text:
\subsection*{Quantum computation and information theory}
Right at the beginning, Artur Eckert gave a keynote review of the
present status of quantum computation theory, which was later extended
with an emphasis to technological and experimental aspects by Seth
Lloyd and Jeff Kimble. Boris Pavlov proposed a new quantum switching
device based on resonance scattering.

In another contribution to the conference,
Elena Calude and Marjo Lipponen took up an idea of Edward Moore to
simulate quantum complementarity by finite automata.

Quantum computation appears to be a major emerging field of basic and
applied research. Like DNA computation, the stakes are high but so are
the difficulties and challenges.
In its present state, quantum computing mostly attempts to make use of
quantum
parallelism. In classical terms, a quantum computation may
pursue ``a very large number'' of sometimes contradictory tasks at a
single
computation step. This is due to the fact that, stated pointedly,
because what physicists
call ``coherent superposition'' of (classically contradictory) {\it
yes-no} bit states, quantum information may represent both classically
mutually exclusive bit states at once. These can then be co-processed
simultaneously by applying reversible one-to-one unitary
transformations. In this sense, any classical system in which $n$
classical
bit values are stored and processed
is outperformed by a quantum bit system by an exponential factor $2^n$,
containing all the $2^n$ classical bit values at once.

As can be expected, the exponential speedup comes with a price tag.
There is the so-called problem of
decoherence. Due to what is conventionally referred to as the wave
function collapse by the interaction of quantum bits with their
classically modeled environment, they decay and degenerate
uncontrollably into classical bits. But there seem to be least a partial
remedy. Very clever error correction strategies attempt to protect the
quantum bits. They too may have their price in terms of  an
increased slowdown. Research in this area is fascinating and far from
being completed.

Due to quantum complementarity, the quantum bits themselves are not
completely
measurable. Therefore, any meaningful result needs to be suitably
encoded into interference patterns. For the same reason, Shor's
prominent quantum algorithm \cite{shor:94} for factoring in quantum
polynomial time is probabilistic.

It may be suspected \cite{gottlob}
that,  due to the necessity of reaching reasonable fringe contrast
in the interference patterns, such methods can only be effectively
applied to the class of problems
UP having a unique solution; or to problems having ``close to unique''
solutions insofar as the number of actual solutions increases ``much
slower''
(e.g., polynomial) than the possible solutions.
Personally, I  confess that, having been deeply impressed by the
so called ``interaction-free schemes of measurement''
\cite{elitzur-vaidman:1,kwhz:95,benn:94}, I am more inclined to consider
``free lunch scenarios'' in quantum information and computation theory
as before.

Both Cristian Calude and Artur Eckert mentioned possible consequences
for proof theory. Additional attention, I believe,
should be given  to a revision
of recursive function theory due to the possibility to represent
classically contradicting bit states by a quantum bit which is in a
coherent
superposition thereof. Any diagonalization argument, for example,
includes a {\it not}-operation which results in the switch of one
classical bit state into the other. Quantum mechanically, this operation
has a fixed point quantum bit state which is a fifty-fifty
coherent mixture of the two classical states. Hence, for instance, the
proof of the recursive undecidability of the halting problem collapses,
because the {\it reductio ad absurdum} does no longer work---the
solution is
a fixed point quantum bit state; equivalent to the flipping of a
fair coin.

DNA-based computation has been developed extensively in the
contributions of
Gordon Alford,
Martin Amos,
Alan Gibbons,
Valeria Mihalache,
Gheorge P\u{a}un,
Animesh Ray,
 John Reif, and
Arto Salomaa, both from the theoretical and from the experimental side.
Again, research in that area is in full progress.

The idea here is to encode a problem into strands of DNA, profiting
again
from heavy parallelizing. This parallelizing might be limited by the
population size conceivable.

Another big topic of the conference was reversibility.
Reversible computation is characterized by one-to-one operations, by a
reversible, bijective evolution map of the computer states onto
themselves.
If only a finite number of such states are involved, this
amounts to their permutation.

In such a scheme, not a single bit
gets lost, and any piece of information (including the trash) remains in
the
computer forever. That may be good news for the case of decay and loss
of information, but it is bad news with respect to waste management.
There is no way of trashing garbage bits other than cleverly
compressing them and pile them ``high and deep.''
Stated differently:
in this restricted regime,
many-to-one operations such as deletion of bits are
not allowed.

Furthermore,
one-to-many operations such as copying are forbidden as well.
From this point of view, it may appear appropriate to develop
algorithmic
information theory in terms of prefix
reversible algorithms.

Classical continuum mechanics and electrodynamics are reversible
``at heart;'' that means that all  equations of motion are invariant
with
respect to reversing the arrow of time. Also quantum
computation is based upon unitary evolution of the quantum bit states
and thus is reversible. The
no-copy feature of reversible computation is for instance reflected by
the no-cloning theorem of quantum theory.

The MIT group headed by Thomas Knight presented silicon prototypes of
reversible computers based
on Fredkin's billiard ball model of universal computation, as well as
reversible gate
technology and memory management. The potential technological advantage
of reversible computers over irreversible ones lies in the fact that
reversible computation is not necessarily associated with energy
consumption and heat dissipation while the latter one is.
And since heat dissipation per computation step can be kept at arbitrary
low levels, when ``scaled up to very large sizes'' reversible
computation outperforms irreversible computation. Moreover, after all,
physics at very small scales {\it is} reversible.

We know since Landauer \cite{landauer:61} and Bennett \cite{bennett-82}
that any computation can be embedded into a reversible one. The trick is
to provide markers in order to make back-tracking possible, which amounts
to
memorizing the past states of the system. If no copying is allowed, this
may amount to large space overheads as compared to irreversible
computations. In
one contribution to the conference, Peter Hertling showed that there in
general  exist no better, more clever way to embed an irreversible
Cellular Automaton into reversible ones than doing just that.

In still another variation of the reversibility topic,
the author of this article constructed reversible finite automata via
permutation matrices.

There is one immediate algorithmic aspect of reversibility.
If reversible computation is just a
rephrasing, a permutation of the input, then what use it anyway?
The ``garbage in-garbage out'' metaphor is particularly pressing here.

Let us consider an example. Why should the product of the prime factors
$3\cdot 5$, obtained for instance by Shor's algorithm, be more
``informative'' than the number
15? And what
is the meaning of the term ``informative'' here? This issue seems to be
somewhat related to an old question in proof theory, in which sense a
proof of a statement is ``better'' then the mere knowledge that this
statement is true.
It seems that the term
``informative'' can only be given a subjective, idealistic meaning
devoid of any formal rigor. Because the number 15 would just be a
trivial rephrasing of its product of prime factors $3\cdot 5$, very much
so as its binary expansion 1111. That might eventually be bad news for
standard encryption techniques based on prime factorization. (Of course,
there might be ``a lot of '' reversible computation steps
necessary for this kind of ``trivial rephrasing,'' making
the equivalence ``very hard'' to verify.)

Christopher Moore gave a very interesting overview of analogue
computation. There he stressed the importance of perceiving the process
of computation as a {\em dynamical} one. In this sense, results from the
theory of dynamical systems can be applied to the theory of computation
and {\it vice versa}.
It seems to remain an interesting question if analogue computation,
which
in principle could make use of the infinite complexity of the objects
involved, could outperform Turing machines.

Herbert Wiklicky gave a very interesting framework for practically all
previously
discussed models of computation in terms of linearized systems or
Hilbert machines.

Speaking about unicorns in the little zoo of unconventional computers,
Zeno's argument against motion pops up here and there. In the context of
computations it appears as a model of computation using a Turing machine
with a geometrical increase of execution speed. In this way,
tasks which would need infinite computation time otherwise could be
completed in finite times.

Let me finish with a speculation.
Assume a pessimistic scenario in which we have convinced ourselves that
we cannot make use of the speedups promised by novel models of
computation. Then it might not be unreasonable to assume
that these alleged speedups emerge from an improper representation
of the natural phenomena, from our incorrect theoretical perception of
physical systems rather than from the phenomena themselves. That, of
course, would have far-reaching consequences for physics.


text:
\section{Introduction}
In a recent work, Merca \cite{Merca} extensively studied the function which he called $A(n)$ which is defined to be the difference between the number of cubic partitions of $n$ into an even
numbers of parts and the number of cubic partitions of $n$ into an odd numbers of
parts.  (Cubic partitions were introduced by Chan \cite{Chan1, Chan2} to be integer partitions in which even parts are allowed to appear in two different colors.  Chen introduced these in connection with Ramanujan's cubic continued fraction.)
Merca notes that the generating function for $A(n)$ is given by
\begin{equation*}
    \sum_{n=0}^{\infty} A(n)q^n  = (q;q^2)_{\infty}(q^2;q^4)_{\infty} = \frac{f_1}{f_4}
\end{equation*}
where the $q$--Pochhammer symbol $(a;q)_\infty$ is defined by
$$
(a;q)_\infty = (1-a)(1-aq)(1-aq^2)(1-aq^3)\dots
$$
and $f_a^b$ is defined by
$$
f_a^b = (q^a;q^a)_\infty^b
$$
for positive integers $a$ and any integer $b.$

In \cite{Merca}, Merca proved the following two Ramanujan--like congruences satisfied by $A(n):$
\begin{theorem}[{\cite[Theorem 1.10]{Merca}}]
\label{Merca1}
For all $n \geq 0$,
\begin{align}
A(9n+5) & \equiv 0 \pmod{3}, \label{m1} \\
A(27n+26) & \equiv 0 \pmod{3}.\label{m2}
\end{align}
\label{T1}
\end{theorem}
Merca's proof of these two congruences relies solely on Smoot's {\tt Mathematica} implementation of Radu's algorithm for proving partition congruences (which relies heavily on the machinery of modular forms).  Indeed, Merca's proof of (\ref{m1}) involves finding a generating function for $A(9n+5)$ which is produced by Smoot's {\tt Mathematica} package. In this case, the generating function in question turns out to be a non--trivial linear combination of four ratios of eta products (in essence, ratios of products whose terms involve $q$--Pochhammer symbols as defined above), while the generating function for $A(27n+26)$ contains a dozen such terms.

Our goal in the work below is to provide elementary proofs of Merca's two congruences via classical generating function manipulations and dissections.  We then significantly extend Merca's work on such divisibility properties for $A(n)$ by proving two infinite families of non--nested Ramanujan--like congruences modulo 3 satisfied by $A(n)$ wherein Merca's two congruences above serve as the initial members of each family. Indeed, we will prove the following:

For all $j\geq 0$ and all $n\geq 0,$
$$
A\left(9^{j+1}n + \frac{39\cdot9^j+1}{8}\right) \equiv 0 \pmod{3}
$$
and
$$
A\left(3\cdot 9^{j+1}n + \frac{23\cdot9^{j+1}+1}{8}\right) \equiv 0 \pmod{3}.
$$
In order to accomplish the above goals, we require a few classical tools.  First, we recall Ramanujan's theta functions
\begin{align}
f(a,b) & :=\sum_{n=-\infty}^\infty a^\frac{n(n+1)}{2}b^\frac{n(n-1)}{2}, \mbox{ for } |ab|<1, \nonumber \\
\phi(q) & := f(q,q) = \sum_{n=-\infty}^{\infty} q^{n^2} = \frac{(q^2;q^2)_{\infty}^5}{(q;q)_{\infty}^2(q^4;q^4)_{\infty}^2},
\nonumber
\\
\psi(q) & := f(q,q^3) = \sum_{n=0}^{\infty} q^{n(n+1)/2} = \frac{(q^2;q^2)_{\infty}^2}{(q;q)_{\infty}}.
\nonumber
\end{align}
These functions satisfy many interesting properties (see Entries 18, 19, and 22 in \cite{Berndt}), including:
\begin{align}
\phi(-q) & = \frac{(q;q)_{\infty}^2}{(q^2;q^2)_{\infty}},
\nonumber
\\
\psi(-q) & = \frac{(q;q)_{\infty}(q^4;q^4)_{\infty}}{(q^2;q^2)_{\infty}}.
\nonumber
\end{align}

As noted above, the generating function for $A(n)$ is given by
\begin{align}
    \sum_{n=0}^{\infty} A(n)q^n & = (q;q^2)_{\infty}(q^2;q^4)_{\infty} \nonumber \\
    & = \frac{f_1}{f_4} \nonumber \\
    & = \frac{f_1^2}{f_2}\frac{f_2}{f_1f_4} \nonumber \\
    & = \frac{\phi(-q)}{\psi(-q)}. \label{gf1}
\end{align}

In order to prove the congruences in question, we require a few well--known $q$--series dissections.

\begin{lemma}
We have
\begin{equation*}
    \phi(-q) = \frac{f_9^2}{f_{18}} -2q\frac{f_3f_{18}^2}{f_6f_9}.
\end{equation*}
\label{lemma1}
\end{lemma}

\begin{proof}
A proof of this identity can be seen in \cite[Eq. 14.3.4]{H}.
\end{proof}

\begin{lemma}
We have
\begin{equation*}
\frac{1}{\psi(-q)} = \frac{f_{18}^9}{f_3^2f_9^3f_{12}^2f_{36}^3} + q\frac{f_6^2f_{18}^3}{f_3^3f_{12}^3} + q^2\frac{f_6^4f_9^3f_{36}^3}{f_3^4f_{12}^4f_{18}^3}.
\end{equation*}
\label{lemma2}
\end{lemma}

\begin{proof}
A proof of this identity appears in \cite[Lemma 2.1]{Toh}.
\end{proof}

\begin{lemma}
We have
\begin{equation*}
\frac{1}{\phi(-q)} = \frac{f_6^4f_{9}^6}{f_3^8f_{18}^3} + 2q\frac{f_6^3f_{9}^3}{f_3^7} + 4q^2\frac{f_6^2f_{18}^3}{f_3^6}.
\end{equation*}
\label{lemma3}
\end{lemma}

\begin{proof}
A proof of this result can be seen in \cite[Theorem 1]{HS}.
\end{proof}

\begin{lemma}
We have
\begin{equation*}
\frac{1}{\psi(q)} = \frac{f_3^2f_{9}^3}{f_6^6} - q\frac{f_3^3f_{18}^3}{f_6^7} + q^2\frac{f_3^4f_{18}^6}{f_6^8f_9^3}.
\end{equation*}
\label{lemma4}
\end{lemma}

\begin{proof}
A proof of this result can be seen in \cite[Lemma 2.2]{HS1}.
\end{proof}

These are all of the tools that we need in order to prove Merca's congruences in an elementary fashion.  We now transition to providing these proofs.

\section{Elementary proof of Theorem \ref{T1}}

Initially, we use Lemmas \ref{lemma1} and \ref{lemma2} to extract the terms involving $q^{3n+2}$ in (\ref{gf1}):
\begin{equation*}
\sum_{n=0}^{\infty} A(3n+2)q^{3n+2} = q^2 \frac{f_6^4f_9^5f_{36}^3}{f_3^4f_{12}^4f_{18}^4} - 2q^2\frac{f_6f_{18}^5}{f_3^2f_9f_{12}^3}.
\end{equation*}
Dividing by $q^2$ and replacing $q^3$ by $q$ yields
\begin{align}
\sum_{n=0}^{\infty} A(3n+2)q^{n} & = \frac{f_2^4f_3^5f_{12}^3}{f_1^4f_{4}^4f_{6}^4} - 2\frac{f_2f_{6}^5}{f_1^2f_3f_{4}^3} \nonumber \\
& \equiv \frac{f_2}{f_1f_4} \frac{f_3^4f_{12}^2}{f_6^3} - 2\frac{f_2}{f_1^2} \frac{f_6^5}{f_3f_{12}} \pmod{3}. \label{eq2.1}
\end{align}
Using Lemmas \ref{lemma2} and \ref{lemma3} we extract the terms of the form $q^{3n+1}$ from both sides of the last congruence to obtain
\begin{align*}
\sum_{n=0}^{\infty} A(9n+5)q^{3n+1}
& \equiv q\frac{f_3f_{18}^3}{f_6f_{12}} - 4q\frac{f_6^8f_9^3}{f_3^8f_{12}} \pmod{3}\\
& \equiv  q\frac{f_3f_{18}^3}{f_6f_{12}} - 4q\frac{f_3}{f_6f_{12}} \frac{f_6^9f_9^3}{f_3^9} \pmod{3}\\
& \equiv -3q\frac{f_3f_{18}^3}{f_6f_{12}} \pmod{3} \\
& \equiv 0 \pmod{3},
\end{align*}
which proves \eqref{m1}.

Using Lemmas \ref{lemma2} and \ref{lemma3} we extract the terms of the form $q^{3n+2}$ from both sides of \eqref{eq2.1}:
\begin{align*}
\sum_{n=0}^{\infty} A(9n+8)q^{3n+2}
& \equiv  q^2\frac{f_6f_{9}^3f_{36}^3}{f_{12}^2f_{18}^3} - 8q^2\frac{f_6^7f_{18}^3}{f_3^7f_{12}} \pmod{3}.
\end{align*}
Dividing by $q^2$ and replacing $q^3$ by $q$, we obtain
\begin{align}
\sum_{n=0}^{\infty} A(9n+8)q^{n}
& \equiv  \frac{f_2f_3^3f_{12}^3}{f_{4}^2f_{6}^3} - 8\frac{f_2^7f_{6}^3}{f_1^7f_{4}} \pmod{3} \nonumber \\
& \equiv \frac{f_2}{f_4^2} \frac{f_3^3f_{12}^3}{f_6^3} -8\frac{f_2}{f_1f_4} \frac{f_6^5}{f_3^2} \pmod{3}. \label{eq2.2}
\end{align}
Now we employ Lemmas \ref{lemma2} and \ref{lemma4} to extract the terms of the form $q^{3n+2}$ from the congruence above. The resulting congruence after division by $q^2$ and replacing $q^3$ by $q$ is given by
\begin{align*}
\sum_{n=0}^{\infty} A(27n+26)q^{n}
& \equiv  -\frac{f_1^3f_{12}^3}{f_{4}^4} - 8\frac{f_2^9f_3^3f_{12}^3}{f_1^6f_{4}^4f_6^3} \pmod{3}\\
& \equiv -\frac{f_1^3f_{12}^3}{f_4^4}  -8\frac{f_1^3f_{12}^3}{f_4^4} \pmod{3}  \\
& \equiv 0 \pmod{3},
\end{align*}
which proves \eqref{m2}.

\section{Infinite families of congruences modulo 3}
While the elementary proofs of Merca's original congruences that we have provided above are very satisfying, it turns out that much more is true about $A(n)$ modulo 3, and our elementary approach to proving these congruences yields the insights needed in order to see this.  In this light, we now proceed to proving two infinite families of non--nested Ramanujan--like congruences modulo 3 satisfied by $A(n).$

We begin by proving the following internal congruence satisfied by $A(n)$ which will serve as an important component in our remaining proofs.

\begin{theorem}
For all $n \geq 0$,
\begin{equation}
    A(27n+8) \equiv A(3n+1) \pmod{3}.
    \label{m4}
\end{equation}
\end{theorem}

\begin{proof}
We prove this result by showing that both $A(27n+8)$ and $A(3n+1)$ have the same generating function modulo 3. Using Lemmas \ref{lemma2} and \ref{lemma4} we extract the terms of the form $q^{3n}$ from \eqref{eq2.2}:
\begin{align*}
\sum_{n=0}^{\infty} A(27n+8)q^{3n}
& \equiv \frac{f_3^3f_{18}^3}{f_6f_{12}^3}  -8\frac{f_6^5f_{18}^9}{f_3^4f_9^3f_{12}^2f_{36}^3} \pmod{3},
\end{align*}
which after replacement of $q^3$ by $q$ yields
\begin{align}
\sum_{n=0}^{\infty} A(27n+8)q^{n}
& \equiv \frac{f_1^3f_{6}^3}{f_2f_{4}^3}  -2\frac{f_2^5f_{6}^9}{f_1^4f_3^3f_{4}^2f_{12}^3} \pmod{3}.
\label{eq2.3}
\end{align}

The generating function for $A(3n+1)$ is obtained in the same way using \eqref{gf1} and Lemmas \ref{lemma1} and \ref{lemma2}:
\begin{align*}
\sum_{n=0}^{\infty} A(3n+1)q^{3n+1}
& = q\frac{f_6^2f_9^2f_{18}^2}{f_3^3f_{12}^3}  -2q\frac{f_{18}^{11}}{f_3f_6f_9^4f_{12}^2f_{36}^3}.
\end{align*}
Dividing this expression by $q$ and replacing $q^3$ by $q$, we are left with
\begin{align*}
\sum_{n=0}^{\infty} A(3n+1)q^{n}
& = \frac{f_2^2f_3^2f_{6}^2}{f_1^3f_{4}^3}  -2 \frac{f_{6}^{11}}{f_1f_2f_3^4f_{4}^2f_{12}^3} \\
& \equiv \frac{f_1^3f_2^3f_{6}^3}{f_2f_4^3f_6}  -2 \frac{f_1^3f_2^5f_{6}^{11}}{f_1^4f_2^6f_3^4f_{4}^2f_{12}^3} \pmod{3} \\
& \equiv \frac{f_1^3f_{6}^3}{f_2f_{4}^3}  -2 \frac{f_2^5f_{6}^{9}}{f_1^4f_3^3f_{4}^2f_{12}^3} \pmod{3},
\end{align*}
which coincides with \eqref{eq2.3} and the proof is complete.
\end{proof}

We can now prove the following two theorems.

\begin{theorem}
\label{infinite_family1}
For all $j\geq 0$ and all $n\geq 0,$
$$
A\left(9^{j+1}n + \frac{39\cdot9^j+1}{8}\right) \equiv 0 \pmod{3}.
$$
\end{theorem}
\begin{proof}
This theorem follows from a straightforward proof by induction on $j.$
First, we note that the basis case, $j=0,$ is simply the statement that, for all $n\geq 0,$
$$
A(9n+5) \equiv 0 \pmod{3}.
$$
This is Theorem \ref{Merca1} equation (\ref{m1}) above, the first of Merca's original congruences.

Next, we assume that the statement is true for some $j\geq 0.$  We then wish to prove that
$$
A\left(9^{j+2}n + \frac{39\cdot9^{j+1}+1}{8}\right) \equiv 0 \pmod{3}
$$
for all $n\geq 0$ as well.  Note that
\begin{align*}
9^{j+2}n + \frac{39\cdot9^{j+1}+1}{8}
&=
27\left(3\cdot 9^jn+\frac{13\cdot 9^j}{8} - \frac{7}{24}\right) + 8.
\end{align*}
Thanks to (\ref{m4}), we know that, for all $n\geq 0,$
$$A(27n+8)\equiv A(3n+1) \pmod{3}.$$
Thus,
\begin{align*}
A\left(9^{j+2}n + \frac{39\cdot9^{j+1}+1}{8}\right)
&=
A\left(27\left(3\cdot 9^jn+\frac{13\cdot 9^j}{8} - \frac{7}{24}\right) + 8\right) \\
& \equiv
A\left(3\left(3\cdot 9^jn+\frac{13\cdot 9^j}{8} - \frac{7}{24}\right) + 1\right) \pmod{3}\\
&=
A\left( 9^{j+1}n+\frac{39\cdot 9^{j}+1}{8} \right) \\
&\equiv 0 \pmod{3}
\end{align*}
by the induction hypothesis.  The result follows.
\end{proof}

\begin{theorem}
\label{infinite_family2}
For all $j\geq 0$ and all $n\geq 0,$
$$
A\left(3\cdot 9^{j+1}n + \frac{23\cdot9^{j+1}+1}{8}\right) \equiv 0 \pmod{3}.
$$
\end{theorem}
\begin{proof}
As above, this theorem follows from a straightforward proof by induction on $j.$ Note that  the basis case, $j=0,$ is simply the statement that, for all $n\geq 0,$
$$
A(27n+26) \equiv 0 \pmod{3}.
$$
This is Theorem \ref{Merca1} equation (\ref{m2}) above, the second of Merca's original congruences.

Next, we assume that the statement is true for some $j\geq 0.$  We then wish to prove that
$$
A\left(3\cdot 9^{j+2}n + \frac{23\cdot9^{j+2}+1}{8}\right) \equiv 0 \pmod{3}
$$
for all $n\geq 0$ as well.  Note that
\begin{align*}
3\cdot 9^{j+2}n + \frac{23\cdot9^{j+2}+1}{8}
&=
27\left(9^{j+1}n+\frac{3(23)\cdot 9^j}{8} - \frac{7}{24}\right) + 8.
\end{align*}
Thanks to (\ref{m4}), we know that, for all $n\geq 0,$
$$A(27n+8)\equiv A(3n+1) \pmod{3}.$$
Thus,
\begin{align*}
A\left(3\cdot 9^{j+2}n + \frac{23\cdot9^{j+2}+1}{8}\right)
&=
A\left(27\left(9^{j+1}n+\frac{3(23)\cdot 9^j}{8} - \frac{7}{24}\right) + 8\right) \\
& \equiv
A\left(3\left(9^{j+1}n+\frac{3(23)\cdot 9^j}{8} - \frac{7}{24}\right) + 1\right) \pmod{3}\\
&=
A\left( 3\cdot 9^{j+1}n + \frac{23\cdot9^{j+1}+1}{8} \right) \\
&\equiv 0 \pmod{3}
\end{align*}
by the induction hypothesis.  The result follows.

\end{proof}

redpajama_set_name: RedPajamaArXiv


text:
\section{Outline}
Heavy Quark Effective Theory (HQET)~\cite{HQET} is a powerful approach to
the processes involving heavy
flavor hadrons and, in particular, to the study of form factors of
semileptonic decays (see
reviews~\cite{Neub}).
From a field-theoretical standpoint, the HQET is a particular example of construction based
on a more general notion of Wilson  Operator Product Expansion (OPE) or Wilson effective
action~\cite{wilson}, to which I mainly refer.

The outline of the talk is as follows:
\begin{itemize}
\item{
 Heavy flavor decays as short distance probes. Total semileptonic widths}
\item{
 From inclusive to exclusive: bounds for $B\rightarrow D^*$ leading form factor}
\item{
Subleading $B\rightarrow D^*$  form factors}
\item{
 Some recent developments}
\item{
Conclusions}
\end{itemize}

\section{Heavy Flavor Decays as Short Distance Probes}

The quark picture of semileptonic $B$ meson decay $B \rightarrow D + e+\bar \nu_e$ is presented by
Fig.~1. (Better to put  ${\bar B}$ but I will systematically neglect by bar on $B$.)
\begin{figure}[ht]
\epsfxsize=10.0cm
\centerline{\epsfbox{sl.eps}}
\caption{Semileptonic  $B \rightarrow D + e+\bar \nu_e$ decay}
\end{figure}
The specific feature  of heavy flavor states is that they contain a heavy quark $Q$
which could be viewed as an almost static object (in the rest frame of hadron). Thus these
states can be considered as ground states of light flavor QCD in the presence of static source of
gluon filed. A relevant theoretical parameter is
\begin{equation}
\frac{\Lambda_{QCD}}{m_Q} \ll 1\;.
\label{hm}
\end{equation}
An immediate consequence is so called spin-flavor symmetry, i.e., an independence on  heavy
quark spin  and flavor  in the large mass limit where the relation (\ref{hm}) holds.

However, the characteristic size of heavy flavor hadron is of order $1/ \Lambda_{QCD}$  and
the structure of  heavy flavor states  as well as that of the QCD
vacuum is governed by nonperturbative dynamics in strong coupling regime.
 To see a simple underlying  quark-gluon structure at short distances, where the strong
interaction becomes  weaker one needs a short distance probe. Heavy  flavor decays, particularly
the ones inclusive over final hadrons, provide such a probe. This probe is similar to the famous
ratio
$R$ for the inclusive cross  section of hadron production with a substitution of QCD vacuum
by  a heavy flavor hadron. This comparison is illustrated by Figs.~2 and 3.
\begin{figure}[ht]
\epsfxsize=10cm
\centerline{\epsfbox{annih.eps}}
\caption{$e^+ e^-$ annihilation into hadrons (inclusive)}
\end{figure}
Fig.~2 is for the process of $e^+e^-$ inclusive annihilation into hadrons, its cross section is
given by
\begin{equation}
R=\frac{\sigma(e^+ e^- \rightarrow {\rm hadrons})}{\sigma(e^+ e^- \rightarrow
\mu^+
\mu^-)} = N_{{\rm colors}}\sum_q Q_q^2 \left[1+ O [\alpha_S (W)] + \dots \right],
\label{ee}
\end{equation}
in the limit of high total energy $W$, i.e., when  $W\gg \Lambda_{QCD}$.
\begin{figure}[ht]
\epsfxsize=10cm
\centerline{\epsfbox{width.eps}}
\caption{Inclusive width of $B \rightarrow X_u e {\bar \nu}$ decays}
\end{figure}
Fig.~3 presents  semileptonic decays of $B$ meson inclusive over  nonstrange hadrons in the
final state ($b\rightarrow u$ transition in terms of quarks),
\begin{equation}
\Gamma_{SL}^{ub}=\sum_{X_u}\Gamma (B \rightarrow X_u  e {\bar \nu})=
\frac{G_F^2}{192 \pi^3}
m_b^5 |V_{ub}|^2\,\left[
1+ O [\alpha_S (m_b)] + \dots \right]
\label{sl}
\end{equation}
in the limit of large mass of b quark, $m_b\gg \Lambda_{QCD}$.

In both examples the leading term is given by partonic description of the process
where only hard   momenta count and an influence
of soft background of quarks and gluons in the vacuum  and in B meson is strongly
suppressed.  Theoretical basis of quark-hadron duality is not trivial and based on two notions:

\vspace{0.2cm}

\noindent
(i) Wilson Operator Product Expansion~\cite{wilson}. The expansion takes place at large {\em
Euclidean}  momenta where no hadron can be produced. The OPE means  a factorization of
short and large distances in Euclidean domain.

\vspace{0.2cm}

\noindent
(ii) Dispersion relations. Based on causality  dispersion relations express certain integrals
over observable cross sections (in {\em Minkowski} domain) as correlators at  Euclidean
momenta.

\vspace{0.2cm}

The construction described allows for calculation of corrections to the leading partonic term, both
perturbative, $[\alpha_S (m_b)]^n$, and nonperturbative, $[\Lambda_{QCD}/m_b]^k$.
For the semileptonic total width (\ref{sl}) it results in
\begin{equation}
\Gamma_{SL}^{ub}=\frac{G_F^2 m_b^5 |V_{ub}|^2}{192 \pi^3}
 \left[1- \frac{2\,\alpha_S}{3\pi}-\frac{\mu^2_\pi}{2m_b^2} -
\frac{3\,\mu^2_G}{2 m_b^2}
\right]
\label{btou}
\end{equation}
where power corrections~\cite{buv} are defined by parameters $\mu^2_\pi$ and $\mu^2_G$.
The first one, $\mu^2_\pi$, has the meaning of mean value for momentum square of $b$
quark inside of $B$ meson,
\begin{equation}
\mu^2_\pi = \langle B|  {\bar b} {\vec \pi}^2 b |B\rangle \approx 0.6 \pm 0.1 \;{\rm GeV}^2\;.
\end{equation}
The value given is due to theoretical estimate~\cite{bb}, the direct determination  is  not done
yet. The second parameter, $\mu^2_G$, measures chromomagnetic interaction with the spin
of heavy quark, so its value  is fixed by spin splitting in $B,\;B^*$ system,
\begin{equation}
\mu_G^2=\langle B|{\bar b}\frac{i}{2} \sigma^{\mu\nu} G_{\mu\nu} b
|B\rangle =\langle {\vec \sigma}{\vec B} \rangle
=\frac{3}{4}(M_{B^*}^2 - M_B^2) \approx 0.36\,{\rm GeV}^2\;.
\label{mugnum}
\end{equation}

\section{From  Inclusive to Exclusive}

The decays $B\rightarrow D^* l {\bar \nu_l}$ and $B\rightarrow D l {\bar \nu_l}$ produce about 2/3 of
the total semileptonic width. The suppression of production of higher states can be understood
as due to a closeness to the point of zero recoil. In other words, hadrons in the final state are slow
moving. The point of zero recoil, where the spatial momentum ${\vec q}=0$, is a special  one
in the case of heavy quark in the final state.  The quark-hadron
duality takes place at this point in spite of  small momentum transfer.
 It is a remarkable  case where  OPE works up to large distances (in case of light
quarks a similar phenomena shows up in the axial current anomaly and in the certain term
of OPE for the product of two axial currents).

The theoretical knowledge of the amplitude of $B\rightarrow D^* l {\bar \nu_l}$ decay at zero
recoil is used, in particular, for an extraction of $|V_{cb}|$ from the data. The crucial
 for this extraction question is about a theoretical accuracy of HQET predictions.

Let us start with kinematics of the decay. There are four invariant form factors describing
the transition~\cite{Neub},
\begin{eqnarray}
{\langle D^*|{\vec V} - {\vec A}|B\rangle}&= &\sqrt{m_B m_{D^*}}\, h_{A_1}(w)
\left[ -(w+1) {\vec e}_{D^*}+ \right. \nonumber\\
& & \left. wR_1(w) i {\vec v}_{D^*} \times e_{D^*} +wR_3(w) {\vec v}_{D^*}({\vec v}_{D^*}
\cdot{\vec e}_{D^*})\right]\;,
\label{vecV}
\end{eqnarray}
\begin{eqnarray}
{\langle D^*| V_0 -  A_0|B\rangle}&= &\sqrt{m_B m_{D^*}}\, h_{A_1}(w)\,
 ({\vec v}_{D^*}\cdot{\vec e}_{D^*})
\left[-(w+1) + ~~~~~~~~\right. \nonumber\\
& & \left. R_2(w) \,\frac{m_B}{m_{D^*}} -R_3(w) \left(\frac{m_B}{m_{D^*}}-1\right)\right]\;,
\end{eqnarray}
where matrix elements of $V-A$ current are written separately for spatial
${\vec V}-{\vec A}$ and timelike  $V_0-A_0$ components and
\begin{equation}
w=\frac{E_{D^*}}{m_{D^*}}=\sqrt{1+\frac{{\vec q}^2}{m_{D^*}^2}}\;,~~~~~~~
{\vec v}_{D^*}=-\frac{{\vec q}}{m_{D^*}\,w}\;.
\end{equation}
The $ h_{A_1}(w)$ form factor is the only one contributing at zero recoil. Subleading
(in terms of their contribution to the amplitude  near zero recoil) form factors are
introduced in the form of their ratios
$R_i$ to the leading one. In the HQET limit  these ratios go 1,
\begin{equation}
R_1(w)=R_2(w)=R_3(w)=1 ~~~~~~ ({\rm at }~~m_b,m_c\rightarrow \infty)
\end{equation}

\subsection{Zero Recoil Sum Rule}
The amplitude at zero recoil is defined by $ h_{A_1}(1)$ as discussed above.  To get its value
let us write down the sum rule~\cite{suv} for {\em inclusive}
transitions $B\rightarrow X_c l {\bar \nu_l}$ at zero recoil, ${\vec q}=0$,
\begin{equation}
|h_{A_1}(1)|^2 + \sum_{m_{D^*}<m_{X_c}<m_{D^*}+\mu} |h_{n}(1)|^2 \,=\, \xi_A(\mu)
-\Delta_{1/m^2} + O\left( \frac{1}{m^3}\right)\;.
\label{sr}
\end{equation}
The states $X_c$ produced at zero recoil are $D^*$ and its radial
excitations and
 $h_{n}(1)$ are analogs of $h_{A_1}(1)$ for radial excitations. The production of
excitation is suppressed in the heavy quark limit, $h_{n}(1)\sim 1/m$ and the sum in the l.h.s.
 of Eq.~(\ref{sr}) is of
$1/m^2$ order. At the r.h.s. the quantity
$\xi_A(\mu)$ gives a perturbative part and has the form
\begin{equation}
\xi_A(\mu)=1 +2\frac{\alpha_S}{\pi}\left[ \frac{m_b+m_c}{m_b-m_c}
\ln\frac{m_b}{m_c}- \frac{8}{3}\right]+
\frac{2}{3}\frac{\alpha_S}{\pi} \mu^2\left[ \frac{1}{m_c^2}+\frac{1}{m_b^2}
+ \frac{2}{3m_b m_c}\right]
\end{equation}
Let me note here that the full calculation of the $\alpha_S^2$ corrections was
 recently performed~\cite{cmu}. It
turned out to be a small change as compared with  the value taken from the BLM procedure.
Numerically the result for $\xi_A$ is
\begin{equation}
 \xi_A(\mu=0.5 m_c)=0.99 \pm 0.02
\end{equation}

The quantity $\Delta_{1/m^2}$ presents nonperturbative effects which are of the
second order in $1/m$, Eq.~(\ref{sr}) contains no corrections of the first order in $1/m$ in
correspondence with Luke theorem~\cite{Luke},
\begin{equation}
\Delta_{1/m^2}=\frac{\mu_G^2}{3m_c^2}+ \frac{\mu_\pi^2-\mu_G^2}{4}\left[
\frac{1}{m_c^2}+\frac{1}{m_b^2} + \frac{2}{3m_b m_c}\right]\;.
\end{equation}
Numerically it is about 0.1 which is not a small effect, particular for the quantity of the
second order in $1/m$.

The most uncertain in the use of sum rule (\ref{sr}) for theoretical evaluation  of
$h_{A_1}(1)$ is the sum over excitation. In the absence of direct experimental information
the natural assumption would be to say that the sum is comparable with
$\Delta_{1/m^2}$. Indeed, the origin of $\Delta_{1/m^2}$ can be traced to the
contribution of the highly excited states with the excitation energies of order of $m_c$ while
the sum under consideration refers to less excitation energies. If some kind of continuity
exists then an assumption
\begin{equation}
\sum_{m_{D^*}<m_{X_c}<m_{D^*}+\mu} |h_{n}(1)|^2 =(0.5\pm 0.5)\,\Delta_{1/m^2}
\end{equation}
seems reasonable for the range of $\mu \sim 0.5\, m_c$.

Summarizing numerical estimates we get (see Ref.~\cite{bsu} for details)
\begin{equation}
h_{A_1}(1)= 0.91 -0.013 \,\,\frac{\mu_\pi^2 -0.5\,{\rm GeV}^2}{0.1\,{\rm GeV}^2}
\pm 0.02_{\rm excit} \pm 0.01_{\rm pert} \pm 0.025_{1/m^3}
\end{equation}
where the uncertainties are marked by the their sources.
Overall,
\begin{equation}
h_{A_1}(1)= 0.91\pm 0.06\;,
\end{equation}
what leads to the 6\% uncertainty in the $|V_{cb}|$ extraction from the exclusive
$D^*$ production at zero recoil. The theoretical uncertainty  is less by
about a factor of two in the
determination of  $|V_{cb}|$  from inclusive production~\cite{bsu}. The  difference
is  due to the fact that the inclusive approach is much less sensitive to power
in $1/m_c$ corrections which are the main source of corrections discussed above.
Indeed, for inclusive widths an expansion in powers of $1/m_b$ exists even for a small
mass of final quark.

\section{Subleading Form Factors}
In difference with $h_{A_1}(1)$ form factors $R_1(1)$, $R_2(1)$ and $R_3(1)$ are not
protected against corrections of the first order in 1/m at zero recoil. For this reason they
can deviate from unity (HQET limit) quite considerably.  Theoretical estimates of
Ref.~\cite{neub94} for $R_1(1)$, $R_2(1)$ are
\begin{equation}
 R_1(1)\approx  1 + \frac{m_D* - m_c}{2m_c}+ \frac{m_B* - m_b}{2m_b}\approx 1.35\;;~~~~~~
R_2(1)\approx 0.79\;.
\label{R}
\end{equation}
The theoretical expression for $ R_1(1)$ is presented in somewhat simplified form  to
demonstrate that the  quite considerable deviation from the HQET value comes from the
$1/m_c$ correction.
The
predictions (\ref{R}) are consistent with CLEO experiment~\cite{CLEO} where form factors were
measured,
\begin{equation}
R_1=1.18\pm0.30\pm0.12\;; ~~~~~R_2=0.71\pm0.22\pm0.07\;.
\end{equation}

If corrections of the first order in $1/m_c$ are so large,then the natural question to ask is about
a magnitude of $1/m_c^2$ corrections. The method of sum rules discussed in the previous
part for $h_{A_1}(1)$ can be applied to get information about other form factors.  Using a set of
sum rules from Ref.~\cite{bsuv} I came to the following estimate for the $1/m^2$ correction
to $R_2(1)$
\begin{equation}
\left[\Delta R_2(1) \right]_{1/m^2} \approx \frac{\mu_G^2}{m_c^2\left(\frac{m_B}{m_{D^*}}-
1\right)}\left( 1+ \frac{m_c}{3\,m_b}\right) \approx 0.15
\end{equation}
The size of the correction is larger that the one of the first order in $1/m$ given by Eq.~(\ref{R}),
thus the series in powers of $1/m$ does not look as a convergent expansion. The lesson is that one
should be cautious with powers of $1/m_c$.
\section{Some Recent Developments}
This is a brief mention of a  few recent studies relevant to the topic:
\begin{itemize}
\item{Constraints for form factors following from inclusive sum rules are explored
for different semileptonic decays~\cite{BR}.  It results in upper
and lower bounds for transitions of $B$ meson to $D$, $D^*$, $\rho$, $\pi$, $\omega$, $K$ and
 }
\item{Inclusive sum rules following from polarization operator of the semileptonic current
are used to put a new type of bound for the cross channel (like $B\, D$) and in this way to the
form factors~\cite{BGL}.
}
\item{
Extraction of $|V_{ub}|$ from  exclusive modes~\cite{LW}. Using the set of transitions,
$B\rightarrow\rho$,
$B\rightarrow K^*$,  $D\rightarrow\rho$, $D\rightarrow K^*$ it is possible to diminish considerably a model dependence.
}
\item{
Semileptonic $B$ decays to excited states~\cite{LLSW}. This interesting development is presented
at this conference by Zoltan Ligeti (see his talk).
}
\end{itemize}
\section{Conclusions}
\begin{itemize}
\item{The theory of semileptonic decays is in a good shape}
\item{The main source of theoretical uncertainties is in the $(1/m_c)^n$ corrections, not in the perturbative ones}
\item{Convergence of exclusive and inclusive approaches}
\end{itemize}

\section*{Acknowledgments}
I am thankful to Zoltan Ligeti, Mikhail Shifman, Nikolai Uraltsev and Mikhail Voloshin for
helpful discussions. This work was supported in part by DOE under the grant number
DE-FG02-94ER40823.

\section*{References}


text:
\section{Introduction}

The exponential growth of out-of-time-order correlator (OTOC) has been regarded as a indicator of chaos or quantum instability in many quantum
systems\cite{sys1,sys2,sys3,sys4,sys5,sys6,sys7,sys8,sys9,sys10,sys11,sys12,sys13,sys14,sys15}.
In particular, its growth rate is closely associated with the classical Lyapunov exponent(CLE) $\lambda$ in the large N limit\cite{sys12,sys13,sys14,sys15,cqlpv1,cqlpv2,cqlpv3}.
In the context of black holes physics, the OTOC can be considered to as an indicator of chaos in gravity dual theory \cite{gr1} and
the corresponding upper bound of the maximum quantum Lyapunov exponent (QLE) is obtained in  black hole geometry \cite{sys1,gr2,gr3}.
Experimentally, the OTOC has been measured in the ion traps systems\cite{exp1,exp2,exp3,exp4} and in the nuclear magnetic resonance platforms\cite{nmr1,nmr2,nmr3}.

However, an important discovery is that the OTOC in some non-chaotic regular systems also shows exponential growth behavior in early time
and the fast scrambling emerges not only in the chaotic case but also in the regular one\cite{sp1}. In the integrable systems, it is shown that
the OTOC at the saddle points (unstable fixed points) could grow exponentially \cite{sp2} due to the quantum instability existing in the vicinity of these points \cite{sp3}. The similar behavior of OTOC also appears in the inverted harmonic oscillator system with a Higgs potential\cite{sp4}.
It is not surprising because the instability of the unstable fixed point causes its CLE to be positive.
Then, it is natural to ask whether unstable regular orbits lead the OTOC shows exponential growth behavior in early time.
To study the early behavior of OTOC of unstable regular orbits, we consider the simplest unstable but regular system --- the inverted harmonic
oscillator system, described by the Hamiltonian $\hat{H}=p^2/2m-m\omega^2 q^2/2$ \cite{iho1}. This special system has been widely studied in various aspects.
It is shown that this regular system has an
exponential sensitivity to initial conditions as in the chaotic systems\cite{iho11}.
The phase-space volume of the classical inverted harmonic oscillator is unbounded, however, its corresponding volume for quantum inverted oscillator can be bounded by the system photons\cite{ihodicke1,ihodicke2}.
Especially, the inverted harmonic oscillator is not just a pure theoretical model and has been realized experimentally\cite{iho2}. In mathematics, it even challenges the Riemann hypothesis\cite{iho3}.
Moreover, it also shows important significance in both general relativity, quantum mechanics and chaotic domain\cite{sign1,sign2,sign3,sign4,sign5,sign6,sign7}.

Fidelity OTOC (FOTOC) is related to the quantum variance of a Hermitian operator on the initial state and is also shown to be associated with the CLE of chaotic systems\cite{sys13} and
the quantum instability of saddle point in non-chaotic system \cite{sp3}.
The FOTOC provides a method for us to visualize the chaotic or scrambling dynamics of a quantum system in semi-classical phase space.
In Schrodinger picture, the evolution of the quantum wave packet in the phase space will be directly related to the quantum variance of momentum or coordinate operator.
On the other hand, Husimi $Q$ function is one of the quasi-probability distribution functions and was used to visualized the evolution of quantum wave packets in phase space\cite{hs1,hs2,EE1,EE2,EE3}.
In this paper, we will study the early evolution of Husimi quasi-probability wave packets in two inverted harmonic oscillator
systems and to see what Husimi quasi-probability wave packets behave as the OTOCs grow at an exponential rate.

This paper is organized as follows. In section II, we study the relationship between the mean photon number and the bounded region in quantum IHO system.
In section III, we analyze the early behavior of OTOC of unstable regular orbits in IHO system and visualize the OTOC by Husimi Q function.
In section IV, we study the false chaotic signal of stable regular orbit and explain it by Husimi Q function.
Finally, we present results and a brief summary.

\section{Mean photon number in inverted harmonic oscillator}
In this section, we study the classical one-dimensional IHO system with unstable regular orbits, namely,
\begin{eqnarray}
H &=& \frac{p^2}{2m} + \hat{V},  \ \ \ \ \ \ V = -\frac{1}{2}m\omega^2 q^2,   \label{iho1}
\end{eqnarray}
\begin{figure}[htp]
\begin{center}
\includegraphics[width=6cm]{phap1.eps}
\caption{The classical dynamics phase diagram of IHO. The coordinate $(q, p)$ of points $O, A, B, C, D$ and $E$ are $(0, 0), (5, -5), (3, 3), (-4.267, 5.643), (-3, 3)$ and $(-4.267, -5.643)$, respectively.
Point $O$ is the saddle point.}
\label{f11}
\end{center}
\end{figure}
where $\omega$ and $m$ are the frequency and mass of IHO, respectively.
In contrast to harmonic oscillator, the potential $V$  tends to infinity in the limit $q\rightarrow \infty$.
In Fig. \ref{f11}, we present the classical dynamics phase diagram of IHO system.
For the classical system (\ref{iho1}), there is a saddle point $(q=0, p=0)$ in phase space. The particle starting from initial point on the asymptote ($p=-q$) moves to the saddle point and the corresponding orbit is stable, which forms a so-called stable manifold. The particle starting from other points moves to infinity and its orbit is unstable, but its CLE is the same as that from the saddle point, i.e. $\lambda_O=1$.

To study the quantum dynamics of IHO, we introduce the quadratic quantization form of the position and momentum operators and set $\hbar=m=\omega=1$,
\begin{eqnarray}
x &=& (a^{\dag} + a)/\sqrt{2} ,  \ \ \ \ \ \ p = i(a^{\dag} - a)/\sqrt{2},   \label{qq1}
\end{eqnarray}
and then the quantum Hamiltonian of IHO becomes
\begin{eqnarray}
\hat{H} &=& -\frac{1}{2}(a^2 + a^{\dag 2}).   \label{iho2}
\end{eqnarray}
Notice that the Hamiltonian (\ref{iho2}) is equivalent to the Hamiltonian (\ref{iho1}) in the classical limit where the system photon number $N_p\rightarrow \infty$.
As in Refs.\cite{hs2,EE4}, we take the photon coherent state as the initial state of system
\begin{eqnarray}
|\psi\rangle &=& e^{-\beta\beta^{*}/2}e^{\beta a^{\dagger}}|0\rangle, \label{bcoh1}
\end{eqnarray}
with
\begin{eqnarray}
\beta &=& (q + ip)/\sqrt{2}.   \label{tb1}
\end{eqnarray}
where $|0\rangle$ is the ground state of light field, $q$ and $p$ are generalized coordinates and momentum.
According to quantum theory, the mean photon number on the initial coherent state is
\begin{eqnarray}
\langle a^{\dag}a\rangle &=& \langle\psi|a^{\dag}a|\psi\rangle=\beta\beta^{\dag}=(q^2 +p^2)/2.   \label{np1}
\end{eqnarray}
Eq.(\ref{np1}) implies that the mean photon number is related to the center position of coherent state in phase space. In other words,
if the photon number of the system $N_p$ is given, the coordinate and momentum parameters of coherent states in phase space are bounded in the region $(q^2 +p^2)/2\leq N_p$.
This shows that there is a natural boundary in phase space of the quantum IHO system with finite photon number, while the corresponding classical IHO system is unbounded because in the classical limit $N_p\rightarrow \infty$.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=5cm]{expv1.eps}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
\includegraphics[width=5cm]{expv2.eps}
\caption{Time evolution of the mean photon number. (a) Plots of the mean photon number for point B in Fig. \ref{f11} with different system photon number.
(b) Plots of the mean photon number for six different points [as marked in Fig. \ref{f11}] with fixed system photon number $N_p = 300$.}
\label{f2}
\end{center}
\end{figure}

Taking a coherent state centering at the point $B$ in Fig. \ref{f11} as the initial value, we present the mean photon number
evolution with time for different initial photon number $N_p$, as shown in Fig. \ref{f2}(a).
We find that the mean photon number $\langle a^{\dag}a\rangle$ keeps increasing with the time in the classical limit case $N_p\rightarrow\infty$.
For the case with finite initial photon number $N_p$, the curve of mean photon number changing with time (i.e., $\langle a^{\dag}a\rangle-t$ ) first overlaps that in the classical limit. After a time $t_p$, the mean photon number decreases and does not change with time along the line in the classical limit. This means that during the period $0<t\leq t_p$ there exists the so-called classical-quantum correspondence for the system (\ref{iho1}) because as $t> t_p$ the quantum behavior in the system differs from that in the classical system. We also note that the time $t_p$ increases with the initial photon number $N_p$, which is understandable because the classical-quantum correspondence becomes clearer in the system with larger photon number. Moreover, the largest mean photon number $N_{lm}$ maintaining the classical-quantum correspondence is less than the initial photon number $N_p$. Strictly, this also implies that the maintaining of classical-quantum correspondence is determined by the largest mean photon number $N_{lm}$ rather than the initial photon number $N_p$.

In Fig. \ref{f2}(b), we exhibit the mean photon number $\langle a^{\dag}a\rangle$ evolution with time for different initial points in Fig. \ref{f11}.
It is shown that the change of mean photon number depends on the center position of the quantum wave packets in phase space.
For the starting points located in the stable manifold of the saddle point, such as point $A$, $C$,and $D$,
the mean photon number $\langle a^{\dag}a\rangle$ decreases firstly and then increases to the maximum value.
For the starting points located in the unstable manifold, the mean photon number directly grows to the maximum. Moreover,
Fig. \ref{f2} also implies that the length of time maintaining classical-quantum correspondence in the quantum IHO depends else on the center positions of the initial coherent states of system in the phase space.

\section{Exponential growth of OTOC in inverted harmonic oscillator}
In this section, we study the quantum variance derived from the OTOC in the IHO system and analyse the Husimi quasi-probability wave packet during the OTOC growth exponentially.
The OTOC is defined as\cite{definition}
\begin{eqnarray}
C(t) &=& \langle[\hat{W}(t),\hat{V}(0)]^{\dagger}[\hat{W}(t),\hat{V}(0)]\rangle,   \label{otoc1}
\end{eqnarray}
where $\langle...\rangle$ denotes the expectation values and  $\hat{W}(t)=e^{i\hat{H}t}\hat{W}e^{-i\hat{H}t}$. $\hat{H}$ is a quantum Hamiltonian,
$\hat{W}$ and $\hat{V}$ are two arbitrary local operators.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=6cm]{ocp.eps}
\caption{Time evolution of the OTOC for point B in Fig. \ref{f11} with different system photon number.}
\label{f3}
\end{center}
\end{figure}
Here, we choose $\hat{W}=\hat{P}=i(a^{\dag} - a)/\sqrt{2}$ and $\hat{V}$ as a projection operator onto the initial state $|\psi\rangle$, i.e., $\hat{V}=|\psi\rangle\langle\psi|$.
Substituting the operators $\hat{V}=|\psi\rangle\langle\psi|$ and $\hat{W}=\hat{P}$ into the Eq. (\ref{otoc1}),
one has
\begin{eqnarray}
C(t) &=& \langle\psi(t)|\hat{P}^{2}|\psi(t)\rangle - \langle\psi(t)|\hat{P}|\psi(t)\rangle^2\equiv Var[\hat{P}(t)], \label{otoc3}
\end{eqnarray}
\begin{figure}[ht]
\begin{center}
\includegraphics[width=5cm]{oadc1.eps}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
\includegraphics[width=5cm]{obcec2.eps}
\caption{Time evolution of the OTOC for the initial coherent states centered at different points. The gray dashed line in the left panel corresponds to the exponential growth rate given by twice the classical Lyapunov exponent. Here, we set the system photon number $N_p = 300$.}
\label{f4}
\end{center}
\end{figure}
where $Var[\hat{P}(t)]$ is the quantum variance of the momentum operator $\hat{P}$.
This relation is similar to the definition of FOTOC\cite{sys13} and enable us to visualize the OTOC in semiclassical phase space.
In Fig. \ref{f3}, we find that the exponential growth rate $\tilde{\lambda}$ of OTOC is independent of system photon number $N_p$,
and the time to keep $Var[\hat{P}(t)]$ growing exponentially increases with $N_p$.
For the initial state centered at the point in the stable manifold, the evolution of OTOC with time is completely consistent before the Ehrenfest time $\tau=\frac{1}{\tilde{\lambda}}\ln N_p$,
and the exponential growth rate is twice the CLE of saddle point, i.e $\tilde{\lambda}=2\lambda_O$,
as shown in Fig. \ref{f4}(a). For the initial state centered at the point in the unstable manifold,
in Fig. \ref{f4}(b), we find that the time of the OTOC owning the same exponential growth rate as the saddle point depends on the position of the initial state in the phase space.
From the view of the evolution of the OTOC, Figs. \ref{f3} and \ref{f4}(b) further illustrate that both the photon number of system and the position of the initial coherent state affect the classical-quantum correspondence in the IHO system.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=12cm]{h1.eps}
\caption{Change of Husimi quasi-probabilistic wave packets with time for fixed system photon number $N_p = 300$.
The top, middle and bottom panels denote respectively the case in which the initial coherent state centered at the points
$O, A$ and $B$ in Fig. \ref{f11}.}
\label{f5}
\end{center}
\end{figure}

Husimi Q function is one of the quasi-probability distribution functions, and it was used to distinguish chaotic and periodic orbits in quantum systems\cite{hs1,hs2,EE1,EE2,EE3,EE4}.
The definition of Husimi Q function is
\begin{eqnarray}
Q(q_1,p_1) &=& \frac{1}{\pi}\langle q_1,p_1|\rho|q_1,p_1\rangle,    \label{hq1}
\end{eqnarray}
where $|q_1,p_1\rangle$ is photon coherent state and $\rho$ is the density matrix. In Fig. \ref{f5}, we present the change of Husimi Q distribution in phase
space for the IHO system with different initial values.
Figs. \ref{f5}(a)-\ref{f5}(c) and \ref{f5}(e)-\ref{f5}(g) show that the change of quasi-probability wave packets for the initial points
in the stable manifold own similar behaviors during the OTOC grows exponentially.
This means that the evolution of OTOC associated with the initial states centre on stable manifold are highly consistent before the Ehrenfest time,
which is also shown in Fig. \ref{f4}(a).
In Figs. \ref{f5}(i)-\ref{f5}(l), we show that the evolution of Husimi quasi-probabilistic wave packet with the initial state centered at point $B$ located on the unstable manifold.
It is found that the time  reduces for the wave packet owning the behaviors similar to that on stable manifold.
Actually, during the OTOC grows exponentially, it is easy to find that in the IHO system, the corresponding quantum wave packets spread along
the phase trajectory  in the classical phase space.
In addition, we present the Husimi quasi-probabilistic wave packets after the OTOC grows exponentially in Figs. \ref{f5}(d), \ref{f5}(h) and \ref{f5}(l) for points $O, A$ and $B$. We find that some discrete wave packets appear outside the primary wave packet and there is no the corresponding classical states, which is different from those during the OTOC exhibits exponential growth behavior.

\section{Exponential growth of OTOC in inverted harmonic oscillator with Higgs potential}
In this section, we focus on the inverted harmonic oscillator with a Higgs potential\cite{sp4},
which is classically non-chaotic but a nonzero CLE appears at the the unstable maximum of the potential.
We find that the OTOC of stable regular orbit far from the saddle point also grows exponentially in early time.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=6cm]{phap2.eps}
\caption{The classical dynamics phase diagram of HIHO.
Point $T$ is the saddle point and point $F$ (8, 9) is a point on a periodic orbit far away from the saddle point. Here, we set $\gamma = 3$ and $g = 1/25$. }
\label{f6}
\end{center}
\end{figure}
The one-dimensional inverted harmonic oscillator including the Higgs potential (HIHO) is described by the Hamiltonian
\begin{eqnarray}
H &\equiv& P^2 + \hat{V},  \ \ \ \ \ \ V \equiv -\frac{1}{4}\gamma^2 Q^2 + gQ^4 + \frac{\gamma^4}{64g},   \label{hiho1}
\end{eqnarray}
where $\gamma$ and $g$ are constant parameters and the term $Q^4$ ensures the existence of the potential boundary.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=6cm]{expv3.eps}
\caption{Time evolution of the mean photon number for the initial state centering at the point $F$ in Fig. \ref{f6} with different system photon number. Here, we set $\gamma = 3$ and $g = 1/25$. }
\label{f66}
\end{center}
\end{figure}
There is a saddle point $T$. According to Jacobian matrix theory\cite{sm1} and the tangent-space method\cite{tsm}, we calculate the CLE for the saddle point $T$ and the other point $F$ far from the point $T$, and find that $\lambda_T=\gamma$ and $\lambda_F=0$, respectively.
Substituting the quadratic quantization form of position and momentum operators into Hamiltonian (\ref{hiho1}),
\begin{figure}[ht]
\begin{center}
\includegraphics[width=6cm]{iho1c1.eps}
\caption{Time evolution of the OTOC for the initial state entered at the saddle point $T$ and the point $F$ in Fig. 6. The gray dotted line is the fitting curve.
Here, we set $N_p = 250$, $\gamma = 3$ and $g = 1/25$.}
\label{f7}
\end{center}
\end{figure}
one can obtain the quantum Hamiltonian of HIHO
\begin{eqnarray}
\hat{H}^{'} &=& -\frac{1}{2}(a^{\dag} - a)^2  -\frac{1}{8}\gamma^2 (a^{\dag} + a)^2 + \frac{g}{4} (a^{\dag} + a)^4 + \frac{\gamma^4}{64g}.   \label{hiho2}
\end{eqnarray}
In Fig. \ref{f66}, we find that the mean photon number in HIHO first oscillates and then tends to the saturate.
Moreover, the evolution of the mean photon number with time is indistinguishable when the system photon number is larger than $250$.
Thus, in the HIHO system with $\gamma = 3$ and $g = 1/25$, the system photon number $N_p=250$ can satisfy the requirement of photon number of quantum dynamics evolution.
Similarly, the OTOC at saddle point shows exponential behavior at early time due to the inherently instability as in IHO.
However, for the regular point $F$, the evolution of the OTOC also grows exponentially and its growth rate $\tilde{\lambda}_F\approx25.52$, which is larger than that of the saddle point, as shown in Fig. \ref{f7}.
In particular, the exponential growth behavior of OTOC for the initial state entered at point $F$ also occurs before the corresponding Ehrenfest time $\tau= \ln(N_p)/\tilde{\lambda}_F\approx0.22$,
which exacerbates the difficulty of discriminating chaos through OTOC.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=16cm]{h2.eps}
\caption{Change of Husimi Q distribution with time for fixed system photon number $N_p = 250$. Parts (a)-(d) denote the case in which the initial coherent state center at the saddle point T in Fig. \ref{f6}.
Parts (e)-(h) denote the case in which the initial coherent state center at stable point F in Fig. \ref{f6}. Here, we set $\gamma = 3$ and $g = 1/25$.}
\label{f8}
\end{center}
\end{figure}

We also study the early evolution behavior of the corresponding Husimi quasi-probabilistic wave packet in phase space.
In Fig. \ref{f8}, we find that during the OTOC grows exponentially, the quantum wave packets initially entered at points
$T$ and $F$ spread along the phase trajectory in the classical phase space.
Particularly, for the stable point $F$, the Husimi quasi-probabilistic wave packet diffuse rapidly along the vertical axis in phase space as the OTOC grows at an exponential rate, as shows in Figs. \ref{f8}(e)-\ref{f8}(h).
This rapid change of wave packet in a short time leads the quantum variance of momentum operator increases sharply, which lead the OTOC grows exponentially.

\section{Conclusion}
We have studied the early behavior of OTOC in two inverted harmonic oscillator system IHO and HIHO.
It is shown the OTOC starting from the coherent states centred on the regular orbits possess the behaviors of
exponential growth in these two systems.
For the IHO system, the OTOC from of the coherent states centred on stable manifold has the same growth rates as that at the saddle point at the early time, and their exponential growth rates
is twice the CLE of the saddle point.
For the coherent states centred on the unstable manifolds of IHO, the
behavior of OTOC deviating from exponential growth form appears more earlier than that in the stable manifold.
Moreover, we find that both the system photon number and
the position of the initial coherent state affect the classical-quantum correspondence relationship in IHO system.
We analyze the Husimi quasi-probability wave packets of different initial states during the
OTOC grows exponentially, and find that in both inverted harmonic oscillator systems, the corresponding
quantum wave packets spread along the phase trajectory in the classical phase space.
Particularly, in HIHO system, the quantum wave packets of stable regular orbits diffuse rapidly along the vertical axis in phase space,
which leads the OTOC shows false chaotic signals.
Our results could help to further understand the OTOC and correspondence principle.

\section{\bf Acknowledgments}

\vspace*{0.2cm}


text:
\section{Introduction}

Recently there has been a resurgence of interest in scale (Weyl) invariant theories as a possible solution to the hierarchy problem -- the need to keep the Brout, Englert, Higgs (BEH) boson \cite{Englert:1964et,Higgs:1964pj,Higgs:1964ia} light in the presence of gravity or large mass scales associated with physics beyond the Standard Model, such as Grand Unification. Such scale invariant theories have to generate all dimension-full scales through spontaneous breaking of the symmetry, including the Planck scale and the electroweak breaking scale associated with the BEH boson. To generate a period of inflation it is also necessary for the spontaneous scale symmetry breaking to give rise to the inflation scale and it has been shown that this is possible in the context of a model with two scalar fields non-minimally coupled to the Ricci scalar, $R$  \cite{ShapoZen,ShapoZen2,ShapoBlas,ShapoBell,Bezrukov:2012hx,GarciaBellido:2012zu,Rubio:2014wta,Trashorras:2016azl,Karananas:2016kyt,Casas:2017wjh,Casas:2018fum,Ferreira:2016vsc,Ferreira:2016wem,Ferreira:2016kxi,Ferreira:2018itt,Ferreira:2018qss,Ghilencea:2018thl,Ghilencea:2018dqd}.
Such models can naturally lead to an acceptable period of inflation and, for the case in which one scalar is identified with the BEH boson, are similar to ``Higgs" inflation \cite{Bezrukov:2007ep}.

In this paper we are interested in a simpler possibility, generalising $R + R^2$ inflation to a scale independent form \cite{Maeda:1987xf,Wang:1990pr,Herrera:1995me,Rinaldi:2015uvu,Bamba:2015uxa,Tambalo:2016eqr,Karam:2018mft,Kubo:2018kho,Wetterich:2019qzx}. The Planck constraints on the inflationary observables are in remarkable agreement with the $R+R^2$ model of Starobinsky \cite{Starobinsky:1980te}, proposed in 1980 and with a spectrum of CMB perturbations as analysed by Mukhanov and Chibisov \cite{Mukhanov:1981xt} shortly afterwards. In particular the model predicts a spectrum with scalar perturbation index $n_s \sim 0.96$ and tensor to scalar ratio $r \sim 0.004$.

As an $R^2$ term is already scale invariant, to build a fully scale invariant form it is only necessary to add a single ``Jordan Brans Dicke" \cite{Jordan:1959,Brans:1961sx} scalar field that, after inflation, is the dominant source of spontaneous breaking of the scale symmetry and is responsible for generating the Planck mass. Since the $R^2$ term involves fourth order derivatives it implies the existence of an additional propagating scalar degree of freedom similar to the two scalar model mentioned above. In the locally symmetric case one scalar is eliminated, providing the longtitudinal degree of freedom of a massive''Weyl photon" \cite{Utiyama:1973nq,Smolin:1979uz,Nishino:2011zz} and this has recently been extended to include an $R^2$ term in \cite{Ghilencea:2019jux}.

In this paper we analyse the inflationary properties of both the global and local scale invariant $R + R^2$ models. Our analysis simplifies previous studies of scale invariant $R^2$ inflation \cite{Maeda:1987xf,Wang:1990pr,Herrera:1995me,Rinaldi:2015uvu,Bamba:2015uxa,Tambalo:2016eqr,Karam:2018mft,Kubo:2018kho,Wetterich:2019qzx}. through the observation that ``inertial" spontaneous Weyl symmetry breaking leads to the decoupling of the dilaton/Weyl photon and results in single field inflation with the second scalar being either a massless dilaton or providing the longitudinal component of a massive Weyl photon. In the global case the decoupling of the dilaton eliminates the usual ``fifth-force" constraints on Jordan Brans Dicke models \cite{Ferreira:2016kxi,Brax:2014baa}. Moreover the recently developed frame-independent approach \cite{Karamitsos:2017elm,Ferreira:2018qss} allows us to determine the inflationary parameters directly in the Jordan frame where the scale invariance is manifest. As a result we are able to give analytic forms for the inflationary parameters. We find that the inflationary predictions of both the global and local Weyl invariant models are the same but differ from the original $R+R^2$ Starobinsky model, except in a particular limit.

In Section \ref{global} we construct the model that is globally invariant under scale (Weyl) symmetry. The origin of spontaneous ``inertial" Weyl symmetry breaking is discussed in Section \ref{inertial} where we show that this occurs independently of the scalar potential. In Section \ref{dilaton} we identify the dilaton, the Goldstone boson associated with the spontaneous breaking of the Weyl symmetry, and show that it decouples from the inflaton sector. Section \ref{local} analyses the locally Weyl symmetric model, and shows that it also undergoes inertial spontaneous symmetry breaking that generates a mass for the ``Weyl photon". The Weyl photon decouples from the inflaton sector which has the same form as found in the globally symmetric case. In Section \ref{inflation} it is shown that in a significant region of parameter space, slow-roll inflation occurs in both the globally and locally Weyl symmetric cases and gives an analytic form for the inflationary observables. The results are compared to the original Starobinsky model. Finally in Section V we present a summary and our conclusions.

\section{Globally Weyl invariant $R+R^2$ model.}\label{global}

\subsection{A Minimal Model}

The model consists of a real Jordan-Brans-Dicke scalar, $\phi$ non-minimally coupled to the Ricci scalar \cite{Jordan:1959,Brans:1961sx} plus the $R^2$ term introduced by Starobinsky \cite{Starobinsky:1980te}.
The action is given by
\bea
\label{R2S}
S &=& \int d^4x \sqrt{-g} \left( {1\over 2}g^{\mu\nu}\partial_\mu\phi\partial_\nu\phi  \right.
        \nonumber \\
&& \qquad   \left.
-\frac{\lambda}{4}\phi^4-{1\over 12}\alpha_1\phi^2R+ \frac{1}{6f_0^2} R^2 \right)
\eea
This theory is the most general one relevant to the inflationary era that can be constructed from these fields that is invariant under the global Weyl (scale) transformation
\bea
\phi&\rightarrow&e^{-\epsilon} {\phi}\nonumber \\
g_{\mu\nu}&\rightarrow&e^{2\epsilon} { g}_{\mu\nu}
\label{WT}
\eea
Note that other, quadratic, terms could be included (involving the Ricci and Riemann tensors) but which are absent (or do not contribute to the overall dynamics) during the inflationary regime.

Since the $R^2$ term involves fourth order derivatives it contains an additional (scalar) degree of freedom.  To make this explicit it is convenient to reduce the fourth order derivatives to second order by introducing the
auxiliary field $\eta$\footnote{By redefining the fields on moving to the Einstein frame one obtains an explicit kinetic energy term for the new field $\eta$ with the canonical form for a scalar field coupled to gravity.}  \cite{Hindawi:1995an,Whitt:1984pd}  with the action now given by
\bea
\label{R3S}
   S &=&  \int d^4x \sqrt{-g}
        \left( {1\over 2}g^{\mu\nu}\partial_\mu\phi\partial_\nu\phi -{\lambda\over 4}\phi^4
        \right. \nonumber \\
      &&\qquad \left.  -{1\over 12}\alpha_1\phi^2R
       - {1\over 12}\alpha_2\eta^2 R-{\xi\over 4}\eta^4 \right)

   \eea
Here the equation of motion for $\eta$ gives
\be
\eta^2=-{\alpha_2\over 6\xi}R
\label{eta0}
\ee
which, substituted in the eq.(\ref{R3S}), gives eq.(\ref{R2S}) for
\be
\xi=6f_0^2({\alpha_2\over 12})^2
\label{B}
\ee
The action is very close to that found in the 2-scalar model analysed by several groups \cite{ShapoZen,ShapoZen2,ShapoBlas,ShapoBell,Ferreira:2016vsc,Ferreira:2016wem,Ferreira:2016kxi,Ferreira:2018itt,Ferreira:2018qss,Ghilencea:2018thl}; in the Jordan frame
the difference is the absence of a kinetic term for $\eta$.

\subsection{Inertial symmetry breaking}\label{inertial}

Under the Weyl transformation, $\eta\rightarrow e^{-\epsilon}\eta$ the Weyl current is given by
\be
K_\mu\equiv  \frac{1}{\sqrt{-g}}  {\delta S\over \delta \partial^\mu \epsilon}=(1-\alpha_1)\phi\partial_\mu\phi-\alpha_2\;\eta\partial_\mu\eta
\ee
It can be written as $K_\mu = \partial_\mu K$, where the kernel $K$ is given by
\be
K={1\over 2}\left((1-\alpha_1)\phi^2-\alpha_2\eta^2\right)
\label{kernel}
\ee
and is covariantly conserved
\be
D^\mu K_\mu=0
\label{cons}
\ee
Consider now a patch of the universe which can be described as approximately spatially constant, but time dependent. These regions can be described by the Friedman- Robertson-Walker (FRW) metric corresponding to
$g_{\mu\nu}=[1,-a^2(t),-a^2(t),-a^2(t)]$
 \ In terms of the kernel the conservation law of eq.(\ref{cons}) becomes
 \be
 \ddot K+3H\dot K=0
 \ee
 where $H={\dot a}/a$,
giving
\be
K(t)=c_1+c_2\int_{t_0}^t{dt'\over a^3(t')}
\ee
 where $c_1$ and $c_2$ are constants. Thus in an expanding universe $K(t)$ will evolve to a constant value, $\bar{K} = {K}(t\rightarrow\infty)$ that, from eq.(\ref{kernel}) implies the scalar fields acquire constant vacuum expectation values, spontaneously breaking Weyl symmetry \cite{Ferreira:2018itt}.

 Note that this inertial Weyl symmetry breaking does not rely on a scalar potential, the value of the kernel being determined by the chaotic initial conditions. Note also that a constant kernel implies a relation between the scalar fields which is the reason the model results in single field inflation.

\subsection{Dilaton Decoupling}\label{dilaton}

To identify the Goldstone mode associated with the spontaneous breaking of the global Weyl symmetry we change variables to
\bea
\phi&=&e^{-\sigma(x)/f} \hat\phi\nonumber \\
\eta&=&e^{-\sigma(x)/f} \hat\eta\nonumber \\
g_{\mu\nu}&=&e^{2\sigma(x)/f}  \hat g_{\mu\nu}
\label{variable}
\eea
 where $f$ has dimensions of mass. Note that $\hat\phi,\;\hat\eta$ and $\hat g_{\mu\nu}$ are invariant under the global Weyl symmetry and only $\sigma$ transforms, $\sigma\rightarrow\sigma +c$ where $c=\epsilon f$. In terms of the new metric
 \be
R(g)=e^{2\sigma/f}\left( R(\hat g)+6\hat D^2\sigma/f+6(\hat D\sigma/f)^2\right)
\ee
giving
\bea
S &=&  \int d^4x \sqrt{-g}
\left( - \frac{1}{12}({\alpha _1}{{\hat \phi }^2} + {\alpha _2}{{\hat \eta }^2})\hat R
+ \frac{1}{2}{\partial _\mu }\hat \phi {\partial ^\mu }\hat \phi
\right.
\nonumber \\
&& \left.
 + \frac{1}{{f^2}}\hat K{\partial _\mu }\sigma {\partial ^\mu }\sigma  + \frac{1}{f}{\partial _\mu }\sigma {\partial ^\mu }\hat K - \frac{\lambda }{4}{{\hat \phi }^4} - \frac{\xi }{4}{{\hat \eta }^4}\right.\nonumber\\
&&\left. +\lambda_LC(\hat\phi,\hat\eta)\right)
\eea
where we have added a Lagrange multiplier, $\lambda_L$, and the constraint
\be
C(\hat\phi,\hat\eta)=\bar K -\frac{1}{2}\left( {(1 - {\alpha _1}){{\hat \phi }^2} - {\alpha _2}{{\hat \eta }^2}} \right)
\label{khat}
\ee
For the constant $\bar K$ this gives the ``Einstein frame'' form of the theory
\bea
S &=& \int d^4x \sqrt { -\hat g} \left( { - \frac{1}{{12}}({\alpha _1}{{\hat \phi }^2} + {\alpha _2}{{\hat \eta }^2})\hat R} + \frac{1}{2}{\partial _\mu }\hat \phi {\partial ^\mu }\hat \phi \right. \nonumber \\
&& \left.
 - \frac{\lambda }{4}{{\hat \phi }^4} - \frac{\xi }{4}{{\hat \eta }^4} + \frac{1}{2}{\partial _\mu }\sigma {\partial ^\mu }\sigma +\lambda_LC(\hat\phi,\hat\eta) \right)
\label{EFaction}
\eea
where the dilaton is canonically normalized with the choice $f^2=2\bar {K} $.

The field $\sigma$ is the massless dilaton. We see that it decouples from the hatted scalar fields. The same is true if one adds vector and fermion fields to the theory \cite{Ferreira:2018itt}. As a result one avoids the severe astrophysical constraints on the fifth force normally associated with the dilaton.

Moreover, note that the dilaton now yields the Weyl current in the Einstein frame.
We see that under the Weyl
transformation $\delta\sigma/f = \delta\epsilon$, where all hatted variables are invariant,
the Weyl current is now:
\bea
K_\mu\equiv  \frac{1}{\sqrt{-g}} \frac{\delta S}{ \delta \partial^\mu \epsilon} = f\partial_\mu\sigma
\eea
which is the familiar form of the current of a Nambu-Goldstone boson.
Inertial symmetry breaking can be understood in the ``Einstein frame''
as a chaotic initial $K(x)$ parameterized by an arbitrary constant $\bar{K}$ and an initially
chaotic dilaton field $\sigma(x)$, as $K=e^{2\sigma(x)/f}\bar{K}$, with $f^2=2\bar{K} $.
This dilaton form of the current,  $f\partial_\mu\sigma $, is covariantly
conserved, equivalent to the massless dilaton equation of motion, $D^2 \sigma =0$. This
implies that the dilaton redshifts to a constant, yielding the spontaneously broken
scale symmetry.

\section{ LOCALLY WEYL INVARIANT $R + R^2$ MODEL}\label{local}
For the case of local Weyl transformations, $\epsilon=\epsilon(x)$ in eq.(\ref{WT}), the action, eq.(\ref{R2S}), is no longer invariant. To correct this it is necessary to introduce a vector field, $A_\mu$, the ``Weyl photon" that transforms as
\be
A_\mu\rightarrow A_\mu-\partial_\mu \epsilon (x)
\ee
With this one can define a Weyl derivative acting on scalar fields, $\phi$, by
${\tilde D_\mu }\phi  \equiv ({\partial _\mu } - {A_\mu })\phi $ that transforms covariantly
\be
\tilde D_\mu\phi\rightarrow e^{-\epsilon(x)}\tilde D_\mu \phi
\ee
and replaces the Riemannian derivative when constructing a locally invariant version of eq.(\ref{R2S}). Similarly it is necessary to replace the Riemannian Ricci scalar, $R$, with its Weyl geometric version, $\tilde R$ defined by
\be
\tilde R = R - 6{D_\mu }{A^\mu } - 6{A_\mu }{A_\mu }
\label{tildeR}
\ee
 where $D_\mu$ is the Riemannian covariant derivative. Now $\tilde R$ transforms covariantly under local Weyl transformations
 \be
\tilde R\rightarrow e^{-2\epsilon(x)}\tilde R
\ee
 Using these covariant forms the locally invariant version of eq.(\ref{R2S}) is \cite{Ghilencea:2019jux,Ghilencea:2018dqd}
\bea
{S_L} &=&
\int {d^4}x\sqrt {- g} \left( \frac{1}{2}{g^{\mu \nu }}{{\tilde D}_\mu }\phi {{\tilde D}_\nu }\phi  - \frac{\lambda }{4}{\phi ^4}
\right. \nonumber \\
&& \left.
- \frac{1}{{12}}{\alpha _1}{\phi ^2}\tilde R + \frac{1}{{6f_0^2}}{{\tilde R}^2} - \frac{1}{{4{e^2}}}{F^{\mu \nu }}{F_{\mu \nu }}\right)
\eea
where we have added to the Lagrangian the Weyl invariant Weyl photon kinetic term, $-{1\over 4e^2}F^{\mu\nu}F_{\mu\nu}$ where $F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$. Introducing an auxiliary field as above this can be rewritten as
\bea
S_L &\equiv & \int d^4 x \sqrt { - g}
\left(\frac{1}{2}g^{\mu \nu }{{\tilde D}_\mu }\phi {{\tilde D}_\nu }\phi  - \frac{\lambda }{4}{\phi ^4}
\right. \nonumber \\
& & \!\!\!\!\!\!\!\!\!   \left.
- \frac{1}{12}({\alpha _1}{\phi ^2} + {\alpha _2}{\eta ^2})\tilde R - \frac{\xi }{4}{\eta ^4} - \frac{1}{4{e^2}} {F^{\mu \nu }}{F_{\mu \nu }} \right)
\eea
Substituting $\tilde R$, using eq.(\ref{tildeR}), integrating by parts,
and rescaling  $A_\mu \rightarrow eA_\mu$ gives
\bea
\label{SL0}
{S_L} & =& \int {d^4}x\sqrt { - g} \left( {\frac{1}{2}{\partial _\mu }\phi {\partial ^\mu }\phi  - \frac{1}{{4}}{F^{\mu \nu }}{F_{\mu \nu }} - \frac{1}{2}e{A_\mu }{K^\mu } }\right.
\nonumber \\
& & \!\!\!\!\!\!\!\!\!  \!\!\!\!\! \!\!\!\!\!   \left.
 + \frac{1}{2}{e^2}K {A_\mu }{A^\mu } - \frac{1}{{12}}({\alpha _1}{\phi ^2} + {\alpha _2}{\eta ^2})R - \frac{\lambda }{4}{\phi ^4} - \frac{\xi }{4}{\eta ^4} \right)
\eea
where
\be
{K_\mu } = {\partial _\mu }K,\qquad K = \frac{1}{2}\left((1 + {\alpha _1}){\phi ^2}
+ {\alpha _2}{\eta ^2}\right)
\ee
Note that the variation of the action or eq.(\ref{SL0}) with respect to $\partial_\mu\epsilon(x)$,
with  $\delta eA_\mu= \partial_\mu \epsilon (x)$,
now yields the vanishing identity:
\bea
\frac{1}{{\sqrt {- g} }}\frac{{\delta S_L}}{{\delta {\partial^\mu\epsilon }}}
= K_\mu -  {\partial _\mu }K=0
\eea
In any local gauge theory the variation of the full action
with respect to the local gauge angle always
produces zero, since this is the very definition of the symmetry.

The conserved Weyl current in the global case
is now replaced by the Weyl photon equation of motion
\be
  \frac{1}{{\sqrt { - g} }}\frac{{\delta S_L}}{{\delta {A^\mu }}}
  = D_\nu F^{\mu\nu} - 2{A_\mu }{e^2}K- e{K_\mu }=0
\label{weylcurrent}
\ee
  Changing variables as in eq.(\ref{variable}), for constant $\bar K$, the action becomes
\bea
S_L &=& \int \sqrt g \left( { - \frac{1}{{12}}({\alpha _1}{{\hat \phi }^2} + {\alpha _2}{{\hat \eta }^2})\hat R + \frac{1}{2}{\partial _\mu }\hat \phi {\partial ^\mu }\hat \phi
}- \frac{\lambda }{4}{{\hat \phi }^4}\right. \nonumber \\
&& \left. - \frac{\xi }{4}{{\hat \eta }^4} - \frac{1}{{4}}{F^{\mu \nu }}{F_{\mu \nu }} + \frac{1}{2}e^2\bar K\;{B_\mu }{B^\mu+\lambda_L C } \right)
\eea
where $B_\mu=A_\mu-{e\over f}\partial_\mu\sigma$.

The spontaneous breaking of the Weyl symmetry has generated a mass for the Weyl photon,
$M^2=e^2\bar{K}$, with the would-be Goldstone mode, $\sigma$, providing its longitudinal component. As for the dilaton in the global case the Weyl photon decouples from the hatted fields.  In the present local case the dilaton, $\sigma$
and the gauge field, $A_\mu$,  are not separately physical. They are replaced by the combined
(Stueckelberg) field $B_\mu$ which is physical and invariant under local Weyl transformations.

Note that the conserved Weyl current is now given by the divergence
of the field $B_\mu$ itself, which satisfies
the Weyl Maxwell equation:
\be
D_\nu F^{\mu\nu} - M^2{B_\mu }=0  \qquad M^2=e^2\bar{K}
\ee
A massive spin-1 field necessarily obeys the Lorentz gauge condition
$D_\mu B^\mu =0$ owing to the anti-symmetry of $F_{\mu\nu}$ (note that a massive
spin-1 field has mass term $-M^2B_\mu$, contrary to a scalar $+M^2\phi$).
Hence an arbitrary initial physical $B_\mu$ will redshift to zero, yielding the
spontaneously broken scale symmetry vacuum, with $B_\mu=0$, and  which is the analogue
of the redshifting of $K_\mu\rightarrow 0$ and  $K_\mu\rightarrow \bar{K}$ in the global case.
The vacuum is analogous to a superconductor, where the massive $B_\mu$ has the solution
$B_\mu=0$, which is the analogue of the London equation for a superconductor,
$e\vec{A}-\vec{j}=0$.

\section{INFLATION}\label{inflation}

It is now straightforward to analyse the possibility that there is a period of inflation in both the global and local cases. The decoupled fields, the dilaton in the global case or the Weyl photon in the local case, play no role in the inflationary era. In their absence the action for the hatted fields is the same for both cases and so their inflationary structure is the same.

In the Jordan frame the Klein Gordon equations resulting from eq.(27) are given by
\bea
{D^2}\hat \phi  + \lambda {\hat \phi ^3} + \frac{1}{6}{\alpha _1}\hat \phi R &=& 0\nonumber\\
 \xi {\hat \eta ^3} + \frac{1}{6}{\alpha _2}\hat \eta R &=& 0
\eea
Thus
\be
{\alpha _2}\frac{{{D^2}\hat \phi }}{{\hat \phi }} =  - {\alpha _2}\lambda {\hat \phi ^2} + {\alpha _1}\xi {\hat \eta ^2}
\label{roll1}
\ee
Note that there is an infrared stable fixed point that determines the ratio of the field vacuum
expectation values (VEVs) given by
\be
{\hat\eta^2\over\hat\phi^2}={\alpha_2\lambda\over\alpha_1\xi}
\ee
that is approached after any period of inflation \cite{Ferreira:2016vsc} .

Taking the slow-roll limit with the ``inflation derivative" (i.e. in terms of the number of e-foldings) $ {D^2}\hat \phi  \to 3H\dot \hat \phi  = 3{H^2}{\partial _N}\hat \phi $ where $ N = \ln a(t) $ gives
\be
{3\over 2}H^2\alpha_2{\partial_N\hat\phi^2\over\hat\phi^2}=-\alpha_2\lambda\hat\phi^2+\alpha_1\xi\hat\eta^2
\ee
$H^2$ may be eliminated using the (00) Einstein equation in the slow-roll limit
\be
G_{00}=3H^2\approx-{3\over 2}g_{00}{\lambda\hat\phi^4+\xi\hat\eta^4\over\alpha_1\hat\phi^2+\alpha_2\hat\eta^2}
\ee
From eq.(\ref{khat}) one sees that the constraint that the Kernel should be constant corresponds to the fields being constrained to move on an ellipse. Without loss of generality we can choose the ellipse $\bar K = 1$ and map the first quadrant of the ellipse into the variables
\be
x=(1-\alpha_1)\hat\phi^2,\;\;y=-\alpha_2\hat\eta^2,
\label{roll2}
\ee
so that the ellipse becomes the line segment $x + y = 1$. Using eqs(\ref{roll1}-\ref{roll2}) we find
  \be
  \partial_Nx={S(x)\over W(x)}x(1-x)(x-x_0)
  \label{xevolution}
  \ee
    where
    \be
    S(x)={1\over 3}\times{-\alpha_2 \lambda+\alpha_1(1-\alpha_1)\xi\over (1-\alpha_1)^2\alpha_2^2} \times{1-x-\alpha_1\over 1-x}
    \ee

    \be
    x_0={\xi(1-\alpha_1)\over \xi(1-\alpha_1)-\alpha_2^2\lambda}
    \ee
    and
    \be
    W(x)={\lambda x^2\over 4(1-\alpha_1)^2}+{\xi(1-x)^2\over 4\alpha_2^2}
     \ee
    There are various parameter choices that can give slow roll inflation. For the case in which one coupling dominates, $\partial_Nx$ is independent of the coupling. For the case in which $\xi$ dominates,  $\partial_Nx\propto {\alpha_1\over 1-\alpha_1}$ and slow roll can occur for $\alpha_1$ small. For the case in which $\lambda$ dominates  $\partial_Nx\propto {1-x-\alpha_1\over \alpha_2}$ and slow roll can occur for $\alpha_2$ large.
Here we examine the former case with $\lambda\ll\xi$ and $\alpha_1$ small which makes it easy to compare to the 2-scalar case that assumed the same limit \cite{Ferreira:2018qss}. Keeping only $\xi$ non-zero we have
\be
{\partial _N}x=  - \frac{4}{3}{\alpha _1}x\ee
  \be
     H^2\approx {\xi y^2\over 2\alpha_2^2(y-\alpha_1 x)}
     \label{hubble}
      \ee
 and $x(t)$ will roll from an initial $x(0)$ toward $x(t_E) \equiv  x_E \approx 1$. Here $t_E$ is the end of inflation given by the inflation parameter, $\epsilon$, approaching unity: $\epsilon=-{1\over 2}{d\ln H^2\over dN}\approx 1$ corresponding to $y\approx -{2\over 3}\alpha_1$ or, equivalently, $\alpha_1\hat\phi^2\approx\alpha_2\hat\eta^2$.
Thus
  \be \label{phisol}
     x=x_Ee^{-\nu N_J},\;\;y=1-x_Ee^{-\nu N_J},\;\;\nu=-{4\over 3}\alpha_1
     \ee
where $N_J$ is the number of e-foldings until the end of inflation in the Jordan frame. This is the same form as was found for the 2-scalar model.

For clarity, the discussion to date has been in the Jordan frame. To determine the slow roll parameters we use the frame independent formalism introduced in \cite{Karamitsos:2017elm} and as applied in  \cite{Ferreira:2018qss}. Denoting $\theta_1=\hat\phi,\;\theta_2=\hat\eta$, the frame-invariant field-space metric $G_{AB}$ and potential $U$ are given by
\be \label{EffDef}
{G_{AB}} = \frac{{{\delta _{11}}}}{f} + \frac{3}{2}\frac{{{f_{,A}}{f_{,B}}}}{{{f^2}}},\;\;\;U = \frac{{\lambda \;\theta _1^4 + \xi \;\theta _2^4}}{{4{f^2}}},
\ee
{where the key difference to the two-field model considered in \cite{Ferreira:2018qss} is the occurrence of $\delta_{11}$ (instead of $\delta_{AB}$) and where}
\be \label{fdef}
f \equiv M_P^2 \equiv  - \frac{1}{6}\sum\limits_{A = 1}^2 {{\alpha _A}} \theta _A^2.
\ee

Defining the scalar (tensor) spectral index, $n_S$ ($n_T$), the running of the scalar (tensor) spectral index, $\alpha_S$ ($\alpha_T$) and the tensor to scalar ratio, $r$, we have that the frame independent analysis proceeds as before, giving for the inflation observables
\bea
n_S &=& 1 + \frac{4\alpha_1 (e^{-\nu N_J}+1)}{3(1 - e^{-\nu N_J})} +  {\cal O}(\alpha_1^2), \nonumber \\
r &=& \frac{64 \alpha_1^2  e^{-\nu N_J}}{3  (e^{-\nu N_J} - 1)^2} + {\cal O}(\alpha_1^3), \nonumber \\
\alpha_S &=&- \frac{32 \alpha_1^2 e^{-\nu N_J}}{9 (e^{-\nu N_J}-1)^2} + {\cal O}(\alpha_1^3), \nonumber \\
n_T &= &-  \frac{8 \alpha_1^2  e^{-\nu N_J}}{3  (e^{-\nu N_J} - 1)^2}  + {\cal O}(\alpha_1^3) , \nonumber \\
\alpha_T &=& - \frac{32 \alpha_1^3  e^{-\nu N_J} (1 + e^{-\nu N_J})}{9 (e^{-\nu N_J} - 1)^3}  + {\cal O}(\alpha_1^3),
\label{obsExp}
\eea
 where the relation between the number of e-foldings in the Einstein and Jordan frames is given by
 \begin{align} \label{Nsol2}
N_E &= N_J + \frac{1}{2} \ln\left(\frac{2\alpha_1 }{(e^{-\nu N_J} - 1 )}\right)
+ {\cal O}(\alpha_1) \\
&= {N_J + \frac{1}{2} \ln \left(\frac{3}{2N_J}\right) + {\cal O}(\alpha_1)}, \nn
\end{align}
{where we have expanded the exponential for small $\alpha_1$ in the second line and assume $N_J \gg 1$. For $N_J = 60$, we therefore obtain $N_E \sim 58$.}
{One feature that is immediately apparent is the absence of $\alpha_2$ in \eqref{obsExp}. Indeed we have explicitly checked up to $10$th order in $\alpha_1$ that no such dependence is present. In Appendix \ref{app:limit} we show why this is the case, but here we simply emphasise that despite the appearance of four parameters $(\alpha_1,\alpha_2,\lambda,\xi)$, in the $\xi \gg \lambda$ limit we are considering here only one single parameter controls all of the above observables. Working in this limit then also enforces a tight observational bound on $\alpha_1$: Current constraints on $n_S$ (Planck 2018 \cite{Aghanim:2018eyx,Akrami:2018odb} finds $n_S=0.9649 \pm 0.0042$) enforce $|\alpha_1| < 0.019$.\footnote{For the model considered here there is also an upper bound on $n_S$, namely $n_S \lesssim 1-2/N_J\simeq 0.9663$ (where we are assuming $N_J\simeq60$), which is saturated as $\alpha_1\rightarrow 0$.} The tensor-to-scalar ratio $r$, on the other hand, is always comfortably within current observational bounds ($r \lesssim 0.064$ -- see \cite{Aghanim:2018eyx,Akrami:2018odb}). In fact, the precise value for $r$ can be accurately predicted for our model: $0.0033 > r > 0.0026$, where the lower bound is a consequence of the $n_S$ induced bound on $\alpha_1$ and the upper bound is saturated for $\alpha_1 \sim 0$ (and we again assume $N_J\simeq60$))

The remaining observables in \eqref{obsExp} satisfy $a_S \sim -2/N_J^2 \sim -5 \cdot 10^{-4}$, $a_T \sim - 10^{-5}$ and $n_T \sim - 4 \cdot 10^{-4}$, for all allowed choices of $\alpha_1$. Also, to reproduce the observed magnitude of the fluctuations it is neccessary to choose ${\xi\over\alpha_2^2}=O(10^{-10})$.
}

{
The above results may be compared to the original Starobinsky model. Focusing on $n_S$ and $r$, there one finds
\be
n_s-1\approx-{2\over N_E},\;\;r\approx{12\over N_E^2}.
\ee
Expanding the observables in \eqref{obsExp} in the limit $\alpha_1\rightarrow 0$ for general $N_J$ (assumed to satisfy $N_J \gg 1$), the scale invariant model gives
\be
n_s-1\approx-{2\over N_J},\;\;r\approx{12\over N_J^2}
\ee
{at leading order in $N_J^{-1}$}, so that the predictions are very close to the original Starobinsky model (recall that $N_J$ and $N_E$ only differ by $\sim 3\%$ in the scale invariant model investigated here, assuming close to 60 e-folds of inflation).
}

{
Finally a note on observables beyond the 2-point function and on isocurvature perturbations. The essential features here are identical to the ones of the scale-independent two field models considered in \cite{Ferreira:2018qss}: The local non-Gaussian parameter $f_{NL}$ satisfies \cite{Elliston:2012ab,Byrnes:2012sc}
\begin{align}
f_{NL}^{\rm local} &\approx \frac{5}{6}\frac{N^{,A}N^{,B}(\nabla_A\nabla_BN)}{(N_{,C}N^{,C})^2},
\end{align}
where $N$ is the frame-covariant number of e-folds ($N = N_E$) \cite{Karamitsos:2017elm}. Evaluating this expression and expanding for small $\alpha_1$ we find
\begin{equation}
f_{NL}^{\rm local} \approx \frac{5 \alpha_1 (e^{-\nu N_J}+1)}{9 (e^{-\nu N_J}-1)} + {\cal O}(\alpha_1^2) \sim 1.5 \times 10^{-2},
\end{equation}
so no sizeable non-Gaussian signature. Indeed this is to be expected from the existence of a physically equivalent single scalar theory with a canonical kinetic term (see appendix \ref{app:single}). The effective single field nature then mandates suppressed local non-Gaussianity \cite{Maldacena:2002vr,Acquaviva:2002ud}, while the canonical nature of the kinetic term ensures the absence of sizeable equilateral non-Gaussianity (because, for general single-field models, $f_{NL}^{\rm equil} \lesssim 1/c_s^2$).
Significant isocurvature perturbations are also absent, since the acceleration vector between paths in the geodesic flow
\begin{eqnarray}
\omega^A&=&X^B\nabla_B\left[\frac{X^A}{\sqrt{X_C X^C}}\right] = 0
\label{isoparams1}
\end{eqnarray}
for the models considered here (up to second order in slow-roll), where we have defined $X_A \equiv (\ln U)_{,A}$. When $\omega^A = 0$, the transfer function that converts curvature perturbations into isocurvature perturbations vanishes \cite{Karamitsos:2017elm,Ferreira:2018qss}, so no sizeable isocurvature perturbations are generated for these models.
}

\section{ SUMMARY AND CONCLUSIONS}
We have studied the scale free version of $R^2$ inflation, setting it within the framework of the scale-invariant theories extensively studied in \cite{ShapoZen,ShapoZen2,ShapoBlas,ShapoBell,Bezrukov:2012hx,GarciaBellido:2012zu,Rubio:2014wta,Trashorras:2016azl,Karananas:2016kyt,Casas:2017wjh,Casas:2018fum,Ferreira:2016vsc,Ferreira:2016wem,Ferreira:2016kxi,Ferreira:2018itt,Ferreira:2018qss,Ghilencea:2018thl,Ghilencea:2019jux,Ghilencea:2018dqd}.  It can be shown to correspond to the two field model but without a canonical kinetic term for one of the scalar fields in the Jordan frame (for a more careful analysis of this limit, see Appendix \ref{app:limit}).
As a consequence, we have shown that scale-free $R^2$ inflation is endowed with inertial symmetry breaking of Weyl invariance and that the techniques that were developed in \cite{Ferreira:2016vsc,Ferreira:2016wem} can be applied seamlessly to this scenario and used to derive predictions for the inflationary observables. The model we have considered here is, to some extent, the simplest non-minimally coupled theory which is scale invariant; it falls into the class of models considered in \cite{KS2014} which are favoured by the Planck constraints.

The effect of radiative corrections on the inflationary era have been briefly discussed in the context of the two scalar model \cite{Ferreira:2016wem,Ferreira:2016kxi}
 and the general structure applies to the model considered here. In particular, if the scale invariance is preserved by quantum corrections by a suitable choice of the regularisation procedure that does not introduce an extrinsic mass scale, inertial symmetry breaking still occurs and the dilaton still decouples. There will still be a period of slow roll inflation but the form of the constraint between the hatted fields will change and the ellipse on which they move will be distorted \cite{Ferreira:2016kxi} giving a small change in the inflationary predictions provided the couplings are in the perturbative  range.

We have shown that the inflationary observables -- $n_S$, $r$, $\alpha_S$, $n_T$ and $\alpha_T$ -- are uniquely determined in terms of the small parameter, $\alpha_1$, which controls the non-minimal coupling of the extra scalar field. As a result, and with current constraints on the scalar spectral index, $n_S$, we can place tight bounds, $|\alpha_1|<0.019$, which in turn leads to a very clear prediction for the tensor to scalar ratio: $0.0033 > r > 0.0026$. This small range of values is within reach of the constraining power of future experiments: while the Simons Observatory \cite{SimonsObs} has a forecast sensitivity of $\sigma(r) = 0.003$, the LITEBIRD Mission  will push the sensitivity to $\sigma(r) = 0.001$, while the S4 CMB consortium will aim for $\sigma(r) = 0.0005$ \cite{rforecasts}. This means that this model offers a characteristic and precise prediction that will be experimentally testable within the next decade.
\\

\textit{Acknowledgements ---} PGF is supported by the ERC and the Beecroft Trust. JN acknowledges support from Dr.\ Max R\"ossler, the Walter Haefner Foundation and the ETH Zurich Foundation.
CTH acknowledges Fermilab, operated by Fermi Research Alliance,
LLC under Contract No. DE-AC02-07CH11359 with the United States Department of
Energy.


text:
\section{Introduction} \label{sec:introduction}
The Quantum Generative adversarial networks (GAN) for the qubit model of quantum computation is described in \cite{qugan1} where the generator and discriminator are modelled as quantum circuits composed of unitary gates. An important challenge for Quantum GANs is to effectively implement affine and non-linear transforms. Attempts have been made to induce non-linearity indirectly \cite{qubitdisadv3, qubitdisadv4} but the resulting circuit has significant failure probability \cite{circuitclassifier, cvnn}. Further, Qubit based quantum computers are only partially suitable for continuous valued problems since the measurement output of the circuits is discrete \cite{cvnn, qubitdisadv1, qubitdisadv2}. Continuous variable (CV) quantum computation \cite{cvbegin1, cvbegin2} is an alternative to the qubit model. The state of the smallest unit of CV computation, \textit{qumode}, is part of an infinite dimensional Hilbert space. Information is encoded in quantum states of fields and the system is suitable for implementation on photonic hardware \cite{cvintro1, cvintro2, cvintro3}.
It is shown in \cite{cvbegin2} that the qubit model can be implemented using such a photonic system without any loss of universality. Moreover, when the continuous quadrature of canonical position ($\hat{x}$) and canonical momentum ($\hat{p}$) operators are used for information processing, the CV model lends itself for constructing neural networks \cite{cvnn}.

We present OpticalGAN, an extension of Quantum GAN for continuous variable quantum computation. We use quantum variational circuits to simulate neural networks. Affine transformations are implemented using Gaussian operations on the initial vacuum state and non-linearity is implemented using the Kerr and Cubic phase gates \cite{quantumoptics, stateprep}. The generator and discriminator are implemented using these variational circuits and trained in an adversarial setting as described in \cite{qugan1}. During training the parameters of the variational circuit are optimized using backpropagation to minimize the adversarial loss \cite{gan}. The contributions of our work are:
\begin{itemize}
  \item We propose \textbf{OpticalGAN}, a generative adversarial network formulated for continuous variable (CV) quantum computation. The quantum circuits of OpticalGAN simulate generator and discriminator networks in CV settings using optical gates. (Section \ref{sec:opticalgan}).

  \item We show the efficacy of OpticalGAN by training it to \textbf{generate fock and coherent states } (Section \ref{sec:expresults}). Our experiments show that during adversarial training, the cross-entropy loss between the generated and target distribution tends to zero.
\end{itemize}

\section{OpticalGAN} \label{sec:opticalgan}

\begin{figure}
\centering
\begin{subfigure}{.5\textwidth}
  \centering
  \includegraphics[width=1\linewidth]{opticalgan}
  \caption{Architecture of OpticalGAN}
    \label{fig:opticalgan}
\end{subfigure}%
\begin{subfigure}{.5\textwidth}
  \centering
  \includegraphics[width=0.5\linewidth]{genmerged.png}
  \caption{Generated coherent state with $\alpha=1$ and Generated Fock state $\ket{9}$}
  \label{fig:coherentgen}
\end{subfigure}

\caption{Architecture and Results}
\label{fig:test}
\end{figure}

The architecture of OpticalGAN is shown in Fig. \ref{fig:opticalgan}. There are four components, a real data source, generator and discriminator quantum circuits and a classical computer. The data source generates quantum states that are representative of the real data distribution. The mapping from classical data to quantum states can be achieved as in \cite{cvnn}. We assume that the quantum state generated can be represented using a single qumode for the sake of simplicity. The following sections first discuss the basic building block of OpticalGAN, the quantum neural network layer \cite{cvnn}. We then describe the generator, discriminator variational circuits and finally the cost function and training paradigm.

In order to implement a quantum neural network layer $\mathcal{L}$, we use \cite{cvnn}. The essential idea is that by using interferometers, squeeze gates (denoted by $S$), displacement gates ($D$) and a non-gaussian Kerr gate ($K$), one can create quantum neural networks. We use the layer circuit described in \cite{cvnn} as a component in the generator and discriminator variational circuits.

\subsection{Generator and Discriminator variational circuits} \label{sec:generator}
The generator is a variational circuit consisting of two qumodes and the output is obtained from one of the two qumodes, as shown in Fig. \ref{fig:generator}. It takes as input a gaussian random variable $z$ as input and produces a quantum state. $z$ is used as a parameter for the displacement gate $D$. The gate is applied on one of the two qumodes that are instantiated in the vacuum state $\ket{0}$. This is followed by variational sub-circuits denoted by $\mathcal{L}$ \cite{cvnn}. Since the $\mathcal{L}$'s can be stacked, the variational circuit can be thought of as a quantum neural network with several layers. $\vec{\theta_g}$ denotes the parameters of the generator circuit. $\vec{\theta_g}$ is tuned during training so that the generator produces an output that matches the distribution of the real data.

The discriminator circuit is similar to the generator. The key difference is that it takes as input the state produced by either the real source or the generator and tries to distinguish them. It uses quantum neural network layers \cite{cvnn} as variational sub-circuits. $\vec{\theta_d}$ denotes the parameters. At the end of the circuit we perform a homodyne measurement to calculate the expected value of the $\hat{x}$ quadrature. Given the expected value ($X$), the probability that the input state is real is:
\begin{equation} \label{eq:prob}
    D(input) = P(input \sim real; \vec{\theta_d}) = \frac{1}{1 + e^{-X}}
\end{equation}

\begin{figure}
\centering
\begin{subfigure}{.5\textwidth}
  \centering
  \includegraphics[width=1\linewidth]{generator}
  \caption{Design of the Generator circuit}
  \label{fig:generator}
\end{subfigure}%
\begin{subfigure}{.5\textwidth}
  \centering
  \includegraphics[width=1\linewidth]{discriminator}
  \caption{Design of the Discriminator circuit}
  \label{fig:discriminator}
\end{subfigure}
\caption{Design of the Generator and Discriminator circuits}
\label{fig:test}
\end{figure}

The loss function is computed using a classical computer. The output of the discriminator, i.e. the expectation value $X$ is used for computing the GAN loss \cite{gan}
\begin{equation}
    L_{D} = \mathbb{E}_{x \sim real}[log D(x)] + \mathbb{E}_{z \sim \mathcal{N}(0,1)[log(1-D(G(z)))]} \quad and \quad
    L_{G} = \mathbb{E}_{z \sim \mathcal{N}(0,1)}[log(D(G(z)))]
\end{equation}
Here $x$ is a data point from the real distribution, $G(z)$ is the output of the generator. $D(x)$ is the probability $P(x \sim real; \vec{\theta_d})$ given in Eq. \ref{eq:prob}. We follow the training protocol proposed in \cite{qugan1}. Gradient calculation is done using automatic differentiation \cite{paramshift}. We use this to perform quantum gradient descent to find the optimal values of $\vec{\theta_g}$ and $\vec{\theta_d}$.

\section{Experiments and Results} \label{sec:expresults}
The experiments have been performed using Pennylane and Strawberryfields \cite{pennylane, strawberryfields}. We use OpticalGAN to generate coherent states which are at unit distance from vacuum state in phase space and to generate energy eigentates (fock basis). Due to limited space, Fig. \ref{fig:coherentgen} shows both the generated coherent state for $\alpha=1$ and the generated fock state $\ket{9}$. For more details and results, refer to the code available at https://github.com/abcd1729/opticalgan. We are currently working on generating images using OpticalGAN.

\clearpage

\bibliographystyle{unsrt}