Research Article TC, tropical cyclone; ENSO, El Niño-Southern Oscillation;
JASO, July-August-September-October; MDR (-A,-C), Main Development Region
(-Atlantic, -Caribbean); GOM, Gulf of Mexico Robert Doane-Solomon, Pembroke
College, University of Oxford, Oxford, OX1 1DW, United Kingdom robert.doane-
<EMAIL_ADDRESS>RDS was funded by the Met Office Academic Partnership
(MOAP). DJB and AW were supported by the EUCP project funded by the European
Commission’s Horizon 2020 programme, Grant Agreement number 776613. KH was
funded as part of the NERC National Centre for Atmospheric Science.

# The link between North Atlantic tropical cyclones and ENSO in seasonal
forecasts

Robert Doane-Solomon  Atmospheric, Oceanic and Planetary Physics, Department
of Physics, University of Oxford, Oxford, United Kingdom  Daniel J. Befort
Atmospheric, Oceanic and Planetary Physics, Department of Physics, University
of Oxford, Oxford, United Kingdom  Joanne Camp  Met Office Hadley Centre,
Exeter, United Kingdom  Kevin Hodges  Department of Meteorology, University of
Reading, Reading, United Kingdom  National Centre for Atmospheric Science,
University of Reading, Reading, RG6 6ET, UK  Antje Weisheimer  National Centre
for Atmospheric Science, Atmospheric, Oceanic and Planetary Physics,
Department of Physics, University of Oxford, Oxford, United Kingdom  European
Centre for Medium-Range Weather Forecasts (ECMWF), Reading, UK

###### Abstract

This study assesses the ability of six European seasonal forecast models to
simulate the observed teleconnection between ENSO and tropical cyclones (TCs)
over the North Atlantic. While the models generally capture the basin-wide
observed link, its magnitude is overestimated in all forecast models compared
to reanalysis. Furthermore, the ENSO - TC relationship in the Caribbean is
poorly simulated. It is shown that incorrect forecasting of wind shear appears
to affect the representation of the teleconnection in some models, however it
is not a completely sufficient explanation for the overestimation of the link.

###### keywords:

North Atlantic, tropical cyclones, ENSO, seasonal forecasts

##  1 Introduction

Tropical cyclones (TCs) cause significant annual losses and casualties, with
North Atlantic tropical cyclones being the most destructive natural disasters
in the United States [16]. Several variables affect the interannual
variability of TC numbers and their location, such as the El Niño–Southern
Oscillation (ENSO) [41, 32, 25] and tropical Atlantic sea surface temperatures
[27]. Years exhibiting La Niña conditions since 2010 have been associated with
record-breaking tropical cyclone activity in the North Atlantic in 2020 and an
extremely active season in 2017 [29, 31]. Indeed, ENSO is thought to be the
most important single factor in seasonal tropical cyclone variability [5, 42,
8, 15]. Global shifts in atmospheric and oceanic weather patterns occur in
response to ENSO, due to the strengthening/weakening of the Walker and Hadley
circulations. Over the North Atlantic, ENSO impacts include changes in sea
surface temperatures [13], tropospheric vorticity [4], vertical wind shear and
humidity [28], all of which can have an impact on TCs. In an El Niño year wind
shear usually increases in the Main Development Region of the tropical
Atlantic (MDR - the area between 10-20N & 20-85W), and vice versa during La
Niña years [5]. TCs can be thought of as heat engines transferring heat from
warm tropical ocean waters to the cold upper atmosphere - wind shear
suppresses a TC by dissipating warm air from the top of the storm and inducing
asymmetries in its circulation, making the heat engine less efficient [11,
40]. Consequently, TC activity usually decreases in the North Atlantic during
El Niño, whereas a La Niña event usually enhances TC activity [2, 28]. Due to
their socio-economic impacts, skillful seasonal predictions of TC activity are
crucial for disaster planning for nations bordering the Atlantic, Caribbean
and Gulf of Mexico. To obtain skillful seasonal forecasts of TC activity it is
important to correctly represent the observed relationship between tropical
Pacific sea surface temperatures and those factors impacting North Atlantic
TCs, e.g. wind shear and local SSTs.
Many operational forecasting centers produce seasonal forecasts with
approximately 6 month lead times from which predictions of TC activity can be
determined. Previous studies show that the interannual variability of NA TCs
can be predicted to some extent using such seasonal forecasts. [35] studied 3
seasonal forecast models from the European Seasonal to Interannual Prediction
(EUROSIP) system, finding a high degree of correlation between predicted and
actual NA TC numbers per season. More recently a study analysed the ability of
6 state-of-the-art operational European seasonal forecast models in simulating
interannual TC variability over the NA and Western North Pacific basins during
1993 to 2014 [1]. They found that these models are skillful in simulating TC
variability over the NA basin and generally poor over the Western North
Pacific, despite significant climatological differences between the models.
However, [1] did not investigate the ability of the models to simulate the
observed TC-ENSO teleconnection in either of these basins. ENSO forecast
accuracy is a crucial factor for the correct prediction of seasonal TC
variations [6]. Analysis of the UK Met Office Glosea5-GA3 model over the
1992-2013 period has shown a statistically significant reduction in NA TC
numbers during periods of El Niño activity [7]. Additionally, [36] analysed
the teleconnection between ENSO and NA TCs in an early version of ECMWF’s
seasonal forecast system. This correctly showed fewer storms in El Niño & more
in La Niña, albeit with a small sample size from 1991-1999.

Here, we expand on the [1] study by analysing the predictability of the TC-
ENSO teleconnection in the North Atlantic for the same seasonal forecast
models. This paper is structured as follows: first we assess the relationship
between central equatorial Pacific SSTs and TC numbers during the active July
to October (JASO) season in the different seasonal forecast models, and
compare them to reanalysis and best track observations. Next, we analyse to
what extent the errors in the simulated ENSO-TC relationship can be explained
by local environmental variables over the tropical NA. This involves
investigating the simulated link between ENSO and the interannual variability
of vertical wind shear & humidity, and how differences between the simulated
relationships may contribute to errors in the overall simulated interannual
variability of TCs. We conclude by summarising the results and discussing the
limitations of the methods and data used.

##  2 Methods and Data

In this study, data from 6 different dynamical seasonal forecast models are
used, which are the same models as in [1]. They are from the following six
European models which contribute to the Copernicus Climate Change Service
(C3S) multi-model seasonal forecasting system.: UK Met Office GloSea5-GC2
(UKMO) [34], German Weather Service GCFS2.0 (DWD) [12], Euro-Mediterranean
Centre on Climate Change SPS3 (CMCC) [17], European Centre for Medium-Range
Weather Forecasts SEAS5 (ECMWF) [26], Météo-France System 5 (MF5) and Météo-
France System 6 (MF6) [9]. Details on the resolution and ensemble size of each
of these models are available in Table S1 in the supplementary material.
Forecasts initialised at the beginning of June during the hindcast period 1993
and 2014 are used, which results in a total of 22 TC seasons. The European
Centre for Medium-Range Weather Forecasts (ECMWF) 5th generation reanalysis
(ERA5) [18] is used as a reference for the ENSO Index and TC numbers, as well
as specific humidity and wind shear. This reanalysis covers the globe using a
31km grid, similar to that of the highest resolution forecast models in this
study (Table S1). The same tracking and detection methods are used to identify
TCs in ERA5 as in the 6 seasonal forecast models. Best Track data from the
International Best Track Archive for Climate Stewardship (IBTrACS) [30] is
also used an additional reference for TC counts. IBTrACS is a dataset of
historical storms’ location and intensity, interpolated to 3-hourly intervals
with 0.1°resolution. We sub-sample this data to 12-hourly intervals to be
consistent with the seasonal forecasting systems.

The tropical cyclones in ERA5 and the seasonal forecast models are tracked
using the methods described in [22]. Potential tropical cyclones are first
identified using the vertical average of vorticity between 850 and 700hPa,
spectrally filtered to T63 with the large scale background removed, and a
threshold of $5\times 10\textsuperscript{-6}s\textsuperscript{-1}$. The
potential cyclones are then used to track the motion of the storms. By
minimizing a cost function subject to adaptive constraints, smooth storm
trajectories are produced [23, 19, 21]. Tropical cyclones are then identified
by using the following criteria, the same as in [1, 22]:

  1. 1.

The T63 relative vorticity at 850hPa must be at least $6\times
10\textsuperscript{-5}s\textsuperscript{-1}$

  2. 2.

The T63 vorticity difference between 850 and 200 hPa must be at least $6\times
10\textsuperscript{-5}s\textsuperscript{-1}$

  3. 3.

There must be a T63 vorticity centre at each level between 850hPa and 200hPa
for which there is data

  4. 4.

The above criteria must apply for a minimum of 2 consecutive timesteps

  5. 5.

Tracks must start between 0N-30N

Gridded track density statistics are calculated using the spherical kernel
estimators described in [20]. The JASO (July-August-September-October) period
is used for all analysis, being the most active season over the NA basin [1].
The Niño 3.4 region (5S-5N, 170W-120W) is used for the ENSO index in our
analysis. The index used here is equivalent to the SST anomaly in this region,
using 1993-2014 as the baseline (Niño 3.4 Index). Study of the North Atlantic
is performed basin-wide, with particular interest paid to the following 3
regions: Main Development Region - Atlantic (MDR-A) (10-20N, 20-60W), Main
Development Region - Caribbean (MDR-C) (10-20N, 60-85W) and Gulf of Mexico
(GOM) (20-30N, 100-80W) [10, 3]. These 3 regions have high tropical cyclone
track densities and/or landfall potential [39]. In all maps of the basin
shown, areas are masked where the tropical cyclone track density in ERA5 is
less than half the basin-wide average. The basin is defined as the regions
(10-20N, 20-85W), (20-30N, 20-100W) & (30-40N, 20-80W). This is to help the
analysis by removing areas with low tropical cyclone density.

This study focuses on the TC - ENSO teleconnection. Connectors and
relationships like TC - wind shear, wind shear - ENSO, TC - humidity &
humidity - ENSO are also analysed. Here, vertical wind shear (WS) is
calculated using the following formula:

$\text{VWS}\ =\sqrt{(u_{200}-u_{850})^{2}+(v_{200}-v_{850})^{2}}$

where $u$ and $v$ refer to zonal and meridional winds, and the subscript
refers to the pressure level. If tropical cyclone numbers are being compared,
IBTrACS is also used as a reference. The forecast model performance is
assessed by comparing linear correlation and regression coefficients from the
forecasts with those from ERA5 and IBTrACS. Spatial statistics are generated
by calculating temporal correlation coefficients between two variables at each
grid point (e.g. wind shear and TC numbers), or between one variable and the
ENSO Index. Bootstrapping is used to provide an estimate of the uncertainty of
regressions and correlations. This is done by resampling the target data 1000
times with replacement. For the purposes of this study, the values at the 10th
and 90th percentiles of a bootstrapped distribution were used to provide
estimates of upper and lower bounds for that value.
TC track composites of El Niño years and La Niña years (as predicted by that
model) were also used to investigate any nonlinear behaviour. For a given
model, El Niño years are defined as when that model had a JASO ENSO Index
$>0.5$°C, or $<-0.5$°C for La Niña years. These years are shown in
Supplementary Table S2 below. TC track density anomalies for these years are
then standardised. This is done by dividing the anomaly by the standard
deviation of the number of tracks that cross that point at the respective grid
cell.

##  3 Results

###  3.1 Impact of ENSO on TC numbers

We initially investigate the relationship between ENSO and North Atlantic TCs
on a basin-wide scale (Figure 1). As shown in Figure 1a, a negative
relationship between the ENSO Index and TC numbers is found when using IBTrACS
or tracks identified in ERA5, which is in line with previous studies [2, 28].
All 6 forecast models capture this negative relationship between ENSO and TC
numbers, despite the fact that some models (DWD, CMCC) have significant
negative biases in TC numbers. The magnitude of the regression coefficient
(shown in the legend of Fig. 1a) between ENSO and TC numbers, ranges from
approximately -2.5 to -3 in IBTrACS and ERA5 and approximately -1 to -2 in the
models or similar. Figure 1b shows the correlation coefficient, with the error
bars corresponding to the correlation coefficients at the 10th & 90th
percentiles of the bootstrapped distribution. We see that for all the seasonal
forecast models, the central correlation value is larger in magnitude than for
ERA5 and IBTrACS, however the regression coefficients are all smaller in
magnitude. This is likely due to the interannual variability of the ensemble
mean number of storms predicted by the seasonal forecasts being much lower
than in ERA5 and IBTrACS. However, the central correlation value of all the
models are consistent with the uncertainty range of ERA5, except for CMCC
whose which lies just outside. The similarity of the correlation and
regression values (and their respective uncertainty) between ERA5 and IBTrACS
in Figure 1 motivates the use of ERA5 alone as the comparison in the following
plots.

We next analyse the effect of ENSO on the TC spatial pattern across the basin.
First, we show average JASO track densities across the basin in Figure 2. The
ERA5 data shows the MDR-A (the central Atlantic between 10 & 20N) is the most
active region for TC development. Many storms there track north-westwards and
recurve in the open Atlantic, while some move westwards into the MDR-C (the
Caribbean). In the western part of MDR-C and the GOM (Gulf of Mexico), an area
of enhanced activity is also seen. A similar pattern in MDR-A is seen in the
models, however some models (UKMO, MF6) have too large a proportion of storms
on a recurving track while others appear slightly too zonal (MF5). As in
Figure 1a, we see that the DWD and CMCC models produced far fewer storms than
ERA5.

The track density changes in ENSO years (SI Table S2) are shown in the
composites of standardised track density anomalies for warm and cold events
(Figure 3). For El Niño years in ERA5 (Fig. 3a), we find two main areas of
lower track density: one over the MDR-A and recurving northwards, and another
in the western MDR-C & GOM. These two areas are separated by an area of
increased TC activity in the eastern MDR-C and near Florida. The length of the
period used here (1993-2014) does not appear to affect the robustness of the
pattern in ERA5, as we find similar results when analysing the period from
1979 to 2017 (Fig. S2 in SI). This is consistent with previous studies [38,
25, 4]. All models capture the basin-wide reduction in TC activity during El
Niño years, but are unable to capture the finer spatial pattern described
above. During El Niño years, no model predicts an increase anywhere in MDR-C.
For La Niña years in ERA5 (Fig. 3h), we find TC track density anomalies are
essentially the opposite of the response during El Niño years. In La Niña, we
see an increase in storm count over the MDR-A and in the western MDR-C & GOM,
and a decrease in the eastern MDR-C and near Florida. The models do not
capture this pattern, but the basin-wide average response is captured well
during La Niña years.

In Fig 1 we saw that the interannual variability between basin-wide TC counts
and ENSO is well represented in the models. Here we analyse the spatial
structure of this relationship across the Atlantic basin. Figure 4 shows the
correlation between ENSO Index & TC numbers in different areas of the basin.
ERA5 displays a statistically significant negative correlation in MDR-A and
the western MDR-C & GOM regions. This is in line with the basin-wide results
in Figure 1. Weak negative correlation is seen in the majority of the MDR-C
region & near Florida. Most forecast models in general predict well the strong
negative correlation in MDR-A (except for MF6). However, many models show a
stronger negative correlation near Florida and in MDR-C, which is not found in
ERA5. We also note that many models predict an area of significant negative
correlation around 30N, though the size and location of this differs between
them. Such a signal is not seen in ERA5. Along with the results from Figure 3,
this provides further evidence that despite the good overall basin-wide
response to different ENSO phases, all analysed forecast models have
difficulties representing the details of the spatial pattern of the ENSO - TC
teleconnection over the North Atlantic basin.

###  3.2 Relationship between ENSO, TC numbers and environmental variables

To explore what might be causing some of the errors found in the TC-ENSO
relationship in the different models, we examine the effect of environmental
variables that link the ENSO SST with the atmospheric conditions required for
the formation of TCs in the NA basin. Wind shear (WS) and humidity are among
these variables heavily influenced by ENSO. For example, during El Niño the
Walker circulation shifts eastwards, leading to stronger high-altitude
westerly winds within the Hadley cell, and descending dry air over the
Caribbean. This in turn suppresses the formation of TCs [5, 28].
The correlations between ENSO and wind shear, and between wind shear and TC
numbers, are shown in Figure 5. The ERA5 reanalysis (Fig. 5a) shows that the
ENSO Index is positively correlated with the wind shear across most of the
Atlantic basin, except for the wider region around Florida. Wind shear is
generally negatively correlated with TC numbers across the basin, as expected
[14]. South of 25N, this is replicated in the models, though with slight
shifts in location and intensity. The magnitude of the correlation is
overestimated in the MDR-A for all models, except for MF6 which shows a large
region with correlations around 0. However, the observed ENSO-WS/WS-TC
relationship north of 25N is uniformly poorly simulated by the models - they
display a sharp transition in the sign of the correlation coefficient, in
contrast to ERA5.
As moist air is another crucial element for TC development and is known to be
affected by ENSO, we performed the same analysis for 500hPa specific humidity.
The results are available in the supplementary material (Figure refHumidity
map), however it was judged that errors in forecast models’ representation of
the ENSO - TC teleconnection was more likely due to wind shear than humidity.
This is because in the locations with the largest errors in the ENSO - TC
correlation, the ENSO-humidity and humidity - TC correlations are generally
well simulated (around 30N and in the MDR-C).

##  4 Conclusions and Discussion

We expand on a recent study by [1], analysing the performance of six European
seasonal forecast models at predicting the TC-ENSO teleconnection in this
region. Basin-wide, it is found that the observed negative correlation between
the Niño 3.4 ENSO index and the total number of TC counts over the season is
well predicted by all of the models studied. The correct sign of the
correlation coefficient is usually forecast correctly in the MDR and GOM;
occasionally no signal is seen (as in MF6). However in the eastern Caribbean,
the models simulate a statistical relationship that is not present in ERA5.
Errors in this region are also seen in the ENSO composite analysis (Figure 3).
Similar errors are found by [38] when analysing the response of global climate
models to Central Pacific El Niño events (their Figure 6), which they
partially attribute to errors in simulating wind shear. However in our
analysis, we see no firm evidence that wind shear and humidity drive the
spatial differences in the MDR-C between the models and ERA5. However, the
magnitude of the correlation is generally too large in the forecasts,
especially in between 10 & 20N: this is due to models consistently
overestimating the relationship between local variables and TC numbers.
The sharp transition in the sign of the correlation coefficient in the
forecast models north of 25N may be due to errors simulating the Hadley cell’s
response to ENSO. It is believed that the northern edge of the Hadley cell
usually moves south during El Niño and north during La Niña [37, 24, 33]. In
theory this would lead to a strong correlation between ENSO and wind shear in
the tropics inside the Hadley cell, with a smooth decrease to an area of no
correlation on the north side of the boundary. As this is not seen in the
models, the sharp decrease may be due them failing to alter the width of the
Hadley cell in response to ENSO, keeping the edge location fixed. This
incorrect modelling of the edge of the Hadley cell may cause the pattern seen
in the simulated correlation between wind shear & TCs, which is strongly
positive in the subtropics despite the physical effect of wind shear causing
storms to weaken and dissipate. This is not seen in ERA5 which has a weak
negative correlation almost throughout the NA basin. Together this may explain
the areas of negative correlation seen in the subtropics in the overall ENSO -
TC correlation (Figure 4).
In general, it is found that the forecast models simulate statistical
relationships between the environmental variables and TC numbers (as opposed
to ENSO and environmental variables) for larger areas of the North Atlantic
than can be detected in reanalysis data, especially in the subtropics. One
reason for this is the fact that the seasonal forecast models use a coarse
horizontal resolution of 50-80 km. Secondly, it is also likely due to the fact
that there is only 1 realization for each year in ERA5. However, there are
tens of ensemble model runs for each year in the forecasts which are combined
into the ensemble mean. So it is likely the area of significant correlations
with TC numbers would be larger in the forecasts, as the sample size is much
larger. To address this issue we computed the distribution of individual
correlations from randomly sampled ensemble members before taking the mean.
However, the TC locations vary considerably between ensemble members and so
significant correlations are only seen where the TCs are identified. So when
the mean is taken, the average correlation in any one location is very low,
leading to a poor signal to noise ratio. However, by bootstrapping the data as
in figures 1, we can still estimate the uncertainty of the correlations and
whether the forecast models are consistent with ERA5.
This analysis then shows that the basin-wide TC-ENSO teleconnection is well
represented by these 6 seasonal forecasts between 1993 and 2014. However, due
to the limitations of a small sample size, the uncertainty around the notable
spatial and intensity errors is large. Further research including longer
hindcasts will allow for more robust results.

##  5 Conflicts of Interest

All authors declare they have no conflicts of interest.

##  6 Acknowledgements

We are grateful for the support received from the Met Office Academic
Partnership (MOAP). Robert Doane-Solomon received funding from MOAP. Daniel J.
Befort and Antje Weisheimer received funding from the European Union under
Horizon 2020 (Grant Agreement 776613. Kevin Hodges was funded as part of the
NERC National Centre for Atmospheric Science. All reanalysis and seasonal
forecast data is available from the C3S website
(https://climate.copernicus.eu). Tropical cyclone track data is available upon
request for the seasonal forecast models and reanalysis. IBTrACS data is
available at https://www.ncei.noaa.gov/products/international-best-track-
archive.

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##  7 Figures

Figure 1: a): Scatterplot of number of TCs in the North Atlantic basin for
JASO from 1993-2014 vs NINO 3.4 SST Index for IBTracks (grey), reanalysis
(black) and the seasonal forecast models (colours). Values in the legend are
the regression coefficient between the two variables, and the values in
brackets are the 10th & 90th percentiles of the bootstrapped regression
distribution. b): Bar plot of the correlation between the number of TCs in the
North Atlantic basin for JASO from 1993-2014 and NINO 3.4 SST index. Error
bars are the 10th & 90th percentiles of the bootstrapped correlation
distribution.

Figure 2: Map of TC track densities from reanalysis and forecast model data.
Densities are shown where there is on average $\geq 1$ storm per season.

Figure 3: ENSO track composites for El Niño (a-g) & La Niña (h-n). Map of
standardised track density anomalies during El Niño & La Niña years predicted
by the respective model. Panels a) and h) for reanalysis data and all other
panels for seasonal forecast models. Areas are masked where there is less than
half the basin average TC track density in ERA5.

Figure 4: ENSO-TC correlation map. Correlation coefficient of interannual
variability between NINO 3.4 SST index and number of TCs in the North Atlantic
basin for JASO from 1993-2014. Panel a) for ERA5 data and panels b)-g) for
seasonal forecast models. Areas are masked where there is less than half the
basin average TC track density in ERA5. Shaded areas correspond to
correlations significant at the 5% level.

Figure 5: ENSO - Wind Shear & Wind Shear - TC correlations. Left column:
Correlation coefficients of interannual variability between NINO 3.4 SST index
and wind shear in the North Atlantic basin for JASO from 1993-2014. Right
column: Correlation coefficients of interannual variability between wind shear
and total number of TCs in the North Atlantic basin for the same periods.
Panels a) and h) for reanalysis data and all other panels for seasonal
forecast models. Areas are masked where there is less than half the basin
average TC track density in ERA5. Shaded areas correspond to correlations
significant at the 5% level.

Supplementary Material

##  1 Model Biases

We conducted an analysis of forecast model biases and found significant
effects. The models have considerably different average SSTs in the Niño 3.4
region, although they all have fairly similar levels of variability. We
conclude that many of the forecast models have cold biases. They also predict
very different numbers of storms, ranging from an average of $2.9\pm 0.6$ per
year (DWD) to $18.1\pm 2.4$ (UKMO), compared to an ERA5 count of $14.5\pm
4.4$. However there appears to be no correspondence between the average SST
and average number of storms formed. The variation in storm numbers is lower
in the forecasts than best track data.

Figure S1: a) shows the JASO average SST in the Niño 3.4 region, b) shows the
JASO average TC number in the North Atlantic. Error bars in both plots
correspond to values between the 10th & 90th percentiles of the distribution
of SST indices/TC number over the 1993-2014 period.

##  2 Model Data

Table S1: Seasonal forecast datasets used in this study

Model Atmosphere resolution Ocean resolution Ensemble size UKMO N216/L85
$\approx$ 90 km 0.25°/L75 28 DWD T127/L91 $\approx$ 100 km 0.4°/L70 30 CMCC
$\approx$ 110 km/L46 0.25°/L50 40 ECMWF TCO319/L91 $\approx$ 32 km 0.25°/L75
25 MF5 TL255/L91 $\approx$ 80 km 1°/42 levels 15 MF6 TL359/L91 $\approx$ 50 km
1°/75 levels 25

##  3 TC-ENSO Composite Analysis

Figure S2: ERA5 standardised track density anomaly composites. The columns
shows the anomalies for both ENSO phases, and the rows show how the pattern
changes depending on the years studied. Table S2: Years used for composite
analysis

Model El Niño Years La Niña Years UKMO 1993, 1994, 1997, 2002, 2006, 2009,
2012, 2014 1995, 1996, 1998, 1999, 2003, 2007, 2010 DWD 1993, 1997, 2014 1996,
1999, 2000, 2007 CMCC 1997, 2002, 2006, 2009, 2012, 2014 1995, 1996, 1998,
1999, 2003, 2007, 2010 ECMWF 1997, 2002, 2009, 2012, 2014 1996, 1998, 1999,
2003, 2007, 2010 MF5 1997, 2002, 2006, 2009, 2012, 2014 1995, 1996, 1998,
1999, 2000, 2003, 2005, 2007, 2010 MF6 1993, 1997, 2002, 2006, 2009, 2012
1998, 1999, 2000, 2003, 2007, 2010 ERA5/IBTrACS 1997, 2002, 2004, 2006, 2009,
2012 1995, 1998, 1999, 2007, 2010, 2011

##  4 Humidity Correlation Analysis

In Figure S3 ERA5 shows a negative correlation between humidity and ENSO
across all of the extended tropical NA basin, with a large area of
statistically significant negative correlations in the MDR-C. The USA
coastline near Florida shows no clear signal. All models capture the
correlation across the tropics, and most have an area of significance in
MDR-C. However around Florida some models show stronger correlations, which is
not replicated in ERA5 and is possibly related to errors in this region shown
in Figs. 3 & 4. For the relationship between humidity and TC numbers, ERA5
shows a large area of strong positive correlation across the tropics, and no
correlation in the subtropics. Generally, the models simulate this pattern,
although in most it is too strong and widespread.

Figure S3: ENSO - Humidity & Wind Shear - Humidity correlations. Left column:
Correlation coefficients of interannual variability between NINO 3.4 SST index
and 500hPa specific humidity in the North Atlantic basin for JASO from
1993-2014. Right column: Correlation coefficients of interannual variability
between 500hPa specific humidity and total number of TCs in the North Atlantic
basin for the same periods. Panels a) and h) for reanalysis data and all other
panels for seasonal forecast models. Areas are masked where there is less than
half the basin average TC track density in ERA5. Shaded areas correspond to
correlations significant at the 5% level.


Alternative Approaches to Solve Simple Harmonic Motion

Zhiwei Chong $\,{}^{1}$ and Yajun Wei $\,{}^{2}$

<EMAIL_ADDRESS>
International Division, Experimental School Affiliated with Zhuhai No.1 High
School, Zhuhai, Guangdong, China
<EMAIL_ADDRESS>Zhuhai No.1 High School, Zhuhai, Guangdong, China

###### Abstract

This paper presents two alternative approaches to solve simple harmonic motion
(SHM) without resorting to differential equations. In one approach, the
distance between the equilibrium position and the maximal displacement is
divided into $N$ equal segments. In each segment, the motion is approximated
as one with constant acceleration under the average of two forces at each end
of the segment. Summing up the time to cover each segment and taking a
large-$N$ limit reproduce one quarter of the period for SHM. In the other
approach, the time moving from the maximal displacement to the equilibrium
position is divided into $N$ equal intervals. The motion during each interval
is approximated as one with constant acceleration. A second order recurrence
relation for displacement is obtained. The large-$N$ limit of its solution
results in the same solution obtained from solving differential equation.

##  1 Introduction

Simple harmonic motion (henceforth SHM, with a spring-mass system in mind) is
treated in all introductory physics textbooks[2, 3, 5, 7] and some calculus
textbooks [9]. Other than the approaches in these textbooks, this paper
presents two new approaches without resorting to differential equations but
recurrence relations or difference equations instead. They are conceptually
easy but technically slightly challenging. Nevertheless, the relevant
mathematics is still within the reach of most first year undergraduate
students or even good high school students.

In textbooks, mainly there are three approaches to treat the SHM problem.
Firstly, a connection is established between SHM and circular motion, and the
period of SHM is found [2, 3]. The merit of this approach is that it is
accessible to students without knowledge in calculus. Moreover, it provides a
rigorous derivation of the period for SHM. Secondly, the solution to the
differential equation for SHM is presented and students are invited to verify
the solution [2, 5]. This approach does not involve solving differential
equation but only requires students to have some experience in
differentiation. Lastly, the differential equation is solved either by
integrating twice [5] or using techniques in the theory of differential
equations [7, 9]. In the literature, there are a few theoretical articles on
spring-mass system, however, the focus is on the effect incurred by the mass
of the spring [11, 13, 15, 17]. Some experimental explorations on SHM include
[19, 21, 23].

This paper originates from teaching physics to Year-10 students. The period
$T=2\pi\sqrt{m/k}$ for the motion of a spring-mass system, where $m$ is mass
and $k$ is spring constant, seems rather mysterious to them. By dimensional
analysis, it is easy to convince them of the reasonability of the square-root
part of the expression; there is no alternative combination of $m$ and $k$,
which produces the dimension of time. However, the coefficient $2\pi$ remains
to be a mystery.

Solving differential equations is not an option for high school students and
first year undergraduate students either. Nevertheless, it is worthwhile to
invite them to solve a new problem as much as they can with whatever knowledge
they have at hand. For example, finding an approximate solution to SHM using
_average_ force, speed, and acceleration still constitutes a good exercise for
them. Note that Year-10 students are familiar with the average approach, which
is probably the only weapon at their hands for the moment. It will soon be
seen that the notion of average, though seems primitive, may become quite
powerful if executed appropriately.

The first exercise is easy for them: Find the time it takes the mass to move
from rest at the maximum displacement $x=A$ back to its equilibrium at $x=0$
under the _average_ of two forces exerted by the spring at these two
positions. The answer is $2\sqrt{m/k}$, which is larger than one quarter of
the period $T_{{1\over 4}}\equiv T/4={1\over 2}\,\pi\sqrt{m/k}$ and the ratio
between them is 1.27. There is a byproduct here, the students found that the
quantity they were after was again proportional to $\sqrt{m/k}$.

In the second exercise, the maximum displacement is divided into two equal
segments. Then the students are asked to find and sum the time to cover each
segment under the average of two forces exerted by the spring at the two ends
of each segment. The answer is ${4\over 3}(3-\sqrt{3})\sqrt{m/k}$, which is
closer to $T_{{1\over 4}}$ and the ratio between them is $1.08<1.27$, where
the latter is the ratio obtained in the first exercise. It can be expected
that the difference from the exact value will decrease as the number of
segments increases. More importantly, the exact value can be recovered as the
number of segments goes to infinity. A full development along this direction
is presented in Section 2.

In Section 3, another approach in which the time instead is divided into $N$
equal intervals and a system of difference equations is obtained for SHM. By
solving the system of difference equations and taking large-$N$ limit, the
solution obtained from solving differential equation can be reproduced. We
conclude and discuss in Section 4.

##  2 Dividing Distance into Equal Segments: Riemann Sum

A spring-mass system consists of a mass $m$ and a massless spring with
stiffness $k$. One end of the spring is fixed and the other is attached with a
mass $m$ which lies on a horizontal surface without friction. It is stretched
to the right by distance $A$. Its equilibrium position is denoted as $x=0$ and
the initial position $x=A$. For simplicity, we focus on the motion from its
initial position back to equilibrium, that is, the first quarter of a complete
cycle.

The mass is released from rest. The goal is to find the position $x$ of the
mass at instant $t$, that is, the function $x(t)$. Moreover, we denote the
time it takes the mass to move from $x$ back to its equilibrium as $\tau$.
Clearly, $t+\tau=T_{{1\over 4}}$. We will first find $\tau$, then $t$ readily
results. Figure 1 helps to illustrate the idea.

Figure 1: Illustration of the idea.

The following is the key to our approach: The distance $x$ covered by the mass
during the time interval $\tau$ is divided into $N$ equal segments. See Figure
2 for details. The motion in each segment is considered as one moving with
constant acceleration under the _average_ of two forces exerted by the spring
at each end of the segment under consideration. The quantity $\tau$ can be
obtained by calculating and summing over the time to cover each segment. The
sum appears to be the Riemann sum of a familiar integral which can be readily
integrated, and the same result as that obtained from solving differential
equation can be recovered.

Figure 2: Divide the distance into $N$ equal segments.

The size of each segment is denoted as $\Delta x=x/N$. The equilibrium is
denoted as $x_{0}=0$ and the right end of the $j$-th segment is
$x_{j}=j\,\Delta x$. In particularly, $x_{N}=x$. The time it takes the mass to
cover the $j$-th segment is denoted as $\tau_{j}$ and
$\tau=\sum_{j=1}^{N}\tau_{j}$.

By mechanical energy conservation, the speed $v_{j}$ at position $x_{j}$
satisfies

$\displaystyle{1\over 2}\,m\,v_{j}^{2}\,+\,{1\over
2}\,k\,x_{j}^{2}\,=\,{1\over 2}\,k\,A^{2},\quad j=0,1,\dots,N,$

then $v_{j}$ is obtained as

$\displaystyle v_{j}=\omega A\left(1-\delta^{2}\,j^{2}\right)^{{1\over
2}},\quad\omega^{2}\equiv k/m,\quad\delta\equiv\Delta x/A.$ (1)

Recursively,

$\displaystyle v_{j-1}=\omega A\left(1-\delta^{2}\,(j-1)^{2}\right)^{{1\over
2}}.$ (2)

Both $v_{j}$ and $v_{j-1}$ together with the average acceleration found below
will be used to find the time taken to cover the $j$-th segment.

In the $j$-th segment, the mass moves with an average acceleration under the
average force on it. Applying Newton’s second law and recalling
$x_{j}=j\,\Delta x,\Delta x=x/N$ and $\delta=\Delta x/A$ we have

$\displaystyle m\overline{a}_{j}\,=\,{1\over
2}\,k\,(x_{j-1}+x_{j})\,=\,{1\over 2}\,k\,\Delta x\,(2j-1)\,=\,{1\over
2}\,k\,A\,\delta\,(2j-1),$

or

$\displaystyle\overline{a}_{j}\,=\,{1\over 2}\,\omega^{2}\,A\,\delta\,(2j-1).$
(3)

Note that $\omega\,A$ in (1) and $\omega^{2}\,A$ in (3) are maximum speed and
acceleration for SHM, respectively.

With (1), (2), and (3), we are ready to calculate the time it takes the mass
to cover the $j$-th segment, that is,

$\displaystyle\tau_{j}={v_{j-1}-v_{j}\over\overline{a}_{j}}={2\over\omega\,\delta}\,{\left(1-\delta^{2}\,(j-1)^{2}\right)^{{1\over
2}}-\left(1-\delta^{2}\,j^{2}\right)^{{1\over 2}}\over 2j-1}$
$\displaystyle={2\,\delta\over\omega}\,{1\over\left(1-\delta^{2}\,(j-1)^{2}\right)^{{1\over
2}}+\left(1-\delta^{2}\,j^{2}\right)^{{1\over 2}}}.$

Equivalently,

$\displaystyle\omega\,A\,\tau_{j}\,=\,{2\,\Delta
x\over\left[1-(x_{j-1}/A)^{2}\right]^{{1\over
2}}+\left[1-(x_{j}/A)^{2}\right]^{{1\over 2}}}.$ (4)

Note the fact that $x_{j-1}<x_{j}$, and the ineqalities below follow from (4).

$\displaystyle{\Delta x\over\left[1-(x_{j-1}/A)^{2}\right]^{{1\over
2}}}<\omega\,A\,\tau_{j}<{\Delta x\over\left[1-(x_{j}/A)^{2}\right]^{{1\over
2}}}.$

Summing up $j$ from 1 to $N$ and recalling $\tau=\sum_{N=1}^{N}\tau_{j}$ gives

$\displaystyle\sum_{j=1}^{N}{\Delta
x\over\left[1-(x_{j-1}/A)^{2}\right]^{{1\over
2}}}<\omega\,A\,\tau<\sum_{j=1}^{N}{\Delta
x\over\left[1-(x_{j}/A)^{2}\right]^{{1\over 2}}}.$

Note the left sum is the so-called lower sum and the right the upper sum in
Riemann integral [9, 25]. Then

$\displaystyle\omega\,A\,\tau=\int_{0}^{x}{dx\over\sqrt{1-(x/A)^{2}}}=A\,\arcsin(x/A).$

Equivalently,

$\displaystyle x=A\sin\omega\tau.$ (5)

For the purpose to express the solution in terms of $t$, we note the following
fact. For $x=A$, the time it takes the mass to arrive at its equilibrium is
just $T_{{1\over 4}}$ which is determined by $\sin\omega\,T_{{1\over 4}}=1$,
or $\omega\,T_{{1\over 4}}=\pi/2$ and $T_{{1\over 4}}={\pi\over 2\omega}$.
Therefore, $t=T_{{1\over 4}}-\tau$, and (5) becomes

$\displaystyle x=A\sin\omega\,(T_{{1\over 4}}-t)=A\,\sin\omega\,({\pi\over
2\omega}-t)=A\,\cos\omega\,t,$ (6)

which is exactly the same as that obtained from solving the differential
equation for SHM [7, 9].

##  3 Dividing Time into Equal Intervals: Difference Equations

Another approach is to divide the time $t$ after the mass is released into $N$
equal intervals. See Figure 3 for details. A recurrence relation or a system
of difference equations will be obtained. In the end, a large $N$ limit is
taken to recover the solution obtained from solving differential equations.

Figure 3: Divide the time into $N$ equal intervals.

To be specific, let $x_{i}$ and $v_{i}$ denote the position and speed of the
mass at the end of the $i$-th time interval, respectively. The size of each
time interval is denoted as $\Delta t=t/N$, and the distance covered in the
$i$-th time interval is denoted as $\Delta x_{i}=x_{i-1}-x_{i}$, where $x_{i}$
is the position of the mass at the end of the $i$-th time interval. Note that
$x_{0}=A$ and $x_{i}<x_{i-1}$. The average speed during the $i$-th interval is

$\displaystyle{1\over 2}\,(v_{i}+v_{i-1})={\Delta x_{i}\over\Delta t}.$ (7)

The key step in our approach is the following. Within each time interval, the
motion is considered as one with constant acceleration under the average of
two elastic forces with spring extensions corresponding to the two ends of the
time interval. That is,

$\displaystyle v_{i}-v_{i-1}=\overline{a}_{i}\,\Delta t,$ (8)

where $\overline{a}_{i}$ is the average acceleration in the $i$-th time
interval which can be obtained by applying Newton’s second law.

$\displaystyle m\overline{a}_{i}\,=\,{1\over 2}\,k\,(x_{i}+x_{i-1}),$

or

$\displaystyle\overline{a}_{i}={1\over
2}\,\omega^{2}\,(x_{i}+x_{i-1}),\quad\omega^{2}\equiv k/m.$ (9)

Solving $v_{i}$ and $v_{i-1}$ from (7) and (8) gives

$\displaystyle 2v_{i}=2\,{\Delta x_{i}\over\Delta t}+\overline{a}_{i}\,\Delta
t,$ (10) $\displaystyle 2v_{i-1}=2\,{\Delta x_{i}\over\Delta
t}-\overline{a}_{i}\,\Delta t.$ (11)

Recursively, an alternative expression for $v_{i}$ can be obtained from (11)

$\displaystyle 2v_{i}=2\,{\Delta x_{i+1}\over\Delta
t}-\overline{a}_{i+1}\,\Delta t.$ (12)

Equating (10) with (12) gives

$\displaystyle\Delta x_{i+1}-\Delta x_{i}={1\over
2}\,(\overline{a}_{i}+\overline{a}_{i+1})\,\Delta t^{2}$ (13)

With the expression for average acceleration in (9), a system of difference
equations for $x_{i}$ is obtained.

$\displaystyle(1+\alpha^{2})\,x_{i+1}-2\,(1-\alpha^{2})\,x_{i}+(1+\alpha^{2})\,x_{i-1}=0,\quad\alpha\equiv{1\over
2}\,\omega\Delta t.$ (14)

To solve (14) we need two initial conditions. One is obviously the initial
position of the mass, that is, $x_{0}=A$. The other can be found as follows.
The distance covered in the first time interval is $A-x_{1}={1\over
2}\,\overline{a}_{1}\,\Delta t^{2}={1\over 4}\,\omega^{2}(A+x_{1})\,\Delta
t^{2}=\alpha^{2}\,(A+x_{1})$, where the second equality follows from (9) and
the third from the definition of $\alpha$ in (14). Then the two initial
conditions are

$\displaystyle x_{0}=A,\quad x_{1}={1-\alpha^{2}\over 1+\alpha^{2}}\,A$ (15)

The system of difference equations (14) with initial conditions (15) can be
solved by following a standard procedure which resembles the technique in
solving second order ordinary differential equations. The characteristic
equation is

$\displaystyle(1+\alpha^{2})\,r^{2}-2\,(1-\alpha^{2})\,r+(1+\alpha^{2})=0.$
(16)

Its two roots are

$\displaystyle r_{+}={1+i\,\alpha\over 1-i\,\alpha},\quad
r_{-}={1-i\,\alpha\over 1+i\,\alpha},\quad i^{2}=-1.$ (17)

The general solution to the system of difference equations is

$\displaystyle x_{k}=\lambda_{+}\,r_{+}^{k}\,+\,\lambda_{-}\,r_{-}^{k},\quad
k=0,1,\cdots,N,$ (18)

where $\lambda_{+}$ and $\lambda_{-}$ are constants to be determined by
initial conditions (15). They are

$\displaystyle\lambda_{+}\,=\,\lambda_{-}\,=\,{1\over 2}\,A.$ (19)

Finally, we obtain the solution to (14) and (15)

$\displaystyle x_{k}={1\over 2}\,A\,(r_{+}^{k}\,+\,r_{-}^{k}),\quad
k=0,\cdots,N.$ (20)

In particular, the position of the mass at instant $t$, or equivalently, at
the end of the $N$-th time interval is

$\displaystyle x_{N}={1\over 2}\,A\,(r_{+}^{N}\,+\,r_{-}^{N}).$ (21)

The solution to the SHM can be recovered by taking the large $N$ limit of
(21). For this purpose, we write

$\displaystyle r_{+}^{N}=\left({1+i\,\alpha\over
1-i\,\alpha}\right)^{N}={\left(1+{1\over 2}\,i\,\omega\,\Delta
t\right)^{N}\over\left(1-{1\over 2}\,i\,\omega\,\Delta
t\right)^{N}}={\left(1+{1\over 2}\,i\,\omega\,t\,{1\over
N}\right)^{N}\over\left(1-{1\over 2}\,i\,\omega\,t\,{1\over N}\right)^{N}}.$
(22)

In the large-$N$ limit, the numerator becomes $e^{i\,\omega\,t/2}$ and the
denominator $e^{-i\,\omega\,t/2}$. Thereby, the limit for $r_{+}^{N}$ is
$\lim_{N\to\infty}r_{+}^{N}=e^{i\,\omega\,t}$ and similarly
$\lim_{N\to\infty}r_{-}^{N}=e^{-i\,\omega\,t}$. Finally, the large-$N$ limit
of (21) is

$\displaystyle x(t)=\lim_{N\to\infty}x_{N}={1\over
2}\,A\,\left(e^{i\,\omega\,t}\,+\,e^{-i\,\omega\,t}\right)=A\,\cos\omega\,t,$
(23)

which is exactly the same result obtained from solving differential equation
for SHM [7, 9].

##  4 Summary

This paper presents two alternative approaches to solve the problem of SHM.
Both use mathematics within the reach of undergraduate students taking an
introductory physics course or even good high school students. The key idea is
dividing the distance or time into $N$ segments or intervals. After obtaining
relevant physical quantities, a large-$N$ limit is taken. Then, exactly the
same result can be obtained as that from solving differential equations, which
is beyond the reach of most students taking the course.

The value of our approaches lies in two aspects. On one hand, they enable
students to attack a problem at an early stage using elementary physics and
mathematics rather than after they have learned enough calculus or even theory
of differential equations at a much later stage. On the other hand, even this
paper is presented to students after they are able to solve differential
equations, they can still learn quite some mathematics and problem solving
skills from this paper. Moreover, by comparing various approaches, they have a
good chance to further appreciate the power, simplicity, and elegance of
calculus.

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# Comparing classicality of qutrits from Hilbert-Schmidt, Bures and
Bogoliubov-Kubo-Mori ensembles

Arsen Khvedelidze  A. Razmadze Mathematical Institute, Iv. Javakhishvili
Tbilisi State University, Tbilisi, Georgia  Institute of Quantum Physics and
Engineering Technologies, Georgian Technical University, Tbilisi, Georgia
Laboratory of Information Technologies, Joint Institute for Nuclear Research,
Dubna, Russia  Astghik Torosyan  Laboratory of Information Technologies, Joint
Institute for Nuclear Research, Dubna, Russia

###### Abstract

In the report we analyze the indicator/measure of classicality of quantum
states defined as the probability to find a state with a positive Wigner
function within a unitary invariant random ensemble. The indicators of
classicality of three ensembles associated with the Hilbert-Schmidt, Bures and
Bogoliubov-Kubo-Mori metrics on the space of quantum states of 3-level system
are computed. Their dependence on a moduli parameter of the Wigner function is
studied for all strata of a qutrit state space stratified in accordance with
the unitary group action.

###### Contents

  1. 1 Introduction
  2. 2 Unitary invariant ensembles of qudits
  3. 3 Wigner function positivity and classicality
  4. 4 Examples
    1. 4.1 $\boldsymbol{N=2}\,,$ Qubit
    2. 4.2 $\boldsymbol{N=3}\,,$ Qutrit
  5. 5 Summary

##  1 Introduction

It is natural to expect that some states of a quantum system are more
“quantum” than the others. To transform this intuitive thought into a
qualitative concept, we use the conventional statistical interpretation of
quantum mechanics. The quasiprobability distribution functions will be
regarded as a source of information about the classicality/quantumness of a
state. Our consideration is based on the ideas borrowed from the geometric
probability theory [1] and a commonly accepted opinion that if
quasiprobability functions attain negative values, then it is a certain sign
of quantum nature (see [2]-[5] and [6] with references therein). This
observation allows one to specify the notion of “classical states à la Wigner”
as the states whose Wigner function is positive semidefinite everywhere in the
phase space. Based on this definition, several measures of
classicality/quantumness have been constructed [7]-[13]. When dealing with an
ensemble of random states, the probability to find a “classical state” among
the members of an ensemble is an example of these kind of measures [14]-[16].

In the present article, after the introduction of classicality indicator
$\mathcal{Q}$ as the geometric probability, we will compute it for a 3-level
quantum system, a qutrit. We will compare the characteristics of classicality
of qutrits from three random ensembles: the Hilbert-Schmidt and two other
ensembles, associated with the monotone Riemannian metrics – the Bures and the
Bogoliubov-Kubo-Mori metrics (cf. [17]-[19]). To make the presentation self-
consistent, in the next sections necessary notions and definitions related to
these random ensembles and the Wigner function of a finite-dimensional quantum
system will prelude calculations of the corresponding probabilities.
Calculating the probabilities for different varieties of states, we analyze
the dependence of the classicality measure on the moduli parameter of a qutrit
Wigner function.

##  2 Unitary invariant ensembles of qudits

Let us consider a qudit – a quantum system associated with an
$N\\--$dimensional Hilbert space. The quantum state space $\mathfrak{P}_{N}$
of an $N\\--$level qudit is defined as:

$\mathfrak{P}_{N}=\\{\,\varrho\in M_{N}(\mathbb{C})\ |\
\varrho=\varrho^{\dagger}\,,\quad\varrho\geq
0\,,\quad\mbox{tr}\left(\varrho\right)=1\,\\}\,.$ (1)

The unitary $U(N)$ automorphism of the Hilbert space of an $N\\--$level
quantum system induces the adjoint $SU(N)$ transformations of density matrices
$\varrho\in\mathfrak{P}_{N}$ :

$g\cdot\varrho=g\varrho g^{\dagger}\,,\qquad g\in SU(N)\,.$ (2)

For a closed system it is assumed that the probability density function of the
corresponding ensemble of $N\\--$ dimensional qudits is invariant under (2):

$P(\varrho)=P(g\varrho g^{\dagger})\,,\qquad\forall\ g\in SU(N)\,.$ (3)

Further in the report three ensembles of random states respecting this unitary
symmetry will be used for evaluation of the measure of classicality. Namely,
we will consider the unitary invariant ensembles associated with the following
Riemannian metrics on state space:

  * -

the Hilbert-Schmidt metric $\mathrm{g_{{}_{\mathrm{HS}}}}$ ;

  * -

the Bures metric $\mathrm{g_{{}_{\mathrm{B}}}}$ ;

  * -

the Bogoliubov-Kubo-Mori metric $\mathrm{g_{{}_{\mathrm{BKM}}}}$ .

Before dealing with a specific ensemble, it is worth drawing attention to a
common property of each of these ensembles emerging due to $SU(N)$ invariance
(3).

#### Stratification and factorization of probability distribution on
$\mathfrak{P}_{N}$ .

The invariance property (3) leads to a certain factorization of the
probability distribution functions $P(\varrho)$ into two factors, one
depending on $SU(N)\\--$invariants solely, and the other being a universal
function of the “angular variables”. Moreover, the structure of this
factorization is universal for all states whose unitary orbits are
characterised by the same isotropy group, $H_{\alpha}\subset SU(N)$ , i.e.,
belong to a class with the same “orbit type” 111 Subgroup $H_{x}\subset SU(N)$
is the isotropy group (stabilizer) of point $x\in\mathfrak{P}_{N}$ and is
defined as  $H_{x}=\\{\,g\in SU(N)\ |\ g\cdot x=x\,\\}\,.$ If the conjugacy
class of $H$ is denoted by $[H]$ , then we say that the type of the orbit is
$[H]$, if the stabilizer $H_{x}$ of some/any point $x$ in the orbit belongs to
$[H]\,.$ . Isotropy groups $H_{\varrho}$ of any point
$\varrho\in\mathfrak{P}_{N}$ are determined by the algebraic degeneracy of the
spectrum of $\varrho$ and are in one-to-one correspondence with the Young
diagrams of all possible decompositions of $N$ into non-negative integers.
Hence, we associate the given partition of $N$ with the stratum
$\mathfrak{P}_{[H_{\alpha}]}\,,$ defined as the set of all points of
$\mathfrak{P}_{N}\,,$ whose stabilizer is conjugate to subgroup $H_{\alpha}$ :

$\mathfrak{P}_{[H_{\alpha}]}:=\big{\\{}\,x\in\ \mathfrak{P}_{N}|\
H_{x}\mbox{~{}is~{}conjugate~{}to}\ H_{\alpha}\,\big{\\}}\,,$ (4)

where $\ \alpha=1,2,\dots,p(N)$ . 222The partition function $p(N)$ gives a
number of possible partitions of a non-negative integer ${N}$ into natural
numbers.  The union of $\mathfrak{P}_{[H_{\alpha}]}$ results in the state
space $\mathfrak{P}_{N}$ :

$\mathfrak{P}_{N}=\bigcup_{\mbox{orbit
types}}{\mathfrak{P}}_{[H_{\alpha}]}\,,$ (5)

with each component of the decomposition (5) consisting of density matrices
with a fixed algebraic degeneracy,

$\mathfrak{P}_{[H_{\alpha}]}=\bigcup_{\omega\in
S_{s}}{\mathfrak{P}_{k_{\omega(1)},k_{\omega(2)},\dots,k_{\omega(s)}}}\,.$ (6)

In (6) $S_{s}$ is a symmetric group acting on a given partition of $N$ into
$s$ natural numbers $k_{1},k_{2},\dots,k_{s}\,.$ Algebraically,
$\mathfrak{P}_{k_{1},k_{2},\dots,k_{s}}$ being a set of states with a fixed
degeneracy is defined via the characteristic polynomial of a density matrix:
333Note that in (7) the condition of summing up the degrees of degeneracy to
$N$ means that only the maximal rank states are considered.

$\mathfrak{P}_{k_{1},k_{2},\dots,k_{s}}=\\{\,\varrho\in\mathfrak{P}_{N}\,,k_{i}\in\mathbb{Z}_{+}\,|\,\det(\varrho-\lambda)=\prod_{i=1}^{s}(r_{i}-\lambda)^{k_{i}}\,,\quad\sum_{i=1}^{s}k_{i}=N\,\\}\,.$
(7)

Geometrically, the set $\mathfrak{P}_{k_{1},k_{2},\dots,k_{s}}$ with
$k_{1}=k_{2}=\cdots=k_{N}=1$ represents the interior of an
$(N-1)\\--$dimensional simplex $C_{N-1}$ of eigenvalues:

$C_{N-1}:=\\{\,\boldsymbol{r}\in\mathbb{R}^{N}\,\biggl{|}\,\sum_{i=1}^{N}r_{i}=1\,,\quad
1\geq r_{1}\geq r_{2}\geq\dots\geq r_{N-1}\geq r_{N}\geq 0\,\\}\,,$ (8)

while for all other admissible tuples
$\boldsymbol{k}=(k_{1},k_{2},\dots,k_{s})$ each
$\mathfrak{P}_{k_{1},k_{2},\dots,k_{s}}$ represents the union of the faces and
edges of the $(N-1)\\--$simplex parameterized by the barycentric coordinates
of the following kind:

$\boldsymbol{r}^{\downarrow}(\varrho)=\\{r_{1}\overbrace{(1,\dots,1)}^{k_{1}}\,;\,r_{2}\overbrace{(1,\dots,1)}^{k_{2}}\,;\,\dots\,;\,r_{s}\overbrace{(1,\dots,1)}^{k_{s}}\\}\,.$
(9)

Now, bearing in mind the above described stratification of
$\mathfrak{P}_{N}\,,$ it is easy to show the factorization of
$SU(N)\\--$invariant measures. Indeed, one can be convinced that the Singular
Value Decomposition (SVD) of the density matrix from a stratum
$\mathfrak{P}_{[H_{\alpha}]}$ with spectrum of the form (9):

$\varrho=U\mathrm{diag}\left(r_{1},r_{2},\dots
r_{s}\right)U^{\dagger}\,,\qquad U\in SU(N)/H_{\alpha}\,,$ (10)

reveals the following factorization of the invariant probability distribution
(3):

$\mathrm{P}(\varrho)=P(r_{1},\dots,r_{s})\,\mathrm{d}r_{1}\wedge\cdots\wedge\mathrm{d}r_{N}\wedge\mathrm{d}\mu_{U(N)/H}\,,$
(11)

where the first factor $P(r_{1},\dots,r_{s})$ represents a measure on subset
$\mathfrak{P}_{k_{1},k_{2},\dots,k_{s}}$ of the simplex $\mathcal{C}_{N-1}$,
while the second factor is the measure on coset $U(N)/H$ .

After a preliminary exposition of this generic property of unitary invariant
ensembles, we will now specify the form of the distribution
$P(r_{1},\dots,r_{N})$ for the Hilbert-Schmidt metric and for an important
class of the monotone metrics.

#### The Hilbert-Schmidt ensemble of qudits.

Let us consider the metric corresponding to the distance between two
infinitesimally close matrices $\varrho-\mathrm{d}\varrho$ and
$\varrho+\mathrm{d}\varrho$ calculated with respect to the Frobenius norm,

$\mathrm{g_{{}_{\mathrm{HS}}}}\propto\mathrm{Tr}\left(\mathrm{d}\varrho\otimes\mathrm{d}\varrho\right)\,.$
(12)

If a density matrix belongs to the interior of the simplex $C_{N-1}$ , i.e.,
the matrix has $N$ distinct non-zero eigenvalues
$(k_{1}=k_{2}=\cdots=k_{N}=1)$ , then the metric (12) defines the standard
Hilbert-Schmidt ensemble of random full rank $N\\--$qudits. A straightforward
computation shows that the joint probability distribution of eigenvalues reads

$P^{\rm
HS}(r_{1},\dots,r_{N})\propto\,\delta(1-\sum_{j=1}^{N}r_{j})\prod_{j<k}^{N}(r_{j}-r_{k})^{2}\,,$
(13)

and unitary random factors $U$ in SVD decomposition are distributed according
to the Haar measure on the coset $U(N)/U(1)^{N}$ .

#### Degenerate Hilbert-Schmidt qudits

If the full rank density matrix has a spectrum of the form (9) with an
arbitrary algebraic degeneracy, then the joint probability distribution of
eigenvalues is reduced to the following expression:

$P^{\mathrm{HS}}_{k_{1},\dots,k_{s}}(r_{1},\dots,r_{s})\propto\delta(1-\sum_{i=1}^{s}k_{i}r_{i})\prod_{i<j}^{1\dots
s}(r_{i}-r_{j})^{2k_{i}k_{j}}\,.$ (14)

At the same time the angles in the SVD are distributed according to the Haar
measure on the coset $U(N)/U(k_{1})\times\cdots\times U(k_{s})$ .

#### Monotone metrics and monotone ensembles of $N\\--$qudits

Two of the above-mentioned metrics, the Bures and Bogoliubov-Kubo-Mori ones,
are members of a special class of unitary covariant monotone metrics.
According to [18], any monotone metric can be written as 444 Due to the
unitary covariance of the Riemannian metric, it is sufficient to describe
monotone metrics evaluated for diagonal matrices $D$ .

$\mathrm{g}_{{}_{D}}(X,Y)\propto\mathrm{Tr}\left(XK^{-1}_{D}Y\right)\,,$ (15)

where $K_{D}$ is an operator,

$K_{D}=R_{D}^{1/2}f(L_{D}R^{-1}_{D})R_{D}^{1/2}\,,$

constructed out of the left and right multiplication operators $L_{D}$ and
$R_{D}$ , i.e., $L_{D}X=DX$ and $R_{D}X=XD$ for $X\in M_{N}(\mathbb{C})\,,$
and operator monotone function $f$ , which is symmetric, i.e., $f(t)=tf(1/t)$
, and normalized, $f(1)=1$ .

Assuming that $\varrho$ is a full rank density matrix with a simple spectrum
and using the 1-form coordinate basis for simplex, $\mathrm{d}r_{i}\,,$ and
non-coordinate basis on $SU(N)$ group,
$w_{ij}:=\left(U^{\dagger}\mathrm{d}U\right)_{ij}$ , the non-degenerate
monotone metrics can be written as

$\mathrm{g}_{f}=\frac{1}{4}\sum_{i=1}^{N}\,\frac{\mathrm{d}r_{i}\otimes\mathrm{d}r_{i}}{r_{i}}\,+\frac{1}{2}\,\sum_{i<j}^{N}\,c_{f}(r_{i}\,,r_{j}){(r_{i}-r_{j})^{2}}\,\omega_{ij}\otimes\omega_{ij}\,.$
(16)

In (16) by $c_{f}(x,y)=\frac{1}{yf(x/y)}$ we denote the Morozova-Chentsov
function corresponding to a monotone function $f(t)$ . Note that the Bures and
Bogoliubov-Kubo-Mori metrics are associated with the following choice of
monotone function:

$f_{{}_{\mathrm{B}}}(t)=\frac{1+t}{2}\,,\qquad
f_{{}_{\mathrm{BKM}}}(t)=\frac{t-1}{\ln{t}}\,,$ (17)

and corresponding Morozova-Chentsov functions,

$c_{{}_{\mathrm{B}}}(x,y)=\frac{2}{x+y}\,,\qquad
c_{{}_{\mathrm{BKM}}}(x,y)=\frac{\ln{x}-\ln{y}}{x-y}\,.$ (18)

#### Probability measures from monotone metrics.

For an arbitrary monotone metric evaluated for degenerate qudits, calculations
of the joint probability distribution of eigenvalues give:

$P^{f}_{k_{1},\dots,k_{s}}(r_{1},\dots,r_{s})\propto\frac{\delta\left(1-\sum_{i=1}^{s}k_{i}r_{i}\right)}{\sqrt{r_{1}\cdot
r_{2}\cdot\ldots\cdot
r_{s}}}\,\prod_{i<j}^{s}c_{f}^{k_{i}k_{j}}(r_{i},r_{j})\,(r_{i}-r_{j})^{2k_{i}k_{j}}\,.$
(19)

##  3 Wigner function positivity and classicality

Here, for the reader’s convenience, before giving a definition of the
indicator of classicality, we present the basic settings of the Wigner
function of a mixed state of a finite-dimensional quantum system.

#### Wigner function settings

The Wigner quasiprobability distribution
$W^{(\boldsymbol{\nu})}_{\varrho}(\Omega_{N})$ of an $N\\--$level qudit is
constructed via dual pairing [20, 21],

$W^{(\boldsymbol{\nu})}_{\varrho}(\Omega_{N})=\mbox{tr}\left[\varrho\,\Delta(\Omega_{N}\,|\,\boldsymbol{\nu})\right]\,,$
(20)

of a density matrix $\varrho$ with the Stratonovich-Weyl (SW) kernel
$\Delta(\Omega_{N}\,|\,\boldsymbol{\nu})\in\mathfrak{P}^{\ast}_{N}\,$ from the
dual space:

$\mathfrak{P}^{\ast}_{N}=\\{\,X\in M_{N}(\mathbb{C})\ |\
X=X^{\dagger}\,,\quad\mbox{tr}\left(X\right)=1\,,\quad\mbox{tr}\left(X^{2}\right)=N\,\\}\,.$
(21)

For $N\geq 3$ , algebraic equations (21) admit a family of solutions. As a
result, the generic Wigner function depends on $N-2$ real parameters
$\boldsymbol{\nu}=(\nu_{1},\nu_{2},\dots,\nu_{N-2})\,,$ (see details in [21]).
The structure of phase space $\Omega_{N}$ depends on the isotropy group of the
SW kernel. For any given isotropy group $H\in U(N)$ of the form

$H={U(k_{1})\times U(k_{2})\times\ldots\times U(k_{s+1})}\,,$

we identify the phase-space $\Omega_{N}$ with the complex flag manifold,

$\Omega_{N}\,\to\,\mathbb{F}^{N}_{d_{1},d_{2},\dots,d_{s}}={U(N)}/{H}\,,$

where $(d_{1},d_{2},\dots,d_{s})$ is a sequence of positive integers with a
sum $N$ , such that $k_{1}=d_{1}$ and $k_{i+1}=d_{i+1}-d_{i}$ with
$d_{s+1}=N\,.$ After presenting necessary notions, we are ready to introduce
the definition of the classicality of states.

#### Classical states and classicality indicator

The “classical states” form the subset
$\mathfrak{P}_{N}^{(+)}\subset\mathfrak{P}_{N}$ of states whose Wigner
function is non-negative everywhere over the phase space:

$\mathfrak{P}^{(+)}_{N}=\\{\,\varrho\in\mathfrak{P}_{N}\,\ |\
W_{\varrho}(z)\geq 0\,,\quad\forall z\in\Omega_{N}\,\\}\,,$ (22)

and similarly, the “classical states on a fixed stratum”
$\mathfrak{P}_{H_{\alpha}}$ are defined as:

$\mathfrak{P}^{(+)}_{H_{\alpha}}=\mathfrak{P}^{(+)}_{N}\cap\mathfrak{P}_{H_{\alpha}}\,.$
(23)

Based on (22), we define the geometric probability of finding a classical
state in an ensemble as

$\mathcal{Q}_{N}=\frac{\mathrm{Volume(Classical~{}States)}}{\mathrm{Volume(All~{}States)}}\,.$
(24)

Here it is assumed that the Riemannian volume is calculated with respect to
the measure dictated by the probability distribution function of an ensemble.
Probability (24) shall be considered as the global indicator of classicality.
Moreover, for classical states (22) on the fixed stratum
$\mathfrak{P}_{H_{\alpha}}$ we define the $Q\\--$indicator of classicality of
the stratum:

$\mathcal{Q}_{N}[H_{\alpha}]=\frac{\mathrm{Volume(Classical~{}States~{}on~{}\mathfrak{P}_{[H_{\alpha}]})}}{\mathrm{Volume(All~{}States~{}on~{}\mathfrak{P}_{[H_{\alpha}]})}}\,.$
(25)

According to (6), stratum $\mathfrak{P}_{[H_{\alpha}]}$ consists from subsets
of matrices with a certain degeneracy type. Hence, owing to the unitary
covariance of probability distribution functions (3), the
$\mathcal{Q}\\--$indicator depends only on the joint probability distribution
of eigenvalues of the density matrix and can be rewritten as:

$\mathcal{Q}_{N}[H_{\alpha}]=\frac{\sum_{\omega\in
S_{s}}\int_{\mathcal{C}^{\ast}_{N-1}(H_{\alpha})}P^{f}_{k_{\omega}(1),\dots,k_{\omega}(s)}(r_{1},\dots,r_{s})\,\mathrm{d}r_{1}\wedge\dots\wedge\mathrm{d}r_{s}}{\sum_{\omega\in
S_{s}}\
\int_{\mathcal{C}_{N-1}(H_{\alpha})}P^{f}_{k_{\omega}(1),\dots,k_{\omega}(s)}(r_{1},\dots,r_{s})\,\mathrm{d}r_{1}\wedge\dots\wedge\mathrm{d}r_{s}}\,.$
(26)

In (26) the integral in the denominator represents the volume of the orbit
space of stratum $\mathfrak{P}_{[H_{\alpha}]}$ . The subset
$\mathcal{C}_{N-1}(H_{\alpha})$ is a union of faces of the simplex $C_{N-1}$
determined by the isotropy group $[H_{\alpha}]$ . The integration in the
nominator of (26) is over the image of $\mathfrak{P}^{+}_{[H_{\alpha}]}$ under
the canonical quotient map:

$\displaystyle\mathcal{C}_{N-1}^{\ast}(H_{\alpha})=\\{\,p(x)\,\ |\
x\in\mathfrak{P}_{H_{\alpha}}^{(+)}\,\\}\,.$ (27)

According to [21], the subset (27) can be identified with a certain cone in
$\mathbb{R}^{N-1}\,.$ Having denoted by
$\boldsymbol{r}=\\{r_{1},r_{2},\dots,r_{N}\\}$ the eigenvalues of the density
matrix $\varrho$ and by $\boldsymbol{\pi}=\\{\pi_{1},\pi_{2},\dots,\pi_{N}\\}$
the eigenvalues of the SW kernel, both arranged in decreasing order, we obtain
that $\mathcal{C}_{N-1}^{\ast}(H_{\alpha})$ is the following dual cone:

$\mathcal{C}_{N-1}^{\ast}(H_{\alpha})=\ \left\\{\,\boldsymbol{\pi}\in\mbox{\bf
spec}\left(\Delta(\Omega_{N})\right)\ \,|\
\,(\boldsymbol{r}^{\downarrow},\boldsymbol{\pi}^{\uparrow})\geq
0,\quad\forall\,\boldsymbol{r}\in\mathcal{C}_{N-1}(H_{\alpha})\,\right\\}\,,$
(28)

where
$(\boldsymbol{r}^{\downarrow},\boldsymbol{\pi}^{\uparrow})=r_{1}\pi_{N}+r_{2}\pi_{N-1}+\dots+r_{N}\pi_{1}$
.

##  4 Examples

In this section results of the calculations of classicality indicators for the
random ensembles described in Section 2 will be given for qubit and qutrit
cases.

###  4.1 $\boldsymbol{N=2}\,,$ Qubit

For a single qubit the expansion coefficients of the density matrix over the
Pauli $\boldsymbol{\sigma}$-matrices are given by components of the
3-dimensional Bloch vector $\boldsymbol{\xi}=(\xi_{1},\xi_{2},\xi_{3})$:

$\varrho=\frac{1}{2}\left(I+(\boldsymbol{\xi},\boldsymbol{\sigma})\right)\,.$
(29)

Expressing the eigenvalues of (29) in terms of the length $r\in(0\,,1]$ of the
Bloch vector

$r_{1}=\frac{1+r}{2}\,,\quad r_{2}=\frac{1-r}{2}\,,$ (30)

and taking into account that the Wigner function of a qubit is uniquely
constructed with the aid of the SW kernel, whose spectrum is:

$\pi_{1}=\frac{1+\sqrt{3}}{2}\,,\qquad\pi_{2}=\frac{1-\sqrt{3}}{2}\,,$ (31)

we conclude that the Wigner function of a qubit is positive definite inside
the Bloch ball of radius ${1}/\sqrt{3}\,.$ Since all states, except the
maximally mixed state $r_{1}=r_{2}=1/2\,,$ have the torus $T^{2}\in SU(2)$ as
their isotropy group, there is only one indicator $\mathcal{Q}_{[T^{2}]}$ for
all possible ensembles of qubits. Hence, for any random ensemble of qubits,
characterized by a probability distribution $P(r)\,,$ the expression of the
classicality indicator is reduced to the following ratio:

$\mathcal{Q}_{[T^{2}]}=\frac{\displaystyle{\int_{0}^{\frac{1}{\sqrt{3}}}}P(r)\mathrm{d}r}{\displaystyle{\int_{0}^{1}}P(r)\mathrm{d}r}\,.$
(32)

####  Hilbert-Schmidt ensemble.

Noting that for the Hilbert-Schmidt ensemble the probability distribution
function is $P^{\mathrm{HS}}(r)\propto r^{2}\,,$ the calculation of (32)
gives:

$\mathcal{Q}^{\mathrm{HS}}_{[T^{2}]}=\frac{1}{3\sqrt{3}}\approx 0.19245\,.$
(33)

#### Bures ensemble

Using the probability distribution function of the Bures ensemble
$P^{\mathrm{B}}(r)\propto\frac{r^{2}}{\sqrt{1-r^{2}}}\,,$ we find the
classicality indicator:

$\mathcal{Q}^{\mathrm{B}}_{[T^{2}]}=\frac{2}{\pi}\left(\arcsin\frac{1}{\sqrt{3}}-\frac{\sqrt{2}}{3}\right)\approx
0.0917211\,.$ (34)

#### Bogoliubov-Kubo-Mori ensemble

Analogously, calculations with the Bogoliubov-Kubo-Mori measure
$P^{\mathrm{B}}(r)\propto\frac{r\left(\ln{\frac{1+r}{2}}-\ln{\frac{1-r}{2}}\right)}{\sqrt{1-r^{2}}}$
result in the classicality indicator:

$\displaystyle\mathcal{Q}^{\mathrm{BKM}}_{[T^{2}]}=\frac{2}{\pi}\left(\arcsin\frac{1}{\sqrt{3}}-\sqrt{\frac{2}{3}}\operatorname{arcoth}\sqrt{3}\right)\approx
0.0495506\,.$ (35)

###  4.2 $\boldsymbol{N=3}\,,$ Qutrit

Qutrit state space $\mathfrak{P}_{3}$ admits the following orbit type
decomposition:

$\mathfrak{P}_{3}=\mathfrak{P}_{[T^{3}]}\bigcup\mathfrak{P}_{[S(U(2)\times
U(1))]}\bigcup\mathfrak{P}_{[SU(3)]}\,.$ (36)

Three strata in (36) are labeled by the isotropy group or directly by the
degeneracy of the density matrices. For full rank states, putting eigenvalues
of $\varrho$ in decreasing order, $1>r_{1}>r_{2}>r_{3}>0$ , the components of
decomposition (36) are described as follows (see geometrical illustration in
Fig. 1):

  1. 1.

the regular stratum $\mathfrak{P}_{[T^{3}]}$ of maximal dimension 6 consists
of matrices with a simple spectrum, $1>r_{1}\neq r_{2}\neq r_{3}>0$ . The
corresponding orbit space is the face $F_{123}$ of the ordered 2-simplex, the
interior of $\triangle AOB$ ,

  2. 2.

the degenerate 4-dimensional stratum $\mathfrak{P}_{[S(U(2)\times U(1))]}$
with density matrices whose degeneracy is $\boldsymbol{k}=(2,1)$ and
$\boldsymbol{k}=(1,2)$ , i.e., $1>r_{1}\neq r_{2}=r_{3}>0$ and
$1>r_{1}=r_{2}\neq r_{3}>0$ . The corresponding orbit space represents the
union of edges $F_{1|23}$ and $F_{12|3}$ of the 2-simplex, two sides of
$\triangle AOB$ ,

  3. 3.

the 0-dimensional stratum $\mathfrak{P}_{[SU(3)]}$ of the maximally mixed
state with the triple degeneracy $\boldsymbol{k}=(3)$ ,
$r_{1}=r_{2}=r_{3}=1/3$ .

Figure 1: The ordered 2-simplex of qutrit eigenvalues is represented by
$\triangle AOB$ , and the hatched region, $\triangle COD\,,$ corresponds to
the classical states. Edges $AO/\\{A\\}$ and $BO/\\{B\\}$ are locus of
degenerate states $F_{12|3}$ and $F_{1|23}$ , while their parts $CO/\\{O\\}$
and $DO/\\{O\\}$ represent the degenerate classical states $F^{+}_{12|3}$ and
$F^{+}_{1|23}$ .

Taking into account the decreasing order of the eigenvalues, $1\geq r_{1}\geq
r_{2}\geq r_{3}\geq 0$ , the spectrum of qutrit admits the following
parameterization:

$\displaystyle r_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{3}-\frac{2r}{\sqrt{3}}\,\cos\left(\frac{\varphi+2\pi}{3}\right),$
(37) $\displaystyle r_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{3}-\frac{2r}{\sqrt{3}}\,\cos\left(\frac{\varphi+4\pi}{3}\right),$
(38) $\displaystyle r_{3}$ $\displaystyle=$
$\displaystyle\frac{1}{3}-\frac{2r}{\sqrt{3}}\,\cos\left(\frac{\varphi}{3}\right),$
(39)

with $r\in[0,1/\sqrt{3}]$ and the angle $\varphi\in[0,\pi]$ . If $r$ and
$\varphi$ are treated as the polar coordinates on a plane,
$\left(r\cos\varphi\,,r\sin\varphi\right)\,,$ then geometrically the formulae
(37)-(39) can be interpreted as a map between the ordered simplex
$\mathcal{C}_{2}$ and the domain of the upper half-plane outlined by the
Maclaurin trisectrix:

$r(\varphi,1/\sqrt{3})=\frac{1}{2\sqrt{3}\cos({\varphi}/{3})}\,.$ (40)

More precisely, under transformations (37)-(39), the ordered simplex of
eigenvalues $\mathcal{C}_{2}$ maps to the domain (see Fig. 2)

$F_{123}=\,:\ \biggl{\\{}\,r\geq 0\,,\varphi\in[0,\pi]\,\ \biggl{|}\,\
\cos\left(\frac{\varphi}{3}\right)\leq\frac{1}{2\sqrt{3}r}\,\biggl{\\}}\,.$
(41)

Figure 2: The image of the ordered simplex $\mathcal{C}_{2}$ on the plane
$x=r\cos\varphi\,,\,y=r\sin\varphi\,$ under the mapping (37)-(39).

#### Wigner function of a qutrit

The master equations (21) for eigenvalues of the Stratonovich-Weyl kernel of a
qutrit,

$\pi_{1}+\pi_{2}+\pi_{3}=1\,,\qquad\pi^{2}_{1}+\pi^{2}_{2}+\pi^{2}_{3}=3\,,$
(42)

define a one-parametric family of the Wigner functions. Due to the permutation
symmetry of (42), the corresponding moduli space is a unit circle factorised
by the symmetric group $S_{3}\,.$ Let $\mu_{3}$ and $\mu_{8}$ be Cartesian
coordinates of this arc with a polar angle from the interval
$\zeta\in[0,\frac{\pi}{3}]\,,$

$\mu_{3}=\sin\zeta\,,\qquad\mu_{8}=\cos\zeta\,,\qquad$ (43)

then, providing the decreasing order of the SW kernel eigenvalues,
$\pi_{1}\geq\pi_{2}\geq\pi_{3}\,,$ one can represent the whole class of
solutions to (42) as:

$\displaystyle\pi_{1}=\frac{1}{3}+\frac{2}{\sqrt{3}}\,\mu_{3}+\frac{2}{3}\,\mu_{8}\,,\quad\pi_{2}=\frac{1}{3}-\frac{2}{\sqrt{3}}\,\mu_{3}+\frac{2}{3}\,\mu_{8}\,,\quad\pi_{3}=\frac{1}{3}-\frac{4}{3}\,\mu_{8}\,.$
(44)

#### Classical states of qutrit.

The image of classical states from the regular stratum
$\mathfrak{P}_{[T^{3}]}$ to the unitary orbit space is the interior
$F^{+}_{123}$ of a cone which is cut out from the simplex $\mathcal{C}_{2}$ by
the line (see Fig. 1)

$L_{\boldsymbol{\pi}}(\boldsymbol{r})\,:\qquad
r_{1}\pi_{3}+r_{2}\pi_{2}+r_{3}\pi_{1}=0\,,$ (45)

while the orbit space of classical states from the stratum
$\mathfrak{P}_{S(U(2)\times U(1))}$ consists of two pieces: $F^{+}_{1|23}$ and
$F^{+}_{12|3}$ , corresponding to the matrices of degeneracy types $(2,1)$ and
$(1,2)$ respectively. Using the polar form of parameterization of the spectrum
of a density matrix (37)-(39) and expressions (44) for the SW kernel
eigenvalues, the cone of classical states on a regular stratum reads:

$\displaystyle F^{+}_{123}\,:\biggl{\\{}\,r>0\,,\varphi\in(0,\pi)\,\
\biggl{|}\,\
\cos\left(\frac{\varphi}{3}+\zeta-\frac{\pi}{3}\right)\leq\frac{1}{4\sqrt{3}r}\,\biggl{\\}}\,,$
(46)

while the cone of classical states on the degenerate stratum
$\mathfrak{P}^{(+)}_{[S(U(2)\times U(1))]}$ is:

$\displaystyle F^{+}_{1|23}$ $\displaystyle=$
$\displaystyle\left\\{\,\varphi=0,\ r\in(0,\frac{1}{2\sqrt{3}})\ \biggl{|}\
\cos\left(\zeta-\frac{\pi}{3}\right)<\frac{1}{4\sqrt{3}r}\,\right\\}\,,$ (47)
$\displaystyle F^{+}_{12|3}$ $\displaystyle=$ $\displaystyle\left\\{\,\
\varphi=\pi,\ r\in(0,\frac{1}{\sqrt{3}})\ \biggl{|}\
\cos\left(\zeta\right)<\frac{1}{4\sqrt{3}r}\,\right\\}\,.$ (48)

$\zeta=0$ |  $\zeta=\pi/6$ |  $\zeta=\pi/3$
---|---|---

Figure 3: The orbit space $F_{123}$ in blue and its subspace $F^{+}_{123}$ in
red for different values of the moduli parameter: $\zeta=0,\pi/6,\pi/3$ .

####  $\mathcal{Q}_{3}\\--$indicator for Hilbert-Schmidt ensemble of qutrits
from regular stratum.

The regular stratum $\mathfrak{P}_{[T^{3}]}$ consists of density matrices with
a simple spectrum. The expression $\mathcal{Q}_{[T^{3}]}$ comprises the
integrals over the face $F_{123}$ and its subset $F^{+}_{123}$ :

$\mathcal{Q}^{\mathrm{HS}}_{[T^{3}]}=\frac{\mathrm{vol_{HS}}(F^{+}_{123})}{\mathrm{vol_{HS}}(F_{123})}\,.$
(49)

In (49) the expression $\mathrm{vol_{HS}}(X)$ denotes the Riemannian integral
over a region $X$ taken with the measure induced on $X\in\mathcal{C}_{2}$ from
the Hilbert-Schmidt on $\mathfrak{P}_{3}$ ,

$\displaystyle\mathrm{vol}_{\mathrm{HS}}(X)=\int_{X}\,P_{1,1,1}^{\mathrm{HS}}(r_{1},r_{2},r_{3})\,\mathrm{d}r_{1}\wedge\mathrm{d}r_{2}\wedge\mathrm{d}r_{3}\,.$
(50)

Taking into account the expression (12) for the Hilbert-Schmidt measure and
the polar form of the parameterization of the qutrit orbit space (41) and of
(46), we obtain the indicator of classicality as a function of the moduli
parameter $\zeta$ :

$\mathcal{Q}^{\mathrm{HS}}_{[T^{3}]}(\zeta)=\frac{20\cos^{2}{\left(\zeta-{\pi}/{6}\right)}+1}{128\left(4\cos^{2}{\left(\zeta-{\pi}/{6}\right)}-1\right)^{5}}\,.$
(51)

####  $\mathcal{Q}_{3}\\--$indicator for Hilbert-Schmidt ensemble of qutrits
from degenerate stratum.

The stratum $\mathfrak{P}_{[S(U(2)\times U(1))]}$ has two pieces, $F_{1|23}$
and $F_{12|3}$, associated with density matrices with degenerate eigenvalues
$r_{1}=r_{2}\neq r_{3}$ and $r_{1}\neq r_{2}=r_{3}$ , respectively. Hence, the
$\mathcal{Q}_{3}\\--$indicator for the degenerate stratum of a qutrit reads:

$\mathcal{Q}^{\mathrm{HS}}_{[S(U(2)\times
U(1))]}=\frac{\mathrm{vol_{HS}}(F^{+}_{1|23})+\mathrm{vol_{HS}}(F^{+}_{12|3})}{\mathrm{vol_{HS}}(F_{1|23})+\mathrm{vol_{HS}}(F_{12|3})}\,,$
(52)

where we keep the notation previously used for the regular stratum (50),
noticing only that the dimension of integration over the degenerate orbit
state strata has decreased by one:

$\displaystyle\mathrm{vol}_{\mathrm{HS}}(F_{1|23})=\int_{F_{1|23}}\,P_{2,1}^{\mathrm{HS}}(r_{1},r_{2})\,\mathrm{d}r_{1}\wedge\mathrm{d}r_{2}\,.$
(53)

The evaluation of all integrals in (52) gives:

$\mathcal{Q}^{\mathrm{HS}}_{[S(U(2)\times
U(1))]}(\zeta)=\frac{1}{1056}\left({\csc^{5}\left(\zeta+\frac{\pi}{6}\right)+\sec^{5}(\zeta)}\right)\,.$
(54)

The functional dependence of the indicator $\mathcal{Q}_{3}^{\mathrm{HS}}$ for
the regular (51) and degenerate (54) strata is depicted in Fig. 4(a) . Apart
from this, in Fig. 4(b) we present the ratio

$\mathrm{R^{HS}}(\zeta)=\frac{\mathcal{Q}^{\mathrm{HS}}_{[S(U(2)\times
U(1))]}(\zeta)}{\mathcal{Q}^{\mathrm{HS}}_{[T^{3}]}(\zeta)}$ (55)

as a certain measure of the relation between the symmetry of a state and its
classicality.

(a)

(b)

Figure 4:  (a) $\mathcal{Q}_{3}\\--$indicators of a Hilbert-Schmidt qutrit as
functions of $\zeta$ for the regular (gray curve) and degenerate (blue curve)
strata. The absolute minimum of both indicators is attained at $\zeta=\pi/6$ .
(b) The ratio of degenerate to regular $\mathcal{Q}_{3}\\--$indicators.

####  $\mathcal{Q}_{3}\\--$indicator for Bures ensemble of qutrits from
regular stratum.

Using the generic expressions for the joint probability distributions of
eigenvalues for monotone metrics (19) and the technique developed above, we
compute the $\mathcal{Q}_{3}\\--$indicators for the Bures and Bogoliubov-Kubo-
Mori ensembles of qutrits. The results of our calculations are presented in
Fig. 5(a).

(a)

(b)

Figure 5: (a) The plot of $\mathcal{Q}_{3}$ for the Bures (solid curves) and
BKM (dashed curves) ensembles of qutrits from the regular (gray curves) and
degenerate (blue curves) strata. (b) The ratio $R$ of degenerate to regular
$\mathcal{Q}_{3}\\--$indicators for the Bures (solid blue) and the BKM (dashed
blue) ensembles.

##  5 Summary

Bearing in mind the results of the calculations of $\mathcal{Q}_{3}$ , we will
summarize with a few comments. The indicator of classicality $\mathcal{Q}_{3}$
, being a functional of the ensemble probability distribution function, at the
same time depends on two characteristics of the SW kernel: its isotropy group
$H_{\alpha}$ and the moduli parameter $\zeta$ . Our studies of the
$\mathcal{Q}_{3}\\--$indicator reveal several interesting peculiarities
concerning their interrelations:

  * •

There is a certain coherence between the classification of states according to
their classicality and their symmetry properties. In particular, it turns out
that the states with a “larger” symmetry are more classical, cf. Fig. 4 and
Fig. 5. This observation demands further study and we plan to formalize it in
forthcoming publications;

  * •

The character of the dependence of $\mathcal{Q}_{3}$ on the type of the
ensemble is monotone, i.e., the values of $\mathcal{Q}_{3}$ for all strata are
ordered in correspondence with the order of the ensembles, see Fig. 6;

  * •

The $\mathcal{Q}_{3}(\zeta)\\--$indicator of the Hilbert-Schmidt ensemble is a
symmetric function with respect to the global minimum point, $\zeta=\pi/6$ ,
see Fig. 4(a) ;

  * •

For monotone metrics the symmetry possessed by the Hilbert-Schmidt ensemble is
broken. Data specifying the range of violation is given in Table 1.

(a)

(b)

Figure 6: Pairwise ratios of $\mathcal{Q}_{3}\\--$indicators of different
ensembles for the regular (a) and for the degenerate (b) stratum. Global
$\mathcal{Q}_{3}\\--$indicator vs. moduli parameter
---
Ensemble |  $\min{\mathcal{Q}_{3}(\zeta)}$ |  $\zeta_{\min}$ |  $\mathcal{Q}_{3}(0)-\mathcal{Q}_{3}({\pi}/{3})$
Hilbert-Schmidt |  0.0006751 |  $\pi/6\approx 0.523599$ |  0
BKM |  0.0000121609 |  0.527798 |  0.0000216102
Bures |  0.0000891011 |  0.525096 |  0.0000472609
Table 1: Data on symmetry properties of $\mathcal{Q}_{3}\\--$indicators.

## Acknowledgments

The work of A.K. was supported in part by the Shota Rustaveli National Science
Foundation of Georgia, Grant FR-19-034.

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