††thanks: These authors contributed equally††thanks: These authors contributed
equally

# Microscopic Theory of Magnetic Disorder-Induced Decoherence in
Superconducting Nb Films

Evan Sheridan  Materials Science Division, Lawrence Berkeley National
Laboratory, Berkeley, CA 94720, USA  Molecular Foundry, Lawrence Berkeley
National Laboratory, Berkeley, CA 94720, USA  King’s College London, Theory
and Simulation of Condensed Matter, WC2R 2LS London, UK  Thomas F. Harrelson
Materials Science Division, Lawrence Berkeley National Laboratory, Berkeley,
CA 94720, USA  Molecular Foundry, Lawrence Berkeley National Laboratory,
Berkeley, CA 94720, USA  Eric Sivonxay  Energy Technologies Area, Lawrence
Berkeley National Laboratory, Berkeley, CA 94720, USA  Department of Materials
Science and Engineering, University of California, Berkeley, Berkeley, CA
94704, USA  Kristin A. Persson  Energy Technologies Area, Lawrence Berkeley
National Laboratory, Berkeley, CA 94720, USA  Department of Materials Science
and Engineering, University of California, Berkeley, Berkeley, CA 94704, USA
M. Virginia P. Altoé  Molecular Foundry, Lawrence Berkeley National
Laboratory, Berkeley, CA 94720, USA  Irfan Siddiqi  Department of Physics,
University of California, Berkeley, California 94720, USA  Center for Quantum
Coherent Science, University of California, Berkeley, California 94720, USA
D. Frank Ogletree  Molecular Foundry, Lawrence Berkeley National Laboratory,
Berkeley, CA 94720, USA  David I. Santiago  Center for Quantum Coherent
Science, University of California, Berkeley, California 94720, USA  Sinéad M.
Griffin  Materials Science Division, Lawrence Berkeley National Laboratory,
Berkeley, CA 94720, USA  Molecular Foundry, Lawrence Berkeley National
Laboratory, Berkeley, CA 94720, USA

###### Abstract

The performance of superconducting qubits is orders of magnitude below what is
expected from theoretical estimates based on the loss tangents of the
constituent bulk materials. This has been attributed to the presence of
uncontrolled surface oxides formed during fabrication which can introduce
defects and impurities that create decoherence channels. Here, we develop an
ab initio Shiba theory to investigate the microscopic origin of magnetic-
induced decoherence in niobium thin film superconductors and the formation of
native oxides. Our ab initio calculations encompass the roles of structural
disorder, stoichiometry, and strain on the formation of decoherence-inducing
local spin moments. With parameters derived from these first-principles
calculations we develop an effective quasi-classical model of magnetic-induced
losses in the superconductor. We identify d-channel losses (associated with
oxygen vacancies) as especially parasitic, resulting in a residual zero
temperature surface impedance. This work provides a route to connecting atomic
scale properties of superconducting materials and macroscopic decoherence
channels affecting quantum systems.

##  I Introduction

Recent breakthroughs in superconducting qubits, following decades-long
improvements in fabrication and design, have established superconductors as a
leading platform for scalable fault-tolerant quantum computingArute  _et al._
(2019); Blais  _et al._ (2019); Krantz  _et al._ (2019); Müller  _et al._
(2019); Zmuidzinas (2012); Murch  _et al._ (2013a, b); Macklin  _et al._
(2015). Despite this progress, decoherence in superconducting materials
remains a major obstacle for future progress in quantum information
systemsMüller  _et al._ (2019); Krantz  _et al._ (2019), and in other
applications including particle detection and quantum sensors in high-energy
physics Zmuidzinas (2012). Central to diagnosing and mitigating the
macroscopic decoherence channels in superconducting qubits is an understanding
of the behaviour of the constituent materials at an atomic scale, especially
considering inevitable inhomogeneities such as defects, structural disorder,
and interfaces.

Decoherence channels in superconducting materials are broadly classified into
charge noise, spin impurity magnetic flux noise, and non-magnetic
quasiparticle excitations Krantz  _et al._ (2019). Magnetic flux noise arises
through interactions between Cooper pairs and magnetic impurities mediated by
an electromagnetic field, and have been suggested to contribute to 1/f noise
Wellstood  _et al._ (1987); Koch  _et al._ (2007); Weissman (1988); Pershoguba
_et al._ (2015). While BCS superconductors are robust against non-magnetic
impuritiesAnderson (1959), over a critical magnetic impurity density a gapless
superconductor emergesAbrikosov and Gor’kov (1960). Scattering on classical
spins in s-wave superconductors can cause quasiparticle bound states to appear
deep within in the gap, forming so-called Yu-Rusinov-Shiba (YSR) states Yu
(1965); Shiba (1968); Rusinov (1969). Time-reversal symmetry breaking induced
by magnetic impurities results in pair breaking that suppresses
superconductivity. Recently, Pellin et al. Kharitonov  _et al._ (2012)
proposed that magnetic impurities in s-wave superconductors cause the surface
impedance to plateau with decreasing temperature, such that the decoherence
due to magnetic impurities cannot be made arbitrarily small by lowering the
temperature. They also show that the magnetic scattering rate required to
suppress superconductivity is three orders of magnitude less than the
nonmagnetic scattering rate Kharitonov  _et al._ (2012), emphasizing the need
for microscopic models incorporating atomic and chemical specificity of
magnetic-induced decoherence in superconductors.

Previous investigations of local defects in superconducting Al films
identified unpaired spins at the air interface of the superconducting film
from the adsorption of molecular oxygen, or the formation of a metallic oxide
layer De Graaf  _et al._ (2017, 2018); Choi  _et al._ (2009). Electron spin
resonance found free spins on the interface of superconducting films;
subsequent processing steps reduced the impurity concentration and improved
the system’s coherence De Graaf  _et al._ (2018). Applying a DC-magnetic field
in both the in-plane and out-of-plane direction of a NbN coplanar waveguide
resonators during operation alters its quality factor – the authors suggest
this is due to the presence of paramagnetic impurities caused by dangling
bonds on its surfaceZollitsch  _et al._ (2019). Ab initio calculations suggest
that adsorbed molecular oxygen creates a paramagnetic species on aluminium-
based superconductors Adelstein  _et al._ (2017), contributing to 1/f noise
consistent with experiments Wang  _et al._ (2015). Similar calculations also
find that dangling hydroxyl groups on the surface of Al2O3 can generate a
local spins Lee  _et al._ (2014).

In Nb-based superconducting qubits, atomic-scale characterisation with X-ray
photoelectron spectroscopy and resonant inelastic X-ray scattering have found
that a Nb-oxide layer readily forms at the interface Halbritter (1987);
Premkumar  _et al._ (2020); Altoé  _et al._ (2020). Magnetic and charge
defects in this Nb-oxide layer have been suggested as a significant source of
dissipation Halbritter (1987); Proslier  _et al._ (2011); Kharitonov  _et al._
(2012). Recently, Altoé et al. Altoé  _et al._ (2020) found that the SiOx
primarily contributes to the two-level system (TLS) losses, while the Nb oxide
layer dominates non-TLS losses in a coplanar waveguide resonator, both of
which can be mitigated through surface treatments.

In this work we explore the atomistic origins of magnetic-induced dissipation
in superconducting Nb thin films. We develop an ab initio Shiba theory for
surface impedance that incorporates the realistic atomic structure of surface
oxides to describe the origins of dissipative loss channels. We describe
magnetic impurities inside the oxide layer on the surfaces of Nb-based
superconducting devices using density functional theory (DFT) to parameterize
an Yu-Shiba-Rusinov theory of magnetic impurities for BCS superconductivity.
We consider the range of disorder and off-stoichiometry found in the surface
oxide by considering crystalline and amorphous phases of Nb-oxides (NbO, NbO2,
and Nb2O5) with and without defects, disorder, and strain. From these
theoretical results, we suggest fabrication considerations to mitigate the
most destructive magnetic decoherence channels in Nb-based superconductors.

##  II Properties of Niobium Oxide Thin Films and Crystal Phases

Niobium oxides have been studied for potential optoelectronic applications
yielding rich insights into the relevant phases and structures Störmer  _et
al._ (2009); Nico  _et al._ (2011, 2016), in addition to exotic emergent
phenomena including metal-insulator transitionsEyert (2002). In this work we
focus on NbO, NbO2, and Nb2O5 Nico  _et al._ (2011)(Figure 1), which were
identified as the dominant stoichiometries of the Nb-oxides formed on Nb
resonators from elemental STEM EELS mapping in Ref. Altoé  _et al._ (2020).
Transmission electron microscopy measurements from the same work also suggests
primarily amorphous Nb2O5, which has been further confirmed with fluctuation
electron microscopy measurements and is consistent with X-ray Absorption
Spectroscopy performed on the same filmsHarrelson  _et al._ (2021). Informed
by these measurements, we consider both crystalline and amorphous phases of Nb
oxides to calculate the role of structural disorder, defects and strain on the
electronic and magnetic properties from first principles.

NbO adopts a face-centered cubic structure derived from the rocksalt structure
but with ordered defects resulting in square-planar coordination Burdett and
Hughbanks (1984); Bowman  _et al._ (1966). NbO2 exists in either the rutile
(metallic) or tetragonal (semiconducting) phase with a small bandgap of 0.5 eV
Nico  _et al._ (2016), both of which comprise of octahedrally coordinated Nb
forming edge- and corner-sharing networks. Nb2O5 can be formed by oxidizing
NbO, and is polymorphic with as many as eight different phases Nowak and
Ziolek (1999); Nico  _et al._ (2011, 2016), mainly formed from NbO6 octahedra.
The range of crystalline polymorphs are labelled by letters corresponding to
the temperature at which they were obtained including the TT- or T-phase at
low temperatures, the M-, B and N-phases at intermediate temperatures, and the
H-phase at higher temperatures. Other phases are stabilized through different
thermal processing steps Nowak and Ziolek (1999). As expected from a fully-
occupied O-p manifold and an empty Nb-d manifold, the Nb2O5 polymorphs are
wide bandgap semiconductors with gaps varying between 3.4-4.0 eV Nico  _et
al._ (2016).

Figure 1: Crystal structures of the Nb oxides considered in this work with
their nominal valence configuration on the Nb atoms, the Nb coordination
environment(s) and the resulting magnetic order in the stoichiometric case.

The variety of point- and edge-sharing octahedra in Nb pentoxide is a direct
result of the versatility of the nominal $d^{0}$ oxidation state and their
corresponding physical features Nowak and Ziolek (1999). N-Nb2O5 is described
by 4x4 blocks of point-sharing octahdra, where the blocks are connected by
edge-sharing octahedra. Tetragonal M-Nb2O5 can similarly be described as a set
of of 4x4 blocks, but consists of a mix of point-sharing octahdra and
tetrahedra, where adjacent blocks are connected by edge-sharing octahedra.
Monoclinic B-Nb2O5 possesses a TiO2(B)-like structure and is described by a
set 2x2 blocks of edge-sharing octahedrally distorted NbO6 polyhedra connected
by point-sharing octahedra. The pseudo-hexagonal TT-phase of Nb2O5 has also
been identified as a metastable variant of the T-phase, with magnetic
susceptibility measurements suggesting that O vacancies contribute to a
paramagnetic spin density of $2.1-9.9\times 10^{21}$ moments per mole Herval
_et al._ (2015). In some cases, O defects become ordered resulting in several
stable phases that also form magnetic moments Cava  _et al._ (1991); Waldron
_et al._ (2001); Fang  _et al._ (2015). In fact, the slightly O deficient
Nb12O29 is antiferromagnetically orderedCava  _et al._ (1991).

##  III Methods

We calculated the electronic structure and magnetic properties of the Nb
oxides using DFT as implemented in the Vienna Ab initio Simulation Package
(VASP) Kresse and Hafner (1993). Our full DFT calculations details are given
in the SI; here we justify our choice of exchange-correlation functional. We
apply a Hubbard-U of 4 eV to the Nb-d states to account for the localization
of these transition-metal orbitals, which is consistent with previous DFT+U
studies on Nb-oxidesKoçer  _et al._ (2019). We further test the sensitivity of
our calculated magnetic moments to this choice of U (see SI
LABEL:fig:u_study), finding a very slight increase in the calculated projected
magnetic moments as a result of the increased localization of the d orbitals.
Finally, we calculated the bandgap of B-Nb2O2 to be 2.89 eV (Figure
LABEL:fig:SI_HSE), underestimating the experimental value of 3.8-3.9 eV Pinto
_et al._ (2017), as expected for semilocal DFT exchange-correlation
functionalsPerdew (2009). Improvements to the treatment of electron
delocalization using a screened hybrid functional can give closer results to
experimentKrukau  _et al._ (2006); in our case HSE06 opens the bandgap to 3.86
eV. However, we find that the qualitative features of the calculated density
of states (DOS) do not change between DFT+U and HSE06 (see SI Figure
LABEL:fig:SI_HSE), so we use DFT+U with a U of 4 eV on the Nb-d states due to
the increased computational cost of hybrid functionals.

Amorphous configurations were generated from high-temperature ab initio
molecular dynamics implemented within VASP. We used scikit-learn to perform
the statistical analysis with a Random Forest ClassifierPedregosa  _et al._
(2011), cross-validation was performed with the KFold module in scikit-learn
with 5 folds, and features were generated with Pymatgen Ong  _et al._ (2013).
The formation of YSR states and the associated impedance of the
superconducting state in the SRF limit was calculated using our own code which
is available on github111https://github.com/Evan1415/impedance.

##  IV Results

###  IV.1 First-principles calculations of Nb oxides

####  IV.1.1 Crystalline Phases: Nb2O5, NbO2, and NbO

We first performed DFT calculations of the electronic and magnetic properties
of crystalline phases; NbO, the two phases of NbO2 and three polymorphs of
Nb2O5. As summarized in Figure 1, NbO and NbO2 have partially filled d
manifolds resulting in paramagnetism. Magnetic susceptibility measurements of
NbO show it to be a weakly paramagnetic metal Pollard Jr. (1968), consistent
with this nominal $d^{3}$ oxidation state. Our DFT calculations suggest a
paramagnetic metal ground state with the spin-polarized calculation being 1
meV/formula unit (f.u.) lower in energy than the non-spin-polarized case. We
calculate a ferrimagnetic ordering to be slightly lower in energy (0.07
meV/f.u) than a fully ferromagnetic state with locally projected magnetic
moments of 0.06, -0.05, 0.02 $\mu_{B}$ on the three Nb in the unit cell.
Finally, we investigate the influence of hydrostatic pressure on the magnetic
properties by uniformly scaling the lattice parameters and allowing the
internal atomic positions to optimize any residual forces. We find that
negative pressure (a proxy for tensile strain) increases the tendency to spin
polarizee in NbO – the ferrimagnetic calculation becomes 3.3 meV/f.u. lower in
energy than a non-magnetic case for 4% negative pressure. We also find the
largest local spin moment increases from 0.06 $\mu_{B}$ at equilibrium
pressure to 0.14 $\mu_{B}$ at 4% negative pressure as a result of the
increased localization of the Nb-d orbitals from the extended Nb-O bonds.

NbO2 exists in two stable structures: a high-temperature metallic rutile phase
that undergoes a metal-to-insulator transition to a tetragonal insulator,
which has a dilute density of paramagnetic moments Sakata (1969). In the
rutile phase, our DFT calculation converges to a non-spin-polarized ground
state at equilibrium volume. However, this is likely due to the high symmetry
imposed in the primitive cell calculation which does not allow for highly-
disordered site-specific magnetic moments consistent with
paramagnetismTrimarchi  _et al._ (2018). On the other hand, we calculate the
insulating tetragonal phase to have an antiferromagnetically ordered ground
state at 0 K, which is 155 meV/f.u. lower in energy than the non-spin-
polarizeed case, with a local magnetic moment of 0.93 $\mu_{B}$ at
equilibrium. Similar to NbO, we find that hydrostatic pressure causes this
moment to increase slightly from 0.93 $\mu_{B}$ at equilibrium pressure to
0.96 $\mu_{B}$ at 4% negative pressure.

To explore the role of off-stoichiometry on the electronic and magnetic
properties of the crystalline structures, we use both a rigid-band
approximation and explicit doping cases. Firstly, we include electron and hole
doping using a rigid-band model which add a compensating background charge to
the calculation with our results for the N-Nb2O5 plotted in Figure 2. For the
stoichiometric case with Nb5+ having an empty d-manifold, and fully filled O-p
we find no unpaired spins as expected. However, electron doping the system
shifts the Fermi level into the bottom of the conduction band, which comprises
Nb-d states, resulting in non-zero magnetic moments on Nb. Likewise, for hole
doping, we find a moment is induced on the O-p states, consistent with the
Fermi level shift into the valence band. We find similar qualitative trends
for all Nb-oxides considered (SI Figure LABEL:fig:SI_RB).

We calculated explicit defects considering both representative O vacancies and
O interstitials in the monoclinic N-Nb2O5 phase. N-Nb2O5 is chosen for its
favourable thermodynamical stability Valencia-Balvín  _et al._ (2014) and
variety of point- and edge-sharing octahedral linkages Nowak and Ziolek
(1999), providing a good point of reference to the amorphous phase. We
constructed a 1x2x1 supercell of N-Nb2O5 containing 32 formula units and fully
relaxed all internal coordinates. For each O vacancy chosen at point-sharing
NbO6 sites, the nominal charge of Nb decreases to Nb+4.98. In the case of
interstitials, additional O ions were placed at the midpoint of four point-
sharing NbO6 octahedras which correspond to a nominal charge reduction of
Nb+5.02 per interstitial where the separation between the defect images in the
supercell is $7.58$ Å . Figure 2(a) illustrates the structural rearrangement
for the case of a single oxygen vacancy, where mid-gap states are projected as
a real space charge density. Single O vacancies near the boundary between
point- and edge-sharing octahedra result in tetrahedrally coordinated Nb-sites
causing an increase the average Nb-O bond length. A spin-polarizeed charge
density (dangling bond) is then localized on the Nb-d states near the vacancy.
For a single interstitial oxygen as shown in Figure 2(b), the charge density
localizes on the dangling O 2p state. In fact, the interstitial creates a
creates a seven coordinated Nb bonded environment where the average Nb-O bond
length is the same as the stoichiometric scenario.

We next calculated the total and orbital resolved density of states (DOS) in
Figure 3. For the stoichiometric N-Nb2O5, we confirm a diamagnetic electronic
structure comprising a filled Nb-d manifold with a bandgap of 2.74 eV.
Including an oxygen vacancy donates two electrons which localize on the Nb
atoms, creating Nb-d impurity states within the gap (Figure 3, top panel).
These localized magnetic impurities reduce the overall bandgap between the top
of d-channel impurity states and the bottom of the conduction band Nb-d states
to 1.73 eV. For an interstitial O, the resulting hole doping causes O-p states
to form within the gap, reducing the bandgap between O-p localized impurity
states and bulk Nb-d states to 1 eV. Finally, we calculate that the doping-
dependent exchange interaction, estimated by the difference between the
magnetic and non-magnetic configurations. For Nb2O5.02 it is $J_{d}=45.8$ meV
, while for Nb2O4.98 we find that $J_{p}=8.5$ meV, approximately six times
smaller than the d-channel case.

Figure 2: (a) Real space charge density projection of mid-gap states for
N-Nb2O5 in the case of (a) an O-vacancy and (b) an O-interstitial. The
calculated magnetic moments per formula unit (per Nb2O5) are shown as a
function of the change in O stoichiometry (defined as the value of $x$ in
Nb2O5+x) for the rigid band approximation (c) and explicit O doping (d) for
the crystalline N-Nb2O5 and the amorphous Nb2O5 (configuration 5) structure.

To better understand the defect-induced spin polarizeation, we calculate a net
magnetic polarizeation as:

$P(E)=[N_{\uparrow}(E)-N_{\downarrow}(E)]/[N_{\uparrow}(E)+N_{\downarrow}(E)],$
(1)

where $N(E)$ is the density of states of majority $\uparrow$ or minority
$\downarrow$ electrons (Figure 3(b)). We find that O vacancies favour a strong
spin polarizeation in the d-channel, while O interstitials promote p-channel
spin polarizeed electronic states, consistent with our real space charge
density projection of mid-gap states in Figure 2. We find similar qualitative
behaviour of the formation of localized d-channel and p-channel midgap states
in other pentoxides, however for the case of defects in a 2$\times$2$\times$2
supercell of monoclinic B-Nb2O5 magnetism in the p-channel is suppressed. Such
discrepancies can be attributed to the differing structural characteristics in
the local bonding environments of the Nb pentoxides. Thus, the key structural
motif for a robust magnetic oxide is maximally separated under-coordinated Nb
sites, which also plays a key role for amorphous Nb2O5.

####  IV.1.2 Amorphous Nb2O5

We now turn to amorphous Nb2O5, which is suggested to be the dominant
stoichiometry and structural makeup of the Nb-oxides thin films from
experimentAltoé  _et al._ (2020); Harrelson  _et al._ (2021). We generate
amorphous Nb2O5 structures using quenched molecular dynamics, with details
given in the SI Section LABEL:sec:si_amorph, and all subsequent discussions
are for nine such generated amorphous structures. Importantly, we find that
our ab initio generated structures are consistent with the Nb-Nb distances
observed in the fluctuation electron microscopy measurements, and consistent
with X-ray absorption spectraHarrelson  _et al._ (2021).

In Figure 1 (far right panel), we show an example of a quenched amorphous
structure, which has a combination of 4-, 5- and 6-coordinated Nb. The mean
coordination number of the amorphous Nb atoms in all stoichiometric
configurations is 5.62 with a standard deviation of 0.70, while the Nb atoms
in the N-, M-, and B-phases have a coordination number of 5.88, 5.50, and 6,
respectively. Thus, the Nb atoms are less coordinated in the amorphous phase
than the crystalline phases, with the exception of the M-phase, which has
significant tetrahedral Nb coordination. The calculated average density of the
amorphous phase is 4.21 g/cm3, which agrees well with the experimentally
determined density (4.28 g/cm3) Venkataraj  _et al._ (2002). This amorphous
density is very similar to the N- and M-phases (4.29 g/cm3 and 4.30 g/cm3,
respectively), while the B-phase is significantly denser (5.29 g/cm3). We find
that this trend in density correlates well with doubly coordinated oxygen,
which we is 60%, 70%, 72.5%, for the B-, N-, and M-phases, respectively. With
the average doubly coordinated oxygen being 66.7% for the amorphous phase, we
study the N-phase in more detail given its similarity both in density and
oxygen coordination.

We calculated the spin-polarizeed DOS for amorphous Nb2O5 by averaging over
the DOS of the nine configurations with the result plotted in Figure 1(a). To
account for vacuum level shifts between different configurations, we matched
the valence band edges for each of the nine structures (ignoring midgap
states). We find that our calculated bandgaps of Nb2O5 phases (amorphous is
2.72 eV, N is 2.10 eV, M is 2.15 eV, and B is 2.89 eV) consistent with the
ranges reported in experiment given the expected underestimation of PBE
bandgapsSathasivam  _et al._ (2017). In the amorphous phase we define the
bandgap as the energy gap between the top of the bulk valence band and the
bottom of the bulk conduction band, neglecting midgap states. We find three
classes of midgap states in our amorphous configurations of Nb2O5: non-
magnetic O p-states, magnetic Nb d-states, and non-magnetic Nb d-states. We
plot the real-space projection of the example midgap states in Figure 3
representing (c) non-magnetic p-orbitals, and (d) magnetic d-orbitals. The
midgap O p-states comprise of bonded adjacent O atoms in the structure with
zero magnetic moment. The magnetic Nb d-state in Figure 1(d) has a projected
moment of $1.65$ $\mu_{B}$ that is highly localized on the Nb atom, similar to
what is found with Nb dangling bonds in the crystalline case previously
discussed. In fact, for the Nb sites where local spin moments form, we find
the average coordination of Nb to be 5.5, which is lower than the average
coordination number of all Nb ions in the dataset (5.62). The average Nb-O
bond length for these five Nb atoms is 2.105 Å, where the average Nb-O bond
length for all Nb atoms in the set is 2.028 Å (standard deviation of 0.0148
Å). Therefore in the amorphous configurations, we find local spin moments
formed where the coordination number is lower, and hence the average Nb-O bond
length is longer, resulting in increased Nb orbital localization.

Two of the nine amorphous structures have spin-polarizeed ground states with
local unpaired spins as discussed above. However, the overall energy of the
two magnetic configurations is at least 1 eV greater than the seven other
configurations (see SI Section LABEL:sec:si_amorph), indicating that the spin-
polarized configurations are a result of local minima in the material’s
potential energy. In the case where structures with magnetic moments are
kinetically trapped, we estimate the expected magnetic moment by averaging
over all considered configurations, which we find to be $0.0296$ $\mu_{B}$ per
formula unit. We note that the calculated magnetic moment per formula unit
depends on the radius of the sphere over which the projected moments are
integrated, which introduces a systematic uncertainty in this result. We find
that the sum of the projected moments is smaller than the total observed
moment in the entire primitive cell in all amorphous configurations,
suggesting that the value of $0.0296$ $\mu_{B}$ per formula unit is an
underestimation. Prior experiments indicate that the observed magnetic moment
per f.u. for the T- and TT-phase of Nb2O5 with a dilute concentration of O
vacancies is between 0.00349-0.0164 $\mu_{B}$ per formula unit Herval  _et
al._ (2015). This is comparable to, but lower than, our amorphous predictions
which is explained by the increased disorder in the amorphous phase.

Figure 3: (a) Spin-polarizeed density of states for crystalline N-Nb2O5 for
dilute electron-doped (N-Nb2O4.98), stoichiometric (N-Nb2O5), and hole-doped
(Nb2O5.02) crystalline, and averaged amorphous Nb2O5 structures. The total DOS
(shaded) is plotted along with the orbital projected states of Nb-d states
(red) and O-p states (cyan). The Fermi level is marked with a dashed line in
all cases and set to 0 eV. (b) The corresponding atomic resolved magnetic
polarizeation $P(E)$ (see text for definition) of N-Nb2O4.98 and N-Nb2O5.02.
(c) and (d) Charge density isosurfaces are shown for midgap O-p and Nb-d
orbitals from amorphous calculations.

We now consider deviations from stoichiometry in amorphous Nb2O5 using both
the rigid band approximation and explicit doping calculations. (Full details
given in the SI Section LABEL:sec:si_dft_details). With a rigid band
approximation in Figure 2(c), we find that any deviation from stoichiometry
induces local moments in a previously non-magnetic configuration, as in the
crystalline case. Explicitly considering O vacancies and interstitials has
similar qualitative behaviour where we find that any O vacancies or two O
interstitials produce magnetic impurities in a previously non-magnetic
configuration. Electron doping (or explicitly including O vacancies) creates
spins localized on Nb atoms, while hole doping (O interstitials) result in
local spins on O atoms. In the explicit doping case this influence is less
pronounced than in the rigid band approximation as structural rearrangements
around the defect suppresses the creation of local spin polarizeation. In
summary, we find that deviations from stoichiometry in both crystalline and
amorphous Nb2O5 that result in the non-zero Nb-d manifold, induce local
moments similar in magnitude.

Local Descriptor | Feature Importance
---|---
Minimum Bond Distance | 28.0%
Maximum Bond Distance | 19.5%
Average Bond Distance | 16.2%
Polyhedron Volume | 13.2%
Std. Dev. of Polyhedron Components | 11.7%
Coordination | 11.4%
Table 1: Feature importance of local descriptors used to build the random
forest classification model.

Finally, we investigate correlations between local structural descriptors
(e.g. bond lengths, coordination number and polyhedral volume) and the
formation of local spin moments using a random forest classifier, with details
given in the SI Section LABEL:sec:si_ml. Confirming our insights in assessing
the coordination number and bond lengths surrounding our set of configurations
with local moments above, we find that the minimum bond distance (a bond
length above a threshold) is the most important descriptor for determining the
presence of a magnetic moment (Table 1). Inspection of individual decision
trees reveals that ions with a minimum bond length above a value set by the
decision tree are categorised as magnetic, which suggests that covalent
bonding suppresses the formation of magnetic moments. In addition to the local
bonding descriptors, we find that polyhedron shape and coordination are
important secondary indicators of magnetism.

###  IV.2 Ab-initio Shiba theory for surface impedance

To connect atomic-scale disorder and defects to resonator performance, we
solve for the superconducting DOS in the presence of magnetic impurities with
parameters and structural information derived from our ab initio calculations.
From this updated superconducting DOS, we calculate the resulting response
function to estimate the impedance changes as a result of our specific profile
of magnetic impurities.

We first develop an ab initio Shiba theory to describe the interaction of
dilute spins in a superconducting host. In our Shiba model, we treat magnetic
impurities as classical spins that are randomly distributed, unpolarizeed, and
of finite concentration in a superconducting host. The impurity problem is
solved within the infinite dimensional limit and for a finite exchange
coupling $J$, where the self energy of the lattice becomes momentum
independent allowing for a full summation of the perturbation series. The
quasi-classical BCS Green’s function for the impurity problem is

$\mathcal{G}(v(\varepsilon))=\frac{v(\varepsilon)^{2}}{\sqrt{1-v(\varepsilon)^{2}}},$
(2)

where $v(\varepsilon)$ is a self-consistent parameter obtained through solving

$\phi(v)=\frac{\varepsilon}{|\Delta|}=v\left(1-\frac{1}{\tau_{s}|\Delta|}\frac{\sqrt{1-v^{2}}}{((1-\gamma)^{2}-v^{)}}\right),$
(3)

and $|\Delta|$ is the superconducting gap order parameter. $\tau_{s}$ is the
scattering time on magnetic impurities, $\varepsilon$ is the energy and
$\gamma\propto(1-J^{2})/(1+J^{2})+1$ is the parameter of the Shiba theory that
quantifies the magnetic exchange coupling. Parametrizing Equation (3) from our
ab initio calculations allows for a self-consistent solution representing the
realistic microscopic energy scales participating in defect-induced
decoherence. Estimates from the non-stoichiometric crystalline calculations
indicate that the ratio of exchange couplings are in the range
$6\leq\gamma^{d}/\gamma^{p}\leq 9$ per u.c (unit-cell), where $\gamma^{p,d}$
is the Shiba exchange coupling constant for p\- or d-channel magnetic
impurities, calculated from $J=E^{p,d}-E_{\text{nm}}^{p,d}$, the energy
difference between the magnetic and non-magnetic configurations (see SI
Section LABEL:sec:si_shiba). In addition, we calculate the ratio between the
scattering time due to magnetic impurities to be $\tau_{s}^{p}\sim
2\tau_{s}^{d}$ per u.c, finding double the number of p-channel magnetic
impurities than d-channel on average. Using these parameter estimates from
first-principles calculations, we consider a strong exchange coupling and low
impurity density, i.e $\gamma=0.9$ per u.c and $\tau_{s}^{-1}=0.05$ per u.c
for the case of d-channel magnetic impurities. For p-channel magnetic
impurities we consider the low exchange coupling but high impurity density
limit, i.e $\gamma=0.1$ per u.c and $\tau_{s}^{-1}=0.1$ per u.c.

The resulting DOS for these limits are illustrated in Figure 4(a). For
d-channel magnetism an impurity band centres near the Fermi level resulting in
the onset of gapless superconductivity where the impurity band is distinct
from the continuum. In contrast to this, in the case of p-channel magnetism
there is no isolated impurity band as it merges with the continuum far from
the Fermi level. Thus, the nature of the magnetic carriers (d-channel or
p-channel) – and hence the local atomic structure – determines whether gapless
superconductivity arises.

Figure 4: (a) The normalised superconducting density of states for a Nb film
with magnetic impurities in the strong (red) [$\gamma=0.9,\tau_{s}^{-1}=0.05$]
and weak (teal) [$\gamma=0.1,\tau_{s}^{-1}=0.1$] coupling limits in the
vicinity of the Fermi level. (b) The real part of the current dissipation
function in the corresponding limits, where the inset shows the region near
the Fermi level which participates in the dissipation. For strong coupling,
the decoherence is associated to d-channel losses, while for weak coupling it
is compatible with p-channel losses.

Next we explore how this distinction of p\- versus d-channel magnetism can be
useful in determining the spin contribution to decoherence in the limit of low
frequency superconducting resonators. An interesting case emerges in the low
frequency limit ($\omega\sim 0.01$ meV and $|\Delta|\sim 1$ meV) when the
current response function can be approximated to be dependent only on its
dissipative contribution Kharitonov  _et al._ (2012). The Fermi distribution
effectively constrains only quasiparticles within $T$ of the Fermi level. If
$\bar{Q}_{2}(\varepsilon,k)\neq 0$ then there must be a gapless
superconductor, i.e $N(\varepsilon=0)\neq 0$. Our analysis confirms this in
the two limits considered (Figure 4(b)). For d-channel magnetic impurities,
quasiparticle excitations near the Fermi level create a gapless state and
cause dissipation through the finite contribution of
$\bar{Q}_{2}(\varepsilon,k)$ at the Fermi level. However, for p-channel
impurities, the opposite is observed where $\bar{Q}_{2}(\varepsilon,k)$
remains zero at the Fermi level, indicating a critical impurity density is
required for p-channel impurities to contribute to the finite dissipation at
the Fermi level. In the low frequency limit the surface resistance $R(\omega)$
is quadratic in frequencyKharitonov  _et al._ (2012), i.e
$R(\omega)\propto\omega^{2}$. Because of this, gapless superconductivity –
regardless of the strength – always results in a finite dissipation, provided
there is a finite DOS at the Fermi level:

$R(\omega)|_{T}=0\Leftrightarrow N(\varepsilon=0)>0.$ (4)

As a result, we can attribute an ohmic loss in superconducting films to the
structural disorder and defects in the the surface oxide formed on these
films. This residual resistance is primarily associated with O deficient
growth, where the primary magnetic carrier is the Nb ion. We note that while
our results indicate that O surplus growth conditions do not significantly
participate in the magnetic-impurity-induced decoherence processes for our Nb-
oxide films, it does not rule out this is general, with details depending on
the precise atomic-scale structure and chemistry of the surface layers.

In Figure LABEL:fig:SI_shiba (SI Section LABEL:sec:si_shiba) we present a set
of plausible impurity densities that are compatible with p-channel decoherence
losses. We notice that for a considerable increase in the amount of off-
stoichiometry at the surface oxide corresponding to an O surplus, then it is
possible to induce p-channel magnetic decoherence losses at an impurity
density $\tau_{s}^{-1}\sim 0.5$ per u.c and a higher exchange coupling of
$\gamma=0.3$. As $\gamma$ is increased, so does the corresponding exchange
energy between the magnetic O ions, as a result of the increased magnetic
impurity density. This cooperative effect results in the onset of a gapless
phase due to the broadening of the continuum into the gap, which eventually
crosses the Fermi level and a concomitant dissipation.

##  V Discussion and Conclusion

Our DFT calculations confirm the presence of paramagnetic moments in NbO and
NbO2, as expected from their unfilled d manifold, and consistent with
experiments. However, for diamagnetic Nb2O5 with its $d^{0}$ configuration, we
find that structural distortions (considered here in the amorphous limit),
strain, and defects can induce and enhance the formation of local spin
moments. In the crystallines phases of Nb2O5 considered here, electron doping
results in localized Nb-d moments, while hole doping induces O-p moments,
consistent with a simple rigid-band model of shifting the chemical potential
into the conduction/valence band respectively. Similar trends with electron
and hole doping are also observed in the amorphous Nb2O5 configurations. In
amorphous phases of Nb2O5, we find that local spin moments form even in the
nomimally $d^{0}$ stoichiometry as a result of the formation of dangling bonds
in the disordered phases. We observe that local spin moment formation is
favored in undercoordinated Nb atoms with miminal covalent bonding, consistent
with Ref. Streltsov and Khomskii (2016) where the authors suggest that local
moments are suppressed in $4d$ transition metal complexes in favor of bonding
orbitals.

We classify the local spin moments in Nb2O5 into two types : d-channel states
localized on Nb-d primarily formed from O vacancies or under-coordinated Nb,
and p-channel states localized on O-p formed from O interstitials. While the
latter hole-doping scenario is difficult to achieve at equilibrium, we expect
chemical and structural inhomogeneities to have local regions with excess
oxygen. Furthermore, our estimates for the magnitude of the magnetic moments
are comparable to other studies of wide bandgap insulating systems that
exhibit defect-induced magnetism Herval  _et al._ (2015); Esquinazi  _et al._
(2013).

Using these atomistic calculations to inform our Shiba theory, we find that
regardless of composition, d-channel magnetic impurities have a more sizeable
influence on the resulting non-TLS loss channels for Nb-resonators. This is
due to the much greater d-channel magnetic moments and exchange interaction
compared to p-channel values. While the d-channel impurity density is on
average about 50% that of p-channel, the exchange coupling is approximately
one order of magnitude greater. Our surface impedance calculations reveal that
the combination of these three factors results in a finite residual resistance
for d-channel magnetic carriers that cannot be reduced by lowering the
temperature, and a dissipationless current for p-channel impurities. Our
results suggest that oxygen-deficient superconducting films, as suggested by
valence band X-Ray Photoemission Spectroscopy in Ref. Harrelson  _et al._
(2021), (with d-channel magnetism) are a significant decoherence channel in
Nb-based superconducting qubits. We note, however, that it is possible that
p-channel carriers also induce a dissipation, however this requires a
significant increase in magnetic impurity density.

As a result, improvement of Nb-based superconducting quantum systems relies on
the mitigation of the decoherence processes caused by the NbOx layer. Complete
removal of the oxide layer is preferable, and may be possible through surface
passivation after complete layer removal, or by changing fabrication steps to
avoid the formation of the oxide layer altogether. Such surface removal and
treatments were recently carried out in Nb thin films, leading to a
significant improvement in quality factor of over 5 millionAltoé  _et al._
(2020) and consistent with this theoretical study. Without surface oxide
removal, our results suggest that annealing oxide layers to increase its
crystallinity and remove oxygen vacancies will reduce the number of magnetic
impurities, which agrees with the work by Proslier et. al Proslier  _et al._
(2011). Our work uncovers the microscopic origins of non-TLS magnetic losses
in native oxides on superconducting resonators and how they influence
performance. We further suggest routes to reducing the most destructive
atomistic decoherence channels through fabrication procedures.

##  VI Acknowledgments

We thank John Clarke, John Vinson, Aritoki Suzuki, Adam Schwartzberg and Tess
Smidt for insightful discussions and manuscript feedback. This work was funded
by the U.S. Department of Energy, Office of Science, Office of Basic Energy
Sciences, Materials Sciences and Engineering Division under Contract No. DE-
AC02-05-CH11231 “High-Coherence Multilayer Superconducting Structures for
Large Scale Qubit Integration and Photonic Transduction program (QIS-LBNL)”.
This research used resources of the National Energy Research Scientific
Computing Center (NERSC), a U.S. Department of Energy Office of Science User
Facility located at Lawrence Berkeley National Laboratory, operated under
Contract No. DE-AC02-05CH11231. E.S. acknowledges support from the US-Irish
Fulbright Commission, the Air Force Office of Scientific Research under award
number FA9550-18-1-0480 and the EPSRC Centre for Doctoral Training in Cross-
Disciplinary Approaches to Non-Equilibrium Systems (EP/L015854/1). This work
also used the Extreme Science and Engineering Discovery Environment (XSEDE),
which is supported by National Science Foundation grant number ACI-1548562.
Work at the Molecular Foundry was supported by the Office of Science, Office
of Basic Energy Sciences, of the U.S. Department of Energy under Contract No.
DE-AC02-05CH11231.

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# Necessary Conditions on Effective Quantum Entanglement Catalysts

Yu-Min Guo<EMAIL_ADDRESS>School of Mathematical Sciences, Capital Normal
University, Beijing 100048, China  Yi Shen  School of Mathematical Sciences,
Beihang University, Beijing 100191, China  Li-Jun Zhao  School of Mathematical
Sciences, Beihang University, Beijing 100191, China  Lin Chen  School of
Mathematical Sciences, Beihang University, Beijing 100191, China
International Research Institute for Multidisciplinary Science, Beihang
University, Beijing 100191, China  Mengyao Hu  School of Mathematical
Sciences, Beihang University, Beijing 100191, China  Zhiwei Wei  School of
Mathematical Sciences, Capital Normal University, Beijing 100048, China  Shao-
Ming Fei<EMAIL_ADDRESS>(corresponding author) School of Mathematical
Sciences, Capital Normal University, Beijing 100048, China  Max-Planck-
Institute for Mathematics in the Sciences, Leipzig 04103, Germany

###### Abstract

Quantum catalytic transformations play important roles in the transformation
of quantum entangled states under local operations and classical
communications (LOCC). The key problems in catalytic transformations are the
existence and the bounds on the catalytic states. We present the necessary
conditions of catalytic states based on a set of points given by the Schmidt
coefficients of the entangled source and target states. The lower bounds on
the dimensions of the catalytic states are also investigated. Moreover, we
give a detailed protocol of quantum mixed state transformation under
entanglement assisted LOCC.

##  I Introduction

Quantum entangled states are important physical resources and play significant
roles in the burgeoning field of quantum information PhysRevLett.122.190502 ;
RevModPhys.91.025001 ; PhysRevA.98.042320 ; PhysRevLett.121.190503 . The
transformation of entangled states is widely used in numerous remarkable
information processing tasks such as quantum secret communication
RevModPhys.81.865 , quantum key distribution BB84 , quantum super-secret
coding PhysRevA.71.012316 , quantum teleportation PhysRevLett.70.1895 ,
quantum computing PhysRevLett.86.5188 ; doi:10.1098/rspa.1985.0070 and the
dynamics in quantum resource theory RevModPhys.81.865 . Nevertheless, a
quantum entangled state can not be transformed into another arbitrary given
state under local operations and classical communications (LOCC) in general.
It is of great significance to study the conditions of such state
transformations PhysRevLett.83.436 .

Let
$|{\psi}\rangle_{AB}=\sum_{i=1}^{d}\sqrt{p_{i}}|{a_{i}}\rangle|{b_{i}}\rangle$
and
$|{\phi}\rangle_{AB}=\sum_{j=1}^{d}\sqrt{q_{j}}|{c_{i}}\rangle|{d_{i}}\rangle$
be two bipartite states in Schmidt form with the Schmidt coefficients $p_{i}$
and $q_{j}$ in the decreasing order, shared between Alice and Bob,
respectively. Denote ${\bf p}$ (${\bf q}$) the Schmidt vectors composed of
$p_{i}$’s ($q_{j}$’s). It is proved that $|{\psi}\rangle_{AB}$ can be
transformed into $|{\phi}\rangle_{AB}$ under LOCC if and only if ${\bf p}$ is
majorized by ${\bf q}$, namely, $|{\psi}\rangle\rightarrow|{\phi}\rangle$ if
and only if ${\bf p}\prec{\bf q}$, where $\prec$ denotes the majorization
relation between ${\bf p}$ and ${\bf q}$ Marshall1979Inequalities ;
Uhlmann1982book ,

$\displaystyle\sum_{i=1}^{l}p_{i}\leq\sum_{i=1}^{l}q_{i},\quad\forall\,l\in\\{1,2,\cdots,d\\},$

with equality for $l=d$. Two bipartite states $|{\psi}\rangle$ and
$|{\phi}\rangle$ are said to be incomparable if
$|{\psi}\rangle\nrightarrow|{\phi}\rangle$ and
$|{\phi}\rangle\nrightarrow|{\psi}\rangle$ under LOCC.

Different entangled states are not necessarily convertible according to
quantum resource theory. Without additional resources, it is usually
impossible to transform one entangled state into another one under LOCC.
Jonathan and Plenio PhysRevLett.83.3566 showed that at the presence of another
state called a catalyst, it is possible to transform two incomparable states
from one to another under LOCC without changing the catalyst. For incomparable
$|{\psi}\rangle$ and $|{\phi}\rangle$, there could exist a bipartite state
$|{\chi}\rangle=\sum_{x=1}^{k}\sqrt{r_{x}}|{x_{A^{\prime}}}\rangle|{x_{B^{\prime}}}\rangle\in{\cal
H}^{A^{\prime}B^{\prime}}$ in auxiliary systems $A^{\prime}$ and $B^{\prime}$
such that
$|{\psi}\rangle\otimes|{\chi}\rangle\rightarrow|{\phi}\rangle\otimes|{\chi}\rangle$,
namely, $\quad\bf{p}\otimes\bf{r}\prec\bf{q}\otimes\bf{r}$, where ${\bf r}$ is
the Schmidt vector of the state $|{\chi}\rangle$, and the state
$|{\chi}\rangle$ in this transformation process is called the catalyst state.
The dimension of vector ${\bf r}$ (the Schmidt rank of $|{\chi}\rangle$) is
also called the dimension of the catalytic state. Such state transformation
assisted by an ancillary state (catalyst) is called entanglement assisted LOCC
(ELOCC) transformations PhysRevLett.83.3566 .

Although the existence of catalyst states for any given transformation has
been proven by using the Renyi entropies and power means Turgut_2007 , the
properties of the catalysts for a given transformation are still less known.
In fact, it is difficult to find a suitable catalyst for any given
transformations. As mentioned in PhysRevLett.83.3566 and Sun2005The , it seems
very hard in general to solve this problem analytically, due to lack of
suitable mathematical tools to deal with the majorization of tensor product
vectors, especially, the identification of Schmidt descending vector of the
composite state. In PhysRevA.65.042306 Bandyopadhyay and Roychowdhury
presented a useful algorithm to decide the existence of catalysts.
Nevertheless, to determine whether there exists a $k\times k$ catalyst for two
$n\times n$ incomparable states, the algorithm runs in exponential time with
complexity $O\left([(nk)!]^{2}\right)$, which shows also the difficulty of
solving this problem even numerically. The lower bound on the dimensions of a
possible catalyst for a given pure entangled state transformation has been
proposed in PhysRevA.79.054302 . The upper bound on the dimensions of the
required catalyst is nevertheless unbounded in general PhysRevA.64.042314 .

Catalysis is also useful to increase the maximal transformation probability in
probabilistic entanglement transformation. A sufficient and necessary
condition on catalysts has been derived for certain probabilistic
transformations PhysRevA.69.062310 . It has also been shown that any entangled
target state can be embezzled (up to a small amount of error) by using only a
family of bipartite catalysts PhysRevA.67.060302 . However, embezzling with
small error requires that the catalyst Schmidt number diverges to infinity.
For some mixed entangled states, there also exist some mixed catalyst states
which can improve the transformation of mixed entangled states prl.85.437 .

In this paper we focus on finite-dimensional ELOCC transformations in which
embezzling is not possible with a high degree of accuracy. Based on the
Schmidt coefficients of the initial, target and catalyst states, we present
two important bounds on the entanglement of a potential catalyst state for any
pure state transformations. We analyze the ELOCC conversion between $4\times
4$ entangled quantum states catalyzed by $2\times 2$ catalyst, and give both
necessary and sufficient conditions of catalytic states. Detailed examples are
given to show that our criteria work is better than the existing ones
PhysRevA.99.052348 , .

##  II Transformation of entangled states with different Schmidt ranks

Let
$|{\psi}\rangle_{AB}=\sum_{i=1}^{n}\sqrt{p_{i}}|{a_{i}}\rangle|{b_{i}}\rangle$
and
$|{\phi}\rangle_{AB}=\sum_{j=1}^{t}\sqrt{q_{j}}|{c_{i}}\rangle|{d_{i}}\rangle$
be two bipartite states in Schmidt form with Schmidt ranks n and t, with the
Schmidt coefficients $p_{i}$ and $q_{j}$ in decreasing order, respectively. In
the following, we always add zeros to a vector with lower dimension so that it
has the same dimension as the one with higher dimension. Clearly, if $n<t$,
$|{\psi}\rangle_{AB}$ can not be transformed into $|{\phi}\rangle_{AB}$ by
LOCC. When $n>t$ and $p_{1}<\frac{1}{t}$, ${\bf p}$ is majorized by
$(\frac{1}{t},\frac{1}{t},\dots,\frac{1}{t},0,\dots,0)$ and the
$t$-dimensional vector $(\frac{1}{t},\frac{1}{t},\dots,\frac{1}{t})$ is
majorized by any $t$-dimensional vector. Hence, there always exist an LOCC
operation to accomplish the transformation
$|{\psi}\rangle\rightarrow|{\phi}\rangle$ in this case. Therefore, we have the
following Lemmas.

###### Lemma 1

$|{\psi}\rangle_{AB}$ can be transformed into $|{\phi}\rangle_{AB}$ under LOCC
if and only if $p_{1}<\frac{1}{t}$ for $t<n$, where $n$ is the Schmidt rank of
$|{\psi}\rangle_{AB}$.

###### Lemma 2

For any $1<t<n$ and $1\leq s<n$, if

$\frac{s(t-1)}{t(n-1)}<\sum_{i=n-s+1}^{n}p_{i}<\frac{s}{n},$ (1)

$|{\psi}\rangle_{AB}$ can be transformed into $|{\phi}\rangle_{AB}$ under
LOCC.

###### Proof.

Since $\frac{s(t-1)}{t(n-1)}<\sum_{i=n-s+1}^{n}p_{i}$, we have
$\sum_{i=1}^{n-s}p_{i}<1-\frac{s(t-1)}{t(n-1)}$ and
$p_{n-s+1}>\frac{t-1}{t(n-1)}$, which implies

$p_{1}\geqslant p_{2}\geqslant\dots\geqslant p_{n-s}\geqslant
p_{n-s+1}>\frac{t-1}{t(n-1)}.$

According to that

$p_{1}=1-\sum_{i=2}^{n}p_{i}<1-[(n-s-1)\frac{t-1}{t(n-1)}+\frac{s(t-1)}{t(n-1)}]=\frac{1}{t},$

One completes the proof of Lemma by Lemma 1.  $\sqcap$$\sqcup$

Lemma 2 gives the condition on the smallest $s$ Schmidt coefficients of
$|{\psi}\rangle_{AB}$ under LOCC state transformation. In particular, when
$s=1$ we have

$\frac{t-1}{t(n-1)}<p_{n}<\frac{1}{n}.$

The above results show that, interestingly, when one only knows part of the
information about the a bipartite entangled quantum state, namely, a set of
the smallest Schmidt coefficients, one can determine what entanglement
transformations are allowed under LOCC. As important channels in information
processing such as quantum teleportation, the capacities of quantum entangled
states are related to their dimensions. Our results show that the dimension of
the target states would be reduced under LOCC transformation, which results in
a reduction of the quantum channel’s capacity. In the following, we consider
entangled state transformation with the same dimensions and with the help of
catalysts.

##  III Bounds on entanglement catalyst states

Let r be the Schmidt vector of a catalyst state. For two incomparable Schmidt
vector $\textbf{p},\,\textbf{q}\in\mathbb{R}^{d}$, if
$\textbf{p}\otimes\textbf{r}\prec\textbf{q}\otimes\textbf{r}$, the following
inequalities hold,

$p_{1}\leqslant q_{1},~{}~{}~{}~{}p_{d}\geqslant
q_{d},~{}~{}~{}~{}\sum_{i=1}^{d-1}p_{i}\leqslant\sum_{i=1}^{d-1}q_{i}.$ (2)

We define the set $\mathcal{L}$ to be all the values of $l$ such that
$\sum_{i=1}^{l}p_{i}>\sum_{i=1}^{l}q_{i}$,

$\displaystyle\mathcal{L}=\\{l\in\\{1,2\cdots
d\\}|\sum_{i=1}^{l}(p_{i}-q_{i})>0\\}.$ (3)

Denote $m_{{\cal L}}\equiv\mathop{\rm min}\\{l\big{|}l\in{\cal L}\\}$ and
$M_{{\cal L}}\equiv\max\\{l\big{|}l\in{\cal L}\\}$. From (2) we have that
$\\{1,(d-1),\,d\\}\notin\mathcal{L}$. We call two incomparable vectors ${\bf
p}$ and ${\bf q}$ solvable incomparable vectors if ${\bf p}$ and ${\bf q}$
satisfy the conditions that $1,(d-1),d\notin\mathcal{L}$. The main problem is
to find the catalyst states for two solvable incomparable states, see Fig. 1.
For a fixed dimension of catalytic system, there exist two solvable
incomparable vectors that cannot be catalyzed. For instance, ${\bf
p}=(0.414047778,0.31764445,0.18499118,0.083316592)$ and ${\bf
q}=(0.428610282,0.289194489,0.212421079,0.06977415)$ are solvable incomparable
vectors. But there do not exist two dimensional catalyst vectors for ${\bf p}$
and ${\bf q}$ Sun2005The .

Figure 1: Two solvable and non-comparable pure states $|{\psi}\rangle$ and
$|{\phi}\rangle$ could be transformed under LOCC by using auxiliary catalytic
states.

In the following, we derive first time some general conditions on catalyst-
state transformation under LOCC.

###### Theorem 1

For arbitrary incomparable Schmidt vectors
$\mathbf{p},\mathbf{q}\in\mathbb{R}^{d}$, if there exists a catalyst vector
$\mathbf{r}\in\mathbb{R}^{k}$, such that
$\mathbf{p}\otimes\mathbf{r}\prec\mathbf{q}\otimes\mathbf{r}$, then
$\mathbf{r}$ must satisfy the following inequality,

$\displaystyle\frac{r_{1}}{r_{k}}>\max_{l\in\mathcal{L}}(\mathop{\rm
min}\\{\frac{p_{l}}{p_{l+1}},\frac{q_{l}}{q_{l+1}}\\}).$ (4)

###### Proof.

We prove it by contradiction. Suppose

$\displaystyle\frac{r_{1}}{r_{k}}\leqslant\mathop{\rm
min}\\{\frac{p_{l}}{p_{l+1}},\frac{q_{l}}{q_{l+1}}\\}.$

Then $p_{l+1}r_{1}\leqslant p_{l}r_{k}$ and $q_{l+1}r_{1}\leqslant
q_{l}r_{k}$, the first $lk$ largest elements of $\mathbf{p}\otimes\mathbf{r}$
and $\mathbf{q}\otimes\mathbf{r}$ are given by $\\{p_{i}r_{j}\\}$ and
$\\{q_{i}r_{j}\\}$ with $1<i<l$ and $1<j<k$, respectively. Consequently one
has the following relation:

$\displaystyle\begin{aligned}
\sum_{i=1}^{kl}(\mathbf{p}\otimes\mathbf{r})_{i}^{\downarrow}&=\left(\sum_{j=1}^{l}p_{j}\right)\left(\sum_{i=1}^{k}r_{i}\right)=\sum_{j=1}^{l}p_{j}\\\
&>\sum_{j=1}^{l}q_{j}=\left(\sum_{j=1}^{l}q_{j}\right)\left(\sum_{i=1}^{k}r_{i}\right)\\\
&=\sum_{i=1}^{kl}(\mathbf{q}\otimes\mathbf{r})_{i}^{\downarrow},\end{aligned}$

where $(\mathbf{x})_{i}^{\downarrow}$ means that the components of the vector
$\mathbf{x}$ are arranged in decreasing order, which contradicts the condition
$\mathbf{p}\otimes\mathbf{r}\prec\mathbf{q}\otimes\mathbf{r}$.
$\sqcap$$\sqcup$

The inequality (4) has another equivalent expression,

$\displaystyle\frac{r_{k}}{r_{1}}<\mathop{\rm
min}_{l\in\mathcal{L}}(\max\\{\frac{p_{l+1}}{p_{l}},\frac{q_{l+1}}{q_{l}}\\}).$

Moreover, due to the definitions of $m_{{\cal L}}$ and $M_{{\cal L}}$, we have

$\displaystyle\begin{aligned} \frac{r_{1}}{r_{k}}&>\mathop{\rm
min}\\{\frac{p_{m_{{\cal L}}}}{p_{m_{{\cal L}}+1}},\frac{q_{m_{{\cal
L}}}}{q_{m_{{\cal L}}+1}}\\},\\\\[2.84526pt] \frac{r_{1}}{r_{k}}&>\mathop{\rm
min}\\{\frac{p_{M_{{\cal L}}}}{p_{M_{{\cal L}}+1}},\frac{q_{M_{{\cal
L}}}}{q_{M_{{\cal L}}+1}}\\}.\end{aligned}$

Let $l_{1},l_{2},\dots,l_{m}$; $l_{i}<l_{i+1}$ be the elements of the set
$\mathcal{L}$ defined in (3), where $m=\\#\mathcal{L}$ is the number of the
elements in $\mathcal{L}$.

###### Theorem 2

Let $\mathbf{p},\mathbf{q}\in\mathbb{R}^{d}$ be arbitrary two solvable
incomparable Schmidt vectors. If
$\mathbf{p}\otimes\mathbf{r}\prec\mathbf{q}\otimes\mathbf{r}$, we have

$\mathop{\rm
min}_{\tau=1,\dots,m+1}\\{r_{j_{\tau}}q_{l_{\tau}}\\}<\max_{\tau=1,\dots,m+1}\\{r_{j_{\tau}+1}q_{l_{\tau-1}+1}\\},$

where $l_{0}=0$, $l_{m+1}=d$, and all the terms with $r_{0}$ or $r_{k+1}$ are
neglected. The indices $\\{j_{s}\\}_{s=1}^{m+1}$ satisfy $0\leqslant
j_{i}\leqslant k$ for $i=1,2,\dots,m+1$, $j_{i+1}\leqslant j_{i}$ for
$i=1,2,\dots,m$, and $j_{i+1}<j_{i}$ for one $i=1,2,\dots,m+1$ at least.

###### Proof.

We prove it by contradiction. Denote
$\Xi=\sum_{i=1}^{m}(j_{i}-j_{i+1})l_{i}+j_{m+1}d$. If

$\mathop{\rm
min}_{\tau=1,\dots,m+1}\\{r_{j_{\tau}}q_{l_{\tau}}\\}\geqslant\max_{\tau=1,\dots,m+1}\\{r_{j_{\tau}+1}q_{l_{\tau-1}+1}\\},$

then the $\Xi$ largest elements of $\mathbf{q}\otimes\mathbf{r}$ are
$r_{1}q_{1}$, $r_{1}q_{2}$, $\dots$, $r_{1}q_{d}$, $r_{2}q_{1}$, $r_{2}q_{2}$,
$\dots$, $r_{2}q_{d}$, $\dots$, $r_{j_{m+1}}q_{d}$, $r_{j_{m+1}+1}q_{1}$,
$r_{j_{m+1}+1}q_{2}$, $\dots$, $r_{j_{m+1}+1}q_{l_{m}}$, $r_{j_{m+1}+2}q_{1}$,
$\dots$, $r_{j_{m+1}+2}q_{l_{m}}$,$\dots$, $r_{j_{m}}q_{l_{m}}$,
$r_{j_{m}+1}q_{l_{1}}$, $\dots$, $r_{j_{m}+1}q_{l_{m_{1}}}$, $\dots$,
$r_{j_{1}}q_{1}$, $\dots$, $r_{j_{1}}q_{l_{1}}$. By the definition of
$\mathcal{L}$, one has for any $\tau$,
$\sum_{i=1}^{l_{\tau}}p_{i}>\sum_{i=1}^{l_{\tau}}q_{i}$.

Since

$\displaystyle\sum_{s=1}^{\Xi}(\mathbf{q}\otimes\mathbf{r})_{s}^{\downarrow}$
$\displaystyle=\sum_{\tau=1}^{m+1}(\sum_{u=j_{\tau+1}+1}^{j_{\tau}}r_{u})\cdot(\sum_{v=1}^{l_{\tau}}q_{v})$
$\displaystyle<\sum_{\tau=1}^{m+1}(\sum_{u=j_{\tau+1}+1}^{j_{\tau}}r_{u})\cdot(\sum_{v=1}^{l_{\tau}}p_{v})$
$\displaystyle\leqslant\sum_{s=1}^{\Xi}(\mathbf{p}\otimes\mathbf{r})_{s}^{\downarrow},$

where $j_{m+2}=0$, the sum of the largest $\Xi$ elements of
$\mathbf{q}\otimes\mathbf{r}$ is strictly less than the sum of the largest
$\Xi$ elements of $\mathbf{p}\otimes\mathbf{r}$.  $\sqcap$$\sqcup$

Remark 1 Theorem 2 has another useful equivalent statement: Let $\mathbf{p}$
and $\mathbf{q}\in\mathbb{R}^{d}$ be two fixed solvable incomparable Schmidt
vectors, and $\mathbf{r}\in\mathbb{R}^{k}$ the Schmidt vector of a catalyst
state. Denote $\\{j_{s}\\}_{s=1}^{m+1}$ a sequence of natural numbers
satisfying $0<j_{m^{\prime}+1}<j_{m^{\prime}}<\dots<j_{1}<k$, where
$m^{\prime}\leqslant m=\\#\mathcal{L}$, $m$ is the number of elements in
$\mathcal{L}$. Consider a subset $\bar{\mathcal{L}}$ of $\mathcal{L}$ with
$m^{\prime}$ elements,
$\bar{\mathcal{L}}=\\{l_{1},l_{2},\dots,l_{m^{\prime}}|l_{i}\in\mathcal{L},~{}~{}l_{i}<l_{i+1}\\}$.
Set $l_{0}=0$ and $l_{m+1}=d$. If
$\mathbf{p}\otimes\mathbf{r}\prec\mathbf{q}\otimes\mathbf{r}$, then

$\mathop{\rm
min}_{\tau=1,\dots,m^{\prime},m^{\prime}+1}\\{r_{j_{\tau}}q_{l_{\tau}}\\}<\max_{\tau=1,\dots,m^{\prime},m^{\prime}+1}\\{r_{j_{\tau}+1}q_{l_{\tau-1}+1}\\}.$

To elucidate their practical applications in actual catalytic processes, we
enumerate several special corollaries of the Theorem 2 below.

We consider the case that $\\{j_{i}\\}_{i=1}^{m+1}$ take two different values,
say, $0\leqslant c_{1}=j_{m+1}=\dots=j_{s}<j_{s-1}=\dots=j_{1}=c_{2}\leqslant
k$. In this case, Theorem 2 gives rise to the following simple form:

###### Corollary 1

If $\mathbf{p}\otimes\mathbf{r}\prec\mathbf{q}\otimes\mathbf{r}$, then

$\mathop{\rm
min}\\{r_{c_{1}}q_{d},r_{c_{2}}q_{l_{s}}\\}<\max\\{r_{c_{1}+1}q_{l_{s}+1},r_{c_{2}+1}q_{1}\\},$

where $l_{0}=0$, $l_{m+1}=d$, $1\leq c_{1}<c_{2}\leq k$, and the terms with
$r_{k+1}$ are ignored.

Corollary 1 still gives rise to a series of conditions one catalyst. In
particular, for $c_{2}=c_{1}+1=s$, we have the following conclusions, see
proof in Appendix A:

###### Proposition 1

If $\mathbf{p}\otimes\mathbf{r}\prec\mathbf{q}\otimes\mathbf{r}$, we have

$\frac{q_{1}}{q_{d}}>\frac{r_{s-1}}{r_{s+1}},~{}~{}or~{}~{}\frac{q_{1}}{q_{l}}>\frac{r_{s}}{r_{s+1}}~{}~{}or~{}~{}\frac{q_{l+1}}{q_{d}}>\frac{r_{s-1}}{r_{s}}$
(5)

for any $l\in\mathcal{L}$, where $s=2,3,\cdots,k-1$. If $s=1$ or $s=k$, then

$\frac{q_{d}}{q_{l+1}}<\frac{r_{k}}{r_{k-1}},~{}~{}~{}\frac{q_{1}}{q_{l}}>\frac{r_{1}}{r_{2}}.$
(6)

The above Proposition gives detailed criteria on catalytic vectors. In
particular, it is often difficult to determine the values of all components of
the catalytic vectors for high dimensional case. The inequality (6) says that
one may first simply verify the first two and the last two components of the
catalytic vectors, which could greatly simplify the construction of qualified
catalytic states.

Recently, by using a very nice idea the authors in PhysRevA.99.052348 derived
elegant results on the minimum and maximum bounds about the auxiliary
catalysts under ELOCC. It is showed that (Theorem 1 in PhysRevA.99.052348 ) if
$\mathbf{p}\otimes\mathbf{r}\prec\mathbf{q}\otimes\mathbf{r}$, $\mathbf{r}$
satisfies

$\begin{array}[]{rcl}\displaystyle\max_{v\in(1,2,\ldots,k-1)}(\frac{r_{v}}{r_{v+1}})&<&\displaystyle\mathop{\rm
min}(\frac{q_{1}}{q_{m_{{\cal L}}}},\frac{q_{M_{{\cal
L}}+1}}{q_{d}}),\\\\[11.38109pt]
\displaystyle\frac{r_{1}}{r_{k}}&>&\displaystyle\max_{l\in\mathcal{L}}(\frac{q_{l}}{q_{l+1}}).\end{array}$
(7)

As the Proposition 1 takes into account additionally two adjacent components
of the vector $\mathbf{r}$, it gives rise to finer constraints on the catalyst
vector. The proposition 1 is generally better than (7) since, instead of that
all the inequalities are simultaneously true, (5) only requires that at least
one of the inequalities is true, see also the remark in Appendix B.

Furthermore, for a special case of corollary 1 we have, see proof in Appendix
C,

###### Corollary 2

Let $c_{1}=k-1$, $c_{2}=k$ and $s=m$, we have

$\displaystyle\frac{q_{d}}{q_{M_{{\cal L}}+1}}$
$\displaystyle<\frac{r_{k}}{r_{k-1}}$ (8)

if $\mathbf{p}\otimes\mathbf{r}\prec\mathbf{q}\otimes\mathbf{r}$.

Remark 2 If we take $c_{1}=0$, $c_{2}=1$ and $s=1$, then we get another
special case of corollary 1,

$\displaystyle\frac{q_{1}}{q_{m_{{\cal L}}}}$
$\displaystyle>\frac{r_{1}}{r_{2}}.$ (9)

The inequality (8) can be also written in a symmetric form:

$\displaystyle\frac{r_{k}}{r_{k-1}}>\mathop{\rm
min}\\{\frac{p_{d}}{p_{M_{{\cal L}}+1}},\frac{q_{d}}{q_{M_{{\cal L}}+1}}\\}.$

Corollary 2 can be used, for example, to test whether a 2-dimensional
catalytic state can be used to catalyze a given pair of solvable incomparably
entangled states. We can also the corollary 1 to verify the availability of
higher dimensional catalytic states. For the case of 3-dimensional catalytic
states, we can directly check whether each component of the Schmidt vector
satisfies the condition given in corollary 1. For higher dimensional case, we
can take any 3 components of the Schmidt vectors to test whether they violate
the above corollary.

The following example illustrates that our result (8) behaviors better than
the result (9) from PhysRevA.99.052348 in determining whether a catalytic
state can complete a certain catalytic process.

###### Example 1

Consider two bipartite incomparable pure states with the corresponding Schmidt
vectors $\mathbf{p}=(0.4,0.35,0.15,0.1)$ and $\mathbf{q}=(0.5,0.2,0.2,0.1)$,
respectively. Obviously , these two states can not transformed into each other
under LOCC. Now consider a two-dimensional catalyst state with Schmidt vector
$\mathbf{r}=(0.7,0.3)$. (8) says that if $r_{1}/r_{2}\geqslant q_{3}/q_{4}$,
then $\mathbf{p}\otimes\mathbf{r}\nprec\mathbf{q}\otimes\mathbf{r}$. For this
example, one has $q_{1}/q_{2}=2.5>r_{1}/r_{2}=2.\dot{3}>q_{3}/q_{4}=2$, where
$\dot{3}$ stands for repeated decimal. Simple calculation gives rise to
$\mathbf{p}\otimes\mathbf{r}\nprec\mathbf{q}\otimes\mathbf{r}$.

##  IV bounds on the dimension of catalyst

According to the resource theory, the dimensions of catalytic states
introduced on auxiliary systems should be as small as possible. For fixed
incomparable Schmidt vectors $\mathbf{p}$ and $\mathbf{q}\in\mathbb{R}^{d}$,
it is of significance to find the minimum dimension of their catalyst states.
It has been proved in PhysRevA.64.042314 that all non-uniform vectors are
potentially useful as catalyst states. Y. R. Sanders and G. Gour give a lower
bound on the dimensions of catalyst states based on the $k$-th concurrence
$C_{d-1}$ PhysRevA.79.054302 , which works for the case that
$C_{d-1}(|{\psi}\rangle)<C_{d-1}(|{\phi}\rangle)$. However, generally for
large Schmidt dimensions of $|{\psi}\rangle$ and $|{\phi}\rangle$, the $k$-th
concurrences are often difficult to calculate.

Next, we analyze the relationship between the dimensions of the auxiliary
systems and the set $\mathcal{L}$. As the minimum dimension of the catalytic
state is 2, we can derive the following corollary by Theorem 2.

###### Corollary 3

Let ${\bf p}$ and ${\bf q}$ be two solvable incomparable Schmidt vectors with
dimension $n$, associated with the states $|{\psi}\rangle$ and
$|{\varphi}\rangle$, respectively. Denote ${\cal
L}=\\{l_{1},l_{2},\dots,l_{m}\\}$. Let $0<r<0.5$ be a real number. If
$q_{l_{i}+1}r<q_{m}(1-r)$ for any integer $1\leqslant i\leqslant m$, and
$q_{l_{i}+1}r<q_{l_{j}+1}(1-r)$, $q_{l_{j}+1}(1-r)<q_{l_{i}}r$ for any
integers $i,j:1\leqslant i<j\leqslant m$, then the two-dimensional vector
${\bf r}=(1-r,r)$ cannot make ${\bf p}\otimes{\bf r}\prec{\bf q}\otimes{\bf
r}$ true. Namely, the catalytic state $|{\chi}\rangle$ with Schmidt vector
${\bf r}=(1-r,r)$ cannot catalyze the states $|{\psi}\rangle$ and
$|{\varphi}\rangle$ under LOCC.

In other words, the dimension of catalytic state (if it exists) for such
$|{\psi}\rangle$ and $|{\varphi}\rangle$, as mentioned in the above corollary,
must be strictly greater than two. For arbitrary dimensions of the catalytic
states, we have

###### Corollary 4

For any pair of solvable incomparable Schmidt vectors ${\bf p}$ and ${\bf q}$
with dimension $n$, associated with the states $|{\psi}\rangle$ and
$|{\varphi}\rangle$, respectively, if any $k$-dimensional probability vector
${\bf r}=(r_{1},r_{2},\dots,r_{k})$ does not satisfy the conditions in Theorem
2, $|{\psi}\rangle$ and $|{\varphi}\rangle$ cannot be catalyzed by a state
with $k$-dimensional Schmidt vector.

The above corollary helps to reduce the difficulty of finding the minimum
dimensions for catalytic system. In addition, we can utilize the Proposition 1
to give a lower bound on the dimension of catalyst states for arbitrary
solvable incomparable Schmidt vectors ${\bf p}$ and ${\bf q}$. Denote

$\displaystyle a$
$\displaystyle\equiv\max(\frac{q_{1}}{q_{m_{\mathcal{L}}}},\frac{q_{M_{\mathcal{L}}+1}}{q_{d}}),$
$\displaystyle b$
$\displaystyle\equiv\max_{l\in\mathcal{L}}(\frac{q_{l}}{q_{l+1}})\geqslant 1,$
$\displaystyle c$ $\displaystyle\equiv\max_{l\in\mathcal{L}}(\mathop{\rm
min}\\{\frac{p_{l}}{p_{l+1}},\frac{q_{l}}{q_{l+1}}\\}).$

###### Theorem 3

For any two solvable incomparable Schmidt vectors
$\mathbf{p},\mathbf{q}\in\mathbb{R}^{d}$ and for any Schmidt vector
$\mathbf{r}\in\mathbb{R}^{k}$, if
$\mathbf{p}\otimes\mathbf{r}\prec\mathbf{q}\otimes\mathbf{r}$, then the
dimension $k$ of $\mathbf{r}$ must satisfy

$k>\frac{\ln(c)}{\ln(a\sqrt{b})}+1.$ (10)

###### Proof.

For arbitrary $r_{s}$, $s=2,3,\dots,k-1$, from Proposition 1 at least one of
the inequalities in (5) holds. If
$\frac{q_{1}}{q_{d}}>\frac{r_{s-1}}{r_{s+1}}$ holds for some $s$, rewriting
$\frac{q_{1}}{q_{d}}=\frac{q_{1}}{q_{l}}\frac{q_{l+1}}{q_{d}}\frac{q_{l}}{q_{l+1}}$,
we have $\frac{r_{s-1}}{r_{s+1}}<a^{2}b$. Otherwise,

$\displaystyle\frac{r_{s}}{r_{s+1}}$ $\displaystyle<\max\\{\mathop{\rm
min}_{l\in\mathcal{L}}(\frac{q_{1}}{q_{l}}),\mathop{\rm
min}_{l\in\mathcal{L}}(\frac{q_{l+1}}{q_{d}})\\}$
$\displaystyle=\max(\frac{q_{1}}{q_{m_{\mathcal{L}}}},\frac{q_{M_{\mathcal{L}}+1}}{q_{d}})\leqslant
a\sqrt{b}.$

Taking into account (6), we get

$\frac{r_{s}}{r_{s+1}}<a\sqrt{b},~{}~{}\forall s=1,2,\dots,k-1.$ (11)

Therefore,
$\frac{r_{1}}{r_{k}}=\prod_{v=1}^{k-1}\frac{r_{v}}{r_{v+1}}<(a\sqrt{b})^{k-1}$.
On the other hand, we have from Theorem 1 that $\frac{r_{1}}{r_{k}}>c$. Hence,

$c<\frac{r_{1}}{r_{k}}<(a\sqrt{b})^{k-1}.$

The above relations give rise to (10).  $\sqcap$$\sqcup$

The bound (10) depends only on the parameters $a$, $b$ and $c$, which are
exclusively determined by the Schmidt vectors $\mathbf{p}$ and $\mathbf{q}$
and can be easily calculated.

##  V maximum probability of transformation and majorization distance

For any two incomparable states $|{\psi}\rangle$ and $|{\phi}\rangle$, the
maximum probability of transforming $|{\psi}\rangle$ to $|{\phi}\rangle$ under
local operations is given by PhysRevLett.83.1046

$P_{\max}^{\prime}(|{\psi}\rangle\rightarrow|{\phi}\rangle)=\mathop{\rm
min}_{l\in\\{1,2,\cdots,d\\}}\frac{E_{l}(|{\psi}\rangle)}{E_{l}(|{\phi}\rangle)},$

where $E_{l}(|{\psi}\rangle)=\sum_{i=l}^{d}p_{i}$.

To characterize the transformation of two incomparable states $|{\psi}\rangle$
and $|{\phi}\rangle$ assisted with a catalytic state $\chi$ of Schmidt vector
${\bf r}=(r_{1},r_{2},\cdots,r_{k})$, we consider the modified maximum
probability
$P_{\max}(|\psi\rangle\otimes|\chi\rangle\rightarrow|\phi\rangle\otimes|\chi\rangle)$
PhysRevLett.83.1046 ,

$\displaystyle\begin{aligned}
&P_{\max}(|\psi\rangle\otimes|\chi\rangle\rightarrow|\phi\rangle\otimes|\chi\rangle)\\\
&=\mathop{\rm
min}_{l^{\prime}\in\\{1,2,\cdots,kd\\}}\frac{E_{l^{\prime}}(|\psi\rangle\otimes|\chi\rangle)}{E_{l^{\prime}}(|\phi\rangle\otimes|\chi\rangle)}\end{aligned}$

where
$E_{l^{\prime}}(|\psi\rangle\otimes|\chi\rangle)=1-\sum_{i=1}^{l^{\prime}-1}(\mathbf{p}\otimes\mathbf{r})_{i}^{\downarrow}$.

###### Proposition 2

Let
$\delta(|\psi\rangle\otimes|\chi\rangle\rightarrow|\phi\rangle\otimes|\chi\rangle)$
be the modified majorization distance Horodecki_2018 between the product
states $|\psi\rangle\otimes|\chi\rangle$ and
$|\phi\rangle\otimes|\chi\rangle$,

$\displaystyle\begin{aligned}
&\delta(|\psi\rangle\otimes|\chi\rangle\rightarrow|\phi\rangle\otimes|\chi\rangle)\\\
&=2\max_{l^{\prime}\in\\{1,\cdots,kd\\}}\sum_{i=1}^{l^{\prime}}\left((\mathbf{p}\otimes\mathbf{r})_{i}^{\downarrow}-(\mathbf{q}\otimes\mathbf{r})_{i}^{\downarrow}\right).\end{aligned}$

We have $P_{\max}=1$ if and only if $\delta=0$.

###### Proof.

We have

$\displaystyle\begin{aligned}
\delta=0&\Leftrightarrow\max_{l^{\prime}\in\\{1,\cdots,kd\\}}\sum_{i=1}^{l^{\prime}}\left((\mathbf{p}\otimes\mathbf{r})_{i}^{\downarrow}-(\mathbf{q}\otimes\mathbf{r})_{i}^{\downarrow}\right)=0\\\
&\Leftrightarrow\forall
l^{\prime},\sum_{i=1}^{l^{\prime}}(\mathbf{p}\otimes\mathbf{r})_{i}^{\downarrow}\leqslant\sum_{i=1}^{l^{\prime}}(\mathbf{q}\otimes\mathbf{r})_{i}^{\downarrow}\\\
&\Leftrightarrow\mathbf{p}\otimes\mathbf{r}\prec\mathbf{q}\otimes\mathbf{r}.\end{aligned}$

Then we have

$\displaystyle\begin{aligned} P_{\max}=1&\Leftrightarrow\forall
l^{\prime},~{}E_{l^{\prime}}(|\psi\rangle\otimes|\chi\rangle)\geqslant
E_{l^{\prime}}(|\phi\rangle\otimes|\chi\rangle)\\\
&\Leftrightarrow\sum_{i=1}^{l^{\prime}-1}(\mathbf{p}\otimes\mathbf{r})_{i}^{\downarrow}\leqslant\sum_{i=1}^{l^{\prime}-1}(\mathbf{q}\otimes\mathbf{r})_{i}^{\downarrow}\\\
&\Leftrightarrow\mathbf{p}\otimes\mathbf{r}\prec\mathbf{q}\otimes\mathbf{r}.\end{aligned}$

To sum up, we have
$\delta=0\Leftrightarrow\mu(|\psi\rangle\otimes|\chi\rangle)\prec\mu(|\phi\rangle\otimes|\chi\rangle)\Leftrightarrow
P_{\max}=1$.  $\sqcap$$\sqcup$

##  VI Protocols for mixed state transformation assisted by catalysts

We consider the LOCC transformations for mixed states with catalysts. Let
$\rho_{A}=\sum_{i}p_{i}|{i}\rangle\langle{i}|$,
$\sigma_{A}=\sum_{j}q_{j}|{j}\rangle\langle{j}|$ and $\alpha_{A^{\prime}}$ be
the reduced density matrices of the states $|\psi^{AB}\rangle$,
$|\phi^{AB}\rangle$ and $|{\chi}\rangle_{A^{\prime}B^{\prime}}$, respectively.
By matrix majorization relations, we have
$\rho_{A}\otimes\alpha_{A^{\prime}}\rightarrow\sigma_{A}\otimes\alpha_{A^{\prime}}$
if and only if $\textbf{p}\otimes\textbf{r}\prec{\bf q}\otimes{\bf r}$.

Consider a mixed state $\rho_{s}$ with ensemble
$\\{p_{i},|{\psi_{i}}\rangle\\}_{i=1}^{m}$, and a catalyst state
$|{\chi}\rangle=\sqrt{1-t}|{0}\rangle+\sqrt{t}|{1}\rangle$ with Schmidt rank
$2$. Assume that $|{\psi_{i}}\rangle$ and $|{\varphi}\rangle$ have the same
Schmidt rank, and there exists common $t$ such that
$|{\psi_{i}}\rangle\otimes|{\chi}\rangle\longrightarrow|{\varphi}\rangle\otimes|{\chi}\rangle$
holds. Namely, there exists LOCC operation $U_{i}^{LOCC}$ such that
$U_{i}^{LOCC}(|{\psi_{i},\chi}\rangle)=|{\varphi,\chi}\rangle$ for all
$i=1,...,m$. Then the following protocol transforms the state
$\rho_{s}\otimes|{\chi}\rangle$ to $|{\varphi}\rangle\otimes|{\chi}\rangle$
under LOCC:

1) Introducing an ancillary $m$-dimensional system ${\cal H}_{a}$ with
orthonormal basis $\\{|{i}\rangle|i=1,2,\dots,m\\}$. Prepare the initial state
of the system, catalyst and ancillary to be

$\rho_{sca}=\sum_{i=1}^{m}p_{i}|{\psi_{i}}\rangle\langle{\psi_{i}}|\otimes|{\chi}\rangle\langle{\chi}|\otimes|{i}\rangle\langle{i}|.$

2) Constructing a quantum channel $\Lambda_{sca}$ on the initial state,
$\Lambda_{sca}(\cdotp)=\sum_{i=1}^{m}K_{i}(\cdotp)K_{i}^{\dagger}$, with the
Kraus operators given by $K_{i}=U_{i}^{LOCC}\otimes|{i}\rangle\langle{i}|$
satisfying $\sum_{i=1}^{m}K_{i}^{\dagger}K_{i}=\bf{I}$.

3) Applying the quantum channel $\Lambda_{sca}$, we obtain

$\displaystyle\begin{aligned}
\Lambda_{sca}(\rho_{sca})&=\sum_{i=1}^{m}K_{i}(\rho_{sca})K_{i}^{\dagger}\\\
&=\sum_{j=1}^{m}U_{j}^{LOCC}\otimes|{j}\rangle\langle{j}|(\sum_{i=1}^{m}p_{i}|{\psi_{i},\chi,i}\rangle\langle{\psi_{i},\chi,i}|)(U_{j}^{LOCC})^{\dagger}\otimes|{j}\rangle\langle{j}|\\\
&=\sum_{i=1}^{m}p_{i}U_{i}^{LOCC}(|{\psi_{i},\chi}\rangle\langle{\psi_{i},\chi}|)(U_{i}^{LOCC})^{\dagger}\otimes|{i}\rangle\langle{i}|\\\
&=\sum_{i=1}^{m}p_{i}|{\varphi,\chi}\rangle\langle{\varphi,\chi}|\otimes|{i}\rangle\langle{i}|\\\
&=|{\varphi,\chi}\rangle\langle{\varphi,\chi}|\otimes\sum_{i=1}^{m}p_{i}|{i}\rangle\langle{i}|,\end{aligned}$

which realizes the conversion from a mixed state
$\sum_{i}p_{i}|{\psi_{i}}\rangle\langle{\psi_{i}}|$ to a pure state
$|{\varphi}\rangle$ assisted by a catalyst state $|{\chi}\rangle$ and the
ancillary system under LOCC quantum channel. Tracing over the catalyst and the
ancillary systems, one gets the pure state $|{\varphi}\rangle$.

##  VII concluding discussions

We have investigated the bounds on catalytic states and presented general
necessary conditions for the existence of catalytic states. To the open
question whether any two solvable non-comparably entangled pure states have a
catalytic state which catalyzes the transformation, our intuition is that the
existence of the catalytic states is related to the set
$\mathcal{L}=\\{l\in\\{1,2\cdots d\\}|\sum_{i=1}^{l}(p_{i}-q_{i})>0\\}$ given
by the Schmidt coefficients of the two states. Our bounds on catalytic states
give effective ways in entanglement catalyst for not only pure but also mixed
states. These results on catalysts would deepen our understanding on quantum
entanglement transformations under LOCC and highlight further the related
investigations.

## Acknowledgments

YMG and SMF acknowledges the support from NSF of China under Grant No.
12075159, Beijing Natural Science Foundation (Z190005), Academy for
Multidisciplinary Studies, Capital Normal University, and Shenzhen Institute
for Quantum Science and Engineering, Southern University of Science and
Technology, Shenzhen 518055, China (No. SIQSE202001). YS, MYH, LJZ and LC were
supported by the NNSF of China (Grant Nos. 11871089 and 11947241), and the
Fundamental Research Funds for the Central Universities (Grant Nos. KG12080401
and ZG216S1902).

## Appendix

###  VII.1 Proof of Proposition 1

###### Proof.

If the conclusion is not true, one would have $q_{1}r_{s+1}\leq q_{l}r_{s}$,
$q_{l+1}r_{s}\leq q_{d}r_{s-1}$ or $q_{1}r_{s+1}\leq q_{d}r_{s-1}$, which
imply that the largest $d(s-1)+l$ elements of $\mathbf{q}\otimes\mathbf{r}$
are $q_{1}r_{1},\cdots,q_{d}r_{1}$, $q_{1}r_{2},\cdots,q_{d}r_{2}$, $\cdots$,
$q_{1}r_{s},\cdots,q_{l}r_{s}$. The summation of these largest $d(s-1)+l$
elements of $\mathbf{q}\otimes\mathbf{r}$ gives

$\displaystyle\sum_{i=1}^{d(s-1)+l}(\mathbf{q}\otimes\mathbf{r})_{i}$
$\displaystyle=\sum_{i=1}^{d}\sum_{j=1}^{s-1}q_{i}r_{j}+\sum_{t=1}^{l}q_{t}r_{s}$
$\displaystyle=\sum_{i=1}^{s-1}r_{i}+r_{s}\sum_{j=1}^{l}q_{j}.$

Meanwhile, the largest $d(s-1)+l$ elements of $\mathbf{p}\otimes\mathbf{r}$
are not less than the sum of the terms $p_{1}r_{1},\cdots,p_{d}r_{1}$,
$p_{1}r_{2},\cdots,p_{d}r_{2}$, $\cdots$, $p_{1}r_{s},\cdots,p_{l}r_{s}$,

$\displaystyle\sum_{i=1}^{d(s-1)+l}(\mathbf{p}\otimes\mathbf{r})_{i}^{\downarrow}$
$\displaystyle\geq\sum_{i=1}^{d}\sum_{j=1}^{s-1}p_{i}r_{j}+\sum_{t=1}^{l}p_{t}r_{s}$
$\displaystyle=\sum_{i=1}^{s-1}r_{i}+r_{s}\sum_{j=1}^{l}p_{j}.$

On the other hand, since $l\in\mathcal{L}$ and
$\sum_{j=1}^{l}p_{j}>\sum_{j=1}^{l}q_{j}$, we have

$\displaystyle\sum_{i=1}^{d(s-1)+l}(\mathbf{p}\otimes\mathbf{r})_{i}^{\downarrow}\geqslant$
$\displaystyle\sum_{i=1}^{s-1}r_{i}+r_{s}\sum_{j=1}^{l}p_{j}$ $\displaystyle>$
$\displaystyle\sum_{i=1}^{s-1}r_{i}+r_{s}\sum_{j=1}^{l}q_{j}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{d(s-1)+l}(\mathbf{q}\otimes\mathbf{r})_{i}^{\downarrow}$

for any $s=2,3,\cdots,k-1$, which contradicts the assumption that
$\mathbf{p}\otimes\mathbf{r}\prec\mathbf{q}\otimes\mathbf{r}$.

Next we prove the first inequality in (6). Since $l\in\mathcal{D}_{1}$, we
have $p_{1}+\cdots+p_{l}>q_{1}+\cdots+q_{l}$ and $p_{1}+\cdots+p_{l+1}\leq
q_{1}+\cdots+q_{l+1}$, which give rise to $p_{l+1}<q_{l+1}$. Taking into
account $p_{d}>q_{d}$, we have $\frac{p_{d}}{p_{l+1}}>\frac{q_{d}}{q_{l+1}}$
as $\mathbf{p}$ and $\mathbf{q}$ are solvable incomparable.

We use proof by contradiction. If
$\frac{q_{d}}{q_{l+1}}\geq\frac{r_{k}}{r_{k-1}}$, i.e., $q_{d}r_{k-1}\geq
q_{l+1}r_{k}$, then the smallest $d-l$ elements of
$\mathbf{q}\otimes\mathbf{r}$ are
$q_{l+1}r_{k},q_{l+2}r_{k},\cdots,q_{d}r_{k}$. Moreover, the sum of the
smallest $d-l$ elements of $\mathbf{p}\otimes\mathbf{r}$ is not larger than
the sum of $p_{l+1}r_{k},p_{l+2}r_{k},\cdots,p_{d}r_{k}$. Because of
$l\in\mathcal{L}$, we get $p_{l+1}+\cdots+p_{d}<q_{l+1}+\cdots+q_{d}$.
Therefore, we have the following relation,

$\displaystyle\sum_{i=dk-d+l+1}^{dk}(\mathbf{p}\otimes\mathbf{r})_{i}^{\downarrow}\leqslant\sum_{i=l+1}^{d}p_{i}r_{k}$
$\displaystyle<\sum_{i=l+1}^{d}q_{i}r_{k}=\sum_{i=dk-d+l+1}^{dk}(\mathbf{q}\otimes\mathbf{r})_{i}^{\downarrow},$

which contradicts the assumption that
$\mathbf{p}\otimes\mathbf{r}\prec\mathbf{q}\otimes\mathbf{r}$, and proves the
first inequality in (6).

Concerning the second inequality in (6), we use again the proof by
contradiction. If $\frac{q_{1}}{q_{l}}\leq\frac{r_{1}}{r_{2}}$, the largest
$l$ elements of $\mathbf{q}\otimes\mathbf{r}$ are
$q_{1}r_{1},q_{2}r_{1},\cdots,q_{l}r_{1}$. Since the sum of the largest $l$
elements of $\mathbf{p}\otimes\mathbf{r}$ is not lees than the sum of
$p_{1}r_{1},p_{2}r_{1},\cdots,p_{l}r_{1}$ for $l\in\mathcal{L}$, we have
$p_{1}+\cdots+p_{l}>q_{1}+\cdots+q_{l}$. Then we have the relation,

$\displaystyle\sum_{i=1}^{l}(\mathbf{p}\otimes\mathbf{r})_{i}^{\downarrow}\geqslant\sum_{i=1}^{l}p_{i}r_{1}$
$\displaystyle>\sum_{i=1}^{l}q_{i}r_{1}=\sum_{i=1}^{l}(\mathbf{q}\otimes\mathbf{r})_{i}^{\downarrow},$

which contradicts the assumption that
$\mathbf{p}\otimes\mathbf{r}\prec\mathbf{q}\otimes\mathbf{r}$ and proves the
second inequality in (6).  $\sqcap$$\sqcup$

###  VII.2 A remark on the relationship between Proposition 1 and the Theorem
1 in PhysRevA.99.052348

In the proof of Theorem 1 in PhysRevA.99.052348 , it is mentioned that
“$q_{m}r_{v^{\prime}}\geqslant q_{1}r_{v^{\prime}+1}$ implies that the first
$\left(v^{\prime}-1\right)d+m$ elements of
$(\mathbf{q}\otimes\mathbf{r})^{\downarrow}$ consist of $q_{x}r_{y}$, $\forall
x\in\\{1,2,\ldots,d\\}$ and $\forall
y\in\left\\{1,2,\ldots,v^{\prime}-1\right\\}$, along with
$q_{x^{\prime}}r_{v^{\prime}}$ $\forall x^{\prime}\in\\{1,2,\ldots,m\\}$”.
Generally this could be not true because some elements of $q_{x}r_{y}$,
$x\in\\{1,2,\ldots,d\\}$, $y\in\left\\{1,2,\ldots,v^{\prime}-1\right\\}$,
along with $q_{x^{\prime}}r_{v^{\prime}}$ $\forall
x^{\prime}\in\\{1,2,\ldots,m\\}$ might less than some elements of
$q_{x^{\prime}}r_{v^{\prime}}$, $x^{\prime}\in\\{1,2,\ldots,d\\}$,
$y^{\prime}\in\\{v^{\prime}+1,\ldots,k\\}$, along with
$q_{x^{\prime}}r_{v^{\prime}}$ $\forall x^{\prime}\in\\{m+1,\ldots,d\\}$, when
$q_{d}r_{v^{\prime}-1}<q_{m+1}r_{v^{\prime}}$ or
$q_{d}r_{v^{\prime}-1}<q_{1}r_{v^{\prime}+1}$. This implies that the first
$\left(v^{\prime}-1\right)d+m$ elements of
$(\mathbf{q}\otimes\mathbf{r})^{\downarrow}$ do not consist of $q_{x}r_{y}$,
$\forall x\in\\{1,2,\ldots,d\\}$ and $\forall
y\in\left\\{1,2,\ldots,v^{\prime}-1\right\\}$, along with
$q_{x^{\prime}}r_{v^{\prime}}$ $\forall x^{\prime}\in\\{1,2,\ldots,m\\}$.

From Proposition 1, we have that the first $\left(s-1\right)d+l$ elements of
$(\mathbf{q}\otimes\mathbf{r})^{\downarrow}$ consist of $q_{x}r_{y}$, $\forall
x\in\\{1,2,\ldots,d\\}$ and $\forall y\in\left\\{1,2,\ldots,s-1\right\\}$,
along with $q_{x^{\prime}}r_{s}$ $\forall x^{\prime}\in\\{1,2,\ldots,l\\}$
when $q_{l}r_{s}\geqslant q_{1}r_{s+1}$, $q_{d}r_{s-1}\geqslant q_{l+1}r_{s}$
and $q_{d}r_{s-1}\geqslant q_{1}r_{s+1}$, which is true for $\forall
l\in\cal{L}$. Hence, Proposition 1 fills such loopholes in the Theorem 1 in
PhysRevA.99.052348 , and also gives finer characterizations of $\mathbf{r}$ by
increasing the number of constraints on $\mathbf{r}$.

###  VII.3 Proof of corollary 2

###### Proof.

By the definition of $M_{{\cal L}}$, we have:

$\displaystyle\begin{aligned} \sum_{i=M_{{\cal
L}}+1}^{d}p_{i}\leqslant\sum_{i=M_{{\cal L}}+1}^{d}q_{i},\\\ \sum_{i=M_{{\cal
L}}+2}^{d}p_{i}>\sum_{i=M_{{\cal L}}+2}^{d}q_{i}.\end{aligned}$

This implies that $p_{M_{{\cal L}}+1}<q_{M_{{\cal L}}+1}$. Combining with
$p_{d}\geqslant q_{d}$, we get

$\displaystyle\frac{p_{d}}{p_{M_{{\cal L}}+1}}>\frac{q_{d}}{q_{M_{{\cal
L}}+1}}.$

If $\frac{q_{d}}{q_{M_{{\cal L}}+1}}\geqslant\frac{r_{k}}{r_{k-1}}$, we have
the following partial order the vector $\mathbf{q}\otimes\mathbf{r}$,

$\displaystyle q_{d}r_{k}\leqslant q_{d-1}r_{k}\leqslant\dots\leqslant
q_{M_{{\cal L}}+1}r_{k}\leqslant q_{d}r_{k-1}.$ (12)

For $\frac{p_{d}}{p_{M_{{\cal L}}+1}}>\frac{q_{d}}{q_{M_{{\cal L}}+1}}$, we
have the following partial order the vector $\mathbf{p}\otimes\mathbf{r}$,

$\displaystyle p_{d}r_{k}\leqslant p_{d-1}r_{k}\leqslant\dots\leqslant
p_{M_{{\cal L}}+1}r_{k}\leqslant p_{d}r_{k-1}$ (13)

Combining (12) and (13), we have that the last $d-M_{{\cal L}}$ elements of
$(\mathbf{p}\otimes\mathbf{r})^{\downarrow}$ and
$(\mathbf{q}\otimes\mathbf{r})^{\downarrow}$ are $p_{d}r_{k}$, $p_{d-1}r_{k}$,
$\dots$, $p_{M_{{\cal L}}+1}r_{k}$ and $q_{d}r_{k}$, $q_{d-1}r_{k}$, $\dots$,
$q_{M_{{\cal L}}+1}r_{k}$, respectively. Therefore, we have

$\displaystyle\begin{aligned} &\sum_{i=dk-d+M_{{\cal
L}}+1}^{dk}(\mathbf{p}\otimes\mathbf{r})_{i}^{\downarrow}=r_{k}\sum_{i=M_{{\cal
L}}+1}^{d}p_{i}\\\ &\leqslant r_{k}\sum_{i=M_{{\cal
L}}+1}^{d}q_{i}=\sum_{i=dk-d+M_{{\cal
L}}+1}^{dk}(\mathbf{q}\otimes\mathbf{r})_{i}^{\downarrow},\end{aligned}$

which implies that

$\displaystyle\sum_{i=1}^{dk-d+M_{{\cal
L}}}(\mathbf{p}\otimes\mathbf{r})_{i}^{\downarrow}>\sum_{i=1}^{dk-d+M_{{\cal
L}}}(\mathbf{p}\otimes\mathbf{r})_{i}^{\downarrow},$

and contradicts the assumption that
$\mathbf{p}\otimes\mathbf{r}\prec\mathbf{q}\otimes\mathbf{r}$.
$\sqcap$$\sqcup$

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