#  Plasma-Parameter Dependence of Ro-Vibrational Temperatures
for $\mathrm{H}_{2}$ in LHD Divertor

Keisuke Fujii<EMAIL_ADDRESS>Tsubasa Oshioka  Atsushi Niihama  Kuzmin
Arseniy  Taiichi Shikama  Masahiro Kobayashi  Masahiro Hasuo  the LHD
Experiment Group  Oak Ridge National Laboratory, Oak Ridge, TN 37831-6169,
United States of America  Department of Mechanical Engineering and Science,
Graduate School of Engineering, Kyoto University Kyoto 615-8540, Japan
National Institute for Fusion Science, Toki, Gifu, 5909-5292, Japan

###### Abstract

We analyzed a thousand visible spectra of Fulcher-$\alpha$ band measured for
divertor plasmas in Large Helical Device. With a coronal model and Baysian
inference, the population distribution of hydrogen molecule in the
electronical ground state were estimated. The non-thermal population
distribution was recovered with a two-temperature model, which has two sets of
rotational and vibrational temperatures, as well as their mixture coefficient.
The lower rotational temperature significantly changes according to the plasma
parameters. Its nearly linear dependence on the electron density was found,
which is consistent with previous works. The lower vibrational temperature
also shows a small density dependence, as reported by a previous work. On the
other hand, the higher rotational and vibrational temperatures as well as the
mixture coefficient only show slight changes over the broad range of plasma
parameters. These population parameters show a significant correlation; with
higher electron density, all the temperatures and the fraction of the higher-
temperature component increase simultaneously. This suggests that the
electron-impact plays an important role to determine the population
distribution.

##  1 Introduction

Hydrogen molecules and their isotopes play a key role on chemical reactions in
the divertor of magnetic fusion devices. The rates of many molecular relevant
processes, such as the formation of negative ions [1, 13], dissociation and
ionization [21], and dissociative attachment [5, 19, 18], are known to be
significantly influenced by the rotational and vibrational population
distribution in the electronic ground state of hydrogen molecules. Rotational
and vibrational populations in the electronic ground state ($X$ state) have
been estimated for variety of plasmas, based on the direct measurement with
vaccuum ultraviolet lasers [20, 26, 31, 12] or the combination of visible
emission observations and population-kinetics models (e.g., a corona model and
collisional-radiative model) [33, 7, 4]. In many cases, non-thermal population
distribution has been observed, where the vibrational temperature is
significantly higher than the rotational temperature [33] and highly excited
rotational states are excessively populated than expected from a Boltzmann
distribution [31, 12, 4]. A Monte-Carlo simulation has also predicted the
nonthermal population distribution of hydrogen molecules [23].

The rotational temperature, which is estimated from the population at the low-
rotational-quantum-number states, has been thought close to the wall
temperatures [27, 32], because the energy gap among the rotational states is
in the similar scale to the room temperature. Addition to the wall effect,
cascade from upper electronic levels may also change the rotational
temperature. This effect has been attributed to account the electron-density
dependence of the rotational temperature found in the spectroscopic
observation of magnetic plasma confinement devices [2, 30, 15].

A surface effect has been considered as the origin of the higher vibrational
temperature. The surface assisted recombination of two hydrogen atoms results
in the highly excited molecules [5]. Although, a plasma-condition dependence
on the vibrational temperatures have been studied [8], the connection to the
elementary processes are still unknown.

Despite the understandings of such microscopic processes, the full
understandings of population kinetics are still lacking. Experimental
evaluations of the population kinetics in variety of plasmas may help us to
reveal the macroscopic behavior. In this work, we study the emission spectra
of hydrogen Fulcher-$\alpha$ band, which are routinely obtained from the
divertor region of LHD. The spectra were observed with an echelle spectrometer
that has both high wavelength resolution $\approx$ 0.1 nm and wide wavelength
bandwidth 400–760 nm. We analyzed a thousand of spectra observed for a hundred
of LHD discharge experiments, covering wide range of plasma parameters. By
assuming a two temperature model and utilizing the Bayesian inference
framework, we robustly estimated the population distribution of the electronic
ground state of hydrogen molecules for these experiments. We found that the
lower rotational temperature shows a significant dependence on the plasma
parameters, as reported in previous works [2, 30]. On the other hand, the
other parameters show only a little change over the parameter range we
studied. Furthermore, significant correlation among these temperatures is
found, suggesting an existence of a simple mechanism generating the non-
thermal population distribution. For example, Sawada et al have pointed out
that with the electron-impact excitation similar nonthermal population
distribution can be formed in plasmas [22]. Our result is qualitatively
consistent with their simulation.

The rest of this paper is organized as follows. In section 2, we briefly
present our experimental setup, although the same setup was used in our
preceding paper [16]. In section 3, the analysis method and our assumption are
described. The result for typical spectra are also shown in this section. In
section 4, we present our full results for the thousands frames. The detailed
correlation among the estimated parameters are presented.

##  2 Experimental Setup

###  2.1 Large Helical Device

LHD is a helical-type magnetic-plasma-confinement machine. With a pair of
helical coils, plasmas with the major radius $R=$ 3.5–4.0 m and the minor
radius $\approx 0.6$ m are generated. This helical coil also makes helical
divertor structures. A poloidal cross section of the LHD plasma is shown in
Fig. 1. In the figure, the closed magnetic flux surfaces are indicated by gray
curves. The cross points of open magnetic field lines are shown by gray dots.
The intrinsic helical divertor structure and the stochastic layers can be
seen. Inside the last closed flux surface (LCFS), ions and electrons are
confined. Once these charged particles diffuse out the LCFS, they are guided
by an open magnetic field line to divertor plates through the divertor leg
structure.

Depending on the coil current, LHD can take several magnetic configurations.
The magnetic axis position $R_{ax}$ is an important parameter representing a
configuration. With the inner shifted configuration $R_{ax}=3.60$ m (Fig. 1
(b)), the LHD plasma has thinner ergodic layer and a less helical ripple than
those with $R_{ax}=3.65$ m (Fig. 1 (a)).

Figure 1: Poloidal crosssections of LHD plasmas for (a) the configuration
$R_{ax}=3.65$ m and (b) $R_{ax}=3.60$ m. Some closed magnetic flux surfaces
are shown by solid curves while the stochastic layer encircling the last
closed flux surface is shown by gray dots. The stochastic layer also provides
helical divertor structure, which connects the main plasma and divertor
located in the inner and outer boards.

The plasma is heated by neutral-beam-injections (NBIs) and electron cyclotron
heating (ECH). LHD equips several gas injection systems. By varying the gas
pressure, heating power, and the magnetic configuration, variety of plasmas
spanning broad range of parameter space can be generated.

An example of the temporal sequence of a discharge experiment is shown in Fig.
2. This plasma is initiated by ECH and sustained by NBI (Fig. 2 (a)). In Fig.
2 (b), the temporal evolutions of the electron temperature ($T_{e}$) and
density ($n_{e}$) measured by Thomson scattering method are shown. In this
experiment, helium gas is injected at $t$ = 4.5 s, resulting in an increase in
$n_{e}$ and decrease in $T_{e}$, followed by their gradual recoveries.

This Thomson scattering system measures the radial distributions of $T_{e}$
and $n_{e}$ with a spatial resolution of $\approx 10$ mm and temporal
resolution of $\approx 30$ ms. The spatial distributions of $T_{e}$ and
$n_{e}$ measured at $t$ = 4.00 and 4.75 s are shown in Fig. 3. Although it is
difficult to precisely determine the position of the LCFS due to the
stochastic nature of the magnetic field lines at the edge region, typically
this is located at $r_{eff}\approx 0.6$ m, where $r_{eff}$ is the effective
minor radius. $T_{e}^{ax}$ and $n_{e}^{ax}$ shown in Fig. 2 (b) are the values
of $T_{e}$ and $n_{e}$ at the magnetic axis $r_{eff}$ = 0 m, while
$T_{e}^{LCFS}$ and $n_{e}^{LCFS}$ indicate these values at $r_{eff}$ = 0.6 m.

Many other diagnostics have been installed in LHD. In this work, we consider
some of old-established diagnostics, namely the ion saturation current onto
divertor plates $I_{is}$ measured by Langmuir probes, and the gas pressure
measured in the divertor region $P_{H_{2}}$ by a fast-ionization-guage, in
addition to the $n_{e}$ and $T_{e}$ values measured by the Thomson scattering
method. The temporal evolutions of the $I_{is}$ and $P_{H_{2}}$ are shown in
Fig. 2 (c). Because of the gas injection, the gas pressure increases. The ion
saturation current shows a non-monotonic behavior, indicating the interplay
between the increase in $n_{i}$ and decrease in $T_{i}$ in the diverotr
region.

Figure 2: Temporal evolution of a typical experiment (shotnumber #142897). (a)
Heating power by neutral beam injection (NBI) and electron cyclotron heating
(ECH). (b) Temporal evolutions of $T_{e}$ and $n_{e}$ on the plasma axis
($T_{e}^{ax}$ and $n_{e}^{ax}$, respectively) and last closed flux surface
($T_{e}^{LCFS}$ and $n_{e}^{LCFS}$, respectively). Because of the gas
injection at $t$ = 4.5 s (indicated by the thick vertical bar), $n_{e}$
increases and $T_{e}$ decreases. (c) Temporal evolutions of the ion saturation
current measured at the divertor plate ($I_{is}$) and neutral gas pressure
measured near the divertor region ($P_{H2}$). (d) Measured emission
intensities of hydrogen atomic Balmer-$\beta$ ($I_{H_{\beta}}$) and the
Fulcher-$\alpha$ $(0-0)Q1$ transition. The estimated result of the excitation
flux ($n_{X}\,R\,n_{e}$, see the text for its definition) is also shown. (e)
Estimated rotational and vibrational temperatures of the $X$-state of hydrogen
molecule. Estimation uncertainties are also shown either by the error bars and
width of the curves.  Figure 3: Radial distributions of (a) $T_{e}$ and (b)
$n_{e}$ measured for #142897 at $t=$ 4.00 s and 4.75 s with Thomson scattering
method. Rough position of the LCFS is shown by vertical dashed lines.

###  2.2 Visible spectroscopy

We observed the visible emission from the divertor region of the LHD. The
experimental setup is the same with that shown in Ref.[16], where the emission
at the inner divertor region was collected by an optical lens, focused on an
optical fiber edge, transferred to the entrance slit of an echelle
spectrometer which have been developed by our group [29, 14]. This
spectrometer measures the spectrum in the wavelength range of 400–780 nm with
the wavelength resolution of $\approx 0.1$ nm simultaneously. 33 ms exposure
time and 4 Hz frame rate are used for all the results shown in this work.

Figure 4 shows the visible spectra measured by this spectrometer at $t$ = 4.00
and 4.75 s for the experiment shown in Fig. 2. The vertical bars in the figure
show the central wavelengths of the Q branches of the hydrogen
Fulcher-$\alpha$ band. The Fulcher-$\alpha$ band is the emission lines from
$d^{3}\Pi_{u}^{-}$ state to $a^{3}\Sigma_{g}^{+}$ state. The transition
notations $(v^{\prime}-v^{\prime\prime})QN^{\prime}$ are shown in the figure,
where $v^{\prime}$ and $v^{\prime\prime}$ indicate the vibrational quantum
numbers of the upper and lower states, respectively, while $N^{\prime}$
indicates the rotational quantum number of the upper state. As it is in the
$Q$ branch, $N^{\prime}=N^{\prime\prime}$. For example, $(0-0)Q1$ indicates
$v^{\prime}=0,v^{\prime\prime}=0$, and $N^{\prime}$ = $N^{\prime\prime}$ = 1.

Figure 4: The emission spectra observed at $t$ = 4.0 and 4.75 s with the
echelle spectrometer. The central wavelengths of Fulcher-$\alpha$ band are
shown by the vertical bars. The transitions for the emission lines used in the
analysis are also indicated.

After the gas injection, the intensities of these lines change. The intensity
of the $(0-0)Q1$ line decreases while that of the $(0-0)Q11$ line increases.
We estimate the emission intensities by fitting the spectra by a Gaussian
function. The temporal evolution of the $(0-0)Q1$ line intensity is shown in
Fig. 2 (d). In this panel, we also plot the intensity evolution of the
hydrogen atom Balmer-$\beta$ line. Balmer-$\beta$ line intensity increases
according to the gas injection while the $(0-0)Q1$ line intensity decreases.

We estimate the population of the $d$ states from the intensity
$I_{(v^{\prime}-v^{\prime\prime})QN^{\prime}}$ based on the following relation

$\displaystyle
I_{(v^{\prime}-v^{\prime\prime})QN^{\prime}}=h\nu^{dv^{\prime}N^{\prime}}_{av^{\prime\prime}N^{\prime\prime}}\;A^{dv^{\prime}N^{\prime}}_{av^{\prime\prime}N^{\prime\prime}}\;n_{dv^{\prime}N^{\prime}},$
(1)

where $\nu^{dv^{\prime}N^{\prime}}_{av^{\prime\prime}N^{\prime\prime}}$ is the
photon energy of the
$(d,v^{\prime},N^{\prime})\rightarrow(a,v^{\prime\prime},N^{\prime\prime})$
transition and $n_{dv^{\prime}N^{\prime}}$ is the line-integrated population
density of the upper state.
$A^{dv^{\prime}N^{\prime}}_{av^{\prime\prime}N^{\prime\prime}}$ is the
Einstein coefficient for the transition $dv^{\prime}N^{\prime}\rightarrow
av^{\prime\prime}N^{\prime\prime}$,which are computed by [28]

$\displaystyle
A^{dv^{\prime}N^{\prime}}_{av^{\prime\prime}N^{\prime\prime}}=\frac{16\pi^{3}}{3h^{4}\epsilon_{0}c^{3}}(h\nu^{dv^{\prime}N^{\prime}}_{av^{\prime\prime}N^{\prime\prime}})^{3}\,\overline{R_{e}}^{2}\,q^{dv^{\prime}}_{av^{\prime\prime}}\,\frac{S_{N^{\prime}N^{\prime\prime}}}{2N^{\prime}+1}$
(2)

with the vecuum permittivity $\epsilon_{0}$, the planck constant $h$, the
light speed $c$, the dipole transition moment $\overline{R_{e}}$ between $a$
and $d$ states, the Franck-Condon factor between the upper and lower
vibrational states $q^{dv^{\prime}}_{av^{\prime\prime}}$, and the Hönl-London
factor for the $Q$ branch $S_{N^{\prime}N^{\prime\prime}}=(2N^{\prime}+1)/2$.
The values of $q^{dv^{\prime}}_{av^{\prime\prime}}$ are taken from Ref. [9].

Figure 5: The population of $d$-state observed at $t$ = 4.0 and 4.75 s. The
solid curves show the best fit by the coronal model (Eq. (5)) with the two-
temperature Boltzmann distribution for $X$ state (Eq. (7)). The estimated $X$
state populations are shown in the inset.

Figure 5 shows the population of the upper state of the Fulcher-$\alpha$ band,
normalized by the statistical weight of the upper state
$g_{N^{\prime}}=(2N^{\prime}+1)g_{as}$ with the nucleus statistical weight
$g_{as}=2-(-1)^{N^{\prime}}$. The rotational population for each $v^{\prime}$
state shows the decreasing trend as a function of the excited energy. The
slope in the vertical logarithmic scale is steeper at the lower density plasma
($t$ = 4.00 s) and more flat at the higher density plasma ($t$ = 4.75 s).

##  3 Estimation of the Population Distribution of the $X$ states

Only the upper-state population can be directly obtained from the observed
emission intensities. In order to estimate the population distribution in the
$X$ state, we utilize the coronal model, which has been widely used in the
literature [33, 7, 4, 8]. Note that the coronal model holds only when the
electron density is sufficiently small. In high-density plasmas, the
population influx from excited states becomes important, however, the validity
criteria is not yet established. Thus, in this work we ignore the effect and
assume that the coronal model is valid in our parameter range.

###  3.1 Coronal model

With the assumption of the coronal equilibrium, the population distribution of
the $d$ state is determined by the population influx from the $X$ state
($\Gamma_{XvN}^{dv^{\prime}N^{\prime}}$) and the outflux to the $a$ state
($\Gamma_{dv^{\prime}N^{\prime}}^{av^{\prime\prime}N^{\prime\prime}}$).
$\Gamma_{XvN}^{dv^{\prime}N^{\prime}}$ is computed with the Born-Oppenheimer
approximation,

$\displaystyle\Gamma_{XvN}^{dv^{\prime}N^{\prime}}\approx
q^{Xv}_{dv^{\prime}}\delta(N-N^{\prime})\,n_{XvN}\,R(T_{e})\,n_{e}.$ (3)

where $q^{Xv}_{dv^{\prime}}$ is the Franck-Condon factor between $X$ and $d$
states, and $R(T_{e})$ is the excitation rate coefficient from $X$ to $d$
states. The population outflux
$\Gamma_{dv^{\prime}N^{\prime}}^{av^{\prime\prime}N^{\prime\prime}}$ is

$\displaystyle\Gamma_{dv^{\prime}N^{\prime}}^{av^{\prime\prime}N^{\prime\prime}}\approx
A^{dv^{\prime}N^{\prime}}_{av^{\prime\prime}N^{\prime\prime}}\delta(N^{\prime}-N^{\prime\prime})\;n_{dv^{\prime}N^{\prime}}.$
(4)

From the steady state condition
$\sum_{v,N}\Gamma_{XvN}^{dv^{\prime}N^{\prime}}=\sum_{v^{\prime\prime}}\Gamma_{dv^{\prime}N^{\prime}}^{av^{\prime\prime}N^{\prime\prime}}$,
the population of the $d$ state can be written by

$\displaystyle
n_{dv^{\prime}N^{\prime}}=\frac{\sum_{v,N}\Gamma_{XvN}^{dv^{\prime}N^{\prime}}}{\sum_{v^{\prime\prime}}A^{dv^{\prime}N^{\prime}}_{av^{\prime\prime}N^{\prime}}}.$
(5)

###  3.2 Ro-Vibrational Temperature Estimation of the $X$ state

Because Eq. (5) involves more number of unknown parameters ($n_{XvN}$) than
the measured values of $n_{dv^{\prime}N^{\prime}}$, an appropriate
parameterization is necessary to estimate the distribution of $n_{XvN}$ [8].
Although Boltzmann’s distribution has been assumed for $n_{XvN}$ in several
works [33, 7, 4, 8], it has been also known that the distribution deviates
from the Boltzmann distribution, particularly in the high rotational-quantum-
number states. In fact, our observation of the population distribution shown
in Fig. 5 also presents a deviation from the Boltzmann distribution, where
highly rotational states are excessively populated. According to the preceding
works which report the direct observations of the $X$-state population [20,
26, 31, 12], the distribution may have the following properties:

  * 1.

The distribution of the population in the state with small rotational quantum
number $N$ in each vibrational state follows the Boltzmann distribution with
temperature $T_{rot}^{lo}$.

  * 2.

The large $N$ states are populated more than the Boltzmann distribution and
can be approximated by another Boltzmann distribution with higher temperature
$T_{rot}^{hi}>T_{rot}^{lo}$.

  * 3.

The total population for each $v$ state follows yet another Boltzmann
distribution with $T_{vib}$.

Based on these observations, we assume the following two-temperature form for
the $X$-state population,

$\displaystyle
n_{XvN}=n_{X}\biggl{(}(1-\alpha)f(E_{XvN}|T_{vib}^{lo},T_{rot}^{lo})+\alpha
f(E_{XvN}|T_{vib}^{hi},T_{rot}^{hi})\biggr{)},$ (6)

where $f(E_{XvN}|T_{vib},T_{rot})$ is the Boltzmann distribution with
vibrational and rotational temperatures, $T_{vib}$ and $T_{rot}$,
respectively,

$\displaystyle
f(E_{XvN}|T_{vib},T_{rot})=\frac{1}{T_{vib}T_{rot}}\exp\left[-\frac{E_{Xv0}-E_{X00}}{T_{vib}}-\frac{E_{XvN}-E_{Xv0}}{T_{rot}}\right].$
(7)

Here, $E_{XvN}$ is the excited energy of $X$ state with vibrational and
rotational quantum nubers $v$ and $N$. Thus, $E_{X00}$ is the ground state
energy. Equation (6) and its substitution to Eq. (5) have six adjustable
parameters, i.e., the excitation flux $n_{X}\,R\,n_{e}$, the weight of the two
distributions $\alpha$, and a pair of vibrational and rotational temperatures
$T_{rot}^{lo},T_{vib}^{lo},T_{rot}^{hi},T_{vib}^{hi}$. Here, $n_{X}\,R\,n_{e}$
represents the product of the total hydrogen density $n_{X}=\sum_{vN}n_{XvN}$,
the electronic-state-resolved excitation rate coefficient $R(R_{e})$, and the
electron density. Because these three terms are unknown and impossible to
resolve from the analysis, we treat it as a single adjustable parameter. We
fit the distribution of $n_{dv^{\prime}N^{\prime}}$ by adjusting these
parameters. In order to make a robust inference not only for these two frames
but also for more than $10^{3}$ frames of data, we adopt a hierarchical
Bayesian model, the detail of which will be described in the Appendix.

The bold curves in Fig. 5 show the fit result of $n_{dv^{\prime}N^{\prime}}$
for $t$ = 4.00 and 4.75 s. The change in the populations is well captured. The
reconstructed population in the $X$ state is also shown in the inset. The
estimated values of these parameters are listed in Table 1.

Table 1: Estimated parameters for #142897 at $t$ = 4.00 and 4.75 s. The $\pm$ range indicates the 16–84% confidential interval of these parameters.  $t$ |  $T_{rot}^{lo}$ (eV) |  $T_{rot}^{hi}$ (eV) |  $T_{vib}^{lo}$ (eV) |  $T_{vib}^{hi}$ (eV) | $\alpha$
---|---|---|---|---|---
4.00 s | $0.153^{+0.006}_{-0.006}$ | $0.56^{+0.05}_{-0.05}$ | $0.65^{+0.03}_{-0.05}$ | $0.88^{+0.05}_{-0.04}$ | $0.48^{+0.02}_{-0.02}$
4.75 s | $0.289^{+0.014}_{-0.013}$ | $0.77^{+0.04}_{-0.09}$ | $0.77^{+0.04}_{-0.04}$ | $0.90^{+0.04}_{-0.04}$ | $0.51^{+0.03}_{-0.03}$

The value of $T_{rot}^{lo}$ changes significantly in these two timings, while
the changes in $T_{rot}^{hi},T_{vib}^{lo},T_{vib}^{hi}$, and $\alpha$ are less
significant. The values of the three temperatures,
$T_{rot}^{hi},T_{vib}^{lo},T_{vib}^{hi}$, are estimated to the similar range.
As shown in Fig. 5 inset, this results in the convergence into a single
Boltzmann’s distribution for the $X$ state in the highly excited levels. This
behavior is consistent with the previous works, where the population in the
$X$ state has been directly measured [20, 26, 31, 12].

##  4 Parameter dependence of the $X$ state population

We collect the spectra obtained for 120 discharge experiments (74 experiments
with $R_{ax}$ = 3.65 m and 46 experiments with $R_{ax}$ = 3.60 m) totalling
1145 frames. The summary of the experiments is shown in Table 2. There are
also some variations in ECH and NBI powers. The same inference of the $X$
state population is carried out for all of these frames. Figure 6 shows the
distribution among several parameters, $n_{e}^{LCFS}$, $T_{e}^{LCFS}$,
$P_{H_{2}}$, $I_{is}$, $I_{H\beta}$, $n_{X}\,R\,n_{e}$, $T_{rot}^{lo}$, and
$T_{vib}^{lo}$ at these frames. The parameters for the $R_{ax}=3.65$-m- and
3.60-m-experiments are shown in different colors.

Table 2: Summary of the experiments we analyze in this work. The ECH column indicates the typical ECH power for these experiments. In the column of NBI, the indices of the neutral beam injectors used for the experiments are shown. #1, 2 and 3 are negative-ion based NBI with 180 keV injection energy, which mainly heat the electrons. #4, 5 are the positive-ion based NBI with 40 keV injection energy, mainly heating the ions.  $R_{ax}$ (m) | shotnumber |  $B_{t}$ (T) | ECH (MW) | NBI
---|---|---|---|---
3.65 | 142857-142917 | -2.712 | 75 | #1,2,3
| 143523-143560 | 2.712 | 60 | #1,2,3
3.60 | 143293-143297 | -1.640 | 0 | #1,2,3,4,5
| 143306-143307 | -2.750 | 60 | #1,2,3,4,5
| 143387-143415 | -2.750 | 60 | #1,2,3
| 143943-143950 | -2.750 | 250-550 | #2,3
| 143973-143988 | -2.750 | 75 | #2,3
Figure 6: A pair plot for plasma parameters and $T_{rot}^{lo}$ and
$T_{vib}^{lo}$. The observation results for different magnetic configurations
($R_{ax}$ = 3.60 m and 3.65 m) are shown in different colors.

###  4.1 Plasma parameters

$n_{e}^{LCFS}$ varies over 0.06–$1\times 10^{20}\mathrm{\;m^{-3}}$, while
$T_{e}^{LCFS}$ varies 0.08–0.9 keV. $T_{e}^{LCFS}$ has a negative correlation
against $n_{e}^{LCFS}$. The values of $n_{e}^{LCFS}$ and $T_{e}^{LCFS}$ are
dependent on each other, i.e., the most of the points in the top-left panel in
Fig. 6 stay on a single curve. Positive correlations among $n_{e}^{LCFS}$,
$P_{H_{2}}$, and $I_{is}$ are apparent. This may be intuitive, as higher gas
pressure often gives more dense plasma and dense plasmas often result in more
ion flux to the divertor. However, the scatter plots show their diffuse
relations compared with the $n_{e}^{LCFS}$-$T_{e}^{LCFS}$ relation. This
suggests that other factor also affects their relations. The similar trend can
be seen in Fig. 2 (a), (b), where $n_{e}^{LCFS}$ and $T_{e}^{LCFS}$ changes
accordingly, while the change in the $P_{H_{2}}$ and $I_{is}$ are not
monotonic against the change of $n_{e}^{LCFS}$.

The values of $I_{H\beta}$ and $n_{X}\,R\,n_{e}$ also show the positive
correlations against $n_{e}^{LCFS}$. This is consistent with the positive
correlation between $P_{H_{2}}$ and $n_{e}^{LCFS}$, i.e., the atom and
molecule densities should have a positive dependence on $P_{H_{2}}$ and the
emission rate is almost linearly proportional to $n_{e}$. Their relations show
a large scatter, suggesting existence of another process to affect the
molecular density in front of the divertor.

###  4.2 The rotational temperature

Figure 7: (a-d) Dependence of $T_{rot}^{lo}$ on some plasma parameters. The
results for $R_{ax}$ = 3.65 m and 3.6 m are shown in the upper and lower
panels, respectively. Scatters of $T_{rot}^{lo}$-$n_{e}^{LCFS}$ and
$T_{rot}^{lo}$-$T_{e}^{LCFS}$ are smaller than the rest of plots. In (a), the
empirical relation of $T_{rot}$ with $n_{e}$ (Eq. (8)) is shown by a solid
curve. The width of the curve indicates the uncertainty in the wall
temperature ($300$–$600$ K) at the desorption location. The dashed curve is
the same relation, but $n_{e}$ at the dievertor plate is used instead of
$n_{e}^{LCFS}$, which is estimated from Eq. (9). (e) The histogram of
$T_{rot}^{lo}$. The prior distribution optimized by Bayesian inference is also
shown by the solid curve. The median and the 68% range of the prior are shown
by solid and dashed lines, respectively. See Appendix for the details of
Bayesian inference.

The values of $T_{rot}^{lo}$ show a clear dependence on $n_{e}^{LCFS}$ and
$T_{e}^{LCFS}$. Expanded correlation plots are shown in Fig. 7. The scatter
plot between $T_{rot}^{lo}$ and $n_{e}^{LCFS}$ is closest to a single curve,
while the scatter of $T_{rot}^{lo}$-$P_{H_{2}}$ and $T_{rot}^{lo}$-$I_{is}$
are larger. This suggests the direct dependence of the rotational temperature
on the electron density, rather than the gas pressure and the ion flux to the
divertor. This interpretation is consistent with that of the previous works
[2, 30, 15]. However, it should be noted that we only consider the electron
density at $r_{eff}$ = 0.6 m and those exactly at the emission location are
unavailable.

The solid curve in Fig. 7 (a) shows the empirical dependence of $T_{rot}$ [24,
3, 15],

$\displaystyle T_{rot}\approx 280+T_{s}+6\times
10^{-17}n_{e}\;\;\mathrm{[K]},$ (8)

where $T_{s}$ is the surface temperature (assuming no extra excitation
mechanism due to the surface recombination) in K, and $n_{e}$ is in
$\mathrm{m^{-3}}$. We assume the wall temperature at the desorption location
as $T_{s}\approx 300$–$600$ K. The width of the curve shows this uncertainty.
Although our result consistently shows the positive $n_{e}$ dependence with
convergence to $\approx$ 600 K at $n_{e}\rightarrow 0$, a significant
discrepancy is found; our results lie at the larger density side of the solid
curve. This discrepancy may be originated from the difference in
$n_{e}^{LCFS}$ and the $n_{e}$ values at the emission location.

In order to estimate the electron density on the divertor plate $n_{e}^{div}$,
we use the relation

$\displaystyle\frac{n_{e}^{div}}{10^{19}\,\mathrm{m^{-3}}}\approx
0.08\times\left[\frac{n_{e}^{LCFS}}{10^{19}\,\mathrm{m^{-3}}}\right]^{1.5},$
(9)

which has been suggested in Ref. [17]. The dashed curves in Fig. 7 (a) show
Eq. (8) but with this relation, the width of which again indicates
$T_{s}\approx 300$–$600$ K. Our data point is in between the two curves. This
may be understood that the dominant emission location is between the divertor
plate and the LCFS. This is consistent with the emission location observation
based on Zeeman spectroscopy in [10, 11].

###  4.3 The vibrational temperature

Figure 8: (a-d) Dependence of $T_{vib}^{lo}$ on some plasma parameters. The
results for $R_{ax}$ = 3.65 m and 3.6 m are shown in the upper and lower
panels, respectively. In (a), the empirical relation of $T_{vib}$ with $n_{e}$
(Eq. (8)) is shown by a solid curve. The dashed curve is the same relation,
but $n_{e}$ at the dievertor plate is used instead of $n_{e}^{LCFS}$, which is
estimated from Eq. (9). (e) The histogram of $T_{vib}^{lo}$. The prior
distribution optimized by Bayesian inference is also shown by the solid curve.
The median and the 68% range of the prior are shown by a solid and dashed
lines, respectively. See Appendix for the details of Bayesian inference.

$T_{vib}^{lo}$ also shows a dependence on $n_{e}^{LCFS}$ and $T_{e}^{LCFS}$.
Expanded correlation plots are shown in Fig. 8. A similar positive
$n_{e}$-dependence of $T_{vib}^{lo}$ has been reported in [2]. From their
plot, we extract the dependence as

$\displaystyle T_{vib}\approx 2400+2.6\times 10^{-16}\;n_{e}\;\;\mathrm{[K]},$
(10)

with $n_{e}$ is again in $\mathrm{m^{-3}}$. The solid curves in Fig. 8 (a)
represent this relation, and the dashed curves show the same relation with the
assumption of Eq. (9). The dependence of $T_{vib}^{lo}$ is in between the two
curves. This is consistent with the above discussion for $T_{rot}^{lo}$, where
the emission location of the molecules is suggested between the divertor plate
and the LCFS.

These observations for $T_{rot}^{lo}$ and $T_{vib}^{lo}$ indicate that Eq. (8)
and Eq. (10) hold universally, although the geometry of the original
experiment is different from that in this work; Brezinsek et al have measured
the emission in front of the graphite limiter of TEXTOR [2], while we measured
the emission from LHD divertor. This suggests that these parameters are mostly
determined by the electron density, and the effect of the wall and electron /
ion temperatures is small in this parameter range.

###  4.4 The other population parameters

Figure 9: A pair plot for the population parameters. The observation results
for different magnetic configurations ($R_{ax}$ = 3.60 m and 3.65 m) are shown
in different colors. The relations among these parameters except for
$n_{X}\,R\,n_{e}$ stay on single curves, suggesting a simple mechanism to
determine the population distribution.

Correlation among the population parameters, $n_{X}\,R\,n_{e}$,
$T_{rot}^{lo}$, $T_{vib}^{lo}$, $T_{rot}^{hi}$, $T_{vib}^{hi}$, and $\alpha$,
are shown in Fig. 9. $T_{rot}^{lo}$ and $n_{X}\,R\,n_{e}$ vary over the factor
of 5 and 10, respectively. On the other hand, the variations of
$T_{rot}^{hi}$, $T_{vib}^{lo}$, $T_{vib}^{hi}$, and $\alpha$ are smaller. For
example, $T_{rot}^{hi}$ only changes by a factor of $\approx$ 1.5.
Furthermore, the scatter plots among these population parameters except for
$n_{X}\,R\,n_{e}$ show a significant correlation. For example, the pair plot
of $T_{rot}^{lo}$ and $n_{X}\,R\,n_{e}$ (the top left panel) shows a large
scatter, while $T_{rot}^{lo}$-$T_{vib}^{lo}$ (the second panel from the top
and the left) is close to a single curve. This suggests that a simple process
determines the entire population distribution.

There are several processes have been proposed for the $X$ state population of
hydrogen molecules, such as electron impact, proton impact, and neutral
particle impact [22]. From Fig. 6, we see high correlation between
$T_{rot}^{lo}$ and $n_{e}$. On the other hand, the correlation between
$T_{rot}^{lo}$ and $n_{X}\,R\,n_{e}$ shows larger scatter, although the
$T_{e}$-dependence of $R$ is known to small. This indicates that the
dependence on $n_{X}$ is smaller, suggesting the unimportance of the
molecular-collision effect on the population distribution. As such, a further
study based on our data may help identification of important processes.

##  5 Conclusion

We analyzed a thousand visible spectra of Fulcher-$\alpha$ band measured for
divertor plasmas in the LHD. With the coronal model and Baysian inference, the
population distribution of the electronical ground state was estimated from
these spectra. A nearly linear $n_{e}$-dependences of $T_{rot}^{lo}$ and
$T_{vib}^{lo}$ were found, which is consistent with previous works. On the
other hand, the higher rotational and vibrational temperatures as well as the
mixture coefficient only show slight changes over the broad range of plasma
parameters, and these parameters show a significant correlation. This suggests
an importance of electron-impact excitation to determine the non-thermal
population distribution of molecular hydrogen in divertor plasmas.

##  Appendix A Bayesian inference of the $X$-state population

The two-temperature model we assume in Eq. (7) is not always robust. For
example, the two temperature model has many optima that fit equally well for a
single-temperature distribution; $T^{lo}=T^{hi}$ with any value of $\alpha$ is
one optimum, but $\alpha=0$ with any value of $T^{hi}$ is another optimum. In
order to carry out a robust inference on the population parameters in Eq. (7)
for thousands of frames, we utilized Bayesian inference framework. In this
section, we show the details of the probabilistic model we have adopted in
this work.

Let
$\mathbf{n}=\\{n_{t,l}|f\in\\{1,2,\cdots,N_{t}\\},l\in\\{1,2,\cdots,N_{l}\\}\\}$
be the observed populations at the $d$ state and $\sigma_{t,l}$ be its
uncertainty from the measurement, where $N_{t}$ is the total numbef of frames
and $N_{l}$ is the total number of molecular lines.for these spectra For each
frame $t$, we define the population parameters of the $X$ state,
$\theta_{t}=\\{[n_{X}Rn_{e}]_{t},T_{rot}^{lo}{{}_{t}},T_{vib}^{lo}{{}_{t}},T_{rot}^{hi}{{}_{t}},T_{vib}^{hi}{{}_{t}},\alpha_{t}\\}$.
Our aim is to infer the posterior distribution of $\theta_{t}$ with given the
observation $\mathbf{n}$, $p(\theta_{t}|\mathbf{n})$.

With a given value of $\theta_{t}$, we can compute the expected populations
$\overline{n}_{l}(\theta_{t})$ in the $d$ state based on Eq. (5). We assume
the gaussian noise for $n_{t,l}$ with the standard deviation
$\gamma\sigma_{t,l}$,

$\displaystyle
p(n_{t,l}|\theta_{t})=\mathcal{N}(n_{t,l}|\overline{n}_{l}(\theta_{t}),\gamma\sigma_{t,l})$
(11)

where
$\mathcal{N}(x|\mu,\sigma)=\exp[-(x-\mu)^{2}/2\sigma^{2}]/\sqrt{2\pi}\sigma$
is the normal distribution with mean $\mu$ and the standard deviation
$\sigma$. Here, we additionally assume the excess noise factore $\gamma$,
which is a single parameter common for all the frames and upper states to
capture the possible unidentified error in the measurement, e.g. overlap by
other lines. $\gamma$ is estimated and merginalized later.

We also assume prior distributions for the population parameters $\theta_{t}$,
which makes the inference robust. As a prior for
$[n_{X}Rn_{e}]_{t},T_{rot}^{lo}{{}_{t}},T_{vib}^{lo}{{}_{t}},T_{rot}^{hi}{{}_{t}}$,
and $T_{vib}^{hi}{{}_{t}}$, we assume the inverse gamma distribution, which is
a typical distribution for a nonnegative random variable,

$\displaystyle p([n_{X}Rn_{e}]_{t}|a_{n},b_{n})$
$\displaystyle=\mathcal{IG}([n_{X}Rn_{e}]_{t}|a_{n},b_{n})$ (12)
$\displaystyle p(T_{rot}^{lo}{{}_{t}}|a_{rot},b_{rot}^{lo})$
$\displaystyle=\mathcal{IG}(T_{rot}^{lo}{{}_{t}}|a_{rot},b_{rot}^{lo})$ (13)
$\displaystyle p(T_{rot}^{hi}{{}_{t}}|a_{rot},b_{rot}^{hi})$
$\displaystyle=\mathcal{IG}(T_{rot}^{hi}{{}_{t}}|a_{rot},b_{rot}^{hi})$ (14)
$\displaystyle p(T_{vib}^{lo}{{}_{t}}|a_{vib},b_{vib}^{lo})$
$\displaystyle=\mathcal{IG}(T_{vib}^{lo}{{}_{t}}|a_{vib},b_{vib}^{lo})$ (15)
$\displaystyle p(T_{vib}^{hi}{{}_{t}}|a_{vib},b_{vib}^{hi})$
$\displaystyle=\mathcal{IG}(T_{vib}^{hi}{{}_{t}}|a_{vib},b_{vib}^{hi}),$ (16)

and for $\alpha_{t}$ we assume beta distribution, which is a typical
distribution for a variable in $[0,1]$,

$\displaystyle p(\alpha_{t}|a_{n},b_{n})$
$\displaystyle=\mathcal{B}(\alpha_{t}|a_{\alpha},b_{\alpha})$ (17)

where $\mathcal{IG}(x|a,b)=x^{-a-1}e^{-b/x}\,b^{a}/\Gamma(a)$ is the inverse
gamma distribution with the shape parameter $a$ and the scale parameter $b$
with $\Gamma(a)=\int_{0}^{\infty}x^{a-1}e^{-x}dx$ the gamma function.
$\mathcal{B}(x|a,b)=x^{a-1}(1-x)^{b-1}/B(a,b)$ with the beta function
$B(a,b)=\int_{0}^{1}x^{a-1}(1-x)^{b-1}dx$. The hyperparameters,
$\Theta=\\{\gamma,a_{n}$, $b_{n}$, $a_{rot}$, $b_{rot}^{lo}$, $b_{rot}^{hi}$,
$a_{vib}$, $b_{vib}^{lo}$, $b_{vib}^{hi}$, $a_{\alpha}$, $b_{\alpha}\\}$ are
common for all the frames and will be inferred simultaneously. The graphical
representation for our model is shown in Fig. 10.

Based on the Bayes rule, the posterior distribution can be found by
integrating the hyperparameters,

$\displaystyle p(\theta_{t}|\mathbf{n},\boldsymbol{\sigma})\propto\int
d\Theta\prod_{t,l}p(n_{t,l}|\overline{n_{t}}(\theta_{t}),\gamma\sigma_{t,l})p(\theta_{t}|\Theta)p(\Theta),$
(18)

where $p(\Theta)=\Gamma(\Theta|1,1)$ is the hyperprior distribution for
$\Theta$. The inference (i.e., the integration) was carried out with Markov
Chain Monte Carlo (MCMC) method with Stan Modeling Language [6, 25].

The median values of the hyperparameters are shown in Table 3. The prior
distributions of $T_{rot}^{lo}$ and $T_{vib}^{lo}$ are shown in Fig. 7 (e) and
Fig. 8 (e), respectively, by solid curves. The histogram of the estimated
$T_{rot}^{lo}$ and $T_{vib}^{lo}$ are also shown in the same panel. It can be
seen that these prior distributions are optimized so that they fit the entire
distributions of $T_{rot}^{lo}$ and $T_{vib}^{lo}$. In this panel, the median
value as well as the $68$% intervals are also shown.

Table 3: Inferred values of the hyperprior parameters for our Bayesian model. The range for the 68% confidential interval for the corresponding parameters are also tabulated.  parameter | $a$ | $b$ | 16% – 50% – 84%
---|---|---|---
$T_{rot}^{lo}$ (eV) | 7.0 | 0.88 | 0.09 – 0.13 – 0.20
$T_{rot}^{hi}$ (eV) |  | 4.5 | 0.47 – 0.67 – 1.01
$T_{vib}^{lo}$ (eV) | 20 | 12 | 0.49 – 0.61 – 0.77
$T_{vib}^{hi}$ (eV) |  | 18 | 0.74 – 0.92 – 1.15
$n$ (arb.) | 0.85 | 3 | 2 – 6 – 30
$\alpha$ | 36 | 54 | 0.35 – 0.40 – 0.45
Figure 10: A graphical representation of our hierarchical Bayesian model. The
measured quantities are indicated by filled circles, while the inferred random
variables are shown by open circles. A variable shown with a dot ($n(\theta)$)
represents a fixed parameter with given input parameters. Square panels
indicate that variables inside a panel have vector-valued parameters. For
example, population parameters, such as $T_{rot}^{lo}$, are shared by all the
upper states $l$, while hyperparameters such as $a_{rot}$ are shared for all
the frames $t$.

## acknowledgments

This work was supported by JSPS KAKENHI Grant Number 19K14680.

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# Reversible Entanglement Beyond Quantum Operations

Xin Wang<EMAIL_ADDRESS>Thrust of Artificial Intelligence,
Information Hub,The Hong Kong University of Science and Technology
(Guangzhou), Guangdong 511453, China Yu-Ao Chen<EMAIL_ADDRESS>Thrust of Artificial Intelligence, Information Hub,The Hong Kong University of
Science and Technology (Guangzhou), Guangdong 511453, China Lei Zhang
<EMAIL_ADDRESS>Thrust of Artificial Intelligence,
Information Hub,The Hong Kong University of Science and Technology
(Guangzhou), Guangdong 511453, China Chenghong Zhu<EMAIL_ADDRESS>gz.edu.cn Thrust of Artificial Intelligence, Information Hub,The Hong Kong
University of Science and Technology (Guangzhou), Guangdong 511453, China

###### Abstract

We introduce a reversible theory of exact entanglement manipulation by
establishing a necessary and sufficient condition for state transfer under
trace-preserving transformations that completely preserve the positivity of
partial transpose (PPT). Under these free transformations, we show that
logarithmic negativity emerges as the pivotal entanglement measure for
determining entangled states’ transformations, analogous to the role of
entropy in the second law of thermodynamics. Previous results have proven that
entanglement is irreversible under quantum operations that completely preserve
PPT and leave open the question of reversibility for quantum operations that
do not generate entanglement asymptotically. However, we find that going
beyond the complete positivity constraint imposed by standard quantum
mechanics enables a reversible theory of exact entanglement manipulation,
which may suggest a potential incompatibility between the reversibility of
entanglement and the fundamental principles of quantum mechanics.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries and Entanglement Manipulation
    1. 2.1 Preliminaries
    2. 2.2 Entanglement manipulation
  3. 3 Reversibility of Exact Entanglement Manipulation
    1. 3.1 Necessary and sufficient condition for entanglement transformations
    2. 3.2 Reversibility of exact entanglement manipulation under PPTq operations
  4. 4 Discussions
  5. A Appendix: Inequality chain for entanglement measures under PPTq

##  1 Introduction

Reversibility is a fundamental concept in many areas of physics, including
thermodynamics and quantum mechanics. The second law of thermodynamics governs
the direction of heat transfer and the efficiency of energy conversion. With
the existence of heat reservoirs, the second law allows for a reversible
exchange of work and heat, as exemplified in the Carnot cycle [Car79]. Based
on axiomatic approaches and idealized conditions, it has been shown that
entropy is the unique function that determines all transformations between
comparable equilibrium states [LY99, Gil16].

Within the realm of quantum information science, reversibility is crucial
because it allows for the efficient manipulation of quantum resources. If a
process is reversible, it implies that no quantum resources are irretrievably
lost during transformations. Through the development of quantum information
processing, quantum entanglement has been recognized as an essential resource
for various applications, including quantum communication [BSST99], quantum
computation [BM11, JL03], quantum sensing [DRC17], and cryptography [Eke91].
Understanding the reversibility of entanglement is thus pivotal for quantum
information and fuels the debate surrounding the axiomatization of
entanglement theory. This discourse is largely driven by the parallels drawn
with thermodynamics, which have fostered the potential proposition of a single
entanglement measure, analogously to entropy, that could potentially govern
all entanglement transformations. Such progress in understanding entanglement
reversibility would not only mirror thermodynamic properties but also
contribute significantly to the axiomatization of entanglement theory.

Reversibility of entanglement pertains to the process of asymptotic
entanglement manipulation. For pure quantum states, this process is
reversible, indicating that entanglement can be manipulated and then restored
to its original state [BBPS96] through local operations and classical
communication (LOCC). However, this asymptotic entanglement reversibility does
not apply to mixed states [VC01, VDC02, VWW04, CdOF11, YHHSR05], meaning that
once entanglement is manipulated, it cannot be perfectly restored to its
initial state under LOCC. The irreversibility of quantum entanglement under
LOCC presents a stark contrast to the principles of thermodynamics [HOH02],
where certain processes are inherently reversible. This irreversibility also
underscores the impossibility of developing a single measure [HHO03] capable
of governing all entanglement transformations under LOCC, suggesting that a
deeper understanding of entanglement manipulation is essential for possible
reversibility.

In the pursuit of a reversible entanglement theory, broader classes of
operations beyond LOCC can be considered to potentially reduce the gap between
entanglement cost and distillable entanglement. However, this approach has yet
to yield success as Wang and Duan [WD17a] prove that entanglement is
irreversible under quantum operations that completely maintain the positivity
of partial transpose (PPT), a meaningful set of quantum operations that
includes all LOCC operations. This result [WD17a] also implies the
irreversibility in the resource theory of entanglement with non-positive
partial transpose (NPT). Furthermore, Lami and Regula [LR23] show that
entanglement theory is irreversible under all non-entangling transformations,
which are positive maps that do not produce entanglement. Notably, even for
asymptotically entanglement non-generating operations [BP10], the
reversibility of entanglement is not known [BBG+23, FGW21]. The question of
whether a reversible entanglement theory could exist remains a vital open
problem in quantum information theory [Ple05b].

In this paper, we introduce a reversible theory of exact entanglement
manipulation, showing a possible counterpart of entanglement manipulation to
the second law of thermodynamics. This reversible theory operates under
transformations that completely preserve the positivity of partial transpose,
which are called PPT quasi-operations (PPTq operations) throughout the paper.
Our key result is that logarithmic negativity fully determines entangled
transformations under PPTq operations, i.e.,

$\displaystyle\rho\xrightarrow{{\operatorname{PPTq}}}\sigma\iff
E_{N}(\rho)\geq E_{N}(\sigma),$ (1)

which means logarithmic negativity plays an analogous role of entropy in the
second laws of thermodynamics. Based on this necessary and sufficient
condition of state transformation (cf. Theorem 1), we prove that the
logarithmic negativity determines exact distillable entanglement and exact
entanglement cost. We further show the reversibility of exact entanglement
manipulation under PPTq operations (cf. Theorem 3), showing that

$\displaystyle
R_{{\operatorname{PPTq}}}(\rho_{AB}\to\sigma_{A^{\prime}B^{\prime}})=\frac{E_{N}(\rho_{AB})}{E_{N}(\sigma_{A^{\prime}B^{\prime}})},$
(2)

where $R_{{\operatorname{PPTq}}}(\rho_{AB}\to\sigma_{A^{\prime}B^{\prime}})$
is the asymptotic ratio for exact state transformation under PPTq operations.
We further establish an inequality chain of the entanglement manipulation
rates for PPTq operations (cf. Theorem 4), presenting a distinction between
this reversible theory of entanglement beyond quantum operations and standard
entanglement theory.

While our research establishes a reversible theory of exact entanglement
manipulation, it’s crucial to note that the allowed transformations in this
theory are beyond the boundaries of quantum mechanics. But for operations in
quantum mechanics, the reversibility under operations that do not generate
entanglement asymptotically remains an unresolved enigma in the field [BBG+23,
FGW21]. The above reversibility of exact entanglement manipulation under PPT
quasi-operations may suggest that the coexistence of entanglement
reversibility and quantum mechanics might be mutually incompatible.

Fig 1: Exactly reversible interconvention between quantum states $\rho_{AB}$
and $\sigma_{A^{\prime}B^{\prime}}$. Consider $n$-copies of
$\rho_{AB}^{\otimes n}$ and $m$-copies of
$\sigma_{A^{\prime}B^{\prime}}^{\otimes m}$, the forward direction indicates
the exact conversion from $\rho_{AB}^{\otimes n}$ to
$\sigma_{A^{\prime}B^{\prime}}^{\otimes m}$ with rate
$R_{\Omega}(\rho_{AB}^{\otimes n}\to\sigma_{A^{\prime}B^{\prime}}^{\otimes
m})=m/n$, while the backward direction shows the exact conversion from
$\sigma_{A^{\prime}B^{\prime}}^{\otimes m}$ to $\rho_{AB}^{\otimes n}$ with
rate $R_{\Omega}(\sigma_{A^{\prime}B^{\prime}}^{\otimes
m}\to\rho_{AB}^{\otimes n})=n/m$. Two states can be exactly interconverted
reversibly if $R_{\Omega}(\rho_{AB}^{\otimes
n}\to\sigma_{A^{\prime}B^{\prime}}^{\otimes
m})R_{\Omega}(\sigma_{A^{\prime}B^{\prime}}^{\otimes m}\to\rho_{AB}^{\otimes
n})=1$.

##  2 Preliminaries and Entanglement Manipulation

###  2.1 Preliminaries

#### Notations.

We will use the symbols $A$ and $B$ to denote the finite-dimensional Hilbert
spaces ${\cal H}_{A}$ and ${\cal H}_{B}$ associated with Alice and Bob
systems, respectively. We denote the dimension of ${\cal H}_{A}$ and ${\cal
H}_{B}$ as $d_{A}$ and $d_{B}$. A quantum state on system $A$ is a positive
semidefinite operator $\rho_{A}$ with trace one. The trace norm of $\rho$ is
denoted as $\|\rho\|_{1}=\operatorname{Tr}(\sqrt{\rho^{\dagger}\rho})$. Let
$\\{|i\rangle\\}_{i=1}^{d}$ be a standard computational basis, then a standard
maximally entangled state of Schmidt rank $d$ is
$\Phi^{d}=1/d\sum_{i,j=1}^{d}|ii\rangle\\!\langle jj|$. A Bell state
$\Phi^{2}$ is alternatively called an ebit. A bipartite quantum state
$\rho_{AB}$ is said to be a PPT state, if it admits positive partial
transpose, i.e., $\rho_{AB}^{T_{B}}\geq 0$, where $T_{B}$ denotes the partial
transpose on the system B. Throughout the paper, we take the logarithm to be
base two unless stated otherwise.

#### Properties of linear maps.

Let ${\cal N}$ be a linear map. ${\cal N}$ is Hermitian-preserving (HP) if it
maps any Hermitian operator to another Hermitian operator; ${\cal N}$ is
positive if it maps any positive semi-definite operator to another positive
semi-definite operator, and is called completely positive (CP) if this
positivity is preserved on any extended reference system. ${\cal N}$ is also
trace-preserving (TP) if it preserves the traces of input operators.

#### Properties of bipartite linear maps.

Let ${\cal N}$ be a bipartite linear map. ${\cal N}$ is a local operation and
classical communication (LOCC) if it is a composition of a (finite) sequence
of quantum instruments. ${\cal N}$ is separability-preserving if it preserves
the set of separable states. ${\cal N}$ is positivity-of-the-Partial-
Transpose-preserving (PPT-preserving) if it preserves the set of PPT states
[HHH96, Per96], and is considered completely PPT-preserving (PPT) if this
property holds on any extended reference system. We note that free operations
beyond LOCC are of importance to advance our understanding of quantum
entanglement (see, e.g., [Rai97, Rai00, EVWW01, BP10, CdVGG20, RFWG19]).

#### Quasi-states.

A quasi-state is a mathematical entity that encapsulates the intricacies of a
probabilistic quantum system. Despite the non-physicality in quantum
mechanics, the class of quasi-states is the largest possible set that can be
represented by physical quantum states under a statistical meaning. Formally,
the definition of this class is given as

$\widetilde{\mathcal{S}}=\left\\{\sum\nolimits_{j}c_{j}\rho_{j}\mathrel{\mathop{\mathchar
58\relax}}\rho_{j}\in\mathcal{S},c_{j}\in{{\mathbb{R}}}\textrm{ s.t.
}\sum\nolimits_{j}c_{j}=1\right\\},$ (3)

where $\mathcal{S}$ is the set of quantum states. Note that
$\widetilde{\mathcal{S}}$ is the set of all Hermitian matrices with trace one.

###  2.2 Entanglement manipulation

#### Quasi-state transformations.

Throughout this paper, we allow both state and quasi-state transformations,
viewing them as operator transformations under linear maps. Specifically, we
call a bipartite quasi-state $\rho$ can be transformed into another bipartite
quasi-state $\sigma$ under $\Omega$ operations, provided there exists a
bipartite linear map ${\cal N}\in\Omega$ such that ${\cal N}(\rho)=\sigma$.
This broader framework of quasi-state transformation enables us to define
distillable entanglement and entanglement cost also for bipartite quasi-
states, similar to how we would for standard quantum states. This approach
offers a fresh perspective on understanding the limits of manipulating quantum
entanglement.

#### Entanglement distillation.

The maximally entangled state plays a role as the currency in quantum
information since it has become a key ingredient in many quantum information
processing tasks (e.g., teleportation [BBC+93], superdense coding [BW92], and
quantum cryptography [Eke91]). It is important to understand how many
maximally entangled states we can obtain from a source of less entangled
states using free operations. Imagine that Alice and Bob share a large supply
of identically prepared states, and they want to convert these states to high-
fidelity Bell pairs. Let $\Omega$ represent a set of free operations or
allowed transformations. The one-shot zero-error, or exact distillable
entanglement, under $\Omega$ operations of quantum state or quasi-state
$\rho_{AB}$ is defined as
$E^{(1)}_{0,D,\Omega}(\rho_{AB})=\sup_{\Lambda\in\Omega}\left\\{\log
d\mathrel{\mathop{\mathchar
58\relax}}\Phi^{d}_{\hat{A}\hat{B}}=\Lambda_{AB\to\hat{A}\hat{B}}\left(\rho_{AB}\right)\right\\}$,
where $\Phi^{d}_{\hat{A}\hat{B}}=[1/d]\sum_{i,j=1}^{d}|ii\rangle\\!\langle
jj|_{\hat{A}\hat{B}}$ represents the standard maximally entangled state of
Schmidt rank $d$. The zero-error, or exact distillable entanglement, of a
bipartite state or quasi-state state $\rho_{AB}$, under $\Omega$ operations is
defined as

$\displaystyle
E_{D,\Omega}^{{\operatorname{exact}}}(\rho_{AB})=\lim_{n\to\infty}\frac{1}{n}E^{(1)}_{0,D,\Omega}\left(\rho_{AB}^{\otimes
n}\right).$ (4)

For entanglement distillation with asymptotically vanishing error, the rate is
quantified via distillable entanglement. The distillable entanglement of a
bipartite state or quasi-state $\rho_{AB}$, under the $\Omega$ operations, is
defined as

$\displaystyle
E_{D,\Omega}\left(\rho_{AB}\right)=\sup\left\\{r\mathrel{\mathop{\mathchar
58\relax}}\lim_{n\to\infty}\left[\inf_{\Lambda\in\Omega}\lVert\Lambda\left(\rho_{AB}^{\otimes
n}\right)-\Phi^{2rn}_{\hat{A}\hat{B}}\rVert_{1}\right]=0\right\\}.$ (5)

As distillable entanglement quantifies the fundamental limit of entanglement
distillation and related task of quantum communication, substantial efforts
have been made to obtain its accurate estimation and fundamental properties
[Rai00, EVWW01, HHH00, CW04, WD16, WD17b, LDS17, Wan21, ZZW23]. As exact
entanglement distillation is more restricted, it holds that

$\displaystyle E_{D,\Omega}^{{\operatorname{exact}}}(\rho)\leq
E_{D,\Omega}(\rho).$ (6)

#### Entanglement dilution.

The reverse task of entanglement distillation is called entanglement dilution.
At this time, Alice and Bob share a large supply of Bell pairs and they are to
convert $rn$ Bell pairs to $n$ high fidelity copies of the desired state
$\rho^{\otimes n}$ using suitable free operations. Let $\Omega$ represent a
set of free operations, which for example can be LOCC or PPT. The one-shot
zero-error, or exact entanglement cost, of a bipartite state or quasi-state
$\rho_{AB}$ under the $\Omega$ operations is defined as
$E^{(1)}_{0,C,\Omega}\left(\rho_{AB}\right)=\inf_{\Lambda\in\Omega}\left\\{\log
d\mathrel{\mathop{\mathchar 58\relax}}\rho_{AB}=\Lambda_{\hat{A}\hat{B}\to
AB}\left(\Phi^{d}_{\hat{A}\hat{B}}\right)\right\\}$. The zero-error, or exact
entanglement cost, of $\rho_{AB}$ under the $\Omega$ operations is defined as

$\displaystyle
E^{{\operatorname{exact}}}_{C,\Omega}\left(\rho_{AB}\right)=\lim_{n\to\infty}\frac{1}{n}E^{(1)}_{0,C,\Omega}\left(\rho_{AB}^{\otimes
n}\right).$ (7)

Previous works [TH00, APE03, MW08, WW20] have shown progresses towards
understanding the exact entanglement cost of quantum states.

For the case of asymptotically vanishing error of entanglement dilution, the
rate is quantified via entanglement cost. The concise definition of
entanglement cost using $\Omega$ operations is given as follows:

$\displaystyle
E_{C,\Omega}\left(\rho_{AB}\right)=\inf\left\\{r\mathrel{\mathop{\mathchar
58\relax}}\lim_{n\to\infty}\inf_{\Lambda\in\Omega}\|\rho_{AB}^{\otimes
n}-\Lambda\left(\Phi^{2rn}_{\hat{A}\hat{B}}\right)\|_{1}=0\right\\},$ (8)

When LOCC is free, entanglement cost is given by the regularization of the
entanglement of formation [HHT01], which is shown to be non-additive [Has09].
Further efforts have been made to improve understanding of the entanglement
cost in specific and general quantum states. [CSW12, WD17a, Wil18, WJZ23]. As
exact entanglement dilution is more restricted, it holds that

$\displaystyle E_{C,\Omega}(\rho)\leq
E_{C,\Omega}^{{\operatorname{exact}}}(\rho).$ (9)

#### Exact entanglement transformations.

The ratio of state conversion plays an integral role in the manipulation of
quantum resources such as entanglement. In this paper, we mainly focus on the
asymptotic exact state conversion. It describes the process of converting one
state into another exactly as the number of copies approaches infinity under
certain free operations. Let $\Omega$ represent a set of free operations. The
asymptotic conversion ratio of exact entanglement transformation from a state
or quasi-state $\rho_{AB}$ to another state or quasi-state
$\sigma_{\hat{A}\hat{B}}$ is defined as

$\displaystyle
R_{\Omega}\left(\rho_{AB}\to\sigma_{A^{\prime}B^{\prime}}\right)=\sup\left\\{\frac{m}{n}\mathrel{\mathop{\mathchar
58\relax}}\exists n_{0}\mathrel{\mathop{\mathchar 58\relax}}\forall n\geq
n_{0},\exists\Lambda_{n}\in\Omega\mathrel{\mathop{\mathchar
58\relax}}\Lambda_{n}\left(\rho_{AB}^{\otimes
n}\right)=\sigma_{A^{\prime}B^{\prime}}^{\otimes m}\right\\}.$ (10)

Two states can be exactly interconverted reversibly if
$R_{\Omega}\left(\rho_{AB}\to\sigma_{A^{\prime}B^{\prime}}\right)\times
R_{\Omega}\left(\sigma_{A^{\prime}B^{\prime}}\to\rho_{AB}\right)=1$.

Class of Operations | Abbreviation
---|---
Local Operations and Classical Communication | LOCC
Separable Operations | SEP
Completely PPT-preserving Operations | PPT
Positive and Trace-preserving and PPT-preserving Maps | PTP-PPT
Hermitian-preserving and Trace-preserving PPT Maps | PPTq

Fig 2: Schematic hierarchy of operations. The pictured inclusions among LOCC,
SEP, PPT, PTP-PPT, PPTq. The main focus of this paper is entanglement
manipulation under PPTq operations.

##  3 Reversibility of Exact Entanglement Manipulation

###  3.1 Necessary and sufficient condition for entanglement transformations

In this section, our focus is centered in exploring the possibility of
reversible entanglement manipulation. To thoroughly characterize the
fundamental limits of entanglement transformation, we briefly venture beyond
the confines of the quantum realm in the context of this section, focusing on
entanglement transformations beyond quantum operations. Following the idea of
axiomatic approaches, we consider transformations that demand the weakest
possible requirements for entanglement manipulation, relaxing the manipulating
objects from quantum states to quasi-states (c.f. Eq. (3)).

We specifically introduce the PPT quasi-operations (PPTq), which are
Hermitian-preserving and trace-preserving maps that completely preserve the
positive of partial transpose. The underlying intuition is that PPT states are
all bound entanglement [HHH98], which are useless in entanglement
distillation. Furthermore, as depicted in Figure 2, PPTq operations encompass
all SEP, LOCC, and PPT operations, whereas some PTP-PPT operations do not fall
within the scope of ${\operatorname{PPTq}}$.

###### Definition 1

An HPTP bipartite map ${\cal N}_{AB\to A^{\prime}B^{\prime}}$ is called a PPT
quasi-operation (PPTq operation) if $T_{B^{\prime}}\circ{\cal N}_{AB\to
A^{\prime}B^{\prime}}\circ T_{B}$ is completely positive.

We are going to firstly establish the necessary and sufficient condition for
perfect transformations between quasi-states. It turns out surprising that the
logarithmic negativity [VW02, Ple05a] is the key to fully characterize the
transformations under PPTq operations. We also note that the following theorem
directly applies to the more restricted case for transformations between
quantum states.

###### Theorem 1

For two bipartite states or quasi-states $\rho$ and $\sigma$, there exists
${\cal N}_{AB\to A^{\prime}B^{\prime}}\in\operatorname{PPTq}$ such that ${\cal
N}(\rho)=\sigma$ if and only if $E_{N}(\rho)\geq E_{N}(\sigma),$ (11) where
$E_{N}(\rho)=\log\|\rho^{T_{B}}\|_{1}$ is the logarithmic negativity.

###### Proof.

($\implies$) Construct a linear map ${\cal M}=T_{B^{\prime}}\circ{\cal N}\circ
T_{B}$. Since ${\cal N}_{AB\to A^{\prime}B^{\prime}}\in\operatorname{PPTq}$,
we know that ${\cal M}$ is completely positive and trace-preserving. By
construction, it holds that ${\cal
M}\left(\rho^{T_{B}}\right)=\sigma^{T_{B^{\prime}}}$. Since any CPTP operation
does not increase the trace norm, we immediately have that
$\lVert\rho^{T_{B}}\rVert_{1}\geq\lVert\sigma^{T_{B^{\prime}}}\rVert_{1}$ and
hence $E_{N}(\rho)\geq E_{N}(\sigma)$ by the monotonicity of the $\log$
function.

($\impliedby$) The key idea to show this direction is to construct a CPTP map
${\cal M}$ such that ${\cal
M}\left(\rho^{T_{B}}\right)=\sigma^{T_{B^{\prime}}}$ based on the assumption
that $\|\rho^{T_{B}}\|_{1}\geq\|\sigma^{T_{B^{\prime}}}\|_{1}$, which could
guarantee that ${\cal N}=T_{B^{\prime}}\circ{\cal M}\circ T_{B}$ is a HPTP and
PPT map that can successfully transform $\rho$ to $\sigma$. Consider the
spectral decomposition $\rho^{T_{B}}=\sum_{j}r_{j}|j\rangle\\!\langle
j|=R_{+}-R_{-}$ with $R_{+}=\sum_{j\mathrel{\mathop{\mathchar
58\relax}}r_{j}\geq 0}r_{j}|j\rangle\\!\langle j|$ and
$R_{-}=-\sum_{j\mathrel{\mathop{\mathchar
58\relax}}r_{j}<0}r_{j}|j\rangle\\!\langle j|$. Here we denote the projections
for positive and negative parts as

$P_{+}=\sum_{j\mathrel{\mathop{\mathchar 58\relax}}r_{j}\geq
0}|j\rangle\\!\langle j|,\quad P_{-}=\sum_{j\mathrel{\mathop{\mathchar
58\relax}}r_{j}<0}|j\rangle\\!\langle j|,\quad P_{+}+P_{-}=I.$ (12)

Without loss of generality, we assume that $E_{N}(\sigma)>0$. Let us then
denote $\sigma^{T_{B^{\prime}}}=\sum_{n}s_{n}|n\rangle\\!\langle n|$ and
choose some $k\in{{\mathbb{N}}}$ such that $s_{k}\geq 0$. We further could
write
$\sigma^{T_{B^{\prime}}}=\widetilde{S}_{+}-S_{-}+s_{k}|k\rangle\\!\langle k|$
with

$\widetilde{S}_{+}=\sum_{n\mathrel{\mathop{\mathchar 58\relax}}s_{n}\geq
0,\,n\neq k}s_{n}|n\rangle\\!\langle n|,\quad
S_{-}=-\sum_{n\mathrel{\mathop{\mathchar
58\relax}}s_{n}<0}s_{n}|n\rangle\\!\langle n|.$ (13)

We construct the following CPTP map:

$\displaystyle{\cal
M}(\omega)=\frac{\operatorname{Tr}(P_{+}\omega)}{\operatorname{Tr}R_{+}}\widetilde{S}_{+}+\frac{\operatorname{Tr}(P_{-}\omega)}{\operatorname{Tr}R_{-}}S_{-}+\lambda|k\rangle\\!\langle
k|,$ (14)

with
$\lambda=\operatorname{Tr}\omega-{\operatorname{Tr}(P_{+}\omega)\operatorname{Tr}\widetilde{S}_{+}}/{\operatorname{Tr}R_{+}}-{\operatorname{Tr}(P_{-}\omega)\operatorname{Tr}S_{-}}/{\operatorname{Tr}R_{-}}$.
This construction guarantees the condition of TP as the trace of the R.H.S. of
Eq. (14) is equal to $\operatorname{Tr}\omega$. To see that this map is also
CP, we only need to show that $\lambda\geq 0$. Note that the prerequisite
$E_{N}(\rho)\geq E_{N}(\sigma)$ implies that
$\frac{\operatorname{Tr}\widetilde{S}_{+}}{\operatorname{Tr}R_{+}}\leq 1$ and
$\frac{\operatorname{Tr}S_{-}}{\operatorname{Tr}R_{-}}\leq 1$, thus we have
that

$\displaystyle\lambda\geq\operatorname{Tr}\omega-\operatorname{Tr}(P_{+}\omega)-\operatorname{Tr}(P_{-}\omega)=0.$
(15)

Finally, we apply the quantum channel ${\cal M}$ in Eq. (14) to $\rho^{T_{B}}$
and obtain

$\displaystyle{\cal M}(\rho^{T_{B}})={\cal
M}(R_{+}-R_{-})=\widetilde{S}_{+}-S_{-}+s_{k}|k\rangle\\!\langle
k|=\sigma^{T_{B}}.$ (16)

$\sqcap$$\sqcup$

This result implies that logarithmic negativity emerges as the pivotal
entanglement measure for determining the transformations between entangled
states, which is analogous to the role of entropy in the second law of
thermodynamics.

###  3.2 Reversibility of exact entanglement manipulation under PPTq
operations

Entanglement distillation and entanglement dilution are two vital operational
tasks in entanglement manipulation. Entanglement distillation involves the
transformation of a large number of identically and independently distributed
(i.i.d.) copies of a certain state into as many ebits as possible. Conversely,
entanglement dilution is concerned with the reverse process, turning ebits
into as many copies of the original state. These procedures are typically
carried out by means of LOCC. Here, we are going to solve the key rates for
both exact distillable entanglement and exact entanglement cost under PPTq
operations, which are shown to be equal to the logarithmic negativity of the
state.

###### Theorem 2

For any bipartite state or quasi-state $\rho_{AB}$, it holds that
$\displaystyle
E_{D,\operatorname{PPTq}}^{\operatorname{exact}}(\rho_{AB})=E_{C,\operatorname{PPTq}}^{\operatorname{exact}}(\rho_{AB})=E_{N}({\rho_{AB}}).$
(17)

###### Proof.

The proof of this theorem is divided into two parts. We are going to firstly
show $E_{N}(\rho_{AB})\leq
E_{D,\operatorname{PPTq}}^{\operatorname{exact}}(\rho_{AB})$ and then show
$E_{C,\operatorname{PPTq}}^{\operatorname{exact}}(\rho_{AB})\leq
E_{N}(\rho_{AB})$.

Let us consider $n$-copy of the state $\rho_{AB}$ and try to transform it to
as many ebits as possible. By Theorem 1, we know that there exists one
$\Lambda\in{\operatorname{PPTq}}$ and maximally entangled state
$\Phi_{A^{\prime}B^{\prime}}^{d}$ with $d=\lfloor
2^{n{E_{N}(\rho_{AB})}}\rfloor$ such that

$\displaystyle\Lambda(\rho_{AB}^{\otimes n})=\Phi_{A^{\prime}B^{\prime}}^{d},$
(18)

since $E_{N}(\rho_{AB}^{\otimes n})=nE_{N}(\rho_{AB})\geq\log\lfloor
2^{n{E_{N}(\rho_{AB})}}\rfloor=\log d=E_{N}(\Phi_{A^{\prime}B^{\prime}}^{d})$.
Thus, by the definition of exact entanglement distillation, we have

$\displaystyle E^{(1)}_{0,D,{\operatorname{PPTq}}}(\rho_{AB}^{\otimes
n})\geq\log\lfloor 2^{n{E_{N}(\rho_{AB})}}\rfloor,$ (19)

which leads to

$\displaystyle
E_{D,{\operatorname{PPTq}}}^{{\operatorname{exact}}}(\rho_{AB})$
$\displaystyle=\lim_{n\to\infty}\frac{1}{n}E^{(1)}_{0,D,{\operatorname{PPTq}}}(\rho_{AB}^{\otimes
n})\geq\lim_{n\to\infty}\frac{1}{n}\log\lfloor
2^{n{E_{N}(\rho_{AB})}}\rfloor=E_{N}(\rho_{AB}).$ (20)

For the reverse task of entanglement dilution, let us consider $n$-copy of the
state $\rho_{AB}$ and try to transform at least ebits as possible to prepare
$\rho_{AB}^{\otimes n}$. By Theorem 1, we know that there exists one
$\Lambda\in{\operatorname{PPTq}}$ and maximally entangled state
$\Phi_{A^{\prime}B^{\prime}}^{d}$ with $d=\lceil
2^{n{E_{N}(\rho_{AB})}}\rceil$ such that

$\displaystyle\Lambda(\Phi_{A^{\prime}B^{\prime}}^{d})=\rho_{AB}^{\otimes n},$
(21)

since $E_{N}(\rho_{AB}^{\otimes n})=nE_{N}(\rho_{AB})\leq\log\lceil
2^{n{E_{N}(\rho_{AB})}}\rceil=\log d=E_{N}(\Phi_{A^{\prime}B^{\prime}}^{d})$.
Therefore, by the definition of exact entanglement cost, we have

$\displaystyle E^{(1)}_{0,C,{\operatorname{PPTq}}}(\rho_{AB}^{\otimes
n})\leq\log\lceil 2^{n{E_{N}(\rho_{AB})}}\rceil,$ (22)

which leads to

$\displaystyle
E_{C,{\operatorname{PPTq}}}^{{\operatorname{exact}}}(\rho_{AB})$
$\displaystyle=\lim_{n\to\infty}\frac{1}{n}E^{(1)}_{0,C,{\operatorname{PPTq}}}(\rho_{AB}^{\otimes
n})\leq\lim_{n\to\infty}\frac{1}{n}\log\lceil
2^{n{E_{N}(\rho_{AB})}}\rceil=E_{N}(\rho_{AB}).$ (23)

Note that exact transformations between $\rho$ and $\Phi^{2}$ guarantees the
inequality

$\displaystyle E_{D,{\operatorname{PPTq}}}^{\operatorname{exact}}(\rho)\leq
E_{C,{\operatorname{PPTq}}}^{\operatorname{exact}}(\rho).$ (24)

Thus, combining Eq. (24), Eq. (20) and Eq. (23), we arrive at

$\displaystyle
E_{D,\operatorname{PPTq}}^{\operatorname{exact}}(\rho_{AB})=E_{C,\operatorname{PPTq}}^{\operatorname{exact}}(\rho_{AB})=E_{N}({\rho_{AB}}).$
(25)

$\sqcap$$\sqcup$

The above result has already implies a collapse of two important entanglement
measures. As the logarithmic negativity quantifies both exact entanglement
cost and exact distillable entanglement, we want to note that it also explains
why logarithmic negativity has been fruitfully used in the theory of quantum
entanglement. For example, previous works have shown that the logarithmic
negativity is an upper bound to distillable entanglement [VW02] and possesses
an operational interpretation as the exact entanglement cost under PPT
operations for certain classes of quantum states [APE03, Ish04].

Furthermore, using results on exact entanglement cost and exact distillable
entanglement, we are now able to prove the reversibility of asymptotic exact
entanglement manipulation under PPTq operations.

###### Theorem 3

For any two bipartite states $\rho_{AB}$ and $\sigma_{A^{\prime}B^{\prime}}$,
the asymptotic exact entanglement transformation rate is given by
$\displaystyle
R_{\operatorname{PPTq}}(\rho_{AB}\to\sigma_{A^{\prime}B^{\prime}})=\frac{E_{N}(\rho_{AB})}{E_{N}(\sigma_{A^{\prime}B^{\prime}})},$
(26) which implies the reversibility of asymptotic exact entanglement
manipulation under ${\operatorname{PPTq}}$ operations, i.e., $\displaystyle
R_{\operatorname{PPTq}}(\rho_{AB}\to\sigma_{A^{\prime}B^{\prime}})\times
R_{\operatorname{PPTq}}(\sigma_{A^{\prime}B^{\prime}}\to\rho_{AB})=1.$ (27)

###### Proof.

The key idea of this proof is to use the maximally entangled state for
intermediate exchange between states, that is,

$\displaystyle
R_{\operatorname{PPTq}}(\rho_{AB}\to\Phi_{\hat{A}\hat{B}}^{2})=E_{D,{\operatorname{PPTq}}}^{{\operatorname{exact}}}(\rho_{AB})=E_{N}(\rho_{AB}),$
(28) $\displaystyle
R_{\operatorname{PPTq}}(\Phi_{\hat{A}\hat{B}}^{2}\to\sigma_{A^{\prime}B^{\prime}})=E_{C,{\operatorname{PPTq}}}^{{\operatorname{exact}}}(\sigma_{A^{\prime}B^{\prime}})^{-1}=E_{N}(\sigma_{A^{\prime}B^{\prime}})^{-1}.$
(29)

Therefore, we could obtain the transformation ratio as

$\displaystyle
R_{\operatorname{PPTq}}(\rho_{AB}\to\sigma_{A^{\prime}B^{\prime}})$
$\displaystyle=R_{\operatorname{PPTq}}(\rho_{AB}\to\Phi_{\hat{A}\hat{B}}^{2})\times
R_{\operatorname{PPTq}}(\Phi_{\hat{A}\hat{B}}^{2}\to\sigma_{AB})$ (30)
$\displaystyle=E_{N}(\rho_{AB})\times
E_{N}(\sigma_{A^{\prime}B^{\prime}})^{-1}.$ (31)

$\sqcap$$\sqcup$

As we have established the tight connection for exact measures
$E_{C,{\operatorname{PPTq}}}^{\operatorname{exact}}(\rho)$ and
$E_{D,{\operatorname{PPTq}}}^{\operatorname{exact}}(\rho)$ in Theorem 2, we
finally arrive at the inequality chain of the entanglement manipulation rates
for ${\operatorname{PPTq}}$ operations as shown the following theorem.

###### Theorem 4

For any bipartite state $\rho$, it holds that $\displaystyle
E_{N}^{\tau}(\rho)\leq E_{C,{\operatorname{PPTq}}}(\rho)\leq
E_{C,{\operatorname{PPTq}}}^{\operatorname{exact}}(\rho)=E_{N}(\rho)=E_{D,{\operatorname{PPTq}}}^{\operatorname{exact}}(\rho)=E_{D,{\operatorname{PPTq}}}(\rho).$
(32)

_Sketch of Proof._ Note that $E_{C,{\operatorname{PPTq}}}(\rho)\leq
E_{C,{\operatorname{PPTq}}}^{\operatorname{exact}}(\rho)$ and
$E_{D,{\operatorname{PPTq}}}^{\operatorname{exact}}(\rho)\leq
E_{D,{\operatorname{PPTq}}}(\rho)$ follow by Equation (9) and Equation (6),
respectively. Since Theorem 2 bridges the gap between these two exact
measures, it is sufficient to prove the two endpoints in the inequality chain.
A full proof can be found in Appendix A.

Remark 1 In the standard quantum resource theory, the achievable rate of
entanglement dilution typically surpasses that of entanglement distillation.
However, the introduction of ${\operatorname{PPTq}}$ operations reverses this
kind of operational inequality. The uncanny phenomenon is attributed to the
unique property of quasi-operations, which can be decomposed into positive and
negative operation components. Such property allows ${\operatorname{PPTq}}$
operations to “borrow” additional entanglement resources from seemingly out of
nowhere. Specifically, these components can generate states with extra
distillable entanglement. Once the desired transformations are accomplished,
this borrowed resource can then be effectively “returned” by combining the
positive and negative components to form the target state. This unique feature
allows for the achievable rate of entanglement distillation to surpass that of
entanglement dilution, presenting a surprisingly reversal of the usual
operational inequality in standard resource theory.

Notably, even though the proof demonstrates
$E_{C,{\operatorname{PPTq}}}(\rho)\leq E_{D,{\operatorname{PPTq}}}(\rho)$, the
absence of asymptotic continuity in the logarithmic negativity $E_{N}$
suggests that $E_{C,{\operatorname{PPTq}}}(\rho)$ does not necessarily equal
to $E_{N}(\rho)$. This observation is indicative of the fact that the resource
theory in ${\operatorname{PPTq}}$ remains compatible with the existing
entanglement theories under quantum operations.

##  4 Discussions

Our work has demonstrated the reversibility of exact entanglement
transformations under PPTq operations. This reversibility establishes a
parallel between entanglement manipulation and the second law of
thermodynamics, particularly when operating under idealized conditions. The
logarithmic negativity is the key entanglement measure in this reversible
entanglement theory to determine the exact transformation between entangled
states, serving a role analogous to entropy in the realm of thermodynamics.
The advent of a reversible theory of exact entanglement manipulation under
PPTq operations paves the way for further exploration into the smallest subset
of quantum operations or maps nestled within the set of PPTq operations that
could guarantee the reversibility of asymptotic entanglement manipulation. Our
work also opens a possible avenue to study quantum resources beyond free
quantum operations.

Paradigm | Class of Operations | Free Resource | Resource | Reversible?
---|---|---|---|---
Thermodynamics | - | Heat | Work | ✓
Coherence [WY16] | Incoherent | Incoherent States | Coherent States | ✗
Coherence [BaG15] | Maximally Incoherent | Incoherent States | Coherent States | ✓
Entanglement [VC01] | LOCC | Separable States | Entangled States | ✗
NPT Entanglement [WD17a] | PPT | PPT States | NPT States | ✗
Entanglement [LR23] | Non-Entangling | Separable States | Entangled States | ✗
NPT Entanglement (This Work) | PPTq |  Quasi-states w. zero $E_{N}$ |  Quasi-states w. positive $E_{N}$ | ✓

Table 1: Mainstream resource theories. This work presents a reversible theory
of exact entanglement manipulation under PPTq operations, where asymptotic
exact entanglement transformation is reversible.

While our work delineates a reversible theory of exact entanglement
manipulation, it is imperative to remark that these transformations ostensibly
transcend the standard quantum mechanics as the allowed operations extend
beyond quantum operations. Meanwhile, within the domain of quantum mechanics,
the reversibility of entanglement under quantum operations that do not
asymptotically generate entanglement (ANE) remains an open question [BBG+23,
FGW21] due to a gap in the proof of the generalised quantum Stein’s lemma. The
reversibility of exact entanglement manipulation along with the uncertainty of
reversibility under ANE quantum operations may suggest a potential
incompatibility between the foundational principles of quantum mechanics and
the reversibility of quantum entanglement. In addition, as a recent work
[RL23] shows that the reversibility of quantum resources could happen when
relaxing to probabilistic transformations, it will also be interesting to
study the interplay between reversibility, success probability, and positivity
of allowed transformations of quantum resources.

## Acknowledgements

We would like to thank Chengkai Zhu for helpful discussions and thank Benchi
Zhao, Xuanqiang Zhao, Mingrui Jing for their useful comments. This work has
been supported by the Start-up Fund from The Hong Kong University of Science
and Technology (Guangzhou).

## References

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Appendix for Reversible Entanglement Beyond Quantum Operations

##  Appendix A Appendix: Inequality chain for entanglement measures under PPTq

In the appendix, we will rigorously demonstrate how the inequality chain in
Theorem 4

$E_{N}^{\tau}(\rho)\leq E_{C,{\operatorname{PPTq}}}(\rho)\leq
E_{C,{\operatorname{PPTq}}}^{\operatorname{exact}}(\rho)=E_{N}(\rho)=E_{D,{\operatorname{PPTq}}}^{\operatorname{exact}}(\rho)=E_{D,{\operatorname{PPTq}}}(\rho).$
(S1)

is build. We start from the construction of two endpoints in the chain, i.e.
prove $E_{N}^{\tau}(\rho)\leq E_{C,{\operatorname{PPTq}}}(\rho_{AB})$
(Proposition S2) and $E_{D,{\operatorname{PPTq}}}(\rho_{AB})\leq
E_{N}(\rho_{AB})$ (Proposition S3). For the temper logarithm negativity part,
we first need to extend the definition of tempered negativity in Ref. [LR23]
to fit in the regime of quasi-states.

###### Definition S1

Let $\sigma_{AB},\rho_{AB}$ be two quasi-states. The tempered negativity
between $\sigma_{AB}$ and $\rho_{AB}$ is defined as

$N_{\tau}\left(\sigma_{AB}\mid\rho_{AB}\right)=\sup\\{\operatorname{Tr}\left[X\sigma_{AB}\right]\mathrel{\mathop{\mathchar
58\relax}}\lVert X^{T_{B}}\rVert_{\infty}\leq
1,\|X\|_{\infty}=\operatorname{Tr}\left[X\rho_{AB}\right]\\}.$ (S2)

Further, the tempered negativity of $\rho_{AB}$ is denoted by

$N_{\tau}\left(\rho_{AB}\right)=N_{\tau}\left(\rho_{AB}\mid\rho_{AB}\right).$
(S3)

Here, we restate three properties of $N_{\tau}$, and show that the original
proof in Ref. [LR23] remains true after the extension. Then we are ready to
give the proposition of inequality.

###### Lemma S1

For any quasi-state $\omega_{AB}$ and state $\rho_{AB}$,

  1. $\mathbf{(a)}$

$\lVert\sigma_{AB}^{T_{B}}\rVert_{1}\geq N_{\tau}(\sigma_{AB}\mid\rho_{AB})$,

  2. $\mathbf{(b)}$

$\lVert\sigma_{AB}-\rho_{AB}\rVert_{1}\leq\varepsilon\implies
N_{\tau}\left(\sigma_{AB}\mid\rho_{AB}\right)\geq(1-\varepsilon)N_{\tau}\left(\rho_{AB}\right)$,
and

  3. $\mathbf{(c)}$

$N_{\tau}\left(\rho_{AB}^{\otimes n}\right)\geq
N_{\tau}\left(\rho_{AB}\right)^{n}$.

###### Proof.

$\mathbf{(a)}$ This property follows by the fact that
$\lVert\sigma_{AB}\rVert_{1}=\sup\\{\operatorname{Tr}\left[X\sigma_{AB}\right]\mathrel{\mathop{\mathchar
58\relax}}\lVert X^{T_{B}}\rVert_{\infty}\leq 1\\}$.

$\mathbf{(b)}$ Note that the Hölder’s inequality holds for arbitrary complex
matrices. Hence, this properties still holds by Equation (S46) in Ref. [LR23].

$\mathbf{(c)}$ Since $\rho_{AB}$ is a quantum state, this property holds from
the same reasoning in Ref. [LR23].  $\sqcap$$\sqcup$

###### Proposition S2

For any bipartite state $\rho_{AB}$, it holds that

$\displaystyle E_{C,{\operatorname{PPTq}}}(\rho_{AB})\geq E_{N}^{\tau}(\rho).$
(S4)

###### Proof.

This proof mainly follows from Lemma S1 and the idea of chain inequalities in
Ref. [LR23]. For $r=E_{C,{\operatorname{PPTq}}}(\rho_{AB})$, there exists a
sequence of PPT quasi-operations $\\{\Lambda_{n}\\}_{n}$ such that

$\sigma_{n,\hat{A}\hat{B}}=\Lambda_{n}\left(\Phi_{\hat{A}\hat{B}}^{2\lfloor
rn\rfloor}\right),\quad\varepsilon_{n}=\lVert\sigma_{n,\hat{A}\hat{B}}-\rho_{AB}^{\otimes
n}\rVert_{1}\textrm{ and }\lim_{n\to\infty}\varepsilon_{n}=0.$ (S5)

Then for all $n$, Theorem 1 implies
$E_{N}\left(\Phi_{\hat{A}\hat{B}}^{2\lfloor rn\rfloor}\right)\geq
E_{N}\left(\sigma_{n,\hat{A}\hat{B}}\right)$ and hence

$\displaystyle 2^{\lfloor rn\rfloor}$
$\displaystyle\geq\lVert\left(\sigma_{n,\hat{A}\hat{B}}\right)^{T_{B}}\rVert_{1}$
(S6)
$\displaystyle\overset{\mathbf{(a)}}{\geq}N_{\tau}\left(\sigma_{n,\hat{A}\hat{B}}\mid\rho_{AB}^{\otimes
n}\right)\overset{\mathbf{(b)}}{\geq}\left(1-\varepsilon_{n}\right)N_{\tau}\left(\rho_{AB}^{\otimes
n}\right)\overset{\mathbf{(c)}}{\geq}\left(1-\varepsilon_{n}\right)N_{\tau}\left(\rho_{AB}\right)^{n}.$

Taking $n\to\infty$ on both sides gives $r=\lim_{n\to\infty}\lfloor
rn\rfloor/n\geq E_{N}^{\tau}\left(\rho_{AB}\right)$.  $\sqcap$$\sqcup$

For the entanglement distillation part, the proposition can be given as
follows.

###### Proposition S3

For any state or quasi state $\rho_{AB}$, it holds that

$\displaystyle E_{D,{\operatorname{PPTq}}}(\rho_{AB})\leq E_{N}(\rho_{AB}).$
(S7)

###### Proof.

This proof mainly follows the proof from Ref. [VW02]. For
$r=E_{D,{\operatorname{PPTq}}}(\rho_{AB})$, there exists a sequence of PPT
quasi-operations $\\{\Lambda_{n}\\}_{n}$ such that

$\sigma_{n,\hat{A}\hat{B}}=\Lambda_{n}\left(\rho_{AB}^{\otimes
n}\right),\quad\varepsilon_{n}=\lVert\sigma_{n,\hat{A}\hat{B}}-\Phi^{2\lfloor
rn\rfloor}_{\hat{A}\hat{B}}\rVert_{1}\textrm{ and
}\lim_{n\to\infty}\varepsilon_{n}=0.$ (S8)

Then for all $n$, Theorem 1 implies $E_{N}\left(\rho_{AB}^{\otimes
n}\right)\geq E_{N}\left(\sigma_{n,\hat{A}\hat{B}}\right)$ and hence

$\displaystyle\lVert\left(\rho_{AB}^{\otimes n}\right)^{T_{B}}\rVert_{1}$
$\displaystyle\geq\lVert\left(\sigma_{n,\hat{A}\hat{B}}\right)^{T_{B}}\rVert_{1}=\lVert\left(\Phi^{2\lfloor
rn\rfloor}_{\hat{A}\hat{B}}\right)^{T_{B}}+\left(\sigma_{n,\hat{A}\hat{B}}-\Phi^{2\lfloor
rn\rfloor}_{\hat{A}\hat{B}}\right)^{T_{B}}\rVert_{1}$ (S9)
$\displaystyle\geq\lVert\left(\Phi^{2\lfloor
rn\rfloor}_{\hat{A}\hat{B}}\right)^{T_{B}}\rVert_{1}-\lVert\left(\sigma_{n,\hat{A}\hat{B}}-\Phi^{2\lfloor
rn\rfloor}_{\hat{A}\hat{B}}\right)^{T_{B}}\rVert_{1}$ (S10)
$\displaystyle\geq\left(1-\varepsilon_{n}\right)2^{\lfloor rn\rfloor},$ (S11)

where the last inequality follows from the fact $\lVert
X^{T_{B}}\rVert_{1}\leq d\lVert X\rVert_{1}$ for
$X\in\operatorname{Herm}\left({\cal H}_{d}\right)$. Taking $n\to\infty$ on
both sides, the additivity of logarithm negativity gives

$E_{N}\left(\rho_{AB}\right)\geq\lim_{n\to\infty}\frac{1}{n}\log\left(1-\varepsilon_{n}\right)2^{\lfloor
rn\rfloor}=\lim_{n\to\infty}\frac{\lfloor
rn\rfloor}{n}+\frac{\log\left(1-\varepsilon_{n}\right)}{n}=r.$ (S12)

$\sqcap$$\sqcup$

We are ready to construct the whole chain.

###### Theorem 4

For any bipartite state $\rho$, it holds that

$\displaystyle E_{N}^{\tau}(\rho)\leq E_{C,{\operatorname{PPTq}}}(\rho)\leq
E_{C,{\operatorname{PPTq}}}^{\operatorname{exact}}(\rho)=E_{N}(\rho)=E_{D,{\operatorname{PPTq}}}^{\operatorname{exact}}(\rho)=E_{D,{\operatorname{PPTq}}}(\rho).$
(S13)

###### Proof.

For all bipartite state $\rho$,

$\displaystyle E_{N}^{\tau}(\rho)$ $\displaystyle\leq
E_{C,{\operatorname{PPTq}}}(\rho)$ (Proposition S2) (S14) $\displaystyle\leq
E_{C,{\operatorname{PPTq}}}^{\operatorname{exact}}(\rho)$ (Equation (9)) (S15)
$\displaystyle=E_{N}(\rho)=E_{D,{\operatorname{PPTq}}}^{\operatorname{exact}}(\rho)\quad$
(Theorem 2) (S16) $\displaystyle\leq E_{D,{\operatorname{PPTq}}}(\rho)$
(Equation (6)) (S17) $\displaystyle\leq E_{N}(\rho).$ (Proposition S3) (S18)

$\sqcap$$\sqcup$