# Non-locality in high spin systems with tensor correlations

V. Ravishankar  Department of Physics, Indian Institute of Technology, Delhi
R. P. Sandhir  Department of Physics and Computer Science, Dayalbagh
Educational Institute, Agra

###### Abstract

We address the problem of detecting non-locality in coupled $N$ level systems
in the language of spin. Through a number of examples, we show that non-
locality can be detected via a violation of the standard Bell inequality,
irrespective of what $N$ is, with correlations from observables of order
$k\sim 2s$ in the fundamental spin operators. We further show that,
contrarily, if the order $k$ is frozen, then non-locality eludes a detection
when $s\gg k$, leading to a weak classical limit. Armed with these results, we
proceed to characterize observables that ‘genuinely’ reflect non-locality in
higher dimensions, and demonstrate that one needs to go beyond the standard
set-ups such as Stern-Gerlach and those with correlations involving
measurement of spin projections along quantization axes.

Bell Inequality, Classical Limit, N-level Systems, Non-locality, Spin-Spin
Correlations

###### pacs:

03.65.Ud, 03.65.Aa, 03.67.Mn

††preprint: APS/123-QED

##  I Introduction

There exists a fairly large body of work on non-locality in $N$ level
systemsBell (1964); Mermin (1980); Kaszlikowski  _et al._ (2000); Garg and
Mermin (1984); Peres (1992); Braunstein  _et al._ (1992); Gisin and Peres
(1992); Collins  _et al._ (2002a); Wódkiewicz (1995); Wu  _et al._ (2001)
which explore non-locality mainly from the view point of identifying
appropriate correlations, and to explore possible attainment of classical
limit in the large $N$ limit. Generalizations to multipartite systems have
been accomplished by construction of correlation functions for higher
dimensions Fu (2004) and by establishing multi-party Bell inequalities.
Important that these questions are, we argue that they need refinement before
we look for answers. Indeed, if no restrictions are imposed on $N$ level
systems, discovering non-locality is trivial in the following sense Bell
(1964): One merely needs to identify a four dimensional subspace spanned by
four separable states, and the rays in the subspace mimic a two-qubit state.
This leads to a maximal violation of the Bell inequality, which cannot be
improved upon, thanks to the Cirel’son bound Cirel’son (1980). Similar
arguments can be advanced for other measures of inequalities which involve
more complicated inequalities with larger number of correlations. The
refinement that is required is the stipulation that the correlations genuinely
probe the Hilbert space fully and not be trapped in a subspace. This can
considerably complicate the analysis, as reflected in the divergent
conclusions drawn in literature.

The purpose of this paper is to revisit non-locality in $N$ level systems
keeping the stipulation mentioned above in mind, with exclusive emphasis on
spin systems. High spin systems are ubiquitous, and spin dependent
interactions make the preparation and manipulation of states relatively
easier. In contrast, manipulation of multiqubit states to prepare $N$ level
systems, and then couple them to form a bipartite state is much more daunting
though generalised Svetlichny’s inequalities have been proposed for the same
Collins  _et al._ (2002b); Mitchell  _et al._ (2004). In a similar manner,
measurements of correlations also gets complicated. High spin systems do not
suffer from these drawbacks. Further, correlations in high spin systems can be
studied through observables which are of different orders in the fundamental
spin operators. One knows from atomic and nuclear physics that measurement of
observables which are of higher order in spin is more difficult than those of
lower orders. With these in mind, we investigate (i) the ‘simplest’
observables that lead to the largest possible non-locality, (ii) their
behaviour as $N\rightarrow\infty$, and (iii) the precise import of the so
called quantum-classical transition in that limit. Rather than trying to prove
general theorems, we consider representative examples to illustrate the
general trend. For our purposes, it suffices to consider observables which are
upto quartic degree in spin. We make a rather detailed comparison with
existing results, wherever possible.

##  II Preliminaries and the Experimental Setting

  1. 1.

Throughout the paper, we consider a bipartite system of two equal spins, $s$.
We invariably study non-locality in one representative fully entangled (Bell)
state, viz, the singlet state given by

$|0_{s}>=\frac{(-1)^{s}}{\sqrt{2s+1}}\sum_{m=-s}^{s}(-1)^{m}|m,-m>.$ (1)

The notation in the ket in the LHS emphasizes that the state is isotropic,
with total spin zero. The isotropy makes the choice of correlations simpler.

  2. 2.

We employ the Bell inequality in its standard form, formulated for the Bell
function:

$\mathcal{B}\equiv|\mathcal{C}(a,b)-\mathcal{C}(a,b^{\prime})|+|\mathcal{C}(a^{\prime},b)+\mathcal{C}(a^{\prime},b^{\prime})|,$
(2)

where the spin-spin correlation
$C(a,b)=\langle 0_{s}|O_{A}(a)O_{B}(b)|0_{s}\rangle$ is defined in terms of
the expectation values of products of two observables, each belonging to a
subsystem, as indicated. It should be emphasized that the arguments
$\\{a,~{}b,~{}a^{\prime},~{}b^{\prime}\\}$ refer to collective variables that
constitute the parameter space. Indeed, the state exhibits non-locality if the
Bell function violates the inequality ${\mathcal{B}}\leq 2$, i.e., there are
some regions in the parameter space where ${\cal B}>2$.

  3. 3.

We shall denote by $O^{s}_{k}(a)$, observables of order $k$ in the spin
operators, for a spin $s$ particle. Similarly, we denote by $C^{s}_{k}(a,b)$,
a correlation constructed from two observables of the same order $k$ in spin
operators. The corresponding Bell function will be denoted by ${\cal
B}^{s}_{k}$ 111More generally, one could consider correlations of degree
$k~{}(k^{\prime})$ in subsystem $A(B)$, but they are of no consequence to us
here..

  4. 4.

Customarily, and especially in experiments, the parameters are taken to be
quantization axes with respect to the spin observables measured. We adhere to
the same tradition here. Furthermore, the experimental configuration
considered throughout the paper is the standard planar configuration depicted
in Fig. 1, with the configuration:

$\theta_{ab}=\theta_{a^{\prime}b}=\theta_{a^{\prime}b^{\prime}}=\frac{\theta_{ab^{\prime}}}{3},$
(3)

in which the violation of the Bell Inequality, if any, is always maximal. We
may note parenthetically that this choice of the parameters is not general
enough for spins $\geq\frac{1}{2}$, the importance of which we briefly discuss
in the concluding section.

Figure 1: The standard experimental configuration considered for Bell-type
experiments involving quantisation axes $\hat{a},\hat{a^{\prime}},\hat{b}$ and
$\hat{b^{\prime}}$, with angles defined by Eq. 3.

In the simplest and the most studied two qubit case, the choice of the
correlation is essentially unique, and is given by
$C=(\vec{\sigma}_{A}\cdot\hat{a})(\vec{\sigma}_{B}\cdot\hat{b})$. The
parameter spaces are the respective Bloch spheres, and the inequality is
violated by all entangled pure states, and maximally by a Bell state. The
violation is also the maximum allowed for a quantum system with this
particular Bell formulation, with the Bell function taking the value
$2\sqrt{2}$.

##  III Bell inequality in higher spin states: Linear correlations

The number of linearly independent correlations for a spin $s$ state is
$O(N^{4})$, but the Bell function does not follow linearity in correlations,
and that is the crux of the problem. It is, therefore, advisable to consider
specific examples to start with. Consider the linear correlation

$\mathcal{C}_{1}^{s}(\hat{a},\hat{b})=\langle
0_{s}|(\vec{\Sigma}^{s}_{A}\cdot\hat{a})(\vec{\Sigma}^{s}_{B}\cdot\hat{b})|0_{s}\rangle;~{}~{}\vec{\Sigma}^{s}_{A,B}=\frac{\vec{S}_{A,B}}{s}$
(4)

in terms of the normalized spin operators $\vec{\Sigma}^{s}$, which respect
the unit norm bound: they satisfy $|Tr\\{\rho^{s}\Sigma^{s}_{i}\\}|\leq 1$ for
all states. Let $\cos\theta_{ab}\equiv\hat{a}\cdot\hat{b}$. Exploiting
bilinearity in the spin operators and the isotropy of the state, we find

$\mathcal{C}^{s}_{1}(\hat{a},\hat{b})=-\frac{s+1}{3s}\cos\theta_{ab}\equiv\frac{s+1}{3s}\mathcal{C}_{1}^{\frac{1}{2}}(\hat{a},\hat{b}).$
(5)

Note that the spin dependent scaling factor, which is also inherited by the
corresponding Bell function ${\cal B}^{s}_{1}$, decreases rapidly with $s$,
achieving an asymptotic value of $\frac{1}{3}$. It falls by a factor 2/3 for
$s=1$. Since the maximum violation for two qubit case is $2\sqrt{2}$, it
follows that Bell inequality is respected by the correlation for spins $s\geq
1$. So one is obliged to look at correlations which involve higher orders in
spin.

##  IV Biquadratic Correlations

We consider the simplest case, $s=1$, first.

###  IV.1 s=1

Since the linear correlations fail to reveal non-locality, we move on to
biquadratic correlations. It would appear that the choice
$O=(\vec{\Sigma}^{s}\cdot\hat{a})^{2}$ is natural. But that is not to be,
since its expectation value is confined to the subinterval $[0,1]$, with the
corresponding correlation having the form $\frac{1}{3}(1+\cos^{2}\theta)$,
which peaks at $\frac{2}{3}$, which immediately negates the possibility of a
violation. In contrast, consider the trial observable

$O^{(2)}=2(\vec{\Sigma}^{1}\cdot\hat{a})^{2}-1$ (6)

whose expectation values cover the full range $[-1,+1]$. It is easy to verify
that

$\displaystyle\mathcal{C}^{1}_{2}$ $\displaystyle=$ $\displaystyle\langle
0_{1}|\\{2(\vec{\Sigma}^{1}_{A}\cdot\hat{a})^{2}-1\\}\\{2(\vec{\Sigma}^{1}_{B}\cdot\hat{b})^{2}-1\\}|0_{1}\rangle$
(7) $\displaystyle=$ $\displaystyle\frac{1}{3}(4\cos^{2}\theta-1)$

We note that this correlation has a support in the much larger interval
$[-\frac{1}{3},+1]$, though it falls short of the support $[-1,+1]$ for the
correlation in the two qubit case. For this reason, one may anticipate that a
violation of Bell inequality, if any, would be smaller than in the two qubit
case.

The corresponding Bell function is plotted in red in the standard planar
geometry (Fig.2). There is a clear violation of the inequality in the sector
$[0.97,2.17]$ rad. The maximum value attained is 2.55, and the percentage area
of violation is found to be about 14.7%. These numbers may be contrasted with
the two qubit case which shows violation in the intervals $[0.1,0.19]$ and
$[1.95,\pi]~{}rad$ with the maximum value $2\sqrt{2}\approx 2.82$, and a total
percentage area of 16.32%. We return to a critique of these results later.

####  IV.1.1 Experimental determination of $C_{2}^{1}$

We briefly digress to discuss how $C^{1}_{2}(\hat{a},\hat{b})$ may be
determined experimentally through appropriate count rate measurements. Observe
that $\Pi_{0}(\hat{a})\equiv(1-(\vec{S}\cdot\hat{a})^{2})$, is the projection
operator for the state $m=0$ along the quantization axis $\hat{a}$. The
correlation can be written as:

$\displaystyle\mathcal{C}^{1}_{2}$ $\displaystyle=$ $\displaystyle\langle
0_{1}|\\{1-2\Pi^{A}_{0}(\hat{a})\\}\\{1-2\Pi^{B}_{0}(\hat{b})\\}|0_{1}\rangle$
(8) $\displaystyle=$ $\displaystyle-\frac{1}{3}+4\langle
0_{1}|\Pi^{AB}_{0}(\hat{a},\hat{b})|0_{1}\rangle$ $\displaystyle=$
$\displaystyle-\frac{1}{3}+\frac{4N^{AB}_{0}(\hat{a},\hat{b})}{N}$

where the last line gives the measurement prescription explicitly, in terms of
$N^{AB}_{0}(\hat{a},\hat{b})$ which is the joint count rate for the two spins
to be in the state $m=0$ along their respective quantization axes, and $N$ is
the total count rate.

Figure 2: Bell function $\mathcal{B}^{1}_{2}$ for the $spin-1$ singlet state
using quadratic operators given by Eq. 6.

###  IV.2 Quadratic correlations for $s\geq 3/2$

We first note that

$\langle
0_{s}|(\vec{\Sigma}^{s}_{A}\cdot\hat{a})^{2}(\vec{\Sigma}^{s}_{B}\cdot\hat{b})^{2}|0_{s}\rangle=F(s)+G(s)\cos^{2}\theta$
(9)

where the scaling factors,

$\displaystyle F(s)$ $\displaystyle=$
$\displaystyle\frac{2s^{3}+4s^{2}+3s+1}{30s^{3}}$ $\displaystyle G(s)$
$\displaystyle=$ $\displaystyle\frac{4s^{3}+8s^{2}+s-3}{30s^{3}},$ (10)

Figure 3: Green: scaling factor $G(s)$, Red: scaling factor $F(s)$, Yellow:
bound for scaling factor G(s) - $\frac{2}{15}$, Blue: bound for scaling factor
F(s) - $\frac{1}{15}$.

are monotonically decreasing functions of $s$, as may be seen in Fig. 3, and
attain their asymptotic values $F=1/15,~{}G=2/15$ in the limit
$s\rightarrow\infty$.

The correlation $\mathcal{C}_{2}^{s}$, corresponding to
$O_{A}=2(\vec{\Sigma}^{s}_{A}\cdot\hat{a})^{2}-1$ (and similarly for $O_{B}$),
is evaluated to be

$\mathcal{C}_{2}^{s}=4\\{F(s)+G(s)\cos^{2}\theta\\}-\frac{4}{3}\frac{s+1}{s}+1.$
(11)

Fig. 4 depicts the behaviour of the correlation as a function of $\theta$ for
various values of $s$. The range of the correlation is seen to diminish
rapidly as $s$ is increased. The corresponding Bell functions ${\cal
B}_{2}^{s}$ are plotted in Fig. 5 in the standard planar configuration in the
parameter space. It is clear that there is a very mild violation for
$s=\frac{3}{2}$, with a maximum value of 2.09, and none whatsoever for $s\geq
2$. This raises the question whether the correlation chosen is optimal for all
spins, and if higher order observables are required when $s\geq 2$. We seek to
settle this issue through a global (numerical) search in the next section.

Figure 4: Correlation $C_{2}^{s}$ given by Eq. 11 for Red: $s=1$, Green:
$s=\frac{3}{2}$, Blue: $s=2$, Yellow: $s=\frac{5}{2}$. Figure 5: Bell function
$\mathcal{B}_{2}^{s}$ for the $spin-s$ singlet state using quadratic operators
given by Eq. 6. Red: $s=1$, Green: $s=\frac{3}{2}$, Blue: $s=2$, Yellow:
$s=\frac{5}{2}$, Blue dashed: bound of the inequality.

##  V Global study of biquadratic correlations

Our studies have so far been restricted to a few special examples which do
not, however, shed complete light on the extent to which non-locality may be
detected in higher spin systems. To rectify this drawback, We consider a
general quadratic observable:

$O=C_{2}(\vec{\Sigma}^{s}\cdot\hat{a})^{2}+C_{1}(\vec{\Sigma}^{s}\cdot\hat{a})+C_{0}$
(12)

with the proviso that its expectation values be bounded by unit norm. We fix
the optimal values of the constants $C_{i}$ through a numerical search222The
numerical results agree with the analytic results which are easily available
for $s=\frac{1}{2},~{}\frac{3}{2}$.. The global search also determines whether
several of the prescriptions given by the earlier works do really yield the
maximum possible violations or not.

Figure 6: Scatter plot of coefficients $C_{0},C_{1},C_{2}$ for $spin-1$
constraints given by Eq. 13 shown in steps of 0.5 for convenience.

The unit norm bound on the coefficients $C_{i}$ in Eq. 12 simply translates to
the the set of $2s+1$ constraints (in each $m$ value)

$-1\leq\frac{C_{2}m^{2}}{s^{2}}+\frac{C_{1}m}{s}+C_{0}\leq 1~{}~{}\forall
m\in[-s,s]$ (13)

in the three parameters. The data points for the coefficients are generated by
selecting the bounds [-5,5] for each coefficient here and everywhere. Each of
the equalities yields a two dimensional plane in the $(2s+1)$ dimensional
space, and the intersections of the planes gives us the vertices of a
polyhedron within and on which the coefficients $C_{i}$ are constrained to
lie. The additional requirement that the observables admit their maximum value
restricts our search to identifying one of the vertices. We present the
results for various spins in the following subsections.

###  V.1  $s=1$

When $s=1$, we find that the maximum violation occurs when
$C_{2}=2;~{}C_{1}=0;~{}C_{0}=-1$ which is exactly the trial observable that we
constructed in the previous section. The numerical search confirms that the
maximum violation occurs in the planar geometry, with ${\cal B}_{max}=2.55$ as
reported. More detailed results are presented in the scatter plot in Fig. 6
which displays the planar segment in which the coefficients $C_{i}$ are
constrained, and Fig. 7 which shows a histogram plot of the values of the
correlation for all coefficients $C_{i}$ that are consistent with the
constraints. The corresponding histogram for the Bell function, which is of
direct interest, is shown in Fig. 8. From this we estimate that the relative
region over which the violation takes place is $\sim 15\%$.

Figure 7: Correlator $\mathcal{C}_{2}^{1}$ for all possible valid coefficient
and configuration combinations. Figure 8: Bell function $\mathcal{B}^{1}_{2}$
for all possible valid coefficient and configuration combinations.

In short, we may conclude that it is not possible to improve upon the
violation seen for the correlation chosen in Eq. 6. The maximum violation,
pegged at the value 2.55 falls short of the maximum allowed value, 2.88, which
is realized for a two qubit system – notwithstanding the fact that we are
dealing with a completely entangled pure state. This conundrum will be
addressed in the concluding section.

###  V.2  $s\geq\frac{3}{2}$

Consider $s=\frac{3}{2}$ first. The optimal values of the coefficients are
found to be $C_{2}=2.25,C_{1}=0,C_{0}=-1.25$, yielding the observable

$O_{A}=\frac{9}{4}(\vec{\Sigma}_{A}\cdot\hat{a})^{2}-\frac{5}{4}$ (14)

which can be recast into the elegant form

$O_{A}(\hat{a})=\Pi_{3/2}(\hat{a})+\Pi_{-3/2}(\hat{a})-\Pi_{1/2}(\hat{a})-\Pi_{1/2}(\hat{a})$
(15)

in terms of the projection operators $\Pi_{m}\equiv|m\rangle\langle m|$ along
the quantization axis $\hat{a}$, with a similar expression for $O_{B}$. Note
that expectation values of $O_{A,B}$ span the full range $[-1,+1]$. The
correlation also assumes the elegant form

$\mathcal{C}_{2}^{3/2}=P_{2}(\cos\theta)$ (16)

which has its support in $[-\frac{1}{2},1]$, larger than the one obtained for
$s=1$. The Bell function plotted in Fig. 9 clearly shows that non-locality is
exhibited by these correlations. The maximum value is 2.62, which is larger
than the violation for $s=1$. The percentage area of violation is given by
$\approx 7.85\%$ which, however, is smaller than the case for $s=1$.
Incidentally, note that the coincidence count rates to be measured are clear
from the very expression for the observables given in Eqn. 15.

Figure 9: Bell function $\mathcal{B}_{2}^{\frac{3}{2}}$ for the
$spin-\frac{3}{2}$ singlet state using quadratic operators given by Eq. 14.
Blue dashed: bound of the inequality.

The increase in the violation from $s=1$ to $s=\frac{3}{2}$ is anomalous. In
fact, we find that the biquadratic correlation fails to violate Bell
inequality everywhere in the parameter space if $s\geq 2$.

It is getting clear from the studies so far that violation of the Bell
inequality requires correlations of observables which are of maximal order in
the spin variables. To strengthen this qualitative conclusion, and also to
draw more quantitative conclusions for comparison with similar findings, and
claims made so far Mermin (1980); Gisin and Peres (1992); Wódkiewicz (1995);
Banaszek and Wódkiewicz (1998); Peres (1992), we consider two more cases –
generic observables of degree three, and a specific observable of degree four.
This necessitates a discussion of $s=2$ states as well. We address the cubic
case first, and conduct a global search in the parameter space.

##  VI  Global study of Cubic Correlations

The generic form of the observable, say for the subsystem $A$, is given by

$\displaystyle O_{A}=C_{3}(\vec{\Sigma}^{s}_{A}\cdot
a)^{3}+C_{2}(\vec{\Sigma}^{s}_{A}\cdot a)^{2}+C_{1}(\vec{\Sigma}^{s}_{A}\cdot
a)+C_{0}.$ (17)

The usual requirement that $O_{A}$ be bounded by unit norm leads to the set of
$2s+1$ constraints

$-1\leq\frac{C_{3}m^{3}}{s^{3}}+\frac{C_{2}m^{2}}{s^{2}}+\frac{C_{1}m}{s}+C_{0}\leq
1~{}\forall m\in[-s,s].$ (18)

As before, the coefficient data were generated by selecting the bounds [-5,5].
The search in the parameter space yields a maxima in violation of Bell
inequality in two orthogonal subspaces, of even and odd parity in spin
observables. The even parity case has already been disposed off in the
previous section. We address the complementary space.

####  VI.0.1 $s=\frac{3}{2}$

A numerical search yields the maximum violation for the correlation when
$C_{3}=4.5,C_{1}=-3.5,C_{2}=C_{0}=0$, corresponding to the observable

$O_{A}=\frac{9}{2}(\vec{S}\cdot\hat{a})^{3}-\frac{7}{2}(\vec{S}\cdot\hat{a)}$
(19)

For this optimal observable, the maximum violation is pegged at
$\mathcal{B}^{\frac{3}{2}}_{3}=2.45$, with the violating region being $\sim$
6.67% of the total volume. We show the scatter of coefficients
$C_{1},C_{2},C_{3}$ for constraint $C_{0}=0$ in Fig. 10.

Figure 10: Scatter plot of coefficients $C_{1},C_{2},C_{3}$ for $C_{0}=0$ for
$spin-\frac{3}{2}$ constraints given by Eq. 18 shown in steps of 0.5 for
convenience.

It would be premature to draw any conclusion from the reduced relative region
of contribution, as this is only a _partial_ contribution. Combined with the
biquadratic contribution, we see that there is no _significant_ diminution in
the over all area.

####  VI.0.2 $s\geq 2$

We conclude the discussion on cubic correlations with a brief discussion of
higher spins. Similarly to the quadratic case, $s=2$ shows a mild violation,
with a maximum value of the Bell function given by $\mathcal{B}^{2}_{3}=2.03$
for the configuration $C_{3}=4,C_{1}=-3,C_{2}=C_{0}=0$. The histogram of this
Bell function for all possible valid coefficient combinations with the
standard planar experimental configuration is shown in Fig. 11. The Bell
inequality is respected for the cubic correlations involving all spins
$s\geq\frac{5}{2}$.

Figure 11: Bell function $\mathcal{B}^{2}_{3}$ for all possible valid
coefficient and configuration combinations.

##  VII Biquartic correlations for $s=2$

In this last example, we now consider a specific quartic observable for $s=2$:

$O_{A}=2(\vec{\Sigma}\cdot\hat{a})^{4}-1$ (20)

which clearly spans the full range $[-1,+1]$. This observable is nontrivial
for $s\geq 2$, and we restrict ourselves to spin 2 here. A straight forward
evaluation yields the correlation to be

$\mathcal{C}^{2}_{4}=\frac{7}{10}+\frac{45}{32}\sin^{2}\theta\cos^{2}\theta+\frac{39}{640}\sin^{4}\theta+\frac{51}{32}\cos^{4}\theta$
(21)

Fig. 12 plots this correlation. The resulting Bell violation is shown in Fig.
13, and the maximum value attained by the Bell function is
$\mathcal{B}^{2}_{4\max}=2.371$. The percentage area of violation is seen to
be $\sim 2.5\%$. These signatures must be combined with the violation seen
with the cubic correlation for a fuller picture.

Figure 12: The quartic correlator $\mathcal{C}^{2}_{4}$ for the $spin-2$
singlet state given by Eq. 21. Figure 13: Bell function $\mathcal{B}^{2}_{4}$,
for the $spin-2$ singlet state from the correlator given in Eq. 21. Blue:
Bound of the inequality.

##  VIII Summary of results

We first summarize the results in a nutshell. The examples considered above
suggest strongly that

  1. 1.

Non-locality in $N$ level systems will be seen if the observables employed are
of degree $N$, with a small leeway for lower order observables.

  2. 2.

The magnitude of violation does not see any significant diminution with
increasing $N$. The relative region in the parameter space remains roughly
constant if the observables are scaled in this manner.

  3. 3.

Finally, inspite of optimization, and in spite of employing the most nonlocal
state, the violation fails to reach the maximum allowed value, $2\sqrt{2}$.

The results are collected and displayed in table 1.

Table 1: Summary of Results $s$ | $\mathcal{O}(O)$ | $O_{A}$ | $\mathcal{B}_{\text{max}}$ | % Area of Violation
---|---|---|---|---
$\frac{1}{2}$ | 1 | $(\vec{S}\cdot\hat{a})$ | $2\sqrt{2}$ | $\approx 16.32\%$
1 | 2 | $2(\vec{S}\cdot\hat{a})^{2}-1$ | 2.55 | $\approx 14.7\%$
$\frac{3}{2}$ | 2 | $\frac{9}{4}(\vec{S}\cdot\hat{a})^{2}-\frac{5}{4}$ | 2.62 | $\approx 7.85\%$
$\frac{3}{2}$ | 3 | $\frac{9}{2}(\vec{S}\cdot\hat{a})^{3}-\frac{7}{2}(\vec{S}\cdot\hat{a})$ | 2.45 | $\approx 6.67\%$
2 | 3 | $4(\vec{S}\cdot\hat{a})^{3}-3(\vec{S}\cdot\hat{a})$ | 2.03 | $\approx 0.1\%$
2 | 4 | $\frac{(\vec{S}\cdot\hat{a})^{4}}{8}-1$ | 2.371 | $\approx 2.522\%$

##  IX Discussion and comparison with other works

Studies in non-locality have been dominated by three issues: (i) Detection of
non-locality in $N$ dimensional systems, (ii) Extent of non-locality vis-a-vis
the two qubit system, and by extension, (iii) Fate of non-locality in the
large $N$ limit. The three issues are not unrelated to each other.

One hurdle in comparing results is in the different measures of non-locality
employed, not only in different works, but also for different spins in the
same work. Care must therefore be exercised since a naïve normalization may
lead to misleading conclusions on the degree of non-locality. With this
caveat, we first look at the last issue, the large $N$ limit, since it
addresses the all important question of the classical limit of quantum
systems.

###  IX.1 Large $N$ limit

The fate of non-locality in the large $N$ limit was first studied by Mermin
Mermin (1980), who concluded that non-locality persists in all dimensions. He
argued that the classical limit is realized as an asymptote, as
$N\rightarrow\infty$. While the extent of violation was not seen to decrease,
the violating region in the parameter space was seen to shrink to zero. This
result seemed to gain support from a number of other works Ardehali (1991);
Garg and Mermin (1982); Mermin (1980); Scully and Wódkiewicz (1994). These
conclusions were based on specific choices of correlations, and it has been
realized to be erroneous by Garg and Mermin Garg and Mermin (1984).
Subsequently Ardehali Ardehali (1992), and more recently Dagomir et. al.
Kaszlikowski  _et al._ (2000) have argued that there are correlations for
which non-locality, in fact, increase with the increasing dimension of the
Hilbert space. Their results are, of course, not in conflict with the
Cirel’son theorem Cirel’son (1980) and the more general results obtained in
Peres (1992); Gisin and Peres (1992).

Interesting that the above studies are, we observe that – bereft of any other
criterion - neither the value nor the range of violation is, in fact, an issue
for $N$ level systems. As observed by Bell in his seminal paperBell (1964),
any correlated state of two $N$ level systems can show as large a violation of
Bell inequality as the two qubit system. Indeed, all that one needs to do is
to project out a convenient four dimensional subspace spanned by a basis
consisting of four separable pure states. The resultant quasi two qubit system
would exhibit maximum violation, which cannot be improved upon further, thanks
to Cirel’son’s theorem Cirel’son (1980)333That does not render experiments
fruitless because any violation of Cirel’son’s theorem would also negate
quantum mechanics as a fundamental theory.. Hence, non-locality in $N$ level
systems is to be discussed with correlations that genuinely probe the full
state space, and the large $N$ limit needs a rephrasing. The real interest,
therefore, lies in viewing non-locality in $N$ level systems through the prism
of genuine qudit correlations which do not mimic any lower dimensional
correlation. Equivalently, the eigenstates of observables in each $N$ system
must span the full space.

With this condition, for the class of observables under consideration, we now
demonstrate that the so called classical limit can indeed be achieved, but in
a weaker sense. More precisely, we show that the so called classical limit is
a manifestation of limited experimental resources.

####  IX.1.1 Weak classical limit

Suppose that one has resources to measure upto $k^{th}$ order observables in
spin space. Recall that for a given $s$, $k_{max}=2s$. We then consider the
constraints on a $k^{th}$ degree observable
$O_{A}=\sum_{l=0}^{k}C_{l}(\vec{S}^{s}_{A}\cdot\hat{a})^{l}$ by the unit norm
bound. We get $2N$ constraints:

$-1\leq\sum_{l=0}^{k}C_{l}m^{l}\leq 1;-s\leq m\leq+s$ (22)

where $k\leq 2s$. The dimension of the parameter space is $N_{p}=k+1$ which is
to be compared with the number of bounds, $N_{b}=2(2s+1)$.

For a fixed order of the observable, as $s$ increases, $N_{b}\gg N_{p}$,
thanks to which one has to identify the region in the parameter space that is
compatible with all the equations. The allowed region is the common
intersection of regions culled out by each inequality. It is not difficult to
see that as $s\rightarrow\infty$, the allowed region shrinks rapidly to the
allowed to a point $C_{l}\rightarrow\pm\delta_{l,0}$. Bell inequality is
trivially respected.

On the other hand, suppose that the order of the polynomial function in spin
scales with $s$. We would then have a polynomial of degree $2s$ subject to
$2s+1$ equations. $N_{b}\sim N_{p}$ and for that reason the allowed region in
the parameter space does not shrink. The few simple examples that we discussed
in the paper illustrate precisely that (see Table 1).

###  IX.2 Extent of violation in $N$ level systems

We now turn our attention to compare our results with some of the earlier
works. In particular, we now focus on the magnitude of violation.
Unfortunately, almost none of the earlier works give information on the range
of parameters over which violation occurs. Several measures employ
inequalities which depend on $N$, which further makes it difficult to make
meaningful comparisons. Table 2 presents the results obtained in Peres (1992);
Banaszek and Wódkiewicz (1998); Gisin and Peres (1992) since all of them use
the Bell inequality formulation employed in this paper. Note: the work
reported in Banaszek and Wódkiewicz (1998) admits a comparison only when
$N\rightarrow\infty$.

Table 2 gives a summary of the results obtained in Peres (1992); Banaszek and
Wódkiewicz (1998); Gisin and Peres (1992), for $s=1$. Note that the
observables in each of the works are quite different from each other and a
comparison is, therefore, of quite some interest. So are the measures of non-
locality employed. The results in table 2 must be, therefore, interpreted
carefully.

With the approach given by Peres Peres (1992), while the magnitude of the
violation tends to a constant (a claim consistent with that of Banaszak
Banaszek and Wódkiewicz (1998)), the range of parameters for which a violation
is detected becomes vanishingly small as $s\rightarrow\infty$. Contrarily, our
results indicate that once a suitable dimension of parameter space is
established, said region of violation consists of a set of orthogonal
subspaces that do not diminish asymptotically. Finding the right observables
of a suitable order dependent on $s$ is the key.

At this juncture, it is pertinent to note that Peres Peres (1992) also makes a
note of experimental limitations on observing non-locality: provided that
consecutive $m$’s are distinguishable, non-locality _should_ be observable.
This goes hand-in-hand with our notion of a ‘weak’ classical limit being
attained due to limited experimental resources. This argument is strengthened
by that of Gisin Gisin and Peres (1992), who shows that mere selection of
large quantum numbers does not necessarily indicate classical behaviour. Also
provided that one can construct pairs of observables for which violation
occurs, large quantum numbers are of no consequence. The present study, apart
from being in consonance with these observations, brings out the need for
those pairs of observables to be nontrivial. They should not be defined in a
proper subspace and they must be of sufficiently high rank.

Table 2: Summary of Results

Author | Inequality | Behaviour $s(r)\rightarrow\infty$ | Violation
---|---|---|---
Peres | $f(C)\leq 2$ | $f(C)\rightarrow\frac{3\sin x}{x}-\frac{\sin 3x}{3x}$ | 2.55 for $s=1$
Gisin | $f(j)\leq 2$ |  $f(j)\rightarrow 2\sqrt{2}(1-\frac{0.1464}{s}$) | 2.55 for $s=1$
Banaszek | $|\mathcal{B}|\leq 2$ | $\mathcal{B}\approx 2.19$ | 2.19 $\forall$ EPR states

##  X Non-locality and geometry of spin states

A common feature among all studies reporting violation of Bell inequality is
that none of them attain the maximum violation that is allowed by the
Cirel’son bound Cirel’son (1980). It becomes all the more striking if we
remember that there are observables defined in four dimensional subspaces
whose correlations are as good as — in fact equivalent to – the two qubit
case. The reason for this failure merits an answer, which lies in a more
careful analysis of the geometry of spin states.

States of a spin half particle admit a representation on the Bloch sphere,
which essentially means that they can always be looked upon as possessing a
definite value of $S_{z}$ along a suitable quantization axis:
$|\psi\rangle\equiv|m=\frac{1}{2}\rangle_{\hat{a}}$ where $\hat{a}$ is the
quantization axis. This does not hold for higher spins and the set of states
that can be identified as belonging to an eigenvalue of $S_{z}$ with respect
to any quantization axis is a set of measure zero. In other words, the state
of a particle with a spin $s\geq 1$ does not, in general admit a Bloch sphere
representation. Consequently, the observables and the correlations that we
have considered in this paper are not generic in nature.

The most general representation of a pure spin $s$ state is multivector in
nature. Indeed, as shown in Majorana (1932); Ravishankar and Ramachandran
(1987), the state (pure) of a spin $s$ particle can be represented by $2s$
points on a sphere. If the particle is in a state $|s,m\rangle$, then the
description collapses to $s+m$ identical points and diametrically opposite
$s-m$ identical points, along the quantization axis, on the sphere. This means
that in general, an observable for a spin $s$ particle involves $2s$
independent directions, and consequently, a correlation for a coupled system
of two spin $S$ particles involves $4s$ independent directions. Coupled with
the fact that a pure state is determined almost completely – except for
discrete ambiguities – by its highest rank observable, viz, it is not
surprising that we need observables of order $2s$. Correlations involving such
observables exhaust all the possibilities and should, therefore, yield maximal
violation. However, this makes the theoretical analysis - and its experimental
implementation – more complicated. We address this problem elsewhere.

##  XI Conclusion

In conclusion, we have undertaken a comprehensive study of non-locality in
coupled high spin systems with emphasis on (i) suitable observables that
display non-locality, (ii) the so called classical limit when
$N=(2s+1)\rightarrow\infty$, (iii) genuine observables that span the Hilbert
space and also exhibit non-locality, and finally (iv) the need to go beyond
the standard Stern-Gerlach set ups and observables that are tied to bivectors
in the two spin systems. Hopefully, this study will spur further activity
experimentally to detect and characterize non-locality in spin systems.

## References

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  * Note (1) More generally, one could consider correlations of degree $k~{}(k^{\prime})$ in subsystem $A(B)$, but they are of no consequence to us here.
  * Note (2) The numerical results agree with the analytic results which are easily available for $s=\frac{1}{2},~{}\frac{3}{2}$.
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2}\boxtimes\Delta_{18}^{\otimes 2})$, where the shuffle morphism can be seen
to coincide with the morphism corresponding to the permutation giving rise to
the reordering of elements given in the RHS of the computation preceding the
present proof.

To show that twist morphism $\theta^{\alpha}_{2}$ satisfies the compatibility
condition with braiding morphism $c_{2,2}^{\alpha}$, we can proceed similarly.
First we rewrite the two perimeters of the diagram

${X^{\otimes 2}\boxtimes X^{\otimes 2}}$${X^{\otimes 2}\boxtimes X^{\otimes
2}}$${X^{\otimes 2}\boxtimes X^{\otimes 2}}$${X^{\otimes 2}\boxtimes
X^{\otimes
2}}$$\scriptstyle{\theta^{\alpha}_{2}\boxtimes\mathbbm{1}}$$\scriptstyle{c_{2,2}^{\alpha}}$$\scriptstyle{c_{2,2}^{\alpha}}$$\scriptstyle{\mathbbm{1}\boxtimes\theta^{\alpha}_{2}}$

as composition of comultiplications in $X$ and the appropriate shuffle, then
applying the morphisms $T$, $\alpha$ and multiplication $\mu$. Then we can
compare the two expressions by using TSD property of $T$ and categorical
$2$-cocycle condition of $\alpha$. Similar considerations are also applied to
the diagram proving that $(\mathbbm{1}\boxtimes\theta^{\alpha}_{2})\circ
c_{2,2}^{\alpha}=c_{2,2}^{\alpha}\circ(\theta^{\alpha}_{2}\boxtimes\mathbbm{1})$.

We need to prove now that the morphism $c_{2,2}^{\alpha}$ has an inverse in
the category $\mathcal{C}$. Upon defining the morphism
$\hat{c}^{\alpha}_{2,2}:=(\mu\otimes\mathbbm{1}^{\otimes
2}\boxtimes\mathbbm{1}^{\otimes 2})\circ(\tilde{\alpha}^{\otimes
2}\boxtimes\mathbbm{1}^{\otimes 2})\circ[(T\otimes\mathbbm{1})^{\otimes
2}\otimes(T^{\otimes 2}\boxtimes\mathbbm{1}^{\otimes
2})]\circ\shuffle\circ(\Delta_{6}^{\otimes 2}\boxtimes\Delta^{\otimes 2})$
where we have indicated $\tilde{\alpha}:X^{\otimes
3}\longrightarrow\mathbb{I}$ the convolution inverse of $\alpha$, and the
shuffle morphism corresponds to the permutation

$\sigma=\bigl{(}\setcounter{MaxMatrixCols}{18}\begin{smallmatrix}1&2&3&4&5&6&7&8&9&10&11&12&13&14&15&16&17&18\\\
3&4&8&9&12&15&16&17&2&5&7&10&12&18&11&1&6&14\end{smallmatrix}\bigr{)},$

we see that $\hat{c}_{2,2}^{\alpha}$ is a left and right inverse of
$c_{2,2}^{\alpha}$, which gives the required invertibility. It is fundamental
here, that $\alpha$ is convolution invertible.

Once the fact that $c_{2,2}^{\alpha}$ satisfies the braid equation, the
compatibility condition with $\theta_{2}^{\alpha}$ and invertibility, the rest
of the proof proceeds as that of Theorem 4.3, as the category
$\mathcal{R}^{*}(X)$ is defined inductively as in Section 4.

To show that, given equivalent $2$-cocycles $\alpha$ and $\beta$, there exists
an equivalence of ribbon categories
$\mathcal{R}^{*}_{\alpha}(X)\cong\mathcal{R}^{*}_{\beta}(X)$ one proceeds as
in the counterpart of the same proof of Theorem 4.3, where we use that a
morphism $f:X\longrightarrow\mathbb{I}$ giving the equivalence between
$\alpha$ and $\beta$ is convolution invertible. ∎

Since over an algebraically closed fields of characteristic zero
$\mathbbm{k}$, a cocommutative finite dimensional Hopf algebra $X$ is
isomorphic to the group algebra $\mathbbm{k}[G]$ for some group $G$, in a
sense, the construction of Section 4 is general enough for TSD structures
obtained via heaps. The ribbon category of Theorem 7.16 gives new results when
applied to finite dimensional Hopf algebras over a ring/field that is not an
algebraically closed field of characteristic zero.

######  Remark 7.17.

Observe that Theorem 7.16 generalizes the results of Section 4 to the setting
of cohomology coefficients in a non-commutative group $G$, since from a
$2$-cocycle with coefficients in $G$ and a group character
$\chi:G\longrightarrow\mathbbm{k}^{\times}$, we obtain a ribbon category in
the category of vector spaces over $\mathbbm{k}$, where braiding and twisting
are generated by $c^{\alpha}_{2,2}$ and $\theta^{\alpha}_{2}$.

##  8\. Invariants of framed links in symmetric monoidal categories

Lastly, we describe in this section how the construction of Section 7, whose
twisting morphisms are revisited here, gives rise to quantum invariants of
framed links in symmetric monoidal categories. This discussion follows
standard arguments, already introduced in Section 5 for instance.

Recall that twists in a braided category, along with duals, uniquely determine
a pivotal structure, i.e. an isomorphism of objects $X$ with their double
duals $X^{**}$. The family of twists defined above, with the corresponding
choice of pivotal structure can be used to define invariants of framed links.
It is important to note, though, that duality of an object $X$ in a symmetric
monoidal category can be regarded as a finiteness condition that ensures the
existence of quantum trace.

In fact, for an important class of TSD objects, namely that of involutory Hopf
monoids in a symmetric monoidal category, there often exists another natural
choice of twists, and therefore pivotal structure, that can be used to
construct framed link invariants. Before explicitly giving the definition of
link invariants from the category $\mathcal{R}^{*}_{\alpha}(X)$, we discuss
how to obtain new twists for involutory Hopf algebras, and more generally Hopf
monoids, under some extra conditions.

First recall that a Frobenius algebra $(X,\mu,\eta,\Delta,\epsilon)$ in a
symmetric monoidal category $\mathcal{C}$ is a bialgebra such that the
following Frobenius axiom holds

$(\mathbbm{1}\otimes\mu)\circ(\Delta\otimes\mathbbm{1})=\Delta\circ\mu=(\mu\otimes\mathbbm{1})\circ(\mathbbm{1}\otimes\Delta).$

Given a Frobenius algebra $X$, we can define an (associative) pairing and a
coparining, indicated as $\cup:X\otimes X\longrightarrow\mathbb{I}$ and
$\cap:\mathbb{I}\longrightarrow X\otimes X$ respectively, as
$\cup=\epsilon\circ\mu$ and $\cap=\Delta\circ\eta$. Paring and coparing in a
Frobenius algebra satisfy the axiom

$(\mathbbm{1}\otimes\cup)\circ(\cap\otimes\mathbbm{1})=\mathbbm{1}=(\cup\otimes\mathbbm{1})\circ(\mathbbm{1}\otimes\cap),$

which determines that $\cup$ is an associative non-degenerate pairing. It in
fact turns out that a monoid endowed with an associative pairing and a
copairing satisfying the compatibility condition given above is equivalent to
the previous definition of Frobenius algebra.

Suppose we are given an involutory Hopf algebra which is finitely generated
over a ground PID $\mathbbm{k}$. We know that a Frobenius algebra arises from
such a Hopf algebra, from traditional results of Larson and Sweedler [LS]. The
Frobenius structure then gives an identification of $X$ with its dual $X^{*}$
and, consequently, a pivotal structure where $X\cong X^{**}$ is obtained
through the Frobenius pairing. Let us indicate by $\cup$ and $\cap$ the
Frobenius pairings.

Let us now consider an $\mathbb{I}$-linear symmetric monoidal category
$\mathcal{C}$, and let $X$ denote a (cocommutative) Hopf monoid in
$\mathcal{C}$. We define a braiding $c$ using the quantum heap operation of
$X$, as in the previous section. Suppose $X$ has an integral and a cointegral.
These are, by definition, a point $\lambda:\mathbb{I}\longrightarrow X$ and a
copoint $\gamma:X\longrightarrow\mathbb{I}$, respectively, such that

(1) $\displaystyle\mu_{X}\circ(\lambda\otimes\mathbbm{1})$ $\displaystyle=$
$\displaystyle\epsilon\lambda$ (2)
$\displaystyle(\gamma\otimes\mathbbm{1})\circ\Delta_{X}$ $\displaystyle=$
$\displaystyle\eta\gamma.$

Then we can define a Frobenius structure on $X$ with Frobenius pairing
$\cup:=\gamma\mu(\mathbbm{1}\otimes S)$, and copairing $\cap:=\Delta\lambda$.
Such a construction also holds for finitely generated projective Hopf algebras
(with trivial coinvariants), [Par]. More generally, for an involutory Hopf
monoid with (co)integrals in a symmetric monoidal category, under relatively
mild assumptions, such as $\lambda\gamma=\mathbbm{1}$ and $\lambda
S\gamma=\mathbbm{1}$ as in [CD], the same constructions as above are still
valid. It was shown in [SZ2] that under such circumstances, the quantum heap
braiding of $X$ commutes with Frobenius paring and copairing. We summarize
this in the following result.

######  Lemma 8.1.

Let $X$ denote a (cocommutative) Hopf monoid such that there exist an integral
$\lambda$ and a cointegral $\gamma$ such that setting
$\cup=\gamma\mu(\mathbbm{1}\otimes S)$ and $\cap=\Delta\lambda$, we obtain a
Frobenius algebra structure on $X$. Then, we have the following equalities

$\displaystyle\cup\otimes\mathbbm{1}^{\otimes 2}$ $\displaystyle=$
$\displaystyle(\mathbbm{1}^{\otimes 2}\otimes\cup)\circ c$
$\displaystyle(\cup\otimes\mathbbm{1}^{\otimes 2})\circ c$ $\displaystyle=$
$\displaystyle\mathbbm{1}^{\otimes 2}\otimes\cup.$

Similar conditions hold for $\cap$ and $c$.

###### Proof.

The diagrammatic proof in [SZ2], Lemma 4.6, can be seen to apply in this case
as well. ∎

We can now define $\theta:=(\Cup\otimes\mathbbm{1}^{\otimes
2})\circ(\mathbbm{1}^{\otimes 2}\otimes
c)\circ(\Cap\otimes\mathbbm{1}^{\otimes 2})$, where, by definition, we set
$\Cup=\cup\circ(\mathbbm{1}\otimes\cup\otimes\mathbbm{1})$, and a similar
definition holds for $\Cap$. Consequently, it is easy to see
(diagrammatically) that, applying Lemma 8.1, the braiding $c$ and $\theta$
commute, in the sense that
$c\circ(\mathbbm{1}\otimes\theta)=(\theta\otimes\mathbbm{1})\circ c$ and
$c\circ(\theta\otimes\mathbbm{1})=(\mathbbm{1}\otimes\theta)\circ c$. In fact,
although the proofs in [SZ2] are for algebras in the category of modules, the
diagrammatic proofs directly generalize to the case of Hopf-Frobenius algebras
in more general symmetric monoidal categories, such as those of [CD, BSZ].
Now, defining a category $\tilde{R}^{*}(X)$ as in Section 4, but replacing the
twists with the new $\theta$ defined above and where the deforming $2$-cocycle
is taken to be trivial, we have a proof analogous to that of Theorem 7.16
implying that $\tilde{R}^{*}(X)$ is a ribbon category, since braiding and
twist satisfy the same coherence properties of those of Theorem 7.16.

Let now $\mathcal{L}$ be a framed link and let, in the same notation of
Section 5, $b({\mathcal{L}})=t_{1}^{k_{1}}\cdots t_{n}^{k_{n}}\tau$ be a
framed braid whose closure is $\mathcal{L}$, where $\tau\in\mathbb{B}_{n}$.
Decompose $\tau$ into a product of generators $\sigma_{i}$ of
$\mathbb{B}_{n}$, namely
$\tau=\sigma_{i_{1}}^{q_{1}}\cdots\sigma_{i_{r}}^{q_{r}}$ for some positive
integers $r,q_{1},\ldots q_{r}$ and $1\leq i_{1},\ldots,i_{r},\leq n-1$ for
all $r$. We construct an endomorphism $\Phi_{\mathcal{L}}$ of $X^{\otimes 2n}$
assigning to each power of $\sigma_{j}$ appearing in the decomposition of
$\tau$ into product of generators of $\mathbb{B}_{n}$, the morpshism
$\mathbbm{1}^{\otimes 2}\otimes\cdots\otimes
c_{2,2}^{\alpha}\otimes\cdots\otimes\mathbbm{1}^{\otimes 2}$, where
$c_{2,2}^{\alpha}$ is applied to the copies of $X^{\otimes 2}$ at positions
$i$ and $i+1$. From the discussion in the previous paragraph it follows that
$\Phi_{\mathcal{L}}$ so defined is an invariant of the link $\mathcal{L}$.
Analogously, the quantum trace of $\Phi_{\mathcal{L}}$,
$\Psi_{\mathcal{L}}:=tr_{q}(\Phi_{\mathcal{L}})$ is an invariant of
$\mathcal{L}$.

######  Remark 8.2.

Observe that such a construction of quantum invariants also applies to a Hopf-
Frobenius structure where the twist is defined by means of pairing and
copairing morphisms, upon exchanging $c^{\alpha}_{2,2}$ and
$\theta_{2}^{\alpha}$ with $c$ and $\theta$ given above.

We conclude by briefly turning our attention to the categorical version of
compatible TSD systems encountered in Section 4. In fact, due to a current
lack of concrete examples, we have not explicitly mentioned such structures in
Section 7. A possible source of examples, namely that of ternary augmented
racks, is discussed in the appendix below. The definitions of compatible
systems translate the definitions given in Section 4 in the setting of
symmetric monoidal categories and allow to construct a braided monoidal
category with certain base objects $\\{X_{i}\\}_{i\in I}$ generating the
object family, and morpshims obtained inductively by combining all the
possible compositions and tensoring of braidings $X_{i}\otimes X_{i}\otimes
X_{j}\otimes X_{j}$. In order to obtain a quantum invariant of framed links we
proceed as in the basic case given above, but replacing $\Phi_{\mathcal{L}}$
by a superposition of endomorphisms (hence we need to add the additional
requirement of abelian monoidal category, or with $\mathbbm{k}$-linear
structure), where the sum runs over all possible colorings $X_{i}\otimes
X_{i}$ for each double string of the framed braid chosen to represent the link
$\mathcal{L}$.

##  Appendix A Examples from ternary augmented racks

It has been observed in Section 4 that mutually distributive operations
provide examples of compatible systems of self-distributive structures. The
associated ribbon categories as in Theorem 4.12 have larger Hom sets than
ribbon categories arising from a single ternary self-distributive structure,
but consist of the same objects, i.e. a distinguished object $X^{\otimes 2}$
with all the tensor products arising from it. This class of examples still
leaves open the question of whether there exist compatible systems with
different base spaces, i.e. with $X_{i}\neq X_{j}$ for some $i$ and $j$. A
possible answer comes from ternary augmented shelves [ESZ]. We work in the
category of vector spaces, but we observe that our example is obtained via
linearization of a set-theoretic structure as given in Section 4. A related
construction that is not obtained from linearized operations can be performed,
in the setting of symmetric monoidal categories, by adapting the definitions
in Section 4 in a spirit similar to those given in Section7. We believe that
this case is of interest, but unfortunately we are not aware of an example of
such a “non-linearized” structure at this time. We have therefore decided to
include it hereby as an appendix.

Suppose $X_{1}$ and $X_{2}$ are coaglebras and $H$ is a Hopf algebra that acts
on them and, therefore, acts on $X_{i}^{\otimes 2}$ via comultiplication, for
$i=1,2$. Suppose, further, that there exist morphisms of coalgebras
$p_{i}:X_{i}^{\otimes 2}\longrightarrow H$ satisfying

$p_{i}(z\cdot\Delta(h))=S(h^{(1)})p_{i}(z)h^{(2)},$

for every $z\in X_{i}$ and $h\in H$. Then the following result, extending
Theorem B.6 in [ESZ], shows that in this situation a compatible system arises
naturally.

######  Theorem A.1.

Let $X_{1}$, $X_{2}$ and $H$ be as above, with related morphisms $p_{1}$ and
$p_{2}$. Then, defining maps

$T_{ij}:X_{i}\otimes X_{j}\otimes X_{j}\longrightarrow X_{i}$

by extending the assignment $x\otimes y_{1}\otimes y_{2}\mapsto x\cdot
p_{j}(y_{1}\otimes y_{2})$ by linearity, it follows that
$\\{X_{i},T_{ij}\\}_{i,j=1,2}$ is a compatible system of ternary self-
distributive structures. A similar result holds for a finite number of $H$
modules $X_{i}$, $i=1,2\ldots,n$ with morphisms $p_{i}$.

###### Proof.

The proof is analogous to that of Theorem B.6 in [ESZ] and it is included here
to show that having two morphisms $p_{1}$ and $p_{2}$ does not affect the
procedure. It is enough to prove compatibility condition on simple tensors
$x\otimes y_{1}\otimes y_{2}\otimes z_{1}\otimes z_{2}\in X_{i}\otimes
X_{j}\otimes X_{j}\otimes X_{k}\otimes X_{k}$, where $i,j,k=1,2$, as follows

$\displaystyle T_{ij}(T_{ik}(x\otimes z_{1}^{(1)}\otimes z_{2}^{(1)})\otimes
T_{jk}(y_{1}\otimes z_{1}^{(2)}\otimes z_{2}^{(2)}\otimes T_{jk}(y_{2}\otimes
z_{1}^{(3)}\otimes z_{2}^{(3)}))$ $\displaystyle=$ $\displaystyle(x\cdot
p_{k}(z_{1}^{(1)}\otimes z_{2}^{(1)}))\cdot p_{i}(y_{1}\cdot
p_{k}(z_{1}^{(2)}\otimes z_{2}^{(2)})\otimes y_{2}\cdot
p_{k}(z_{1}^{(3)}\otimes z_{2}^{(3)}))$ $\displaystyle=$ $\displaystyle
x\cdot[p_{k}(z_{1}^{(1)}\otimes z_{2}^{(1)})\cdot p_{j}((y_{1}\otimes
y_{2})\cdot\Delta(p_{k}(z_{1}\otimes z_{2})^{(2)})]$ $\displaystyle=$
$\displaystyle x\cdot[p_{k}(z_{1}\otimes z_{2})^{(1)}\cdot
S(p_{k}(z_{1}\otimes z_{2})^{(2)})\cdot p_{j}(y_{1}\otimes y_{2})\cdot
p_{k}(z_{1}\otimes z_{2})^{(3)}]$ $\displaystyle=$ $\displaystyle
x\cdot[\epsilon(p_{k}(z_{1}\otimes z_{2})^{(1)})1\cdot p_{j}(y_{1}\otimes
y_{2})\cdot p_{k}(z_{1}\otimes z_{2})^{2}]$ $\displaystyle=$ $\displaystyle
x\cdot[p_{j}(y_{1}\otimes y_{2})\cdot p_{k}(z_{1}\otimes z_{2})]$
$\displaystyle=$ $\displaystyle(x\cdot p_{j}(y_{1}\otimes y_{2}))\cdot
p_{k}(z_{1}\otimes z_{2})$ $\displaystyle=$ $\displaystyle
T_{ik}(T_{ij}(x\otimes y_{1}\otimes y_{2})\otimes z_{1}\otimes z_{2}),$

where in the second equality has been used the fact that $p_{k}$ is a morphism
of coalgebras, the third equality is obtained applying the defining property
of $p_{j}$ and, finally, antipode axiom, counit axiom, associativity of the
action of $H$ and definition of maps $T_{ij}$ are used in the last four
equalities. Coassociativity and associativity in $H$ are used throughout,
without explicit mention. ∎

######  Example A.2.

Let $G_{n}$ denote the cyclic group of order $n$ in multiplicative notation,
therefore an element of $G_{n}$ is denoted by the symbol $x^{k}$ for some $k$
determined by reduction modulo $n$. Let $m_{1}$ and $m_{2}$ be positive
integers with $(m_{1},m_{2})=1$ and define $G_{nm_{1}}$, $G_{nm_{2}}$ as
above, with generators denoted by the symbols $y_{1}$ and $y_{2}$
respectively. Then the group algebra $H:=\mathbbm{k}[G_{n}]$ acts on
$X_{i}:=\mathbbm{k}[G_{nm_{i}}]$ via the map $x^{k}\mapsto y_{i}^{m_{i}k}$ and
multiplication in $G_{nm_{i}}$. Define maps $p_{i}:X_{i}\otimes
X_{i}\longrightarrow H$ extending by linearity the assignment

$y_{i}^{k_{1}}\otimes y_{i}^{k_{2}}\mapsto x^{k_{2}-k_{1}}.$

Then

$\displaystyle p_{i}(y_{i}^{k_{1}}\otimes y_{i}^{k_{2}}\cdot\Delta(x^{k}))$
$\displaystyle=$ $\displaystyle p_{i}(y_{i}^{nk+k_{1}}\otimes
y_{i}^{nk+k_{2}})$ $\displaystyle=$ $\displaystyle
x^{m_{i}k+k_{2}-m_{i}k-k_{1}}$ $\displaystyle=$ $\displaystyle
x^{-k}x^{k_{2}-k_{1}}x^{k}$ $\displaystyle=$ $\displaystyle
S(x^{k})p_{i}(y_{i}^{k_{1}}\otimes y_{i}^{k_{2}})x^{k}.$

Moreover, since $(m_{1},m_{2})=1$, it follows that $X_{1}$ and $X_{2}$ are not
subrepresentations of each others, so that the compatible system arising from
Theorem A.1 consists of two base objects that are independent as
representations of $H$.

######  Example A.3.

In this example it is shown that the compatible system given in Example A.2
admits nontrivial compatible systems of $2$-cocycles. With same notation as
before, fix $n=2$, $m_{1}=2$ and $m_{2}=3$. The generator of $G_{2}$ is
denoted by $x$, the ones of $G_{4}$ and $G_{6}$ by $y$ and $z$ respectively.
The compatible structure on $X_{1}:=\mathbbm{k}[G_{4}]$,
$X_{2}:=\mathbbm{k}[G_{6}]$ is given, on simple tensors, by

$\displaystyle T_{12}(y^{t}\otimes z^{s_{1}}\otimes z^{s_{2}})$
$\displaystyle=$ $\displaystyle y^{t-2s_{1}+2s_{2}}$ $\displaystyle
T_{21}(z^{s}\otimes y^{t_{1}}\otimes y^{t_{2}})$ $\displaystyle=$
$\displaystyle z^{s-3t_{1}+3t_{2}},$

while $T_{11}$ and $T_{22}$ are determined by linearizing the heap structure
on groups $G_{1}$ and $G_{2}$, respectively. Define now maps
$\alpha_{ij}:X_{i}\times X_{j}\times X_{j}\longrightarrow{\mathbb{Z}}$, where
${\mathbb{Z}}$ is given multiplicative structure with generator $w$,
determined by

$\alpha_{ij}(b_{i}^{k_{i}}\times b_{j}^{s_{1}}\times
b_{j}^{s_{2}})=\begin{cases}w\ \ {\rm if}\ \ s_{1}=s_{2}\\\ 1\ \ {\rm
otherwise}\end{cases}$

where $b_{1}=y$ and $b_{2}=z$. The system so defined is seen to satisfy the
$2$-cocycle condition of Section 4 by direct inspection. In fact since
$(y^{k},y^{t},y^{t})$ and $(z^{r},z^{s},z^{s})$ are $2$-cycles of $G_{4}$ and
$G_{6}$, respectively, with ternary self-distributive differentials, applying
Remark 4.9 it follows that the system of $2$-cocycles $\\{\alpha_{ij}\\}$ is
nontrivial.

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