# Sequence Dependence of Critical Properties for 2-letter Chains

Athanassios Z. Panagiotopoulos<EMAIL_ADDRESS>Department of Chemical and
Biological Engineering, Princeton University, Princeton, NJ 08544, U.S.A.

###### Abstract

Histogram-reweighting grand canonical Monte Carlo simulations are used to
obtain the critical properties of lattice chains composed of solvophilic and
solvophobic monomers. The model is a modification of one proposed by Larson
_et al._ [J. Chem. Phys. 83, 2411 (1985)], lowering the “contrast” between
beads of different type to prevent aggregation into finite-size micelles that
would mask true phase separation between bulk high- and low-density phases.
Oligomeric chains of length between 5 and 24 beads are studied. Mixed-field
finite-size scaling methods are used to obtain the critical properties with
typical relative accuracies of better than $10^{-4}$ for the critical
temperature and $10^{-3}$ for the critical volume fraction. Diblock chains are
found to have lower critical temperatures and volume fractions relative to the
corresponding homopolymers. Addition of solvophilic blocks of increasing
length to a fixed-length solvophobic segment results in a decrease of both
critical temperature and critical volume fraction, with an eventual slow
asymptotic approach to the long-chain limiting behavior. Moving a single
solvophobic or solvophilic bead along a chain leads to a minimum or maximum in
the critical temperature, with no change in critical volume fraction. Chains
of identical length and composition have a significant spread in their
critical properties, depending on their precise sequence. The present study
has implications on understanding biomolecular phase separation and for
developing design rules for synthetic polymers with specific phase separation
properties. It also provides data potentially useful for the further
development of theoretical models for polymer and surfactant phase behavior.

##  I Introduction

The Monte Carlo method for computational determination of condensed-matter
properties was introduced in the seminal paper by Metropolis _et al._
Metropolis  _et al._ (1953) 70 years ago, for determination of the equation of
state of hard disks. In the intervening period, the method has formed the
basis for many major developments in statistical mechanics of fluids and
solids. The original method was designed for sampling configurations in the
canonical ($NVT$) ensemble. Development of Monte Carlo algorithms for the
constant-pressureWood (1968) ($NPT$) and grand canonicalNorman and Filinov
(1969) ($\mu VT$) ensembles took place in the late 1960’s and greatly
facilitated the study of phase transitions. Free-energy calculation methods
for crystal phases became available in the early 1980’sFrenkel and Ladd (1984)
and made possible the determination of solid relative stabilities and their
solubilities in liquids. The Gibbs ensemble Monte Carlo method for direct
determination of fluid-phase equilibriaPanagiotopoulos (1987) and efficient
histogram-reweighting algorithmsFerrenberg and Swendsen (1989) were developed
in the late 1980’s.

Systems of interest for the present study are relatively short chains composed
of two different bead types, solvophilic and solvophobic. It is well-
established that monomer sequence has a significant impact on chain phase and
aggregation behavior. While early work on such systems focused on modeling
surfactant solutions,Larson, Scriven, and Davis (1985); Larson (1988);
Floriano, Caponetti, and Panagiotopoulos (1999) recent interest in the effect
of chain sequence on the phase behavior has been sparked by the discovery of
the important role that liquid-liquid phase separation plays in cell
biology.Brangwynne  _et al._ (2009); Banani  _et al._ (2017) Intrinsically
disordered proteins are often key determinants of biomolecular phase
separation, with the observed behavior being highly sensitive to their overall
composition and precise amino acid sequence.Das  _et al._ (2020); Rana  _et
al._ (2024) There is also significant interest in the properties of synthetic
polymers with well-defined blocks of solvophilic and solvophobic
segments.Destefano  _et al._ (2024); Taylor, Priestley, and Register (2024)
The use of simple, coarse-grained models with only a handful of independent
control parameters helps provide a fundamental understanding of the underlying
physical phenomena by excluding complications arising from the presence of
many different monomer and interaction types present in real chemical or
biological systems.

Relevant prior work in the area of sequence-dependence of critical properties
for coarse-grained chain models includes the study of Statt _et al._ Statt
_et al._ (2020) for a continuum-space 2-letter heteropolymer, using direct
coexistence (interfacial) molecular dynamics simulations. A number of
different 20-mer chains were studied, yielding a range of possible dense-phase
morphologies and a clear dependence of the critical temperature on a
“blockiness” parameter. A limitation of that work was the use of interfacial
simulations for determination of critical parameters, which results in only
approximate values. This limitation was overcome in a subsequent paperRana,
Brangwynne, and Panagiotopoulos (2021) that utilized a 2-letter lattice model
and grand canonical Monte Carlo simulations. That study provided accurate
critical parameters for the model chains, as well as a path for distinguishing
clearly between true phase separation and aggregation into finite-size
clusters. The scaling of chain critical temperature and volume fraction with
respect to a normalized metric for chain blockiness was also determined. In
another recent study,Panagiotopoulos (2023) the same 2-letter lattice chain
model was used to shed light on the interplay between phase separation into
macroscopic (bulk) fluid phase on one hand, and formation of finite-size
micelles on the other. It was found that certain triblock sequences with
solvophobic ends can undergo both micellization and macroscopic phase
separation.

An element missing from these prior studies that I plan to address in the
present work is the quantitative dependence of the critical parameters on
overall chain length, as well as their dependence on the size of the
solvophilic and solvophobic chain segments. In addition, as explained in the
“Model” subsection, a modification of the 2-letter chain model parameters was
implemented to prevent aggregation into finite-size micelles for all sequences
studied and to restore symmetry with respect to the phase behavior for chains
consisting of a single bead type. The overall aim of the study is to provide
accurate data that could be used for the development of theoretical or
machine-learning methods for describing the effects of sequence and chain
length on phase separation for synthetic or biological polymers.

##  II Model and Methods

###  II.1 Model

The model used in this work is a modification of the lattice surfactant model
first proposed by Larson _et al._ Larson, Scriven, and Davis (1985); Larson
(1988) The model consists of linear chains of $r$ beads connected by bonds,
each bead occupying a single site on a simple cubic lattice. The range of
chain lengths studied here was from $r=5$ to $r=24$. The use of shorter chain
lengths restricted the size of sequence space and ensured that there was
sufficient sensitivity of the critical parameters to sequence variations
involving even a single monomer moving along the chain.

Bonds between adjacent beads in the model chains can be in the [0 0 1], [0 1
1], and [1 1 1] directions of the lattice, as well as their allowable
rotations and reflections, resulting in 26 total possible connectivity
vectors. There are two bead types, solvophobic and solvophilic, conventionally
labeled as “T” and “H”, respectively, for “tail” and “head” groups in the
original surfactant-oriented incarnation of the model. Non-bonded beads
interact along the 26 possible lattice connectivity directions. There are no
interactions with empty lattice sites, assumed to contain a monomeric solvent.
Thus, the model has three independent energy parameters, specifically
$\epsilon_{\mathrm{TT}}$, $\epsilon_{\mathrm{HT}}$, and
$\epsilon_{\mathrm{HH}}$. In the original version of the model,Larson,
Scriven, and Davis (1985); Larson (1988) the values of the parameters were
taken to be $\epsilon_{\mathrm{TT}}=-2$ and
$\epsilon_{\mathrm{HT}}=\epsilon_{\mathrm{HH}}=0$. The unit of energy also
sets the temperature scale for the model. Subsequent studies Rana, Brangwynne,
and Panagiotopoulos (2021); Panagiotopoulos (2023) used the trivial change
$\epsilon_{\mathrm{TT}}=-1$,
$\epsilon_{\mathrm{HT}}=\epsilon_{\mathrm{HH}}=0$, which simply rescales the
temperature by a factor of 2. With this choice of model parameters, implying a
strong contrast between solvophobic and solvophilic interactions, many
sequences were found to form micellar finite-size aggregates instead of
macroscopic bulk liquid phasesRana, Brangwynne, and Panagiotopoulos (2021) at
the range of temperatures over which sampling of equilibrium states is
feasible. This is an undesirable feature for the present study, because it
prevents a direct comparison of critical parameters for many candidate
sequences. In order to overcome this limitation, a modified parameter set with
reduced contrast between solvophobic and solvophilic beads is utilized here,
as follows:

$\epsilon_{\mathrm{TT}}=-\frac{3}{4}\ \ \ ;\ \ \
\epsilon_{\mathrm{HT}}=\epsilon_{\mathrm{HH}}=-\frac{1}{4}$ (1)

The relative strength of attractions between solvophobic and solvophilic beads
is now 3 (rather than $\infty$ in the previous version of the model), a factor
that will be used to normalize critical temperatures with respect to
composition variations. This modification also restores symmetry of behavior
for chains at the limit at which they consist of a single monomer type. Thus,
solvophilic Hr chains now have the same behavior as solvophobic Tr chains,
modulo a factor of 3 rescaling energies, chemical potentials, and
temperatures. In the previous version of the model, purely solvophilic chains
are repulsive and do not undergo a phase transition into a condensed liquid
phase at any finite temperature.

###  II.2 Computational Methods

The computational methodologies used in the present work are similar to those
of Ref. 18 and will only be briefly summarized here for completeness. Monte
Carlo simulations in the grand canonical ($\mu VT$) ensemble in cubic boxes of
edge length $L$ were used to obtain all aspects of thermodynamic behavior for
the model. The volume fraction when $N$ chains of length $r$ are present in
the simulation box is defined as $\phi=Nr/L^{3}$. To facilitate insertion and
removal of chains, an athermal Rosenbluth algorithmRosenbluth and Rosenbluth
(1955) was implemented, as detailed in Ref. 20. Source codes, example input
and output files, and a list of the runs performed for each system are
available online, as explained in the Data Availability Statement at the end
of this article. Statistical uncertainties were obtained from the standard
deviation of results from four independent runs at identical conditions, each
with a different random number seed.

Histogram reweighting with the Ferrenberg-Swendsen algorithmFerrenberg and
Swendsen (1989) was used to rescale information from runs at one set of
thermodynamic conditions to another and to combine runs that have a reasonable
degree of overlap in the number of particles $N$ and energy $E$ covered in
each. The combined histograms were then used to obtain distributions of
particle numbers and energy, $P(N,E)$, as a continuous function of $\mu$ and
$T$. To determine critical points, the mixed-field finite-size scaling method
of Bruce and Wilding Bruce and Wilding (1992); Wilding and Bruce (1992) was
used. The critical temperature $T_{\mathrm{c}}$, critical chemical potential,
$\mu_{\mathrm{c}}$, and number-energy field mixing parameter, $s$, were
optimized simultaneously, using histogram data near the expected critical
point, by minimizing deviations between the observed order parameter
distribution and the 3-dimensional Ising universal curve obtained as an
analytical expression by Tsypin and Blöte.Tsypin and Blöte (2000) Phase
coexistence curves (binodals) were determined from the condition of area
equality for the low-$N$ and high-$N$ regions of the $P(N)$ distributions.

For most chain architectures studied, the simulation box length was set at
$L=15$ in lattice units. For systems of chains with $r>8$, larger boxes of
$L=20$ and $L=25$ were used to ensure that the box dimensions were kept much
bigger than chain average dimensions. These simulation box lengths resulted in
average number occupancies of between 100 and 300 chains at the critical
temperature and chemical potential. The choice of system size is a compromise
between using smaller systems (which enable significantly faster calculations)
and matching the universal order parameter distribution closely (which only
happens at sufficiently large systems). There are systematic effects of system
size on the critical parameters,Orkoulas and Panagiotopoulos (1999) but for
the systems studied here these are generally much smaller than the differences
due to varying chain architectures that are of primary interest for the
present work. In addition, comparisons here are made for similar system sizes
across architectures and thus would not substantially change if the critical
parameters were to be systematically extrapolated to infinite simulation
system size, at significant additional computational cost.

##  III Results and Discussion

###  III.1 Chain-length Dependence

The first task of interest was to establish the dependence of the critical
temperature and volume fraction on chain length for homopolymer sequences
between $r=5$ and $r=24$, matching the range of chain lengths of interest.
Results for the critical temperature are shown in Fig. 1. The figure is for
“all solvophilic” Hr sequences. For “all solvophobic” Tr sequences, the
critical temperature would be scaled up by a factor of 3, following the ratio
of interaction parameters $\epsilon_{\mathrm{TT}}/\epsilon_{\mathrm{HH}}$ from
Eq. 1. These critical temperatures are in excellent agreement with prior
results for $r=8$ and $r=16$ using the same model.Panagiotopoulos, Wong, and
Floriano (1998) According to Flory-Huggins theoryFlory (1953), the inverse
critical temperature, $1/T_{\mathrm{c}}$, is expected to scale as:

$\frac{1}{T_{\mathrm{c}}(r)}-\frac{1}{T_{\mathrm{c}}(\infty)}\propto{\frac{1}{\sqrt{r}}+\frac{1}{2r}}$
(2)

As can be seen in the figure, this scaling is followed quite closely, as also
seen for a broader range of chain lengths in Ref. 20. The extrapolated value
is $T_{\mathrm{c}}(\infty)=20.4\pm 0.1$, a bit lower than the estimate
inPanagiotopoulos, Wong, and Floriano (1998) which not surprising, given the
lower range of $r$ values examined here.

Figure 1: Inverse critical temperature, $1/T_{\mathrm{c}}$ as a function of
the parameter ${\frac{1}{\sqrt{r}}+\frac{1}{2r}}$ , for homopolymers of type
Hr. For homopolymers of type Tr, temperature will be higher by a factor of
exactly 3. Points (from right to left) correspond to $r=5,8,9,12,15,16$, and
24. The line is a linear-least-squares fit to the data. Statistical
uncertainties are smaller than symbol size. Figure 2: Critical volume fraction
$\phi_{\mathrm{c}}$ as a function of the chain length $r$, on a log-log scale,
for homopolymers Hr or Tr of length between $r=$5 and $r=$24\. The line is a
linear-least-squares fit to the data. Statistical uncertainties are comparable
to symbol size.

The dependence of the critical volume fraction on chain length for
homopolymers is shown in Fig. 2. The critical volume fraction for homopolymers
does not depend on the energy scale, so it is identical for Hr and Tr chains.
In agreement with Flory theoryFlory (1953), it is seen that the logarithm of
the critical volume fraction depends linearly on the logarithm of chain
length. The slope obtained from the data is $-0.29\pm 0.01$, which is again
lower than the exponent obtained inPanagiotopoulos, Wong, and Floriano (1998)
by fitting the range of chain lengths of $64\leq r\leq 1000$. The chain
lengths studied here are too short to match the asymptotic long-chain length
behavior for the scaling of the critical parameters, but the reason of
obtaining these relationships is to obtain analytical expressions for the
critical temperature and critical volume over the chain lengths of interest,
which in turn are used in comparisons when chain composition is varied as well
as the chain length.

###  III.2 Block-length Dependence for Diblock Chains

The next task was to examine the dependence of model critical parameters on
the length of the solvophilic segment, $h$, for chains of fixed overall length
of 8 beads, as shown schematically in the inset to Fig. 3. The overall
architecture is HhTr-h, with $r=8$. For the full-contrast parameter set
studied in Ref 26, chains with $h$ between 4 and 7 were shown to micellize,
but the reduced-contrast interaction set used here ensures that all chain
lengths undergo normal phase separation. This is confirmed by the match of the
order parameter distribution to the 3-dimensional Ising universality class,
following the procedure of Ref. 18. One complication that arises in comparing
the behavior of different architectures is that chains of variable $h$ have
different compositions. This can be taken into account by normalizing the
temperature as $T_{\mathrm{c}}/(3-2h/r)$, where the factor of 3 is the ratio
of $\epsilon_{\mathrm{TT}}/\epsilon_{\mathrm{HH}}$ from Eq. 1. The expression
linearly interpolates between the limits for the critical temperatures of the
homopolymers, T8 ($h=0$) and H8 ($h=r$). As seen in Fig. 3, the normalized
critical temperature shows a minimum at $h=5$, with the corresponding value
26% lower than for the homopolymer, a rather large effect. This is physically
the result of the solvophilic H beads “screening” interactions between
solvophobic T beads, thus making them effectively weaker. The asymmetry of the
behavior (skewed towards higher $h$ values) results from the greater
efficiency of this screening by longer H segments.

Figure 3: Normalized critical temperature, $T_{\mathrm{c}}/(3-2h/r)$, as a
function of the solvophilic segment length $h$, for diblock chains HhTr-h with
overall length $r=8$. The horizontal line is the expected value based on the
chain length. Statistical uncertainties are smaller than symbol size.

The critical volume fraction also depends strongly on the length of the
solvophilic segment, $h$, as shown in Fig. 4. The minimum here occurs at $h=3$
and is approximately 12% lower than the critical volume fraction for the
homopolymer, once more a rather strong effect. As observed in Ref. 16, lower
volume fractions are associated with structured liquids. For $h=7$, the
critical volume fraction recovers the value for the homopolymers ($h=$0 or 8).
Note that independent simulations were done for the two homopolymers H8 and
T8, explaining the small difference in critical volume fractions at the two
ends. The fitted line from Fig. 2 is also not a perfect match of the limiting
values. Here, there is competition between two opposing effects: high values
of $h$ mean that the critical temperature is lower, as seen in Fig. 3. At
lower temperatures the chains pack more closely, thus increasing the critical
volume fraction. On the other hand, at low values of $h$, the effect of the
critical temperature is less and the behavior is dominated by the less
efficient packing of T segments together because of the intervening H
segments.

Figure 4: Critical volume fraction, $\phi_{c}$, as a function of the
solvophilic segment length $h$, for diblock chains HhTr-h with overall length
$r=8$. The horizontal line is the expected value based on the chain length.
Statistical uncertainties are comparable to symbol size.

###  III.3 Overall Length Dependence with Fixed End Segment

The next set of simulations were done for chains of variable solvophilic block
length $h$, but with a fixed solvophobic end-segment length of 5 beads. The
architecture is now HhT5 and the overall chain length is $r=h+5$. Fig. 5 shows
the measured dependence of the critical temperature as points. In order to
normalize the critical temperature for the varying chain composition, we use
again the scaling $T_{\mathrm{c}}/(3-2h/r)$. However, in this case, the
overall length of the chain is not fixed, so the appropriate comparison for
the critical temperature is the length-dependent value obtained through Eq. 2,
which is shown as a blue curve in Fig. 5. At $h=0$, the critical temperature
matches that for the homopolymer chain of length 5. Addition of solvophilic
beads to the chain results in a rapid decrease of the critical temperature,
since the added H beads “protect” the solvophobic T beads from coming to
contact as readily and thus lower the effective interactions. The critical
temperature gets lower, even though the expectation based on the overall chain
length would be for the critical temperature to increase. At the opposite
limit of large $h$, the presence of a relatively short solvophobic group at
the end of the chain has a diminishing overall effect, so it is expected that
the data will approach asymptotically the homopolymer result. Indeed, the data
points and the line in Fig. 5 become parallel to each other. Convergence
between them would happen for much longer chains that the ones studied here;
for $r=24$ there is still a sizable gap between the actual critical
temperature and that expected based on the chain average composition and
overall length. The total effect on the normalized critical temperature
results in a minimum around $h=4$, similar to that observed in the previous
subsection for diblock chains of fixed overall length.

Figure 5: Normalized critical temperature, $T_{\mathrm{c}}/(3-2h/r)$ as a
function of the solvophilic segment length $h$, for diblock chains of type
HhT5. The line is the expected dependence based on the overall chain length.
Statistical uncertainties are smaller than symbol size.

Results for the critical volume fraction for systems with increasing
solvophilic block length and a fixed solvophobic end segment are shown in Fig.
6 as points, along with the curve obtained by fitting the homopolymer critical
volume fraction data of Fig. 2. The critical volume fraction rapidly
decreases, at a rate much faster than that expected on the basis of the
overall chain length. The gap between the curve and the points is maximum for
$7\leq h\leq 10$. The critical volume fraction then plateaus and converges to
the value expected based on the length, as seen on the right side of the
figure. Once more, the physical explanation for this behavior is the less
efficient packing of chains when a solvophilic block is added.

Figure 6: Critical volume fraction, $\phi_{c}$ as a function of the
solvophilic segment length $h$, for diblock chains of type HhT5. The line is
the expected dependence based on the overall chain length. Statistical
uncertainties are comparable to symbol size.

Calculations of the phase coexistence (binodal) curves were performed for
selected systems, specifically for the architectures H7T5, H10T5, and H19T5,
as shown on Fig. 7. The lines on this figure were obtained by fitting
coexistence data below the critical point to the scaling relationship
appropriate for the 3-dimensional Ising universality class,

$\phi_{\mathrm{l}}-\phi_{\mathrm{g}}=a(T_{\mathrm{c}}-T)^{\beta}$ (3)

where $\phi_{\mathrm{l}}$ and $\phi_{\mathrm{g}}$ are the volume fractions of
the two phases at coexistence, $a$ is a fitted constant, and $\beta=0.326$ is
the scaling exponent for the coexistence width. This expression was combined
with the approximate “law of rectilinear diameters,”

$\phi_{\mathrm{l}}+\phi_{\mathrm{g}}=b(T_{\mathrm{c}}-T)+2\phi_{\mathrm{c}}$
(4)

where $b$ is another fitted constant. Analytical expressions were obtained for
$\phi_{\mathrm{l}}$ and $\phi_{\mathrm{g}}$ as functions of $T$, using the
known values of $T_{\mathrm{c}}$ and $\phi_{\mathrm{c}}$ for each system and
separate optimizations of parameters $a$ and $b$ for $\phi_{\mathrm{l}}$ and
$\phi_{\mathrm{g}}$. One of the reasons for determining these coexistence
curves was to ensure that there are no finite-size aggregates (micelles)
present in the low-density phase. Indeed, simulation runs up to (and even
slightly beyond) the coexistence chemical potential at a range of temperatures
below the critical point displayed monotonic decay of the aggregate size
distributions with no micellar aggregates present. This behavior stands in
contrast to that observed for comparable chain lengths and architectures in
Ref. 18, because of the use of a reduced-contrast interaction set in the
present study that makes finite-size aggregates less stable relative to the
bulk condensed phase.

Figure 7: Temperature $T$ versus coexisting volume fraction, $\phi$, for
diblock chains of type H7T5, H10T5, and H19T5, respectively from top to
bottom. The lines are fitted to the points as explained in the text.
Statistical uncertainties are smaller than symbol size.

###  III.4 Single Bead Position Dependence

Additional calculations were performed for chains of length $r=8$ that
consisted of beads of the same character, except for a single “different” bead
placed at variable locations, from the end to the middle of the chain. Thus,
the architectures were HpTH7-p or TpHT7-p, where $p$ is the number of beads
prior to the position of the bead of different character and can range between
0 and $r-1$. Results for the critical temperature of such systems are shown in
Fig. 8. As seen in the bottom part of the figure, moving a solvophobic bead
from the end of the chain to the middle where it can less readily interact
with beads in other chains, results in a slight decrease of the critical
temperature, of around $-0.8$%. Conversely, as seen in the top part of the
figure, moving a solvophilic bead from the end of a chain to the middle, where
is less disruptive to solvophobic bead attractions across chains, results in
an increase in the critical temperature, of significantly higher magnitude,
around $2.5\%$ for this system. For comparison, in Ref. 16, a 5% increase in
critical temperature was seen for chains of length $r=20$ when a single
solvophilic bead was moved from the end to the middle of an otherwise
solvophobic chain. Thus, the magnitude of the effect seems to be comparable,
even though different models were used in the two studies (continuum versus
lattice model).

The effects on the critical volume fraction of moving a single bead of
opposite character along a chain were quite small, within statistical
uncertainties of the corresponding value in almost all cases. These values are
listed in the additional material provided online, as explained in the Data
Availability Statement.

Figure 8: Critical temperature, $T_{c}$, as a function of the distance from
the end, $p$, of a single “different” bead in chains with overall length
$r=8$. The general structures are HpTH7-p (bottom) and TpHT7-p (top). Data
have been reflected with respect to the midpoint of the chain due to symmetry.
Statistical uncertainties are smaller than symbol size.

###  III.5 Overall Architecture Effects

The final set of comparisons was made using a variety of chain architectures
for chains with $r=8$, all of the same composition, specifically 50%
solvophobic. There are 38 distinct sequences of 8-mers with 4 T beads and 4 H
beads, when one takes into account front-to-back symmetric sequences. Fig. 9
displays the critical volume fraction plotted against the critical temperature
for a selected subset of the possible sequences for such chains. There is a
good amount of spread in both critical temperatures and critical volume
fractions of the different architectures. Certain features emerge clearly: for
example, the lowest by far critical volume fraction corresponds to the diblock
sequence, HHHHTTTT. On the other hand, this sequence has an unremarkable
critical temperature in the “middle of the pack.” The tendency of more blocky
sequences to possess lower critical volume fractions was also observed in Ref.
17, where it was determined that sufficient blocky sequences form finite-size
micelles rather than phase separating, because of the full-contrast
interaction set used in the prior study.

The highest critical temperatures is observed here for the symmetric triblock
sequence, TTHHHHTT, in which the solvophobic beads are at the chain ends, thus
available to interact with solvophobic beads in other chains. Conversely, the
symmetric triblock sequence, HHTTTTHH, has the lowest critical temperature of
all architectures, since it has “burried” solvophobic groups away from the
chain ends. This pattern is seen more generally – H groups at the chain ends
are associated with low critical temperatures, whereas T groups with higher.
Chains with alternating bead types (e.g., HTHTHTHT), or similar (e.g.,
THTHHTHT) have the highest critical volume fractions, suggesting more
efficient packing in the condensed state.

Figure 9: Critical volume fraction, $\phi_{c}$ as a function of the critical
temperature, $T_{c}$, for sequences with 50% H groups, $r=8$. Statistical
uncertainties for the critical volume fractions are shown when larger than
symbol size. Statistical uncertainties for the critical temperatures are
smaller than symbol size.

##  IV Conclusions

This work focused on the accurate determination of critical properties for
short, oligomeric chains consisting of two types of beads, solvophobic and
solvophilic. A previously proposed lattice model was used, with parameters
modified to prevent formation of finite-size clusters that disrupt phase
transitions into bulk macroscopic liquid phases. While conceptually simple,
the model reproduces many features of the experimentally determined behavior
of polymer/ solvent systems, specifically with respect to the scaling of
critical parameter with chain length.Dobashi, Nakata, and Kaneko (1980) The
critical temperature and critical volume fraction pass through a minimum when
a solvophilic segment is added to a solvophobic block of a chain. The location
of this minimum with respect to the size of the solvophilic block is different
for the two properties, reflecting differences in the physical mechanisms
responsible for the reduction in the critical temperature versus the critical
volume fraction. The position of a single bead of solvophilic or solvophobic
character within a chain composed of opposite-type monomers leads to a maximum
or minimum, respectively, in the critical temperature. There is no appreciable
change in the critical volume fraction for these systems. When examining a
range of architectures at a fixed overall chain length and composition, there
are significant variations of the critical temperature and volume fraction.
Architectures with solvophobic beads near the chain ends have higher critical
temperatures and diblock chains have the lowest critical volume fraction.

One open question that could be addressed in future studies is the character
of the transition from simple phase behavior (with no aggregation) observed in
the current study that used the reduced-contrast interaction set, with the
observed formation of finite-size aggregates seen in previous studies of
related models with a higher-contrast interaction set. Most likely, for
specific architectures and interaction parameters, one should be able to
observe both phase separation and aggregation in the same system, as seen in a
prior computational studyPanagiotopoulos (2023) and also
experimentally.Taylor, Priestley, and Register (2024) The scaling of the
location of the transition with respect to chain length, composition, and
strength of interactions would be of interest.

The present work has implications on the design of synthetic polymers with
desired phase separation properties and in understanding biomolecular phase
separation. It could also be useful in providing data for the development of
refined theoretical models for heteropolymer and intrinsically disordered
protein phase behavior. These models could be based on molecularly informed
field theoreticNguyen  _et al._ (2023) or other variational methods.Sawle and
Ghosh (2015) Another possibility would be to utilize machine-learning methods
(e.g.,Taneja and Lasker (2024)) to correlate chain architecture and the phase
behavior. This latter approach would require much larger datasets for training
the machine learning models than what was generated in the present work, but
this should be computationally feasible with the methods used here.

## Acknowledgements

Financial support for this work was provided by the Princeton Center for
Complex Materials (PCCM), a U.S. National Science Foundation Materials
Research Science and Engineering Center (Award DMR-2011750).

## Data Availability Statement

Computer codes used in this work, example input and output files, information
on the runs performed, and numerical data for the critical points and phase
coexistence volume fractions are freely available for download from the
Princeton Data Commons repository, at DOI 10.34770/2xrn-jt41.

## References

## References

  * Metropolis  _et al._ (1953) N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, “Equation of state calculations by fast computing machines,” J. Chem. Phys. 21, 1087 (1953).
  * Wood (1968) W. W. Wood, “Monte Carlo calculations for hard disks in isothermal-isobaric ensemble,” J. Chem. Phys. 48, 415 (1968).
  * Norman and Filinov (1969) G. E. Norman and V. S. Filinov, “Investigations of phase transitions by a Monte-Carlo method,” High Temp. 7, 216 (1969).
  * Frenkel and Ladd (1984) D. Frenkel and A. J. C. Ladd, “New Monte-Carlo method to compute the free-energy of arbitrary solids - application to the fcc and hcp phases of hard-spheres,” J. Chem. Phys. 81, 3188 (1984).
  * Panagiotopoulos (1987) A. Z. Panagiotopoulos, “Direct determination of phase coexistence properties of fluids by Monte-Carlo simulation in a new ensemble,” Molec. Phys. 61, 813 (1987).
  * Ferrenberg and Swendsen (1989) A. M. Ferrenberg and R. H. Swendsen, “Optimized Monte Carlo data analysis,” Phys. Rev. Lett. 63, 1195 (1989).
  * Larson, Scriven, and Davis (1985) R. G. Larson, L. E. Scriven, and H. T. Davis, “Monte Carlo simulation of model amphiphile-oil-water systems,” J. Chem. Phys. 83, 2411 (1985).
  * Larson (1988) R. G. Larson, “Monte-Carlo lattice simulation of amphiphilic systems in 2 and 3 dimensions,” J. Chem. Phys. 89, 1642 (1988).
  * Floriano, Caponetti, and Panagiotopoulos (1999) M. A. Floriano, E. Caponetti, and A. Z. Panagiotopoulos, “Micellization in model surfactant systems,” Langmuir 15, 3143 (1999).
  * Brangwynne  _et al._ (2009) C. P. Brangwynne, C. R. Eckmann, D. S. Courson, A. Rybarska, C. Hoege, J. Gharakhani, F. Jülicher, and A. A. Hyman, “Germline P granules are liquid droplets that localize by controlled dissolution/condensation,” Science 324, 1729 (2009).
  * Banani  _et al._ (2017) S. F. Banani, H. O. Lee, A. A. Hyman, and M. K. Rosen, “Biomolecular condensates: organizers of cellular biochemistry,” Nature Rev. Molec. Cell Bio. 18, 285 (2017).
  * Das  _et al._ (2020) S. Das, Y.-H. Lin, R. M. Vernon, J. D. Forman-Kay, and H. S. Chan, “Comparative roles of charge, pi, and hydrophobic interactions in sequence-dependent phase separation of intrinsically disordered proteins,” Proc. Natl. Acad. Sci. U.S.A. 117, 28795 (2020).
  * Rana  _et al._ (2024) U. Rana, K. Xu, A. Narayanan, M. T. Walls, A. Z. Panagiotopoulos, J. L. Avalos, and C. P. Brangwynne, “Asymmetric oligomerization state and sequence patterning can tune multiphase condensate miscibility,” Nature Chem. (2024), 10.1038/s41557-024-01456-6.
  * Destefano  _et al._ (2024) A. J. Destefano, S. D. Mengel, M. W. Bates, S. Jiao, M. S. Shell, S. Han, and R. A. Segalman, “Control over conformational landscapes of polypeptoids by monomer sequence patterning,” Macromolecules 57, 1469 (2024).
  * Taylor, Priestley, and Register (2024) L. W. Taylor, R. D. Priestley, and R. A. Register, “Control of solution phase behavior through block-random copolymer sequence,” Macromolecules 57, 916 (2024).
  * Statt  _et al._ (2020) A. Statt, H. Casademunt, C. P. Brangwynne, and A. Z. Panagiotopoulos, “Model for disordered proteins with strongly sequence-dependent liquid phase behavior,” J. Chem. Phys. 152, 075101 (2020).
  * Rana, Brangwynne, and Panagiotopoulos (2021) U. Rana, C. P. Brangwynne, and A. Z. Panagiotopoulos, “Phase separation vs aggregation behavior for model disordered proteins,” J. Chem. Phys. 155, 125101 (2021).
  * Panagiotopoulos (2023) A. Z. Panagiotopoulos, “Phase separation and aggregation in multiblock chains,” J. Chem. Phys. 158, 154901 (2023).
  * Rosenbluth and Rosenbluth (1955) M. N. Rosenbluth and A. W. Rosenbluth, “Monte Carlo calculation of the average extension of molecular chains,” J. Chem. Phys. 23, 356 (1955).
  * Panagiotopoulos, Wong, and Floriano (1998) A. Z. Panagiotopoulos, V. Wong, and M. A. Floriano, “Phase equilibria of lattice polymers from histogram reweighting Monte Carlo simulations,” Macromolecules 31, 912 (1998).
  * Bruce and Wilding (1992) A. D. Bruce and N. B. Wilding, “Scaling fields and universality of the liquid-gas critical point,” Phys. Rev. Lett. 68, 193 (1992).
  * Wilding and Bruce (1992) N. B. Wilding and A. D. Bruce, “Density fluctuations and field mixing in the critical fluid,” J. Phys.-Condens. Mat. 4, 3087 (1992).
  * Tsypin and Blöte (2000) M. M. Tsypin and H. W. J. Blöte, “Probability distribution of the order parameter for the three-dimensional Ising-model universality class: A high-precision Monte Carlo study,” Phys. Rev. E 62, 73 (2000).
  * Orkoulas and Panagiotopoulos (1999) G. Orkoulas and A. Z. Panagiotopoulos, “Phase behavior of the restricted primitive model and square-well fluids from Monte Carlo simulations in the grand canonical ensemble,” J. Chem. Phys. 110, 1581 (1999).
  * Flory (1953) P. J. Flory,  _Principles of Polymer Chemistry_ (Cornell University Press, Ithaca, NY, 1953).
  * Panagiotopoulos, Floriano, and Kumar (2002) A. Z. Panagiotopoulos, M. A. Floriano, and S. K. Kumar, “Micellization and phase separation of diblock and triblock model surfactants,” Langmuir 18, 2940 (2002).
  * Dobashi, Nakata, and Kaneko (1980) T. Dobashi, M. Nakata, and M. Kaneko, “Coexistence curve of polystyrene in methylcyclohexane. II. Comparison of coexistence curves observed and calculated from classical free energy,” J. Chem. Phys. 72, 6692 (1980).
  * Nguyen  _et al._ (2023) M. V. T. Nguyen, K. Dolph, K. T. Delaney, K. Shen, N. Sherck, S. Koehler, R. Gupta, M. B. Francis, M. S. Shell, and G. H. Fredrickson, “Molecularly informed field theory for estimating critical micelle concentrations of intrinsically disordered protein surfactants,” J. Chem. Phys. 159, 244904 (2023).
  * Sawle and Ghosh (2015) L. Sawle and K. Ghosh, “A theoretical method to compute sequence dependent configurational properties in charged polymers and proteins,” J. Chem. Phys. 143, 085101 (2015).
  * Taneja and Lasker (2024) I. Taneja and K. Lasker, “Machine-learning-based methods to generate conformational ensembles of disordered proteins,” Biophys. J. 123, 101 (2024).


# Chain rules for quantum channels

Mario Berta  Department of Computing, Imperial College London, UK
AWS Center for Quantum Computing, Pasadena, USA
California Institute of Technology, Pasadena, USA Marco Tomamichel  Center for
Quantum Technologies &
Department of Electrical and Computer Engineering
National University of Singapore, Singapore

###### Abstract

Divergence chain rules for channels relate the divergence of a pair of channel
inputs to the divergence of the corresponding channel outputs. An important
special case of such a rule is the data-processing inequality, which tells us
that if the same channel is applied to both inputs then the divergence cannot
increase. Based on direct matrix analysis methods, we derive several Rényi
divergence chain rules for channels in the quantum setting. Our results
simplify and in some cases generalise previous derivations in the literature.

###### Index Terms:

Quantum information measures, quantum Rényi divergences, quantum entropy
inequalities, quantum statistics, quantum channel discrimination.

##  I Introduction

A family of fundamental information measures is given by the Rényi divergences
[34], defined for probabilit distributions $P,Q$ over finite alphabets
$\mathcal{X}$ and $\alpha\in(1,\infty)$ as

$\displaystyle D_{\alpha}(P\|Q)$ $\displaystyle=\frac{1}{\alpha-1}\log
Q_{\alpha}(P\|Q)\quad\text{with}$ $\displaystyle Q_{\alpha}(P\|Q)$
$\displaystyle=\sum_{x\in\mathcal{X}}P(x)\left(\frac{P(x)}{Q(x)}\right)^{\alpha-1}\,,$
(1)

if $P$ is absolutely continuous with respect to $Q$ and as $+\infty$
otherwise. This definition extends via the respective limits to $\alpha\to 1$
(the Kulback-Leibler divergence) and $\alpha\to\infty$ (the max-divergence).
They are respectively given as

$\displaystyle D(P\|Q)=\ $ $\displaystyle
D_{1}(P\|Q)=\sum_{x\in\mathcal{X}}P(x)\log\frac{P(x)}{Q(x)}\quad\textrm{and}$
(2) $\displaystyle D_{\infty}(P\|Q)=\max_{x}\log\frac{P(x)}{Q(x)}\,.$ (3)

For $\alpha\in(0,1)$ we rewrite the sum as
$\sum_{x\in\mathcal{X}}P(x)^{\alpha}Q(x)^{1-\alpha}$, indicating that
absolutely continuity of $P$ with respect to $Q$ is not necessary to keep
$D_{\alpha}$ finite for $\alpha<1$.

For distributions over bipartite alphabets $\mathcal{X}\times\mathcal{Y}$ the
Rényi divergences enjoy the chain rule property

$\displaystyle D_{\alpha}(P_{XY}\|Q_{XY})$ $\displaystyle\qquad\leq
D_{\alpha}(P_{X}\|Q_{X})+\max_{x\in\mathcal{X}}D_{\alpha}(P_{Y|X=x}\|Q_{Y|X=x}),$
(4)

displaying how to split up Rényi divergences with respect to multipartite
constituents. Eq. (I) is important in statistics because it can be seen as the
crucial entropy inequality to quantify the asymptotic error exponents of
adaptive channel discrimination [22, 33, 7, 42, 35].

In this report, we give various quantum generalizations of Eq. (I), where
probability distributions and conditional probability distributions are
replaced with positive operators and positive maps, respectively. Some of
these quantum chain rules have been derived previously in the context of
adaptive quantum channel discrimination and related tasks [10, 15, 14, 17].
The proofs of these earlier results are based on involved techniques from
either complex interpolation theory [10], smooth entropies [15], or convex
optimization theory [17]. In contrast, our derivations are based on direct
matrix analysis methods.

The remainder of this report is structured as follows. First, we introduce in
Section II our notation and state some mathematical facts from quantum
information theory. We then state our main result in Section III, after which
we sketch applications to quantum channel discrimination in Section IV. We
conclude with some open question in Section V.

##  II Notation and methods

We consider quantum systems described by finite-dimensional inner product
spaces labelled by $\mathcal{H}$. The set $\mathcal{S}(\mathcal{H})$ of
quantum states is given by positive semi-definite trace one operators on
$\mathcal{H}$. For $\rho,\sigma\in\mathcal{S}(\mathcal{H})$, we denote by
$\rho\ll\sigma$ that the support of $\rho$ is contained in the support of
$\sigma$, and inverses of operators are understood as generalized inverses on
the support of the operator. Quantum channels are completely positive and
trace preserving maps from $\mathcal{S}(\mathcal{H})$ to
$\mathcal{S}(\mathcal{H}^{\prime})$. Bipartite quantum systems are described
by tensor product spaces $\mathcal{H}_{A}\otimes\mathcal{H}_{R}$ with
correspondingly labelled quantum states
$\rho_{AR}\in\mathcal{S}(\mathcal{H}_{A}\otimes\mathcal{H}_{R})$.

An important tool to lift entropy inequalities from the classical to the
quantum setting is given by the asymptotic spectral pinching method (see
[21]). Namely, the spectral pinching map $P_{\sigma}(\cdot)$ with respect to
$\sigma\in\mathcal{S}(\mathcal{H})$ is given as

$\displaystyle P_{\sigma}\;:\;\rho\mapsto\sum_{\lambda}P_{\lambda}\rho
P_{\lambda}\,,$ (5)

where $\sigma=\sum_{\lambda}\lambda P_{\lambda}$ with eigenvalues
$\lambda\in\text{spec}(\sigma)$ in the spectrum of $\sigma$ and corresponding
mutually orthogonal eigenprojectors $P_{\lambda}$. An alternative
representation of the pinching map is

$\displaystyle
P_{\sigma}(\cdot)=\int\mu(\mathrm{d}t)\,\sigma^{it}(\cdot)\sigma^{-it}$ (6)

for some probability measure $\mu$ on $\mathbb{R}$ [37, Lemma 2.1]. We have
that $P_{\sigma}(\rho)$ commutes with $\sigma$,
$\text{Tr}[P_{\sigma}(\rho)\sigma]=\text{Tr}[\rho\sigma]$, and the pinching
inequality [21]

$\displaystyle
P_{\sigma}(\rho)\succeq\big{|}\text{spec}(\sigma)\big{|}^{-1}\rho\,,$ (7)

where here and henceforth $\succeq$ denotes the Löwner order, and
$\big{|}\text{spec}(\sigma)\big{|}$ the number of distinct eigenvalues
$\sigma$. Eq. (7) becomes asymptotically powerful as
$\left|\text{spec}\left(\sigma^{\otimes n}\right)\right|=O(\text{poly}(n))$ by
a type-counting argument [12].

The generalization of the Rényi divergences from Eq. (I) to the quantum
setting is not unique. Rather, one asks for any quantum Rényi divergence

$\displaystyle\mathbb{D}_{\alpha}(\rho\|\sigma)=\frac{1}{\alpha-1}\log\mathbb{Q}_{\alpha}(\rho\|\sigma)\quad\text{on
$\rho,\sigma\in\mathcal{S}(\mathcal{H})$}$ (8)

to satisfy the following properties:

  1. 1.

whenever $\rho$ and $\sigma$ commute, $\mathbb{Q}_{\alpha}(\rho\|\sigma)$
simplifies to the corresponding classical functions in Eq. (I) and, moreover,
for classical quantum states $\rho=\sum_{x}p_{x}|x\rangle\\!\langle
x|\otimes\rho_{x}$ and $\sigma=\sum_{x}q_{x}|x\rangle\\!\langle
x|\otimes\sigma_{x}$, we have

$\displaystyle\mathbb{Q}_{\alpha}(\rho\|\sigma)=\sum_{x}p_{x}^{\alpha}q_{x}^{1-\alpha}\mathbb{Q}_{\alpha}(\rho_{x}\|\sigma_{x})\,;$
(9)

  2. 2.

one has the data processing inequality

$\displaystyle\mathbb{D}_{\alpha}(\rho\|\sigma)\geq\mathbb{D}_{\alpha}(\mathcal{E}(\rho)\|\mathcal{E}(\sigma))$
(10)

for quantum channels
$\mathcal{E}:\mathcal{S}(\mathcal{H})\to\mathcal{S}(\mathcal{H}^{\prime})$.

The smallest such quantum divergences for $\alpha>0$ are the measured Rényi
divergence [13, 23, 4, 3]

$\displaystyle
D_{\alpha}^{M}(\rho\|\sigma)=\sup_{\mathcal{M}}D_{\alpha}(\mathcal{M}(\rho)\|\mathcal{M}(\sigma))$
(11)

for the rank-one projective measurement maps
$\mathcal{M}(\cdot)=\sum_{x}\langle{x}|\cdot|{x}\rangle|x\rangle\\!\langle x|$
with the supremum over orthonormal bases $\\{|{x}\rangle\\}_{x}$ of
$\mathcal{H}$.111Including general measurement maps leads to the same
quantities [4, Theorem 4]. On the other hand, for $\alpha\in(0,2]$ the largest
such quantum divergences are the geometric Rényi divergences [29, 14]

$\displaystyle\widehat{D}_{\alpha}(\rho\|\sigma)$
$\displaystyle=\frac{1}{\alpha-1}\log\widehat{Q}_{\alpha}(\rho\|\sigma)\quad\text{with}$
$\displaystyle\widehat{Q}_{\alpha}(\rho\|\sigma)$
$\displaystyle=\text{Tr}\left[\sigma^{1/2}\left(\sigma^{-1/2}\rho\sigma^{-1/2}\right)^{\alpha}\sigma^{1/2}\right]\;\text{for
$\rho\ll\sigma$}$ (12)

and the respective limit for $\alpha\to 1$ as [2, 28]

$\displaystyle\widehat{D}_{1}(\rho\|\sigma)=\mathrm{Tr}\left[\rho\log\left(\rho^{1/2}\sigma^{-1}\rho^{1/2}\right)\right]\,.$
(13)

The formal definition from Eq. (II) can be extended to $\alpha>0$, but only
for $\alpha\in(0,2]$ the geometric Rényi divergences have the following
operational characterisation.

######  Lemma 1 (Matsumoto [28, 29]).

Let $\rho,\sigma\in\mathcal{S}(\mathcal{H})$ and $\alpha\in(0,2]$. Then, we
have

$\displaystyle\widehat{D}_{\alpha}(\rho\|\sigma)=\inf_{(\Gamma,P,Q)}D_{\alpha}(P\|Q)\,,$
(14)

where the infimum is over probability distributions $P,Q$ over finite
alphabets $\mathcal{X}$, and positive trace preserving maps with
$\Gamma(P)=\rho$ and $\Gamma(Q)=\sigma$.

In fact, for the spectral decomposition
$\sigma^{-1/2}\rho\sigma^{-1/2}=\sum_{x\in\mathcal{X}}\lambda_{x}\Pi_{x}$ the
infimum is achieved as (see [39, Sec. 4.2.3])

$\displaystyle\Gamma(\cdot)=\sum_{x\in\mathcal{X}}\frac{\langle{x}|\cdot|{x}\rangle}{Q(x)}\sigma^{1/2}\Pi_{x}\sigma^{1/2}\,,$
(15) $\displaystyle P(x)=\lambda_{x}Q(x),\quad
Q(x)=\text{Tr}[\sigma\Pi_{x}]\,.$ (16)

Additionally, for the axiomatic extension of the classical Rényi divergences
from Eq. (I), the smallest quantum divergences that are also

  1. 3.

additive on product states222Divergences with properties 1) – 3) are also
termed relative entropies [19], also see [26] for more details.

for $\alpha\in[1/2,\infty]$ are the sandwiched Rényi divergences [31, 43]

$\displaystyle\widetilde{D}_{\alpha}(\rho\|\sigma)$
$\displaystyle=\frac{1}{\alpha-1}\log\widetilde{Q}_{\alpha}(\rho\|\sigma)\quad\text{with}$
$\displaystyle\widetilde{Q}_{\alpha}(\rho\|\sigma)$
$\displaystyle=\text{Tr}\left[\left(\sigma^{\frac{1-\alpha}{2\alpha}}\rho\sigma^{\frac{1-\alpha}{2\alpha}}\right)^{\alpha}\right]\;\text{for
$\rho\ll\sigma$}$ (17)

and the respective limits for $\alpha\to 1$ and $\alpha\to\infty$ as

$\displaystyle\widetilde{D}_{1}(\rho\|\sigma)$
$\displaystyle=\mathrm{Tr}\left[\rho\left(\log\rho-\log\sigma\right)\right]\quad\text{and}$
(18) $\displaystyle\widetilde{D}_{\infty}(\rho\|\sigma)$
$\displaystyle=\inf\left\\{\lambda\in\mathbb{R}\middle|\rho\preceq
2^{\lambda}\sigma\right\\}\,,$ (19)

respectively. We note that
$\widetilde{D}_{1}(\rho\|\sigma)\neq\widehat{D}_{1}(\rho\|\sigma)$ unless
$\rho$ and $\sigma$ commute. Moreover, the formal definition from Eq. (II) can
be extended to $\alpha>0$, but as showcased in [4, Theorem 7] only for
$\alpha\in[1/2,\infty]$ one has the data-processing inequality from Eq. (10).

We record the following noteworthy property.

######  Lemma 2.

Let $\rho,\sigma\in\mathcal{S}(\mathcal{H})$, $\mathcal{E},\mathcal{F}$ be
positive maps from $\mathcal{B}(\mathcal{H})$ to
$\mathcal{B}(\mathcal{H}^{\prime})$, and $\alpha\in(0,\infty]$. Then, we have

$\displaystyle\widetilde{Q}_{\alpha}(\mathcal{E}(\rho)\|\mathcal{F}(\sigma))\leq\big{|}\operatorname{spec}(\sigma)\big{|}^{\alpha}\widetilde{Q}_{\alpha}\big{(}\mathcal{E}(\mathcal{P}_{\sigma}(\rho))\big{\|}\mathcal{F}(\sigma)\big{)}\,.$
(20)

###### Proof.

We follow [39, Eq. (4.52)]. From the pinching inequality in Eq. (7) together
with the positivity of $\mathcal{E}$ we get

$\displaystyle\mathcal{E}(\rho)\preceq\big{|}\text{spec}(\sigma)\big{|}\mathcal{E}(\mathcal{P}_{\sigma}(\rho))\,.$
(21)

Multiplying this from the left and the right with the positive operator
$\mathcal{F}(\sigma)^{\frac{1-\alpha}{2\alpha}}$ leads to

$\displaystyle\mathcal{F}(\sigma)^{\frac{1-\alpha}{2\alpha}}\mathcal{E}(\rho)\mathcal{F}(\sigma)^{\frac{1-\alpha}{2\alpha}}$
$\displaystyle\preceq\big{|}\text{spec}(\sigma)\big{|}\mathcal{F}(\sigma)^{\frac{1-\alpha}{2\alpha}}\mathcal{E}(\mathcal{P}_{\sigma}(\rho))\mathcal{F}(\sigma)^{\frac{1-\alpha}{2\alpha}}\,.$
(22)

The claim then follows by the monotonicity of the trace of the monotone
function $t\mapsto t^{\alpha}$. ∎

Finally, other definitions of interest include the Petz Rényi divergences
[32], the $\alpha$-$z$-divergences [1], and divergences based on convex
optimization methods [17, 9]. We do not directly employ these further
definitions in our work, but note that [17, Corollary 5.2] proves quantum
extensions of the chain rule from Eq. (I) in terms of divergences based on
convex optimization methods.

All of aforementioned quantum Rényi divergences are extended to quantum
channels $\mathcal{E},\mathcal{F}$ as [11, 27]

$\displaystyle\mathbb{D}_{\alpha}\big{(}\mathcal{E}\big{\|}\mathcal{F}\big{)}=\sup_{\rho\in\mathcal{S}(\mathcal{H})}\mathbb{D}_{\alpha}\big{(}\mathcal{E}(\rho)\big{\|}\mathcal{F}(\rho)\big{)}$
(23)

or alternatively as the (generally larger) stabilized version [42, 7]

$\displaystyle\mathbb{D}_{\alpha}^{\text{stab}}\big{(}\mathcal{E}\big{\|}\mathcal{F}\big{)}$
$\displaystyle=\sup_{\rho\in\mathcal{S}(\mathcal{H}\otimes\mathcal{H})}\mathbb{D}_{\alpha}\big{(}(\mathcal{E}_{A}\otimes\mathcal{I}_{R})(\rho_{AR})\big{\|}(\mathcal{F}_{A}\otimes\mathcal{I}_{R})(\rho_{AR})\big{)}\,,$
(24)

where $\mathcal{I}_{R}$ denotes the identity channel on
$\mathcal{S}(\mathcal{H}_{R})$, and the supremum is over all
$\rho_{AR}\in\mathcal{S}(\mathcal{H}_{A}\otimes\mathcal{H}_{R})$ and finite
dimensional $\mathcal{H}_{R}$.333One might choose $|R|=|A|$ without loss of
generality as, e.g., discussed after [27, Definition II.2]. In the general
quantum case, the divergences
$\mathbb{D}_{\alpha}\big{(}\mathcal{E}\big{\|}\mathcal{F}\big{)}$ and
$\mathbb{D}_{\alpha}^{\text{stab}}\big{(}\mathcal{E}\big{\|}\mathcal{F}\big{)}$
are not additive on product channels (see, e.g., [20, 15]), and thus one
defines the corresponding regularized versions

$\displaystyle\mathbb{D}_{\alpha}^{\infty}\big{(}\mathcal{E}\big{\|}\mathcal{F}\big{)}$
$\displaystyle=\lim_{n\to\infty}\frac{1}{n}\mathbb{D}_{\alpha}\big{(}\mathcal{E}^{\otimes
n}\big{\|}\mathcal{F}^{\otimes n}\big{)}$ (25)
$\displaystyle\mathbb{D}_{\alpha}^{\text{stab},\infty}\big{(}\mathcal{E}\big{\|}\mathcal{F}\big{)}$
$\displaystyle=\lim_{n\to\infty}\frac{1}{n}\mathbb{D}_{\alpha}^{\text{stab}}\big{(}\mathcal{E}^{\otimes
n}\big{\|}\mathcal{F}^{\otimes n}\big{)}\,.$ (26)

Note that these limits are well-defined whenever
$\widetilde{D}_{\infty}(\mathcal{E}\|\mathcal{F})$ is finite. To see this
consider the sequences
$f_{n}=\frac{1}{n}\mathbb{D}_{\alpha}\big{(}\mathcal{E}^{\otimes
n}\big{\|}\mathcal{F}^{\otimes n}\big{)}$ where $n\in\mathbb{N}$. We first
note that $f_{n}$ is bounded from above by
$\widetilde{D}_{\infty}(\mathcal{E}\|\mathcal{F})$ for all $n$ since
$\mathbb{D}_{\alpha}(\rho\|\sigma)\leq\widehat{D}_{\alpha}(\rho\|\sigma)\leq\widetilde{D}_{\infty}(\rho\|\sigma)$
for any pair of states (see, e.g., [39]) and, furthermore,
$\widetilde{D}_{\infty}(\mathcal{E}\|\mathcal{F})$ is additive for tensor
product channels. We further get $f_{n+1}\geq\frac{n}{n+1}f_{n}$ by choosing
product inputs and hence the function is monotonic asymptotically and there
can only be a single accumulation point.

##  III Main results

Our main result is the following Rényi divergences based simple meta chain
rule for quantum channels.

######  Proposition 3.

Let $\rho,\sigma\in\mathcal{S}(\mathcal{H})$, $\mathcal{E},\mathcal{F}$ be
positive maps from $\mathcal{B}(\mathcal{H})$ to
$\mathcal{B}(\mathcal{H}^{\prime})$, and $\alpha\in(0,\infty]$. Then, we have

$\displaystyle\mathbb{D}_{\alpha}\big{(}\mathcal{E}(\rho)\big{\|}\mathcal{F}(\sigma)\big{)}\leq\widehat{D}_{\alpha}(\rho\|\sigma)+\mathbb{D}_{\alpha}\big{(}\mathcal{E}\big{\|}\mathcal{F}\big{)}\,,$
(27)

where $\mathbb{D}_{\alpha}$ denotes any Rényi divergence of order $\alpha$.

###### Proof.

Using Matsumoto’s construction (Lemma 1), we write

$\displaystyle\mathcal{E}(\rho)$
$\displaystyle=\sum_{x}\hat{p}_{x}\,\mathcal{E}\big{(}\underbrace{\Gamma\big{(}|x\rangle\\!\langle
x|\big{)}}_{=:\rho^{x}}\big{)}\quad\textrm{and}$ (28)
$\displaystyle\mathcal{F}(\sigma)$
$\displaystyle=\sum_{x}\hat{q}_{x}\,\mathcal{F}\big{(}\underbrace{\Gamma\big{(}|x\rangle\\!\langle
x|\big{)}}_{=\rho^{x}}\big{)}\,.$ (29)

We then use the data-processing inequality to get

$\displaystyle\mathbb{D}_{\alpha}\big{(}\mathcal{E}(\rho)\big{\|}\mathcal{F}(\sigma)\big{)}$
$\displaystyle\leq\mathbb{D}_{\alpha}\left(\sum_{x}\hat{p}_{x}|x\rangle\\!\langle
x|\otimes\mathcal{E}\left(\rho^{x}\right)\middle\|\sum_{x}\hat{q}_{x}|x\rangle\\!\langle
x|\otimes\mathcal{F}\left(\rho^{x}\right)\right)$ (30)
$\displaystyle=\frac{1}{\alpha-1}\log\left(\sum_{x}\hat{p}_{x}^{\alpha}\hat{q}_{x}^{1-\alpha}2^{(\alpha-1)\mathbb{D}_{\alpha}\left(\mathcal{E}\left(\rho^{x}\right)\middle\|\mathcal{F}\left(\rho^{x}\right)\right)}\right)$
(31) $\displaystyle\leq
D_{\alpha}(\hat{p}\|\hat{q})+\max_{x}\mathbb{D}_{\alpha}\left(\mathcal{E}\left(\rho^{x}\right)\middle\|\mathcal{F}\left(\rho^{x}\right)\right)\,,$
(32)

which implies the statement we set out to prove. ∎

The corresponding geometric chain rules for $\alpha\in(0,2]$ as

$\displaystyle\widehat{D}_{\alpha}\big{(}\mathcal{E}(\rho)\big{\|}\mathcal{F}(\sigma)\big{)}\leq\widehat{D}_{\alpha}(\rho\|\sigma)+\widehat{D}_{\alpha}\big{(}\mathcal{E}\big{\|}\mathcal{F}\big{)}\,,$
(33)

generalize the stabilized geometric chain rules from [14, Lemma 7] that were
proven with different techniques for completely positive maps
$\mathcal{E}_{A},\mathcal{F}_{A}$ and $\alpha\in(1,2]$ as

$\displaystyle\widehat{D}_{\alpha}\big{(}(\mathcal{E}_{A}\otimes\mathcal{I}_{R})(\rho_{AR})\big{\|}(\mathcal{F}_{A}\otimes\mathcal{I}_{R})(\sigma_{AR})\big{)}$
$\displaystyle\qquad\leq\widehat{D}_{\alpha}(\rho_{AR}\|\sigma_{AR})+\widehat{D}_{\alpha}^{\text{stab}}\big{(}\mathcal{E}_{A}\big{\|}\mathcal{F}_{A}\big{)}\,,$
(34)

as well as their extension to $\alpha\in(0,1)$ [25, Proposition 45]. We note
that the stabilized version of the channel divergence was required in (III),
whereas our Proposition 3 does not require stabilization.

To obtain chain rules without featuring geometric Rényi divergences, we can
apply pre-processing maps in terms of rank-one projective measurements
$\Lambda(\cdot)$. Namely, for $\rho,\sigma\in\mathcal{S}(\mathcal{H})$ we have
that the pre-processed inputs $\Lambda(\rho),\Lambda(\sigma)$ commute and thus
we find the chain rule

$\displaystyle\mathbb{D}_{\alpha}\big{(}\mathcal{E}(\Lambda(\rho))\big{\|}\mathcal{F}(\Lambda(\sigma))\big{)}\leq
D_{\alpha,M}(\rho\|\sigma)+\mathbb{D}_{\alpha}\big{(}\mathcal{E}\big{\|}\mathcal{F}\big{)}\,.$
(35)

We can then again eliminate the pre-processing map for the sandwiched Rényi
divergences with $\alpha\in(1,\infty]$ by choosing the overall rank-one
projective measurement map
$\Lambda(\cdot)=\mathcal{M}_{\sigma}(P_{\sigma}(\cdot))$ with the pinching map
$P_{\sigma}(\cdot)$ and the projective rank-one measurement map
$\mathcal{M}_{\sigma}(\cdot)$ in the joint eigenbasis of $\sigma$ and
$P_{\sigma}(\rho)$ — then leaving those states invariant. Applying Lemma 2 for
$\alpha\in(1,\infty]$ immediately gives

$\displaystyle\widetilde{D}_{\alpha}\big{(}\mathcal{E}(\Lambda(\rho))\big{\|}\mathcal{F}(\Lambda(\sigma))\big{)}\geq$
$\displaystyle\;\widetilde{D}_{\alpha}\big{(}\mathcal{E}(\rho)\big{\|}\mathcal{F}(\sigma)\big{)}$
$\displaystyle-\frac{\alpha}{\alpha-1}\log\big{|}\text{spec}(\sigma)\big{|}$
(36)

and thus we find the following sandwiched chain rule.

######  Corollary 4.

Let $\rho,\sigma\in\mathcal{S}(\mathcal{H})$, $\mathcal{E},\mathcal{F}$ be
positive maps from $\mathcal{B}(\mathcal{H})$ to
$\mathcal{B}(\mathcal{H}^{\prime})$, and $\alpha\in(1,\infty]$. Then, we have

$\displaystyle\widetilde{D}_{\alpha}\big{(}\mathcal{E}(\rho)\big{\|}\mathcal{F}(\sigma)\big{)}\leq$
$\displaystyle\;D_{\alpha,M}(\rho\|\sigma)+\widetilde{D}_{\alpha}\big{(}\mathcal{E}\big{\|}\mathcal{F}\big{)}$
$\displaystyle+\frac{\alpha}{\alpha-1}\log\big{|}\operatorname{spec}(\sigma)\big{|}\,.$
(37)

As a special case, applying above corollary to unital maps $\mathcal{F}$, the
identity matrix $\sigma=1$,444The definition for
$\mathbb{D}_{\alpha}(\rho\|\sigma)$ and its properties straightforwardly apply
to general positive operators $\sigma$ on $\mathcal{H}$ as well. and
$\rho\in\mathcal{S}(\mathcal{H})$, leads for $\alpha\in[1,\infty]$ to

$\displaystyle
H_{\alpha}(\mathcal{E}(\rho))-H_{\alpha}(\rho)\geq-\inf_{\mathcal{F}:\,\text{unital}}\widetilde{D}_{\alpha}(\mathcal{E}\|\mathcal{F})\,,$
(38)

featuring the quantum Rényi entropies
$H_{\alpha}(\rho)=\frac{1}{1-\alpha}\log\mathrm{Tr}\left[\rho^{\alpha}\right]$
and in particular the von Neumann entropy
$H_{1}(\rho)=-\mathrm{Tr}\left[\rho\log\rho\right]$ in the limit $\alpha\to
1$. This generalizes the well-known monotonicity property
$H_{\alpha}(\mathcal{E}(\rho))-H_{\alpha}(\rho)\geq 0$ for unital quantum
channels $\mathcal{E}$ to the Eq. (38) for generic positive maps.

Going back to the general case, but restricting our attention to tensor-stable
positive maps $\mathcal{E}$ and $\mathcal{F}$, such that $\mathcal{E}^{\otimes
n}$ and $\mathcal{F}^{\otimes n}$ are positive maps. Applying Corollary 4 to
$\rho^{\otimes n}$, $\sigma^{\otimes n}$, $\mathcal{E}^{\otimes n}$ and
$\mathcal{F}^{\otimes n}$, and using that
$\left|\operatorname{spec}\left(\sigma^{\otimes
n}\right)\right|\leq\text{poly}(n)$, we arrive at the corresponding
regularized sandwiched chain rule.

######  Corollary 5.

Let $\rho,\sigma\in\mathcal{S}(\mathcal{H})$, $\mathcal{E},\mathcal{F}$ be
tensor-stable positive maps from $\mathcal{B}(\mathcal{H})$ to
$\mathcal{B}(\mathcal{H}^{\prime})$, and $\alpha\in(1,\infty]$. Then, we have

$\displaystyle\widetilde{D}_{\alpha}\big{(}\mathcal{E}(\rho)\big{\|}\mathcal{F}(\sigma)\big{)}\leq\widetilde{D}_{\alpha}(\rho\|\sigma)+\widetilde{D}_{\alpha}^{\infty}\big{(}\mathcal{E}\big{\|}\mathcal{F}\big{)}\,.$
(39)

This can be compared to [10, Theorem III.1] of which a special case states for
the same range $\alpha\in(1,\infty]$ as in Corollary 5 — but for positive
trace preserving maps instead of tensor-stable maps — that

$\displaystyle\widetilde{D}_{\alpha}\big{(}\mathcal{E}(\rho)\big{\|}\mathcal{F}(\sigma)\big{)}\leq\widetilde{D}_{\alpha}(\rho\|\sigma)+\widetilde{D}_{\infty}\big{(}\mathcal{E}\big{\|}\mathcal{F}\big{)}\,.$
(40)

Applied to quantum channels, this is weaker than Corollary 5 as the sandwiched
Rényi divergences are monotone in $\alpha\in[1/2,\infty]$ and one has
$\widetilde{D}_{\infty}^{\infty}\big{(}\mathcal{E}\big{\|}\mathcal{F}\big{)}=\widetilde{D}_{\infty}\big{(}\mathcal{E}\big{\|}\mathcal{F}\big{)}$
[42, Proposition 10]. Corollary 5 also generalizes the stabilized sandwiched
chain rules from [17, Corollary 5.2] that hold for completely positive maps
$\mathcal{E}_{A},\mathcal{F}_{A}$, $|R|=|A|$, and $\alpha\in(1,\infty]$ as

$\displaystyle\widetilde{D}_{\alpha}\big{(}(\mathcal{E}_{A}\otimes\mathcal{I}_{R})(\rho_{AR})\big{\|}(\mathcal{F}_{A}\otimes\mathcal{I}_{R})(\sigma_{AR})\big{)}$
$\displaystyle\qquad\leq\widetilde{D}_{\alpha}(\rho_{AR}\|\sigma_{AR})+\widetilde{D}_{\alpha}^{\text{stab},\infty}\big{(}\mathcal{E}_{A}\big{\|}\mathcal{F}_{A}\big{)}$
(41)

and were proven with different techniques.

Note that for $\alpha\in(0,1]$ it is unfortunately unclear how to eliminate
the pre-processing map and directly regularizing Eq. (35) for general
$\alpha\in(0,\infty]$ just leads to

$\displaystyle\limsup_{n\to\infty}\sup_{\Lambda^{n}}\frac{1}{n}\mathbb{D}_{\alpha}\left(\mathcal{E}^{\otimes
n}\left(\Lambda^{n}\left(\rho^{\otimes
n}\right)\right)\big{\|}\mathcal{F}^{\otimes
n}\left(\Lambda^{n}\left(\sigma^{\otimes n}\right)\right)\right)$
$\displaystyle\qquad\leq\widetilde{D}_{\alpha}(\rho\|\sigma)+\mathbb{D}_{\alpha}^{\infty}(\mathcal{E}\big{\|}\mathcal{F})\,,$
(42)

where the supremum is over all rank-one projective measurement maps
$\Lambda^{n}(\cdot)$ acting on $n$ copies. It is then only for
$\alpha\in(1,\infty]$ that the left-hand side of Eq. (III) is further lower
bounded by
$\widetilde{D}_{\alpha}\big{(}\mathcal{E}(\rho)\big{\|}\mathcal{F}(\sigma)\big{)}$
(Lemma 2), whereas this remains unclear for $\alpha\in(0,1]$.555In this
context, Lemma 2 is not useful for $\alpha\in(0,1]$ as one would have to
divide the logarithm on both sides of Eq. (20) with the pre-factor $\alpha-1$,
which becomes negative in this range. Notwithstanding, at least for $\alpha=1$
it is known that [15, Theorem 3.5]

$\displaystyle\widetilde{D}_{1}\big{(}\mathcal{E}(\rho)\big{\|}\mathcal{F}(\sigma)\big{)}\leq\widetilde{D}_{1}(\rho\|\sigma)+\widetilde{D}_{1}^{\infty}\big{(}\mathcal{E}\big{\|}\mathcal{F}\big{)}\,.$
(43)

However, this does not directly follow from our Rényi divergences chain rules,
as the order of taking the limits in $n\to\infty$ and $\alpha\to 1$ might
ultimately matter when regularizing Eq. (4). We briefly get back to this
question in Section V and present some conjectures.

##  IV Applications

Rényi divergences chain rules such as Eq. (I) and its quantum extensions allow
to characterize the asymptotic error exponents of adaptive channel
discrimination [22, 33, 7, 42, 15, 35, 14, 17]. Namely, the crucial quantities
to study are the so-called amortized channel divergences [42, 7, 40]

$\displaystyle\mathbb{D}_{\alpha}^{\text{a}}\big{(}\mathcal{E}\big{\|}\mathcal{F}\big{)}=\sup_{\rho,\sigma}\Big{\\{}$
$\displaystyle\mathbb{D}_{\alpha}\big{(}(\mathcal{E}_{A}\otimes\mathcal{I})(\rho_{AR})\big{\|}(\mathcal{F}_{A}\otimes\mathcal{I}_{R})(\sigma_{AR})\big{)}$
$\displaystyle-\mathbb{D}_{\alpha}\big{(}\rho_{AR}\big{\|}\sigma_{AR}\big{)}\Big{\\}}\,,$
(44)

where the supremum is over all
$\rho_{AR},\sigma_{AR}\in\mathcal{S}(\mathcal{H}_{A}\otimes\mathcal{H}_{R})$
and finite dimensional $\mathcal{H}_{R}$.666No bound on the dimension $|R|$ is
known. For classical [22, 33] and classical-quantum channels [7, 42, 35] it is
derived from the chain rules in Eq. (I) and its classical-quantum extensions
that for $\alpha>0$ and all the relevant channel divergences

$\displaystyle\mathbb{D}_{\alpha}^{\text{a}}\big{(}\mathcal{E}\big{\|}\mathcal{F}\big{)}=\mathbb{D}_{\alpha}\big{(}\mathcal{E}\big{\|}\mathcal{F}\big{)}\,.$
(45)

This then means that adaptive or correlated input strategies are of no
asymptotic advantage compared to product strategies and one immediately
obtains a full asymptotic characterization of channel discrimination (see
aforementioned references for details). In the general quantum case, one only
has the stabilized and regularized chain rules as in Corollary 5, and for
$\alpha\in[1,\infty]$ one then derives [15, 17]

$\displaystyle\widetilde{D}_{\alpha}^{\text{a}}\big{(}\mathcal{E}\big{\|}\mathcal{F}\big{)}=\widetilde{D}_{\alpha}^{\text{stab},\infty}\big{(}\mathcal{E}\big{\|}\mathcal{F}\big{)}\,,$
(46)

which includes $\alpha=1$ thanks to [15, Theorem 3.5]. Consequently, this
shows that adaptive strategies are of no asymptotic advantage over entangled
input strategies — with the regularization reflecting that the latter are
still needed compared to product input strategies [15]. Nonetheless, for a
full asymptotic characterization of channel discrimination technical question
concerning the $\alpha\to 1$ limit remain [17] (cf. the discussion in Section
V). Alternatively, one might resort to single-letter converse bounds as
provided by the geometric chain rules Eqs. (33)–(III) giving that

$\displaystyle\widehat{D}_{\alpha}^{\text{a}}\big{(}\mathcal{E}\big{\|}\mathcal{F}\big{)}=\widehat{D}_{\alpha}^{\text{stab}}\big{(}\mathcal{E}\big{\|}\mathcal{F}\big{)}\,.$
(47)

We refer to [14] for details. One advantage, of our novel geometric chain rule
in Eq. (33) is that it does not require a stabilization system and hence one
might explore channel discrimination tasks without additional quantum memory
as done in the classical case [22] (or bounded quantum memory of fixed
dimension).

##  V Outlook

We hope that our matrix analysis based techniques will be more broadly useful
to derive entropy inequalities for quantum channels. Many fundamental
questions remain with respect to the goal of giving for $\alpha>0$ the most
general Rényi divergence chain rules for quantum channels. For example, our
main results from Section III immediately raise the question if

$\displaystyle\limsup_{n\to\infty}\sup_{\Lambda^{n}}\frac{1}{n}\widetilde{D}_{1}\left(\mathcal{E}^{\otimes
n}\left(\Lambda^{n}\left(\rho^{\otimes
n}\right)\right)\big{\|}\mathcal{F}^{\otimes
n}\left(\Lambda^{n}\left(\sigma^{\otimes n}\right)\right)\right)$
$\displaystyle\qquad\geq\widetilde{D}_{1}\big{(}\mathcal{E}(\rho)\big{\|}\mathcal{F}(\sigma)\big{)}\,,$
(48)

where the supremum is over all rank-one projective measurement maps
$\Lambda^{n}(\cdot)$ acting on $n$ copies. Choosing the standard pinching
based measurement map $\Lambda^{n}(\cdot)=\mathcal{M}_{\sigma^{\otimes
n}}(P_{\sigma^{\otimes n}}(\cdot))$ in the form of Eq. (6) only leads to

$\displaystyle\frac{1}{n}\sup_{\Lambda^{n}}\widetilde{D}_{1}\left(\mathcal{E}^{\otimes
n}\left(\Lambda^{n}\left(\rho^{\otimes
n}\right)\right)\big{\|}\mathcal{F}^{\otimes
n}\left(\Lambda^{n}\left(\sigma^{\otimes n}\right)\right)\right)$
$\displaystyle\geq\frac{1}{n}\widetilde{D}_{1}\left(\mathcal{E}^{\otimes
n}\left(\int\mu(\mathrm{d}t)\,\left(\sigma^{it}\rho\sigma^{-it}\right)^{\otimes
n}\right)\big{\|}\left(\mathcal{F}\left(\sigma\right)\right)^{\otimes
n}\right)$ (49)
$\displaystyle\geq\min_{t}\widetilde{D}_{1}\left(\mathcal{E}\left(\sigma^{it}\rho\sigma^{-it}\right)\middle\|\mathcal{F}(\sigma)\right)-\frac{1}{n}\log\text{poly}(n)\,,$
(50)

where we used in the last inequality that by Carathéodory’s theorem the
integral $\int\mu(\mathrm{d}t)(\cdot)^{\otimes n}$ is discretized with at most
$\text{poly}(n)$ terms [6, Theorem D.5] and can then by [5, Lemma 2.5] be
taken outside of the divergence at the price of the asymptotically small
$\log\text{poly}(n)$ fudge term. The appearance of unitary operators like
$\sigma^{it}$ is reminiscent of previous work on quantum entropy inequalities
[18, 8, 41, 36, 24, 37, 38, 5], but here we would need to eliminate them.

As the $\alpha\in(1,\infty]$ analogue of Eq. (V) holds by Corollary 4, we
might also ask more generally what happens for the whole parameter range
$\alpha\in(0,1]$. This is strongly connected with the question if
$\widehat{D}_{\alpha}^{\text{stab},\infty}\big{(}\mathcal{E}\big{\|}\mathcal{F}\big{)}$
is continuous for $\alpha\to 1$ [17]. Any insight on this question would have
widespread applications in quantum information theory [16] — starting from the
fundamental statistical task of channel discrimination.

Note added: After submitting our paper, the work [30] on generalised entropy
accumulation appeared. A crucial proof step is their complementary Rényi
divergence chain rule [30, Theorem 3.1] for quantum channels and it would be
interesting to explore the interplay with our purely matrix analysis based
techniques.

### Acknowledgements

This work was completed prior to MB joining the AWS Center for Quantum
Computing. This research is supported by the National Research Foundation,
Prime Minister’s Office, Singapore and the Ministry of Education, Singapore
under the Research Centres of Excellence programme. MT is also supported in
part by NUS startup grants (R-263-000-E32-133 and R-263-000-E32-731). We thank
David Sutter for discussions.

We mention the following, alternative proof of the sandwiched chain rule
(Corollary 4). This directly lifts the classical chain rule from Eq. (I) to
the quantum case.

###### Proof of Corollary 4.

Let $\\{|{x}\rangle\\}_{x}$ be a joint eigenbasis for both $\sigma$ and
$\mathcal{P}_{\sigma}(\rho)$. We write

$\displaystyle\mathcal{E}(\mathcal{P}_{\sigma}(\rho))$
$\displaystyle=\sum_{x}\underbrace{\langle
x|\rho|x\rangle}_{=:p_{x}}\underbrace{\mathcal{E}\big{(}|x\rangle\\!\langle
x|\big{)}}_{=:\rho^{x}}\quad\textrm{and}$ (51)
$\displaystyle\mathcal{F}(\sigma)$ $\displaystyle=\sum_{x}\underbrace{\langle
x|\sigma|x\rangle}_{=:q_{x}}\underbrace{\mathcal{F}\big{(}|x\rangle\\!\langle
x|\big{)}}_{=:\sigma^{x}}$ (52)

and by the data-processing for the partial trace we find

$\displaystyle\widetilde{Q}_{\alpha}\big{(}\mathcal{E}(\mathcal{P}_{\sigma}(\rho))\big{\|}\mathcal{F}(\sigma)\big{)}$
(53)
$\displaystyle\leq\widetilde{Q}_{\alpha}\left(\sum_{x}{p_{x}}|x\rangle\\!\langle
x|\otimes\rho^{x}\,\middle\|\,\sum_{x}{q_{x}}|x\rangle\\!\langle
x|\otimes\sigma^{x}\right)$ (54)
$\displaystyle=\sum_{x}p_{x}^{\alpha}q_{x}^{1-\alpha}\widetilde{Q}_{\alpha}\left(\rho^{x}\middle\|\sigma^{x}\right)$
(55)
$\displaystyle\leq\sum_{x}p_{x}^{\alpha}q_{x}^{1-\alpha}\cdot\max_{x}\widetilde{Q}_{\alpha}\left(\rho^{x}\middle\|\sigma^{x}\right)$
(56) $\displaystyle\leq
2^{(\alpha-1)D_{\alpha,M}(\rho\|\sigma)}\cdot\max_{\omega\in\mathcal{S}}\widetilde{Q}_{\alpha}\big{(}\mathcal{E}(\omega)\big{\|}\mathcal{F}(\omega)\big{)}\,,$
(57)

where in the last inequality we used that $\rho^{x}$ and $\sigma^{x}$ are
channel outputs for the same input. Combining this with Lemma 2 and taking the
logarithm as well as multiplying by $\frac{1}{\alpha-1}$, yields the desired
result. ∎

## References

  * [1] Koenraad M. R. Audenaert and Nilanjana Datta.  $\alpha$-z-relative Renyi entropies.  Journal of Mathematical Physics, 56:022202, 2015.
  * [2] V. P. Belavkin and P. Staszewski.  C*-algebraic generalization of relative entropy and entropy.  Annals Henri Poincaré, 37(1):51–58, 1982.
  * [3] M. Berta, O. Fawzi, and M. Tomamichel.  Exploiting variational formulas for quantum relative entropy.  In IEEE International Symposium on Information Theory (ISIT), pages 2844–2848, 2016.
  * [4] M. Berta, O. Fawzi, and M. Tomamichel.  On variational expressions for quantum relative entropies.  Letters in Mathematical Physics, 107(12):2239–2265, 2017.
  * [5] Mario Berta, Fernando G. S. L. Brandão, and Christoph Hirche.  On composite quantum hypothesis testing.  Communications in Mathematical Physics, 385(1):55–77, 2021.
  * [6] Mario Berta, Matthias Christandl, and Renato Renner.  The quantum reverse Shannon theorem based on one-shot information theory.  Communications in Mathematical Physics, 306(3):579–615, 2011.
  * [7] Mario Berta, Christoph Hirche, Eneet Kaur, and Mark M. Wilde.  Stein’s lemma for classical-quantum channels.  In IEEE International Symposium on Information Theory (ISIT), pages 2564–2568, 2019.
  * [8] Fernando G. S. L. Brandão, Aram W. Harrow, Jonathan Oppenheim, and Sergii Strelchuk.  Quantum conditional mutual information, reconstructed states, and state redistribution.  Physical Review Letters, 115(5):050501, 2015.
  * [9] Peter Brown, Hamza Fawzi, and Omar Fawzi.  Device-independent lower bounds on the conditional von Neumann entropy.  2021\.
  * [10] Matthias Christandl and Alexander Müller-Hermes.  Relative entropy bounds on quantum, private and repeater capacities.  Communications in Mathematical Physics, 353(2):821–852, 2017.
  * [11] Tom Cooney, Milán Mosonyi, and Mark M. Wilde.  Strong converse exponents for a quantum channel discrimination problem and quantum-feedback-assisted communication.  Communications in Mathematical Physics, 344(3):797–829, 2016.
  * [12] Imre Csiszár.  The method of types.  IEEE Transactions on Information Theory, 44(6):2505–2523, 1998\.
  * [13] Matthew J. Donald.  On the relative entropy.  Communications in Mathematical Physics, 105(1):13–34, 1986.
  * [14] Kun Fang and Hamza Fawzi.  Geometric Rényi divergence and its applications in quantum channel capacities.  Communications in Mathematical Physics, 384(3):1615–1677, 2021\.
  * [15] Kun Fang, Omar Fawzi, Renato Renner, and David Sutter.  Chain rule for the quantum relative entropy.  Physical Review Letters, 124(10):100501, 2020.
  * [16] Kun Fang, Gilad Gour, and Xin Wang.  Towards the ultimate limits of quantum channel discrimination.  2021\.
  * [17] Hamza Fawzi and Omar Fawzi.  Defining quantum divergences via convex optimization.  Quantum, 5:387, 2021.
  * [18] Omar Fawzi and Renato Renner.  Quantum conditional mutual information and approximate Markov chains.  Communications in Mathematical Physics, 340(2):575–611, 2015.
  * [19] Gilad Gour and Marco Tomamichel.  Optimal extensions of resource measures and their applications.  Physical Review A, 102(6):062401, 2020.
  * [20] M. B. Hastings.  Superadditivity of communication capacity using entangled inputs.  Nature Physics, 5(4):255–257, 2009.
  * [21] Masahito Hayashi.  Optimal sequence of quantum measurements in the sense of Stein’s lemma in quantum hypothesis testing.  Journal of Physics A: Mathematical and Theoretical, 35(50):10759–10773, 2002.
  * [22] Masahito Hayashi.  Discrimination of two channels by adaptive methods and its application to quantum system.  IEEE Transactions on Information Theory, 55(8):3807–3820, 2009\.
  * [23] Fumio Hiai and Dénes Petz.  The proper formula for relative entropy and its asymptotics in quantum probability.  Communications in Mathematical Physics, 143(1):99–114, 1991.
  * [24] Marius Junge, Renato Renner, David Sutter, Mark M. Wilde, and Andreas Winter.  Universal recovery maps and approximate sufficiency of quantum relative entropy.  Annales Henri Poincaré, 19(10):2955–2978, 2018.
  * [25] Vishal Katariya and Mark M. Wilde.  Geometric distinguishability measures limit quantum channel estimation and discrimination.  Quantum Information Processing, 20(2):78, 2021.
  * [26] Sumeet Khatri and Mark M. Wilde.  Principles of quantum communication theory: A modern approach.  2020\.
  * [27] Felix Leditzky, Eneet Kaur, Nilanjana Datta, and Mark M. Wilde.  Approaches for approximate additivity of the Holevo information of quantum channels.  Physical Review A, 97(1):012332, 2018.
  * [28] Keiji Matsumoto.  Reverse test and characterization of quantum relative entropy.  2010\.
  * [29] Keiji Matsumoto.  A new quantum version of f-divergence.  2014\.
  * [30] Tony Metger, Omar Fawzi, David Sutter, and Renato Renner.  Generalised entropy accumulation.  2022\.
  * [31] Martin Müller-Lennert, Frédéric Dupuis, Oleg Szehr, Serge Fehr, and Marco Tomamichel.  On quantum Rényi entropies: A new generalization and some properties.  Journal of Mathematical Physics, 54(12):122203, 2013.
  * [32] Dénes Petz.  Quasi-entropies for finite quantum systems.  Reports on Mathematical Physics, 23(1):57–65, 1986.
  * [33] Y. Polyanski and S. Verdú.  Binary hypothesis testing with feedback.  In International Conference on Information Theory and Application, 2011.
  * [34] A. Rényi.  On measures of information and entropy.  In Proceedings 4th Berkeley Symposium on Mathematical Statistics and Probability, volume 1, pages 547–561. University of California Press, 1961\.
  * [35] Farzin Salek, Masahito Hayashi, and Andreas Winter.  When are adaptive strategies in asymptotic quantum channel discrimination useful?  2021\.
  * [36] D. Sutter, M. Tomamichel, and A.W. Harrow.  Strengthened monotonicity of relative entropy via pinched Petz recovery map.  IEEE Transactions on Information Theory, 62(5):2907–2913, 2016\.
  * [37] David Sutter, Mario Berta, and Marco Tomamichel.  Multivariate trace inequalities.  Communications in Mathematical Physics, 352(1):37–58, 2017.
  * [38] David Sutter, Mario Berta, and Marco Tomamichel.  Quantum Markov chains and logarithmic trace inequalities.  In IEEE International Symposium on Information Theory (ISIT), pages 1988–1992, 2017.
  * [39] Marco Tomamichel.  Quantum Information Processing with Finite Resources — Mathematical Foundations, volume 5 of SpringerBriefs in Mathematical Physics.  Springer International Publishing, 2016.
  * [40] Xin Wang and Mark M. Wilde.  Resource theory of asymmetric distinguishability for quantum channels.  Physical Review Research, 1(3):033169, 2019.
  * [41] Mark M. Wilde.  Recoverability in quantum information theory.  Proceedings of the Royal Society A, 471(2182):20150338, 2015.
  * [42] Mark M. Wilde, Mario Berta, Christoph Hirche, and Eneet Kaur.  Amortized channel divergence for asymptotic quantum channel discrimination.  Letters in Mathematical Physics, 110(8):2277–2336, 2020.
  * [43] Mark M. Wilde, Andreas Winter, and Dong Yang.  Strong converse for the classical capacity of entanglement-breaking and Hadamard channels via a sandwiched Rényi relative entropy.  Communications in Mathematical Physics, 331(2):593–622, 2014.


Khukhunaishvili, E. Ranken, P. Tan, R. Taus Rutgers, The State University of
New Jersey, Piscataway, USA
B. Chiarito, J.P. Chou, Y. Gershtein, E. Halkiadakis, A. Hart, M. Heindl, E.
Hughes, S. Kaplan, S. Kyriacou, I. Laflotte, A. Lath, R. Montalvo, K. Nash, M.
Osherson, H. Saka, S. Salur, S. Schnetzer, D. Sheffield, S. Somalwar, R.
Stone, S. Thomas, P. Thomassen University of Tennessee, Knoxville, USA
H. Acharya, A.G. Delannoy, J. Heideman, G. Riley, S. Spanier Texas A&M
University, College Station, USA
O. Bouhali78, A. Celik, M. Dalchenko, M. De Mattia, A. Delgado, S. Dildick, R.
Eusebi, J. Gilmore, T. Huang, T. Kamon79, S. Luo, D. Marley, R. Mueller, D.
Overton, L. Perniè, D. Rathjens, A. Safonov Texas Tech University, Lubbock,
USA
N. Akchurin, J. Damgov, F. De Guio, P.R. Dudero, S. Kunori, K. Lamichhane,
S.W. Lee, T. Mengke, S. Muthumuni, T. Peltola, S. Undleeb, I. Volobouev, Z.
Wang, A. Whitbeck Vanderbilt University, Nashville, USA
S. Greene, A. Gurrola, R. Janjam, W. Johns, C. Maguire, A. Melo, H. Ni, K.
Padeken, F. Romeo, P. Sheldon, S. Tuo, J. Velkovska, M. Verweij, Q. Xu
University of Virginia, Charlottesville, USA
M.W. Arenton, P. Barria, B. Cox, R. Hirosky, M. Joyce, A. Ledovskoy, H. Li, C.
Neu, Y. Wang, E. Wolfe, F. Xia Wayne State University, Detroit, USA
R. Harr, P.E. Karchin, N. Poudyal, J. Sturdy, P. Thapa, S. Zaleski University
of Wisconsin - Madison, Madison, WI, USA
J. Buchanan, C. Caillol, D. Carlsmith, S. Dasu, I. De Bruyn, L. Dodd, B.
Gomber80, M. Grothe, M. Herndon, A. Hervé, U. Hussain, P. Klabbers, A. Lanaro,
K. Long, R. Loveless, T. Ruggles, A. Savin, V. Sharma, W.H. Smith, N. Woods †:
Deceased
1: Also at Vienna University of Technology, Vienna, Austria
2: Also at Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State
University, Moscow, Russia
3: Also at IRFU, CEA, Université Paris-Saclay, Gif-sur-Yvette, France
4: Also at Universidade Estadual de Campinas, Campinas, Brazil
5: Also at Federal University of Rio Grande do Sul, Porto Alegre, Brazil
6: Also at Université Libre de Bruxelles, Bruxelles, Belgium
7: Also at University of Chinese Academy of Sciences, Beijing, China
8: Also at Institute for Theoretical and Experimental Physics named by A.I.
Alikhanov of NRC ‘Kurchatov Institute’, Moscow, Russia
9: Also at Joint Institute for Nuclear Research, Dubna, Russia
10: Also at Helwan University, Cairo, Egypt
11: Now at Zewail City of Science and Technology, Zewail, Egypt
12: Now at Cairo University, Cairo, Egypt
13: Also at British University in Egypt, Cairo, Egypt
14: Now at Ain Shams University, Cairo, Egypt
15: Also at Purdue University, West Lafayette, USA
16: Also at Université de Haute Alsace, Mulhouse, France
17: Also at Tbilisi State University, Tbilisi, Georgia
18: Also at Ilia State University, Tbilisi, Georgia
19: Also at CERN, European Organization for Nuclear Research, Geneva,
Switzerland
20: Also at RWTH Aachen University, III. Physikalisches Institut A, Aachen,
Germany
21: Also at University of Hamburg, Hamburg, Germany
22: Also at Brandenburg University of Technology, Cottbus, Germany
23: Also at Institute of Physics, University of Debrecen, Debrecen, Hungary,
Debrecen, Hungary
24: Also at Institute of Nuclear Research ATOMKI, Debrecen, Hungary
25: Also at MTA-ELTE Lendület CMS Particle and Nuclear Physics Group, Eötvös
Loránd University, Budapest, Hungary, Budapest, Hungary
26: Also at IIT Bhubaneswar, Bhubaneswar, India, Bhubaneswar, India
27: Also at Institute of Physics, Bhubaneswar, India
28: Also at Shoolini University, Solan, India
29: Also at University of Visva-Bharati, Santiniketan, India
30: Also at Isfahan University of Technology, Isfahan, Iran
31: Also at Plasma Physics Research Center, Science and Research Branch,
Islamic Azad University, Tehran, Iran
32: Also at Italian National Agency for New Technologies, Energy and
Sustainable Economic Development, Bologna, Italy
33: Also at Centro Siciliano di Fisica Nucleare e di Struttura Della Materia,
Catania, Italy
34: Also at Università degli Studi di Siena, Siena, Italy
35: Also at Scuola Normale e Sezione dell’INFN, Pisa, Italy
36: Also at Kyung Hee University, Department of Physics, Seoul, Korea
37: Also at Riga Technical University, Riga, Latvia, Riga, Latvia
38: Also at International Islamic University of Malaysia, Kuala Lumpur,
Malaysia
39: Also at Malaysian Nuclear Agency, MOSTI, Kajang, Malaysia
40: Also at Consejo Nacional de Ciencia y Tecnología, Mexico City, Mexico
41: Also at Warsaw University of Technology, Institute of Electronic Systems,
Warsaw, Poland
42: Also at Institute for Nuclear Research, Moscow, Russia
43: Now at National Research Nuclear University ’Moscow Engineering Physics
Institute’ (MEPhI), Moscow, Russia
44: Also at St. Petersburg State Polytechnical University, St. Petersburg,
Russia
45: Also at University of Florida, Gainesville, USA
46: Also at P.N. Lebedev Physical Institute, Moscow, Russia
47: Also at California Institute of Technology, Pasadena, USA
48: Also at Budker Institute of Nuclear Physics, Novosibirsk, Russia
49: Also at Faculty of Physics, University of Belgrade, Belgrade, Serbia
50: Also at University of Belgrade: Faculty of Physics and VINCA Institute of
Nuclear Sciences, Belgrade, Serbia
51: Also at INFN Sezione di Pavia a, Università di Pavia b, Pavia, Italy,
Pavia, Italy
52: Also at National and Kapodistrian University of Athens, Athens, Greece
53: Also at Universität Zürich, Zurich, Switzerland
54: Also at Stefan Meyer Institute for Subatomic Physics, Vienna, Austria,
Vienna, Austria
55: Also at Adiyaman University, Adiyaman, Turkey
56: Also at Şırnak University, Sirnak, Turkey
57: Also at Beykent University, Istanbul, Turkey, Istanbul, Turkey
58: Also at Istanbul Aydin University, Istanbul, Turkey
59: Also at Mersin University, Mersin, Turkey
60: Also at Piri Reis University, Istanbul, Turkey
61: Also at Gaziosmanpasa University, Tokat, Turkey
62: Also at Ozyegin University, Istanbul, Turkey
63: Also at Izmir Institute of Technology, Izmir, Turkey
64: Also at Marmara University, Istanbul, Turkey
65: Also at Kafkas University, Kars, Turkey
66: Also at Istanbul University, Istanbul, Turkey
67: Also at Istanbul Bilgi University, Istanbul, Turkey
68: Also at Hacettepe University, Ankara, Turkey
69: Also at Rutherford Appleton Laboratory, Didcot, United Kingdom
70: Also at School of Physics and Astronomy, University of Southampton,
Southampton, United Kingdom
71: Also at IPPP Durham University, Durham, United Kingdom
72: Also at Monash University, Faculty of Science, Clayton, Australia
73: Also at Bethel University, St. Paul, Minneapolis, USA, St. Paul, USA
74: Also at Karamanoğlu Mehmetbey University, Karaman, Turkey
75: Also at Bingol University, Bingol, Turkey
76: Also at Sinop University, Sinop, Turkey
77: Also at Mimar Sinan University, Istanbul, Istanbul, Turkey
78: Also at Texas A&M University at Qatar, Doha, Qatar
79: Also at Kyungpook National University, Daegu, Korea, Daegu, Korea
80: Also at University of Hyderabad, Hyderabad, India