11institutetext:  Group of Complex Systems and Statistical Physics, Department
of Theoretical Physics, University of Havana. La Habana, Cuba.

First pacs description Second pacs description Third pacs description

# Photon and Photon-Added Intelligent States of Coupled Parametric Oscillators

A.E. Neira  R. Mulet

###### Abstract

We study a quantum system of coupled oscillators subject to a periodic
excitation of its parameters. Using Floquet-Lyapunov theory we derive the
linear integrals of motion of the system and relate their covariance matrix to
that for the canonical observables. The operator integrals allows us to
construct the intelligent (minimum uncertainty) states of the system and the
corresponding photon-added states. We found explicit expressions for the
wavefunction, Wigner function and covariance matrix of these states.

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##  1 Introduction

The understanding of the quantum harmonic oscillator is one of the basic ideas
to be mastered by any Physics graduate student. Probably every text book in
Quantum Mechanics devotes at least one full chapter to introduce and discuss
the model. Therefore, it may appear surprising that the introduction of time-
dependent parameters turns the model into an active field of research [1, 2,
3, 4, 5, 6, 7].

Indeed, the study of quantum harmonic oscillators with explicitly time-
dependent parameters[4] has been stimulated by their effectiveness in the
description of a wide range of quantum phenomena, like propagation of light
through variable media [8], the dynamical Casimir effect [9] and the motion of
trapped ions [10]. The latter, also a popular proposal for scalable quantum
computation [11].

Broadly speaking, the explicit time-dependence of the parameters in the model
induces non-vanishing covariances between the canonical operators and changes
the distribution of excitations of the states of the system [7]. Moreover,
when the coupling between two non-stationary oscillators [12] is considered,
the inseparability of the system’s wave-functions takes a stronger character.
In general, a coordinate transformation that gives real uncoupled modes of
oscillation is no longer possible and quantum entanglement may show up [13].
This entanglement turns out to be a property of the algebraic structure of the
model and not just of the states.

In this work we study a system of $N$ parametric oscillators described by the
Hamiltonian

$\hat{H}(t)=\frac{1}{2}\hat{\textbf{x}}^{T}J\Pi(t)\hat{\textbf{x}}$ (1)

where $\hat{\textbf{x}}$ is an $2N$ dimensional vector whose first $N$
components correspond to the position and the last $N$ to the momentum
operator of each oscillator. The parametric excitation is described by the
explicit time-dependence of the frequencies and coupling factors present in
$\Pi(t)$. In particular, we will focus our attention in a class of models in
which $\Pi(t)=\Pi(t+T)$. The canonical commutation relations for the
observables, written in matrix form and using natural units $\hbar\equiv 1$,
read

$\left[\hat{x}_{i},\hat{x}_{j}\right]=iJ_{ij}.$ (2)

The two $2N\times 2N$ matrices appearing on the right hand side of (1) are, in
general, given by

$\begin{array}[]{ccc}J=\left(\begin{array}[]{cc}0&1_{N}\\\
-1_{N}&0\end{array}\right),&&\Pi(t)=\left(\begin{array}[]{cc}C_{q\leftrightarrow
p}&C_{p\leftrightarrow p}\\\ C_{q\leftrightarrow q}&C_{q\leftrightarrow
p}\end{array}\right),\end{array}$ (3)

which means that $J$ is a skew-symmetric (symplectic) matrix $J^{T}=J^{-1}=-J$
and $\Pi(t)$ describes the configuration of the system. That is, its
$C_{q\leftrightarrow q}$ block contains the frequencies and coupling factors
between the coordinates, $C_{p\leftrightarrow p}$ represents the coupling
between momenta and $C_{q\leftrightarrow p}$ the coupling between positions
and momenta.

A major concern in the study of these parametrically excited oscillators is
the obtention of their coherent or classical states. Such classicality is
determined by a positive Wigner function and by the Poissonian character of
the photon distribution. The canonical coherent states [14] of a stationary
harmonic oscillator are known to minimize the Heisenberg uncertainty relation
[15, 16]. But the family of coherent states is larger, they can be constructed
not only on the simple one-dimensional case, but for coupled oscillators [12,
17] or further generalizations of the quadratic Hamiltonian model. For
example, the parametric excitation of a single harmonic oscillator induces
squeezing of the quadrature components and correlations between them. In this
case, the coherent states become squeezed and correlated [7, 17], and minimize
Schrödinger’s uncertainty relation [18]. Such minimization of uncertainty
inequalities [15, 16, 18, 19] is the defining property of the intelligent
states of a quantum mechanical system, of which the coherent states are a
paramount example.

The intelligent states, in general, belong to the Gaussian regime, that is,
their Wigner function is a Gaussian distribution and occupy a privileged
position in quantum information. From the experimental side, these are the
states that can be most easily obtained in the laboratory and this has also
called the attention of many theoreticians. However, the departure from the
Gaussian regime might provide a better playground for quantum information
processing protocols with no classical analog [20] and motivates the search
for new analytic methods and solutions.

The non-Gaussian regime imposes non classical behavior. Non-classical states
are generally defined as the ones with negative values of the Wigner function
and sub- or super-poissonian photon distribution function [21]. Within these
classes we can identify the excited or photon-added coherent states. Their
behavior is intermediate between that of classical and more general quantum
states and provide a scheme to study the transition from particle-like to
wave-like behavior of light [22]. These states were firstly introduced by
Agarwal and Tara [23] and have gained popularity in the last three decades due
to their possible implementations in quantum information processing. In the
context of oscillators systems, they have been previously studied for a single
oscillator with a generic time-dependence of its parameters [24] and for a
generic Gaussian state in the case of a single-photon addition [25].

Our goal in this work is to find the photon and the photon-added intelligent
states for a system of parametrically coupled periodic oscillators. The
remainder of the manuscript is organized in the following way. We first
construct the linear integrals of motion for (1) exploiting well known results
from Floquet theory. We will show that these operators translate into creation
and annihilation operators and they will be used to introduce the non-
stationary Fock basis spanning the Hilbert space of the oscillators. Then, we
show that these Fock states do not minimize, in general, the Robertson
uncertainty relation. We continue introducing a class of intelligent states by
performing a unitary transformation on the ground state, which will be in
general squeezed and correlated. Then, adding excitations to the previously
introduced intelligent states, we obtain the wavefunction, Wigner function and
the corresponding covariance matrix of the excited states. Finally, we present
the conclusions of our work.

##  2 Floquet Theory and Linear Integrals of Motion

In this section we make use of the periodicity in time of the Hamiltonian
operator (1) to employ the Floquet theorem in the construction of the linear
integrals of motion of the quantum mechanical problem. For completeness we
will review the definition of the time-dependent transformation (Floquet-
Lyapunov) and the Heisenberg’s creation and annihilation operators following
reference [3].

Let us start transforming the Hamiltonian (1) to its Heisenberg picture
equivalent

$\hat{H}_{h}(t)=\frac{1}{2}\hat{\textbf{x}}^{T}(t)J\Pi(t)\hat{\textbf{x}}(t),$
(4)

where the $h$ subscript stands for Heisenberg and the components of
$\hat{\textbf{x}}(t)$ are Heisenberg operators constructed from the
$\hat{\textbf{x}}$ of the Schrödinger picture. The equations of motion derived
from (4) take the form

$\dot{\hat{x}}_{k}(t)=-i\left[\hat{x}_{k}(t),\hat{H}_{Hei}(t)\right]=\sum_{l}\Pi_{kl}(t)\hat{x}_{l}(t)$
(5)

This is _formally_ identical to a classical linear dynamical system with
periodic coefficients. Hence, according to Floquet theory [26], they may be
described in terms of independent complex modes of oscillation $\hat{\chi}(t)$
(Floquet modes), related to $\hat{\textbf{x}}(t)$ through the Floquet-Lyapunov
transformation (FLT), $F(t)$:

$\hat{\textbf{x}}_{h}(t)=F(t)\hat{\chi}(t).$ (6)

The time evolution of the operators $\hat{\chi}(t)$ is entirely described, in
the Heisenberg picture, by imaginary exponential functions

$\hat{\chi}_{j}(t)=\hat{\chi}_{j}(0)\exp(i\omega_{j}t)$
$\hat{\chi}_{j+N}(t)=\hat{\chi}_{j+N}(0)\exp(-i\omega_{j}t)$ (7)

where the $j$ index runs from $1$ to $N$ and the $\omega_{j}$ are the Floquet
exponents of the system. These are the quantized Floquet modes of the
classical system. The time-dependent transformation $F(t)$ shares the period
of the Hamiltonian (1), and its elements satisfy

$F_{i,j+N}(t)=F_{i,j}^{*}(t),\hskip 8.5359ptj=1,...,N\hskip
2.84544pti=1,...,2N$ (8)

Using this last expression we can rewrite the Floquet-Lyapunov transformation
in block form

$F(t)=\left(\begin{array}[]{cc}U&U^{*}\\\ V&V^{*}\end{array}\right).$ (9)

These blocks are time-dependent $N\times N$ matrices which share the period of
the parameters. They constitute a generalization of the periodic functions
appearing in the Floquet theorem [27, 26] to the case of several degrees of
freedom.

The time-dependent transformation (6) implicitly depends on the configuration
$\Pi(t)$ of the system. In the simple case of uncoupled oscillators, every
block of $\Pi(t)$ is a diagonal matrix, this implies that the blocks of
$F(t)$, that is, $U$, $V$ and their complex conjugates, are also diagonal,
$F_{ij}=0$ if $i\neq j$. The coupling between the canonical variables would
then be reflected in non-vanishing off-diagonal elements of the blocks. We
assume non zero coupling throughout this work.

If we impose the following condition to the FLT,

$\sum_{l=1}^{N}F_{l+N,k}(0)F_{l,m}(0)=\frac{i}{2}\delta_{lm},$ (10)

or, equivalently, $V^{T}(0)U(0)=\frac{i}{2}I$, then the transformation will be
canonical [3, 28]. The above condition will also ensure that the quantized
Floquet modes (operators in the Heisenberg picture), satisfy the Weyl-
Heisenberg algebra. Written in matrix form, this is

$\left[\hat{\chi}_{i}(t),\hat{\chi}_{j}(t)\right]=-J_{ij},$ (11)

and, taking (7) into account, we can identified the Floquet modes, in the
Heisenberg picture, with creation ($\hat{\chi}_{i}(t)$) and annihilation
($\hat{\chi}_{i+N}(t)$, $i=1,...,N$) operators. We will assume the fulfillment
of (10) from now on.

In the Schrödinger picture,

$\hat{\textbf{x}}=F(t)\hat{\chi}_{S}(t).$ (12)

But the unitarity of the time evolution operator ensures the invariance of the
Weyl-Heisenberg algebra under its action. This, in turns guarantees that also
the operators obtained from the Floquet modes in the Schrödinger picture
satisfy the commutation relations (11). Then, from these Schrödinger operators
we can define the following (explicit) time-dependent operators

$\left(\begin{array}[]{c}\hat{A}^{\dagger}(t)\\\
\hat{A}(t)\end{array}\right)=\left(\begin{array}[]{cc}e^{-i\Omega t}&0\\\
0&e^{i\Omega t}\end{array}\right)\hat{\chi}_{S}(t)\equiv\hat{I}(t)$ (13)

where $\Omega=diag(\omega_{1},...,\omega_{N})$. Substituting (13) in (12) and
using the inverse of the FLT (see [3]) we can express $\hat{A}(t)$ and
$\hat{A}^{\dagger}(t)$ in terms of the time-independent position and momentum
operators

$\hat{A}^{\dagger}(t)=iV^{\dagger}(t)e^{-i\Omega
t}\hat{q}-iU^{\dagger}(t)e^{-i\Omega t}\hat{p},$ (14)
$\hat{A}(t)=-iV^{T}(t)e^{i\Omega t}\hat{q}+iU^{T}(t)e^{i\Omega t}\hat{p}.$
(15)

It is easy to see that the total derivative of this last expression vanishes,
and the following operator equation is satisfied

$\dfrac{\partial\hat{I}_{k}(t)}{\partial
t}=i\left[\hat{I}_{k}(t),\hat{H}(t)\right],$ (16)

where $\hat{H}(t)$ is the Schrödinger Hamiltonian (1). Hence, the time-
dependent non-Hermitian operators $\hat{A}^{\dagger}(t)$ and $\hat{A}(t)$
(group together in $\hat{I}(t)$), are integrals of motion for the quantum
mechanical problem of the Dodonov-Malkin-Man’ko type [29, 7] and can be used
to define an orthonormal basis for the Hilbert space of the oscillators.
Moreover, since they satisfy the boson commutation relations (11), we can
reinterpret each oscillator as a bosonic system in the second quantization
formalism.

The construction of an orthonormal basis is straightforward: We assume that
for $t\leq 0$ the system is a stationary one, in such a situation the
configuration matrix can be redefine as
$\Pi(t)\rightarrow(\Pi(0)\Theta(-t)+\Pi(t)\Theta(t))$, where $\Theta(t)$ is
the Heaviside step function. Then, for $t\leq 0$, the integrals of motion are
just standard creation and annihilation operators and the Fock basis
$|{n}\rangle$ can be constructed in the usual way. For $t\geq 0$ the action of
the time evolution operator
$\hat{\mathcal{U}}(t)=\exp(\frac{i}{\hbar}\int_{0}^{t}H(t^{\prime})dt^{\prime})$
upon the stationary Fock basis defines the _dynamical Fock states_ [10] of the
parametrically excited system,

$|\\{n\\};t\rangle=\hat{\mathcal{U}}(t)|\\{n\\}\rangle$ (17)

where $\\{n\\}$ is the set $\\{n_{1},...,n_{N}\\}$. We define, in the
Schrödinger picture, the number operator for the $i-$th oscillator ($i<N$) as
$\hat{n}_{i}(t)=\hat{A}_{i}^{\dagger}(t)\hat{A}_{i}(t)$. Its eigenvectors are
the states defined in (17) and the explicit eigenvalue problem takes the form

$\hat{n}_{i}(t)|\\{n\\};t\rangle=n_{i}|\\{n\\};t\rangle,$ (18)

This implies that (17) describes the states that evolved from a stationary
state $|\\{n\\}\rangle$ with $n_{i}$, $i=1,...,N$ excitations in the i-th
mode. In a quantum optical setting nonzero coupling will be translated in
entangled modes, and the dynamical Fock states, as well as all those to be
introduced in this letter will be multimode states. For definiteness we will
be referring to the excitations as photons even if the oscillators system is
not necessarily restricted to the modes of the radiation field.

The action of the linear integrals of motion on (17) is analogous to the
action of creation and annihilation operators on the Fock basis for the
stationary problem. The explicit expressions read

$\hat{A}_{i}(t)|\\{n\\};t\rangle=\sqrt{n_{i}}|...n_{i}-1...;t\rangle,$ (19)
$\hat{A}_{i}^{\dagger}(t)|\\{n\\};t\rangle=\sqrt{n_{i}+1}|...n_{i}+1...;t\rangle.$
(20)

We would like to emphasize that, for the previous treatment to hold it is
necessary that the Floquet exponents $\omega_{j}$ of the system are strictly
real numbers. This is equivalent to say that the elements of $\Pi(t)$ have to
produce stable oscillations for the classical system.

##  3 Robertson Inequality and Intelligent States

The mathematical description of the uncertainty principle for a set of $2N$
observables of a certain quantum system can be given through the Robertson
inequality [19],

$\textbf{Det}(\sigma(\hat{x}))\geq\left(\frac{1}{2}\right)^{2N}.$ (21)

In the left-hand side figures the determinant of covariance (dispersion)
matrix $\sigma$ corresponding to the systems’s quantum mechanical state. We
will take the set to be the $2N$ components of the vector $\hat{\textbf{x}}$.
Then, the elements of its covariance matrix, in a state $|\psi;t\rangle$, are
given by

$\sigma(\hat{x}_{i},\hat{x}_{j})=\frac{1}{2}\langle\left\\{\hat{x}_{i},\hat{x}_{j}\right\\}\rangle_{\psi;t}-\langle\hat{x}_{i}\rangle\langle\hat{x}_{j}\rangle_{\psi;t}$
(22)

where $\left\\{\cdot,\cdot\right\\}$ denotes the anti-commutator. Identical
expressions hold for the integrals of motion $\hat{I}_{i}(t)$. In fact, the
covariance matrix of the canonical observables $\sigma(\hat{x}(t))$ can be
related to that of the integrals of motion $\sigma(\hat{I}(t))$ through the
Floquet-Lyapunov transformation

$\sigma(\hat{\textbf{x}})=F(t)\exp\left(iWt\right)\sigma(\hat{I}(t))\exp\left(iWt\right)F^{T}(t),$
(23)

where

$W=\left(\begin{array}[]{cc}\Omega&0\\\ 0&-\Omega\end{array}\right)$

Expression (23) follows directly from (12) and (13). Note that, although
$\hat{I}(t)$ are time-dependent, $\sigma(\hat{\chi}(t))$ is not necessarily
time-dependent too. On the other hand, $\sigma(\hat{x})$ depends explicitly on
time, due to the presence of the Floquet-Lyapunov transformation and
$\exp(iWt)$ in (23).

The columns of the matrix $F\exp(iWt)$ correspond to the solutions of the
classical system of coupled parametric oscillators subject to the initial
conditions (10). This means that the covariances of canonical observables, in
any quantum mechanical state, are determined by the covariances of
$\hat{I}(t)$ and by the classical motion of the system (1). Such a classical
motion is, in turn, completely fixed by the the Floquet modes $\exp(iWt)$ and
the Floquet-Lyapunov transformation $F(t)$.

The intelligent states [30, 31] of a quantum mechanical system can be defined
as those that minimize the Robertson uncertainty relation (21) for all times.
Since this inequality is an extension of Heisenberg and Schrödinger
inequalities, the intelligent states may be interpreted as a generalization of
the canonical coherent states. In what follows we will build the intelligent
states of the N-dimensional non-stationary system (1). But, let us first show
that in this Fock basis, (17), the Robertson’s inequality (21) is not
minimized. Using (19) and (20) to calculate the mean values present in (22)
for the integrals of motion and making use of the orthonormality of the Fock
dynamical basis, we obtain

$\sigma_{Fock}(A_{i}^{\dagger},A_{j})=\langle
A_{i}^{\dagger}A_{j}\rangle-\langle A_{i}^{\dagger}\rangle\langle
A_{j}\rangle+\dfrac{\delta_{ij}}{2}$ (24) $=\langle
A_{i}^{\dagger}A_{j}^{\dagger}\rangle+\dfrac{\delta_{ij}}{2}=(n_{i}+\dfrac{1}{2})\delta_{ij}$

The covariance matrix for $\hat{A}^{\dagger}(t)$ and $\hat{A}(t)$ in the Fock
dynamical states is

$\sigma_{n;t}(\hat{I}(t))=\left(\begin{array}[]{cc}0&diag(\\{n\\})+1_{N}/2\\\
diag(\\{n\\})+1_{N}/2&0\end{array}\right),$

and for the canonical observables

$\sigma_{n;t}(\hat{\textbf{x}})=\frac{1}{2}F(t)F^{\dagger}(t)+Fdiag(\\{n\\})F^{\dagger}.$
(25)

Notice that the determinant of this matrix will increase with the number of
excitations. This means that, as in the non-parametric problem, in general,
these Fock states do not minimize the uncertainty relation and are non-
classical states of the system.

We define then, the coherent states of the system of interacting paramateric
oscillators, through the action of the displacement operator
$\mathcal{D}(\\{\alpha\\})$ on the ground state $|\\{0\\};t\rangle$

$|\\{\alpha\\};t\rangle=\mathcal{D}(\\{\alpha\\})|\\{0\\};t\rangle=$
$=\prod_{i=1}^{N}\exp\left(\alpha_{i}\hat{A}_{i}^{\dagger}(t)+\alpha_{i}^{*}\hat{A}_{i}(t)\right)|\\{0\\};t\rangle.$
(26)

From (26) it follows that $|\\{\alpha\\};t\rangle$ are eigenstates of the
linear integral of motion $\hat{A}(t)$,

$\hat{A}_{i}(t)|\\{\alpha\\};t\rangle=\alpha_{i}|\\{\alpha\\};t\rangle,$ (27)

where $\alpha_{i}$ is equal to the initial value of the classical Floquet mode
$\chi_{i+N}(0)$. This expression can be used to calculate the coordinate
representation wave function [3, 6]. It is easy to see that, as in the simple
harmonic oscillator, the mean values of the positions $\hat{q}_{i}$ and
momenta $\hat{p}_{i}$ on these states can reproduce the behavior of any
solutions to the classical equation of motion. The mean values of the
quadrature operators in coherent states are given by

$\langle\hat{x}_{i}\rangle_{CS}=\sum_{j=1}^{2N}F_{ij}(t)\chi_{j}(t),$

or, more explicitly,

$\langle\hat{q}_{i}\rangle_{CS}=\sum_{j=1}^{N}\left(U_{ij}(t)\chi_{j}(0)e^{i\omega_{j}t}+U_{ij}^{*}(t)\chi_{j+N}(0)e^{-i\omega_{j}t}\right),$
(28)

which is exactly the general Floquet solution to the classical equations of
motion. In these expressions $\chi(t)$ denotes the classical Floquet modes,
they have the same functional form as the operators (7) of the Heisenberg
picture.

We now find an explicit expression for the covariance matrices of the
integrals of motion and canonical observables in the coherent states (26). To
that end we begin by noticing that, since
$\mathcal{D}^{\dagger}(\\{\alpha\\})=\mathcal{D}^{-1}(\\{\alpha\\})$ and

$\mathcal{D}^{\dagger}(\\{\alpha\\})\hat{A}(t)\mathcal{D}(\\{\alpha\\})=\hat{A}(t)+\alpha,$
$\mathcal{D}^{\dagger}(\\{\alpha\\})\hat{A}^{\dagger}(t)\mathcal{D}(\\{\alpha\\})=\hat{A}^{\dagger}(t)+\alpha^{*},$

the covariances between the integrals of motion in a given coherent state are
invariant under the action of the displacement operator. This is a major
simplification in our demonstration, it allows us to consider only the
covariance matrix for the ground state of the system, which is shared by all
coherent states. Then, for any values of $\\{\alpha\\}$, the covariance matrix
will be given by setting $n_{i}=0,\hskip 5.69046pti=1,...,N$ in (25). Written
explicitly, for the integrals of motion, it reads

$\sigma_{\alpha;t}(\hat{I}(t))=\left(\begin{array}[]{cc}0&1_{N}/2\\\
1_{N}/2&0\end{array}\right)$ (29)

and for canonical observables,

$\sigma_{\alpha;t}(\hat{\textbf{x}})=\frac{1}{2}F(t)F^{\dagger}(t)=\sigma_{0;t}(\hat{\textbf{x}})$
(30)

As in the Fock case (25), the Floquet exponents cancel out and $F(t)$ is
sufficient to know the covariances between the position and momentum
operators. This expression shows that the coherent states (26), of the non-
stationary system (1) are squeezed and correlated.

Squeezing occurs when the uncertainty in position is different from the
uncertainty in momentum. Even in the one-dimensional case, the parametric
excitation induces squeezing of the quadratures components [7]. This can be
directly seen from (30): the position uncertainty for the $i-$th oscillator
($i<N$) is given by the $i$-th main diagonal element of $FF^{\dagger}$. On the
other hand, the momentum uncertainty corresponds to the $i+N$ element of the
diagonal, since these two are in general different, so are the uncertainties.

They are correlated in a double sense. Due to the coupling between the
position and momentum operators in the Hamiltonian, it is impossible to
express the coherent state $|\\{\alpha\\};t\rangle$ of the entire system, as a
product of the single oscillators coherent states $|\alpha_{i};t\rangle$. But
also, because the parametric excitation guarantees finite correlations between
position and momentum operators in each independent oscillator.

To finish this section we want to prove the intelligent character of the
(generalized) coherent states (26). In other words, we have to demostrate that
$\det\left(\sigma_{\alpha;t}(\hat{\textbf{x}})\right)=2^{-2N}$. This requires
an expression for the determinant of the Floquet-Lyapunov transformation
subject to the canonical condition (10). This is a rather simple task if one
notices that the matrix

$\tilde{F}(t)=\left(\begin{array}[]{cc}1_{N}&0\\\
0&i1_{N}\end{array}\right)F^{T}(t),$ (31)

is an element of the complex symplectic group $Sp(\mathcal{C},2n)$, which
guarantees that its determinant is equal one. Then

$det[F(t)]=det[\left(\begin{array}[]{cc}1_{N}&0\\\
0&-i1_{N}\end{array}\right)]=(-i)^{N}.$ (32)

It follows directly from (29) that the determinant of the covariance matrix
for the linear integrals of motion, in a coherent state, is equal to
$(-4)^{-N}$. We then substitute last two results in (23) to obtain that the
determinant of $\sigma(\hat{x})$ is indeed equal to $2^{-2N}$. This concludes
the proof that the coherent states $|\\{\alpha\\};t\rangle$ of the
parametrically excited oscillators minimize the Robertson uncertainty relation
(21) for any configuration $\Pi(t)$ of the system.

In summary, we have related the (quantized) Floquet modes of the classical
system with the linear integrals of motion (16) of the quantum mechanical
problem (1), showing that the work done in [3] for a set of coupled Mathieu
equations can be extended to any configuration of oscillators with time-
periodic parameters. The Floquet-Lyapunov transformation can be seen as a
multidimensional and time-dependent generalization of the usual relations
between the creation and annihilation operators (corresponding to Floquet
modes) and the position and momentum operators of a simple harmonic
oscillator. Moreover, a proper use of the displacement operator allows the
derivation of the coherent states of the model. We also showed that the
explicit time-dependence of covariances between canonical operators, in
coherent and Fock states, is completely determined by this periodic
transformation. If we were to consider a stationary system described by
$\Pi(t_{0})$ in (1), for a fixed $t_{0}$, the intelligent states (26) reduce
to the multimode minimum uncertainty squeezed states derived in [32]. In the
next section, we will show that this framework can be used to construct a
special case of non-Gaussian states.

##  4 Adding quanta to intelligent states

The action of the creation operator, or photon addition, on the ground sate of
a harmonic oscillator generates a set of non-classical states, the Fock basis.
In analogy, the photon-added or excited coherent states (PACS) are defined as
the normalized states obtained through the addition of photons to the
canonical coherent states. In the nonstationary setting (1) of coupled
oscillators, the role of the creation operator is played by the integral of
motion $\hat{A}^{\dagger}(t)$, and the dynamical version of the PACS is
constructed through its action on the intelligent states (26),

$|\\{\alpha\\},\\{m\\};t\rangle=\prod_{k=1}^{N}\dfrac{(\hat{A}_{k}^{\dagger}(t))^{m_{k}}}{\sqrt{m_{k}!L_{m_{k}}(-|\alpha_{k}|^{2})}}|\\{\alpha\\};t\rangle$
(33)

where $L_{m_{k}}(-|\alpha_{k}|^{2})$ is the Laguerre polynomial of order
$m_{k}$. If $m_{k}=0$, for all $k$, we recover the coherent states
$|\\{\alpha\\},\\{0\\};t\rangle=|\\{\alpha\\};t\rangle$, and Fock states are
recovered by setting $\alpha_{k}=0$, for all $k$, since $L_{m_{k}}(0)=1$.

The wave function describing these states can be directly computed from their
definition (see supplementary material). The result is

$\psi_{\alpha,m}(\textbf{q},t)=N_{\alpha,m}\psi_{\alpha}(\textbf{q},t)\mathcal{H}_{\textbf{m}}^{U^{\dagger}U^{-T}}\left(U^{-*}\textbf{q}-\vec{\alpha}\right).$
(34)

In this expression $N_{\alpha,m}$ is the normalization constant appearing in
(33), $\psi_{\alpha}$ denotes the wavefunction of the intelligent states [6,
3] and $\mathcal{H}_{\textbf{m}}$ is the multidimensional Hermite polynomial
[33] of order $\textbf{m}=(m_{1},...m_{N})$, defined by the relation

$\mathcal{H}_{\textbf{m}}^{M}\left(\textbf{q}\right)=(-1)^{\Sigma
m_{k}}e^{\frac{1}{2}\textbf{q}^{T}M\textbf{q}}\dfrac{\partial^{\Sigma
m_{k}}}{\partial q_{1}...\partial
q_{N}}e^{-\frac{1}{2}\textbf{q}^{T}M\textbf{q}}$ (35)

Expression (34) is a generalization of single oscillator wavefunction obtained
in [24] for the PACS. Due to the mode coupling, the wavefunction is not
expressible as a product of single oscillators wavefunctions. This can be
directly seen from the definition of the multidimensional Hermite polynomials,
which cannot be expressed as the product of unidimensional polynomials if the
matrix $U^{\dagger}U^{-T}$ is not diagonal.

From equation (34) we can obtain the Wigner function of the photon-added
states (33). The derivation is sketched in the suplementary material. The
result is

$W_{PACS}(\textbf{q},\textbf{p},t)=2^{N}\exp\left(-2|A(\textbf{q},\textbf{p},t)-\alpha|^{2}\right)\times$
$\times\left(\prod_{k=1}^{N}(-1)^{m_{k}}\dfrac{L_{m_{k}}(|2A(\textbf{q},\textbf{p},t)-\alpha|_{k}^{2})}{L_{m_{k}}(-|\alpha_{k}|^{2})}\right)$
(36)

The _function_ $A(\textbf{q},\textbf{p},t)$ has the same form (15) as the
integral of motion $\hat{A}(t)$,

$A(\textbf{q},\textbf{p},t)=-iV^{T}(t)e^{i\Omega
t}\textbf{q}+iU^{T}(t)e^{i\Omega t}\textbf{p}.$ (37)

If we set $m_{k}=0$ for every value of $k$, then (36) reduces to a Gaussian
distribution. Note that, when coupling is present, (36) cannot be written as
the product of single mode Wigner functions.

This fact has an interesting consecuence that does not arise in single mode
cases [23, 24]. To illustrate this we can consider two coupled oscillators
($N=2$) in an intelligent state (26), described by the eigenvalues
$\alpha_{1}$ and $\alpha_{2}$. The Wigner function is given by (36), with
$m_{1}=m_{2}=0$, and it takes no negative values for any combination of
$\alpha_{1}$ and $\alpha_{2}$, such state can then be considered a classical
state. If we add a single excitation ($m_{2}=1$) to only one of them, the
Wigner function will become

$4\frac{|2A_{1}(q_{1},p_{1},t)-\alpha_{1}|^{2}-1}{1+|\alpha_{1}|^{2}}e^{-2|A(\textbf{q},\textbf{p},t)-\alpha|^{2}},$
(38)

and it can take negative values whenever
$|2A_{1}(q_{1},p_{1},t)-\alpha_{1}|^{2}<1$. The mode $\alpha_{1}$, to which we
added no excitations ($m_{1}=0$), cannot be said to be in a classical state,
since its corresponding canonical observables are correlated to those of the
mode that was excited. This is caused by the coupling between the oscillators
and its true for any number of excitations.

To calculate the covariances we employ the following expression for the
expected value of a generic operator $\hat{o}$ on these states

$\langle\hat{o}\rangle_{\\{\alpha\\},\\{m\\};t}=\dfrac{e^{-|\vec{\alpha}|^{2}}}{\prod
m_{k}!L_{m_{k}}}\prod_{i=1}^{N}\sum_{\\{n_{i},\bar{n}_{i}\\}=0}^{\infty}\dfrac{(\alpha_{i})^{n_{i}}(\alpha_{i}^{*})^{\bar{n}_{i}}}{\sqrt{n_{i}!\bar{n}_{i}!}}\times$
$\times\left(\sqrt{n_{i}+1}...\sqrt{n_{i}+m_{i}}\right)\left(\sqrt{\bar{n}_{i}+1}...\sqrt{\bar{n}_{i}+m_{i}}\right)\times$
$\times\langle n_{1}+m_{1},...;t|\hat{o}|\bar{n}_{1}+m_{1},...;t\rangle.$ (39)

This means that to obtain the covariances between the integrals of motion in
photon-added coherent states, we first need to calculate the mean values of
such integrals and the matrix elements of the products between them in the
dynamical Fock basis. For the sake of simplicity we will define

$\langle
n_{1}+m_{1},...;t|\hat{o}|\bar{n}_{1}+m_{1},...;t\rangle=\langle...\rangle_{Fock}$

and we will also omit the explicit time-dependence. The needed expressions are
readily obtained from (19) and (20),

$\langle\hat{A}_{i}\rangle_{Fock}=\sqrt{n_{i}+m_{i}}\delta_{n_{i},n_{i}-1}$
(40)
$\langle\hat{A}^{\dagger}_{i}\rangle_{Fock}=\sqrt{n_{i}+m_{i}+1}\delta_{n_{i},n_{i}+1}$
(41)
$\langle\hat{A}_{i}\hat{A}_{j}\rangle_{Fock}=\left\\{\begin{array}[]{l}\langle\hat{A}_{i}\rangle\langle\hat{A}_{j}\rangle\hskip
8.5359pti\neq j\\\ \sqrt{n_{i}+m_{i}}\sqrt{n_{i}+m_{i}-1}\times\\\
\times\delta_{n_{i},n_{i}-2},\hskip 8.5359pti=j\end{array}\right.$ (42)
$\langle\hat{A}^{\dagger}_{i}\hat{A}^{\dagger}_{j}\rangle_{Fock}=\left\\{\begin{array}[]{l}\langle\hat{A}^{\dagger}_{i}\rangle\langle\hat{A}^{\dagger}_{j}\rangle\hskip
8.5359pti\neq j\\\ \sqrt{n_{i}+m_{i}+1}\sqrt{n_{i}+m_{i}+2}\times\\\
\times\delta_{n_{i},n_{i}+2},\hskip 8.5359pti=j\end{array}\right.$ (43)
$\langle\hat{A}^{\dagger}_{i}\hat{A}_{j}\rangle_{Fock}=\left\\{\begin{array}[]{l}\langle\hat{A}^{\dagger}_{i}\rangle\langle\hat{A}_{j}\rangle\hskip
8.5359pti\neq j\\\ (n_{i}+m_{i}),\hskip 8.5359pti=j\end{array}\right.$ (44)

The corresponding expression for
$\langle\hat{A}_{i}\hat{A}^{\dagger}_{j}\rangle$ is obtained from the
commutation relations (11),
$\langle\hat{A}_{i}\hat{A}^{\dagger}_{j}\rangle=\delta_{ij}-\langle\hat{A}_{j}\hat{A}^{\dagger}_{i}\rangle$.
We then substitute these results in (39) to compute the analogous expressions
in the photon-added coherent states. For clarity, let us calculate the mean
value of the integral of motion $\hat{A}$,

$\langle\hat{A}_{i}\rangle_{PACS}=\dfrac{e^{-|\alpha_{i}|^{2}}(m_{i}!)^{-1}}{L_{m_{i}}(-|\alpha_{i}|^{2})}\sum_{n=0}^{\infty}\dfrac{|\alpha_{i}|^{2n}}{n!}(n+1)...(n+m_{i}+1).$

Introducing the _Pochhammer symbol_ : $(a)_{b}=1\cdot 2\cdot
3...(a+b-1),\hskip 8.5359pta,b\in Naturals,$ it can be written as

$\langle\hat{A}_{i}\rangle_{PACS}=\alpha_{i}\dfrac{e^{-|\alpha_{i}|^{2}}}{L_{m_{i}}(-|\alpha_{i}|^{2})}\sum_{n=0}^{\infty}\dfrac{|\alpha_{i}|^{2n}}{n!}\dfrac{(m_{i}+2)_{n}}{(2)_{n}}.$
(45)

Identifying the sum on the right hand side with the confluent hypergeometric
function ${}_{1}F_{1}(m_{i}+2,2;-|\alpha_{i}|^{2})$,

$\langle\hat{A}_{i}\rangle_{PACS}=\alpha_{i}\dfrac{e^{-|\alpha_{i}|^{2}}{{}_{1}}F_{1}(m_{i}+2,2;-|\alpha_{i}|^{2})}{L_{m_{i}}(-|\alpha_{i}|^{2})}$
(46)

We now make use of Kummer’s transformation,
${}_{1}F_{1}(a,c;x)=e^{x}{{}_{1}}F_{1}(c-a,c;-x),$ and the relation
${}_{1}F_{1}(-n,a+1;x)=\dfrac{n!}{(a+1)_{n}}L_{n}^{a}(x)$ to finally obtain

$\langle\hat{A}_{i}\rangle_{PACS}=\alpha_{i}\dfrac{L_{m_{i}}^{1}(-|\alpha_{i}|^{2})}{L_{m_{i}}(-|\alpha_{i}|^{2})}$
(47)

Proceeding similar for the creation operators:

$\langle\hat{A}^{\dagger}_{i}\rangle_{PACS}=\alpha_{i}^{*}\dfrac{L_{m_{i}}^{1}(-|\alpha_{i}|^{2})}{L_{m_{i}}(-|\alpha_{i}|^{2})}$
(48)

We can now use (47) and (48), together with (6) to calculate the mean value of
the quadrature operators

$\langle\hat{x}_{i}\rangle_{PACS}=\sum_{j=1}^{2N}F_{ij}(t)\chi_{j}(t)\dfrac{L_{m_{j}}^{1}(-|\alpha_{j}|^{2})}{L_{m_{j}}(-|\alpha_{j}|^{2})}$

where $\chi_{j}(t)$ is the $j$th (classical) Floquet mode. If $m=0$ we recover
the mean values in the coherent states. In practice, the effect of this
photon-addition on the mean values of the quadrature components is a
”deformation” of the classical phase space trajectory of the coherent states.
We could see this as a change of the initial conditions of the classical
motion, which is reflected in the constant initial values of $\chi$ as

$\left\\{\chi_{j}(0)\right\\}_{PACS}=\left\\{\chi_{j}(0)\right\\}_{CS}\dfrac{L_{m_{j}}^{1}(-|\alpha_{j}|^{2})}{L_{m_{j}}(-|\alpha_{j}|^{2})}$
(49)

The second order statistical moments of the integrals of motion are calculated
in a similar way:

$\langle\hat{A}_{i}\hat{A}_{j}\rangle_{PACS}=\left\\{\begin{array}[]{l}\langle\hat{A}_{i}\rangle\langle\hat{A}_{j}\rangle\hskip
8.5359pti\neq j\\\
(\alpha_{i})^{2}\dfrac{L_{m_{i}}^{2}(-|\alpha_{i}|^{2})}{L_{m_{i}}(-|\alpha_{i}|^{2})}\hskip
8.5359pti=j\end{array}\right.$ (50)
$\langle\hat{A}^{\dagger}_{i}\hat{A}^{\dagger}_{j}\rangle_{PACS}=\left\\{\begin{array}[]{l}\langle\hat{A}_{i}^{\dagger}\rangle\langle\hat{A}_{j}^{\dagger}\rangle\hskip
8.5359pti\neq j\\\
(\alpha_{i}^{*})^{2}\dfrac{L_{m_{i}}^{2}(-|\alpha_{i}|^{2})}{L_{m_{i}}(-|\alpha_{i}|^{2})}\hskip
8.5359pti=j\end{array}\right.$ (51)
$\langle\hat{A}^{\dagger}_{i}\hat{A}_{j}\rangle_{PACS}=\left\\{\begin{array}[]{l}\langle\hat{A}_{i}^{\dagger}\rangle\langle\hat{A}_{j}\rangle\hskip
8.5359pti\neq j\\\
|\alpha_{i}|^{2}\dfrac{L_{m_{i}}^{1}(-|\alpha_{i}|^{2})}{L_{m_{i}}(-|\alpha_{i}|^{2})}+m_{l}\hskip
8.5359pti=j\end{array}\right.$ (52)

From these relations follow that all non-diagonal elements of each block of
the covariance matrix (for the creation and annihilation operators) are zero,
and we only need to calculate the diagonal ones, using (22). To avoid a
cumbersome notation we define

$P_{m_{i}}^{n}(-|\alpha_{i}|^{2})\equiv\dfrac{L_{m_{i}}^{n}(-|\alpha_{i}|^{2})}{L_{m_{i}}(-|\alpha_{i}|^{2})}-\left(\dfrac{L_{m_{i}}^{1}(-|\alpha_{i}|^{2})}{L_{m_{i}}(-|\alpha_{i}|^{2})}\right)^{2}$
(53)

The elements of the covariance matrix for the integrals of motion $\hat{I}(t)$
in $|\\{\alpha\\},\\{m\\};t\rangle$ are then given by

$\sigma_{PACS}(A_{i}^{\dagger}(t),A_{i}^{\dagger}(t))=(\alpha_{i}^{*})^{2}P_{m_{i}}^{2}(-|\alpha_{i}|^{2}),$
$\sigma_{PACS}(A_{i},A_{i})=(\alpha_{i})^{2}P_{m_{i}}^{2}(-|\alpha_{i}|^{2}),$
$\sigma_{PACS}(A_{i}^{\dagger},A_{i})=|\alpha_{i}|^{2}P_{m_{i}}^{1}(-|\alpha_{i}|^{2})+m_{i}+1/2,$
(54)

The covariance matrix formed with (54) reduces to that of Fock states (25) if
one takes $\alpha_{k}=0$, and to that of coherent states if $m_{k}=0$, for all
values of $k$, as it should be. The explicit appearance of $\alpha_{i}$ and
$m_{i}$ in (54) implies that the covariances depend on both the coherent state
that was excited and the number of excitations added to it. Although all
coherent states $|\\{\alpha\\};t\rangle$ share the same covariance matrix
(30), the states obtained by their excitation $|\\{\alpha\\},\\{m\\};t\rangle$
do not. Since the excitation of the ground state ($\alpha=0$) produces Fock
states, which are non-classical and do not minimize the uncertainty relation
for $n\neq 0$, also the excitation of the coherent states produces states that
are both non-classical and non-intelligent for any combination of
$\\{\alpha\\}$ and $\\{m\\}$ ($m\neq 0$).

The corresponding covariance matrix for the position and momentum operators
$\sigma_{PACS}(\hat{x})$ is given by direct substitution of (54) in (23).
After some algebraic manipulations and using (7) and (8), we find

$\sigma_{PACS}(x_{i},x_{j})=\sum_{k=1}^{2N}F_{ik}F_{jk}\chi_{k}^{2}(t)P_{m_{k}}^{(2)}(-|\alpha_{k}|^{2})+$
$+\sum_{k=1}^{N}\left\\{(F_{ik}F_{j,k+N}+F_{i,k+N}F_{j,k})\times\right.$
$\left.\times(|\alpha_{i}|^{2}P_{m_{i}}^{1}(-|\alpha_{i}|^{2})+m_{i}+1/2)\right\\}.$
(55)

The functions $\chi_{k}(t)$ appearing in the first term of the right hand
side, are the classical Floquet modes $\chi(0)\exp(iWt)$. The initial values
$\chi_{k}(0)$ are equal to $\alpha_{k}^{*}$, for $k<N$ and to $\alpha_{k}$ for
$k>N$. This is a major difference between the covariances in PACS and the
covariances in intelligent states (30), or Fock states (25). In these two
last, the covariances between canonical operators are totally determined by
the Floquet-Lyapunov transformation and, in the case of Fock states, by the
number of excitations. In the photon-added states, on the other hand, to fully
determine the covariances we also need the mean number of photons
$|\alpha_{k}|^{2}$ (of the intelligent states) and the Floquet modes of the
classical system.

It is important to note that even if the covariances between position and
momentum operators in a photon-added coherent state are functions of time, the
_determinant_ of the covariance matrix is $constant$, it is not affected by
the parametric excitation. This implies that the difference between such a
determinant and the minimum of the Robertson relation is only a function of
$\alpha_{i}$ and $m_{i}$ and does not depend on the Floquet-Lyapunov
transformation. This result is a multidimensional generalization of that
obtained in [24] for the uni-dimensional system.

##  5 Conclusions

In this work we have determined the photon-added coherent states of a system
of coupled oscillators whose frequencies and coupling factors change
periodically with time. Exploiting the Floquet-Lyapunov theory we derived
first, the linear integrals of motion of this system and related them to the
quantum mechanical counterpart of the Floquet modes in the classical theory.
The Floquet-Lyapunov transformation was shown to provide a link between the
covariance matrices of the integrals of motion and of canonical observables
through expression (23). In other words, the time evolution of the covariance
matrix for the canonical observables is fully determined by that of the
Floquet-Lyapunov transformation and the Floquet modes of the classical system.

Furthermore, we showed that the integrals of motion of the problem can be
easily translated into operators that behave as the creation and annihilation
operators of the harmonic oscillator and we used this analogy to obtain the
(generalized) coherent states of this system. These were shown to be the
intelligent states, that is, they minimize the Robertson’s inequality.
Finally, this analogy was further extended to construct the photon-added
coherent states and to obtain their wavefunction in coordinate representation
and the corresponding Wigner distribution. These are not-intelligent states,
for which the determinant of the covariance matrix depends on both the complex
eigenvalues of the annihilation operator that describes the coherent state and
the number of excitations added to it. We obtained that, although the elements
of the covariance matrix for canonical observables are explicitly time-
dependent, the determinant of the covariance matrix is a constant strictly
greater than the Robertson’s minimum, for nonzero $m$.

We want to emphasize that these results are valid for any periodic quadratic
Hamiltonian of the form (1), as long as the corresponding classical system is
stable. Moreover, we are confident that our findings might be of interest to
attack continuous-variable quantum information problems, where analytical
results concerning non-Gaussian states are not as common as for the Gaussian
counterpart. For example, the wavefunction (34) can be used to construct phase
space representations of the PACS, photon distribution functions and in
general any quantity of importance for quantum optical applications in time-
periodic settings.

###### Acknowledgements.

We want to thank E. Aurell, A. Buchleitner and M. Walschaers for the careful
reading of the manuscript and useful suggestions.

##  6 Suplementary Material

We show here the detailed calculation of the photon-added states wavefunction
in coordinate representation. That is, we find

$\psi_{\alpha,m}(\textbf{q},t)=\langle\textbf{q}|\\{\alpha\\},\\{m\\};t\rangle$

From (33) it follows that

$\psi_{\alpha,m}(\textbf{q},t)=N_{m}e^{-|\alpha|^{2}/2}\times$
$\times\sum_{n}\prod_{i=1}^{N}\dfrac{(\alpha_{i})^{n_{i}}}{\sqrt{n_{i}!}}\sqrt{(n_{i}+1)...(n_{i}+m_{i})}\langle\textbf{q}|\textbf{n}+\textbf{m}\rangle$
$=N_{n,m}e^{-|\alpha|^{2}/2}e^{\frac{1}{2}i\textbf{q}^{T}U^{-T}V^{T}\textbf{q}}\sum_{n}\prod_{i=1}^{N}\dfrac{(\alpha_{i})^{n_{i}}}{\sqrt{n_{i}!}}\mathcal{H}_{\textbf{m}+\textbf{n}}^{M}(U^{-*}\textbf{q})$

We have to calculate the sum on the right hand side of this expression. Using
the definition of the multidimensional Hermite polynomials, the sum can be
recast as

$(-1)^{\Sigma_{i}m_{i}}e^{\frac{1}{2}\textbf{q}^{T}U^{-\dagger}MU^{-*}\textbf{q}}\dfrac{\partial^{\Sigma_{i}m_{i}}}{\partial
q_{1}^{m_{1}}...\partial q_{N}^{m_{N}}}\times$ (56)
$\times\left(e^{-\frac{1}{2}\textbf{q}^{T}U^{-\dagger}MU^{-*}\textbf{q}}\sum_{\\{n\\}}\prod_{i=1}^{N}\dfrac{(\alpha_{i})^{n_{i}}}{\sqrt{n_{i}!}}\mathcal{H}_{\textbf{n}}^{M}(U^{-*}\textbf{q})\right)$

The term between parenthesis contains the generating function for Hermite
polynomials, so it will transform into

$e^{-\frac{1}{2}\textbf{q}^{T}U^{-\dagger}MU^{-*}\textbf{q}+\alpha^{T}MU^{-*}\textbf{q}-\frac{1}{2}\alpha^{T}M\alpha}$

Completing squares on the exponent we obtain for the sum

$e^{-\frac{1}{2}\alpha^{T}M\alpha+\alpha^{T}MU^{-*}\textbf{q}}\mathcal{H}_{\textbf{n}}^{M}(U^{-*}\textbf{q}-\alpha)$

We substitute this in the expression for $\psi_{\alpha,m}$ and recognize

$\psi_{\alpha}(\textbf{q},t)=N_{\alpha}e^{-|\alpha|^{2}/2}e^{\frac{1}{2}i\textbf{q}^{T}U^{-T}V^{T}\textbf{q}-\frac{1}{2}\alpha^{T}M\alpha+\alpha^{T}MU^{-*}\textbf{q}}$

as the wavefunction of the intelligent states [3, 6]. This will lead us to the
result (34). The Wigner function is constructed from

$W(\textbf{q},\textbf{p},t)=\int_{R^{N}}d^{N}\textbf{r}e^{-i\textbf{p}^{T}\textbf{r}}\psi_{\alpha,m}(\textbf{q}+\frac{\textbf{r}}{2},t)\psi_{\alpha,m}^{*}(\textbf{q}-\frac{\textbf{r}}{2},t)$

This will transform into an integral of Gaussian weight function times the
product of two N-dimensional Hermite polynomials. To solve it, we can write
such a product as

$\mathcal{H}_{\textbf{m}}^{M}\left(\textbf{v}\right)\mathcal{H}_{\textbf{m}}^{M^{*}}\left(\textbf{w}^{*}\right)=\mathcal{H}_{\textbf{m},\textbf{m}}^{Q}\left(\textbf{z}\right)$

where

$Q=\left(\begin{array}[]{cc}M&0\\\ 0&M^{*}\end{array}\right)$

and $\textbf{z}^{T}=(\textbf{v}^{T}\hskip 5.69046pt\textbf{w}^{\dagger})$. We
then introduce 2N arbitrary parameters and make use of the generating function
of multidimensional Hermite polinomials to express the integral as a Gaussian
integral. The result will contain a product of N two-dimensional Hermite
polynomials whos defining matrix is the Pauli matrix

$\left(\begin{array}[]{cc}0&1\\\ 1&0\end{array}\right).$

This will allow us to express them as standard Laguerre polynomials of order
$m_{k}$, and we will obtain (36).

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4. (d)

$c=a$ and $u_{j}=a.u^{\prime}$ for some $\mathrm{BCCSP}_{\scriptsize\|}$ term
$u^{\prime}$ with $x\in\mathrm{var}(u^{\prime})$. In this case we are going to
prove a slightly weaker property, namely that not all summands $u_{j}$ with
$x\in\mathrm{var}(u_{j})$ can be of this form. Despite being weaker, this
property is enough because, as shown above, all other possibilities for an
occurrence of $x$ in a summand $u_{j}$ have already been excluded. To this
end, consider the closed substitution $\sigma^{\prime}$ defined by

$\sigma^{\prime}(y)=\begin{cases}ap_{N}&\text{ if }y=x\\\ \sigma(y)&\text{
otherwise.}\end{cases}$

Then we have that
$\sigma^{\prime}(t_{i})=\sigma^{\prime}(t^{\prime})\mathbin{\|}\sigma^{\prime}(x)+\sigma^{\prime}(t^{\prime\prime\prime})\xrightarrow{\,{a}\,}\sigma(t^{\prime})\mathbin{\|}p_{N}\sim_{\mathtt{PF}}a\mathbin{\|}p_{N}$.
Since $\sigma^{\prime}(t)\sim_{\mathtt{PF}}\sigma^{\prime}(u)$ then there is a
process $r$ such that $\sigma^{\prime}(u)\xrightarrow{\,{a}\,}r$ and
$\mathtt{T}(r)=\mathtt{T}(a\mathbin{\|}p_{N})$. In particular, this means that
$0pt(r)=N+2$. Hence, from the choices of $N,\sigma$ and $\sigma^{\prime}$, we
can infer that such an $a$-move by $\sigma^{\prime}(u)$ can only stem from a
summand $u_{j}$ such that $x\in\mathrm{var}(u_{j})$. Assume, towards a
contradiction, that all such summands $u_{j}$ are of the form
$a.u_{j}^{\prime}$ for some $\mathrm{BCCSP}_{\scriptsize\|}$ term
$u_{j}^{\prime}$ with $x\in\mathrm{var}(u_{j}^{\prime})$ and
$r=\sigma^{\prime}(u_{j}^{\prime})$. As
$0pt(\sigma^{\prime}(u_{j}^{\prime}))=N+2=0pt(\sigma^{\prime}(x))$, by Lemma
32 we get that $u_{j}^{\prime}$ can only be of the form
$u_{j}^{\prime}=x+w_{j}$ for some $\mathrm{BCCSP}_{\scriptsize\|}$ term
$w_{j}$ with $0pt(\sigma^{\prime}(w_{j}))\leq N+2$. Notice that
$\mathtt{T}(\sigma^{\prime}(x))\subset\mathtt{T}(a\mathbin{\|}p_{N})$. Hence
$\sigma^{\prime}(w_{j})\neq\mathbf{0}$. More precisely,
$\sigma^{\prime}(x)=ap_{N}$ implies that $\\{b\alpha\mid
b\alpha\in\mathtt{T}(a\mathbin{\|}p_{N})\\}\subseteq\mathtt{T}(\sigma^{\prime}(w_{j}))\subseteq\mathtt{T}(a\mathbin{\|}p_{N})$.
Clearly, no trace starting with action $b$ can stem from $\sigma^{\prime}(x)$
and we can then infer, in light of Lemma 32, that
$x\not\in\mathrm{var}(w_{j})$, as $0pt(\sigma^{\prime}(w_{j}))\leq N+2$. This
implies that $\sigma^{\prime}(w_{j})=\sigma(w_{j})$ and thus $\\{b\alpha\mid
b\alpha\in\mathtt{T}(a\mathbin{\|}p_{N})\\}\subseteq\mathtt{T}(\sigma(w_{j}))\subseteq\mathtt{T}(a\mathbin{\|}p_{N})$.
In particular, $\sigma(w_{j})$ can perform at least one (completed) trace of
the form $b\alpha$ where $\alpha$ contains two occurrences of action $a$. From
$\sigma(u_{j})=a.(\sigma(x)+\sigma(w_{j}))$, we then get that
$(ab\alpha,\emptyset)\in\mathtt{PF}(\sigma(u))$, namely $\sigma(u)$ can
perform at least one (completed) trace containing 3 occurrences of action $a$.
This gives a contradiction with
$\sigma(u)\sim_{\mathtt{PF}}a\mathbin{\|}p_{N}$.

We have therefore obtained that $x$ does not occur in the scope of prefixing
in (at least one) $u_{j}$. We proceed now by a case analysis on the possible
forms of this summand.

    1. (a)

$u_{j}=x$. Then $\sigma(u)$ has a summand which is possible futures equivalent
to $\sum_{k=1}^{m}b^{i_{k}}a$. We show that this gives a contradiction with
$\sigma(u)\sim_{\mathtt{PF}}a\mathbin{\|}p_{N}$. This follows directly by
noticing that, due to the summand $b^{i_{1}}a$, we have that
$(b^{i_{1}}a,\emptyset)\in\mathtt{PF}(\sigma(u))$. However,
$(b^{i_{1}}a,\emptyset)\not\in\mathtt{PF}(a\mathbin{\|}p_{N})$, since
$a\mathbin{\|}p_{N}$ by performing the trace $b^{i_{1}}a$ can reach either a
process that can perform an $a$ (in case the first $b$-move is performed by
the summand $b^{i_{1}}a$ of $p_{N}$) or a $b$ (in case the first $b$-move is
performed by a summand $b^{i}a$ of $p_{N}$ such that $i>i_{1}$).

    2. (b)

$u_{j}=(x+w)\mathbin{\|}w^{\prime}$, for some terms $w,w^{\prime}$ with
$w^{\prime}\not\sim_{\mathtt{PF}}\mathbf{0}$. From
$\sigma(u)\sim_{\mathtt{PF}}a\mathbin{\|}p_{N}$, we infer that
$\mathtt{CT}(\sigma(u_{j}))\subseteq\mathtt{CT}(a\mathbin{\|}p_{N})$. We
recall that no completed trace of $a\mathbin{\|}p_{N}$ has $b$ as last symbol
and, moreover, in all the completed traces of $a\mathbin{\|}p_{N}$ there are
exactly two occurrences of $a$. Hence, all (nonempty) completed traces of
$\sigma(x),\sigma(w)$ and $\sigma(w^{\prime})$ must have exactly one
occurrence of $a$ and this occurrence must be as the last symbol in the
completed trace.

We now proceed to show that $\sigma(w^{\prime})$ has a summand $a$ and
$a\not\in\mathtt{I}(\sigma(x)+\sigma(w))$. We start by noticing that it cannot
be the case that
$a\in\mathtt{I}(\sigma(x)+\sigma(w))\cap\mathtt{I}(\sigma(w^{\prime}))$, for
otherwise we would have
$a^{2}\in\mathtt{T}(\sigma(u_{j}))\subseteq\mathtt{T}(\sigma(u))$, thus
contradicting $\sigma(u)\sim_{\mathtt{PF}}a\mathbin{\|}p_{N}$. Assume now,
towards a contradiction, that $\mathtt{I}(\sigma(w^{\prime}))=\\{b\\}$. Then,
due to summand $b^{i_{m}}a$ of $\sigma(x)$, we have that
$\sigma(u_{j})\xrightarrow{\,{b^{i_{m}-1}}\,}ba\mathbin{\|}\sigma(w^{\prime})$
and $a\alpha\not\in\mathtt{T}(ba\mathbin{\|}\sigma(w^{\prime}))$ for any trace
$\alpha\in\mathcal{A}^{\ast}$. Clearly,
$(b^{i_{m}-1},\mathtt{T}(ba\mathbin{\|}\sigma(w^{\prime})))\in\mathtt{PF}(\sigma(u_{j}))$,
and thus it is also a possible future of $\sigma(u)$. However,
$(b^{i_{m}-1},\mathtt{T}(ba\mathbin{\|}\sigma(w^{\prime})))\not\in\mathtt{PF}(a\mathbin{\|}p_{N})$,
as the interleaving of $p_{N}$ with $a$ guarantees that after an initial trace
of an arbitrary number of $b$-transitions it is always possible to perform a
trace starting with $a$. This gives a contradiction with
$\sigma(u)\sim_{\mathtt{PF}}a\mathbin{\|}p_{N}$. We have therefore obtained
that $a\in\mathtt{I}(\sigma(w^{\prime}))$. More precisely, from the
constraints on the completed traces of $\sigma(w^{\prime})$, we can infer that
$\sigma(w^{\prime})$ has a summand $a$.

Our order of business will now be to show that
$\sigma(w^{\prime})\sim_{\mathtt{PF}}a$. Since
$\sigma(w^{\prime})\xrightarrow{\,{a}\,}\mathbf{0}$, we have that
$\sigma(u_{j})\xrightarrow{\,{a}\,}(\sigma(x)+\sigma(w))\mathbin{\|}\mathbf{0}\sim_{\mathtt{PF}}\sigma(x)+\sigma(w)$.
Thus, $\sigma(u)\sim_{\mathtt{PF}}a\mathbin{\|}p_{N}$ implies that
$a\mathbin{\|}p_{N}\xrightarrow{\,{a}\,}r$ for some $r$ with
$\mathtt{T}(r)=\mathtt{T}(\sigma(x)+\sigma(w))$. Since $a\mathbin{\|}p_{N}$
has only one possible initial $a$-transition, namely
$a\mathbin{\|}p_{N}\xrightarrow{\,{a}\,}\mathbf{0}\mathbin{\|}p_{N}$, we get
that $r\sim_{\mathtt{PF}}p_{N}$ and thus
$\mathtt{T}(\sigma(x)+\sigma(w))=\mathtt{T}(p_{N})$. In particular, this
implies that $0pt(\sigma(x)+\sigma(w))=N+1$. Therefore, we have

$\displaystyle 1\leq 0pt(\sigma(w^{\prime}))={}$ $\displaystyle
0pt(\sigma(u_{j}))-0pt(\sigma(x)+\sigma(w))$ $\displaystyle={}$ $\displaystyle
0pt(\sigma(u_{j}))-(N+1)$ $\displaystyle\leq{}$ $\displaystyle
0pt(\sigma(u))-(N+1)$ $\displaystyle={}$ $\displaystyle
0pt(a\mathbin{\|}p_{N})-(N+1)$ (by Lem. 28.3) $\displaystyle={}$
$\displaystyle N+2-(N+1)$ $\displaystyle={}1$

and we can therefore conclude that $\sigma(w^{\prime})\sim_{\mathtt{PF}}a$.
Furthermore, it is not difficult to prove that
$\mathtt{CT}(\sigma(x)+\sigma(w))=\mathtt{CT}(p_{N})$, for otherwise we get a
contradiction with $\sigma(u)\sim_{\mathtt{PF}}a\mathbin{\|}p_{N}$.

So far we have obtained that, modulo possible futures equivalence,

$\sigma(u_{j})\sim_{\mathtt{PF}}\left(\sum_{k=1}^{m}b^{i_{k}}a+\sigma(w)\right)\mathbin{\|}a\,\text{
and }\,\mathtt{CT}(\sum_{k=1}^{m}b^{i_{k}}a+\sigma(w))=\\{b^{i}a\mid
i\in\\{1,\dots,N\\}\\}\enspace.$

To conclude the proof, we need to show that
$\sum_{k=1}^{m}b^{i_{k}}a+\sigma(w)\sim_{\mathtt{PF}}p_{N}$. Let
$I_{m}=\\{i_{1},\dots,i_{m}\\}$ and $I_{N}=\\{1,\dots,N\\}$. Assume, towards a
contradiction, that
$\sum_{k=1}^{m}b^{i_{k}}a+\sigma(w)\not\sim_{\mathtt{PF}}p_{N}$. Since
$\mathtt{CT}(\sigma(x)+\sigma(w))=\mathtt{CT}(p_{N})$, from Lemma 26 we can
infer that $\sigma(w)$ does not contain any occurrence of $\mathbin{\|}$. In
particular, $\sigma(w)$ can be written in the general form
$\sigma(w)=\sum_{l\in L}q_{l}$ for some terms $q_{l}$ that do not have $+$ as
head operator nor contain any occurrence of $\mathbin{\|}$. Moreover, as
$\sum_{k=1}^{m}b^{i_{k}}a+\sigma(w)\not\sim_{\mathtt{PF}}p_{N}$, by Lemma 27,
this means that either there is an $i\in I_{N}\setminus I_{m}$ such that
$b^{i}a\not\sim_{\mathtt{PF}}q_{l}$ for any $l\in L$, or that there is a
summand $q_{l}$ of $\sigma(w)$ such that $q_{l}\not\sim_{\mathtt{PF}}b^{i}a$
for any $i\in I_{N}$. In both cases, we obtain that there is (at least) a
summand $q_{l}$ of $\sigma(w)$ such that $b^{k}a,b^{h}a\in\mathtt{CT}(q_{l})$
for some $k\neq h,h,k\in I_{N}$. We can then proceed as in the proof of Lemma
27 to prove that this gives the desired contradiction. We have therefore
obtained that $\sum_{k=1}^{m}b^{i_{k}}a+\sigma(w)\sim_{\mathtt{PF}}p_{N}$.
Hence, since possible futures equivalence is a congruence with respect to
parallel composition, we get that

$\sigma(u_{j})\sim_{\mathtt{PF}}a\mathbin{\|}p_{N}$

and we can therefore conclude that $\sigma(u)$ has the desired summand.

This concludes the proof.

######  Theorem 34.

Let $\mathcal{E}$ be a finite axiom system over
$\mathrm{BCCSP}_{\scriptsize\|}$ that is sound modulo $\sim_{\mathtt{PF}}$.
Let $N$ be larger than the size of each term in the equations in
$\mathcal{E}$. Assume that $p$ and $q$ are closed terms that contain no
occurrences of $\mathbf{0}$ as a summand or factor, and that
$p,q\sim_{\mathtt{PF}}a\mathbin{\|}p_{N}$. If $\mathcal{E}\vdash p\approx q$
and $p$ has a summand possible futures equivalent to $a\mathbin{\|}p_{N}$,
then so does $q$.

######  Proof 6.7.

Assume that $\mathcal{E}$ is a finite axiom system over the language
$\mathrm{BCCSP}_{\scriptsize\|}$ that is sound modulo possible futures
equivalence, and that the statements (1)–(4) below hold, for some closed terms
$p$ and $q$ and positive integer $N$ larger than the size of each term in the
equations in $\mathcal{E}$:

  1. (1)

$E\vdash p\approx q$,

  2. (2)

$p\sim_{\mathtt{PF}}q\sim_{\mathtt{PF}}a\mathbin{\|}p_{N}$,

  3. (3)

$p$ and $q$ contain no occurrences of $\mathbf{0}$ as a summand or factor, and

  4. (4)

$p$ has a summand possible futures equivalent to $a\mathbin{\|}p_{N}$.

We prove that $q$ also has a summand possible futures equivalent to
$a\mathbin{\|}p_{N}$ by induction on the depth of the closed proof of the
equation $p\approx q$ from $\mathcal{E}$. Recall that, without loss of
generality, we may assume that the closed terms involved in the proof of the
equation $p\approx q$ have no $\mathbf{0}$ summands or factors (by Proposition
42, as $\mathcal{E}$ may be assumed to be saturated), and that applications of
symmetry happen first in equational proofs (that is, $\mathcal{E}$ is closed
with respect to symmetry).

We proceed by a case analysis on the last rule used in the proof of $p\approx
q$ from $\mathcal{E}$. The case of reflexivity is trivial, and that of
transitivity follows immediately by using the inductive hypothesis twice.
Below we only consider the other possibilities.

  * •

Case $E\vdash p\approx q$, because $\sigma(t)=p$ and $\sigma(u)=q$ for some
equation $(t\approx u)\in E$ and closed substitution $\sigma$. Since
$\sigma(t)=p$ and $\sigma(u)=q$ have no $\mathbf{0}$ summands or factors, and
$N$ is larger than the size of each term mentioned in equations in
$\mathcal{E}$, the claim follows by Proposition 33.

  * •

Case $E\vdash p\approx q$, because $p=cp^{\prime}$ and $q=cq^{\prime}$ for
some $p^{\prime},q^{\prime}$ such that $E\vdash p^{\prime}\approx q^{\prime}$,
and for some action $c$. This case is vacuous because
$p=cp^{\prime}\not\sim_{\mathtt{PF}}a\mathbin{\|}p_{N}$, and thus $p$ does not
have a summand possible futures equivalent to $a\mathbin{\|}p_{N}$.

  * •

Case $E\vdash p\approx q$, because $p=p^{\prime}+p^{\prime\prime}$ and
$q=q^{\prime}+q^{\prime\prime}$ for some
$p^{\prime},q^{\prime},p^{\prime\prime},q^{\prime\prime}$ such that $E\vdash
p^{\prime}\approx q^{\prime}$ and $E\vdash p^{\prime\prime}\approx
q^{\prime\prime}$. Since $p$ has a summand possible futures equivalent to
$a\mathbin{\|}p_{N}$, we have that so does either $p^{\prime}$ or
$p^{\prime\prime}$. Assume, without loss of generality, that $p^{\prime}$ has
a summand possible futures equivalent to $a\mathbin{\|}p_{N}$. Since $p$ is
possible futures equivalent to $a\mathbin{\|}p_{N}$, so is $p^{\prime}$. Using
the soundness of $\mathcal{E}$ modulo possible futures equivalence, it follows
that $q^{\prime}\sim_{\mathtt{PF}}a\mathbin{\|}p_{N}$. The inductive
hypothesis now yields that $q^{\prime}$ has a summand possible futures
equivalent to $a\mathbin{\|}p_{N}$. Hence, $q$ has a summand possible futures
equivalent to $a\mathbin{\|}p_{N}$, which was to be shown.

  * •

Case $E\vdash p\approx q$, because $p=p^{\prime}\mathbin{\|}p^{\prime\prime}$
and $q=q^{\prime}\mathbin{\|}q^{\prime\prime}$ for some
$p^{\prime},q^{\prime},p^{\prime\prime},q^{\prime\prime}$ such that $E\vdash
p^{\prime}\approx q^{\prime}$ and $E\vdash p^{\prime\prime}\approx
q^{\prime\prime}$. Since the proof involves no uses of $\mathbf{0}$ as a
summand or a factor, we have that
$p^{\prime},p^{\prime\prime}\not\sim_{\mathtt{PF}}\mathbf{0}$ and
$q^{\prime},q^{\prime\prime}\not\sim_{\mathtt{PF}}\mathbf{0}$. It follows that
$q$ is a summand of itself. By our assumptions,
$q^{\prime}\mathbin{\|}q^{\prime\prime}\sim_{\mathtt{PF}}a\mathbin{\|}p_{N}$
which, by Proposition 30 gives that either $q^{\prime}\sim_{\mathtt{S}}a$ and
$q^{\prime\prime}\sim_{\mathtt{S}}p_{N}$, or
$q^{\prime}\sim_{\mathtt{S}}p_{N}$ and $q^{\prime\prime}\sim_{\mathtt{S}}a$.
In both cases, we can conclude that $q$ has itself as summand of the required
form.

This completes the proof of Theorem 34 and thus of Theorem 25.

As the left-hand side of equation $e_{N}$, i.e., the term
$a\mathbin{\|}p_{N}$, has a summand possible futures equivalent to
$a\mathbin{\|}p_{N}$, whilst the right-hand side, i.e., the term
$ap_{N}+\sum_{i=1}^{N}b(a\mathbin{\|}b^{i-1}a)$, does not, we can conclude
that the collection of infinitely many equations $e_{N}$ ($N\geq 1$) is the
desired witness family. This concludes the proof of Theorem 25.

###  6.2. Extending the negative result

It is easy to check that the equations $e_{N}$ ($N\geq 1$) in the witness
family of the negative result for $\sim_{\mathtt{PF}}$ are all sound modulo
bisimilarity, i.e., the largest symmetric simulation. Consequently, they are
also sound modulo any congruence $\,{\mathcal{R}}\,$ such that
${\sim_{\mathtt{B}}}\subseteq{\,{\mathcal{R}}\,}\subseteq{\sim_{\mathtt{PF}}}$.
Hence, the negative result for all these equivalences can be derived from that
for $\sim_{\mathtt{PF}}$, by exploiting this fact and that any finite axiom
system that is sound modulo $\,{\mathcal{R}}\,$ is also sound modulo
$\sim_{\mathtt{PF}}$.

######  Theorem 35.

Assume that ${|\mathcal{A}|}\geq 2$. Let $\,{\mathcal{R}}\,$ be a congruence
such that
${\sim_{\mathtt{B}}}\subseteq{\,{\mathcal{R}}\,}\subseteq{\sim_{\mathtt{PF}}}$.
Then $\,{\mathcal{R}}\,$ has no finite, ground-complete, equational
axiomatisation over the language $\mathrm{BCCSP}_{\scriptsize\|}$.

######  Proof 6.8.

Let $\mathcal{E}$ be a finite equational axiomatisation for
$\mathrm{BCCSP}_{\scriptsize\|}$ that is sound modulo $\,{\mathcal{R}}\,$.
Since $\,{\mathcal{R}}\,$ is included in $\sim_{\mathtt{PF}}$, we have that
the axiom system $\mathcal{E}$ is sound modulo $\sim_{\mathtt{PF}}$. Let $N$
be larger than the size of each term in the equations in $\mathcal{E}$.
Theorem 34 implies that the equation

$a\mathbin{\|}p_{N}\approx ap_{N}+\sum_{i=1}^{N}b(a\mathbin{\|}b^{i-1}a)$

cannot be derived from $\mathcal{E}$. Since this equation is sound modulo
$\,{\mathcal{R}}\,$, namely

$a\mathbin{\|}p_{N}\,\,{\mathcal{R}}\,\,ap_{N}+\sum_{i=1}^{N}b(a\mathbin{\|}b^{i-1}a)$

it follows that $\mathcal{E}$ is not complete modulo $\,{\mathcal{R}}\,$.

Theorem 35 can be applied to establish for $n\geq 2$ that the $n$-nested trace
and simulation semantics have no finite, ground-complete equational
axiomatisation over $\mathrm{BCCSP}_{\scriptsize\|}$. The $n$-nested trace
equivalences were introduced in [22] as an alternative tool to define
bisimilarity. The hierarchy of $n$-nested simulations, namely simulation
relations contained in a (nested) simulation equivalence, was introduced in
[21].

[$n$-nested semantics] For $n\geq 0$, the relation $\sim^{n}_{\mathtt{T}}$
over $\mathcal{P}$, called the $n$-_nested trace equivalence_ , is defined
inductively as follows:

  * •

$p\sim^{0}_{\mathtt{T}}q$ for all $p,q\in\mathcal{P}$,

  * •

$p\sim^{n+1}_{\mathtt{T}}q$ if and only if for all traces
$\alpha\in\mathcal{A}^{*}$:

    * –

if $p\xrightarrow{\,{\alpha}\,}p^{\prime}$ then there is a $q^{\prime}$ such
that $q\xrightarrow{\,{\alpha}\,}q^{\prime}$ and
$p^{\prime}\sim^{n}_{\mathtt{T}}q^{\prime}$, and

    * –

if $q\xrightarrow{\,{\alpha}\,}q^{\prime}$ then there is a $p^{\prime}$ such
that $p\xrightarrow{\,{\alpha}\,}p^{\prime}$ and
$p^{\prime}\sim^{n}_{\mathtt{T}}q^{\prime}$.

For $n\geq 0$, the relation $\sqsubseteq^{n}_{\mathtt{S}}$ over $\mathcal{P}$
is defined inductively as follows:

  * •

$p\sqsubseteq^{0}_{\mathtt{S}}q$ for all $p,q\in\mathcal{P}$,

  * •

$p\sqsubseteq^{n+1}_{\mathtt{S}}q$ if and only if $p\,{\mathcal{R}}\,q$ for
some simulation $\,{\mathcal{R}}\,$, with $\,{\mathcal{R}}\,^{-1}$ included in
$\sqsubseteq^{n}_{\mathtt{S}}$.

$n$-_nested simulation equivalence_ is the kernel of
$\sqsubseteq^{n}_{\mathtt{S}}$, i.e., the equivalence
${\sim^{n}_{\mathtt{S}}}={\sqsubseteq^{n}_{\mathtt{S}}}\cap{(\sqsubseteq^{n}_{\mathtt{S}})^{-1}}$.

Note that $\sim^{1}_{\mathtt{T}}$ corresponds to trace equivalence,
$\sim^{2}_{\mathtt{T}}$ is possible futures equivalence, and
$\sim^{1}_{\mathtt{S}}$ is simulation equivalence. The following theorem is a
corollary of Theorems 25 and 35.

######  Theorem 36.

Assume that ${|\mathcal{A}|}\geq 2$. Let $n\geq 2$. Then, $n$-nested trace
equivalence and $n$-nested simulation equivalence admit no finite, ground-
complete, equational axiomatisation over the language
$\mathrm{BCCSP}_{\scriptsize\|}$.

##  7\. Adding CCS synchronisation

The negative results provided above conclude our analysis of the
axiomatisability of the purely interleaving parallel composition operator
modulo the congruences in the linear time-branching time spectrum.

The most natural extension of our work consists in allowing the parallel
components to synchronise. In particular, we are interested in the _CCS-style
communication_. It presupposes a bijection $\overline{\cdot}$ on $\mathcal{A}$
such that $\overline{\overline{a}}=a$ and $\overline{a}\neq a$ for all
$a\in\mathcal{A}$. Following [26], the special action symbol
$\tau\not\in\mathcal{A}$, will result from the synchronised occurrence of the
complementary actions $a$ and $\bar{a}$. Let
$\mathcal{A}_{\tau}=\mathcal{A}\cup\\{\tau\\}$. Then, we let the metavariables
$\alpha,\beta,\dots$ range over $\mathcal{A}_{\tau}$.

The rules in Table 11 define the operational semantics of the parallel
composition operator when also synchronisation is taken into account.

$\displaystyle\inference{x\xrightarrow{\,{\alpha}\,}x^{\prime}}{x\mathbin{\|}y\xrightarrow{\,{\alpha}\,}x^{\prime}\mathbin{\|}y}\quad\inference{y\xrightarrow{\,{\alpha}\,}y^{\prime}}{x\mathbin{\|}y\xrightarrow{\,{\alpha}\,}x\mathbin{\|}y^{\prime}}\quad\inference{x\xrightarrow{\,{a}\,}x^{\prime}\quad
y\xrightarrow{\,{\bar{a}}\,}y^{\prime}}{x\mathbin{\|}y\xrightarrow{\,{\tau}\,}x^{\prime}\mathbin{\|}y^{\prime}}\quad\inference{x\xrightarrow{\,{\bar{a}}\,}x^{\prime}\quad
y\xrightarrow{\,{a}\,}y^{\prime}}{x\mathbin{\|}y\xrightarrow{\,{\tau}\,}x^{\prime}\mathbin{\|}y^{\prime}}$
Table 11.  Operational semantics of parallel composition with CCS
communication.

Our order of business for this section will then be to show if and how the
analysis we carried out in the previous sections is affected by the addition
of synchronisation à la CCS.

###  7.1. The positive results

It is not difficult to see that the arguments we used to prove the existence
of finite, ground-complete axiomatisations still hold also with
synchronisation. The only changes we need to apply are reported in Table 12.
For $\textrm{Y}\in\\{1,2,3\\}$, the axiom schema ELCY simply adds the terms
related to communication to ELY.

(ELC1) $\alpha x\mathbin{\|}\beta y\approx\alpha(x\mathbin{\|}\beta
y)+\beta(\alpha x\mathbin{\|}y)$ if $\alpha\neq\overline{\beta}$, or
$\alpha=\tau$, or $\beta=\tau$
---
(ELC1$\tau$) $\alpha x\mathbin{\|}\beta y\approx\alpha(x\mathbin{\|}\beta
y)+\beta(\alpha x\mathbin{\|}y)+\tau(x\mathbin{\|}y)$ if
$\alpha=\overline{\beta}$
(ELC2) $\sum_{i\in I}\alpha_{i}x_{i}\mathbin{\|}\sum_{j\in
J}\beta_{j}y_{j}\approx\sum_{i\in I}\alpha_{i}(x_{i}\mathbin{\|}\sum_{j\in
J}\beta_{j}y_{j})+\sum_{j\in J}\beta_{j}(\sum_{i\in
I}\alpha_{i}x_{i}\mathbin{\|}y_{j})+$
$\sum_{i\in I}\alpha_{i}x_{i}\mathbin{\|}\sum_{j\in J}\beta_{j}y_{j}\approx$
$+\sum_{i\in I,j\in
J\atop\alpha_{i}=\overline{\beta_{j}}}\tau(x_{i}\mathbin{\|}y_{j})$
with $\alpha_{i}\neq\alpha_{k}$ whenever $i\neq k$ and
$\beta_{j}\neq\beta_{h}$ whenever $j\neq h$, $\forall\,i,k\in
I,\forall\,j,h\in J$
(ELC3) $\sum_{i\in I}\alpha_{i}x_{i}\mathbin{\|}\sum_{j\in
J}\beta_{j}y_{j}\approx\sum_{i\in I}\alpha_{i}(x_{i}\mathbin{\|}\sum_{j\in
J}\beta_{j}y_{j})+\sum_{j\in J}\beta_{j}(\sum_{i\in
I}\alpha_{i}x_{i}\mathbin{\|}y_{j})+$
$\sum_{i\in I}\alpha_{i}x_{i}\mathbin{\|}\sum_{j\in J}\beta_{j}y_{j}\approx$
$+\sum_{i\in I,j\in
J\atop\alpha_{i}=\overline{\beta_{j}}}\tau(x_{i}\mathbin{\|}y_{j})$
Table 12.  The different instantiations of the expansion law when
communication is considered.

We remark that, by convention, $\sum_{i\in\emptyset}t_{i}=\mathbf{0}$. A
finite, ground-complete axiomatisation over $\mathrm{BCCSP}_{\scriptsize\|}$
modulo $\sim_{\mathtt{X}}$, for
$\mathtt{X}\in\\{\mathtt{T},\mathtt{CT},\mathtt{F},\mathtt{R},\mathtt{FT},\mathtt{RT},\mathtt{S},\mathtt{CS},\mathtt{RS}\\}$
is given by the axiom system $\mathcal{E}^{c}_{\mathtt{X}}$ obtained from
$\mathcal{E}_{\mathtt{X}}$ as follows:

  * •

we include $\\{\textrm{ELC1},\textrm{ELC1}\tau\\}$ instead of EL1;

  * •

we include ELC2 instead of EL2;

  * •

all other axioms are unchanged (although the action variables occurring in
them now range over $\mathcal{A}_{\tau}$).

For instance, a finite, ground-complete axiomatisation over
$\mathrm{BCCSP}_{\scriptsize\|}$ modulo ready simulation equivalence is given
by the axiom system
$\mathcal{E}^{c}_{\mathtt{RS}}=\mathcal{E}_{1}\cup\\{\textrm{RS},\textrm{RSP1},\textrm{RSP2},\textrm{ELC2}\\}$.

###  7.2. The negative results

In [27] it was proved that bisimilarity does not admit a finite, ground-
complete axiomatisation over $\mathrm{BCCSP}_{\scriptsize\|}$ with CCS-style
synchronisation. We now proceed to show that, by applying similar arguments to
those used in Section 6.1, we can obtain the same negative result for possible
futures equivalence. More precisely, we prove the following:

######  Theorem 37.

Assume that ${|\mathcal{A}_{\tau}|}\geq 2$. Possible futures equivalence has
no finite, ground-complete, equational axiomatisation over the language
$\mathrm{BCCSP}_{\scriptsize\|}$ with CCS synchronisation.

To this end, consider the infinite family of equations $\\{e^{c}_{N}\mid N\geq
1\\}$ given by:

$\displaystyle q_{N}=\sum_{i=1}^{N}\tau^{i}a$ $\displaystyle(N\geq
1)\enspace\phantom{.}$ $\displaystyle
e^{c}_{N}\;\colon\;a\mathbin{\|}q_{N}\approx
aq_{N}+\sum_{i=1}^{N}\tau(a\mathbin{\|}\tau^{i-1}a)$ $\displaystyle(N\geq
1)\enspace.$

Clearly, this family of equations coincides with that used to prove the
negative result in Section 6.1 where we have substituted all occurrences of
action $b$ with the special action $\tau$. We notice that the equations
$e^{c}_{N}$ are sound not only modulo possible futures equivalence, but also
modulo bisimilarity (each $e^{c}_{N}$ is in fact a distinct closed instance of
ELC3 in Table 12). This means that if we can obtain the negative result for
possible futures, then we can proceed exactly as in Section 6.2 to extend it
to all the congruences $\sim$ such that
$\sim_{\mathtt{B}}\subseteq\sim\subseteq\sim_{\mathtt{PF}}$.

Since $\tau$ cannot communicate with any action, hence, in particular, it does
not communicate with $a$, the analysis we carried out in Section 6.1 (with $b$
replaced by $\tau$) still applies.

######  Remark 38.

Most results in Section 6.2 rely on the assumption that terms do not contain
$\mathbf{0}$ summands or factors. Notice that due to undefined communications,
the expansion law schemas in Table 12 may actually introduce terms that are
possible futures equivalent to $\mathbf{0}$ in the equational proofs.

However, we remark that this is not an issue. In fact, in general, we could
extend the results in Appendix D to deal with the possible $\mathbf{0}$
summands introduced by the expansion laws. Briefly, whenever a $\mathbf{0}$
summand is introduced, we can assume that axioms A0 and P0 (Table 3) are also
applied in order to get rid of the unnecessary $\mathbf{0}$ summands and
factors. Hence, the proof of Theorem 37 follows from Theorem 34 and the fact
that none of the summands in the right-hand side of the equations $e^{c}_{N}$
is possible futures equivalent to $a\mathbin{\|}p_{N}$.

##  8\. Concluding remarks

We have studied the finite axiomatisability of the language
$\mathrm{BCCSP}_{\scriptsize\|}$ modulo the behavioural equivalences in the
linear time-branching time spectrum. On the one hand we have obtained finite,
ground-complete axiomatisations modulo the (decorated) trace and simulation
semantics in the spectrum. On the other hand we have proved that for all
equivalences that are finer than possible futures equivalence and coarser than
bisimilarity a finite ground-complete axiomatisation does not exist.

Since our ground-completeness proof for ready simulation equivalence proceeds
via elimination of $\|$ from closed terms (Proposition 4), and all behavioural
equivalences in the linear time-branching time spectrum that include ready
simulation have a finite ground-complete axiomatisation over BCCSP, it
immediately follows from the elimination result that all these behavioural
equivalences have a finite ground-complete axiomatisation over
$\mathrm{BCCSP}_{\scriptsize\|}$. Exploiting various forms of distributivity
of parallel composition over choice, we were able to present more concise and
elegant axiomatisations for the coarser behavioural equivalences.

In this paper we have considered both, a parallel composition operator that
implements interleaving without synchronisation between the parallel
components, and a parallel composition operator with CCS-style
synchronisation. It is natural to consider extensions of our result to
parallel composition operators with other forms of synchronisation. We expect
the extensions with ACP-style or CSP-style synchronisation to be less
straightforward than the extension with CCS-style synchronisation presented in
this paper, especially in the case of the negative results, and we leave these
as topics for future investigations.

As previously outlined, in [2] it was proved that the nested semantics admit
no finite axiomatisation over BCCSP. However, our negative results cannot be
reduced to a mere lifting of those in [2], as the presence of the additional
operator $\mathbin{\|}$ might, at least in principle, allow us to finitely
axiomatise the equations over BCCSP processes that are valid modulo the
considered nested semantics. Indeed, auxiliary operators can be added to a
language in order to obtain a finite axiomatisation of some congruence
relation (see, e.g. the classic example given in [6]). Understanding whether
it is possible to lift non-finite axiomatisability results among different
algebras, and under which constraints this can be done, is an interesting
research avenue and we aim to investigate it in future work. A methodology for
transferring non-finite-axiomatisability results across languages was
presented in [4], where a reduction-based approach was proposed. However, that
method has some limitations and thus further studies are needed.

A behavioural equivalence is _finitely based_ if it has a finite equational
axiomatisation from which all valid equations between open terms are
derivable. In [16] and [3] finite bases for bisimilarity with respect to PA
and $\mathrm{BCCSP}_{\scriptsize\|}$ extended with the auxiliary operators
left merge and communication merge were presented. Furthermore, in [12] an
overview was given of which behavioural equivalences in the linear time-
branching time spectrum are finitely based with respect to BCCSP. The negative
results in Section 6 imply that none of the behavioural equivalences between
possible futures equivalence and bisimilarity is finitely based with respect
to $\mathrm{BCCSP}_{\scriptsize\|}{}$. An interesting question is which of the
behavioural equivalences including ready simulation semantics is finitely
based with respect to $\mathrm{BCCSP}_{\scriptsize\|}$.

In [14] an alternative classification of the equivalences in the spectrum with
respect to [19] was proposed. In order to obtain a general, unified, view of
process semantics, the spectrum was divided into layers, each corresponding to
a different notion of constrained simulation [13]. There are pleasing
connections between the different layers and the partitioning they induce of
the congruences in the spectrum, as given in [14], and the relationships
between the axioms for the interleaving operator we have presented in this
study.

## Acknowledgement

This work has been supported by the project ‘ _Open Problems in the Equational
Logic of Processes_ ’ (OPEL) of the Icelandic Research Fund (grant No.
196050-051).

We thank Rob van Glabbeek for a fruitful discussion on the axiomatisability of
failures equivalence.

## References

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##  Appendix A The Mace4 code for $\mathcal{E}_{\mathtt{CS}}$

The following is the Mace4 code we used to generate a model for
$\mathcal{E}_{\mathtt{CS}}$ in which EL2 (as given in formulas(goals) below)
does not hold.

⬇

set(verbose).

assign(max_megs, 1000).

assign(domain_size,5).

op(750,prefix,”a”).

op(750,prefix,”b”).

op(850,infix,”plus”).

op(950,infix,”par”).

formulas(assumptions).

(x plus 0) = x.

(x plus y) = (y plus x).

((x plus y) plus z) = (x plus (y plus z)).

(x plus x) = x.

(x par 0) = x.

(x par y) = (y par x).

% Axiom EL1

(a x par a y) = (a (x par a y) plus a (y par a x)).

(a x par b y) = (a (x par b y) plus b (y par a x)).

(b x par b y) = (b (x par b y) plus b (y par b x)).

% Axiom CS

(a (a x plus (y plus z))) = (a (a x plus (y plus z)) plus a (a x plus z)).

(a (b x plus (y plus z))) = (a (b x plus (y plus z)) plus a (b x plus z)).

(b (a x plus (y plus z))) = (b (a x plus (y plus z)) plus b (a x plus z)).

(b (b x plus (y plus z))) = (b (b x plus (y plus z)) plus b (b x plus z)).

% Axiom CSP1

((a x plus (a y plus u)) par (a z plus (a w plus v))) =

( ((a x plus u) par (a z plus (a w plus v)))

plus ( ((a y plus u) par (a z plus (a w plus v)))

plus ( ((a x plus (a y plus u)) par (a z plus v))

plus ((a x plus (a y plus u)) par (a w plus v))

))).

((a x plus (a y plus u)) par (a z plus (b w plus v))) =

( ((a x plus u) par (a z plus (b w plus v)))

plus ( ((a y plus u) par (a z plus (b w plus v)))

plus ( ((a x plus (a y plus u)) par (a z plus v))

plus ((a x plus (a y plus u)) par (b w plus v))

))).

((a x plus (b y plus u)) par (a z plus (b w plus v))) =

( ((a x plus u) par (a z plus (b w plus v)))

plus ( ((b y plus u) par (a z plus (b w plus v)))

plus ( ((a x plus (b y plus u)) par (a z plus v))

plus ((a x plus (b y plus u)) par (b w plus v))

))).

((a x plus (b y plus u)) par (b z plus (b w plus v))) =

( ((a x plus u) par (b z plus (b w plus v)))

plus ( ((b y plus u) par (b z plus (b w plus v)))

plus ( ((a x plus (b y plus u)) par (b z plus v))

plus ((a x plus (b y plus u)) par (b w plus v))

))).

((b x plus (b y plus u)) par (b z plus (b w plus v))) =

( ((b x plus u) par (b z plus (b w plus v)))

plus ( ((b y plus u) par (b z plus (b w plus v)))

plus ( ((b x plus (b y plus u)) par (b z plus v))

plus ((b x plus (b y plus u)) par (b w plus v))

))).

% Axiom CSP2

(a x par (a y plus (a z plus w))) =

(a(x par (a y plus (a z plus w) ))

plus ( (a x par (a y plus w))

plus ( (a x par (a z plus w))

))).

(a x par (a y plus (b z plus w))) =

(a(x par (a y plus (b z plus w) ))

plus ( (a x par (a y plus w))

plus ( (a x par (b z plus w))

))).

(a x par (b y plus (b z plus w))) =

(a(x par (b y plus (b z plus w) ))

plus ( (a x par (b y plus w))

plus ( (a x par (b z plus w))

))).

(b x par (a y plus (a z plus w))) =

(b(x par (a y plus (a z plus w) ))

plus ( (b x par (a y plus w))

plus ( (b x par (a z plus w))

))).

(b x par (a y plus (b z plus w))) =

(b(x par (a y plus (b z plus w) ))

plus ( (b x par (a y plus w))

plus ( (b x par (b z plus w))

))).

(b x par (b y plus (b z plus w))) =

(b(x par (b y plus (b z plus w) ))

plus ( (b x par (b y plus w))

plus ( (b x par (b z plus w))

))).

end_of_list.

⬇

formulas(goals).

((a x plus b y) par (a z plus b w)) = (a(x par (a z plus b w)) plus (b(y par
(a z plus b w)) plus (a(z par (a x plus b y)) plus b(w par (a x plus b y))))).

end_of_list.

##  Appendix B The Mace4 code for $\mathcal{E}_{\mathtt{RT}}$

The following is the Mace4 code we used to generate a model for
$\mathcal{E}_{\mathtt{RT}}$ in which RSP2 (as given in formulas(goals) below)
does not hold.

⬇

set(verbose).

assign(max_megs, 1000).

op(750,prefix,”a”).

op(750,prefix,”b”).

op(850,infix,”plus”).

op(950,infix,”par”).

formulas(assumptions).

(x plus 0) = x.

(x plus y) = (y plus x).

((x plus y) plus z) = (x plus (y plus z)).

(x plus x) = x.

(x par 0) = x.

(x par y) = (y par x).

% Axiom RT

( a(((a x plus a y) plus (a u plus a v)) plus z) ) =

( a((a x plus a u) plus z)

plus

a((a y plus a v) plus z)

).

( a(((a x plus a y) plus (b u plus b v)) plus z) ) =

( a((a x plus b u) plus z)

plus

a((a y plus b v) plus z)

).

( a(((b x plus b y) plus (b u plus b v)) plus z) ) =

( a((b x plus b u) plus z)

plus

a((b y plus b v) plus z)

).

( b(((a x plus a y) plus (a u plus a v)) plus z) ) =

( b((a x plus a u) plus z)

plus

b((a y plus a v) plus z)

).

( b(((a x plus a y) plus (b u plus b v)) plus z) ) =

( b((a x plus b u) plus z)

plus

b((a y plus b v) plus z)

).

( b(((b x plus b y) plus (b u plus b v)) plus z) ) =

( b((b x plus b u) plus z)

plus

b((b y plus b v) plus z)

).

% Axiom FP

( (a x plus (a y plus w)) par z) = ( ((a x plus w) par z) plus ((a y plus w)
par z) ).

% Axiom EL2

(a x par a y) = (a (x par a y) plus a (y par a x)).

(a x par b y) = (a (x par b y) plus b (y par a x)).

(b x par b y) = (b (x par b y) plus b (y par b x)).

((a x plus b y) par (a z plus b w)) =

( a (x par (a z plus b w))

plus ( b (y par (a z plus b w))

plus ( a ((a x plus b y) par z)

plus ( b ((a x plus b y) par w)

)))).

end_of_list.

⬇

formulas(goals).

( a x par (b u plus (b v plus w) ) ) =

( ( a x par (b u plus w ) )

plus (( a x par (b v plus w ) )

plus a(x par (b u plus (b v plus w) ))

)).

end_of_list.

##  Appendix C The Mace4 codes for $\mathcal{E}_{\mathtt{CT}}$

The following is the Mace4 code we used to generate a model for
$\mathcal{E}_{\mathtt{CT}}$ in which EL2 (as given in formulas(goals) below)
does not hold.

⬇

set(verbose).

assign(max_megs, 1000).

assign(domain_size,5).

op(750,prefix,”a”).

op(750,prefix,”b”).

op(850,infix,”plus”).

op(950,infix,”par”).

formulas(assumptions).

(x plus 0) = x.

(x plus y) = (y plus x).

((x plus y) plus z) = (x plus (y plus z)).

(x plus x) = x.

(x par 0) = x.

(x par y) = (y par x).

% Axiom EL1

(a x par b y) = (a (x par b y) plus b (y par a x)).

(a x par a y) = (a (x par a y) plus a (y par a x)).

(b x par b y) = (b (x par b y) plus b (y par b x)).

% Axiom CT

(a(a x plus z) plus a(a y plus w)) = (a(a x plus (a y plus (z plus w)))).

(a(a x plus z) plus a(b y plus w)) = (a(a x plus (b y plus (z plus w)))).

(a(b x plus z) plus a(a y plus w)) = (a(b x plus (a y plus (z plus w)))).

(a(b x plus z) plus a(b y plus w)) = (a(b x plus (b y plus (z plus w)))).

(b(a x plus z) plus a(a y plus w)) = (b(a x plus (a y plus (z plus w)))).

(b(a x plus z) plus a(b y plus w)) = (b(a x plus (b y plus (z plus w)))).

(b(b x plus z) plus a(a y plus w)) = (b(b x plus (a y plus (z plus w)))).

(b(b x plus z) plus a(b y plus w)) = (b(b x plus (b y plus (z plus w)))).

% Axiom CTP

((a x plus (a y plus w)) par z) = (((a x plus w) par z) plus ((a y plus w) par
z)).

((a x plus (b y plus w)) par z) = (((a x plus w) par z) plus ((b y plus w) par
z)).

((b x plus (b y plus w)) par z) = (((b x plus w) par z) plus ((b y plus w) par
z)).

end_of_list.

⬇

formulas(goals).

((a x plus b y) par (a z plus b w)) = (a(x par (a z plus b w)) plus (b(y par
(a z plus b w)) plus (a(z par (a x plus b y)) plus b(w par (a x plus b y))))).

end_of_list.

Finally, by combining the formulas(assumptions) given above with the
formulas(goals) presented below, we can generate a model for
$\mathcal{E}_{\mathtt{CT}}$ in which CSP2 does not hold.

⬇

formulas(goals).

( a x par (b u plus (b v plus w) ) ) =

( ( a x par (b u plus w ) )

plus (( a x par (b v plus w ) )

plus a(x par (b u plus (b v plus w) ))

)).

end_of_list.

##  Appendix D Simplifying the equational theory: saturated systems

The axioms A0 and P0 in Table 3 (used from left to right) are enough to
establish that each $\mathrm{BCCSP}_{\scriptsize\|}$ term that is possible
futures equivalent to $\mathbf{0}$ is also provably equal to $\mathbf{0}$.

######  Lemma 39.

Let $t$ be a $\mathrm{BCCSP}_{\scriptsize\|}$ term. Then
$t\sim_{\mathtt{PF}}\mathbf{0}$ if and only if the equation
$t\approx\mathbf{0}$ is provable using axioms _A0_ and _P0_ in Table 3 from
left to right.

######  Proof D.1.

The “if” implication is an immediate consequence of the soundness of the
equations A0 and P0 with respect to $\sim_{\mathtt{PF}}$. To prove the “only
if” implication, define, first of all, the collection NIL of
$\mathrm{BCCSP}_{\scriptsize\|}$ terms as the set of terms generated by the
following grammar:

$t::=\mathbf{0}\;\mid\;t+t\;\mid\;t\mathbin{\|}t\enspace,$

We claim that each $\mathrm{BCCSP}_{\scriptsize\|}$ term $t$ is
$\sim_{\mathtt{PF}}$ equivalent to $\mathbf{0}$ if and only if
$t\in\text{NIL}$. Using this claim and structural induction on
$t\in\text{NIL}$, it is a simple matter to show that if
$t\sim_{\mathtt{PF}}\mathbf{0}$, then $t\approx\mathbf{0}$ is provable using
axioms A0 and P0 from left to right, which was to be shown.

To complete the proof, it therefore suffices to show the above claim. To
establish the “if” implication in the statement of the claim, one proves,
using structural induction on $t$ and the congruence properties of
$\sim_{\mathtt{PF}}$, that if $t\in\text{NIL}$, then
$\sigma(t)\sim_{\mathtt{PF}}\mathbf{0}$ for every closed substitution
$\sigma$. To show the “only if” implication, we establish the contrapositive
statement, namely that if $t\not\in\text{NIL}$, then
$\sigma(t)\not\sim_{\mathtt{PF}}\mathbf{0}$ for some closed substitution
$\sigma$. To this end, it suffices only to show, using structural induction on
$t$, that if $t\not\in\text{NIL}$, then $\sigma_{a}(t)\xrightarrow{\,{a}\,}$
for some action $a\in\mathcal{A}$, where $\sigma_{a}$ is the closed
substitution mapping each variable to the closed term $a\mathbf{0}$. The
details of this argument are not hard, and are therefore left to the reader.

In light of the above result, we shall assume, without loss of generality,
that each axiom system we consider includes the equations in Table 3. This
assumption means, in particular, that our axiom systems will allow us to
identify each term that is possible futures equivalent to $\mathbf{0}$ with
$\mathbf{0}$.

We recall that a $\mathrm{BCCSP}_{\scriptsize\|}$ term $t$ has a $\mathbf{0}$
factor if it contains a subterm of the form $t_{1}\mathbin{\|}t_{2}$, where
either $t_{1}$ or $t_{2}$ is possible futures equivalent to $\mathbf{0}$.

It is easy to see that, modulo the equations in Table 3, every
$\mathrm{BCCSP}_{\scriptsize\|}$ term $t$ has the form $\sum_{i\in I}t_{i}$,
for some finite index set $I$, and terms $t_{i}$ ($i\in I$) that are not
$\mathbf{0}$ and do not have themselves the form
$t^{\prime}+t^{\prime\prime}$, for some terms $t^{\prime}$ and
$t^{\prime\prime}$. The terms $t_{i}$ ($i\in I$) will be referred to as the
summands of $t$. Moreover, again modulo the equations in Table 3, each of the
$t_{i}$ can be assumed to have no $\mathbf{0}$ factors.

We can now introduce the notion of _saturated system_ , namely an axiom system
such that if a closed equation that relates two terms containing no
occurrences of $\mathbf{0}$ as a summand or factor, then there is a closed
proof for it in which all of the terms have no occurrences of $\mathbf{0}$ as
a summand or factor (cf. [27, Proposition 5.1.5]).

For each $\mathrm{BCCSP}_{\scriptsize\|}$ term $t$, we define ${t}/\mathbf{0}$
thus:

$\displaystyle{\mathbf{0}}/\mathbf{0}=\mathbf{0}\qquad\qquad{x}/\mathbf{0}=x\qquad\qquad{at}/\mathbf{0}=a({t}/\mathbf{0})$
$\displaystyle({t+u})/\mathbf{0}=\begin{cases}{u}/\mathbf{0}&\text{ if
}t\sim_{\mathtt{PF}}\mathbf{0}\\\ {t}/\mathbf{0}&\text{ if
}u\sim_{\mathtt{PF}}\mathbf{0}\\\ ({t}/\mathbf{0})+({u}/\mathbf{0})&\text{
otherwise}\end{cases}\qquad(t\mathbin{\|}u)/\mathbf{0}=\begin{cases}{u}/\mathbf{0}&\text{
if }t\sim_{\mathtt{PF}}\mathbf{0}\\\ {t}/\mathbf{0}&\text{ if
}u\sim_{\mathtt{PF}}\mathbf{0}\\\
({t}/\mathbf{0})\mathbin{\|}({u}/\mathbf{0})&\text{ otherwise.}\end{cases}$

Intuitively, $t/\mathbf{0}$ is the term that results by removing all
occurrences of $\mathbf{0}$ as a summand or factor from $t$.

The following lemma collects the basic properties of the above construction.

######  Lemma 40.

For each $\mathrm{BCCSP}_{\scriptsize\|}$ term $t$, the following statements
hold:

  1. (1)

The equation $t\approx t/\mathbf{0}$ can be proven using the equations in
Table 3, and therefore $t\sim_{\mathtt{PF}}t/\mathbf{0}$.

  2. (2)

The term $t/\mathbf{0}$ has no $\mathbf{0}$ summands or factors.

  3. (3)

$t/\mathbf{0}=t$, if $t$ has no occurrence of $\mathbf{0}$ as a summand or
factor.

  4. (4)

$\sigma(t/\mathbf{0})/\mathbf{0}=\sigma(t)/\mathbf{0}$, for each substitution
$\sigma$.

######  Proof D.2.

Immediate by structural induction over $t$.

[Saturated system] We say that a substitution $\sigma$ is a
$\mathbf{0}$-substitution if and only if $\sigma(x)\neq x$ implies that
$\sigma(x)=\mathbf{0}$, for each variable $x$. Let $\mathcal{E}$ be an axiom
system. We define the axiom system $\mathrm{cl}(\mathcal{E})$ thus:

$\mathrm{cl}(\mathcal{E})=\mathcal{E}\cup\\{\sigma(t)/\mathbf{0}\approx\sigma(u)/\mathbf{0}\mid(t\approx
u)\in\mathcal{E},~{}\sigma~{}\text{a $\mathbf{0}$-substitution}\\}\enspace.$

An axiom system $\mathcal{E}$ is saturated if
$\mathcal{E}=\mathrm{cl}(\mathcal{E})$.

The following lemma collects some basic sanity properties of the closure
operator $\mathrm{cl}(\cdot)$.

######  Lemma 41.

Let $\mathcal{E}$ be an axiom system. Then the following statements hold.

  1. (1)

$\mathrm{cl}(\mathcal{E})=\mathrm{cl}(\mathrm{cl}(\mathcal{E}))$.

  2. (2)

$\mathrm{cl}(\mathcal{E})$ is finite, if so is $\mathcal{E}$.

  3. (3)

$\mathrm{cl}(\mathcal{E})$ is sound modulo possible futures equivalence, if so
is $\mathcal{E}$.

  4. (4)

$\mathrm{cl}(\mathcal{E})$ is closed with respect to symmetry, if so is
$\mathcal{E}$.

  5. (5)

$\mathrm{cl}(\mathcal{E})$ and $\mathcal{E}$ prove the same equations, if
$\mathcal{E}$ contains the equations in Table 3.

######  Proof D.3.

We limit ourselves to sketching the proofs of statements (1) and (5) in the
lemma.

In the proof of statement (1), the only non-trivial thing to check is that the
equation

$\sigma(\sigma^{\prime}(t)/\mathbf{0}))/\mathbf{0}\approx\sigma(\sigma^{\prime}(u)/\mathbf{0}))/\mathbf{0}$

is contained in $\mathrm{cl}(\mathcal{E})$, whenever $(t\approx
u)\in\mathcal{E}$ and $\sigma,\sigma^{\prime}$ are $\mathbf{0}$-substitutions.
This follows from Lemma 40.(4) because the collection of
$\mathbf{0}$-substitutions is closed under composition.

To show statement (5), it suffices only to argue that each equation $t\approx
u$ that is provable from $\mathrm{cl}(\mathcal{E})$ is also provable from
$\mathcal{E}$, if $\mathcal{E}$ contains the equations in Table 3. This can be
done by induction on the depth of the proof of the equation $t\approx u$ from
$\mathrm{cl}(\mathcal{E})$, using Lemma 40.(1) for the case in which $t\approx
u$ is a substitution instance of an axiom in $\mathrm{cl}(\mathcal{E})$.

We are now ready to state our counterpart of [27, Proposition 5.1.5].

######  Proposition 42.

Assume that $\mathcal{E}$ is a saturated axiom system. Suppose furthermore
that we have a closed proof from $\mathcal{E}$ of the closed equation
$p\approx q$. Then replacing each term $r$ in that proof with $r/\mathbf{0}$
yields a closed proof of the equation $p/\mathbf{0}\approx q/\mathbf{0}$. In
particular, the proof from $\mathcal{E}$ of an equation $p\approx q$, where
$p$ and $q$ are terms not containing occurrences of $\mathbf{0}$ as a summand
or factor, need not use terms containing occurrences of $\mathbf{0}$ as a
summand or factor.

######  Proof D.4.

The proof follows the lines of that of [27, Proposition 5.1.5], and is
therefore omitted.