# Local hidden variable values without optimization procedures

Dardo Goyeneche  Departamento de Física, Facultad de Ciencias Básicas,
Universidad de Antofagasta, Casilla 170, Antofagasta, Chile  Wojciech Bruzda
Institute of Theoretical Physics, Jagiellonian University, ul. Łojasiewicza
11, 30-348 Kraków, Poland  Ondřej Turek  Department of Mathematics, Faculty of
Science, University of Ostrava, 701 03 Ostrava, Czech Republic  Daniel Alsina
School of Electronic and Electrical Engineering, University of Leeds, Leeds
LS2 9JT, UK  Karol Życzkowski  Institute of Theoretical Physics, Jagiellonian
University, ul. Łojasiewicza 11, 30-348 Kraków, Poland  Center for Theoretical
Physics, Polish Academy of Sciences, Al. Lotników 32/46, 02-668 Warsaw, Poland

(March 10, 2022)

###### Abstract

The problem of computing the local hidden variable (LHV) value of a Bell
inequality plays a central role in the study of quantum nonlocality. In
particular, this problem is the first step towards characterizing the LHV
polytope of a given scenario. In this work, we establish a relation between
the LHV value of bipartite Bell inequalities and the mathematical notion of
excess of a matrix. Inspired by the well developed theory of excess, we derive
several results which directly impact the field of quantum nonlocality. As a
consequence, we provide the LHV value for infinite families of bipartite Bell
inequalities, with an unbounded number of measurement settings, without
needing to solve any optimization problems. We also find tight Bell
inequalities for a large number of measurement settings.

##  1 Introduction

In a seminal paper [1], John Bell proved that quantum correlations cannot be
explained from deterministic and local hidden variable (LHV) models [2]. Since
then, an increasing interest in the field has triggered a large progress of
the theory, supported by experimental implementations and practical
applications. A feasible generation and certification of quantum nonlocality
permits to design quantum technological applications having practical
advantage with respect to its classical counterpart. For instance, quantum
nonlocality can be used as a resource to outperform classical communication in
certain distributed computing tasks [3], and enhances the communication power
in the context of information theory [4]. Quantum nonlocality has also led to
the emergence of device-independent protocols, in which the involved parties
do not need to trust their measurement devices [5, 6]. Concrete practical
applications include quantum key distribution [7, 8, 9, 10, 11, 12, 13],
random number generation [14, 15, 16, 17], quantum cryptography [10, 11, 12,
13], and testing quantumness of clouds of quantum computers [18, 19, 20],
among others.

A fundamental open question in quantum nonlocality is the following: _which
Bell inequalities predict a quantum advantage with respect to LHV models?_
Finding the answer to this question is crucial for the development of the
practical applications shown above. For instance, a way to generate genuine
random numbers [21], quantum cryptographic [22] and device independent [23]
protocols rely on the nonlocal essence of Nature. In order to answer the
aforementioned question, one needs to find both the LHV and quantum values.
However, the problem of finding the LHV value of Bell inequalities is NP-hard,
in the sense that the computational time required to find the LHV value
increases faster than any polynomial function of the number of parties,
measurement settings and outcomes. It is thus of fundamental relevance to
search for a way to estimate the LHV value without involving any optimization
procedure. In spite of a considerable effort made during the last decades –
see [24] and references therein – the complete understanding of the boundary
between LHV theories and quantum nonlocality remains open even in bipartite
scenarios composed by a small number of measurement settings and outcomes
[25].

In this work, we introduce a versatile tool that might shed some light to the
above described problem. That is, we link the problem of finding the LHV value
of bipartite Bell inequalities to the problem of finding matrices with maximal
_excess_ [26, 27]. This one-to-one correspondence allows us to find the LHV
value for infinitely many families of bipartite Bell inequalities in a wide
range of scenarios. The observed connection plays an important role, specially
when taking into account that identification of tight Bell inequalities relies
on the study of LHV strategies.

This work is organized as follows. In Section 2, we introduce our main tool,
i.e. the excess of a matrix. In Section 3, we give a brief introduction to
Bell inequalities. In Section 4, we establish the one-to-one connection
between maximal excess and the LHV value of bipartite Bell inequalities. This
connection allows us to derive an upper for the LHV value that is stronger
than known upper bounds for the quantum value. In Section 5, we show how to
find the LHV value of several classes of Bell inequalities without requiring
to implement any optimization procedure. Additionally, in Section 6 we find
seven tight Bell inequalities for the two outcomes scenario with a large
number of measurement settings, ranging between 8 and 20. Proofs of our
results are provided in Appendix A.

##  2 Excess of a matrix

A square matrix $H$ with entries $\pm 1$ is called Hadamard if its columns are
pairwise orthogonal. In 1973, Schmidt [26] asked about the maximal number of
1’s that could be present in a Hadamard matrix of a given order $n$. To this
end he introduced the notion of excess of a Hadamard matrix $H$ as the
difference between the number of the positive and negative entries of $H$.
This value is usually denoted by $\Sigma(H)$.

The maximal possible value of $\Sigma(H)$, among all Hadamard matrices of a
given order $n$, called _maximal excess_ , was analyzed by Best [27], who
found the following bounds:

$n^{2}2^{-n}\binom{n}{\frac{n}{2}}\leq\max\Sigma(H)\leq n\sqrt{n}.$ (1)

where the maximum is taken over all existing Hadamard matrices of order $n$.

Moreover, there are infinitely many orders $n$ for which $\max\Sigma(H)$ is
already known [27, 33, 34, 35, 36, 37]. For a Hadamard matrix of order $n$,
upper bound (1) is attained if and only if $H$ is a _constant row sum Hadamard
matrix_ , i.e. the sum of all entries in each row gives the same value [27].
Let us note that constant row sum matrices are sometimes called _regular_ in
the context of Hadamard matrices [27]. Constant row sum Hadamard matrices of
order $n$ only exist when $n$ is a square number. Thus, the upper bound (1) is
not saturated when $n=2$. As we will see later, this has a direct connection
with the fact that the CHSH Bell inequality [43] has a quantum advantage.

Note that the excess of a Hadamard matrix is also equal to the sum of all
entries of $H$. This allows a straightforward extension of the notion of
excess to any complex square matrix $M$ of order $n$.

######  Definition 1.

The sum of all entries of any square complex matrix $M$ of order $n$ is called
its excess, denoted

$\Sigma(M)=\sum_{j,k=0}^{n-1}M_{jk}.$ (2)

In Section 3, we restrict our attention to a class of matrices satisfying
certain symmetries. In particular, any matrix within this class has real
excess and allows us to introduce the notion of maximal excess. In Section 4,
we will show that this class of matrices is one-to-one related to the full set
of bipartite Bell inequalities.

##  3 Bell inequalities

Suppose a bipartite scenario in which two observers, Alice and Bob, implement
$m$ measurement settings per side having $q$ outcomes each. From an ensemble
of identically prepared quantum states they can estimate a joint probability
distribution $P(a,b|x,y)$, where $a,b\in[0,\dots,q-1]$ denote outcomes for
Alice and Bob, respectively, conditioned to the measurement settings
$x,y\in[0,\dots,m-1]$, respectively. It can be shown that a single correlation
of the form $P(a,b|x,y)$ does not provide enough evidence to ensure a conflict
with LHV models [28]. However, a linear combination of such quantum
probabilities attains values that cannot be reproduced by any LHV model [1].
Such expressions, known as _Bell inequalities_ [24], are defined as follows:

$\sum_{x,y=0}^{m-1}\sum_{a,b=0}^{q-1}S^{ab}_{xy}\,P(a,b|x,y)\leq\mathcal{C},$
(3)

where $S^{ab}_{xy}$ is a real-valued function and $\mathcal{C}$, so called
_classical_ or local hidden variable (LHV) value, is defined as the maximal
achievable value of the left hand side in Eq. (3) in a local deterministic
theory. That is, there is statistical independence between Alice’s and Bob’s
results, $P(a,b|x,y)=P(a|x)P(b|y)$, and the outcomes are deterministic, i.e.
$P(a|x),P(b|y)\in\\{0,1\\}$, for every pair of measurement settings
$x,y\in\\{0,...,m-1\\}$ and outcomes $a,b\in\\{0,...,q-1\\}$. The quantum
value $\mathcal{Q}$ is defined as the maximal possible value of the left hand
side in (3), if optimization is implemented over all joint probability
distributions admissible in quantum theory when observers implement local
measurements and do not communicate their results. Probabilities in quantum
theory take the form
$P(a,b,|x,y)=\mathrm{Tr}[(\Pi_{a}^{x}\otimes\Pi_{b}^{y})\rho_{AB}]$, where
$\\{\Pi_{a}^{x}\\}$ and $\\{\Pi_{b}^{y}\\}$ define Positive-Operator Valued
Measure (POVM), while $\rho_{AB}$ is a bipartite quantum state. The remarkable
observation of Bell is that LHV correlations can be weaker than quantum
correlations under certain conditions, thus it is possible to have
$\mathcal{C}<\mathcal{Q}$. This important result, together with its
experimental verification [29], confirmed the nonlocal behavior of Nature.

Inequality (3), when restricted to projective quantum measurements, can be
represented by expectation values of correlators through the discrete double
Fourier transform of the joint probability distribution [30]:

$P(a,b|x,y)=\frac{1}{q^{2}}\sum_{s,t=0}^{q-1}\omega^{as+bt}\langle
A^{s}_{x}\otimes B^{t}_{y}\rangle,$ (4)

where $\omega=e^{2\pi i/q}$. Here, $A^{s}_{x}$ denotes the $s^{\rm th}$ power
of the quantum observable $A_{x}$ associated to Alice, analogously for Bob. We
assume that every observable has $q$ outcomes, associated with $q$ different
roots of unity as eigenvalues. Note that any bipartite quantum probability
distribution can be written as (4), even those produced by POVM measurements.
This is due to the fact that any POVM can be seen as a projective measurement
in a larger Hilbert space, due to the dilation theorem of Naimark [31].

Note that relation (4) is constrained to quantum joint probability
distributions, thus it agrees the normalization conditions

$\sum_{a,b=0}^{q-1}P(a,b|x,y)=1$ (5)

for every $x,y=0,\dots,m-1$ and the no-signaling principle [32]. That is,

$\begin{split}\sum_{b=0}^{q-1}P(a,b|x,y)&=\sum_{b=0}^{q-1}P(a,b|x,y^{\prime}),\\\
\sum_{a=0}^{q-1}P(a,b|x,y)&=\sum_{a=0}^{q-1}P(a,b|x^{\prime},y),\end{split}$
(6)

for every $a,b=0,\dots,q-1$ and $x,y,x^{\prime},y^{\prime}=0,\dots,m-1$.
Conditions (6) imply that Alice and Bob cannot have faster-than-light
communication. Equation (4) assumes quantum mechanics on the right hand side,
agreeing with the no-signaling principle due to the fact that
$A^{0}_{x}=B^{0}_{y}=\mathbb{I}$, for every $x,y=0,\dots,m-1$.

Due to (4), inequality (3) can be then rewritten as

$\sum_{x,y=0}^{m-1}\sum_{s,t=0}^{q-1}M_{ms+x,mt+y}\,\langle\\\
A^{s}_{x}\otimes B^{t}_{y}\rangle\leq\mathcal{C},$ (7)

where $M$ is a square matrix of order $n=mq$ with entries

$M_{ms+x,mt+y}=\frac{1}{q^{2}}\sum_{a,b=0}^{q-1}\omega^{sa+tb}\,S^{ab}_{xy},$
(8)

for every $0\leq s,t\leq q-1$, $0\leq x,y\leq m-1$. Entries of the matrix $M$
are defined as the Fourier transform (8) of real valued functions. Thus, the
following symmetric conditions hold:

$M_{m\left[q-s\right]_{q}+x,m\left[q-t\right]_{q}+y}=(M_{ms+x,mt+y})^{*},$ (9)

for every $0\leq s,t\leq q-1$, $0\leq x,y\leq m-1$, where the asterisk denotes
complex conjugation and $[x]_{q}$ denotes $x$ modulo $q$. The structure of
matrices $M$ satisfying (9) is described in Appendix B.

The LHV value $\mathcal{C}$ of a bipartite Bell inequality can be
alternatively calculated as follows, based on (7):

$\mathcal{C}=\max_{\begin{subarray}{c}a_{0},\dots,a_{m-1}\in\Omega_{q}\\\
b_{0},\dots,b_{m-1}\in\Omega_{q}\end{subarray}}\sum_{x,y=0}^{m-1}\sum_{s,t=0}^{q-1}M_{ms+x,mt+y}\,a^{s}_{x}\,b^{t}_{y},$
(10)

where $\Omega_{q}=\\{1,\omega,\dots,\omega^{q-1}\\}$. Eq.(10) plays a central
role in our work, as we will see in Section 4.

######  Remark 1.

Notice that if $q=2$, symmetry (9) implies that matrix $M$ is real. In
general, any matrix $M\in\mathbb{C}^{mq\times mq}$ obeying condition (9) has
the following properties:

  * •

If $q$ is even, entries $M_{ms+x,mt+y}$ are real for all $x,y,s,t$ such that
$0\leq x,y\leq m-1$ and $s,t\in\\{0,q/2\\}$.

  * •

If $q$ is odd, entries $M_{x,y}$ are real for all $0\leq x,y\leq m-1$.

In Appendix B, we illustrate the structure of matrices $M$ for odd and even
orders.

We are now ready to introduce a notion of equivalence. We say that two Bell
inequalities (3) are equivalent if they differ in a relabeling of inputs and
outputs only. That is, if their coefficients are given by $S^{ab}_{xy}$ and
$S^{a^{\prime}b^{\prime}}_{x^{\prime}y^{\prime}}$, where
$S^{a^{\prime}b^{\prime}}_{x^{\prime}y^{\prime}}=\mathcal{P}S^{ab}_{xy}$, and
$\mathcal{P}$ is a permutation. This notion of equivalence can be translated
to matrices $M$ through (8).

######  Definition 2.

Two matrices $M$ and $M^{\prime}$, associated to a Bell inequality (7), with
$m$ measurement settings and $q$ outcomes, are equivalent if there exists a
permutation $\mathcal{P}^{a\to a^{\prime},b\to b^{\prime}}_{x\to
x^{\prime},y\to y^{\prime}}$ such that

$M^{\prime}=\mathcal{F}\mathcal{P}\mathcal{F}^{-1}M,$ (11)

where $\mathcal{F}$ is the double Fourier transform defined in (8).

Definition 2 induces the following notion of maximal excess.

######  Definition 3.

Let $q\geq 2$. We say that $M$ of order $mq$ has maximal excess with respect
to $q$-equivalence if $M$ obeys (9) and

$\Sigma(M)=\max\\{\Sigma(M^{\prime});M^{\prime}\text{ is $q$-equivalent to
}M\\}.$

Note that any matrix $M^{\prime}$, $q$-equivalent to $M$, satisfies the
symmetric relation (9). This means that $\Sigma(M^{\prime})$ is by
construction a real quantity.

An important class of inequalities are the so-called _correlation_ Bell
inequalities. These are inequalities of the form (7) such that there are no
marginal terms, i.e. $M_{ms+x,mt+y}=0$ when either $s=0$ or $t=0$. Physically,
this means that every term of the Bell inequality (7) is a linear combination
of correlations existing between Alice and Bob. Let us introduce the entire
set of matrices related to correlation Bell inequalities.

######  Definition 4.

Let $q\geq 2$. We say that a matrix $M$ is a correlation matrix if it
satisfies (9) and $M_{\alpha,\beta}=0$ when either $\alpha=0$ or $\beta=0$.
The nontrivial part of $M$, i.e. the submatrix of order $\mathfrak{n}=m(q-1)$
having entries $M_{\alpha,\beta}$ with $m\leq\alpha,\beta\leq mq-1$ is called
the core of $M$.

Note that the $q$-equivalence notion introduced in Definition 2 is equivalent
to the standard notion of equivalence between two Hadamard matrices when
$q=2$. Furthermore, Definition 3 reduces to the standard notion of the maximal
excess for Hadamard matrices, introduced in Section 2, when $q=2$.

##  4 Excess and LHV models

In this section, we establish a connection between the notion of excess of
certain class of matrices and the LHV value of a Bell inequality. The key
observation is the following. Given a classical value (10) for a matrix $M$,
we can always find an equivalent matrix $M^{\prime}$ such that the optimal LHV
strategy is achieved when all outputs are equal to $+1$, for both Alice and
Bob. Thus, the LHV value of the related Bell inequality is given by the sum of
all the entries of matrix $M^{\prime}$, i.e. its excess. Let us now formalize
this observation.

######  Proposition 1.

If $M$ is a matrix of order $mq$ having maximal excess with respect to
$q$-equivalence, then the LHV value (10) equals the excess of the matrix $M$,
i.e. $\mathcal{C}=\Sigma(M)$. Also, for any Bell inequality (7) induced by a
given matrix $M$, there is a $q$-equivalent matrix $M^{\prime}$ such that
$\mathcal{C}(M)=\Sigma(M^{\prime})$.

The relevance of Prop. 1 consists in the fact that matrices $M$ having maximal
excess have been exhaustively studied by mathematicians during the last five
decades, including a wide range of Hadamard matrices [27, 33, 34, 35, 36, 37,
38], complex Hadamard matrices [39], Hadamard tensors [40] and orthogonal
designs [41]. These results considerably extend the set of Bell inequalities
for which the LHV value is known [24], which is not a minor observation taking
into account the hardness to calculate the LHV value of a bipartite Bell
inequality [42].

For the rest of the work, we restrict our attention to matrices $M$ associated
to Bell inequalities achieving the LHV value when the local deterministic
strategies of both Alice and Bob have outcomes $+1$.

Let us illustrate Prop. 1 with the celebrated CHSH inequality [43], associated
to the matrix

$M=\left(\begin{array}[]{rrrr}0&0&0&0\\\ 0&0&0&0\\\ 0&0&1&1\\\
0&0&1&-1\end{array}\right),$

of order $n=m\times q=2\times 2=4$, whose core is a Hadamard matrix of order
$\mathfrak{n}=m\times(q-1)=2\times 1=2$:

$\langle A_{0}\otimes B_{0}+A_{0}\otimes B_{1}+A_{1}\otimes B_{0}-A_{1}\otimes
B_{1}\rangle\leq 2.$ (12)

Here, $A_{i}$ and $B_{j}$ are dichotomic quantum observables represented by
Hermitian operators having eigenvalues $\pm 1$ each. This Hadamard matrix has
maximal excess $\Sigma(M)=2$ [27], coinciding with the LHV value of the CHSH
inequality [43].

The next result requires the introduction of the numerical radius of a matrix
$M$, $r(M)=\max_{|\psi\rangle\in\mathcal{H}_{n}}|\langle\psi|M|\psi\rangle|$.
Now, let us express an upper bound for excess in terms of the numerical
radius.

######  Proposition 2.

Let $M$ be a matrix of order $n=mq$ having maximal excess with respect to
$q$-equivalence. Then, the following upper bound holds for the LHV value
$\mathcal{C}(M)$:

$\mathcal{C}(M)\leq nr(M).$ (13)

For any matrix $M$, we have that $\rho(M)\leq r(M)\leq\sigma(M)$, where
$\rho(M)=\max_{j}|\lambda_{j}(M)|$ is the spectral radius and
$\sigma(M)=\max_{j}\sqrt{\lambda_{j}(MM^{{\dagger}})}$ is the maximal singular
value of $M$, also called spectral norm. For normal matrices, equalities
$\rho(M)=r(M)=\sigma(M)$ hold. Consequently, the upper bound (13) is more
restrictive than a well-known upper bound for the quantum value, which we
recall in the following Observation.

######  Observation 1 ([44, 45]).

The quantum value of a bipartite Bell inequality, induced by a matrix $M$ of
order $n$, satisfies

$\mathcal{Q}(M)\leq n\sigma(M).$ (14)

The characterization of the entire set of matrices $M$ that saturate the bound
(13) or (14) is open in every scenario. Let us now provide a new upper bound
for which we can fully characterize its tightness, in every bipartite
scenario.

######  Proposition 3.

Let $M$ be a matrix of order $n=mq$ having maximal excess with respect to
$q$-equivalence. Then we have

$\mathcal{C}(M)\leq\sqrt{n}\,\nu(M),$ (15)

where
$\nu(M)=\sqrt{\sum_{i=0}^{n-1}\bigl{|}\sum_{j=0}^{n-1}M_{ij}\bigr{|}^{2}}$.
Moreover, upper bound (15) is saturated if and only if $M$ is a constant row
sum matrix. In such case, $\mathcal{C}(M)=n\Gamma$, where $\Gamma$ is the
constant row sum value.

Notice that the constant row sum value $\Gamma$ occurring in Proposition 3 is
always a positive number due to the assumption that $M$ has maximal excess and
to the symmetry assumption (9), implying that $\Sigma(M)\in\mathbb{R}$.

Let us show a further consequence of Prop. 3. If a matrix $M$ with maximal
excess is both constant row sum and unitary, then $\mathcal{C}(M)=n$,
coinciding with the upper bound (14). Thus, $\mathcal{C}(M)=\mathcal{Q}(M)=n$,
and the related Bell inequality does not have any quantum advantage. In
particular, if $M$ is an unnormalized constant row sum Hadamard matrix for
$q=2$, then the associated correlation Bell inequality satisfies
$\mathcal{C}(M)=\mathcal{Q}(M)=n\sqrt{n}$. That is, we saturate both the Best
bound (1) and the Epping-De Vicente bound (14). Interestingly, inequality (15)
is saturated by all the matrices considered in nonlocal computation [46] and
quantum games having no quantum advantage [47].

To illustrate Prop. 3, note that the correlation Bell inequality associated to
the circulant Hadamard matrix $\mathrm{core}(M)=\mathrm{circ}[(-1,1,1,1)]$
satisfies $\mathcal{C}(M)=\mathcal{Q}(M)=8$, where there is no need to
calculate the quantum value, as the LHV value saturates the bound (14). As a
further example, consider the three inequivalent constant row sum Hadamard
matrices of order $16$ [48], defining three inequivalent correlation Bell
inequalities in the bipartite scenario composed of $16$ measurement settings
per side having $2$ outcomes each. These three inequalities have both LHV and
quantum values equal to 64. The same conclusion does not apply to the Hadamard
matrix $H_{2}$ of order 2, related to CHSH inequality [43], as this Hadamard
matrix does not have constant row sum.

On the other hand, non-unitary constant row sum matrices can imply a
correlation Bell inequality having a quantum advantage. For instance, for
$\mathfrak{n}=3$ the matrix $\mathrm{core}(M)={\rm circ}\left[(0,-1,1)\right]$
satisfies $\mathcal{C}(M)=4<3\sqrt{3}=\mathcal{Q}(M)$. Here, the quantum value
$\mathcal{Q}(M)$ is achieved when considering measurement settings $A_{j}$ and
$B_{k}$ of the form $U_{j}DU^{\dagger}_{j}$, for

$U_{j}=\left[\begin{array}[]{rr}\cos\alpha_{j}&\sin\alpha_{j}\\\
\sin\alpha_{j}&\cos\alpha_{j}\end{array}\right],$ (16)

with $D=\\{\\{1,0\\},\\{0,-1\\}\\}$ and $\alpha_{j}=0$, $\frac{2\pi}{3}$,
$\frac{\pi}{3}$ for $A_{0}$, $A_{1}$, $A_{2}$ and $\alpha_{j}=\frac{\pi}{4}$,
$\frac{7\pi}{12}$, $\frac{11\pi}{12}$ for $B_{0}$, $B_{1}$, $B_{2}$,
respectively.

Note that a weaker version of the bound (15) occurs when
$\nu(M)=\|M\phi\|\leq\|M\|\|\phi\|=\sqrt{n}\sigma(M)$. In such case, the bound
reduces to $\Sigma(M)\leq n\sigma(M)$, coinciding with the upper bound for the
quantum value (14). From here and Prop. 2, an interesting fact arises.

######  Observation 2.

The problem to classify all correlation Bell inequalities with $\mathfrak{n}$
settings for which there is no quantum advantage contains, as a special case,
the problem of finding all constant row sum correlation matrices $M$ of order
$\mathfrak{n}$, for which $\nu(M)$, defined in Prop. 3, equals the maximal
singular value $\sigma(M)$. In particular, this problem contains the -long
standing- circulant Hadamard conjecture [49], which states that such matrices
exist only for orders 1 and 4.

There is a simple property that sometimes reduces the complexity of
calculating excess of a matrix: excess of a tensor product is the product of
excesses, see Observation 2.4 in [50]. Applying this property to Bell
inequalities we have the following result.

######  Proposition 4.

Let $M_{1}$ and $M_{2}$ be matrices of order $n_{1}=m_{1}q$ and
$n_{2}=m_{2}q$, respectively. If both $M_{1}$ and $M_{2}$ have maximal excess
with respect to $q$-equivalence and $M_{2}$ is real, then the LHV value of the
Bell inequality induced by the tensor product matrix $M=M_{1}\otimes M_{2}$,
having $q$ outcomes per setting, is given by
$\mathcal{C}(M)=\mathcal{C}(M_{1})\mathcal{C}(M_{2})$.

Note that the product of matrices producing a tight Bell inequality does not
induce a tight inequality, in general. Indeed, the inequality induced by core
$H_{4}=H_{2}\otimes H_{2}$ is not tight, even thought CHSH inequality, induced
by the core $H_{2}$, is tight. On the other hand, sometimes this occurs, as
the core ${H_{8}=H_{2}\otimes H_{2}\otimes H_{2}}$ induces a tight Bell
inequality, see Section 6.

##  5 LHV value without optimization

In this section, we show how to construct some families of correlation
matrices $M$ of order $\mathfrak{n}=m(q-1)$ that allow us to achieve the LHV
value of Bell inequalities with an unbounded number of measurement settings
each, for $q=2$ outcomes. The following results summarize some relevant
contributions to excess theory achieved in the last four decades. Before
presenting the results, let us recall some required notions. A matrix $M$ is
called skew symmetric if $M^{T}=-M$, where $T$ denotes transposition. A
conference matrix is a square matrix having zero main diagonal, $\pm$1 off-
diagonal entries and orthogonal columns. A Hadamard matrix $H$ is skew-type if
$H-\mathbb{I}$ is skew symmetric. Furthermore, recall that any real matrix $M$
satisfies (9), for $q=2$.

######  Observation 3 ([51]).

If $M$ is a skew-type Hadamard matrix of order $k\equiv 0\,\mathrm{(mod\,4)}$
or a conference matrix of order $k\equiv 2\,(\mathrm{mod}\,4)$, then there is
a Hadamard matrix $M$ of order $\mathfrak{n}=4k(k-1)$ (thus
$\mathfrak{n}\equiv 0\,(\mathrm{mod}\,16)$ and $\mathfrak{n}\equiv
8\,(\mathrm{mod}\,16)$, respectively), with maximal excess

$\Sigma(M)=4(k-1)^{2}(2k+1).$ (17)

Note that none of the Hadamard matrices from Observation 3 are equivalent to
one having constant row sum, as $\mathfrak{n}$ is not a square number. Thus,
the related Bell inequalities are potential candidates to have a quantum
advantage, in the sense that the upper bound (14) is strictly larger than the
LHV value in each case.

######  Observation 4 ([52]).

Let $M$ be a conference matrix of order $\mathfrak{n}$ and let $k$ be an odd
integer such that ${k\leq\sqrt{\mathfrak{n}-1}<k+2}$. Then we have

$\Sigma(M)\leq\frac{\mathfrak{n}(k^{2}+2k+\mathfrak{n}-1)}{2(k+1)}.$

Equality holds if and only if either (i) $\mathfrak{n}-1$ is a square and $M$
has constant row sum equal to $k$, or (ii) $\mathfrak{n}-1$ is not a square
and row sums are either $k$ or $k+2$.

For instance, let us show an infinite family of Bell inequalities, arising
from Hadamard matrices of order $\mathfrak{n}=\ell+1$, with $\ell\equiv
3\,(\mathrm{mod}\,4)$, for which the maximal excess is known. The Hadamard
matrices have the form

$M=\left(\begin{array}[]{cc}-1&\mathbf{1}^{T}_{\ell}\\\
\mathbf{1}_{\ell}&A\end{array}\right),$ (18)

where $\mathbf{1}_{\ell}$ is the all-one vector of length $\ell$ and the
square matrix $A$ of order $\ell$ is given by

$A_{ij}=\left\\{\begin{array}[]{cl}1&\mbox{if $j-i\in C\cup\\{0\\}$}\\\
-1&\mbox{if
$j-i\in\mathbb{F}_{\ell}\setminus(C\cup\\{0\\})$.}\end{array}\right.$ (19)

Here, $\mathbb{F}_{\ell}$ denotes the finite field of order $\ell$ and $C$ is
the set of nonzero squares of $\mathbb{F}_{\ell}$. This leads us to the
following statement.

######  Observation 5 ([53]).

Let $m$ be an integer number such that $\ell=(2m+1)^{2}+2$ is the power of a
prime, and $k$ be an even integer such that $k\leq\sqrt{\mathfrak{n}}<k+2$.
Also, let $t=k$ if $|\mathfrak{n}-k^{2}|<|\mathfrak{n}-(k+2)^{2}|$ and $t=k-2$
otherwise. Then the Hadamard matrix (18) of order $\mathfrak{n}=\ell+1$ has
maximal excess

$\Sigma(M)=\mathfrak{n}(t+4)-4s,$ (20)

where $s$ is the integer part of the following expression:
$\mathfrak{n}\bigl{(}(t+4)^{2}-\mathfrak{n}\bigr{)}/(8t+16)$.

As complementary information, a lower bound for the maximal excess of known
Hadamard matrices has been found up to order $\mathfrak{n}=1000$, in many
cases achieving the maximal excess [51]. There are more explicit results for
the maximal excess of Hadamard matrices than those shown above — see [54, 55,
56, 57] and references therein. The maximal excess has been also found for
weaving Hadamard matrices [38], complex Hadamard matrices [39] and orthogonal
designs [41]. Furthermore, maximal excess for tensors [40] opens the
possibility to naturally extend our results to multipartite systems.
Interestingly, maximal excess for tensors is connected with the so-called
_discrepancy of multivariable functions_ , associated with certain multiparty
communication problems, see [40] and references therein. This connection might
contribute to a better understanding of communication complexity for
multipartite quantum systems [58].

We tend to believe the results and references shown in this section provide a
valuable contribution to the theory of quantum nonlocality. On the one hand,
this considerably extends the set of Bell inequalities for which the classical
value is currently known [24]. On the other hand, the same refined techniques
used in these references might inspire researchers to develop more efficient
ways to find the LHV value for larger classes of Bell inequalities.

##  6 Tight Bell inequalities

Tight Bell inequalities define facets of the LHV polytope. These hyperplanes
completely characterize the set of correlations compatible with a LHV model.
The quantum correlations space, for a bipartite scenario with $m$ settings and
$q$ outcomes per party, is defined by all complex vectors
$v\in\mathbb{C}^{m^{2}q^{2}}$ having entries of the form $\langle
A^{s}_{x}\otimes B^{t}_{y}\rangle$, $x,y\in\\{0,\dots,m-1\\}$ and
$s,t\in\\{0,\dots,q-1\\}$, where $A_{x}$ and $B_{y}$ are unitary operators
having the $q^{\rm th}$ roots of unity as eigenvalues. The no-signaling
conditions (6) restrict quantum correlations to a $d=m^{2}(q-1)^{2}$
dimensional subspace, when single party measurements are not considered.

Number of settings $(m)$ |  Number of vertices |  Number of affine inde- pendent vertices |  Tightness
---|---|---|---
2 | 4 | 3 | Tight
4* | 4 | 3 | Non-tight
8 | 64 | 63 | Tight†
12 | 2640 | 143 | Tight†
16 (1)* | 896 | 105 | Non-tight
16 (2)* | 192 | 81 | Non-tight
16 (3)* | 64 | 45 | Non-tight
16 (4) | 21504 | 255 | Tight†
16 (5) | 21504 | 255 | Tight†
20 (1) | 20064 | 399 | Tight†
20 (2) | 20064 | 399 | Tight†
20 (3) | 20064 | 399 | Tight†
Table 1: Study of tightness for bipartite correlation Bell inequalities of the
form
$\langle\sum_{\alpha,\beta=0}^{\mathfrak{n}-1}H_{\alpha\beta}A_{\alpha}\otimes
B_{\beta}\rangle\leq\mathcal{C}(H)$, with $m=\mathfrak{n}$ measurement
settings and $q=2$ outcomes. Here, $H$ is a Hadamard matrix of order
$\mathfrak{n}=m(q-1)$. Note that $H=\mathrm{core}(M)$ in (7), where $M$ has
order $n=mq$. All entries outside the core of $M$ are zero. The table shows
the number of vertices of the LHV polytope that are touched by each
hyperplane, and the number of affine independent vertices. For a tight Bell
inequality, the latter number has to be equal to $n^{2}-1$, implying that the
hyperplane forms a facet of the LHV polytope. The asterisk in the first column
of the table means that the related Hadamard matrix has constant row sum. The
seven tight cases denoted with symbol ${\dagger}$ in the fourth column are
new, to the best of our knowledge.

Given that facets of the LHV polytope are faces with maximal dimension, they
define hyperplanes in the correlation space. That is, they have dimension
$d-1$. This means that for a two outcomes scenario ($q=2$) there are
$d-1=m^{2}-1$ linearly independent vectors $v$, associated to LHV strategies
defining a tight Bell inequality.

Based on the above findings, in this section we study tightness of correlation
Bell inequalities induced by Hadamard matrices $\mathrm{core}(M)$ up to order
$\mathfrak{n}=20$. Results are summarized in Table 1, which leads us to
establish the following conjecture:

######  Conjecture 1.

A Hadamard matrix of order $\mathfrak{n}$ induces a tight correlation Bell
inequality with $m=\mathfrak{n}$ settings and $q=2$ outcomes per party if and
only if it is not equivalent to a matrix with constant row sum.

Furthermore, in the study described in Table 1, we noted that the set of
optimal LHV values can be arranged as _mutually quasi-unbiased weighing
matrices_ [62, 63], for both Alice and Bob strategies. See Appendix C for
further details.

##  7 Conclusions

We introduced a one-to-one relation between the local hidden variable (LHV)
value of Bell inequalities and the mathematical notion of excess of a matrix,
see Prop. 1. This connection allowed us to obtain -without any optimization-
the LHV value of infinitely many families of Bell inequalities with an
unbounded number of measurement settings per party, see Section 5.

We proposed the conjecture that every Hadamard matrix, not equivalent to a
constant row sum one, induces a tight Bell inequality. We supported this
conjecture with Hadamard matrices up to order 20, see Table 1.

Furthermore, we derived two upper bounds for the LHV value of bipartite Bell
inequalities that are stronger than the upper bound for the quantum value, see
Props. 2 and 3. Tightness of the last bound has been fully characterized in
every scenario, including some remarkable cases like all Bell inequalities
related to nonlocal computation [46] and quantum xor games without quantum
advantage [47].

Finally, we have shown that the problem to characterize the set of bipartite
Bell inequalities with no quantum advantage for two outcomes contains the
circulant Hadamard conjecture, a long-standing open problem in Combinatorics,
see Observation 2.

Our results provide new insights by merging two extensively studied areas of
research coming from different fields. Quantum nonlocality seems to be the
most benefited side, as the mathematical theory of excess is considerably more
advanced than currently known techniques to find local hidden variable values,
as far as we know. We believe our results will deepen the interest in quantum
nonlocality among both physicists and mathematicians.

_Acknowledgements_ Authors kindly acknowledge valuable discussions with A.
Acín, R. Augusiak, A. Cabello, J. Calsamiglia, D. Cavalcanti, J. De Vicente,
P. Horodecki, C. Jebaratnam, R. Ramanathan, O. Reardon-Smith, G. Senno, J.
Tura and A. Winter. D.G. is supported by MINEDUC-UA project code ANT 1855 and
Grant FONDECYT Iniciación number 11180474, Chile. W.B. and K.Ż. are supported
by Narodowe Centrum Nauki under the Maestro grant number
DEC-2015/18/A/ST2/00274, by Foundation for Polish Science under the Team-Net
project no. POIR.04.04.00-00-17C1/18-00. D.A. is supported by UK EPSRC Grant
No. EP/M013472/1.

##  Appendix A Proofs of results

In this section, we provide the proofs of the results presented in previous
sections. For convenience, the propositions are re-written below.

Proposition 1. _If $M$ is a matrix of order $mq$ having maximal excess with
respect to $q$-equivalence, then the Bell inequality (7), having $q$ outcomes
per setting, has LHV value $\mathcal{C}(M)$ equal to the excess of the matrix
$M$, $\Sigma(M)$. Also, for any Bell inequality (7) induced by a matrix $M$,
there is a $q$-equivalent matrix $M^{\prime}$ such that
$\mathcal{C}(M)=\Sigma(M^{\prime})$._

###### Proof.

The optimal LHV strategy is given by all-one outcomes for both Alice and Bob.
Thus, the LHV value is given by the sum of all the entries of matrix $M$.
Conversely, if the local variables producing the LHV value are $\\{a_{x}\\}$
and $\\{b_{y}\\}$, for Alice and Bob, respectively, then
$M^{\prime}_{[sx][ty]}=(a^{s}_{x}b^{t}_{y})^{*}M_{[sx][ty]}$ produces an
equivalent Bell inequality satisfying
$\mathcal{C}(M^{\prime})=\Sigma(M^{\prime})$. Note that $M^{\prime}$ satisfies
the symmetry (9). ∎

Proposition 2. _Let $M$ be a matrix of order $n=mq$ having maximal excess with
respect to $q$-equivalence. Then, the following upper bound holds for the LHV
value $\mathcal{C}(M)$:_

$\mathcal{C}(M)\leq nr(M).$ (21)

###### Proof.

Let $M$ be a matrix having maximal excess. The numerical radius is given by
$r=\max_{|\psi\rangle\in\mathcal{H}}|\langle\psi|M|\psi\rangle|\geq\langle\phi|M|\phi\rangle$,
where
$|\phi\rangle=\frac{1}{\sqrt{n}}\sum_{i=0}^{n-1}|i\rangle=\frac{1}{\sqrt{n}}(1,1,\ldots,1)^{T}$.
Thus
$r\geq\langle\phi|M|\phi\rangle=\frac{1}{n}\Sigma(M)=\frac{1}{n}\mathcal{C}(M)$.
∎

Proposition 3. _Let $M$ be a matrix of order $n=mq$ having maximal excess with
respect to $q$-equivalence. Then we have_

$\mathcal{C}(M)\leq\sqrt{n}\,\nu(M),$ (22)

_where
$\nu(M)=\sqrt{\sum_{i=0}^{n-1}\bigl{|}\sum_{j=0}^{n-1}M_{ij}\bigr{|}^{2}}$.
Moreover, upper bound (15) is saturated if and only if $M$ is a constant row
sum matrix. In such case, $\mathcal{C}(M)=n|s|$, where $s$ is the constant row
sum value._

###### Proof.

Upper bound (15) simply arises from the Cauchy-Schwarz inequality

$\mathcal{C}(M)=\Sigma(M)=n\langle\phi|M|\phi\rangle\leq
n\|M|\phi\rangle\|=n\frac{1}{\sqrt{n}}\,\nu(M).$ (23)

Here, equality is attained if and only if $|\phi\rangle$ and $M|\phi\rangle$
are parallel, which occurs when each row of matrix $M$ has the same sum, i.e.
$M$ has constant row sum. ∎

Proposition 4. _Let $M_{1}$ and $M_{2}$ be matrices of order $m_{1}q$ and
$m_{2}q$, respectively. If both $M_{1}$ and $M_{2}$ have maximal excess with
respect to $q$-equivalence and $M_{2}$ is real, then the LHV value of the Bell
inequality induced by the tensor product matrix $M=M_{1}\otimes M_{2}$, having
$q$ outcomes per setting, is given by
$\mathcal{C}(M)=\mathcal{C}(M_{1})\mathcal{C}(M_{2})$._

###### Proof.

The proof is straightforward from the fact that the excess of a tensor product
of two matrices is equal to the product of excesses, see Obs. 2.4 in [50]. We
only have to demonstrate that the matrix $M=M_{1}\otimes M_{2}$ obeys
condition (9). By the definition of the tensor product, we have $[M_{1}\otimes
M_{2}]_{ij}=[M_{1}]_{k_{1}l_{1}}\cdot[M_{2}]_{k_{2}l_{2}}$, where
$i=k_{1}m_{2}q+k_{2}$, $j=l_{1}m_{2}+l_{2}$, $0\leq k_{1},l_{1}<m_{1}q$,
$0\leq k_{2},l_{2}<m_{2}q$. Let $0\leq s,t<q$ and $0\leq x,y<m_{1}m_{2}q$.
Writing $x=m_{2}qe_{x}+f_{x}$ and $y=m_{2}qe_{y}+f_{y}$ for $0\leq
e_{x},e_{y}<m_{1}$ and $0\leq f_{x},f_{y}<m_{2}q$, we have

$\displaystyle[M_{1}\otimes
M_{2}]_{m_{1}m_{2}q[q-s]_{q}+x,m_{1}m_{2}q[q-t]_{q}+y}$ $\displaystyle=$
$\displaystyle[M_{1}\otimes
M_{2}]_{(m_{1}[q-s]_{q}+e_{x})m_{2}q+f_{x},(m_{1}[q-t]_{q}+e_{y})m_{2}q+f_{y}}$
$\displaystyle=$
$\displaystyle[M_{1}]_{m_{1}[q-s]_{q}+e_{x},m_{1}[q-t]_{q}+e_{y}}\cdot[M_{2}]_{f_{x},f_{y}}\,,$
and since $M_{1}$ obeys (9), we have $\displaystyle=$
$\displaystyle([M_{1}]_{m_{1}s+e_{x},m_{1}t+e_{y}})^{*}\cdot[M_{2}]_{f_{x},f_{y}}.$

Also, it holds

$\displaystyle([M_{1}\otimes M_{2}]_{m_{1}m_{2}qs+x,m_{1}m_{2}qt+y})^{*}$
$\displaystyle=$ $\displaystyle([M_{1}\otimes
M_{2}]_{(m_{1}s+e_{x})m_{2}q+f_{x},(m_{1}t+e_{y})m_{2}q+f_{y}})^{*}$
$\displaystyle=$
$\displaystyle([M_{1}]_{m_{1}s+e_{x},m_{1}t+e_{y}}\cdot[M_{2}]_{f_{x},f_{y}})^{*}$
$\displaystyle=$
$\displaystyle([M_{1}]_{m_{1}s+e_{x},m_{1}t+e_{y}})^{*}\cdot([M_{2}]_{f_{x},f_{y}})^{*}.$

Comparing these two expressions, we see that, if $M_{2}$ is real, then
$M_{1}\otimes M_{2}$ obeys (9), and so $\Sigma(M)$ is real as well. ∎

##  Appendix B Structure of matrices $M$

There are two structure for matrices $M$: _(i)_ when $q$ is odd, the top-left
$m\times m$ square block is real. For instance, for $q=5$ and $m=3$ we have a
matrix of the form

$\left(\begin{array}[]{ccccccccccccccc}\text{R}&\text{R}&\text{R}&.&.&.&.&.&.&.&.&.&.&.&.\\\
\text{R}&\text{R}&\text{R}&.&.&.&.&.&.&.&.&.&.&.&.\\\
\text{R}&\text{R}&\text{R}&.&.&.&.&.&.&.&.&.&.&.&.\\\
.&.&.&.&.&.&.&.&.&.&.&.&.&.&.\\\ .&.&.&.&.&.&.&.&.&.&.&.&.&.&.\\\
.&.&.&.&.&.&.&.&.&.&.&.&.&.&.\\\ .&.&.&.&.&.&.&.&.&.&.&.&.&.&.\\\
.&.&.&.&.&.&.&.&.&.&.&.&.&.&.\\\ .&.&.&.&.&.&.&.&.&.&.&.&.&.&.\\\
.&.&.&.&.&.&.&.&.&.&.&.&.&.&.\\\ .&.&.&.&.&.&.&.&.&.&.&.&.&.&.\\\
.&.&.&.&.&.&.&.&.&.&.&.&.&.&.\\\ .&.&.&.&.&.&.&.&.&.&.&.&.&.&.\\\
.&.&.&.&.&.&.&.&.&.&.&.&.&.&.\\\ .&.&.&.&.&.&.&.&.&.&.&.&.&.&.\\\
\end{array}\right).$

Here, letter $R$ denotes real entries and dots complex entries satisfying (9).
_(ii)_ when $q$ is even, there are 4 $m\times m$ real blocks, associated to
$s,t\in\\{0,q/2\\}$, for $0\leq x,y\leq m-1$. For instance, for $q=4$ and
$m=3$ we have a matrix of the form

$\left(\begin{array}[]{cccccccccccc}\text{R}&\text{R}&\text{R}&.&.&.&\text{R}&\text{R}&\text{R}&.&.&.\\\
\text{R}&\text{R}&\text{R}&.&.&.&\text{R}&\text{R}&\text{R}&.&.&.\\\
\text{R}&\text{R}&\text{R}&.&.&.&\text{R}&\text{R}&\text{R}&.&.&.\\\
.&.&.&.&.&.&.&.&.&.&.&.\\\ .&.&.&.&.&.&.&.&.&.&.&.\\\
.&.&.&.&.&.&.&.&.&.&.&.\\\
\text{R}&\text{R}&\text{R}&.&.&.&\text{R}&\text{R}&\text{R}&.&.&.\\\
\text{R}&\text{R}&\text{R}&.&.&.&\text{R}&\text{R}&\text{R}&.&.&.\\\
\text{R}&\text{R}&\text{R}&.&.&.&\text{R}&\text{R}&\text{R}&.&.&.\\\
.&.&.&.&.&.&.&.&.&.&.&.\\\ .&.&.&.&.&.&.&.&.&.&.&.\\\
.&.&.&.&.&.&.&.&.&.&.&.\\\ \end{array}\right),$

where, again, $R$ denotes real entries and dots represent complex entries
satisfying (9).

##  Appendix C Optimal LHV strategies

This Appendix is devoted to showing that optimal LHV strategies for both Alice
and Bob, when arranged as rows of matrices, define Mutually quasi-unbiased
weighing matrices, for some correlation Bell inequalities arising from
Hadamard matrices of orders $\mathfrak{n}=2$ and $\mathfrak{n}=8$. Each of
these matrices is the core of the matrix $M$ that generates the corresponding
correlation Bell inequality, in the scenario of two parties, $m=\mathfrak{n}$
settings and 2 outcomes each.

A weighing matrix $W$ of weight $k$ is a square matrix of order $m$ having
entries from the set $\\{-1,0,1\\}$, such that $WW^{T}=k\mathbb{I}$. Two
weighing matrices $W_{1}$ and $W_{2}$ of order $m$ and weight $k$ are
_mutually quasi-unbiased weighing matrices_ (MQUWM) [63] for parameters
$(m,k,l,a)$ if there exists positive integers $a$ and $l$ such that
$(1/\sqrt{a})W_{1}W^{T}_{2}$ is a weighing matrix of weight $l=k^{2}/a$. We
denote these matrices as MQUWM$(m,k,l,a)$.

Case $\mathbf{m=2}$ settings (CHSH):

There are 4 optimal LHV strategies in this case. For Alice and Bob, the
strategies define the following two sets of two MQUWM(2,2,1,4):

$W^{A}_{1}=\left(\begin{array}[]{rr}1&1\\\ 1&-1\end{array}\right)\hskip
14.22636ptW^{A}_{2}=\left(\begin{array}[]{rr}1&-1\\\ 1&1\end{array}\right),$
(24)

and

$W^{B}_{1}=\left(\begin{array}[]{rr}1&1\\\ -1&1\end{array}\right)\hskip
14.22636ptW^{B}_{2}=\left(\begin{array}[]{rr}1&-1\\\ 1&1\end{array}\right),$
(25)

respectively. That is, we have two orthonormal bases in each case, all of them
identical up to global phases and reordering of vectors.

Case $\mathbf{m=4}$ settings:

There are 4 optimal LHV strategies. For both Alice and Bob, the strategies can
be arranged in rows as the following weighing matrices of weight 4, i.e.
Hadamard matrices:

$W^{A}_{1}=\left(\begin{array}[]{rrrr}1&-1&-1&-1\\\ 1&-1&1&1\\\ 1&1&-1&1\\\
1&1&1&-1\end{array}\right),$ (26)

and

$W^{B}_{1}=\left(\begin{array}[]{rrrr}-1&1&1&1\\\ 1&-1&1&1\\\ 1&1&-1&1\\\
1&1&1&-1\end{array}\right),$ (27)

respectively. In this case, we also have two orthonormal bases.

Case $\mathbf{m=8}$ settings:

There are 64 optimal LHV strategies. For Alice, her local strategies can be
arranged as the following maximal set of 8 MQUWM(8,8,16,4), along the rows:

$W^{A}_{1}=\left(\begin{array}[]{rrrrrrrr}1&-1&-1&-1&-1&-1&-1&-1\\\
1&-1&-1&-1&1&1&1&1\\\ 1&-1&1&1&-1&-1&1&1\\\ 1&-1&1&1&1&1&-1&-1\\\
1&1&-1&1&-1&1&-1&1\\\ 1&1&-1&1&1&-1&1&-1\\\ 1&1&1&-1&-1&1&1&-1\\\
1&1&1&-1&1&-1&-1&1\\\ \end{array}\right)$

$W^{A}_{2}=\left(\begin{array}[]{rrrrrrrr}1&-1&-1&-1&-1&-1&1&1\\\
1&-1&-1&-1&1&1&-1&-1\\\ 1&-1&1&1&-1&-1&-1&-1\\\ 1&-1&1&1&1&1&1&1\\\
1&1&-1&1&-1&1&1&-1\\\ 1&1&-1&1&1&-1&-1&1\\\ 1&1&1&-1&-1&1&-1&1\\\
1&1&1&-1&1&-1&1&-1\\\ \end{array}\right)$

$W^{A}_{3}=\left(\begin{array}[]{rrrrrrrr}1&-1&-1&-1&-1&1&-1&1\\\
1&-1&-1&-1&1&-1&1&-1\\\ 1&-1&1&1&-1&1&1&-1\\\ 1&-1&1&1&1&-1&-1&1\\\
1&1&-1&1&-1&-1&-1&-1\\\ 1&1&-1&1&1&1&1&1\\\ 1&1&1&-1&-1&-1&1&1\\\
1&1&1&-1&1&1&-1&-1\\\ \end{array}\right)$

$W^{A}_{4}=\left(\begin{array}[]{rrrrrrrr}1&-1&-1&-1&-1&1&1&-1\\\
1&-1&-1&-1&1&-1&-1&1\\\ 1&-1&1&1&-1&1&-1&1\\\ 1&-1&1&1&1&-1&1&-1\\\
1&1&-1&1&-1&-1&1&1\\\ 1&1&-1&1&1&1&-1&-1\\\ 1&1&1&-1&-1&-1&-1&-1\\\
1&1&1&-1&1&1&1&1\\\ \end{array}\right)$

$W^{A}_{5}=\left(\begin{array}[]{rrrrrrrr}1&-1&-1&1&-1&-1&-1&1\\\
1&-1&-1&1&1&1&1&-1\\\ 1&-1&1&-1&-1&-1&1&-1\\\ 1&-1&1&-1&1&1&-1&1\\\
1&1&-1&-1&-1&1&-1&-1\\\ 1&1&-1&-1&1&-1&1&1\\\ 1&1&1&1&-1&1&1&1\\\
1&1&1&1&1&-1&-1&-1\\\ \end{array}\right)$

$W^{A}_{6}=\left(\begin{array}[]{rrrrrrrr}1&-1&-1&1&-1&-1&1&-1\\\
1&-1&-1&1&1&1&-1&1\\\ 1&-1&1&-1&-1&-1&-1&1\\\ 1&-1&1&-1&1&1&1&-1\\\
1&1&-1&-1&-1&1&1&1\\\ 1&1&-1&-1&1&-1&-1&-1\\\ 1&1&1&1&-1&1&-1&-1\\\
1&1&1&1&1&-1&1&1\\\ \end{array}\right)$

$W^{A}_{7}=\left(\begin{array}[]{rrrrrrrr}1&-1&-1&1&-1&1&-1&-1\\\
1&-1&-1&1&1&-1&1&1\\\ 1&-1&1&-1&-1&1&1&1\\\ 1&-1&1&-1&1&-1&-1&-1\\\
1&1&-1&-1&-1&-1&-1&1\\\ 1&1&-1&-1&1&1&1&-1\\\ 1&1&1&1&-1&-1&1&-1\\\
1&1&1&1&1&1&-1&1\\\ \end{array}\right)$

$W^{A}_{8}=\left(\begin{array}[]{rrrrrrrr}1&-1&-1&1&-1&1&1&1\\\
1&-1&-1&1&1&-1&-1&-1\\\ 1&-1&1&-1&-1&1&-1&-1\\\ 1&-1&1&-1&1&-1&1&1\\\
1&1&-1&-1&-1&-1&1&-1\\\ 1&1&-1&-1&1&1&-1&1\\\ 1&1&1&1&-1&-1&-1&1\\\
1&1&1&1&1&1&1&-1\\\ \end{array}\right).$

For Bob, we also have a maximal set of 8 MQUWM(8,8,16,4), with a different
ordering of vectors:

$W^{B}_{1}=\left(\begin{array}[]{rrrrrrrr}-1&1&1&1&1&1&1&1\\\
1&-1&-1&1&-1&1&1&1\\\ 1&1&1&1&-1&-1&-1&1\\\ 1&1&-1&-1&1&1&-1&1\\\
1&-1&1&-1&1&-1&1&1\\\ 1&1&1&-1&-1&1&1&-1\\\ 1&1&-1&1&1&-1&1&-1\\\
1&-1&1&1&1&1&-1&-1\\\ \end{array}\right)$
$W^{B}_{2}=\left(\begin{array}[]{rrrrrrrr}-1&1&-1&1&-1&1&1&1\\\
-1&-1&1&1&1&1&-1&1\\\ 1&1&1&1&-1&1&-1&-1\\\ 1&1&-1&1&1&-1&-1&1\\\
-1&1&1&1&1&-1&1&-1\\\ 1&-1&1&1&-1&-1&1&1\\\ 1&1&1&-1&1&1&1&1\\\
1&-1&-1&1&1&1&1&-1\\\ \end{array}\right)$
$W^{B}_{3}=\left(\begin{array}[]{rrrrrrrr}-1&-1&1&1&-1&1&1&1\\\
-1&1&-1&1&1&-1&1&1\\\ -1&1&1&-1&1&1&-1&1\\\ 1&1&-1&1&-1&1&-1&1\\\
1&1&1&-1&-1&-1&1&1\\\ 1&-1&1&1&1&-1&-1&1\\\ 1&1&1&1&1&1&1&-1\\\
1&-1&-1&-1&1&1&1&1\\\ \end{array}\right)$
$W^{B}_{4}=\left(\begin{array}[]{rrrrrrrr}-1&1&1&-1&-1&1&1&1\\\
1&-1&1&1&-1&1&-1&1\\\ -1&1&1&1&1&1&-1&-1\\\ 1&1&-1&1&1&1&1&1\\\
1&1&1&1&-1&-1&1&-1\\\ -1&-1&1&1&1&-1&1&1\\\ 1&-1&1&-1&1&1&1&-1\\\
1&1&1&-1&1&-1&-1&1\\\ \end{array}\right)$
$W^{B}_{5}=\left(\begin{array}[]{rrrrrrrr}-1&1&1&1&-1&1&1&-1\\\
-1&-1&-1&1&1&1&1&1\\\ 1&1&-1&1&-1&-1&1&1\\\ 1&1&1&1&1&1&-1&1\\\
-1&1&1&-1&1&-1&1&1\\\ 1&-1&1&-1&-1&1&1&1\\\ 1&-1&1&1&1&-1&1&-1\\\
1&1&-1&-1&1&1&1&-1\\\ \end{array}\right)$
$W^{B}_{6}=\left(\begin{array}[]{rrrrrrrr}-1&1&1&1&-1&-1&1&1\\\
1&1&-1&1&-1&1&1&-1\\\ 1&1&1&-1&-1&1&-1&1\\\ -1&1&-1&1&1&1&-1&1\\\
1&-1&1&1&1&1&1&1\\\ -1&1&1&-1&1&1&1&-1\\\ 1&1&-1&-1&1&-1&1&1\\\
1&1&1&1&1&-1&-1&-1\\\ \end{array}\right)$
$W^{B}_{7}=\left(\begin{array}[]{rrrrrrrr}-1&1&1&1&-1&1&-1&1\\\
-1&1&-1&1&1&1&1&-1\\\ 1&1&-1&-1&-1&1&1&1\\\ 1&-1&-1&1&1&1&-1&1\\\
-1&-1&1&-1&1&1&1&1\\\ 1&-1&1&1&-1&1&1&-1\\\ 1&1&1&1&1&-1&1&1\\\
1&1&1&-1&1&1&-1&-1\\\ \end{array}\right)$
$W^{B}_{8}=\left(\begin{array}[]{rrrrrrrr}1&1&1&1&-1&1&1&1\\\
-1&1&1&1&1&-1&-1&1\\\ -1&1&-1&-1&1&1&1&1\\\ 1&1&-1&1&1&1&-1&-1\\\
-1&-1&1&1&1&1&1&-1\\\ 1&-1&-1&1&1&-1&1&1\\\ 1&1&1&-1&1&-1&1&-1\\\
1&-1&1&-1&1&1&-1&1\\\ \end{array}\right)$

Let us now see how these optimal LHV strategies of Alice and Bob are
connected. The $i^{\rm th}$ element of the $j^{\rm th}$ basis $W^{A}_{j}$ is
connected with the $k^{\rm th}$ element of the $l^{\rm th}$ basis $W^{B}_{l}$
through the following ordering $(i,j,k,l)$:

$\displaystyle(1,1,1,1),(2,1,1,8),(3,1,4,4),(4,1,4,5),(5,1,5,6),$
$\displaystyle(6,1,7,7),(7,1,7,2),(8,1,7,3),(1,2,1,2),(2,2,1,7),$
$\displaystyle(3,2,4,6),(4,2,4,3),(5,2,5,1),(6,2,7,5),(7,2,7,4),$
$\displaystyle(8,2,7,8),(1,3,1,3),(2,3,1,6),(3,3,4,1),(4,3,4,8),$
$\displaystyle(5,3,6,4),(6,3,6,2),(7,3,8,5),(8,3,8,7),(1,4,1,4),$
$\displaystyle(2,4,1,5),(3,4,4,7),(4,4,4,2),(5,4,6,8),(6,4,6,3),$
$\displaystyle(7,4,6,6),(8,4,6,1),(1,5,2,5),(2,5,3,1),(3,5,3,8),$
$\displaystyle(4,5,3,2),(5,5,5,7),(6,5,5,4),(7,5,8,3),(8,5,8,6),$
$\displaystyle(1,6,2,3),(2,6,2,4),(3,6,2,7),(4,6,3,6),(5,6,6,5),$
$\displaystyle(6,6,5,2),(7,6,8,8),(8,6,7,1),(1,7,2,2),(2,7,3,5),$
$\displaystyle(3,7,3,7),(4,7,3,4),(5,7,5,8),(6,7,5,3),(7,7,7,6),$
$\displaystyle(8,7,8,1),(1,8,2,1),(2,8,2,8),(3,8,3,3),(4,8,2,6),$
$\displaystyle(5,8,5,5),(6,8,6,7),(7,8,8,2),(8,8,8,4).$

Regarding higher order Hadamard matrices, we studied tightness without
analyzing their geometrical structure, due to the large number of classical
strategies achieving the LHV value.

Case $\mathbf{m=12}$ settings:

For order 12, there is an unique Hadamard matrix, see the catalog of Hadamard
matrices by J. Seberry [64]. In this case, we find a tight Bell inequality,
composed by 12 measurement settings per party and 2 outcomes each.

Case $\mathbf{m=16}$ settings:

We noted that the three constant row sum Hadamard matrices of order 16,
denoted $H_{1},H_{2}$ and $H_{3}$ [64], do not imply tight Bell inequalities,
for two outcomes. This is consistent with the fact that correlation tight Bell
inequalities always have a quantum violation [65]. The remaining two non-
constant row sum Hadamard matrices, $H_{4}$ and $H_{5}$, lead to a quantum
violation and produce inequivalent tight Bell inequalities, composed by 16
measurement settings per party and 2 outcomes each.

Case $\mathbf{m=20}$ settings:

For order $m=20$, we found that the three Hadamard matrices $H_{1}$, $H_{2}$,
$H_{3}$ [64] induce inequivalent tight Bell inequalities, composed by 20
measurement settings per party and 2 outcomes each.

The above analysis include the entire set of classes of Hadamard matrices
existing up to order $m=20$. Recall that two equivalent Hadamard matrices
imply essentially the same Bell inequality, up to a relabeling of inputs and
outputs. See Table 1 and Conjecture 1 for a summary of these results.

##  Appendix D Bell inequalities for any $q\geqslant 2$

In this section, we illustrate our method by presenting an infinite family of
Bell inequalities with $m=q$ settings and $q$ outcomes, for which we find its
LHV for any $q\geq 2$. The family is characterized by the following matrix of
order $q^{2}$:

$M_{ms+x,mt+y}=\omega^{xt-sy},$ (28)

where $0\leq s,t\leq q$, $0\leq x,y\leq m$ and $\omega=e^{2\pi i/q}$. This
hermitian version of the Fourier matrix -only existing for square orders- has
been used to study complex equiangular tight frames [66, 67, 68]. The first
observation is that (28) naturally satisfies the required symmetry (9), thus
implying a Bell inequality (7).

Let us now show that this family of Bell inequalities satisfy
$\mathcal{C}=q^{3}$, for every $q\geq 2$. To do so, let $a_{x}=\omega^{x}$ and
$b_{y}=\omega^{-y}$ be LHV strategies for Alice and Bob, respectively, where
$0\leqslant x,y\leqslant m-1$. Thus, we have

$\sum_{x,y,s,t=0}^{q-1}F_{qs+x,qt+y}a_{x}^{s}b_{y}^{t}=\sum_{x,y,s,t=0}^{q-1}\omega^{x(s+t)-y(s+t)}=q^{3},$

which saturates the upper bound (15). Therefore, the chosen LHV strategy is a
solution to the optimization problem (10), implying that $\mathcal{C}=q^{3}$.

On the one hand, the value $q^{3}$ is also the upper bound for the quantum
value (14), implying that this family of Bell inequalities does not have
quantum advantage for any $q\geq 2$. On the other hand, by using the inverse
of (8) it is simple to see that these Bell inequalities coincide with the
bipartite version of the _Guess Your Neighbour Input game_ [69]:

$\sum_{a,b=0}^{q-1}P(a,b|b,a)\leq q.$ (29)

Inequalities (29) satisfy $\mathcal{C}=\mathcal{Q}=\mathcal{NS}$ for every
$q\geq 2$, where $\mathcal{NS}$ is the maximal value allowed in no-signaling
theories. This implies that these inequalities are not physically interesting,
in the sense that they do not define a border between two kind of
correlations. Nonetheless, it is interesting the fact that these inequalities
are related to an hermitian version of the Fourier transform. We leave as an
open problem to find further Butson type complex Hadamard matrices,
inequivalent to the Fourier matrix of order $q^{2}$, such that they satisfy
the symmetry (9). Finally, note that any real Hadamard matrix of order $mq$
satisfies the symmetry (9), for $q=2$.

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Theory of sounds in He - II

Christian Fronsdal

Bhaumik Institute, Department of Physics and Astronomy

Unversity of California Los Angeles CA USA

A dynamical model for Landau’s original approach to superfluid Helium is
presented, with two velocities but only one mass density. Second sound is an
adiabatic perturbation that involves the temperature and the roton, aka the
notoph. The action incorporates all the conservation laws, including the
equation of continuity. With only 4 canonical variables it has a higher power
of prediction than Landau’s later, more complicated model, with its 8 degrees
of freedom. The roton is identified with the massless notoph. This theory
gives a very satisfactory account of second and fourth sounds.

Second sound is an adiabatic oscillation of the temperature and both vector
fields, with no net material motion. Fourth sound involves the roton, the
temperature and the density.

With the experimental confirmation of gravitational waves the relations
between Hydrodynamics and Relativity and particle physics have become more
clear, and urgent. The appearance of the Newtonian potential in irrotational
hydrodynamics comes directly from Einstein’s equations for the metric. The
density factor $\rho$ is essential; it is time to acknowledge the role that it
plays in particle theory.

To complete the 2-vector theory we include the massless roton mode. Although
this mode too is affected by the mass density, it turns out that the wave
function of the unique notoph propagating mode ${\cal N}$ satisfies the normal
massless wave equation $\mathchoice{\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt} \hrule height=0.4pt}}}{\vbox{\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt,height=4.0pt\kern 4.0pt\vrule
width=0.4pt} \hrule height=0.4pt}}}{\vbox{\vbox{\hrule
height=0.3pt\hbox{\vrule width=0.3pt,height=3.0pt\kern 3.0pt\vrule
width=0.3pt} \hrule height=0.3pt}}}{\vbox{\vbox{\hrule
height=0.3pt\hbox{\vrule width=0.3pt,height=3.0pt\kern 3.0pt\vrule
width=0.3pt} \hrule height=0.3pt}}}{\cal N}=0$; the roton propagates as a free
particle in the bulk of the superfluid without meeting resistance. In this
circumstance we may have discovered the mechanism that lies behind the flow of
He-II through very thin pores.

email<EMAIL_ADDRESS>webpage fronsdal.physics.ucla.edu

Table

I. Introduction

II. The classical action principles

Hydrodynamics. Thermodynamics. Gauge theory. Speeds of sound.

III. Dynamics of first and second sounds.

First sound. Second sound. Interpretation.

IV. Fourth sound.

V. What comes nex?

I. Introduction

The classical action for adiabatic hydro-thermo-dynamics of irrotational fluid
flows allows for the well known calculation of the speed of sound, understood
as an oscillation of the mass density and the velocity potential at fixed,
uniform entropy (Laplace 1825) [1]. Some fluids transmit a second type of
“sound” that has been interpreted as an oscillation of entropy and temperature
at fixed pressure (Tisza 1938) [2]. Experiments have confirmed that the
temperature is oscillating (Peshkov 1946) [3] and that the pressure is only
weakly involved.

This paper presents an alternative interpretation of second and fourth sounds,
within Landau’s 2 - flow theory [4] of phonons and rotons, as an adiabatic
oscillation of the temperature and the dynamical roton mode, with fixed
density and entropy. The theory is an application of adiabatic thermodynamics,
formulated as an action principle.

The dynamics of the roton field (${\dot{\vec{X}}}$) was identified with the
notoph (Rasetti and Regge 1972) [5], providing the link to Special Relativity
and Quantum Theory that is needed in any mature, physical field theory.

A 2-form gauge field $Y$ is related to ${\dot{\vec{X}}}$ by
$Y_{ij}=\epsilon_{ijk}X^{k}$. The dynamical roton is the massless field [7]

${\cal N}=\rho(\vec{\bigtriangledown}\cdot\vec{X}+{\rm const}).$ $None$

The principal new discovery that is reported here is that second sound is an
adiabatic oscillation of the temperature and ${\cal N}$.

Section II is a brief introduction to the ideas that have led to a dynamical
formulation of Landau’s theory. The speed of second sound is calculated in
Section III and fourth sound is tackled in Section IV.

II. The classical action principles

Hydrodynamics

The essence of classical hydrodynamics is expressed by two equations, the
equation of continuity,

$\dot{\rho}+\vec{\bigtriangledown}\cdot\rho\vec{v}=0$ $None$

and the Bernoulli equation

${\partial\over\partial
t}\vec{v}=-\vec{\bigtriangledown}\vec{v}^{2}/2-{1\over\rho}\vec{\bigtriangledown}p-\vec{\bigtriangledown}\varphi.$
$None$

Here $\rho$ is the (mass) density and $p$ is the pressure. It applies only to
irrotational flows, when the velocity takes the form
$\vec{v}=-\vec{\bigtriangledown}\Phi.$ The two equations of motion are the
Euler - Lagrange equations of a classical action principle. The field
$\varphi$ is the Newtonian potential.111This theory is what remains of a
relativistic theory when the relativistic scalar $\psi$ is expanded as
$\psi=c^{2}t+\Phi+O(1/c^{2})$ and, $g_{00}=c^{2}+2\phi+O(1/c^{2})$ and the
other components are Lorenzian.

For some of the most elementary flows another branch of hydrodynamics must be
invoked. In a popular, didactic experiment a glass of water is placed on a
turntable. After some time the water is seen to be turning with the glass like
a solid body, the surface rising towards the edge to form a meniscus. In the
theory that is used to explain this phenomenon the velocity is a time
derivative, ${\dot{\vec{X}}}$, and the ‘Bernoulli equation’ takes a different
form,

$\vec{\bigtriangledown}{\dot{\vec{X}}}^{2}/2-\vec{\bigtriangledown}\varphi-{1\over\rho}\vec{\bigtriangledown}p=0.$
$None$

This theory, by itself, is not an alternative to the irrotational theory. It
does not have an equation of continuity; instead ${\dot{\vec{X}}}$ is subject
to constraints, as expected of a vector field. The inclusion of the Newtonian
potential in this equation is ad hoc, it can not be justified by an
application of General Relativity and the vector field ${\dot{\vec{X}}}$ is
not affected by the transformations of the Galilei group. In conclusion, we
need both types of vector fields to explain some of the simplest experiments.

The study of lementary applications like these are incontrovertible evidence
that two kinds of flow are needed in hydrodynamis. A satisfactory description
of the waterglass - on - turntable and the whorls seen in the wake of ships
was proposed by Onsager. (Onsager 1962) [6].

Both theories can be expressed as action principles; the irrotational
Lagrangian density is

${\cal
L}_{1}[\rho,\Phi,\varphi]=\rho(\dot{\Phi}-\vec{\bigtriangledown}\Phi^{2}/2-\varphi)-W_{1}[\rho]$
$None$

and Eq. (2.3) - without $\varphi$ \- is the Euler-Lagrange equation of

${\cal
L}_{2}[\rho,\vec{X}]=\rho(\vec{\bigtriangledown}{\dot{\vec{X}}}^{2}/2)-W_{2}[\rho].$
$None$

The density factor is traditional in ${\cal L}_{1}$, less so in ${\cal
L}_{2}$; its appearance in both is crucial. 222 That compressibilty of air is
what makes flight possible was understood by Leonardo da Vinci in the 15th
century.

Current hydrodynamics results from adding (2.4) and (2.5),

${\cal L}_{Hydro}[\rho.\Phi,\vec{X}]={\cal L}_{1}[\rho,\Phi]+{\cal
L}_{2}[\rho,\vec{X}]+{\kappa\rho\over 2}d\psi dY.$ $None$

The last term will be explained below.

The idea of two independent vector fields was already introduced by Landau [4]
in his theory of superfluid Helium, his phonon and roton velocities fields are
$-\vec{\bigtriangledown}\Phi$ and ${\dot{\vec{X}}}$.

The classical theory of ordinary sound is derived from the Lagrangian (2.6),

${\cal L}[\rho.\Phi,\vec{X}]=\rho(\dot{\Phi}-K-\varphi)-W[\rho],)$ $None$

with the kinetic potential

$K=(\vec{\bigtriangledown}\phi)^{2}/2-(\vec{\bigtriangledown}{\dot{\vec{X}}})^{2}/2-\kappa{\dot{\vec{X}}}\cdot\vec{\bigtriangledown}\Phi.$
$None$

This Lagrangian is invariant under the transformations of the Galilei group.
(The field $\vec{X}$ is inert, up to a change of gauge.) The flow
$\rho(\kappa{\dot{\vec{X}}}-\vec{\bigtriangledown}\Phi)$ is identified by the
property of being conserved, as expressed by equation of continuity, derived
from the Lagrangian by variation of $\Phi$.

As we shall show, this is a suitable action for Landau’s phonons and rotons
and a wide range of other applications of adiabatic hydro-thermo-dynamics. In
the literature inspired by Landau’s work on superfluids one finds that the
applications make little use of roton dynamics; instead the field
${\dot{\vec{X}}}$ is more or less fixed. It is, therefore, not surprising to
find that the theory, in its original, non - relativistic context, is
characterized by strong constraints, as has been revealed by completion of the
theory (Rasetti and Regge 1972). This is what brings the number of independent
variables of hydrodynamics down to just 4.

The completed roton theory is a relativistic gauge theory. Like
electrodynamics, it was completed with its development as a quantized gauge
theory (Ogievetskij and Polubarinov 1963) [7] and Green, Schwartz and Witten
(1987) [9]. Both relativity and quantum theory are needed for the formulation
of unitarity. We return to this topic below.

Both ${\cal L}_{1}$ and ${\cal L}_{2}$ are non-relativistic limits of
relativistic field theories; the former is a limit of

${1\over 2}\rho(g^{\mu\nu}\psi_{,\mu}\psi_{,\nu}-c^{2})-W[\rho].$ $None$

The non-relativistic limit includes the Newtonian potential defined by

$g_{00}=c^{2}t+2\varphi+O(1/c^{2}),~{}~{}~{}\psi=c^{2}+\Phi+O(1/c^{2})$ $None$

This is the origin of the Newtonian potential in Eq.(2.4) [9]. Its appearence
in (2.5) cannot be justified by General Relativity.

Thermodynamics

The use of a variational principle for (adiabatic) thermodynamics is not often
seen in the literature. There follows a resumé that shows that the basic
equations are the Euler-Lagrange equations of a simple action. This
reformulation of adiabatic thermodynamics contains nothing that is unfamiliar.
What it does is to set the limits of the applications; 333 It is evidently
incompatible with an oscillating entropy.  it fixes the Hamiltonian, the
kinetic potential and the angular momentum and it puts us in a better position
to confront new applications, such as gravitational waves [11] and the speed
of second sound.

The equations that define adiabatic thermodynamics of a uniform system at rest
are

${\partial F(T,V)\over\partial V}+P=0,~{}~{}~{}{\partial
F(T,\rho)\over\partial T}+S=0,$ $None$

where $V$ is the volume, $F$ is the Helmholtz free energy and $P$ is the
pressure. We prefer to formulate the theory in terms of densities,

$s=\rho S,~{}~{}~{}f(T,\rho)=\rho F(T,V).$

Following Callen we set the local version of Eq.s (2.11)

$\rho{\partial f\over\partial\rho}-f=p,~{}~{}~{}{\partial f\over\partial
T}+s=0.$

Consider the action

$A_{1}[\Phi,\rho,T,S,P]=\int dt\bigg{(}\int_{\Sigma}{\cal
L}_{1}-\int_{\partial\Sigma}P\bigg{)}.$ $None$

Here $P$ is the 3-form of pressure on the multifaceted boundary.

The Lagrangian density is

${\cal
L}_{1}=\rho(\dot{\Phi}-\vec{\bigtriangledown}\Phi^{2}/2-\varphi)-f(T,\rho)-sT.$
$None$

Assume that the specific entropy density $S$ is fixed, constant and uniform.
Vary this Lagrangian with respect to local variations of $\rho$ and $T$, with
$S$, $P$ and - temporarily - the boundary $\partial\Sigma$, fixed, then the
Euler - Lagrange equations are as follows.

Variation of $A_{1}$ with respect to $\Phi$ gives the equation of continuity,
with $\vec{v}=-\vec{\bigtriangledown}\Phi$;

$\dot{\rho}+\vec{\bigtriangledown}\cdot(\rho\vec{v})=0,$ $None$

Variation with respect to $T$ gives the adiabatic relation:

${\partial\over\partial T}f+s=0;$ $None$

it can be used to eliminate the temperature.

Theorem. When $s=\rho S$, $S$ fixed, constant and uniform, then

$\vec{\bigtriangledown}{\partial\over\partial\rho}(f+sT)={1\over\rho}\vec{\bigtriangledown}p.$
$None$

Local variation of ${\cal L}_{1}$, Eq.(2.13), by $\rho$, followed by the
elimination of $T$, leads to the Bernoulli equation in the original form, Eq.
(2.2).

$\vec{\bigtriangledown}{\dot{\Phi}}-\vec{\bigtriangledown}(\vec{\bigtriangledown}\Phi)^{2}/2-\vec{\bigtriangledown}\varphi-{1\over\rho}\vec{\bigtriangledown}p=0.$

There is a proof in Fronsdal (2020) [10].

Finally, variation of the boundary gives

${\cal L}_{1}|_{\partial\Sigma}=P.$ $None$

On-shell, on the boundary,

$p=\rho{\partial f\over\partial\rho}-f={\cal L}_{1}-\rho{\partial{\cal
L}_{1}\over\partial\rho}=P.$ $None$

The first equality agrees with the first of Eq.s (2.11), but since it is taken
to hold in a wider context it may be regarded as a definition; the second is a
consequence of the fact that $-f$ is the only term in the Lagrangian density
that is not linear in $\rho$. The last equality confirms the identification of
$p$ as the pressure, an extrapolation of $P$ from the boundary to the
interior.

We shall replace ${\cal L}_{1}$ by ${\cal L}_{1}+{\cal L}_{2}$, as in Eq.
(2.6).

Gauge theory

The gauge theory behind the roton field ${\dot{\vec{X}}}$ was discovered by
Rasetti and Regge [5]. It is the theory of a massless 2-form, components
$(Y_{\mu\nu})$. The free action density is $\rho dY^{2}$. For hydrodynamics
the complete action density is

${\cal L}_{2}[\rho,Y]=\sqrt{-g}{c^{2}\over 12}\rho\,dY^{2}+{\kappa\over
2}\rho\,dYd\psi;$ $None$

The 2-form $Y_{\mu\nu}$ is related to $\vec{X}$ and $\psi$ is related to
$\Phi$ by (2.10). The non relativistic Lagrangian in Eq. (2.12) is derived
from ${\cal L}_{1}$ in Eq. (2.9) and ${\cal L}_{2}$ is a limit of ${\cal
L}[\rho,Y]$ in (2.19).

1\. The field $\psi$ is a scalar field with a vacuum expectation value,
$\psi=\Phi+c^{2}t$. The field $\Phi$ transforms, together with the velocity
$-\vec{\bigtriangledown}\Phi$, under the Galilei group, in the usual way,
making ${\cal L}_{1}$ invariant under this group.

2\. The components of the 2-form are

$Y_{ij}=\epsilon_{ijk}X^{k},~{}~{}~{}Y_{0i}=:\eta_{i}.$ $None$

The vector field $\vec{\eta}$ is a gauge field, variation of the action with
respect to $\vec{\eta}$ gives the constraint - the gauge condition -

$\vec{\bigtriangledown}\wedge\vec{m}=0,~{}~{}~{}\vec{m}:=\rho({\dot{\vec{X}}}+\kappa\vec{\bigtriangledown}\Phi),$
$None$

with the general solution

$\vec{m}=-\vec{\bigtriangledown}\tau.$

A special choice for the gauge parameter $\tau$ is required for a massless
mode to be recognized. This mode is

${\cal N}:=\rho(\vec{\bigtriangledown}\cdot\vec{X}+\kappa).$ $None$

It is the only propagating field of this gauge theory. The free field equation
is (Ref.s [11]).

$\mathchoice{\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt} \hrule
height=0.4pt}}}{\vbox{\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt,height=4.0pt\kern 4.0pt\vrule width=0.4pt} \hrule
height=0.4pt}}}{\vbox{\vbox{\hrule height=0.3pt\hbox{\vrule
width=0.3pt,height=3.0pt\kern 3.0pt\vrule width=0.3pt} \hrule
height=0.3pt}}}{\vbox{\vbox{\hrule height=0.3pt\hbox{\vrule
width=0.3pt,height=3.0pt\kern 3.0pt\vrule width=0.3pt} \hrule
height=0.3pt}}}\,{\cal N}=0.$ $None$

Remark. Non-relativistic electrodynamics, a limit when the speed of light -
$c$ \- tends to infinity, makes sense only in the absence of the magnetic
field. And non-relativistic hydrodynamics is the regime where ${\cal N}=0$,
for this field, like $\vec{B}$, enters the Lagrangian and the equations of
motion multiplied by $c^{2}$. Consequently, any study of a steady
configuration will be one in which ${\cal N}$ is negligible.

III. Dynamics of first and second sound

First sound

The classical theory of (first) sound propagation rests on the Eulerian
theory, with the Lagrangian density ${\cal L}_{1}$ and the two equations of
motion (2.1) and (2.2). The speed of propagation is expressed in terms of the
adiabatic derivative of the pressure

$C_{1}=\sqrt{dp(\rho,S)\over d\rho}.$ $None$

This equation is used in the quoted sources (Arp et al [12], Brooks and
Donnelly [13], Maynard [14] ) to determine the equation of state. We shall
obtain similar formulas for second and fourth sounds.

First sound is an oscillation of the density and the velocity potential,
$\rho$ and $\Phi$, and $S$, fixed. The speed of first (ordinary) sound is
usually calculated for a plane wave, a first order perturbation of a static
configuration with uniform density. The two Euler - Lagrange equations are

$\dot{\rho}-\vec{\bigtriangledown}\cdot\rho\vec{\bigtriangledown}\Phi=0,~{}~{}~{}~{}\dot{\Phi}-{\partial(f+sT)\over\partial\rho}\bigg{|}_{T}=0.$
$None$

Eliminating $T$, or using Eq. (2.16) and differentiations leads, in first
order perturbation theory, to

$\ddot{\rho}=\rho\vec{\bigtriangledown}\cdot\dot{\vec{v}}(\rho,S),~{}~{}~{}~{}~{}~{}~{}~{}\vec{\bigtriangledown}\cdot\dot{\vec{v}}={1\over\rho}{\partial
p(\rho,S)\over\partial\rho}\bigg{|}_{S}\Delta\rho$ $None$

and to (3.1). Note: in this case $\vec{v}=-\vec{\bigtriangledown}\Phi$, since
${\dot{\vec{X}}}=0$.

Second sound

Since the notoph is a massless particle, we add a Stefan-Boltzmann term to the
enternal energy density:

$u(T,\rho)\to\tilde{u}(T,\rho,{\cal N})=u+{\alpha\over k}T^{k}\
N.~{}~{}~{}{\cal N}:=\rho(\vec{\bigtriangledown}\cdot{\dot{\vec{X}}}+\kappa).$
$None$

Let $(T_{0},\rho_{0},\vec{X}_{0},{\cal N}_{0})$ be a stationary solution of
the Euler - Lagrange equations for the Lagrangian

${\cal L}=\rho(\dot{\Phi}-K)-f-sT-{\alpha\over k}{\cal N}T^{k}$
$K=-\beta{\dot{\vec{X}}}^{2}/2+\vec{v}^{2}/2,~{}~{}~{}~{}\beta=1+\kappa^{2};$
$None$

this is an alternative expression for Eq.(2.6) and
$\rho\vec{v}=\rho(\kappa{\dot{\vec{X}}}-\vec{\bigtriangledown}\Phi$ is the
conserved current.

Second sound is a first order, adiabatic deformation

$(T_{0},{\dot{\vec{X}}}_{0})\to(T_{0}+dT,\vec{X}_{0}+d\vec{X}),$

with ${\cal N}_{0}=0,\vec{v}_{0}=0$ and $dp=0,d\rho=0,d\vec{v}=0$.

To the experimenter second sound is excited by a forced oscillation of the
temperature at the boundary and that is not registered by a pressure sensitive
microphone, hence $dp\approx 0$.

We must review the equations that govern these oscillations.

1\. The relevant part of the internal energy density is

$-f-sT-{\alpha\over k}{\cal N}T^{k}+\rho\beta{\dot{\vec{X}}}^{2}/2$

In adiabatic thermodynamics, for any fixed value of S, the theory is an
isolated Lagrangian action principle and $u$ is the Hamiltonian density. For
any adiabatic variation the integrated internal energy is at a minimum.

Variation with respect to $T$ and $\vec{X}$, $\rho$ fixed and uniform, gives
the two Euler - Lagrange equations. From variation of $T$:

$\int dT{\partial(\tilde{f}\ +sT)\over dT}\bigg{|}_{S,\rho,{\cal N}}=\int
dT\bigg{(}\rho{\partial(\tilde{F}+ST)\over\partial
T}\bigg{|}_{S,\rho}+{\alpha\over k}T^{k}{d{\cal N}\over dT}\bigg{)}$ $None$

In the first order of the perturbation this quantity is zero,

$d(\rho C_{V})-\alpha T^{k-1}d{\cal N}=0.$ $None$

Explanation: The internal energy density is $\tilde{u}$ and this has values
that are measured and that give the recorded values of $C_{V}$; but in the
cited papers ${\cal N}$ does not represent another variable; it is just a
function of $T$, so their $\rho C_{V}$ includes a term
$-(\alpha/k)T^{k}(d{\cal N}/dT)$.

Derivation with respect to the time gives

$\rho{\partial C_{V}\over\partial T}\dot{T}-\alpha T^{k-1}d\dot{\cal N}=0.$

This is valid to first order in perturbation theory if ${\cal N}_{0}=0$, as is
natural under the circumstances. Under the same conditions,

$\rho{\partial C_{V}\over\partial T}\ddot{T}-\alpha T^{k-1}\ddot{\cal N}=0.$
$None$

2\. From variation of $\vec{X}$,

$\rho\beta d{\dot{\vec{X}}}\cdot{\dot{\vec{X}}}-{\alpha\over k}T^{k}d{\cal
N}=0,$

or

$d\vec{X}\cdot(\beta\,d\ddot{\vec{X}}-\alpha
T^{k-1}\vec{\bigtriangledown}T)=0.$ $None$

From (3.7) and the divergence of (3.8) follows that

$\rho{\partial C_{V}\over\partial T}\bigg{|}_{p}\ddot{T}-\alpha
T^{k-1}d\ddot{\cal N}=0,~{}~{}~{}~{}\beta d\ddot{\cal N}-\alpha\rho
T^{k-1}\Delta T=0$ $None$

and the speed $C_{2}$ is

$C_{2}={\alpha\over\sqrt{\beta}}T^{k-1}{\bigg{(}{\partial C_{V}\over\partial
T}\bigg{|}_{p}}\bigg{)}^{-1/2}.$ $None$

The values of $\partial C_{V}/\partial T|_{p}$ will be taken from experimental
data, for $0<p<25MPa$.

Direct comparison with experimental values

Arp et al [12], Brooks and Donnelly [13] and Maynard [14] have collected
results from many experiments. Their results for $C_{V}$ are plotted in Fig.1,
along with our very simple interpolation.444Our interpolation formula was
needed in the lowest interval of temperature only, there was no need for the
elaborate interpolation used by Arp. The logarithmic singularity was placed on
the $\lambda$ line.

Our interpolation for $C_{V}$ is, for $p=0$, $0.4<T<2.4$

$C_{V}=-\ln((2.18-T)^{5/2}+10^{-30})+3.35-4.5T+1.6T^{2},~{}~{}~{}.$ $None$

Units are joules/gram.

Fig.1. The lower part shows values of $C_{V}$ (in joules) determined by
measurements. The solid curve is our simple interpolation of the data. This
interpolation was used to calculate the speed $C_{2}$ of second sound, using
Eq. (3.13), showed for $k=3~{}(units~{}$m/s$)$; it is our prediction for the
speed of second sound in He-II.

The curves $C_{V}(T)$ and $C_{2}(T)$ are shown in Fig. 1. The lowest value of
$C_{V}$ on the interpolation curve is -.033 at $T$ = .8282. The calculation
stops at $T=.7476$, near the point where the experimenters loose their signal
(Williams and Rosenbaum 1979) [15] and peaks at $T$= 2.18046. Only the overall
factor, 17.5 in Eq. (3.16) could be adjusted for a best fit.

Similar fits were obtained for $p=2MPa$ after a small adjustment of the
parameters:

$C_{V}=-\ln((2.165-T)^{2.5}+10^{-30})+4.0-5.85x+2.1x^{2}.$ $None$

The minimum of the interpolation curve is .00389 at $T$ = .700. The curve
begins at $T$ = .8287, and peaks at $T=2.17875$, at the $\lambda$ line.

Calculations have verified similar agreement for pressure 5, 10, 15, 20 and 25
MPa.

Given the experimental data for the values of $C_{V}$, the theory predicts the
speed of second sound to be given up to a multiplicative constant by Eq.
(3.16). These formulas, with $k=3$, give very good fits from the $\lambda$
line down to $T=.8$, where $C_{V}$ has a local minimum and the signal is lost.
Fitting the overall constant factor to the experiment we find that, for $p=0$
and for $p=2$, 555 In the quoted reviews velocities are given in m/sec, energy
densities in joules.  the final result is

$C_{2}={\alpha\over\sqrt{\beta}}~{}T^{2}\bigg{(}{\partial C_{V}\over\partial
T}\bigg{)}^{-1/2},~{}~{}~{}.8<T<T_{\lambda}.$ $None$

with ${\alpha/\sqrt{\beta}}=17.5$ in the units m/s and joules. Here $C_{V}$
was taken from the tables in terms of joules and the velocites were expressed
in terms of $m/s$.

Interpretation

In the term $(\alpha/k)T^{k}{\cal N}$ that we have included in the internal
energy, ${\cal N}$ is the notoph amplitude. The power $k$ in $T^{k}$ was left
open to be determined by measurements. The experimental value is $k=3$. The
factor $T^{3}$ is proportional to the number of quanta predicted by Planck’s
theory. The new term in the internal energy density is thus identified as the
Stefan - Boltzmann term associated with the notoph.

Why include ${\cal N}T^{3}$ instead of $aT^{4}$. The notoph is a new
experience and the wisest course is to accept the value provided by the
experiments, which fixes the value of $K$ at 3..

We have set the parameter ${\cal N}_{0}$ equal to zero. This is because the
experiments were made in vessels of a size such that boundary effects, the
origin of capillary effects, are expected to be weak. The effect of varying
this parameter away from zero is insignificant.

IV. Fourth sound

Fourth sound is observed in thin films and in containers packed with silicon
wafers. The usual interpretation is that the “normal component” remains at
rest; we shall assume that

$\vec{v}_{0}=0,~{}~{}~{}\vec{\bigtriangledown}\Phi_{1}=0,$

that $S$ is fixed, constant and uniform, while $\vec{X},\rho$ and $T$
oscillate together. Of the four equations of motion these three are relevant
for the determination of the speed;

1\. Equation of continuity:

$d\dot{\rho}+\kappa\rho d(\vec{\bigtriangledown}\cdot{\dot{\vec{X}}})=0.$

To zero order in perturbation theory the system is taken to be stationary ,
and both terms are zero. For simplicity we shall replace $\vec{X}$ by ${\cal
N}$ as the independent variable. Consider a plane wave perturbation, then to
first order the equation reduces to

$\dot{\rho}_{1}+\kappa\rho_{1}(\vec{\bigtriangledown}\cdot{\dot{\vec{X}}}_{0})+\kappa\rho_{0}(\vec{\bigtriangledown}\cdot{\dot{\vec{X}}}_{1})=0.$
$None$

It is clearly vital to know something about the zeroth approximation.

In the bulk of the fluid the field ${\dot{\vec{X}}}_{0}$ is stationary and
$(\vec{\bigtriangledown}\cdot{\dot{\vec{X}}}_{0})$ is expected to vanish; in
that case, in first order of perturbation,

$d\rho+\kappa d{\cal N}=\rho_{0},~{}~{}~{}{\rm constant};$

this will allow us to eliminate the density from the Bernoulli equation.

2\. The adiabatic condition:

${\partial(\tilde{f}+sT)\over\partial T}\bigg{|}_{\rho,{\cal
N},S}dT=\rho{\partial(\tilde{F}+ST\over\partial T}\bigg{|}_{S}dT$
$=-{\alpha\over 3}T^{3}{\cal
N}-{\partial(\tilde{f}+sT)\over\partial\rho}\bigg{|}_{\,T{\cal N},S}d\rho$

and using Eq. (2.16):

${\partial(\tilde{f}+sT)\over\partial\rho}\bigg{|}_{T,{\cal
N},S}={1\over\rho}{\partial p\over\partial\rho}$

to get

${\partial(\tilde{f}+sT)\over\partial T}\bigg{|}_{\rho,{\cal
N},S}dT=-{\alpha\over 3}T^{3}d{\cal N}-{\partial p\over\partial\rho}d\rho,$

or

$dC_{V}=(\alpha T^{2}+\kappa C_{1}^{2})d{\cal N}-\rho_{0}C_{1}^{2}$

The time derivatives:

${\partial C_{V}\over\partial T}\dot{T}+\rho_{0}{\partial
C_{1}^{2}\over\partial T}\dot{T}=(\alpha T^{2}+\kappa C_{1}^{2})\dot{N}$

3\. The Bernoulli equation is,

$d\ddot{\vec{X}}=\alpha T^{k-1}\vec{\bigtriangledown}T,$ $None$

or

$\ddot{\cal N}-\alpha T^{2}\Delta T=0,$

and together they give

${C_{4}}^{2}={\ddot{T}\over\Delta T}={\ddot{T}\over\ddot{\cal N}}{\ddot{\cal
N}\over\Delta T}={\bigg{(}{\alpha\over
100}T^{2}+\kappa{C_{1}}^{2}\bigg{)}\over{1\over 10}{\partial
C_{V}\over\partial T}+\rho_{0}{\partial\over\partial T}C_{1}^{2}}{{\alpha
T^{2}\over 100}}.$ $None$

The unit of velocity is here $10^{4}cm/sec,$ and $C_{V}$ is in joules. In the
numerator $\alpha=17.5\beta$ as in the calculation of $C_{2}$ and $1/100$
converts $\alpha$ to the new unit of speed. The factor 1/10 in the denominator
is valid when $C_{V},$ is expressed in joules, as taken from the tables.

Fig.2. The relation (4.4). Red line: Interpolation by the author of
measurements of $C_{4}$. Blue line: Values of $C_{4}$ calculated from
experimental values of $C_{1},C_{2}$ and $C_{V}$ reported in ref.s [13-15].

Fig. 2 summarizes the result for $C_{4}$. The red line is the square of fourth
sound, interpolated from the experimental data [13-15]. It is almost covered
by the blue line, the value given by Eq. (4.4). The fit was made with only one
free parameter, the physical parameter $\kappa$ of the fluid, for the first
time determined experimentally. The vertical coordinate is the square of the
speed of fourth sound in units of $(10^{4}cm/sec)^{2}$.

The value of the parameter $\alpha/\sqrt{\beta}$ was determined in Section
III. That leaves the value of $\kappa$ as the sole free parameter; the value
determined by using the earlier value of $\alpha$ is

$\kappa=.556\pm.0005.$

If instead Eq. (4.4) is used to determine both parameters then
$\alpha/\sqrt{\beta}$ is bracketed between 17.0 and 18.0.

That gives a unique theory, with no free parameters, that can be used to
predict the strength of capillary effects and other properties of He - II.

The analytic interpolations used for $C_{1}$ and $C_{4}$ were

${C_{1}}^{2}=5.63+0.05*[1.2-x]-(0.77*[x-1.2]^{2})$
${C_{4}}^{2}=54000-(50000*[x-1.2]^{2})$

V. What comes next?

This paper reports another application of a version of Landau’s 2-vector
theory of superfluids. It should be pointed out that the alternative idea of
two densities has found no direct experimental support. The number
“$\rho_{s}/\rho_{n}$” is fixed in terms of $\rho,T$, and $p$; it is not an
independent variable. [13]

The need for extra variables, besides a velocity potential, a density and the
temperature, was demonstrated at the end of the 17th century and yet the first
viable suggestion in that direction was Landau’s idea of two velocities, at
first in a very narrow context.

The two ‘versions’ of hydrodynamics date from the beginning. They have been
said to be equivalent, but that is evidently not the case. The rotons are
strongly associated with the socalled ‘Lagrangian version’ of hydrodynamics.
This was pointed out in an important paper by Rasetti and Regge, but that
paper had repercussions in string theory only. [16]

Today we see the roton-notoph identification as the coming-of age of
hydrodynamics, with applications to a large class of fluid phenomena,
including capillary action, flight and gravitational waves. The recently
confirmed unification of General Relativity with Particle Theory has given new
impetus to bringing hydrodynamics into contact with both. We hope that the
present paper will stimulate a more unified approach to these important
branches of physics, by showing that hydrodynamics can be approached by
methods that have been proper to Particle Physics, and profit from it. The
approach assumes a precise model of fluids and makes detailed predictions on
its own, without special adaptations in each special case.

To end where we began, superfluids still pose challenges. The existence of
spin is obvious but details need to be examined. The spectacular ability of
He-II to penetrate very fine pores is probably a manifestation of capillary
phenomenon, related to the properties of the massless notoph, but this needs
to be clarified. Finally, the growing importance of notoph = roton makes it
urgent to discover how to detect it, in the CMB and in the laboratory.

Computation codes are available on request from the author.

Acknowledgements

I thank Gary Williams for discussions and information, and Joe Rudnick for
conversations. I also wish to acknowledge the crucial reference to the paper
[5], by Alexander Zheltukhin. I also thank Chair David Saltzberg for support.

References

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[6] Onsager, L. Private conversation, 1962

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