase as much and therefore the radius of a planet does not increase as
much near $3R(E)$, leading to a low population of planets with radius greater
than $3R(E)$.

For a sub-Neptune with 2 wt% H2 atmosphere and an Earth-like interior
composition, the pressure at the atmosphere-magma boundary ($P(\mathrm{AMB})$)
can be estimated for a planet radius using the equations in (8, 45) (Text 5
for detail). From the calculation, we found that sub-Neptunes with radii of
$2.52$, $3.03$, and $3.48R(E)$ can have pressures of 15, 20, and 25 GPa at the
AMB, respectively. We observed the formation of Mg2FeH6 below 15 GPa.
Therefore, for a smaller sub-Neptune $(R_{p}\lesssim 2.5R(E))$, Mg2FeH6 may
crystallize at the AMB and remain there, because of its lower density than
silicate melt (Fig. S13). Its high concentration at the AMB could regulate the
H ingassing to the magma ocean beneath. At pressures higher than 25 GPa, our
experiments found that Mg2FeH6 is no longer stable. Instead, the detection of
the Mg–H vibration and the bubble-shaped structures found in the recovered
samples point toward much enhanced chemical mixing between Mg and H at the
higher pressure range (Fig. 5b). Therefore, if a sub-Neptune type planet can
grow large enough to $R_{p}\gtrsim 3R(E)$, the pressure at the AMB can become
sufficiently high for the effective chemical mixing between Mg and H, and
therefore efficient H ingassing. If such a process indeed occurs, similar to
the proposed process in (14), large sub-Neptune type planet may not grow
efficiently beyond a certain size, i.e., ${\sim}3R(E)$, as more H is added to
the interior where its molar volume is much smaller than that for H in
atmosphere. Therefore, it is feasible that the sharp decrease in the planets’
occurrence at ${\sim}3R(E)$ (i.e., radius cliff) is related to the chemical
process we identified in this study. An important difference from Kite et al.
(14) is that they assumed the physical mixing between H2 and magma for the
radius cliff. It is, however, important to note that they extrapolated
experimental results measured below 3 GPa (46), accuracy of which is debatable
(10). Our experiments conducted directly at the pressure conditions relevant
to the larger sub-Neptunes’ interiors show that the chemical mixing through
the Mg–H bond formation could be an important process to consider for the
possible enhancement of the H ingassing at the radius cliff.

If the reaction we identified in this study also occurs at lower pressures, it
would be important to consider such chemical interactions in early rocky
planets with H-rich primary atmosphere (12). It is of particular interest how
the potential formation of hydride minerals on the surface can impact the
habitability of the early rocky planets.

The on-going space telescope (e.g., JWST) and large ground-based telescope
missions aim to measure the atmosphere chemistry of exoplanets (including sub-
Neptunes) which will be an important step toward answering a fundamental
question: are we alone in the universe (47)? Because hydrogen-rich atmosphere
should be reactive particularly with refractory materials (silicates and
oxides) underneath, the atmosphere-interior interaction is key to understand
such astrophysical measurements. However, models so far considered mostly
physical mixing of H2 (or H0) in silicate magma (46, 9). Latest experiments
begin to reveal other important interactions between hydrogen-rich atmosphere
and magma: H2O formation through the Fe2+ reduction (11), and H0 alloying with
Fe metal (21). Here we demonstrated that anionic hydrogen, H- (or hydride),
can exist in the setting expected for sub-Neptune exoplanets. Furthermore, our
experiments reveal that hydride can form together with water, both of which
can partition to the atmosphere and the interior of sub-Neptunes (Fig. 5).
Therefore, it is important to consider these reactions in the interpretation
of exoplanet atmosphere spectra. In fact, a recent astrophysical study
detected CrH in the H-rich atmosphere of hot Jupiter WASP-31b together with
H2O (48), demonstrating the importance of understanding the type of chemical
interaction we report here even for the less abundant heavy elements. Hydrides
have been important materials to study in condensed matter physics for
hydrogen storage (49) and high-temperature superconductivity (50). Our study
demonstrates that such studies of hydrides can also impact our understanding
on the evolution and dynamics of sub-Neptune exoplanets, which are common in
our galaxy (5).

We thank two anonymous reviewers for their constructive criticism and helpful
comments, which improved our paper. This work was supported by National
Science Foundation (NSF) grants AST2108129 (T.K. and S.-H.S.) and AST2005567
(X.W. and S.-H.S). This work was performed at GSECARS (The University of
Chicago, Sector 13), APS, Argonne National Laboratory (ANL). GSECARS is
supported by the NSF - Earth Sciences (EAR-1634415). This research used
resources of the APS, a U.S. Department of Energy (DOE) Office of Science User
Facility operated for the DOE Office of Science by ANL under Contract No. DE-
AC02-06CH11357. Use of the GSECARS Raman Lab System was supported by the NSF
MRI Proposal (EAR-1531583). We also acknowledge the use of facilities at the
Eyring Materials Center at ASU supported in part by NNCI-ECCS-1542160.

## References

  * (1) Lambrechts M, Johansen A (2012) Rapid growth of gas-giant cores by pebble accretion.  A&A 544:A32.
  * (2) Lammer H, et al. (2018) Origin and evolution of the atmospheres of early Venus, Earth and Mars.  Astron Astrophys Rev 26(1):2.
  * (3) Venturini J, Ronco MP, Guilera OM (2020) Setting the Stage: Planet Formation and Volatile Delivery.  Space Sci Rev 216(5):86.
  * (4) Owen JE, Shaikhislamov IF, Lammer H, Fossati L, Khodachenko ML (2020) Hydrogen Dominated Atmospheres on Terrestrial Mass Planets: Evidence, Origin and Evolution.  Space Sci Rev 216(8):129.
  * (5) Bean JL, Raymond SN, Owen JE (2021) The nature and origins of sub-Neptune size planets.  Journal of Geophysical Research: Planets 126\.
  * (6) Vazan A, Ormel CW, Noack L, Dominik C (2018) Contribution of the Core to the Thermal Evolution of Sub-Neptunes.  ApJ 869(2):163.
  * (7) Kite ES, Fegley Jr. B, Schaefer L, Ford EB (2020) Atmosphere Origins for Exoplanet Sub-Neptunes.  ApJ 891(2):111.
  * (8) Zeng L, et al. (2021) New Perspectives on the Exoplanet Radius Gap from a Mathematica Tool and Visualized Water Equation of State.  ApJ 923(2):247.
  * (9) Olson P, Sharp ZD (2018) Hydrogen and helium ingassing during terrestrial planet accretion.  Earth and Planetary Science Letters 498:418–426.
  * (10) Schlichting HE, Young ED (2022) Chemical Equilibrium between Cores, Mantles, and Atmospheres of Super-Earths and Sub-Neptunes and Implications for Their Compositions, Interiors, and Evolution.  Planet. Sci. J. 3(5):127.
  * (11) Horn HW, Prakapenka V, Chariton S, Speziale S, Shim SH (2023) Reaction between Hydrogen and Ferrous/Ferric Oxides at High Pressures and High Temperatures—Implications for Sub-Neptunes and Super-Earths.  Planet. Sci. J. 4(2):30.
  * (12) Young ED, Shahar A, Schlichting HE (2023) Earth shaped by primordial H2 atmospheres.  Nature 616:306–311.
  * (13) Fulton BJ, Petigura EA (2018) The California- _Kepler_ Survey. VII. Precise Planet Radii Leveraging _Gaia_ DR2 Reveal the Stellar Mass Dependence of the Planet Radius Gap.  AJ 156(6):264.
  * (14) Kite ES, Bruce Fegley Jr., Schaefer L, Ford EB (2019) Superabundance of Exoplanet Sub-Neptunes Explained by Fugacity Crisis.  ApJ 887(2):L33.
  * (15) Shinozaki A, et al. (2014) Formation of SiH4 and H2O by the dissolution of quartz in H2 fluid under high pressure and temperature.  American Mineralogist 99(7):1265–1269.
  * (16) Stökl A, Dorfi EA, Johnstone CP, Lammer H (2016) Dynamical accretion of primordial atmospheres around planets with masses between 0.1 and 5 $m(e)$ in the habitable zone.  The Astrophysical Journal 825:86.
  * (17) Kimura T, Ohfuji H, Nishi M, Irifune T (2017) Melting temperatures of MgO under high pressure by micro-texture analysis.  Nat Commun 8(1):15735.
  * (18) Prakapenka VB, et al. (2008) Advanced flat top laser heating system for high pressure research at GSECARS: Application to the melting behavior of germanium.  High Pressure Res. 28(3):225–235.
  * (19) Goncharov AF, et al. (2010) X-ray diffraction in the pulsed laser heated diamond anvil cell.  Review of Scientific Instruments 81(11):113902.
  * (20) Fu S, Chariton S, Prakapenka VB, Shim SH (2023) Core origin of seismic velocity anomalies at Earth’s core–mantle boundary.  Nature 615(7953):646–651.
  * (21) Piet H, et al. (2023) Superstoichiometric Alloying of H and Close-Packed Fe-Ni Metal Under High Pressures: Implications for Hydrogen Storage in Planetary Core.  Geophysical Research Letters 50(5):e2022GL101155.
  * (22) Fu S, Chariton S, Prakapenka VB, Chizmeshya A, Shim SH (2022) Stable hexagonal ternary alloy phase in Fe-Si-H at 28.6–42.2 GPa and 3000 K.  Physical Review B 105:104111.
  * (23) Dorogokupets PI, Dewaele A (2007) Equations of state of MgO, Au, Pt, NaCl-B1, and NaCl-B2: Internally consistent high-temperature pressure scales.  High Pressure Research 27(4):431–446.
  * (24) Holtgrewe N, Greenberg E, Prescher C, Prakapenka VB, Goncharov AF (2019) Advanced integrated optical spectroscopy system for diamond anvil cell studies at GSECARS.  High Pressure Research 39(3):457–470.
  * (25) Okuchi T (1997) Hydrogen Partitioning into Molten Iron at High Pressure: Implications for Earth’s Core.  Science 278(5344):1781–1784.
  * (26) Sakamaki K, et al. (2009) Melting phase relation of FeH up to 20 GPa: Implication for the temperature of the Earth’s core.  Physics of the Earth and Planetary Interiors 174(1-4):192–201.
  * (27) Shinozaki A, et al. (2013) Influence of H2 fluid on the stability and dissolution of Mg2SiO4 forsterite under high pressure and high temperature.  American Mineralogist 98(8-9):1604–1609.
  * (28) Efimchenko VS, et al. (2019) Destruction of fayalite and formation of iron and iron hydride at high hydrogen pressures.  Phys Chem Minerals 46(8):743–749.
  * (29) Kim T, et al. (2021) Atomic-scale mixing between MgO and H2O in the deep interiors of water-rich planets.  Nat Astron 5(8):815–821.
  * (30) Duffy TS, Hemley RJ, Mao HK (1995) Structure and bonding in hydrous minerals at high pressure: Raman spectroscopy of alkaline earth hydroxides in AIP Conference Proceedings.  (AIP, Pasadena, California (USA)), Vol. 341, pp. 211–220.
  * (31) Zha CS, Tse JS, Bassett WA (2016) New Raman measurements for H2O ice VII in the range of 300 cm-1 to 4000 cm-1 at pressures up to 120 GPa.  J. Chem. Phys. 145(12):124315.
  * (32) Shantilal Gangrade A, Aditya Varma A, Kishor Gor N, Shriniwasan S, Tatiparti SSV (2017) The dehydrogenation mechanism during the incubation period in nanocrystalline MgH2.  Phys. Chem. Chem. Phys. 19(9):6677–6687.
  * (33) Kuzovnikov M, et al. (2013) Raman scattering study of $\alpha$-MgH2 and $\gamma$-MgH2.  Solid State Communications 154:77–80.
  * (34) Badding J, Hemley R, Mao H (1991) High-pressure chemistry of hydrogen in metals: In situ study of iron hydride.  Science 253(5018):421–424.
  * (35) Yang SZ, et al. (2019) Direct cation exchange in monolayer MoS2 via recombination-enhanced migration.  Physical review letters 122(10):106101.
  * (36) Paillat O, Elphick SC, Brown WL (1992) The solubility of water in NaAlSi3O8 melts: a re-examination of Ab-H2O phase relationships and critical behaviour at high pressures.  Contributions to Mineralogy and Petrology 112:490–500.
  * (37) Nettelmann N, Fortney JJ, Kramm U, Redmer R (2011) Thermal Evolution and Structure Models of the Transiting Super-Earth GJ 1214b.  ApJ 733(1):2.
  * (38) George L, Drozd V, Durygin A, Chen J, Saxena SK (2009) Bulk modulus and thermal expansion coefficient of mechano-chemically synthesized Mg2FeH6 from high temperature and high pressure studies.  International Journal of Hydrogen Energy 34(8):3410–3416.
  * (39) Didisheim JJ, et al. (1984) Dimagnesium iron(II) hydride, Mg2FeH6, containing octahedral FeH64- anions.  Inorg. Chem. 23(13):1953–1957.
  * (40) Retuerto M, et al. (2015) Neutron Powder Diffraction, x-ray absorption and Mössbauer spectroscopy on Mg2FeH6.  International Journal of Hydrogen Energy 40(30):9306–9313.
  * (41) Zareii SM, Sarhaddi R (2012) Structural, electronic properties and heat of formation of Mg2FeH6 complex hydride: An _ab initio_ study.  Phys. Scr. 86(1):015701.
  * (42) Jackson JM, Sinogeikin SV, Bass JD (1999) Elasticity of MgSiO3 orthoenstatite.  American Mineralogist 84(4):677–680.
  * (43) Rahim W, et al. (2020) Geometric Analysis and Formability of the Cubic A2BX6 Vacancy-Ordered Double Perovskite Structure.  Chem. Mater. 32(22):9573–9583.
  * (44) Hsu DC, Ford EB, Ragozzine D, Ashby K (2019) Occurrence Rates of Planets Orbiting FGK Stars: Combining _Kepler_ DR25, _Gaia_ DR2, and Bayesian Inference.  AJ 158(3):109.
  * (45) Otegi J, Bouchy F, Helled R (2020) Revisited mass-radius relations for exoplanets below 120 m(e).  Astronomy & Astrophysics 634:A43.
  * (46) Hirschmann M, Withers A, Ardia P, Foley N (2012) Solubility of molecular hydrogen in silicate melts and consequences for volatile evolution of terrestrial planets.  Earth and Planetary Science Letters 345–348:38–48.
  * (47) Decadal Survey on Astronomy and Astrophysics 2020 (Astro2020), Space Studies Board, Board on Physics and Astronomy, Division on Engineering and Physical Sciences, National Academies of Sciences, Engineering, and Medicine (2021) Pathways to Discovery in Astronomy and Astrophysics for the 2020s.  (National Academies Press, Washington, D.C.), p. 26141.
  * (48) Braam M, van der Tak FF, Chubb KL, Min M (2021) Evidence for chromium hydride in the atmosphere of hot jupiter wasp-31b.  Astronomy & Astrophysics 646:A17.
  * (49) Sakintuna B, Lamaridarkrim F, Hirscher M (2007) Metal hydride materials for solid hydrogen storage: A review.  International Journal of Hydrogen Energy 32(9):1121–1140.
  * (50) Duan D, et al. (2017) Structure and superconductivity of hydrides at high pressures.  National Science Review 4(1):121–135.


# Quantum coherence with generalized wave packets under Lorentz boost

Arnab Mukherjee<EMAIL_ADDRESS>Soham Sen
sensohomhary<EMAIL_ADDRESS>Sunandan Gangopadhyay
<EMAIL_ADDRESS>Department of Astrophysics and High Energy
Physics
S. N. Bose National Centre for Basic Sciences, JD Block, Sector-III, Salt
Lake, Kolkata 700106, India

###### Abstract

In this paper we consider a single particle, apin-momentum entangled state and
measure the effect of relativistic boost on quantum coherence. The effect of
the relativistic boost on single-particle generalized Gaussian wave packets is
studied. The coherence of the wave function as measured by the boosted
observer is studied as a function of the momentum and the boost parameter.
Using various formulations of coherence, it is shown that in general the
coherence decays with the increase in momentum of the state, as well as the
boost applied to it. A more prominent loss of coherence due to relativistic
boost is observed for a single particle electron than that of a neutron. The
analysis is carried out with generalized Gaussian wave packet of the form
$\sim p^{n}\exp(-\frac{p^{2}}{\sigma^{2}})$. We also obtain a bound on the
unknown parameter $n$ appearing in the wave packet. It is found to have a
dependence on the mass of the particle and the width of the Gaussian wave
packet.

##  I Introduction

A very few fundamental questions raised in the previous century [1, 2, 3, 4]
are considered to be the origin of quantum information science. Over the last
few decades, this particular branch of science has gone through an
unprecedented amount of development among which the majority of the research
considers a nonrelativistic background. However, we know that the special
theory of relativity and quantum mechanics both hold up to unbelievable
accuracy in real-world observations. On one hand, special relativistic
analysis becomes crucial at relative velocities (between inertial frames) to
be of the order of the speed of light and on the other hand, non-relativistic
quantum mechanics considers physical phenomena at velocities much smaller than
the speed of the light. Interestingly quantum entanglement, built up based on
non-relativistic considerations, do have dependence on the reference frame of
the observer. It was observed in [5, 6] that the spin of a massive particle
depends on the momentum change caused by Lorentz transformation. This occurs
due to the dependence of the Lorentz transformation of the spin of a particle
on its momentum [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]. In [7], the
relativistic generalization of the Einstein-Podolsky-Rosen (EPR) experiment
was first considered. Later in [19, 20, 21], the observer dependence of
entanglement was considered for both Klein-Gordon and Dirac particles (for
observers near the event horizon of a black hole). This relativistic
description of quantum information theory also helps in enhancing the black
hole information paradox [22, 23] and quantum gravity phenomenology [24], and
cosmology [25, 26] as well. One of the most unique property that a quantum
system may carry is quantum coherence which is, by default, not present in a
classical system. It has been observed in [27] that quantum correlations arise
in composite systems due to quantum coherence. One can also exploit this
quantum coherence to complete complicated quantum tasks as well [28, 29, 30,
31, 32, 33, 34, 35, 36]. Although there are no unique description of
coherence, there have been several basis dependent [37] and independent [38,
39] measures of coherence. In [40], we observe an endeavour to devise a
relativistic description of this quantum coherence for single particle,
entangled states. In [40], the authors analyzed the change in coherence of a
single-particle Gaussian states when Lorentz boost is applied. A generic loss
of coherence was also observed for a relativistic observer. In this work, our
main motivation is to extend this analysis of relativistic coherence to much
generalized Gaussian-like wave packets. We have considered here generic
Gaussian wave packet structure (centred at the origin) where the exponential
function is being multiplied by $p^{n}$ (for general $n$ values). The main
motivation behind this consideration are the Fourier transformed form of the
Hermite polynomials which have similar structure to that of the wave function
being considered in this work. Our analysis revolves around a basis
independent analysis.

The construction of the paper goes as follows. In section (II), we provided a
basic representation of the single-particle quantum state under Lorentz boost.
In section (III), we analyze the coherence of a
spin-$\left(\frac{1}{2}\right)$ particle in $(1+1)$ and $(3+1)$-dimensions. In
subsection (III.1), we compute the change in coherence due to fermions with
very small rest mass energy (electron) and in subsection (III.2), we run the
same analysis for a fermion (neutron) with comparatively much heavier rest
mass energy than the electron. In contrast to the electron, the neutron has a
narrow-uncertainty wave packet. In section (IV), we provide some plots
considering the results in earlier sections. We summarize our results in
section (V).

##  II Representation of the single-particle quantum state under Lorentz boost

In the domain of relativistic quantum mechanics one of the most pertinent
questions is how a quantum state behaves under Lorentz boosts. To see the
effects of Lorentz boosts on the quantum states at first, we consider a single
particle quantum state $|\Psi\rangle$. It is well known that an inertial
reference frame $\mathcal{O}$ is related to another inertial frame
$\mathcal{O^{\prime}}$ through a Poincare transformation

$\displaystyle x^{\prime\mu}=T(\Lambda,a)x^{\nu}=\Lambda^{\mu}_{\hskip
5.69046pt\nu}\,x^{\nu}+a^{\mu}$ (1)

where $x=(x^{0},\,\textbf{x})$ is the spacetime coordinates of $\mathcal{O}$,
and similarly for $\mathcal{O^{\prime}}$. $T(\Lambda,a)$ induces a unitary
transformation on quantum states as

$\displaystyle|\Psi\rangle\rightarrow\mathcal{U}(\Lambda,a)|\Psi\rangle\,.$
(2)

Now we can define the single particle quantum state as

$\displaystyle|\Psi\rangle=|\mathbf{p}\rangle\otimes|\sigma\rangle\equiv|\mathbf{p},\sigma\rangle$
(3)

where $\mathbf{p}$ labels the spatial components of the 4-momentum $p^{\mu}$
with $p^{0}=\sqrt{\mathbf{p}^{2}+m^{2}}$ and $\sigma$ labels the spin for
massive particles. The quantum states $|p,\sigma\rangle$ are eigenvectors of
the momentum operator $P^{\mu}$ with eigenvalues $p^{\mu}$ and satisfies the
relation

$\displaystyle
P^{\mu}|\mathbf{p},\sigma\rangle=p^{\mu}|\mathbf{p},\sigma\rangle\,.$ (4)

Under pure Lorentz boosts $\Lambda$, a single particle quantum state
$|\mathbf{p},\sigma\rangle$ transforms as111A detailed derivation of eq.(II)
has been presented in an Appendix.

$\displaystyle\mathcal{U}(\Lambda)|\mathbf{p},\sigma\rangle=\sqrt{\frac{(\Lambda
p)^{0}}{p^{0}}}\displaystyle\sum_{\sigma^{\prime}}\mathcal{D}_{\sigma^{\prime}\sigma}(W(\Lambda,\mathbf{p}))|\Lambda\mathbf{p},\sigma^{\prime}\rangle\,.$

In the above equation $\Lambda\mathbf{p}$ is the spatial part of the momentum
after Lorentz transformation and $W(\Lambda,\mathbf{p})$ denotes one of the
elements of the unitary representation of little group
$\mathcal{D}(W(\Lambda,\mathbf{p}))$. For massive particles, the little group
is the rotation group in three dimensions and hence $W(\Lambda,\mathbf{p})\in
SO(3)$. However, its unitary representation takes the form
$\mathcal{D}(W(\Lambda,\mathbf{p}))\in SU(2)$.

We consider the four-momentum of the particle in the form

$\displaystyle p^{\mu}=(m\cosh\zeta,m\sinh\zeta\hat{f})\,$ (6)

where $m$ is the mass of the particle, and the velocity of the moving frame
$\mathcal{O^{\prime}}$, $\vec{v}=\tanh\alpha\,\hat{e}$. Then the unitary
representation of the little group $\mathcal{D}(W(\Lambda,\mathbf{p}))$ takes
the form [40]

$\displaystyle\mathcal{D}(W(\Lambda,\mathbf{p}))=\cos\frac{\phi}{2}\mathds{1}+i\sin\frac{\phi}{2}(\Sigma\cdot\hat{n})$
(7)

where

$\displaystyle\cos\frac{\phi}{2}$ $\displaystyle=$
$\displaystyle\frac{\cosh\frac{\alpha}{2}\cosh\frac{\zeta}{2}+\sinh\frac{\alpha}{2}\sinh\frac{\zeta}{2}(\hat{e}\cdot\hat{f})}{\sqrt{\frac{1}{2}+\frac{1}{2}\cosh\alpha\cosh\zeta+\frac{1}{2}\sinh\alpha\sinh\zeta(\hat{e}\cdot\hat{f})}}$
(8) $\displaystyle\sin\frac{\phi}{2}\hat{n}$ $\displaystyle=$
$\displaystyle\frac{\sinh\frac{\alpha}{2}\sinh\frac{\zeta}{2}(\hat{e}\times\hat{f})}{\sqrt{\frac{1}{2}+\frac{1}{2}\cosh\alpha\cosh\zeta+\frac{1}{2}\sinh\alpha\sinh\zeta(\hat{e}\cdot\hat{f})}}$
(9)

with $\phi\text{ and }\hat{n}$ being the angle and axis of the Wigner
rotation.

In an inertial frame $\mathcal{O}$, a pure state can be written as

$\displaystyle|\Psi\rangle$ $\displaystyle=$
$\displaystyle\displaystyle\sum_{\sigma}\int
d^{3}\mathbf{p}\,\psi(\mathbf{p})|\mathbf{p}\rangle\otimes
c_{\sigma}|\sigma\rangle$ (10) $\displaystyle=$
$\displaystyle\displaystyle\sum_{\sigma}\int
d^{3}\mathbf{p}\,\psi(\mathbf{p})c_{\sigma}|\mathbf{p},\sigma\rangle\,.$

The density matrix of the corresponding state becomes

$\displaystyle\varrho$ $\displaystyle=$
$\displaystyle|\Psi\rangle\langle\Psi|$ $\displaystyle=$
$\displaystyle\displaystyle\sum_{\sigma_{1},\sigma_{2}}\int
d^{3}\mathbf{p}_{1}\int
d^{3}\mathbf{p}_{2}\,\psi(\mathbf{p}_{1})\psi^{\ast}(\mathbf{p}_{2})c_{\sigma_{1}}c^{\ast}_{\sigma_{2}}$
$\displaystyle\times$
$\displaystyle|\mathbf{p}_{1},\sigma_{1}\rangle\langle\mathbf{p}_{2},\sigma_{2}|~{}.$

Taking partial trace over the momentum degrees of freedom, the reduced density
matrix for the spin reads

$\displaystyle\rho$ $\displaystyle=$
$\displaystyle\text{Tr}_{p}[|\Psi\rangle\langle\Psi|]=\int
d^{3}\mathbf{p}\langle\mathbf{p}|\Psi\rangle\langle\Psi|\mathbf{p}\rangle$
(12) $\displaystyle=$
$\displaystyle\displaystyle\sum_{\sigma_{1},\sigma_{2}}\int
d^{3}\mathbf{p}\int d^{3}\mathbf{p}_{1}\int
d^{3}\mathbf{p}_{2}\,\delta^{(3)}(\mathbf{p}-\mathbf{p}_{1})\,\delta^{(3)}(\mathbf{p}-\mathbf{p}_{2})$
$\displaystyle\,\times\,\psi(\mathbf{p}_{1})\psi^{\ast}(\mathbf{p}_{2})c_{\sigma_{1}}c^{\ast}_{\sigma_{2}}|\sigma_{1}\rangle\langle\sigma_{2}|$
$\displaystyle=$ $\displaystyle\displaystyle\sum_{\sigma_{1},\sigma_{2}}\int
d^{3}\mathbf{p}\,\psi(\mathbf{p})\psi^{\ast}(\mathbf{p})c_{\sigma_{1}}c^{\ast}_{\sigma_{2}}|\sigma_{1}\rangle\langle\sigma_{2}|\,.$

With respect to a moving frame $\mathcal{O^{\prime}}$ which is connected with
the inertial frame $\mathcal{O}$ through a Lorentz transformation $\Lambda$,
this state can be represented as

$\displaystyle|\Psi_{\Lambda}\rangle$ $\displaystyle=$
$\displaystyle\displaystyle\sum_{\sigma}\int
d^{3}\mathbf{p}\,\psi(\mathbf{p})c_{\sigma}\sqrt{\frac{(\Lambda
p)^{0}}{p^{0}}}\displaystyle$ $\displaystyle\times$
$\displaystyle\sum_{\sigma^{\prime}}\mathcal{D}_{\sigma^{\prime}\sigma}(W(\Lambda,\mathbf{p}))|\Lambda\mathbf{p},\sigma^{\prime}\rangle\,.$
(13)

Therefore, the density matrix in this boosted reference frame is given by

$\displaystyle\varrho_{\Lambda}$ $\displaystyle=$
$\displaystyle\displaystyle\sum_{\sigma_{1},\sigma_{2}}\int
d^{3}\mathbf{p}_{1}\int d^{3}\mathbf{p}_{2}\,\sqrt{\frac{(\Lambda
p_{1})^{0}(\Lambda
p_{2})^{0}}{p_{1}^{0}p_{2}^{0}}}\psi(\mathbf{p}_{1})\psi^{\ast}(\mathbf{p}_{2})$
(14) $\displaystyle\times$ $\displaystyle
c_{\sigma_{1}}c^{\ast}_{\sigma_{2}}\displaystyle\sum_{\sigma^{\prime}_{1},\sigma^{\prime}_{2}}\mathcal{D}_{\sigma^{\prime}_{1}\sigma_{1}}(W(\Lambda,\mathbf{p}_{1}))|\Lambda\mathbf{p}_{1},\sigma^{\prime}_{1}\rangle\langle\Lambda\mathbf{p}_{2},\sigma^{\prime}_{2}|$
$\displaystyle\times$
$\displaystyle\mathcal{D}_{\sigma^{\prime}_{2}\sigma_{2}}^{\dagger}(W(\Lambda,\mathbf{p}_{2}))~{}.$

Hence, the corresponding reduced density matrix is given by

$\displaystyle\rho_{\Lambda}$ $\displaystyle=$
$\displaystyle\text{Tr}_{p}[|\Psi_{\Lambda}\rangle\langle\Psi_{\Lambda}|]$
(15) $\displaystyle=$
$\displaystyle\displaystyle\sum_{\sigma_{1},\sigma_{2},\sigma^{\prime}_{1},\sigma^{\prime}_{2}}\int
d^{3}\mathbf{p}\int d^{3}\mathbf{p}_{1}\int
d^{3}\mathbf{p}_{2}\,\sqrt{\frac{(\Lambda p_{1})^{0}(\Lambda
p_{2})^{0}}{p_{1}^{0}p_{2}^{0}}}$ $\displaystyle\times$
$\displaystyle\delta^{(3)}(\mathbf{p}-\Lambda\mathbf{p}_{1})\,\delta^{(3)}(\mathbf{p}-\Lambda\mathbf{p}_{2})\mathcal{D}_{\sigma^{\prime}_{1}\sigma_{1}}(W(\Lambda,\mathbf{p}_{1}))$
$\displaystyle\times$
$\displaystyle\psi(\mathbf{p}_{1})\psi^{\ast}(\mathbf{p}_{2})c_{\sigma_{1}}c^{\ast}_{\sigma_{2}}|\sigma^{\prime}_{1}\rangle\langle\sigma^{\prime}_{2}|\mathcal{D}_{\sigma^{\prime}_{2}\sigma_{2}}^{\dagger}(W(\Lambda,\mathbf{p}_{2}))$
$\displaystyle=$
$\displaystyle\displaystyle\sum_{\sigma_{1},\sigma_{2},\sigma^{\prime}_{1},\sigma^{\prime}_{2}}\int
d^{3}\mathbf{p}\,\psi(\mathbf{p})\psi^{\ast}(\mathbf{p})c_{\sigma_{1}}c^{\ast}_{\sigma_{2}}\mathcal{D}_{\sigma^{\prime}_{1}\sigma_{1}}(W(\Lambda,\mathbf{p}))$
$\displaystyle\times$
$\displaystyle|\sigma^{\prime}_{1}\rangle\langle\sigma^{\prime}_{2}|\mathcal{D}_{\sigma^{\prime}_{2}\sigma_{2}}^{\dagger}(W(\Lambda,\mathbf{p}))\,,$

where we have used

$\displaystyle\delta^{(3)}(\Lambda\mathbf{p}_{1}-\Lambda\mathbf{p}_{2})=\frac{p_{1}^{0}}{(\Lambda
p_{1})^{0}}\delta^{(3)}(\mathbf{p}_{1}-\mathbf{p}_{2})\,.$ (16)

Here we observe that the state in eq.(10) and the density matrix in
eq.(LABEL:varrho) is separable in spin and momentum, however, in eq.(s)(13,
14) spin and momentum has been coupled due to momentum dependent rotation and
is known as spin-momentum entanglement [40, 41].

##  III Frobenius-norm measure of coherence

In this section, we shall obtain the Frobenius-norm measure of coherence for
spin 1/2 particles with spin-momentum entanglement.

###  III.1 Spin 1/2 particles under Lorentz boost in 1+1-d

The state of a single spin-1/2 particle going in the $\hat{x}$-direction with
respect to an inertial observer $\mathcal{O}$ is given by

$|\Psi\rangle=\frac{1}{\sqrt{2}}\int
dp_{x}\psi(p_{x})|p_{x}\rangle\otimes\left(|0\rangle+|1\rangle\right)$ (17)

with $p^{\mu}=(m\cosh\zeta,m\sinh\zeta\hat{x})$ being the two-momentum of the
particle and $m$ being the mass of the particle. The density matrix
corresponding to this state $|\Psi\rangle$ is therefore given by

$\rho=\frac{1}{2}\int dp_{x_{1}}\int
dp_{x_{2}}\psi(p_{x_{1}})\psi^{*}(p_{x_{2}})|p_{x_{1}}\rangle\langle
p_{x_{2}}|\otimes(\mathbb{1}+\Sigma_{1})~{}.$ (18)

Hence, the reduced density matrix is given by

$\rho=\frac{1}{2}\int
dp_{x}\psi(p_{x})\psi^{*}(p_{x})(\mathbb{1}+\Sigma_{1})~{}.$ (19)

In the frame of $\mathcal{O^{\prime}}$ moving with velocity
$\vec{v}=\tanh\alpha\,\hat{z}$, the state of the particle is given by

$\begin{split}|\Psi_{\Lambda}\rangle&=\frac{1}{\sqrt{2}}\int
dp_{x}\psi(p_{x})\sqrt{\frac{(\Lambda
p)^{0}}{p^{0}}}\mathcal{D}(W(\Lambda,p_{x}))(|\Lambda p_{x},0\rangle\\\
&+|\Lambda p_{x},1\rangle)\,,\end{split}$ (20)

where

$\mathcal{D}(W(\Lambda,p_{x}))=\cos\frac{\phi}{2}\mathds{1}+i\sin\frac{\phi}{2}(\Sigma_{2})\,$
(21)

with

$\displaystyle\cos\frac{\phi}{2}$ $\displaystyle=$
$\displaystyle\frac{\cosh\frac{\alpha}{2}\cosh\frac{\zeta}{2}}{\sqrt{\frac{1}{2}+\frac{1}{2}\cosh\alpha\cosh\zeta}}$
$\displaystyle\sin\frac{\phi}{2}$ $\displaystyle=$
$\displaystyle\frac{\sinh\frac{\alpha}{2}\sinh\frac{\zeta}{2}}{\sqrt{\frac{1}{2}+\frac{1}{2}\cosh\alpha\cosh\zeta}}~{}.$
(22)

Here we have employed $\hat{z}\cdot\hat{x}=0\text{ and
}\hat{z}\times\hat{x}=\hat{y}$. As $\hat{n}=\hat{y}$ with $\hat{n}$ denoting
the alignment of the direction of the axis about which the Wigner rotation
takes place. It is quite straightforward to say that the axis of rotation is
directed along the $y$ axis.
Substituting eq.(s)(21, 22) in eq.(20), we get

$\displaystyle|\Psi_{\Lambda}\rangle=$ $\displaystyle\frac{1}{\sqrt{2}}$
$\displaystyle\int dp_{x}\,\sqrt{\frac{(\Lambda
p)^{0}}{p^{0}}}\psi(p_{x})\Bigg{[}\left(\cos\frac{\phi}{2}+\sin\frac{\phi}{2}\right)\Bigg{.}$
(23) $\displaystyle\times$ $\displaystyle\Bigg{.}|\Lambda
p_{x},0\rangle+\left(\cos\frac{\phi}{2}-\sin\frac{\phi}{2}\right)|\Lambda
p_{x},1\rangle\Bigg{]}\,.$

The density matrix corresponding to the above state is given by

$\begin{split}\varrho_{\Lambda}&=\frac{1}{2}\int dp_{x_{1}}\int
dp_{x_{2}}\,\sqrt{\frac{(\Lambda p_{1})^{0}(\Lambda
p_{2})^{0}}{p_{1}^{0}p_{2}^{0}}}\psi(p_{x_{1}})\psi^{\ast}({p}_{x_{2}})\\\
&\times\bigg{[}\mathcal{A}_{1}\mathcal{A}_{2}|\Lambda{p}_{x_{1}},0\rangle\langle\Lambda{p}_{x_{2}},0|+\mathcal{A}_{1}\mathcal{B}_{2}|\Lambda{p}_{x_{1}},0\rangle\langle\Lambda{p}_{x_{2}},1|\bigg{.}\\\
&\times\bigg{.}\mathcal{A}_{2}\mathcal{B}_{1}|\Lambda{p}_{x_{1}},1\rangle\langle\Lambda{p}_{x_{2}},0|+\mathcal{B}_{1}\mathcal{B}_{2}|\Lambda{p}_{x_{1}},1\rangle\langle\Lambda{p}_{x_{2}},1|\bigg{]}\end{split}$
(24)

where
$\mathcal{A}_{i}=\left(\cos\frac{\phi_{i}}{2}+\sin\frac{\phi_{i}}{2}\right)$
and
$\mathcal{B}_{i}=\left(\cos\frac{\phi_{i}}{2}-\sin\frac{\phi_{i}}{2}\right);i=1,2$.

The reduced density matrix corresponding to $\varrho_{\Lambda}$ is given by

$\displaystyle\rho_{\Lambda}=$ $\displaystyle\frac{1}{2}$ $\displaystyle\int
dp_{x}\,\psi(p_{x})\psi^{\ast}(p_{x})\big{[}\mathcal{A}^{2}|0\rangle\langle
0|+\mathcal{A}\mathcal{B}\bigl{(}|0\rangle\langle 1|\big{.}\bigl{.}$ (25)
$\displaystyle\big{.}\bigr{.}+|1\rangle\langle
0|\bigr{)}+\mathcal{B}^{2}|1\rangle\langle 1|\big{]}$

One can now consider particular forms of $\psi(p_{x})$ to calculate the
reduced density $\rho_{\Lambda}$.
Let us consider $\psi(p_{x})$ in the following form222If one starts with a
three dimensional construction where $\rho_{\Lambda}=\frac{1}{2}\int
d^{3}\mathbf{p}\,\psi(\mathbf{p})\psi^{\ast}(\mathbf{p})\big{[}\mathcal{A}^{2}|0\rangle\langle
0|+\mathcal{A}\mathcal{B}\bigl{(}|0\rangle\langle
1|\big{.}\bigl{.}\,\big{.}\bigr{.}+|1\rangle\langle
0|\bigr{)}+\mathcal{B}^{2}|1\rangle\langle 1|\big{]}$, then one needs to just
consider
$\psi(\mathbf{p})=\psi(p_{x})\delta(p_{y})\delta(p_{z})=f(p_{x})\delta(p_{y})\delta(p_{z})$.
This particular choice of $\psi(\mathbf{p})$ restores the form of the reduced
density matrix in eq.(25).

$\displaystyle\psi(p_{x})=f(p_{x})$ (26)

with $f(p_{x})$ being given by

$\displaystyle
f(p_{x})=\frac{1}{\sqrt{\sigma^{2n+1}\Gamma(n+\frac{1}{2})}}p_{x}^{n}\,\mathrm{e}^{-\frac{1}{2}\left(\frac{p_{x}}{\sigma}\right)^{2}}~{}.$
(27)

Note that the case corresponding to $n=0$ was investigated in [40]. In this
work, we consider a more general form of the function $f(p_{x})$. Using
eq.(s)(26, 27) in eq.(25), we get

$\displaystyle\rho_{\Lambda}=$ $\displaystyle\frac{1}{2}$ $\displaystyle\int
dp_{x}\,|f(p_{x})|^{2}\big{[}\mathcal{A}^{2}|0\rangle\langle
0|+\mathcal{A}\mathcal{B}\bigl{(}|0\rangle\langle 1|\big{.}\bigl{.}$ (28)
$\displaystyle\big{.}\bigr{.}+|1\rangle\langle
0|\bigr{)}+\mathcal{B}^{2}|1\rangle\langle 1|\big{]}$

Taking
$\sinh\zeta=\frac{p_{x}}{m},\cosh\zeta=\sqrt{1+\left(\frac{p_{x}}{m}\right)^{2}},\sinh\alpha=a,\text{
and }\cosh\alpha=b$, we get

$\displaystyle\mathcal{A}^{2}$ $\displaystyle=$ $\displaystyle
1+\frac{a\,\frac{p_{x}}{m}}{1+b\,\sqrt{1+\left(\frac{p_{x}}{m}\right)^{2}}}$
$\displaystyle\mathcal{B}^{2}$ $\displaystyle=$ $\displaystyle
1-\frac{a\,\frac{p_{x}}{m}}{1+b\,\sqrt{1+\left(\frac{p_{x}}{m}\right)^{2}}}$
(29) $\displaystyle\mathcal{A\,B}$ $\displaystyle=$
$\displaystyle\frac{b+\sqrt{1+\left(\frac{p_{x}}{m}\right)^{2}}}{1+b\,\sqrt{1+\left(\frac{p_{x}}{m}\right)^{2}}}~{}.$

Therefore, the components of the reduced density matrix $\rho_{\Lambda}$ reads

$\displaystyle\rho_{\Lambda_{11}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int
dp_{x}\,|f(p_{x})|^{2}\left(1+\frac{a\,\frac{p_{x}}{m}}{1+b\,\sqrt{1+\left(\frac{p_{x}}{m}\right)^{2}}}\right)\,,$
$\displaystyle\rho_{\Lambda_{22}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int
dp_{x}\,|f(p_{x})|^{2}\left(1-\frac{a\,\frac{p_{x}}{m}}{1+b\,\sqrt{1+\left(\frac{p_{x}}{m}\right)^{2}}}\right)\,,$
(30) $\displaystyle\rho_{\Lambda_{12}}$ $\displaystyle=$
$\displaystyle\rho_{\Lambda_{21}}=\frac{1}{2}\int
dp_{x}\,|f(p_{x})|^{2}\left(\frac{b+\sqrt{1+\left(\frac{p_{x}}{m}\right)^{2}}}{1+b\,\sqrt{1+\left(\frac{p_{x}}{m}\right)^{2}}}\right)\,.$

Considering the width of the wave function to be small enough compared to the
mass of the particle, that is $\left(\frac{\sigma}{m}\right)\ll 1$, the
components of the density matrix can be calculated and reads

$\displaystyle\rho_{\Lambda_{11}}$ $\displaystyle=$
$\displaystyle\rho_{\Lambda_{22}}=\frac{1}{2}$ (31)
$\displaystyle\rho_{\Lambda_{12}}$ $\displaystyle=$
$\displaystyle\rho_{\Lambda_{21}}=\frac{1}{2}-\frac{2n+1}{8}\left(\frac{\cosh\alpha-1}{\cosh\alpha+1}\right)\left(\frac{\sigma}{m}\right)^{2}~{}.$

For $n=0$, the above results reduce to those given in [40]. Our results show
the effect of $n$ coming through the generalized wave packet that we have
considered in this work. The matrix form of the reduced density matrix can
therefore be written as

$\displaystyle\rho_{\Lambda}=\begin{pmatrix}\frac{1}{2}&\frac{1}{2}-\mathcal{F}\\\
\frac{1}{2}-\mathcal{F}&\frac{1}{2}\end{pmatrix}$ (32)

with
$\mathcal{F}=\frac{2n+1}{8}\left(\frac{\cosh\alpha-1}{\cosh\alpha+1}\right)\left(\frac{\sigma}{m}\right)^{2}$.

With the above result in hand, we would now like to obtain the Frobenius-norm
measure of coherence for the above reduced density matrix. The Frobenius-norm
measure of coherence is defined as [38]

$\displaystyle
C_{F}(\rho)\equiv\sqrt{\frac{d}{d-1}\displaystyle\sum_{i=1}^{d}\left(\lambda_{i}-\frac{1}{d}\right)^{2}}$
(33)

where $d$ is the dimension of the Hilbert space and $\\{\lambda_{i}\\}$ are
the eigenvalues of the density matrix $\rho$. Hence we need to first calculate
the eigenvalues of $\rho_{\Lambda}$ (in eq.(32)). Calculating the eigenvalues
of the reduced density matrix $\rho_{\Lambda}$ from the eq.(32) and using it
in eq.(33), we get

$\displaystyle
C_{F}(\rho_{\Lambda})=\left[1-\frac{2n+1}{4}\left(\frac{\cosh\alpha-1}{\cosh\alpha+1}\right)\left(\frac{\sigma}{m}\right)^{2}\right]~{}.$
(34)

Now as $\alpha$ is the rapidity parameter for the boosted observer, therefore
we can write

$\displaystyle\cosh\alpha=\gamma\equiv\frac{1}{\sqrt{1-(\frac{v}{c})^{2}}}$
(35)

where $v$ is the velocity of the boosted observer with respect to the inertial
one. Therefore, when $v\rightarrow 0$, $\cosh\alpha\rightarrow 1$. In this
limit, eq.(34) becomes

$\displaystyle C_{F}(\rho_{\Lambda})=1\,.$ (36)

On the other hand, when $v\rightarrow c,~{}\text{ then
}\cosh\alpha\rightarrow\infty$. In this limit, eq.(34) becomes

$\displaystyle
C_{F}(\rho_{\Lambda})=\left[1-\frac{2n+1}{4}\left(\frac{\sigma}{m}\right)^{2}\right]\,.$
(37)

Since $C_{F}(\rho_{\Lambda})\geq 0$, hence we get an upper bound of the
parameter $n$to be

$\displaystyle
n\leq\left[2\left(\frac{m}{\sigma}\right)^{2}-\frac{1}{2}\right]~{}.$ (38)

Eq.(s)(37,38) are the main findings in this work.

###  III.2 Spin 1/2 particles under Lorentz boost in 3+1-dimensions

In this subsection, we consider that the particle can move in any arbitrary
direction. In this scenario, the representation of the Wigner’s little group
takes the form

$\displaystyle\mathcal{D}(W(\Lambda,\mathbf{p}))=\frac{\mathcal{M}}{\sqrt{\mathcal{I}\mathcal{J}}}+i\frac{\mathcal{N}}{\sqrt{\mathcal{I}\mathcal{J}}}(\sigma_{y}p_{x}-\sigma_{x}p_{y})\,,$
(39)

where the quantities $\mathcal{I},~{}\mathcal{J},~{}\mathcal{M}$, and
$\mathcal{N}$ are given by

$\displaystyle\mathcal{I}$ $\displaystyle=$
$\displaystyle(p^{0}+m),\qquad\mathcal{J}=(p^{0}\cosh\alpha+p_{z}\sinh\alpha+m),$
$\displaystyle\mathcal{M}$ $\displaystyle=$
$\displaystyle(p^{0}+m)\cosh\frac{\alpha}{2}+p_{z}\sinh\frac{\alpha}{2},\qquad\mathcal{N}=\sinh\frac{\alpha}{2}\,.$

For making the calculation simpler, we choose the state of a single spin-1/2
particle with respect to an inertial observer $\mathcal{O}$ as

$\displaystyle|\Phi\rangle=\int
d^{3}\mathbf{p}\,\phi(\mathbf{p})\,|\mathbf{p}\rangle\otimes|0\rangle\,.$ (41)

where

$\phi(\mathbf{p})=\frac{1}{\sqrt{2\pi\sigma^{2n+3}\Gamma(n+\frac{3}{2})}}\mathbf{p}^{n}\,\mathrm{e}^{-\frac{1}{2}\left(\frac{\mathbf{p}}{\sigma}\right)^{2}}\,,$

$\mathbf{p}=(p_{x},p_{y},p_{z})$ is the three-momentum of the particle and
$p^{0}=\sqrt{\mathbf{p}^{2}+m^{2}}$. The above form is the three dimensional
generalization of the form taken in eq.(27).
Therefore, in the boosted frame this state will be

$\displaystyle|\Phi_{\Lambda}\rangle=\int
d^{3}\mathbf{p}\,\phi(\mathbf{p})\sqrt{\frac{(\Lambda
p)^{0}}{p^{0}}}\mathcal{D}(W(\Lambda,\mathbf{p}))|\Lambda\mathbf{p},0\rangle\,.$
(42)

Putting eq.(39) in eq.(42), we get

$\displaystyle|\Phi_{\Lambda}\rangle$ $\displaystyle=$ $\displaystyle\int
d^{3}\mathbf{p}\,\sqrt{\frac{(\Lambda
p)^{0}}{p^{0}}}\phi(\mathbf{p})\Bigg{[}\frac{\mathcal{M}}{\sqrt{\mathcal{I}\mathcal{J}}}|\Lambda\mathbf{p},0\rangle-\frac{\mathcal{N}}{\sqrt{\mathcal{I}\mathcal{J}}}(p_{x}+ip_{y})\Bigg{.}$
(43) $\displaystyle\Bigg{.}\times|\Lambda\mathbf{p},1\rangle\Bigg{]}\,.$

The reduced density matrix corresponding to the above state takes the form

$\displaystyle\rho_{\Lambda}=$ $\displaystyle\int
d^{3}\mathbf{p}\,|\phi(\mathbf{p})|^{2}\Bigg{[}\frac{\mathcal{M}^{2}}{\mathcal{I}\mathcal{J}}|0\rangle\langle
0|-\frac{\mathcal{M}\mathcal{N}}{\mathcal{I}\mathcal{J}}(p_{x}-ip_{y})|0\rangle\langle
1|\Bigg{.}\Bigg{.}-\frac{\mathcal{M}\mathcal{N}}{\mathcal{I}\mathcal{J}}(p_{x}+ip_{y})|1\rangle\langle
0|+\frac{\mathcal{N}^{2}}{\mathcal{I}\mathcal{J}}(p_{x}^{2}+p_{y}^{2})|1\rangle\langle
1|\Bigg{]}~{}.$ (44)

Therefore the components of the reduced density matrix $\rho_{\Lambda}$ turn
out to be

$\displaystyle\rho_{\Lambda_{11}}$ $\displaystyle=\int
d^{3}\mathbf{p}\,|\phi(\mathbf{p})|^{2}\Bigg{[}\frac{\mathcal{M}^{2}}{\mathcal{I}\mathcal{J}}\Bigg{]}\,,$
$\displaystyle\rho_{\Lambda_{22}}$ $\displaystyle=\int
d^{3}\mathbf{p}\,|\phi(\mathbf{p})|^{2}\Bigg{[}\frac{\mathcal{N}^{2}}{\mathcal{I}\mathcal{J}}(p_{x}^{2}+p_{y}^{2})\Bigg{]}\,,$
$\displaystyle\rho_{\Lambda_{12}}$ $\displaystyle=-\int
d^{3}\mathbf{p}\,|\phi(\mathbf{p})|^{2}\Bigg{[}\frac{\mathcal{M}\mathcal{N}}{\mathcal{I}\mathcal{J}}(p_{x}-ip_{y})\Bigg{]}\,,$
$\displaystyle\rho_{\Lambda_{21}}$ $\displaystyle=-\int
d^{3}\mathbf{p}\,|\phi(\mathbf{p})|^{2}\Bigg{[}\frac{\mathcal{M}\mathcal{N}}{\mathcal{I}\mathcal{J}}(p_{x}+ip_{y})\Bigg{]}\,.$
(45)

From mass-shell condition, the component of the four-momentum $p^{\mu}$ can be
obtained as

$\displaystyle\left(\frac{p^{0}}{m}\right)$
$\displaystyle=\sqrt{1+\left(\frac{p}{m}\right)^{2}}\,,$
$\displaystyle\left(\frac{p_{x}}{m}\right)$
$\displaystyle=\left(\frac{p_{y}}{m}\right)=\left(\frac{p_{z}}{m}\right)=\left(\frac{p}{\sqrt{3}m}\right)\,.$
(46)

Using these values one can calculate the following quantities (appearing in
eq.(45))

$\displaystyle\mathcal{I}\mathcal{J}$
$\displaystyle=(p^{0}+m)(p^{0}\cosh\alpha+p_{z}\sinh\alpha+m)$
$\displaystyle=m^{2}\left(1+\frac{p^{0}}{m}\right)\left(1+\frac{p^{0}}{m}\cosh\alpha+\frac{p_{z}}{m}\sinh\alpha\right)$
$\displaystyle=m^{2}\left(1+\sqrt{1+\left(\frac{p}{m}\right)^{2}}\right)\left[1+b\sqrt{1+\left(\frac{p}{m}\right)^{2}}+\frac{a}{\sqrt{3}}\frac{p}{m}\right]~{},$
(47) $\displaystyle\mathcal{M}^{2}$
$\displaystyle=(p^{0}+m)\left[(p^{0}+m)\cosh^{2}\frac{\alpha}{2}+p_{z}\sinh\alpha\right]+p_{z}^{2}\sinh^{2}\frac{\alpha}{2}$
$\displaystyle=m^{2}\left(1+\frac{p^{0}}{m}\right)\biggl{[}\left(1+\frac{p^{0}}{m}\right)\cosh^{2}\frac{\alpha}{2}+\frac{p_{z}}{m}\sinh\alpha+\frac{(p_{z}/m)^{2}}{\left(1+\frac{p^{0}}{m}\right)}\sinh^{2}\frac{\alpha}{2}\biggr{]}$
$\displaystyle=m^{2}\left(1+\sqrt{1+\left(\frac{p}{m}\right)^{2}}\right)\left[\frac{1}{2}\left(1+\sqrt{1+\left(\frac{p}{m}\right)^{2}}\right)(\cosh\alpha+1)+\frac{1}{\sqrt{3}}\frac{p}{m}\sinh\alpha+\frac{1}{6}\frac{(\cosh\alpha-1)(p/m)^{2}}{\left(1+\sqrt{1+\left(\frac{p}{m}\right)^{2}}\right)}\right]$
$\displaystyle=m^{2}\left(1+\sqrt{1+\left(\frac{p}{m}\right)^{2}}\right)\left[\frac{(b+1)}{2}\left(1+\sqrt{1+\left(\frac{p}{m}\right)^{2}}\right)+\frac{a}{\sqrt{3}}\frac{p}{m}+\frac{(b-1)}{6\left(1+\sqrt{1+\left(\frac{p}{m}\right)^{2}}\right)}\left(\frac{p}{m}\right)^{2}\right]~{}.$
(48)

Using eq.(s)(47, 48) we get

$\displaystyle\frac{\mathcal{M}^{2}}{\mathcal{I}\mathcal{J}}=\left[\frac{\frac{(b+1)}{2}\left(1+\sqrt{1+\left(\frac{p}{m}\right)^{2}}\right)+\frac{a}{\sqrt{3}}\frac{p}{m}+\frac{(b-1)}{6\left(1+\sqrt{1+\left(\frac{p}{m}\right)^{2}}\right)}\left(\frac{p}{m}\right)^{2}}{\left(1+b\sqrt{1+\left(\frac{p}{m}\right)^{2}}+\frac{a}{\sqrt{3}}\frac{p}{m}\right)}\right]~{}.$
(49)

Similarly, the other coefficients appearing in eq.(45) are given by

$\displaystyle\frac{\mathcal{N}^{2}}{\mathcal{I}\mathcal{J}}(p_{x}^{2}+p_{y}^{2})$
$\displaystyle=\left[\frac{(b-1)\left(\frac{p}{m}\right)^{2}}{3\left(1+\sqrt{1+\left(\frac{p}{m}\right)^{2}}\right)\left(1+b\sqrt{1+\left(\frac{p}{m}\right)^{2}}+\frac{a}{\sqrt{3}}\frac{p}{m}\right)}\right]~{},$
(50)
$\displaystyle\frac{\mathcal{M}\mathcal{N}}{\mathcal{I}\mathcal{J}}(p_{x}\mp
ip_{y})$
$\displaystyle=\left[\frac{\left(\frac{a}{2\sqrt{3}}+\frac{(b-1)}{6\left(1+\sqrt{1+\left(\frac{p}{m}\right)^{2}}\right)}\left(\frac{p}{m}\right)\right)\left(\frac{p}{m}\right)(1\mp
i)}{\left(1+b\sqrt{1+\left(\frac{p}{m}\right)^{2}}+\frac{a}{\sqrt{3}}\frac{p}{m}\right)}\right]~{}.$
(51)

Now taking an approximation upto
$\mathcal{O}(\left(\frac{\sigma}{m}\right)^{2})$, eq.(s)(48 \- 51) take the
following form

$\displaystyle\frac{\mathcal{M}^{2}}{\mathcal{I}\mathcal{J}}=$ $\displaystyle
1-\frac{1}{6}\left(\frac{b-1}{b+1}\right)\left(\frac{p}{\sigma}\right)^{2}\left(\frac{\sigma}{m}\right)^{2}~{},$
(52)
$\displaystyle\frac{\mathcal{N}^{2}}{\mathcal{I}\mathcal{J}}(p_{x}^{2}+p_{y}^{2})=$
$\displaystyle\frac{1}{6}\left(\frac{b-1}{b+1}\right)\left(\frac{p}{\sigma}\right)^{2}\left(\frac{\sigma}{m}\right)^{2}~{},$
(53)
$\displaystyle\frac{\mathcal{M}\mathcal{N}}{\mathcal{I}\mathcal{J}}(p_{x}\mp
ip_{y})=$ $\displaystyle(1\mp
i)\biggl{[}\frac{1}{2\sqrt{3}}\sqrt{\frac{b-1}{b+1}}\left(\frac{p}{\sigma}\right)\left(\frac{\sigma}{m}\right)-\frac{1}{12}\left(\frac{b-1}{b+1}\right)\left(\frac{p}{\sigma}\right)^{2}\left(\frac{\sigma}{m}\right)^{2}\biggr{]}$
(54)

where we have used $a=\sqrt{b^{2}-1}$.

Now substituting eq.(s) (52 \- 54) into eq.(45) and carrying out the
integrations, the components of the reduced density matrix takes the following
form

$\displaystyle\rho_{\Lambda_{11}}$
$\displaystyle=\left[1-\frac{2n+3}{12}\left(\frac{\cosh\alpha-1}{\cosh\alpha+1}\right)\left(\frac{\sigma}{m}\right)^{2}\right]~{},$
(55) $\displaystyle\rho_{\Lambda_{22}}$
$\displaystyle=\frac{2n+3}{12}\left(\frac{\cosh\alpha-1}{\cosh\alpha+1}\right)\left(\frac{\sigma}{m}\right)^{2}~{},$
(56) $\displaystyle\rho_{\Lambda_{12}}$
$\displaystyle=-(1-i)\left[\frac{1}{2\sqrt{3}}\sqrt{\frac{\cosh\alpha-1}{\cosh\alpha+1}}\frac{\Gamma(n+2)}{\Gamma(n+3/2)}\left(\frac{\sigma}{m}\right)-\frac{2n+3}{24}\left(\frac{\cosh\alpha-1}{\cosh\alpha+1}\right)\left(\frac{\sigma}{m}\right)^{2}\right]~{},$
(57) $\displaystyle\rho_{\Lambda_{21}}$
$\displaystyle=-(1+i)\left[\frac{1}{2\sqrt{3}}\sqrt{\frac{\cosh\alpha-1}{\cosh\alpha+1}}\frac{\Gamma(n+2)}{\Gamma(n+3/2)}\left(\frac{\sigma}{m}\right)-\frac{2n+3}{24}\left(\frac{\cosh\alpha-1}{\cosh\alpha+1}\right)\left(\frac{\sigma}{m}\right)^{2}\right]~{}.$
(58)

Defining

$\displaystyle\mathcal{G}\equiv$
$\displaystyle\frac{2n+3}{12}\left(\frac{\cosh\alpha-1}{\cosh\alpha+1}\right)\left(\frac{\sigma}{m}\right)^{2}~{},$
(59) $\displaystyle\mathcal{H}\equiv$
$\displaystyle\biggl{[}\frac{1}{2\sqrt{3}}\sqrt{\frac{\cosh\alpha-1}{\cosh\alpha+1}}\frac{\Gamma(n+2)}{\Gamma(n+3/2)}\left(\frac{\sigma}{m}\right)$
$\displaystyle-\frac{2n+3}{24}\left(\frac{\cosh\alpha-1}{\cosh\alpha+1}\right)\left(\frac{\sigma}{m}\right)^{2}\biggr{]},$
(60)

we can recast the spin-reduced density matrix as

$\displaystyle\rho_{\Lambda}=\begin{pmatrix}1-\mathcal{G}&-(1-i)\mathcal{H}\\\
-(1+i)\mathcal{H}&\mathcal{G}\end{pmatrix}~{}.$ (61)

Hence calculating the eigenvalues of the reduced density matrix
$\rho_{\Lambda}$ from eq.(61), and substituting it in eq.(33) and
approximating upto $\mathcal{O}(\left(\frac{\sigma}{m}\right)^{2})$, we
finally obtain the coherence measure to be

$\displaystyle C_{F}(\rho_{\Lambda})=$
$\displaystyle\biggl{[}1-\frac{(2n+3)}{6}\left[1-\frac{2}{(2n+3)}\left\\{\frac{\Gamma(n+2)}{\Gamma(n+3/2)}\right\\}^{2}\right]\left(\frac{\cosh\alpha-1}{\cosh\alpha+1}\right)\left(\frac{\sigma}{m}\right)^{2}\biggr{]}~{}.$
(62)

For $n=0$, the above result reduces to

$\begin{split}C_{F}(\rho_{\Lambda})&=\left[1-\frac{1}{2}\left(1-\frac{8}{3\pi}\right)\left(\frac{\cosh\alpha-1}{\cosh\alpha+1}\right)\left(\frac{\sigma}{m}\right)^{2}\right]\end{split}~{}.$
(63)

The above result differes from the result in [40] by the numerical factor
appearing in the second term. The reason for this discripancy is that in [40],
the off-diagonal terms in the reduced density matrix were considered to be
small and hence were dropped as an approximation.

##  IV Frobenius-norm measure of coherence and its plots

In this section, we shall plot the Frobenius-norm measure of coherence that we
have obtained in the earlier section with respect to the change in Gaussian
width for different values of the boost parameter. At first we are going to
consider the coherence measure obtained in eq.(34) for electron with rest mass
$m\approx 0.5\text{ MeV}$. For this case, the measure of coherence is valid
upto $\sigma<0.5\text{ MeV}$. In eq.(34),
$\cosh\alpha=\frac{1}{\sqrt{1-\beta^{2}}}$ where $\beta=\frac{v}{c}$ is the
boost parameter. The plot of $C_{F}(\rho)$ (for electrons) vs $\sigma$ for
different values of $\beta$ and a fixed value of $n=2$ is given in Figure(1).
From Figure(1), we observe that there is a substantial amount of loss in
coherence for large boosts (e.g. $\beta=0.80,0.99$) with the change in the
parameter $\sigma$ in contrast to lower values of the boost parameter(e.g.
$\beta=0.3,0.0$).

Figure 1: $C_{F}(\rho)$ vs $\sigma$ (for electron) with
$\beta~{}=~{}0.99,~{}0.8,~{}0.3,~{}0.0$ and $n=2$.

The Frobenius-norm measure of coherence obtained in eq.(62) for a neutron with
a rest mass energy $m=939.36\text{ MeV}$. As before we will need to consider
$\sigma<939.36\text{ MeV}$ is plotted in Figure (2). As opposed to Figure(1),
we observe a relatively lesser loss of coherence for identical boost
parameters in Figure(2).

Figure 2: $C_{F}(\rho)$ vs $\sigma$ (for neutron) with
$\beta~{}=~{}0.99,~{}0.8,~{}0.3,~{}0.0$ and $n=2$. Figure 3: $C_{F}(\rho)$ vs
$\sigma$ with $\beta~{}=~{}0.99$ and $n=2$ for electron and neutron.

In Figure(3), we plot $C_{F}(\rho)$ for both electron and neutron ($n=2$) and
with the boost parameter fixed at $\beta=0.99$. We have plotted upto
$\sigma=0.49\text{ MeV}$ as beyond $\sigma=0.5\text{ MeV}$ the analytical form
of $C_{F}(\rho)$ for the electron does not hold any more (as it violates the
$\sigma<m~{}(0.5\text{ MeV})$ condition). We observe a considerable amount of
loss in the measure of coherence for an electron (than the neutron) even for a
high boost parameter ($\beta=0.99$). Next, we shall look at the dependence of
the Frobenius norm measure of coherence on the parameter $n$ in case of an
electron. With $\beta=0.99$, $\sigma=0.49\text{ MeV}$ and $m=0.5MeV$ (along
with the condition $C_{F}(\rho)\geq 0$), we get $n<2.27$ from the relation in
eq.(34). Therefore, we plot $C_{F}(\rho)$ vs $\sigma$ for $n=0,~{}1$ and $2$
in Figure(4). We observe from Figure(4) that with the increase in the
parameter $n$, the loss of coherence becomes more significant than before.

Figure 4: $C_{F}(\rho)$ vs $\sigma$ (for electron) with $n~{}=~{}0,~{}1,~{}2$
and $\beta=0.99$.

##  V Conclusion

In this work, we consider a single particle entangled state and its quantum
coherence under Lorentz boost. At first, we considered a generalized form of
the Gaussian wave packet (with its peak at the origin), where the general
Gaussian wave function (in momentum space) is multiplied by $n$-th power of
the momentum $p$. For a given boost in one direction ($(1+1)$-dimensional
consideration) we obtained the Frobenius-norm measure of coherence with a
dependence on the boost parameter $\beta$, Gaussian width $\sigma$, the mass
of the fermion, and the arbitrary parameter $n$. We then obtain a bound on
this parameter $n$ in the $\beta\rightarrow 1$ limit. From Figure(1) in
section(IV), we observe that for a zero Lorentz boost there will be no loss in
the Frobenius-norm measure of coherence in the case of an electron. We also
observe that for a relativistic observer there will be a significant amount of
loss in the measure of coherence depending on the value of the Lorentz boost.
Due to the Wigner rotation [41] of a single particle entangled state under
Lorentz boost, the spin and momentum degrees of freedom couple with each other
resulting in such an effect. We also plot this measure of coherence with
respect to the Gaussian width parameter for a fixed value of Lorentz boost and
different values of the arbitrary parameter $n$ in the case of an electron. We
observe that with the increase in the parameter $n$, the loss in the measure
of coherence becomes much more severe than before. For the next part of our
analysis, we have considered a single particle along with a Lorentz boost in
all three spatial directions. In this case, our main motivation is to consider
a neutron as a single particle. Finally, we also observe a similar loss in the
measure of coherence from the perspective of a relativistic observer and the
loss in the Frobenius-norm measure of coherence was negligible in this case
with respect to the case of an electron.

##  Appendix A Single particle quantum state under Lorentz boost

In this Appendix, we shall derive the form of the single particle quantum
state after Lorentz boost has been applied333We shall essentially follow the
analysis in [41].. Under a pure Lorentz boost $\Lambda$,
$\mathcal{U}(\Lambda,a)$ reads

$\displaystyle\mathcal{U}(\Lambda,a)|\mathbf{p},\sigma\rangle\rightarrow\mathcal{U}(\Lambda,0)|\mathbf{p},\sigma\rangle\equiv\mathcal{U}(\Lambda)|\mathbf{p},\sigma\rangle\,.$
(64)

New eigenvalues of the momentum operator $P^{\mu}$ become

$\displaystyle
P^{\mu}\big{[}\mathcal{U}(\Lambda)|\mathbf{p},\sigma\rangle\big{]}$
$\displaystyle=$
$\displaystyle\left(\mathcal{U}(\Lambda)\,\mathcal{U}^{-1}(\Lambda)\right)P^{\mu}\big{[}\mathcal{U}(\Lambda)|\mathbf{p},\sigma\rangle\big{]}$
(65) $\displaystyle=$
$\displaystyle\mathcal{U}(\Lambda)\big{[}\mathcal{U}^{-1}(\Lambda)P^{\mu}\mathcal{U}(\Lambda)\big{]}|\mathbf{p},\sigma\rangle$
$\displaystyle=$
$\displaystyle\mathcal{U}(\Lambda)\big{[}\Lambda^{\mu}_{\hskip
5.69046pt\nu}\,P^{\nu}\big{]}|\mathbf{p},\sigma\rangle$ $\displaystyle=$
$\displaystyle\Lambda^{\mu}_{\hskip
5.69046pt\nu}\,p^{\nu}\big{[}\mathcal{U}(\Lambda)|\mathbf{p},\sigma\rangle\big{]}~{}.$

To derive the previous equation, we have used the following relationship

$\displaystyle\mathcal{U}^{-1}(\Lambda)P^{\mu}\mathcal{U}(\Lambda)=\Lambda^{\mu}_{\hskip
5.69046pt\nu}\,P^{\nu}\,.$ (66)

Therefore, under pure Lorentz boost, eigenvalues of the momentum operator
$P^{\mu}$ changes as $p^{\mu}\rightarrow\Lambda^{\mu}_{\hskip
5.69046pt\nu}\,p^{\nu}$. To find out the effect of the Lorentz boosts on the
spin degrees of freedom $\sigma$, it can be assumed that the spin degrees of
freedom $\sigma$ form a complete basis, such as any transformation of the
state $|\mathbf{p},\sigma\rangle$ with a specific value of $\sigma$ can be
represented as a linear superposition of all the possible $\sigma$ states.
Therefore, $\mathcal{U}(\Lambda)|\mathbf{p},\sigma\rangle$ can be represented
as

$\displaystyle\mathcal{U}(\Lambda)|p,\sigma\rangle=\displaystyle\sum_{\sigma^{\prime}}\mathcal{C}_{\sigma^{\prime}\sigma}(\Lambda,\mathbf{p})|\Lambda\mathbf{p},\sigma\rangle\,$
(67)

where the $\mathcal{C}_{\sigma^{\prime}\sigma}$ are complex numbers and
$\Lambda\mathbf{p}$ is the spatial component of the Lorentz transformed four-
momentum.
To determine the structure of the $\mathcal{C}_{\sigma^{\prime}\sigma}$ and
the effect of the Lorentz boosts $\Lambda$ on a quantum state
$|p,\sigma\rangle$, we use two invariant quantities, namely

$\displaystyle p^{2}\equiv p^{\mu}p_{\mu}=\vec{p}\cdot\vec{p}-E^{2}$
$\displaystyle=$ $\displaystyle-m^{2}$ $\displaystyle p^{0}$ $\displaystyle=$
$\displaystyle E$ (68)

Hence, we can use these two invariant quantities to classify states into
specific classes. For each value of $p^{2}$ and for each sign ( $p^{0}$), it
is possible to choose a ‘standard’ 4-momentum $k^{\mu}$ that identifies a
specific class of quantum states [41]. For massive particles, we can fix the
standard 4-momentum $k^{\mu}$ to be the particle’s momentum in the rest frame,
which is $k^{\mu}=(m,0,0,0)$. Then, any 4-momenta $p^{\mu}$ can be expressed
in terms of the standard momentum as

$\displaystyle p^{\mu}=(L(p)k)^{\mu}=L^{\mu}_{\hskip 5.69046pt\nu}(p)k^{\nu}$
(69)

where $L(p)$ is a Lorentz transformation that depends on $p^{\mu}$ and takes
$k^{\mu}\rightarrow p^{\mu}$. Therefore, quantum states
$|\mathbf{p},\sigma\rangle$ can be defined in terms of the standard momentum
state $|\mathbf{k},\sigma\rangle$ as

$\displaystyle|\mathbf{p},\sigma\rangle=N(p)\,\mathcal{U}(L(p))|\mathbf{k},\sigma\rangle$
(70)

where $N(p)$ is a normalization constant and $\mathbf{k}$ is the spatial
component of the standard 4-momentum.
Applying a Lorentz boost $\Lambda$ on $|\mathbf{p},\sigma\rangle$, we get

$\displaystyle\mathcal{U}(\Lambda)|\mathbf{p},\sigma\rangle$ $\displaystyle=$
$\displaystyle
N(p)\,\mathcal{U}(\Lambda)\,\mathcal{U}(L(p))|\mathbf{k},\sigma\rangle$ (71)
$\displaystyle=$ $\displaystyle N(p)\,\mathcal{U}(I)\,\mathcal{U}(\Lambda
L(p))|\mathbf{k},\sigma\rangle$ $\displaystyle=$ $\displaystyle
N(p)\,\mathcal{U}(L(\Lambda p))\,\mathcal{U}^{-1}(L(\Lambda
p))\,\mathcal{U}(\Lambda L(p))|\mathbf{k},\sigma\rangle$ $\displaystyle=$
$\displaystyle N(p)\,\mathcal{U}(L(\Lambda p))\,\mathcal{U}(L^{-1}(\Lambda
p))\,\mathcal{U}(\Lambda L(p))|\mathbf{k},\sigma\rangle$ $\displaystyle=$
$\displaystyle N(p)\,\mathcal{U}(L(\Lambda p))\,\mathcal{U}(L^{-1}(\Lambda
p)\Lambda L(p))|\mathbf{k},\sigma\rangle$ $\displaystyle=$ $\displaystyle
N(p)\,\mathcal{U}(L(\Lambda
p))\,\mathcal{U}(W(\Lambda,p))|\mathbf{k},\sigma\rangle$

where $W(\Lambda,\mathbf{p})=L^{-1}(\Lambda\mathbf{p})\Lambda L(\mathbf{p})$
is called Wigner rotation, which leaves the standard momentum $\mathbf{k}$
invariant, and only acts on the internal degrees of freedom of
$|\mathbf{k},\sigma\rangle:\,k\xrightarrow{L}p\xrightarrow{\Lambda}\Lambda
p\xrightarrow{L^{-1}}k$. The set of Wigner rotations forms a group known as
the little group, which is a subgroup of the Poincare group. Hence, the final
momentum in the rest frame is different from the original one by a Wigner
rotation,

$\displaystyle\mathcal{U}(W(\Lambda,\mathbf{p}))|\mathbf{k},\sigma\rangle=\displaystyle\sum_{\sigma^{\prime}}\mathcal{D}_{\sigma^{\prime}\sigma}(W(\Lambda,\mathbf{p}))|\mathbf{k},\sigma^{\prime}\rangle$
(72)

where $\mathcal{D}(W(\Lambda,\mathbf{p}))$ is the unitary representation of
the little group.
On the other hand, $\mathcal{U}(L(\Lambda\mathbf{p}))$ takes
$\mathbf{k}\rightarrow\Lambda\mathbf{p}$ without affecting the spin, by
definition. Therefore,

$\displaystyle\mathcal{U}(\Lambda)|\mathbf{p},\sigma\rangle$ $\displaystyle=$
$\displaystyle
N(p)\,\mathcal{U}(L(\Lambda\mathbf{p}))\,\mathcal{U}(W(\Lambda,\mathbf{p}))|\mathbf{k},\sigma\rangle$
(73) $\displaystyle=$ $\displaystyle
N(p)\,\mathcal{U}(L(\Lambda\mathbf{p}))\,\displaystyle\sum_{\sigma^{\prime}}\mathcal{D}_{\sigma^{\prime}\sigma}(W(\Lambda,\mathbf{p}))|\mathbf{k},\sigma^{\prime}\rangle$
$\displaystyle=$ $\displaystyle
N(p)\,\displaystyle\sum_{\sigma^{\prime}}\mathcal{D}_{\sigma^{\prime}\sigma}(W(\Lambda,\mathbf{p}))\,\mathcal{U}(L(\Lambda\mathbf{p}))|\mathbf{k},\sigma^{\prime}\rangle$
$\displaystyle=$ $\displaystyle\frac{N(p)}{N(\Lambda
p)}\displaystyle\sum_{\sigma^{\prime}}\mathcal{D}_{\sigma^{\prime}\sigma}(W(\Lambda,\mathbf{p}))|\Lambda\mathbf{p},\sigma^{\prime}\rangle\,$

where we have used $|\Lambda\mathbf{p},\sigma^{\prime}\rangle=N(\Lambda
p)\,\mathcal{U}(L(\Lambda\mathbf{p}))|\mathbf{k},\sigma^{\prime}\rangle$.
The ratio $\frac{N(p)}{N(\Lambda p)}$ can be fixed by using appropriate
normalization condition. Hence, under pure Lorentz boosts $\Lambda$, a single
particle quantum state $|\mathbf{p},\sigma\rangle$ transforms as

$\displaystyle\mathcal{U}(\Lambda)|\mathbf{p},\sigma\rangle=\sqrt{\frac{(\Lambda
p)^{0}}{p^{0}}}\displaystyle\sum_{\sigma^{\prime}}\mathcal{D}_{\sigma^{\prime}\sigma}(W(\Lambda,\mathbf{p}))|\Lambda\mathbf{p},\sigma^{\prime}\rangle\,$

which is eq.(II) in the main text.

## References

  * [1] A. Einstein, D. Podolsky, and N. Rosen, Phys. Rev. 47 (1935) 777.
  * [2] E. Schrodinger, Math. Proc. Cambridge Philos. Soc. 31 (2008) 555.
  * [3] J. S. Bell, Phys. Physique Fiz. 1 (1964) 195.
  * [4] J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, Phys. Rev. Lett. 23 (1969) 880; Erratum: Phys. Rev. Lett. 24 (549) 1970.
  * [5] R. M. Gingrich and C. Adami, Phys. Rev. Lett. 89 (2002) 270402.
  * [6] A. Peres and D. R. Terno, Rev. Mod. Phys. 76 (2004) 93.
  * [7] M. Czachor, Phys. Rev. A 55 (1997) 72.
  * [8] A. Peres, P. F. Scudo, and D. R. Terno, Phys. Rev. Lett. 88 (2002) 230402.
  * [9] P. M. Alsing and G. J. Milburn, Quantum Inf. Comput. 2 (2002) 487.
  * [10] D. Ahn, H. J. Lee, Y. H. Moon, and S. W. Hwang, Phys. Rev. A 67 (2003) 012103.
  * [11] P. M. Alsing and G. J. Milburn, Phys. Rev. Lett. 91 (2003) 180404.
  * [12] A. Peres and D. R. Terno, J. Mod. Opt. 50 (2003) 1165.
  * [13] J. Pachos and E. Solano, Quantum Inf. Comput. 3 (2003) 115.
  * [14] C. Gonera, P. Konsiński, and P. Maślanka, Phys. Rev. A 70 (2004) 034102.
  * [15] L. Lamata, M. A. Martin-Delgado, and E. Solano, Phys. Rev. Lett. 97 (2006) 250502.
  * [16] S. D. Bartlett and D. R. Terno, Phys. Rev. A 71 (2005) 012302.
  * [17] T. F. Jordan, A. Shaji, and E. C. G. Sudarshan, Phys. Rev. A 73 (2006) 032104.
  * [18] T. F. Jordan, A. Shaji, and E. C. G. Sudarshan, Phys. Rev. A 75 (2007) 022101.
  * [19] I. Fuentes-Schuller and R. B. Mann, Phys. Rev. Lett. 95 (2005) 120404.
  * [20] P. M. Alsing, I. Fuentes-Schuller, R. B. Mann, and T. A. Tessier, Phys. Rev. A 74 (2006) 032326.
  * [21] G. Adesso, I. Fuentes-Schuller, and M. Ericsson, Phys. Rev. A 76 (2007) 062112.
  * [22] S. Lloyd, Phys. Rev. Lett. 96 (2006) 061302.
  * [23] J. Eisert, M. Cramer, and M. B. Plenio, Rev. Mod. Phys. 82 (2010) 277.
  * [24] E. Livine and D. Terno, Phys. Rev. D 75 (2007) 084001.
  * [25] A. S. Majumdar, D. Home, and S. Sinha, Phys. Lett. B 679 (2009) 167.
  * [26] J. Maldacena, Prog. Phys. 64 (2016) 10.
  * [27] T. Baumgratz, M. Cramer, and M. B. Plenio, Phys. Rev. Lett. 113 (2014) 140401.
  * [28] M. Horodecki and J. Oppenheim, Int. J. Mod. Phys. B 27 (2013) 1345019.
  * [29] F. G. S. L. Brandão and G. Gour, Phys. Rev. Lett. 115 (2015) 070503; Erratum: Phys. Rev. Lett. 115 (2015) 199901.
  * [30] G. Gour and R. W. Spekkens, New J. Phys. 10 (2008) 033023.
  * [31] G. Gour,I. Marvian, and R. W. Spekkens, Phys. Rev. A 80 (2009) 012307.
  * [32] I. Marvian and R. W. Spekkens, New J. Phys. 15 (2013) 033001.
  * [33] I. Marvian and R. W. Spekkens, Phys. Rev. A 90 (2015) 062110.
  * [34] I. Marvian and R. W. Spekkens, Nat. Commun. 5 (2014) 3821.
  * [35] I. Marvian, R. W. Spekkens, and P. Zanardi, Phys. Rev. A 93 (2016) 052331.
  * [36] I. Marvian and R. W. Spekkens, Phys. Rev. A. 94 (2016) 052324.
  * [37] E. Chitambar and G. Gour, Phys. Rev. A 94 (2016) 052336.
  * [38] Y. Yao, G. H. Dong, X. Xiao, and C. P. Sun, Sci. Rep. 6 (2016) 32010.
  * [39] W.-C. Wang, M.-F. Fang, and M. Yu, arXiv: 1701.05110.
  * [40] R. Chatterjee and A. S. Majumdar, Phys. Rev. A 96 (2017) 052301.
  * [41] S. Weinberg, The Quantum Theory of Fields: Volume I: Foundations, Cambridge University Press (1995), Cambridge.