# Comparison of Conformational Phase Behavior for Flexible and Semiflexible
Polymers

Dilimulati Aierken<EMAIL_ADDRESS>Soft Matter Systems Research Group, Center
for Simulational Physics, The University of Georgia, Athens, GA 30602, USA
Michael Bachmann<EMAIL_ADDRESS>http://www.smsyslab.org Soft Matter
Systems Research Group, Center for Simulational Physics, The University of
Georgia, Athens, GA 30602, USA

###### Abstract

We employ the recently introduced generalized microcanonical inflection point
method for the statistical analysis of phase transitions in flexible and
semiflexible polymers and study the impact of the bending stiffness upon the
character and order of transitions between random-coil, globules, and
pseudocrystalline conformations. The high-accuracy estimates of the
microcanonical entropy and its derivatives required for this study were
obtained by extensive replica-exchange Monte Carlo simulations. We observe
that the transition behavior into the compact phases changes qualitatively
with increasing bending stiffness. Whereas the $\Theta$ collapse transition is
less affected, the first-order liquid-solid transition characteristic for
flexible polymers ceases to exist once bending effects dominate over
attractive monomer-monomer interactions.

flexible polymers, semiflexible polymers, conformational phases, phase
transitions, Monte Carlo simulations, microcanonical analysis

##  I Introduction

Statistical analysis methods have long been used for the identification,
characterization, and classification of phase transitions in complex systems.
Advanced computer simulation techniques, mostly based on Markov chain Monte
Carlo algorithms, helped advance the field beyond the deadlocks mathematical
approaches had run into. Nonetheless, the general idea remained the same.
Phase transitions were generally only considered in the thermodynamic limit,
which was historically introduced to make studies of macroscopic systems
mathematically tractable. In computational statistical physics, finite-size
scaling analysis provided the tool for extrapolating the results obtained in
simulations of systems of finite size toward this hypothetical limit. These
methods proved to be extremely successful, and for decades the scientific
community did not see any reason to change paradigms and consider a more
general perspective despite the fact that all systems in nature are finite and
systems on nanoscales moved into the focus of substantial interest in
microbiology and technology. Ignoring surface effects was also convenient.

Consequently, conventional canonical statistical analysis of phase transitions
rests on the search for catastrophic behavior of thermodynamic state variables
and response functions in thermodynamic parameter spaces. Hence, significant
effort has been dedicated to the evaluation of critical exponents in
continuous phase transitions and the grouping of systems in universality
classes.

In recent years, however, the growing interest in systems on microscopic and
mesoscopic scales, which do not satisfy the criteria for the thermodynamic
limit, has shifted the perspective. Alternative approaches that enable studies
of phase transitions in systems of any size have turned out to be promising.
Modern interdisciplinary research of systems on nanometer scales cannot ignore
the fact that the surface of the system is at least as important for the
structural behavior of the system as the bulk effects. For example, the
functionality of biomacromolecules like proteins depends on the folding
process into stable geometric conformations. These processes, which occur in a
thermal environment, resemble transitions between disordered and ordered
phases known from macroscopic systems. In fact, for very long polymers, these
structure formation processes _are_ phase transitions, even in the strict
conventional sense. However, many types of heterogeneous polymers like
proteins cannot be scaled up, but still show clear transition features. This
makes the extension of the established theory of phase transitions a
necessity.

However, canonical statistical methods create ambiguities, which render the
unique characterization of phase transitions problematic. A prominent example
is the peak in the specific heat curve of the one-dimensional Ising model. As
it is also finite in the thermodynamic limit, one does not associate a phase
transition with it. But how to interpret a peak in response quantities, if the
thermodynamic limit for this system does not exist? In recent years, it has
nonetheless become common to consider peaks or “shoulders” transition signals
in curves of response quantities for finite systems. This appears to be an
inconsistent approach and renders, in fact, the choice of the temperature as
the basic thermodynamic state variable questionable.

Combining results of previous studies of the microcanonical Boltzmann entropy
[1] and the principle of minimal sensitivity [2, 3], we recently developed a
systematic approach to phase transitions in systems of any size by identifying
least-sensitive inflection points in the entropy and its derivatives [4]. This
analysis method even allows for the classification of phase transitions in
analogy to Ehrenfest’s classification scheme in thermodynamics, which is based
on derivatives of thermodynamic potentials [5].

In this paper, we extend previous Monte Carlo computer simulation studies of a
coarse-grained model for flexible polymers [6] and analyze structural
transitions in semiflexible polymers by means of the generalized
microcanonical least-sensitive inflection point method. We thoroughly
investigate the changes of phase behavior of a coarse-grained model for
flexible and semiflexible polymers.

The paper is organized as follows. In Sec. II, we describe the flexible and
semiflexible variants of the coarse-grained polymer model used in this study,
as well as the numerical methods for the stochastic simulation of the model
and for the statistical analysis of the data obtained in the simulations. The
results of the canonical and microcanonical analysis are discussed and the
phase behavior of the polymers is interpreted in Sec. III. This section also
contains a detailed comparison of geometric properties of the lowest-energy
conformations found for each system. Eventually, the paper is concluded by the
summary in Sec. IV.

##  II Models and Methods

For this study we employed a generic coarse-grained model for polymers in
implicit solvent that allows us to investigate and compare the phase behavior
of flexible and semiflexible polymer chains by means of advanced Monte Carlo
computer simulation methodologies. Model, simulation methods, and the
microcanonical inflection-point technique we used for the statistical analysis
of the conformational transitions are described in the following.

###  II.1 Generic model for flexible and semiflexible polymers

The total energy of a polymer chain with $N$ monomers is composed of
contributions from bonded and non-bonded interactions between monomers. For
the interactions between non-bonded monomers, we employ the standard 12-6
Lennard-Jones (LJ) potential:

$V_{\mathrm{LJ}}(r)=4\epsilon_{\mathrm{LJ}}\left[\left(\dfrac{\sigma}{r}\right)^{12}-\left(\dfrac{\sigma}{r}\right)^{6}\right].$
(1)

Here, $r$ is the monomer-monomer distance, $\sigma=2^{-1/6}r_{0}$ is the van
der Waals distance associated with the potential minimum at $r_{0}$, and
$\epsilon_{\mathrm{LJ}}$ is the energy scale. For computational efficiency, we
introduce a cutoff at $r_{c}=2.5\sigma$. Shifting the potential by the
constant $V_{\mathrm{shift}}=V_{\mathrm{LJ}}(r_{c})$ avoids a discontinuity at
$r_{c}$. Thus the potential energy of non-bonded monomers is given by

$V_{\mathrm{NB}}(r)=\begin{cases}V_{\mathrm{LJ}}(r)-V_{\mathrm{shift}},&r<r_{c},\\\
0,&\text{otherwise.}\end{cases}$ (2)

The elastic bond between two neighboring monomers is modeled by a potential
which combines a Lennard-Jones potential and the finitely extensible nonlinear
elastic (FENE) potential [7, 8, 9]:

$V_{\mathrm{B}}(r)=-\frac{1}{2}K_{\mathrm{f}}R^{2}\mathrm{ln}\left[1-\left(\frac{r-r_{0}}{R}\right)^{2}\right]+V_{\mathrm{LJ}}(r)-V_{\mathrm{shift}},$
(3)

where the FENE parameters are fixed to standard values $R=(3/7)r_{0}$ and
$K_{\mathrm{f}}=(98/5)\epsilon_{\mathrm{LJ}}/r_{0}^{2}$ [10]. The bond length
$r$ is restricted to fluctuations within the range $[r_{0}-R,r_{0}+R]$. With
these parameters the minimum of $V_{\mathrm{B}}$ is located at $r_{0}$.

For semiflexible polymers, an additional potential related to the bending
stiffness is introduced. Any deviation from the reference angle $\theta_{0}$
between neighboring bonds is subject to an energy penalty of the form:

$V_{\mathrm{bend}}(\theta)=\kappa\left[1-\cos(\theta-\theta_{0})\right].$ (4)

If we represent a conformation of a polymer chain with $N$ monomers by
$\boldsymbol{X}=\left(\boldsymbol{r}_{1},...,\boldsymbol{r}_{N}\right)$, where
$\boldsymbol{r}_{i}$ is the position vector of the $i$th monomer, the total
energy of the polymer is given by

$E(\boldsymbol{X})=\sum_{i>j+1}V_{\mathrm{NB}}(r_{i,j})+\sum_{i}V_{\mathrm{B}}(r_{i,i+1})+\sum_{l}V_{\mathrm{bend}}(\theta_{l}),$
(5)

where $r_{i,j}=|\boldsymbol{r}_{i}-\boldsymbol{r}_{j}|$ is the distance
between monomers $i$ and $j$, and $\theta_{l}$ is the bond angle between two
adjacent bonds. The parameter $\kappa\geq 0$ controls the stiffness of the
polymer chain. For $\kappa=0$, the model describes flexible polymers.

In simulations and statistical analysis of the results, we set the basic
scales to the following values: $k_{\mathrm{B}}=1$ (Boltzmann constant),
$\epsilon_{\mathrm{LJ}}=1$, and $r_{0}=1$. For the reference bending angle, we
chose $\theta_{0}=0$. The flexible chain with $N=55$ monomers has already been
studied extensively in the past and serves as the reference for the comparison
with the semiflexible model. This chain length is sufficiently short to
recognize finite-size effects, but long enough for the polymer to form a
stable solid phase at low temperatures [11].

###  II.2 Replica-Exchange Monte Carlo method and multiple-histogram
reweighting

For the simulations of the polymer, we employed the replica-exchange (parallel
tempering) Monte Carlo method [12, 13, 14, 15, 16]. Replicas of the systems
were simulated at different temperatures $T_{k}\in[0.1,5.0]$ with
$k=1,2,...,K$. The total number of temperature threads $K$ ranged from 40 to
50 in individual simulations. Up to ten independent runs were performed for
all $\kappa$ values studied, which allowed for the estimation of numerical
errors in all microcanonical quantities.

At each temperature $T_{k}$, Metropolis sampling was performed, which is based
on the acceptance probability:

$P(E_{k},T_{k};E^{\prime}_{k},T_{k})=\mathrm{min}\left\\{e^{-(E^{\prime}_{k}-E_{k})/{k_{\mathrm{B}}T_{k}}},1\right\\}.$
(6)

Here $k_{\mathrm{B}}T_{k}$ is the the thermal energy at the simulation
temperature $T_{k}$, $E^{\prime}_{k}$ is the energy of the proposed state, and
$E_{k}$ is the energy of the current conformation.

Local structural changes were achieved by displacement moves. In this Monte
Carlo update, a monomer $i$ is randomly picked and its position is modified by
a random shift $\Delta\boldsymbol{r}_{i}$ within a cubic box with edge lengths
$r_{d}$ surrounding the monomer. Before measurements can be performed, $r_{d}$
is determined adaptively for each temperature thread to achieve a Metropolis
acceptance rate of about $50\%$.

In order to explore the conformation space more efficiently, rotational pivot
updates were also used. After 20 to 70 sweeps of displacement updates
(depending on the temperature), we performed a sweep of pivot rotational
updates. In a pivot update, a monomer $i$ is randomly chosen as the pivot
point and a rotation axis vector $\boldsymbol{s}_{i}$. Then the section of the
chain following the pivot monomer was rotated about $\boldsymbol{s}_{i}$ by a
randomly chosen angle taken from the interval $[0,2\pi]$.

Replicas were swapped between neighboring threads $k$ and $k+1$ after every
1500 sweeps with the exchange probability:

$\displaystyle P(E_{k},T_{k};E_{k+1},T_{k+1})=$
$\displaystyle\min\left\\{\exp\left[\left(E_{k}-E_{k+1}\right)\left(\frac{1}{k_{\mathrm{B}}T_{k}}-\frac{1}{k_{\mathrm{B}}T_{k+1}}\right)\right],1\right\\},$
(7)

where $E_{k}$ and $E_{k+1}$ are the energies of the replicas before the swap
at temperatures $T_{k}$ and $T_{k+1}$, respectively.

In order to estimate the density of states of the system, we use the multi-
histogram reweighting method [17, 18]. For this purpose, we measured the
canonical histograms in each thread. By utilizing the canonical distribution
function $P_{\mathrm{can}}(E;T_{k})\sim g(E)\exp\\{-E/k_{\mathrm{B}}T_{k}\\}$,
estimates for the density of states can be obtained by reweighting the energy
histograms $h(E;T_{k})$ measured in the different simulation threads $k$,
$\overline{g}(E)=h(E;T_{k})\exp\\{E/k_{\mathrm{B}}T_{k}\\}$. Using a single
histogram only covers a narrow range of energies effectively. Combining the
histograms obtained at different temperatures by employing the multi-histogram
reweighting method yields an estimator for the density of states that covers
the entire energy range [17, 18]:

$\widehat{g}(E)=\frac{\sum_{k=1}^{K}h(E;T_{k})}{\sum_{k=1}^{K}M_{k}Z_{k}^{-1}e^{-E/k_{\mathrm{B}}T_{k}}}.$
(8)

Here $M_{k}$ is the total number of sweeps in the thread $k$ and $Z_{k}$ is an
estimator for the partition function:

$Z_{k}=\sum_{E}\widehat{g}(E)e^{-E/k_{\mathrm{B}}T_{k}}.$ (9)

Equations (8) and (9) are solved iteratively until reasonable convergence has
been achieved.

###  II.3 Generalized microcanonical inflection-point analysis method

Historically, phase transitions have been identified and classified by means
of discontinuities or divergences in thermodynamic state variables or response
functions. However, systems of finite size do not exhibit such obvious
signals, as these only occur in the thermodynamic limit. In this study, we use
the recently introduced generalized microcanonical inflection-point analysis
method [4] for the systematic identification and classification of transitions
in systems of any size. This method, which combines microcanonical
thermodynamics [1] and the principle of minimal sensitivity [2, 3], has
already led to novel insights into the nature of phase transitions. Even the
Ising model, which has been excessively studied in almost a century, possesses
a more complex phase structure than previously known, as our most recent
analysis showed [19].

Adopting Boltzmann’s formula, the microcanonical entropy can be written as

$S(E)=k_{\mathrm{B}}\;\mathrm{ln}\;g(E),$ (10)

where $g(E)$ is the density of states with energy $E$. When not experiencing
phase transitions, the curves of entropy $S(E)$ and its derivatives exhibit
well-defined concave or convex monotony [4]. From canonical statistical
analysis of first- and second-order transitions, it is known that entropy
and/or internal energy rapidly change, if the temperature is varied near the
transition point. This behavior corresponds to least-sensitive dependencies of
microcanonical quantities in the space of system energies. Thus, a phase
transition causes a least-sensitive inflection point in the entropy or its
higher derivatives and impacts their monotonic behavior. By systematically
analyzing these alterations, different types of transitions can be identified
and classified.

In this scheme, a first-order transition is signaled by a least-sensitive
inflection point with energy $E_{\mathrm{tr}}$ in $S(E)$. Therefore, the first
derivative, which is the inverse microcanonical temperature $\beta(E)$,
possesses a positive valued minimum at $E_{\mathrm{tr}}$,

$\beta(E_{\mathrm{tr}})=\dfrac{\mathrm{d}S(E)}{\mathrm{d}E}\bigg{|}_{E=E_{\mathrm{tr}}}>0.$
(11)

Consequently, if there is a least-sensitive inflection point in $\beta(E)$,
the phase transition is classified as a second-order transition. The
derivative of $\beta(E)$ has a negative-valued peak at the transition energy
$E_{\mathrm{tr}}$,

$\gamma(E_{\mathrm{tr}})=\dfrac{\mathrm{d}^{2}S(E)}{\mathrm{d}E^{2}}\bigg{|}_{E=E_{\mathrm{tr}}}<0.$
(12)

More specifically, we call this an independent second-order transition. This
implies that there is another transition type: dependent transitions. As the
name suggests, dependent transitions are associated with an independent
transition of lower-order rank. It is important to note that not all
independent transitions have a dependent companion. However, if it exists -
and it can only exist at a higher transition energy than its independent
partner — it servers as a precursor of the independent transition in the
disordered phase. For example, if this inflection point is located within the
convex region of the entropy, which is associated with the first-order
transition, the transition is a dependent second-order transition.
Consequently, there is a positive-valued minimum in the derivative of
$\beta(E)$ at the transition energy $E_{\mathrm{tr}}$,

$\gamma(E_{\mathrm{tr}})=\dfrac{\mathrm{d}^{2}S(E)}{\mathrm{d}E^{2}}\bigg{|}_{E=E_{\mathrm{tr}}}>0.$
(13)

Generally, for an independent transition of odd order $(2k-1)$ ($k$ is a
positive integer), the $(2k-1)$th derivative of $S(E)$ possesses a positive-
valued minimum,

$\dfrac{\mathrm{d}^{(2k-1)}S(E)}{\mathrm{d}E^{(2k-1)}}\bigg{|}_{E=E_{\mathrm{tr}}}>0$
(14)

and an independent transition of even order $2k$ is characterized by a
negative-valued peak in $2k$th derivative,

$\dfrac{\mathrm{d}^{2k}S(E)}{\mathrm{d}E^{2k}}\bigg{|}_{E=E_{\mathrm{tr}}}<0.$
(15)

A dependent transition of even order $2k$ is identified by a positive-valued
minimum in the $2k$th derivative,

$\dfrac{\mathrm{d}^{2k}S(E)}{\mathrm{d}E^{2k}}\bigg{|}_{E=E_{\mathrm{tr}}^{\mathrm{dep}}}>0,$
(16)

while for odd order $2k+1$:

$\dfrac{\mathrm{d}^{(2k+1)}S(E)}{\mathrm{d}E^{(2k+1)}}\bigg{|}_{E=E_{\mathrm{tr}}^{\mathrm{dep}}}<0.$
(17)

Dependent transitions are less common than independent transitions. In this
polymer study we did not identify any dependent transitions.

We used the Bézier method [20, 21, 22] to smooth curves and calculate
derivatives.

##  III Results

###  III.1 Canonical analysis

In the past, the common approach to the characterization of phase transitions
has been canonical statistical analysis. For infinitely large systems, one or
more derivatives of an appropriate thermodynamic potential such as the free
enthalpy or the free energy would exhibit nonanalytic behavior, if one of the
potential’s natural variables was altered at the transition point. Because it
is easy to control in experiment, the analysis has typically been done in the
space of the canonical (or heat bath) temperature $T_{\mathrm{can}}$.
Discontinuities in the canonical entropy and specific heat help identify and
characterize discontinuous and continuous phase transitions, respectively, in
the thermodynamic limit. Later, this approach was simply extended to systems
of finite size that do not allow for the hypothetical extrapolation toward the
thermodynamic limit (for example, heterogeneous polymers such as proteins).

However, transition features in finite systems cannot be linked to
nonanalyticities. Hence, _peaks_ and _shoulders_ in response quantities like
heat capacity or order parameter fluctuations have usually been considered
signals of pseudotransitions in finite systems. The problem with this approach
is that extremal points in fluctuating quantities are not safe indicators of
phase transitions. The probably most prominent example is the specific-heat
curve for the one-dimensional Ising model, which exhibits a pronounced maximum
at a certain temperature, but there is no obvious feature that links this
extremal thermal activity to a phase transition. In the thermodynamic limit,
this peak does not develop into a divergence.

Interestingly, if one plots the heat capacity for the 1D Ising system as a
function of the _inverse temperature_ $\beta=1/T$, though, the peak vanishes
and there is no pseudotransition signal. Note that in the microcanonical
analysis the inverse temperature $\beta(E)=dS(E)/dE$ is the more obvious
thermodynamic variable than the temperature. Consequently, microcanonical
inflection-point analysis does not show any transition signal for the 1D Ising
system in equilibrium [19].

Figure 1: Canonical mean energy $\langle E\rangle$ as a function of
$\beta_{\mathrm{can}}$ for flexible ($\kappa=0$) and variants of semiflexible
polymers with different bending stiffness ($\kappa=1,2$). Error bars are shown
but smaller than the symbol sizes.

Figure 2: Thermal fluctuations of (a) energy (heat capacity $C_{V}=d\langle
E\rangle/dT_{\mathrm{can}}$) and (b) square radius of gyration ($d\langle
R^{2}_{\mathrm{gyr}}\rangle/dT_{\mathrm{can}}$), plotted as functions of
$\beta_{\mathrm{can}}$.

In fact, historically the (canonical) temperature was introduced when the need
arose to assign a reliable scale to the thermoscope to quantify the
environmental heat content, which ultimately lead Galilei to the introduction
of the thermometer at the end of the 16th century. It is ironic that the
nowadays most commonly used centigrade scale, the Celsius scale, was
introduced in 1742 by Celsius in _inverted_ form, i.e., Celsius chose 0∘ as
the boiling point of water under norm conditions and 100∘ as the reference
point for the melting of ice. At about the same time, French scientist
Christin independently used a centigrade scale that essentially inverted the
original scale by Celsius, which is what we nowadays refer to as the _Celsius
Scale_ [23].

This brief historic recollection shows that the temperature and its scale were
introduced in an era, when it was still believed to be a quantitative measure
for heat, and the scale was ultimately linked to the instrument and substances
used for its measurement. Thus, there is no physical reason that favors the
historic quantity temperature over the inverse temperature, which is the more
natural variable in the microcanonical analysis. To distinguish the
microcanonical temperature from the heat bath temperature, we denote the
latter by $T_{\mathrm{can}}$ in the following canonical analysis of structural
transitions.

Consequently, we plot the canonical mean energy $\langle E\rangle$ as a
function of $\beta_{\mathrm{can}}=1/T_{\mathrm{can}}$ in Fig. 1 for the three
models studied: $\kappa=0$ (flexible chain) and $\kappa=1,2$ (semiflexible
variants). All curves show a sharp drop at $\beta_{\mathrm{can}}\approx 0.7$.
Around $\beta_{\mathrm{can}}\approx 3.2$, the flexible chain ($\kappa=0$)
experiences another significant drop in energy, which is less pronounced for
the semiflexible polymer with bending stiffness $\kappa=1$. No obvious signal
is visible on this level for the stiffer chain with $\kappa=2$.

The plots of fluctuating quantities, such as the heat capacity $C_{V}=d\langle
E\rangle/dT_{\mathrm{can}}$ and the fluctuations of the mean square radius of
gyration, $d\langle R^{2}_{\mathrm{gyr}}\rangle/dT_{\mathrm{can}}$, allow for
a more detailed analysis. Both quantities are shown as functions of
$\beta_{\mathrm{can}}$ in Figs. 2(a) and 2(b), respectively. Whereas we
certainly see changes in the curvature of the $C_{V}$ curves at
$\beta_{\mathrm{can}}\approx 0.7$, only the plots of the structural
fluctuations show sharp peaks. These indicate the well-known $\Theta$ collapse
transition between extended, random-coil structures and compact globular
conformations. The transition signal is more pronounced for the stiffer
chains. Since this transition is more entropy than energy driven, it is not
surprising that it shows up more prominently in the structural rather than the
energetic fluctuations. This is different for the freezing transition at about
$\beta_{\mathrm{can}}\approx 3.2$, which is strongest for the flexible chain
($\kappa=0$). Large error bars at very small temperatures
($\beta_{\mathrm{can}}\approx 8.5$) prevent a conclusion about another
separate transition for the semiflexible polymer with $\kappa=2$, which, if it
exists, marks the transition into the global energy minimum basin.

###  III.2 Microcanonical analysis of phase transitions

In this section, we perform a full-scale microcanonical inflection-point
analysis of the three models that aims at the identification of all structural
transitions in these systems up to third order. Figures 3, 4, and 5 show the
microcanonical entropy and its derivatives up to third order as functions of
the reduced energy $\Delta E^{(\kappa)}=E-E_{\mathrm{min}}^{(\kappa)}$, i.e.,
we subtracted from the energy the respective putative global energy minima
$E_{\mathrm{min}}^{(\kappa)}$ obtained for each system in the parallel
tempering simulations and verified by simulated annealing. This shift allows
for an easier comparison of the results.

Figure 3: (a) Microcanonical entropy $S$ and its derivatives (b)
$\beta=dS/dE$, (c) $\gamma=d\beta/dE$, (d) $\delta=d\gamma/dE$ for the
different models with $\kappa=0,1,2$ plotted as functions of the energy
difference from the $\kappa$-dependent global energy minimum estimate. In this
figure, we focus on the high-energy (or high-temperature) regime. Least-
sensitive inflection points are marked by a dot.

As in the canonical analysis, we first discuss the low-$\beta$ (or high-
temperature) regime in the relevant energy space. The entropies plotted in
Fig. 3(a) for all cases do not possess least-sensitive inflection points and
thus there is no first-order transition, as expected. However, least-sensitive
inflection points in the first derivative ($\beta$) signal second-order phase
transitions in all three systems, which reflect the mostly entropic $\Theta$
collapse from random-coil to globular polymer structures. The corresponding
peak locations in the second entropy derivative ($\gamma$), shown in Fig.
3(c), allow for a unique determination of the transition points in energy
space and thus also in $\beta$, thereby rendering the ambiguous canonical
analysis of response functions in the previous section obsolete. Since there
are no least-sensitive inflection points in the $\gamma$ plots, none of the
systems undergoes a third-order transition in this energy region.

Figure 4 shows the same microcanonical quantities as plotted in Fig. 3, but
for an intermediate energy range that covers the inverse temperatures in the
interval $\beta=(2,4)$. As expected for flexible polymers, the entropy curve
for $\kappa=0$ does exhibit a least-sensitive inflection point, which
corresponds to the minimum in the backbending region found in the $\beta$ plot
[Fig. 4(b)] at about $\Delta E^{(\kappa)}=36.5$. The inverse temperature
associated with it is approximately $\beta=2.95$, which confirms earlier
results for flexible polymers [6]. The polymer undergoes a freezing transition
from the liquid-globular states into the solid phase, which for this model is
dominated by icosahedral structures. Remarkably, for the semiflexible polymers
with $\kappa=1,2$, the situation changes. For $\kappa=1$, there is still a
weak inflection point in the entropy, but the $\beta$ curve has already almost
plateaued at about the same energy value, where we found the first-order
transition for the flexible polymer. Consequently, the peak value of the next
derivative ($\gamma$) is virtually zero at this energy [Fig. 4(c)]. Therefore,
the freezing transition seems to turn from first order for the flexible
polymer ($\kappa=0$) to second order for the semiflexible case with
$\kappa=1$. However, even more surprisingly, the transition behavior changes
again for the stiffer chain with $\kappa=2$: In fact, the transition signal
has completely vanished. The subsequent structural analysis of ground-state
properties will lend deeper insight into the reasons for these changes.

The second derivative $\gamma$ in Fig. 4(c) shows signs of independent third-
order transitions for $\kappa=0$ and $\kappa=1$ in the ordered phase, but not
for $\kappa=2$. These results suggest that freezing into a unique and
characteristic global energy minimum state is not possible if bending effects
overwhelm non-bonded interactions that are commonly responsible for tertiary
structure formation in polymeric systems. Local bending effects turn into
restraints that inhibit or at least suppress the formation of symmetries.
Previous studies have shown that bending restraints play an important role in
the formation of stable conformations in tertiary assemblies of helical
segments [24].

Figure 4: Same quantities as in Fig. 3, but plotted for an intermediate energy
region.

Figure 5: Plots of the same microcanonical quantities as in Figs. 3 and 4 in
the lowest-energy regions of the three models.

Figure 6: Representatives of lowest-energy conformations for $\kappa=0,1,2$.
Labels help to follow the chain from the first monomer (1, blue) to the last
(55, red).

Eventually, the sequence of figures shown in Fig. 5 covers the lowest-energy
regime for all models compared in this study. In the flexible case, no
additional transitions are expected and, correspondingly, we do not see
signals in any of the derivatives up to third order that would indicate a
transition. The situation is potentially different for the systems with
bending restraint. Whereas we do not find indications of transition signals in
the entropy and inverse temperature curves for $\kappa=1$ and $\kappa=2$ in
Figs. 5(a) and 5(b), respectively, the $\gamma$ results shown in Fig. 5(c)
might tell a different story. The error bars are too large for an ultimate
conclusion, but it looks like a third-order transition develops close to the
ground state for the semiflexible polymer models.

The quantitative results of the microcanonical inflection-point analysis
obtained for the polymer systems studied here are listed in Table 1.

Table 1: Transitions found from microcanonical analysis for different flexible
($\kappa=0$) and semiflexible ($\kappa=1,2$) polymers. Transition energy
$E_{\mathrm{tr}}$, distance from putative ground-state energy $\Delta
E^{(\kappa)}$, inverse microcanonical transition temperature
$\beta_{\mathrm{tr}}$, and the order of the transition (classification) are
listed.

Bending stiffness $E_{\mathrm{tr}}$ $\Delta E^{(\kappa)}$
$\beta_{\mathrm{tr}}$ Classification -231.2 30.5 3.20 3 $\kappa=0$ -225.2 36.5
2.95 1 -68.2 193.5 0.66 2 -205.8 25.1 3.63 3 $\kappa=1$ -197.0 33.9 3.20 1
-24.6 206.3 0.68 2 $\kappa=2$ 5.0 207.4 0.70 2

###  III.3 Structural analysis of lowest-energy conformations

The lowest energies found in the simulations, which we consider best estimates
of the ground state energies, are $E_{\mathrm{min}}^{(0)}=-261.7$ for the
flexible polymer ($\kappa=0$), $E_{\mathrm{min}}^{(1)}=-230.9$ for the
semiflexible chain with $\kappa=1$, and $E_{\mathrm{min}}^{(2)}=-212.4$ if
$\kappa=2$. All values are given in units of the energy scale of the Lennard-
Jones potential, $\epsilon_{\mathrm{LJ}}$. Since the bonded interactions (bond
vibrations) are negligible near the ground state, the differences in the
ground-state energy estimates for the different models must be attributed to
the penalty paid for chain bending. The ground-state conformation for the
flexible polymer that does not experience these restraints is formed by
optimizing the non-bonded interaction and results, for this model, in an
ideally icosahedral structure. Turning on the bending restraint and setting
$\kappa=1$, the competition between nonlocal attraction and local repulsion
increases the tension in the icosahedral structure, although it still stays in
shape. For $\kappa=2$, however, it cannot maintain the optimal icosahedral
arrangement of monomers and rather forms an entangled structure of longer,
less bent, segments. As we have seen in the thermodynamic analysis of the
transitions, the phase behavior changes significantly with increasing bending
stiffness.

Figure 7: Pair distribution functions for the lowest-energy states shown in
Fig. 6 for $\kappa=0,1,2$.

Representative lowest-energy conformations are shown in Fig. 6. For a more
quantitative analysis let us take a look at the pair distribution functions
and the contact maps. We introduce the pair distribution function as

$P(r)=\sum\limits_{i<j}\Delta(r-r_{i,j}),$ (18)

where

$[h]\Delta(r-r_{i,j})=\left\\{\begin{array}[]{ll}1,&|r-r_{i,j}|<r_{t},\\\
0,&\mathrm{otherwise.}\end{array}\right.$ (19)

As a robust threshold for the necessary binning of the $r$ space, we chose
$r_{t}=0.01$. The histograms for the lowest-energy conformations shown in Fig
6 are plotted in Fig. 7. The perfect icosahedral structure is only found for
$\kappa=0$, whereas its decay is already visible for the weaker semiflexible
polymer ($\kappa=1$). The maximum number of nearest-neighbor contacts
($r_{i,j}\approx 1$) found for $\kappa=0$ is not reached in the semiflexible
cases. The broadening of the peaks and additional spikes not present for
$\kappa=0,1$ are clear indicators that the ground-state structure for the
semiflexible polymer with $\kappa=2$ is not icosahedral. Bending restraints
prevent the formation of perfect symmetries and thus a characterization of the
$\kappa=2$ ground-state conformation as a distinct crystalline or
quasicrystalline structure is not possible. It rather resembles tertiary folds
of protein conformations, where effective bending restraints and the local
stable secondary segments purposefully prevent symmetric arrangements of
monomers. This enables different heteropolymers of similar size to form
distinct and functional individual conformations.

Figure 8: Contact maps for the lowest-energy states shown in Fig. 6 for
$\kappa=0,1,2$

This is obvious from the plots of the contact maps for the lowest-energy
states shown in Fig. 8. In the grid spanned by the monomer labels, we mark
pairs of monomers with distances $r_{i,j}<1.2$. Bonded monomer pairs are not
included. For the flexible polymer, the contact map does not exhibit
particularly remarkable structural features. Since there are no bond angle
restraints, each monomer tries to maximize its number of nearest neighbors for
the energetic benefit. Icosahedral conformations with one of the end monomers
in the center are optimal. For the semiflexible polymer with $\kappa=1$, we
already see that changes occur. Monomers do not only try to maximize the
number of contacts, but three monomers connected by two adjacent bonds now
have to cooperate to minimize bending. Close to the diagonal, we find that
short anti-diagonal streaks form. These are clear indicators of turns with two
linear strands in contact with each other (hairpins). In protein folds, these
would be referred to as building blocks for $\beta$-sheets. For stronger
bending rigidity ($\kappa=2$), there are fewer, but longer such segments. We
also observe the formation of streaks parallel to the diagonal, which can be
associated with helical alignments. Both types of these secondary structure
elements can be seen in the geometric representation in Fig. 6. The hairpin-
like sections are located in the interior and the helical segments wrap around
it. Of course, under these conditions, energetically favored structures are
not icosahedral. In fact, no global symmetries can be identified in this
tertiary fold. Although it is tempting to think of these structures in analogy
to protein folds, the lack of dihedral (torsion) constraints, which separates
secondary structure elements from each other in the tertiary fold, prevents a
more direct comparison.

##  IV Summary

The statistical analysis of structural transitions by the recently developed
generalized microcanonical inflection-point method [4] had already yielded
promising results in studies of flexible polymers [6]. For this study, we
extended the coarse-grained model by incorporating bending restraints of
different strengths to investigate the impact of bending stiffness on the
microcanonical entropy and its derivatives, which are used as indicator
functions for the systematic identification and classification of phase
transitions in systems of any size. In this coarse-grained model for
semiflexible polymers, the bending restraint is controlled by the bending
stiffness parameter $\kappa$.

Despite its simplicity, the model required a careful numerical treatment. For
this purpose, we performed extensive parallelized replica-exchange computer
simulations and verified the structural behavior in the lowest-energy basin by
means of stochastic methods optimized for this purpose such as simulated
annealing. Microcanonical thermodynamic properties were obtained from the
density of states, which is the main output of the multiple-histogram
reweighting method applied to the raw simulation data.

Based on the microcanonical results obtained in the simulations, we compared
the phase behavior of three model variants: purely flexible polymers
($\kappa=0$) and semiflexible polymers with $\kappa=1$ (identical energy
scales of attractive nonbonded monomer-monomer interactions and repulsive
bending strength) and $\kappa=2$ (bending restraints dominate over monomer-
monomer attraction). Our simulations reproduced previous results for the
flexible reference system very well, creating sufficient confidence for the
subsequent studies of the semiflexible polymers. Because of the additional
restraints, it was significantly more challenging to achieve in simulations of
semiflexible polymers the data quality necessary to enable an accurate
microcanonical analysis.

For the flexible polymer, we identified the known structural transitions,
i.e., the characteristic second-order transition associated with the $\Theta$
collapse from extended random coils to liquid globules and the first-order
liquid-solid transition, which is accompanied by an independent third-order
transition. The lowest-energy conformation is icosahedral, as expected. The
behavior does not change qualitatively for the weaker semiflexible case with
$\kappa=1$, where the effectively repulsive bending effects are competitive to
attractive monomer-monomer interaction. However, the larger number of possible
attractive monomer-monomer contacts represents an entropic advantage and thus
the icosahedral solid phase is still maintained. This is not the case anymore
if the chain is further stiffened by choosing $\kappa=2$. Whereas the $\Theta$
collapse remains widely unaffected, there is no obvious transition into the
solid phase anymore. Signals at very low energies at the third-order
transition level are too noisy to provide a clear picture. However, it is
obvious that the lowest-energy conformation found is a compromise of
satisfying both compactness and bending restraint. Longer, only slightly bent
segments form and assemble in a well-organized, coil-like structure. It is an
intriguing future task to extend our microcanonical study to stronger bending
restraints and to construct the complete hyperphase diagram in the spaces of
bending stiffness and inverse temperature and to apply it to other modern
applications in polymer science like hyperbranched polymers [25].

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# A brief note on the compatibility of quantum instruments: A conceptual
problem

Arindam Mitra1,2 1Optics and Quantum Information Group, The Institute of
Mathematical Sciences, C. I. T. Campus, Taramani, Chennai 600113, India.
2Homi Bhabha National Institute, Anushakti Nagar, Mumbai 400094, India.

###### Abstract

The compatibility of quantum instruments is defined as the existence of a
joint instrument through the implementation of which one can simultaneously
implement the individual instruments by doing classical post-processing. In
this paper, we discuss the concept of parallel compatibility of instruments
and claim that the traditional definition of compatibility of instruments is
conceptually incomplete and provide a re-definition of the compatibility of
quantum instruments. We discuss the proper reason for the re-definition. We
show that the modified definition of the compatibility of quantum instruments
can capture the idea of observable-observable compatibility, observable-
channel compatibility and channel-channel compatibility.

##  I Introduction

Incompatibility is a basic feature of quantum mechanics [1]. A set of quantum
devices is compatible if those devices are simultaneously implementable given
a single input state. Incompatibility of observables is necessary for the
demonstration of several quantum information theoretic tasks or to get an
advantage in different quantum information theoretic tasks [2, 3, 4, 5]. On
the other hand, from the incompatibility of quantum channel, one can arrive at
no-cloning theorem [6, 7]. Similarly, incompatibility between the identity
channel and any non-trivial observable is consistent with the uncertainty
principal [8].
In this paper, we study the compatibility of quantum instruments. We
investigate whether the traditional definition of the compatibility of quantum
instruments is conceptually complete.
The rest of this paper is organised as follows. In Sec. II, we discuss the
preliminaries. In Sec. III, we discuss the main results. More specifically, in
Sec. III.1, we discuss the concept of parallel compatibility of instruments
and show that this concept is different than the traditional definition of
compatibility of instruments. In Sec. III.2, we claim that the traditional
definition of compatibility of instruments is conceptually incomplete and
provide a re-definition of the compatibility of quantum instruments and we
discuss the proper reason for the re-definition. In Sec. III.3, we show that
the modified definition of compatibility of quantum instruments can capture
the idea of observable-observable compatibility, observable-channel
compatibility and channel-channel compatibility. Finally, in Sec. IV, we
summarize our work and discuss the future outlook.

##  II Preliminaries

In this section, we discuss the preliminaries.

###  II.1 Observables

In the quantum mechanics, an observable $A$ which is acting on Hilbert space
$\mathcal{H}$, is defined by a set of positive Hermitian matrices
$A=\\{A(x)\\}$ such that $\sum_{x}A(x)=\mathbb{I}$ where $\mathbb{I}$ is a
$d\times d$ identity matrix where $d$ is the dimension of the Hilbert space
$\mathcal{H}$ [9, 10]. The probability of obtaining a outcome $x$ is
$p_{A}(x)=\text{tr}[\rho A(x)]$ for any quantum state
$\rho\in\mathcal{S}(\mathcal{H})$ where $\mathcal{S}(\mathcal{H})$ is the
state space (i.e., the set of all density matrices). We denote the outcome set
of an observable $A$ as $\Omega_{A}$.

###  II.2 Quantum channels

A quantum channel
$\Lambda:\mathcal{S}(\mathcal{H})\rightarrow\mathcal{S}(\mathcal{K})$ is a
completely positive trace preserving map (CPTP map) [9, 10]. Any quantum
channel $\Lambda$ can be written as $\Lambda(\rho)=\sum_{i}K_{i}\rho
K^{\dagger}_{i}$ where $K_{i}$’s are the Kraus operators and satisfies
$\sum_{i}K_{i}^{\dagger}K_{i}=\mathbb{I}$. This is known as the Kraus
representation of $\Lambda$.

Suppose $\Phi:\mathcal{S}(\mathcal{H})\rightarrow\mathcal{L}^{+}(\mathcal{K})$
is a completely positive map (CP map). Then if
$\Phi^{*}:\mathcal{L}(\mathcal{K})\rightarrow\mathcal{L}(\mathcal{H})$ is the
dual (i.e., in Heisenberg picture) of the CP map $\Phi$ then
$\text{tr}[\Phi(\rho)X]$=$\text{tr}[\rho\Phi^{*}(X)]$ for all quantum state
$\rho\in\mathcal{S}(\mathcal{H})$ and operator $X\in\mathcal{L}(\mathcal{K})$
where $\mathcal{L}(\mathcal{K})$ is the set of the linear operators in the
Hilbert space $\mathcal{K}$ and $\mathcal{L}^{+}(\mathcal{K})$ is the set of
the positive linear operators in the Hilbert space $\mathcal{K}$. Then
clearly, for any CPTP map
$\Lambda:\mathcal{S}(\mathcal{H})\rightarrow\mathcal{S}(\mathcal{K})$,
$\Lambda^{*}(\mathbb{I}_{\mathcal{K}})=\mathbb{I}_{\mathcal{H}}$ [10]. If
$\Gamma:\mathcal{S}(\mathcal{H})\rightarrow\mathcal{S}(\mathcal{K}_{1}\otimes\mathcal{K}_{2})$
is a CP map and
$\Lambda:\mathcal{S}(\mathcal{H})\rightarrow\mathcal{S}(\mathcal{K}_{1})$ is a
CP map such that $\Lambda(\rho)=\text{tr}_{\mathcal{K}_{2}}[\Gamma(\rho)]$ for
all quantum state $\rho$ then
$\text{tr}_{\mathcal{K}_{1}}[\Lambda(\rho)]=\text{tr}[\Gamma(\rho)]$ for all
quantum state $\rho$ which implies
$\text{tr}_{\mathcal{K}_{1}}[\rho\Lambda^{*}(\mathbb{I}_{\mathcal{K}_{1}})]=\text{tr}[\rho\Gamma^{*}(\mathbb{I}_{\mathcal{K}_{1}\otimes\mathcal{K}_{2}})]$
for all quantum state $\rho$. Therefore,
$\Lambda^{*}(\mathbb{I}_{\mathcal{K}_{1}})=\Gamma^{*}(\mathbb{I}_{\mathcal{K}_{1}\otimes\mathcal{K}_{2}})$.

###  II.3 Quantum Instruments

A quantum instrument $\mathcal{I}$ is defined as a set of completely positive
maps (CP maps)
$\\{\Phi_{x}:\mathcal{S}(\mathcal{H})\rightarrow\mathcal{L}^{+}(\mathcal{K})\\}$
such that $\sum_{x}\Phi_{x}=\Phi^{\mathcal{I}}$ is a CPTP map [10]. An
observable $A$ can be measured through implementing an instrument if
$\text{tr}[\Phi_{x}(\rho)]=\text{tr}[\rho A(x)]$. Such instruments are known
as $A$-compatible instrument. In this case, if
$\Phi_{x}(\rho)=\sum_{j}K_{xj}\rho K_{xj}^{\dagger}$, then
$\sum_{j}K_{xj}^{\dagger}K_{xj}=A(x)$ and therefore, clearly
$\sum_{xj}K_{xj}^{\dagger}K_{xj}=\mathbb{I}$. A lot of works have been already
done to understand the different properties of the quantum instruments [11,
12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25].

###  II.4 Three kinds of compatibility in quantum mechanics

Depending on the types of the quantum devices, there are mainly three kinds of
compatibility in quantum mechanics [1]. Those are-

  1. 1.

_Observable-Observable compatibility:_ Two observables $A=\\{A(x)\\}$ and
$B=\\{B(y)\\}$ are compatible if there exists an observable
$\mathcal{G}=\\{G(x,y)\\}$ with outcome set
$\Omega_{\mathcal{G}}=\Omega_{A}\times\Omega_{B}$ such that

$A(x)=\sum_{y}G(x,y);~{}B(y)=\sum_{x}G(x,y)$ (1)

for all $x\in\Omega_{A}$ and $y\in\Omega_{B}$. Through measuring the
observable $\mathcal{G}$, one can simultaneously measure both of the
obervables $A$ and $B$.

  2. 2.

_Observable-Channel compatibility:_ An observable $A=\\{A(x)\\}$ acting on the
Hilbert space $\mathcal{H}$ and a quantum channel
$\Lambda:\mathcal{S}(\mathcal{H})\rightarrow\mathcal{S}(\mathcal{K})$ is
compatible if there exists a quantum instrument
$\mathcal{I}=\\{\Phi_{x}:\mathcal{S}(\mathcal{H})\rightarrow\mathcal{L}^{+}(\mathcal{K})\\}$
such that $\text{tr}[\Phi_{x}(\rho)]=\text{tr}[\rho A(x)]$ for all
$x\in\Omega_{A}$ and $\rho\in\mathcal{S}(\mathcal{H})$ and
$\sum_{x}\Phi_{x}=\Lambda$. Through implementing the quantum instrument
$\mathcal{I}$, one can simultaneously measure the observable $A$ and implement
the channel $\Lambda$.

  3. 3.

_Channel-Channel compatibility:_ Two quantum channels
$\Lambda_{1}:\mathcal{S}(\mathcal{H})\rightarrow\mathcal{S}(\mathcal{K}_{1})$
and
$\Lambda_{2}:\mathcal{S}(\mathcal{H})\rightarrow\mathcal{S}(\mathcal{K}_{2})$
are compatible if there exists a quantum channel
$\Lambda:\mathcal{S}(\mathcal{H})\rightarrow\mathcal{S}(\mathcal{K}_{1}\otimes\mathcal{K}_{2})$
such that $\Lambda_{1}(\rho)=\text{tr}_{\mathcal{K}_{2}}[\Lambda(\rho)]$ and
$\Lambda_{2}(\rho)=\text{tr}_{\mathcal{K}_{1}}[\Lambda(\rho)]$ for all
$\rho\in\mathcal{S}(\mathcal{H})$. Through implementing the channel $\Lambda$,
one can simultaneously implement both of the channels $\Lambda_{1}$ and
$\Lambda_{2}$.

##  III Compatibility of quantum Instruments

###  III.1 Definitions and concepts

######  Definition 1 (Traditional compatibility).

Two quantum instruments
$\mathcal{I}_{1}=\\{\Phi^{1}_{x}:\mathcal{S}(\mathcal{H})\rightarrow\mathcal{L}^{+}(\mathcal{K})\\}$
and
$\mathcal{I}_{2}=\\{\Phi^{2}_{y}:\mathcal{S}(\mathcal{H})\rightarrow\mathcal{L}^{+}(\mathcal{K})\\}$
are (traditionally) compatible if there exists an instrument
$\mathcal{I}=\\{\Phi_{xy}:\mathcal{S}(\mathcal{H})\rightarrow\mathcal{L}^{+}(\mathcal{K})\\}$
such that $\sum_{y}\Phi_{xy}(.)=\Phi^{1}_{x}(.)$ and
$\sum_{x}\Phi_{xy}(.)=\Phi^{2}_{y}(.)$ for all $x,y$.

The definition 1 is the traditional definition of the compatibility of quantum
instruments and is given in the Sec. 2.2.2 in page no. 33 in [15] as
Definition 2.5. The same definition is given in the Sec. 2 in page no. 15 in
[16], but the term ”coexistence” has been used instead of the term
”compatibility”. This means that for traditionally compatible instruments,
there exists an joint instrument, implementation of which is equivalent to the
simultaneous implementation of both the instruments and individual instruments
can be obtained using the classical post-processing of that joint instrument.
Fig. 1(a) shows the traditional compatibility of the instruments
$\mathcal{I}_{1}$ and $\mathcal{I}_{2}$.

######  Definition 2 (Weak compatibility).

Two quantum instruments
$\mathcal{I}_{1}=\\{\Phi^{1}_{x}:\mathcal{S}(\mathcal{H})\rightarrow\mathcal{L}^{+}(\mathcal{K})\\}$
and
$\mathcal{I}_{2}=\\{\Phi^{2}_{y}:\mathcal{S}(\mathcal{H})\rightarrow\mathcal{L}^{+}(\mathcal{K})\\}$
are weakly compatible if $\sum_{x}\Phi^{1}_{x}=\Lambda$ and
$\sum_{y}\Phi^{2}_{y}=\Lambda$.

It is known that if a set of instrument is compatible then it is also weakly
compatible, but it is easily understood that converse is not true, in general
[15].

Now we provide a new concept of compatibility which we call as ”the parallel
compatibility”. The definition of parallel compatibility is given below-

######  Definition 3 (Parallel compatibility).

Two quantum instruments
$\mathcal{I}_{1}=\\{\Phi^{1}_{x}:\mathcal{S}(\mathcal{H})\rightarrow\mathcal{L}^{+}(\mathcal{K}_{1})\\}$
and
$\mathcal{I}_{2}=\\{\Phi^{2}_{y}:\mathcal{S}(\mathcal{H})\rightarrow\mathcal{L}^{+}(\mathcal{K}_{2})\\}$
are parallelly compatible if there exists an instrument
$\mathcal{I}=\\{\Phi_{xy}:\mathcal{S}(\mathcal{H})\rightarrow\mathcal{L}^{+}(\mathcal{K}_{1}\otimes\mathcal{K}_{2})\\}$
such that $\sum_{y}\text{tr}_{\mathcal{K}_{2}}\Phi_{xy}(.)=\Phi^{1}_{x}(.)$
and $\sum_{x}\text{tr}_{\mathcal{K}_{1}}\Phi_{xy}(.)=\Phi^{2}_{y}(.)$ for all
$x,y$.

Two instruments $\mathcal{I}_{1}$ and $\mathcal{I}_{2}$ which are parallelly
compatible, can be implemented parallelly (i.e., there will be two output
subsystems (on $\mathcal{S}(\mathcal{K}_{1})$ and on
$\mathcal{S}(\mathcal{K}_{2})$ respectively) and effectively $\mathcal{I}_{1}$
will be implemented on the first subsystem and effectively $\mathcal{I}_{2}$
will be implemented on the second subsystem) . Clearly, if
$\mathcal{I}_{1}=\\{\Phi^{1}_{x}:\mathcal{S}(\mathcal{H})\rightarrow\mathcal{L}^{+}(\mathcal{K}_{1})\\}$
and
$\mathcal{I}_{2}=\\{\Phi^{2}_{y}:\mathcal{S}(\mathcal{H})\rightarrow\mathcal{L}^{+}(\mathcal{K}_{2})\\}$
are parallelly compatible, then the channels $\Phi^{1}=\sum_{x}\Phi^{1}_{x}$
and $\Phi^{2}=\sum_{y}\Phi^{2}_{y}$ are compatible (the joint channel is
$\Phi=\sum_{xy}\Phi_{xy}$). Next example (and also Fig. 1(b)) will clarify
this concept.

$\rho$$\mathcal{I}$Classical post-processingof
outcomes$\mathcal{I}_{1}$$\mathcal{I}_{2}$ (a) Traditional compatibility: This
figure shows the traditional way to implement two instruments simultaneously
which is mentioned in th Definition 1. This can be done through two steps- (i)
implementing the joint instrument $\mathcal{I}$ on the state $\rho$ and (ii)
then performing the post-processing of outcomes i.e, taking the marginal over
the outcomes either $x$ or $y$ as mentioned in th Definition 1.

$\rho$Joint channel $\Lambda$(Approx. asymm.
cloning)$\Lambda_{1}$$\mathcal{I}_{1}$$\Lambda_{2}$$\mathcal{I}_{2}$$\mathcal{J}_{1}$$\mathcal{J}_{2}$
(b) Parallel compatibility: This figure shows one specific way of parallel
simultaneous implementation(defined in the Definition 3) of two instruments
which is mathematically described in Example 1. This can be done through the
following steps- (i) implementing the channel $\Lambda$ on the state $\rho$
which is the joint channel of the compatible channels $\Lambda_{1}$ and
$\Lambda_{2}$ (where $\Lambda_{1}(\rho)$ and $\Lambda_{2}(\rho)$ can be
considered as the approximate unequal clones (unless
$\Lambda_{1}=\Lambda_{2}$) of the state $\rho$, in general and therefore, it
can be considered as approximate asymmetric cloning), in general and then (ii)
applying the instruments $\mathcal{J}_{1}$ and $\mathcal{J}_{2}$ on
$\Lambda_{1}(\rho)$ and $\Lambda_{2}(\rho)$ respectively such that
$\mathcal{J}_{1}\circ\Lambda_{1}=\mathcal{I}_{1}$ and
$\mathcal{J}_{2}\circ\Lambda_{2}=\mathcal{I}_{2}$. The existances such channel
$\Lambda$ and such instruments $\mathcal{J}_{1}$ an $\mathcal{J}_{2}$ implies
the parallel compatibility of the instruments $\mathcal{I}_{1}$ and
$\mathcal{I}_{2}$

Figure 1: The Fig. 1(a) and Fig. 1(b) shows two ways of simultaneous
implementation of two instruments. These two ways are inequivalent which is
evident from the Proposition 1 and the Proposition 2.

######  Example 1 (An example of parallelly compatible instruments).

Consider two compatible quantum channels
$\Lambda_{1}:\mathcal{S}(\mathcal{H})\rightarrow\mathcal{S}(\mathcal{H}_{1})$
and
$\Lambda_{2}:\mathcal{S}(\mathcal{H})\rightarrow\mathcal{S}(\mathcal{H}_{2})$
with a joint channel
$\Lambda:\mathcal{S}(\mathcal{H})\rightarrow\mathcal{S}(\mathcal{H}_{1}\otimes\mathcal{H}_{2})$.
Therefore, since from the No-signaling principle, implementation of local
quantum channel on one side does not change the density matrix on the other
side, for any arbitrary quantum channel
$\Gamma_{1}:\mathcal{S}(\mathcal{H}_{1})\rightarrow\mathcal{S}(\mathcal{K}_{1})$
and
$\Gamma_{2}:\mathcal{S}(\mathcal{H}_{2})\rightarrow\mathcal{S}(\mathcal{K}_{2})$,
$\text{tr}_{\mathcal{K}_{2}}(\mathbb{I}\otimes\Gamma_{2})\circ\Lambda(.)=\Lambda_{1}(.)$
and
$\text{tr}_{\mathcal{K}_{1}}(\Gamma_{1}\otimes\mathbb{I})\circ\Lambda(.)=\Lambda_{2}(.)$.
Now consider a pair of arbitrary quantum instruments
$\mathcal{J}_{1}=\\{\Phi^{\prime
1}_{x}:\mathcal{S}(\mathcal{H}_{1})\rightarrow\mathcal{L}^{+}(\mathcal{K}_{1});~{}\sum_{x}\Phi^{\prime
1}_{x}=\Gamma_{1}\\}$ and $\mathcal{J}_{2}=\\{\Phi^{\prime
2}_{y}:\mathcal{S}(\mathcal{H}_{2})\rightarrow\mathcal{L}^{+}(\mathcal{K}_{2});~{}\sum_{y}\Phi^{\prime
2}_{y}=\Gamma_{2}\\}$. Consider another pair of instruments
$\mathcal{I}_{1}=\mathcal{J}_{1}\circ\Lambda_{1}=\\{\Phi^{1}_{x}=\Phi^{\prime
1}_{x}\circ\Lambda_{1}:\mathcal{S}(\mathcal{H})\rightarrow\mathcal{L}^{+}(\mathcal{K}_{1})\\}$
and
$\mathcal{I}_{2}=\mathcal{J}_{2}\circ\Lambda_{2}=\\{\Phi^{2}_{y}=\Phi^{\prime
2}_{y}\circ\Lambda_{2}:\mathcal{S}(\mathcal{H})\rightarrow\mathcal{L}^{+}(\mathcal{K}_{2})\\}$.
Now again consider a giant instrument $\mathcal{I}=\\{\Phi_{xy}=(\Phi^{\prime
1}_{x}\otimes\Phi^{\prime
2}_{y})\circ\Lambda:\mathcal{S}(\mathcal{H})\rightarrow\mathcal{L}^{+}(\mathcal{K}_{1}\otimes\mathcal{K}_{2})\\}$.

Clearly, for all $y$

$\displaystyle\Phi^{1}_{x}(.)$ $\displaystyle=\Phi^{\prime
1}_{x}\circ\Lambda_{1}(.)$ $\displaystyle=\Phi^{\prime
1}_{x}\circ\text{tr}_{\mathcal{K}_{2}}(\mathbb{I}\otimes\Gamma_{2})\circ\Lambda(.)$
$\displaystyle=\Phi^{\prime
1}_{x}\circ\text{tr}_{\mathcal{K}_{2}}(\mathbb{I}\otimes\sum_{y}\Phi^{\prime
2}_{y})\circ\Lambda(.)$
$\displaystyle=\text{tr}_{\mathcal{K}_{2}}\sum_{y}(\Phi^{\prime
1}_{x}\otimes\Phi^{\prime 2}_{y})\circ\Lambda(.)$
$\displaystyle=\sum_{y}\text{tr}_{\mathcal{K}_{2}}\Phi_{xy}(.)$ (2)

Similarly, $\Phi^{2}_{y}=\sum_{x}\text{tr}_{\mathcal{K}_{1}}\Phi_{xy}$ for all
$x$. Hence, $\mathcal{I}_{1}$ and $\mathcal{I}_{2}$ are parallelly compatible.

Next Lemma relates observable-observable compatibility and parallel
compatibility of instruments-

######  Lemma 1.

If an $A$-compatible quantum instrument
$\mathcal{I}_{A}=\\{\Phi^{1}_{x}:\mathcal{S}(\mathcal{H})\rightarrow\mathcal{L}^{+}(\mathcal{K}_{1})\\}$
and a $B$-compatible quantum instrument
$\mathcal{I}_{B}=\\{\Phi^{2}_{y}:\mathcal{S}(\mathcal{H})\rightarrow\mathcal{L}^{+}(\mathcal{K}_{2})\\}$
are parallelly compatible then $A$ and $B$ are compatible.

###### Proof.

Suppose an $A$-compatible quantum instrument
$\mathcal{I}_{A}=\\{\Phi^{1}_{x}:\mathcal{S}(\mathcal{H})\rightarrow\mathcal{L}^{+}(\mathcal{K}_{1})\\}$
and a $B$-compatible quantum instrument
$\mathcal{I}_{B}=\\{\Phi^{2}_{y}:\mathcal{S}(\mathcal{H})\rightarrow\mathcal{L}^{+}(\mathcal{K}_{2})\\}$
are parallelly compatible. Then there exists an instrument
$\mathcal{I}=\\{\Phi_{xy}:\mathcal{S}(\mathcal{H})\rightarrow\mathcal{L}^{+}(\mathcal{K}_{1}\otimes\mathcal{K}_{2})\\}$
such that $\sum_{y}\text{tr}_{\mathcal{K}_{2}}\Phi_{xy}(.)=\Phi^{1}_{x}(.)$
and $\sum_{x}\text{tr}_{\mathcal{K}_{1}}\Phi_{xy}(.)=\Phi^{2}_{y}(.)$ for all
$x,y$. Clearly, as $\mathcal{I}_{A}$ an $A$-compatible quantum instrument
$\text{tr}[\rho
A(x)]=tr[\Phi^{1}_{x}(\rho)]=\text{tr}[\rho)(\Phi^{1}_{x})^{*}(\mathbb{I}_{\mathcal{K}_{1}})]$
all quantum state $\rho$ and all outcomes $x$ of the observable $A$. This
implies
$\sum_{y}\Phi_{xy}^{*}(\mathbb{I}_{\mathcal{K}_{1}\otimes\mathcal{K}_{2}})=(\Phi^{1}_{x})^{*}(\mathbb{I}_{\mathcal{K}_{1}})=A(x)$
for all outcomes $x$ of the observable $A$. Similarly
$\sum_{x}\Phi_{xy}^{*}(\mathbb{I}_{\mathcal{K}_{1}\otimes\mathcal{K}_{2}})=(\Phi^{2}_{y})^{*}(\mathbb{I}_{\mathcal{K}_{2}})=B(y)$
for all outcomes $y$ of the observable $B$.

Now we define the observable
$\mathcal{G}=\\{G(x,y)=\Phi_{xy}^{*}(\mathbb{I}_{\mathcal{K}_{1}\otimes\mathcal{K}_{2}})\\}$
where $x$’s are the outcomes of $A$ and $y$’s are the outcomes of $B$. As,
$p_{ij}=\text{tr}[\Phi_{xy}(\rho)]=\text{tr}[\rho\Phi^{*}_{xy}(\mathbb{I}_{\mathcal{K}_{1}\otimes\mathcal{K}_{2}})]\geq
0$ (since those are probabilities ) for all quantum state $\rho$ for all $x,y$
implies $G(x,y)\geq 0$ for all $x,y$. As $\Phi(\rho)=\sum_{xy}\Phi_{xy}(\rho)$
is a quantum channel (a CPTP map),
$\sum_{xy}G(x,y)=\sum_{xy}\Phi^{*}_{xy}(\mathbb{I}_{\mathcal{K}_{1}\otimes\mathcal{K}_{2}})=\mathcal{I}_{\mathcal{H}}$.
Therefore, $\mathcal{G}$ is indeed an observable acting on the Hilbert space
$\mathcal{H}$. Now,
$\sum_{y}G(x,y)=\sum_{y}\Phi_{xy}^{*}(\mathbb{I}_{\mathcal{K}_{1}\otimes\mathcal{K}_{2}})=(\Phi^{1}_{x})^{*}(\mathbb{I}_{\mathcal{K}_{1}})=A(x)$
and
$\sum_{x}G(x,y)=\sum_{x}\Phi_{xy}^{*}(\mathbb{I}_{\mathcal{K}_{1}\otimes\mathcal{K}_{2}})=(\Phi^{1}_{y})^{*}(\mathbb{I}_{\mathcal{K}_{2}})=B(y)$.
Therefore, $\mathcal{G}=\\{G(x,y)\\}$ is joint observables. Therefore, $A$ and
$B$ are compatible. ∎

Next, we show that the traditional compatibility of instrument and the
parallel compatibility of instruments are conceptually different.

######  Proposition 1.

There exist pairs of quantum instruments which are parallelly compatible, but
not traditionally compatible.

###### Proof.

In the Example (1), $\Gamma_{1}$ and $\Gamma_{2}$ can be arbitrary and
therefore, $\Gamma_{1}\circ\Lambda_{1}$ and $\Gamma_{2}\circ\Lambda_{2}$ can
be arbitrary and not necessarily same. Therefore, for the case where
$\Gamma_{1}\circ\Lambda_{1}\neq\Gamma_{2}\circ\Lambda_{2}$, $\mathcal{I}_{1}$
and $\mathcal{I}_{2}$ are parallelly compatible and are not weakly compatible
and therefore are not traditionally compatible. ∎

Similarly, we have our next proposition.

######  Proposition 2.

There exist pairs of quantum instruments which are traditionally compatible,
but not parallelly compatible.

###### Proof.

Consider two quantum instruments
$\mathcal{I}^{p}=\\{\Phi^{p}_{1}=p_{1}\mathfrak{I},\Phi^{p}_{2}=p_{2}\mathfrak{I}\\}$
and
$\mathcal{I}^{q}=\\{\Phi^{q}_{1}=q_{1}\mathfrak{I},\Phi^{q}_{2}=q_{2}\mathfrak{I}\\}$
with $\mathfrak{I}$ is an identity channel and $p_{i}=\sum_{j}r_{ij}$ and
$q_{j}=\sum_{i}r_{ij}$ where $\\{r_{ij}\geq 0\\}_{ij=\\{1,2\\}}$ and
$\sum_{ij}r_{ij}=1$. Clearly, $\mathcal{I}^{p}$ and $\mathcal{I}^{q}$ are
traditionally compatible and the joint instrument is
$\mathcal{I}^{r}=\\{r_{ij}\mathfrak{I}\\}_{ij=\\{1,2\\}}$. But as discussed
before if $\mathcal{I}^{p}$ and $\mathcal{I}^{q}$ are parallelly compatible,
then $\Phi^{p}$ and $\Phi^{q}$ must be compatible and we know that the
identity channel $\mathfrak{I}$ is not compatible with itself (no-cloning
theorem).

Therefore as $\Phi^{p}=\sum_{i}\Phi^{p}_{i}=\mathfrak{I}$ and
$\Phi^{q}=\sum_{j}\Phi^{q}_{j}=\mathfrak{I}$ are not compatible,
$\mathcal{I}^{p}$ and $\mathcal{I}^{q}$ are also not parallelly compatible. ∎

###  III.2 Reason for re-definition of the compatibility of Quantum
Instruments

_The basic conceptual idea behind the compatibility of any set of quantum
devices, is the simultaneous implementation of those devices, given a single
copy of input state._
These devices can be measurements, channels or instruments. In the Sec. III.1,
it has been clearly pointed out that there are two ways of simutaneous
implementation of quantum instruments and those ways are inequivalent.
Therefore, one of those two ways alone can not capture the notion of
compatibility of quantum instruments. As a result, the traditional definition
of compatibility or the definition of parallel compatibility alone is not
enough to define the compatibility of quantum instruments. Therefore, the
above-said argument suggests us to re-define the compatibility of quantum
instruments. The definition is given below.

######  Definition 4.

A pair of instruments is compatible if either it is traditionally compatible
or it is parallelly compatible. Otherwise it is incompatible.

In the next section, we show that the above-said definition of compatibility
of quantum instruments is universal i.e., applicable to or can capture
incompatibility of all three kinds.

###  III.3 Universality of the compatibility of quantum instruments

In this subsection, we show that the compatibility of quantum instruments
captures the idea of observable-observable compatibility, observable-channel
compatibility and channel-channel compatibility. We start with our next
theorem-

######  Theorem 1.

We provide three statements here-

  1. 1.

Two observables $A$ and $B$ are compatible iff there exist an $A-$compatible
instrument $\mathcal{I}_{A}$ and a $B-$compatible instrument $\mathcal{I}_{B}$
such that $\mathcal{I}_{A}$ and $\mathcal{I}_{B}$ are traditionally
compatible.

  2. 2.

Two quantum channels are
$\Phi^{1}:\mathcal{S}(\mathcal{H})\rightarrow\mathcal{S}(\mathcal{K}_{1})$ and
$\Phi^{2}:\mathcal{S}(\mathcal{H})\rightarrow\mathcal{S}(\mathcal{K}_{2})$
compatible iff there exist two instruments
$\mathcal{I}_{1}=\\{\Phi^{1}_{x}:\mathcal{S}(\mathcal{H})\rightarrow\mathcal{L}^{+}(\mathcal{K}_{1})\\}$
and
$\mathcal{I}_{2}=\\{\Phi^{2}_{y}:\mathcal{S}(\mathcal{H})\rightarrow\mathcal{L}^{+}(\mathcal{K}_{2})\\}$
where $\sum_{x}\Phi^{1}_{x}=\Phi^{1}$ and $\sum_{y}\Phi^{2}_{y}=\Phi^{2}$ that
are parallelly compatible.

  3. 3.

If an $A-$compatible instrument
$\mathcal{I}_{A}=\\{\Phi^{A}_{x}:\mathcal{S}(\mathcal{H})\rightarrow\mathcal{L}^{+}(\mathcal{K}_{1})\\}$
and a $B-$compatible instrument
$\mathcal{I}_{B}=\\{\Phi^{B}_{y}:\mathcal{S}(\mathcal{H})\rightarrow\mathcal{L}^{+}(\mathcal{K}_{2})\\}$
where $\sum_{x}\Phi^{A}_{x}=\Phi^{A}$ and $\sum_{y}\Phi^{B}_{y}=\Phi^{B}$, are
parallelly compatible then $A$ and $B$ both are compatible with both
$\Phi^{A}$ and $\Phi^{B}$.

###### Proof.

We prove the three statements below-

  1. 1.

Suppose $A-$compatible instrument
$\mathcal{I}_{A}=\\{\Phi^{A}_{x}:\mathcal{S}(\mathcal{H})\rightarrow\mathcal{L}^{+}(\mathcal{K}_{1})\\}$
and $B-$compatible instrument
$\mathcal{I}_{B}=\\{\Phi^{B}_{y}:\mathcal{S}(\mathcal{H})\rightarrow\mathcal{L}^{+}(\mathcal{K}_{2})\\}$
are traditionally compatible. Here $\mathcal{K}_{1}=\mathcal{K}_{2}$. Then
there exists an instrument
$\mathcal{I}=\\{\Phi_{xy}:\mathcal{S}(\mathcal{H})\rightarrow\mathcal{L}^{+}(\mathcal{K}_{1})\\}$
such that $\sum_{y}\Phi_{xy}(.)=\Phi^{1}_{x}(.)$ and
$\sum_{x}(.)\Phi_{xy}(.)=\Phi^{2}_{y}(.)$ for all $x,y$. Let the observable
$\mathcal{G}=\\{G(x,y)\\}$ be associated with $\mathcal{I}$ (i.e.,
$\mathcal{I}$ is a $\mathcal{G}$-compatible instrument). Then for any
$\rho\in\mathcal{S}(\mathcal{H})$, $\text{tr}[\rho
G(x,y)]=\text{tr}[\Phi_{xy}(\rho)]$ for all $x,y$ and for all
$\rho\in\mathcal{S}(\mathcal{H})$. Therefore, $\sum_{y}\text{tr}[\rho
G(x,y)]=\text{tr}[\Phi^{A}_{x}(\rho)]=\text{tr}[\rho A(x)]$ and
$\sum_{x}\text{tr}[\rho G(x,y)]=\text{tr}[\Phi^{B}_{xy}(\rho)]=\text{tr}[\rho
B(y)]$ for all $x,y$ and for all $\rho\in\mathcal{S}(\mathcal{H})$. Therefore,
$A$ and $B$ are compatible and the joint observable
$\mathcal{G}=\\{G(x,y)\\}$.
Now suppose $A$ and $B$ are compatible and the joint observable is
$\mathcal{G}=\\{G(x,y)\\}$ and $\mathcal{I}=\\{\Phi_{xy}\\}$ be a
$\mathcal{G}-$compatible instrument. Then it is easy to check that the
instrument $\mathcal{I}^{A}=\\{\Phi^{A}_{x}=\sum_{y}\Phi_{xy}\\}$ is an
$A-$compatible instrument and
$\mathcal{I}^{B}=\\{\Phi^{B}_{x}=\sum_{y}\Phi_{xy}\\}$ is a $B-$compatible
instrument.

  2. 2.

A quantum channel $\Lambda$ can be considered as a single outcome quantum
instrument $\mathcal{I}_{\Lambda}$. Therefore, compatibility of two quantum
channels $\Phi^{1}$ and $\Phi^{2}$ implies the parallel compatibility of two
instruments $\mathcal{I}_{1}=\\{\Phi^{1}\\}$ and
$\mathcal{I}_{2}=\\{\Phi^{2}\\}$.
From Definition 3, it is trivial to show that parallel compatibility of
$\mathcal{I}_{1}$ and $\mathcal{I}_{2}$ implies the compatibility of
$\Phi^{1}$ and $\Phi^{2}$.

  3. 3.

By definition, $A$ is compatible with $\Phi^{A}$ and $B$ is compatible with
$\Phi^{B}$.
Now as $\mathcal{I}_{A}$ and $\mathcal{I}_{B}$ are parallelly compatible,
there exists a quantum instrument
$\mathcal{I}_{AB}=\\{\Phi_{xy}:\mathcal{S}(\mathcal{H})\rightarrow\mathcal{L}^{+}(\mathcal{K}_{1}\otimes\mathcal{K}_{2})\\}$
such that $\Phi^{A}_{x}(.)=\sum_{y}\text{tr}_{\mathcal{K}_{2}}[\Phi_{xy}(.)]$
and $\Phi^{B}_{y}(.)=\sum_{x}\text{tr}_{\mathcal{K}_{1}}[\Phi_{xy}(.)]$. Now,
as $\mathcal{I}_{A}$ is an $A$-compatible instrument, for all
$\rho\in\mathcal{S}(\mathcal{H})$ and $x\in\Omega_{A}$,

$\displaystyle\text{tr}_{\mathcal{H}}[\rho A(x)]$
$\displaystyle=\text{tr}_{\mathcal{K}_{1}}[\Phi^{A}_{x}(\rho)]$
$\displaystyle=\text{tr}_{\mathcal{K}_{1}}[\sum_{y}\text{tr}_{\mathcal{K}_{2}}[\Phi_{xy}(\rho)]$
$\displaystyle=\sum_{y}[\text{tr}_{\mathcal{K}_{1}}\text{tr}_{\mathcal{K}_{2}}[\Phi_{xy}(\rho)]$
$\displaystyle=\text{tr}_{\mathcal{K}_{2}}[\sum_{y}[\text{tr}_{\mathcal{K}_{1}}[\Phi_{xy}(\rho)]]$
$\displaystyle=\text{tr}_{\mathcal{K}_{2}}[\Phi^{\prime A}_{x}(\rho)]$

where $\Phi^{\prime
A}_{x}(\rho)=\sum_{y}[\text{tr}_{\mathcal{K}_{1}}[\Phi_{xy}(\rho)]$. Clearly,
for all $x\in\Omega_{A}$, $\Phi^{\prime
A}_{x}:\mathcal{S}(\mathcal{H})\rightarrow\mathcal{L}^{+}(\mathcal{K}_{2})$.
Now,

$\displaystyle\sum_{x}\Phi^{\prime A}_{x}(.)$
$\displaystyle=\sum_{x}\sum_{y}[\text{tr}_{\mathcal{K}_{1}}[\Phi_{xy}(.)]$
$\displaystyle=\sum_{y}\Phi^{B}_{y}(.)$ $\displaystyle=\Phi^{B}(.).$ (4)

Therefore, $\mathcal{I}^{\prime}_{A}=\\{\Phi^{\prime
A}_{x}:\mathcal{S}(\mathcal{H})\rightarrow\mathcal{L}^{+}(\mathcal{K}_{2})\\}$
is a quantum instrument. Hence, $A$ and $\Phi^{B}$ are compatible through the
instrument $\mathcal{I}^{\prime}_{A}=\\{\Phi^{\prime
A}_{x}:\mathcal{S}(\mathcal{H})\rightarrow\mathcal{L}^{+}(\mathcal{K}_{2})\\}$.
Similarly one can prove that $B$ and $\Phi^{A}$ are compatible.

∎

Hence, we have proved that the traditional compatibility and the parallel
compatibility together can capture the compatibility of all three kinds i.e.,
observable-observable compatibility, observable-channel compatibility and
channel-channel compatibility. Next, we discuss that the traditional
compatibility is also incomplete in this sense.

_The traditional compatibility of quantum instruments can not capture
compatibility of quantum channels -_ If an $A-$compatible instrument
$\mathcal{I}_{A}=\\{\Phi^{A}_{x}:\mathcal{S}(\mathcal{H})\rightarrow\mathcal{L}^{+}(\mathcal{K}_{1})\\}$
and a $B-$compatible instrument
$\mathcal{I}_{B}=\\{\Phi^{B}_{y}:\mathcal{S}(\mathcal{H})\rightarrow\mathcal{L}^{+}(\mathcal{K}_{2})\\}$
with $\sum_{x}\Phi^{A}_{x}=\Phi^{A}$ and $\sum_{y}\Phi^{B}_{y}=\Phi^{B}$ are
traditionally compatible, then $\Phi^{A}=\Phi^{B}$. Therefore, it directly can
not capture channel-channel compatibility since two different channels can be
compatible with each other.
Therefore, it seems that traditional compatibility and parallel compatibility
together or in other words Definition 4 is conceptually complete.
From Lemma 1 and Theorem 1, we got that if two instruments are parallelly
compatible corresponding observables are compatible as well as corresponding
channels are compatible. We provide here one more interesting example-

######  Example 2 (Two parallelly incompatible instruments associated with the
same observable).

A trivial observable $J=\\{J(x)=p_{x}\mathbb{I}\\}$ is compatible with any
quantum channel $\Lambda$ [1]. The corresponding instrument is
$\mathcal{I}_{J,\Lambda}=\\{p_{x}\Lambda\\}$. Now take channel $\Gamma$ which
incompatible with $\Lambda$. Now $J$ is also compatible with the quantum
channel $\Gamma$ and the corresponding instrument is
$\mathcal{I}_{J,\Gamma}=\\{p_{x}\Gamma\\}$. Now from the Theorem 1 we know
that two instruments $\mathcal{I}_{J,\Gamma}$ and $\mathcal{I}_{J,\Lambda}$
are parallelly compatible the corresponding channels are compatible. But since
$\Gamma$ and $\Lambda$ are not compatible, the instruments
$\mathcal{I}_{J,\Gamma}$ and $\mathcal{I}_{J,\Lambda}$ are not parallelly
compatible

##  IV Conclusion

In this paper, we have discussed the concept of parallel compatibility of
instruments and have shown that this concept is different than the traditional
definition of compatibility of instruments. We have claimed that the
traditional definition of compatibility of instruments is conceptually
incomplete and have provided a re-definition of the compatibility of quantum
instrument. We have discussed the proper reason for the re-definition. We have
shown that the modified definition of compatibility of quantum instruments can
capture the idea of observable-observable compatibility, observable-channel
compatibility and channel-channel compatibility. The parallel compatibility
has importance in different information-theoretic tasks. For example- (i)
Suppose, Charlie have to simultaneously transfer information to Alice and Bob
who are separated by a long distance. The information for Alice is encoded in
the probability distribution of the observable $A$ and the information for Bob
is encoded in the probability distribution of the observable $B$. Clearly Fig.
1(b) suggests that it can be done through parallel compatibility of
$\mathcal{I}_{A}$ and $\mathcal{I}_{B}$, (ii) Fig. 1(b) also suggests that if
Bob (i.e., $B$) is an eavesdropper then the parallel compatibility of quantum
instruments allows him to do cloning attack in Quantum Key Distribution.
In future, it will be interesting to completely characterise the compatibility
of quantum instrument.

##  V Acknowledgements

I would like to thank my advisor Prof. Sibasish Ghosh for his valuable
comments on this work.

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# Is pluralism in the history of mathematics possible?

Jacques Bair  HEC-ULG, University of Liege, 4000 Belgium<EMAIL_ADDRESS>,
Alexandre Borovik  School of Mathematics, University of Manchester, Oxford
Street, Manchester, M13 9PL, United Kingdom<EMAIL_ADDRESS>,  Vladimir
Kanovei  IITP RAS, Moscow, Russia<EMAIL_ADDRESS>,  Mikhail G. Katz
Department of Mathematics, Bar Ilan University, Ramat Gan 5290002 Israel
<EMAIL_ADDRESS>,  Semen S. Kutateladze  Sobolev Institute of
Mathematics, Novosibirsk State University, Russia<EMAIL_ADDRESS>,  Sam
Sanders  Department of Philosophy 2, RUB Bochum, Bochum, Germany
http://sasander.wix.com/academic<EMAIL_ADDRESS>,  David Sherry  Department
of Philosophy, Northern Arizona University, Flagstaff, AZ 86011, US
<EMAIL_ADDRESS>,  Monica Ugaglia  Il Gallo Silvestre, Località Collina
38, Montecassiano, Italy<EMAIL_ADDRESS>and  Mark van Atten
Archives Husserl (CNRS / ENS), 45 rue d’Ulm, 75005 Paris, France
<EMAIL_ADDRESS>
###### Abstract.

Leibniz scholarship is currently an area of lively debate. We respond to some
recent criticisms by Archibald et al.

This is a brief response to the article “A Question of Fundamental
Methodology: Reply to Mikhail Katz and his coauthors,” by Archibald et al. in
_The Mathematical Intelligencer_ [1]. The article by Archibald et al. was in
reaction both to our earlier article “Two-track depictions of Leibniz’s
fictions” [3] in the same journal, and to other work of ours.

We have argued that, in addition to procedures that can be adequately
described in purely Archimedean settings, Leibniz (as well as Cauchy and
others) used procedures that exploit genuine infinitesimals, that is, what to
him were mathematical entities.

Archibald’s coauthors Arthur and Rabouin have argued that the term
“infinitesimal” as used by Leibniz does not refer to a mathematical entity,
and is, rather, stenography for exhaustion-type arguments in the style of
Archimedes. We have compared the two approaches in [3] and, in particular,
presented evidence for our interpretation.

Archibald et al. make a number of false claims concerning both [3] and other
publications of ours. Lack of space prevents us from responding in full. A
more detailed response appears at [2].

In closing, it is ironic that Archibald et al. should claim that

  1. [O]ver the years, it became clearer and clearer that our interlocutors do not care much about rational discussion and scientific dialogue from _different perspectives_ , … The latest example of that approach is provided by a paper … “Two-Track Depictions of Leibniz’s Fictions.” [1, p. 2] (emphasis on “different perspectives” added)

“Two-track depictions” [3] is devoted specifically to making explicit a pair
of _different perspectives_ on Leibniz’s calculus, so as to stimulate rational
discussion and scientific dialogue.

Archibald et al. do little to clarify the “Question of Fundamental
Methodology” of their title, namely that the history of mathematics, like
mathematics itself, could benefit from a plurality of approaches.

## References

  * [1] Archibald, Tom; Arthur, Richard T. W.; Ferraro, Giovanni; Gray, Jeremy; Jesseph, Douglas; Lützen, Jesper; Panza, Marco; Rabouin, David; Schubring, Gert. A Question of Fundamental Methodology: Reply to Mikhail Katz and His Coauthors. _The Mathematical Intelligencer_ 44 (2022), no. 4, 360–363. https://doi.org/10.1007/s00283-022-10217-7 MR 4526076
  * [2] Bair, J.; Borovik, A.; Kanovei, V.; Katz, M.; Kutateladze, S.; Sanders, S.; Sherry, D.; Ugaglia, M. Historical infinitesimalists and modern historiography of infinitesimals. _Antiquitates Mathematicae_ 16 (2022), 189–257. https://arxiv.org/abs/2210.14504, https://doi.org/10.14708/am.v16i1.7169
  * [3] Katz, M.; Kuhlemann, K.; Sherry, D.; Ugaglia, M.; van Atten, M. Two-track depictions of Leibniz’s fictions. _The Mathematical Intelligencer_ 44 (2022), no. 3, 261–266. https://doi.org/10.1007/s00283-021-10140-3, https://arxiv.org/abs/2111.00922 MR 4480193


11institutetext: The European Organization for Nuclear Research, CERN
22institutetext: CiTIUS, Universidade de Santiago de Compostela, 15782
Santiago de Compostela, Spain

# Model Performance Prediction for Hyperparameter Optimization of Deep
Learning Models Using High Performance Computing and Quantum Annealing

Juan Pablo GarcÃa Amboage 1122<EMAIL_ADDRESS>Eric Wulff 11
<EMAIL_ADDRESS>Maria Girone 11 TomÃ¡s F. Pena 22

###### Abstract

Hyperparameter Optimization (HPO) of Deep Learning (DL)-based models tends to
be a compute resource intensive process as it usually requires to train the
target model with many different hyperparameter configurations. We show that
integrating model performance prediction with early stopping methods holds
great potential to speed up the HPO process of deep learning models. Moreover,
we propose a novel algorithm called Swift-Hyperband that can use either
classical or quantum Support Vector Regression (SVR) for performance
prediction and benefit from distributed High Performance Computing (HPC)
environments. This algorithm is tested not only for the Machine-Learned
Particle Flow (MLPF), model used in High-Energy Physics (HEP), but also for a
wider range of target models from domains such as computer vision and natural
language processing. Swift-Hyperband is shown to find comparable (or better)
hyperparameters as well as using less computational resources in all test
cases.

##  1 Introduction

Training and Hyperparameter Optimization (HPO) of Deep Learning (DL) models is
often compute resource intensive and calls for the use of large-scale High
Performance Computing (HPC) resources as well as scalable and resource
efficient Hyperparameter (HP) search and evaluation algorithms MlpfHPO .
Current state-of-the-art HPO algorithms such as Hyperband hyperband , ASHA
asha , and BOHB bohb , rely on a method of early termination. Badly performing
trials are automatically terminated to allocate compute resources to more
promising ones. Such methods have been successfully applied to optimize
Machine-Learned Particle Flow (MLPF), a particle flow reconstruction Neural
Network (NN) mlpf1 . Using this technique led to o a reduction of $\sim$44% in
the validation loss of MLPF MlpfHPO .

In this context, performance prediction emerges as a potential approach to
accelerate the HPO process. This involves using meta-models, referred to as a
performance predictors, that can estimate the performance of a given
configuration at a particular epoch by leveraging information from its partial
learning curve. By employing performance prediction, it is possible to
prioritize the training of the most promising configurations based on their
predicted performance, while avoiding the need to fully train configurations
with poorer predicted performance. Consequently, this approach holds great
potential for reducing the time and computational resources required for the
HPO process.

This work explores novel techniques based on performance prediction to
accelerate the HPO process of MLPF and other NN architectures that leverage
the use of HPC resources for training the target model and quantum computing
for training the performance predictors. Moreover, a new HPO algorithm is
proposed, Swift-Hyperband, that integrates Hyperband with the use of model
performance predictors.

##  2 Related work

Baker et al. perfpred  demonstrated that Support Vector Regression (SVR)
models can effectively serve as performance predictors for various NN
architectures. In addition to showing a good predictive capability for
performance prediction tasks, SVR models offer the advantage of having
negligible training and inference times, even when using a consumer-grade
laptop CPU. Hence, using SVRs prevents the training of performance predictors
from becoming a bottleneck for the potential resource savings expected from
this technique.

In the European Center of Excellence in Exascale Computing "Research on AI-
and Simulation-based Engineering at Exascale" (CoE RAISE) the capability of
SVRs to predict the loss of MLPF after 100 training epochs in the Delphes
dataset data_delphes  was successfully shown acat , achieving $R^{2}$ values
of around 0.9 when using 25% of the target learning curve as input. Here,
$R^{2}$ is the so-called coefficient of determination, defined as

$R^{2}=1-\frac{\sum_{i}(y_{i}-f_{i})^{2}}{\sum_{i}(y_{i}-\bar{y})^{2}}=1-\frac{\sum_{i}e_{i}^{2}}{\sum_{i}(y_{i}-\bar{y})^{2}}$
(1)

where $y_{i}$ is the ground truth for data point $i$, $f_{i}$ is prediction
$i$, $\bar{y}$ is the mean of all $y_{i}$ and $e_{i}$ is the error of
prediction $i$. Furthermore, the Quantum Annealer at the JÃlich Supercomputer
Centre, was used to train Quantum Support Vector Regression (QSVR) pasetto
models on MLPF learning curves. While no significant performance benefit was
expected from the use of quantum resources, and the idea of employing quantum
computers for the task of performance prediction was primarily a proof of
concept of integrating this technology into the HPO workflow, the QSVRs
achieved comparable performance to that obtained with classical SVRs acat .
Note that the number of training samples in this case had to be reduced to 20
due to limitations in problem size arising from the current state of quantum
technologies.

There are several strategies that can be considered for integrating
performance predictors with the HPO process perfpred . A straightforward
approach is to generate a certain amount of random HP configurations, fully
train $M$ configurations and partially train $N>M$ configurations. The final
loss of the fully trained configurations and part of their learning curves are
used to train a performance predictor. Then, this performance predictor is
used to predict the final loss of the partially trained configurations and
only those whose predicted loss is below a certain threshold are selected to
complete training. This approach was tested using different types of
performance predictors, including QSVRs, in aachhybrid .

Another, more sophisticated example of the use of performance predictors for
HPO is the algorithm Fast-Hyperband perfpred . This algorithm is a modified
version of the well-known Hyperband algorithm that uses performance predictors
which are trained on the fly during the execution of the algorithm to save
training epochs of the target model with respect to Hyperband. More precisely,
Fast-Hyperband adds an extra decision point 111We understand a decision point
as a certain epoch in which an early stopping HPO algorithm discards some
configurations to keep training only the most promising ones according to a
certain criterion. based on performance prediction for every epoch in each
Hyperband round. Decisions in these new intermediate points use a probability
threshold, computed from an estimate of the standard deviation of every
predictor used. The proposed method of computing this estimate in Fast-
Hyperband is leave one out cross validation. The main drawback of this
approach is that it requires to train many performance predictors, which makes
it impractical to use QSVRs. This is partly due to runtime limitations on the
quantum machine and partly due to the time needed to formulate the regression
problem in a suitable way for the quantum annealer. In addition, the time
spent to connect to the quantum machine needs to be taken into account.
Furthermore, Fast-Hyperband, as it is defined in perfpred , is a sequential
algorithm which makes it unable to benefit from running in a distributed
manner on multiple nodes in an HPC environment.

##  3 Swift-Hyperband

Figure 1: Graphic representation of a Swift-Hyperband bracket. This is an
illustrative example, not the result of an actual execution the algorithm.

To overcome the limitations of Fast-Hyperband mentioned in the previous
section we propose a different way to integrate performance predictors with
Hyperband by a new algorithm called Swift-Hyperband. Similarly to Fast-
Hyperband, Swift-Hyperband adds extra decision points to the classical
Hyperband algorithm that are based on performance prediction. However, Swift-
Hyperband adds only one extra decision point per Hyperband round. When a round
starts, Swift-Hyperband trains a certain number of trials until the end of the
round and then uses their loss at the end of the round to define a threshold.
The other trials in the round are partially trained to the extra decision
point, where the performance predictors are used to predict their losses at
the end of the round. These predictions are compared to the previously defined
threshold to decide if each trial should be terminated or not.

Figure 1 illustrates what happens inside each bracket of Swift-Hyperband. The
vertical dashed lines represent the new decision points in which some
configurations or trials are discarded based on their predicted performance.
The trials that did not complete their round are represented using dashed
learning curves. That is, the dashed learning curves represent training epochs
of the target model that Swift-Hyperband saved with respect to the classical
Hyperband algorithm. The trials that complete their training until the end of
the round are the ones that are always represented by continuous lines. The
decision of which of these trials are promoted to the next round is made as it
is made in the classical Hyperband algorithm.

Swift-Hyperband not only needs to train less performance predictors than Fast-
Hyperband but can also be easily parallelized, as all the initial full and
partial trainings inside a round can be executed in parallel. For these two
reasons, our algorithm can potentially use QSVRs and benefit from HPC
environments. A schematic comparison between Fast-Hyperband and Swift-
Hyperband is shown in Figure 2.

Figure 2: Key differences between Fast-Hyperband and Swift-Hyperband.

##  4 Results

To compare Hyperband, Fast-hyperband, Swift-Hyperband and Quantum-Swift-
Hyperband (Swift-Hyperband using QSVRs) for different NN architectures we
simulate 10 runs of each algorithm using the datasets of learning curves
derived from the following model-dataset combinations:

  * •

MLPF acat  trained on the Delphes dataset data_delphes

  * •

An image recognition Convolutional Neural Network (CNN) modified from asha
trained on the Cifar10 dataset

  * •

An image recognition CNN trained on the SVHN dataset used in perfpred

  * •

A natural language processing Long Shot-Term Memory (LSTM) NN trained in the
PTB dataset

The result of these simulated runs can be seen in Figure 3 and a summary of
the learning curve datasets used for the simulation is available in Table 1.

Neural Network | Evluation metric |  | HPs search space
---
dimension
| Target epoch
---
for HPO
MLPF for Delphes | Loss | 7 | 100
LSTM for PTB | Perplexity | 2 | 60
CNN for Cifar10 | Accuracy | 5 | 100
CNN for SVHN | Accuracy | 9 | 100
Table 1: Summary of learning curve datasets.

(a) MLPF trained on Delphes

(b) LSTM trained on PTB

(c) CNN trained on Cifar10

(d) CNN trained on SVHN

Figure 3: Average, best, and worst performance of the best configuration found
as well as the total number of epochs consumed by the different HPO algorithms
for different NNs and datasets. Figure 4: Performance of the best
configuration found and total runtime needed for different HPO algorithms.

Beyond the simulated runs, we test the speedup provided by the parallelization
of Swift-Hyperband along with the achieved accuracies by running Hyperband,
Fast-Hyperband, Swift-Hyperband, and a parallel version of Swift-Hyperband
that uses MPI to coordinate one CPU node and two GPU worker nodes. For these
runs, the HPO target was a simple 6-layer CNN (different to the CNN used in
the simulated runs) trained on Cifar10 using a 3-dimensional search space
consisting of learning rate, weight decay, and dropout. This network was
chosen because it was relatively fast to train. The results can be seen in
Figure 4.

The results in Figure 3 and Figure 4 show that both Swift-Hyperband and its
version using QSVRs achieve accuracies comparable to classical Hyperband while
needing considerably fewer epochs in all cases. In comparison to Fast-
Hyperband, Swift-Hyperband (SVR and Q-SVR) is faster in all cases except on
the SVHN problem. When it comes to the non-simulated runs we observe that all
algorithms achieve accuracies around 87$\%$, with both Swift-Hyperband and
Parallel-Swift-Hyperband slightly beating Fast-Hyperband. Note that in several
cases the version of Swift-Hyperband that uses QSVRs finds better performing
configurations than the original Hyperband algorithm, something that would not
happen if the QSVRs made perfect predictions. This may indicate that using
performance predictors, aside from saving compute resources, has some type of
regularizing effect that prevents some of the errors that Hyperband makes when
terminating configurations at the end of each round.

##  5 Conclusions

We proposed a new promising parallelizable HPO algorithm integrating Hyperband
and performance predictors that can be used in combination with SVRs or QSVRs.
This leaves the door open for the use of Swift-Hyperband in later HPO cycles
of MLPF. Furthermore, it was shown that, despite the current limitations of
quantum computers, it is possible to execute hybrid Quantum/HPC workflows for
HPO, achieving comparable performance to fully classical workflows. We
consider that there is a need for further studies on the speedup achieved by
the parallelization of Swift-Hyperband when using a greater number of nodes as
Hyperband is known to suffer from straggler issues asha . Hence, developing a
version of the HPO algorithm ASHA that integrates performance predictors,
potentially to be named Swift-ASHA, is one of the identified lines to continue
this work. In addition to this, conducting more empirical tests of Swift-
Hyperband on a wider range of target models would provide valuable insights on
the behavior of the algorithm. Finally, conducting theoretical studies on SVRs
and QSVRs could provide a deeper understanding of why Swift-Hyperband
occasionally outperforms the original Hyperband algorithm and shed light on
the differences observed when employing quantum or classical SVRs.

## Acknowledgements

Eric Wulff and Juan Pablo GarcÃa Amboage was supported by CoE RAISE. The CoE
RAISE project has received funding from the European Unionâs Horizon 2020 â
Research and Innovation Framework Programme H2020-INFRAEDI-2019-1 under grant
agreement no. 951733.

The authors gratefully acknowledge the computing time granted through JARA on
the supercomputer JURECA jureca  at Forschungszentrum JÃlich. The authors
gratefully acknowledge the JÃlich Supercomputing Centre for funding this
project by providing computing time through the JÃlich UNified Infrastructure
for Quantum computing (JUNIQ) on the D-Wave Advantageâ System JUPSI.

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