# Apparent Heating due to Imperfect Calorimetric Measurements

Brecht Donvil  University of Helsinki, Department of Mathematics and
Statistics P.O. Box 68 FIN-00014, Helsinki, Finland<EMAIL_ADDRESS>Joachim Ankerhold  Institute for Complex Quantum Systems and IQST, Ulm
University - Albert-Einstein-Allee 11, D-89069 Ulm, Germany
<EMAIL_ADDRESS>
###### Abstract

Performing imperfect or noisy measurements on a quantum system both impacts
the measurement outcome and the state of the system after the measurement. In
this paper we are concerned with imperfect calorimetric measurements. In
calorimetric measurements one typically measures the energy of a thermal
environment to extract information about the system. The measurement is
imperfect in the sense that we simultaneously measure the energy of the
calorimeter and an additional noise bath. Under weak coupling assumptions, we
find that the presence of the noise bath manifests itself by modifying the
jump rates of the reduced system dynamics. We study an example of a driven
qubit interacting with resonant bosons calorimeter and find increasing the
noise leads to a reduction in the power flowing from qubit to calorimeter and
thus an apparent heating up of the calorimeter.

###### pacs:

03.65.Yz, 42.50.Lc

##  I Introduction

Measurements unavoidably come with noise. In quantum mechanics the effect of a
noisy measurement is double, as it does not only influence the outcome but
also the state of the system after the measurement.

Perfect continuous measurements lead to the quantum Zeno effect, where the
system gets stuck in a state with a small probability to jump away Misra and
Sudarshan (1977). For imperfect measurements the dynamics are much richer. In
case measurement times are sufficiently short and the distribution of
measurement outcomes is much broader than the state vector, the dynamics of
the measured system are described by a non-linear stochastic Liouville
equation Belavkin (1987), see also Jacobs and Steck (2006) for a pedagogic
introduction. Intermediate regimes regimes have been successfully studied with
path-integral methods Mensky et al. (1993); Audretsch and Mensky (1997). The
dynamics of a system weakly coupled to an environment under continuous perfect
measurements can be described by quantum jump equations Hudson and
Parthasarathy (1984); Barchielli and Belavkin (1991); Gardiner et al. (1992);
Dalibard et al. (1992); Carmichael (1993), see also Breuer and Petruccione
(2002); Wiseman and Milburn (2010). If the perfect measurement is disturbed by
the presence of another bath and one considers a coarse grained time scale on
which many jumps happen, the system dynamics undergo quantum state diffusion
Breuer and Petruccione (2002); Wiseman and Milburn (2010).

In quantum calorimetric measurement schemes one continuously measures the
energy, or temperature, of the environment in contact with a system of
interest. These indirect measurements allow to indirectly extract information
about the system. Quantum calorimetry has been used for example to detect
cosmic x-rays Stahle et al. (1999) and in quantum circuit measurements Ronzani
et al. (2018); Kokkoniemi et al. (2019); Senior et al. (2020). Recent
experiments have shown that quantum calorimeters form a promising tool for
single microwave photon detection in quantum circuits Karimi et al. (2020).

Earlier studies of experimental setups such as Ronzani et al. (2018) have
modelled the calorimeter-system dynamics as coupled jump processes of the
energy Suomela et al. (2016) or temperature of calorimeter Kupiainen et al.
(2016); Donvil et al. (2018, 2019) and state of the system. The approaches of
Suomela et al. (2016); Kupiainen et al. (2016); Donvil et al. (2018, 2019)
differ from the usual quantum jump schemesHudson and Parthasarathy (1984);
Barchielli and Belavkin (1991); Gardiner et al. (1992); Dalibard et al.
(1992); Carmichael (1993) in that they explicitly update the state of the
environment with the measured value of the energy or temperature. Concretely,
this means that the jump rates depend on the measured state of the
calorimeter. The previous coupled jump equations Suomela et al. (2016);
Kupiainen et al. (2016); Donvil et al. (2018, 2019) all require that the
calorimeter was under continuous perfect energy or temperature measurements.
In the current work, we study quantum calorimetric measurement set-ups such as
Ronzani et al. (2018) in the case of imperfect detection.

The authors of Warszawski and Wiseman (2002) developed quantum trajectories
for realistic photon detection. They take into account noise on the
measurement outcome and delay in obtaining it. Recently these ideas were
applied to model single photon measurements in quantum circuit calorimetric
measurements Karimi and Pekola (2020). Our approach differs from Warszawski
and Wiseman (2002); Karimi and Pekola (2020) in the sense that we do not
consider noise or delay on the outcome, i.e. imperfect information, but noise
on the actual projective operator applied to the calorimeter during the
measurement.

We consider a simple model for the imperfect measurement by introducing
another bath, the noise bath, that does not interact with calorimeter or
system. The only influence of the noise bath is that its energy is measured
simultaneously with the calorimeter energy, instead of just the calorimeter
energy. This is similar to the setup where quantum state diffusion applies
Breuer and Petruccione (2002); Wiseman and Milburn (2010), although we are not
interested in coarse graining time. Under the presence of the noise bath we
derive a hybrid master equation and corresponding coupled jump process for the
measured energy and system state. Comparing out result to the dynamics
described in Suomela et al. (2016); Kupiainen et al. (2016); Donvil et al.
(2018, 2019), we find that the coupled jump equations have similar structures
but the jump rates are modified due to the presence of the noise bath. Note
that our approach is not limited to energy measurements but applies to any
macroscopic property of the finite size environment such as for example
magnetism.

The paper is structured as follows: in Section II we introduce our model, a
qubit interacting with the calorimeter and additional noise bath, and how we
model the imperfect measurements. In Section III we derive the hybrid master
equation for the measured calorimeter-noise bath energy and the qubit state.
We also give the corresponding energy-qubit state jump process. We consider
the specific example of a calorimeter consisting of resonant bosons in Section
IV. We study the effect of the noise bath on the rates and on the power
flowing to the calorimeter. In Section V we discuss experimental setups in
which the effects of imperfect measurements could play a role. Finally, in
Section VI we discuss the results and provide an outlook.

##  II Model

We consider a system interacting with a calorimeter (see fig. 1) modelled as a
finite size environment. A detector permanently measures the combined energy
of this calorimeter and an additional noise bath which, however, is not
directly coupled to system and calorimeter. Here we introduce the model and
below, in Sec. III, we derive a joint master equation for the state of the
system and the energy monitored by the detector.

Figure 1: Schematic representation of the system-calorimeter-noise bath setup.
The system and calorimeter interact while the energy of the calorimeter and
noise bath is simultaneously measured.

While the framework developed below can be applied to any finite dimensional
system interacting with arbitrary reservoirs, for the sake of simplicity and
being explicit, we here consider a paradigmatic set-up: a qubit linearly
coupled a bosonic calorimeter. The total Hamiltonian then consist of

$H_{Q}=\omega\sigma_{+}\sigma_{-}$ (1)

for the system (qubit),

$H_{B}=\sum_{k}\omega_{k}b_{k}^{\dagger}b_{k}\,$ (2)

for the boson reservoir, i.e. the calorimeter, and

$H_{I}=\sum_{k}g_{k}(\sigma_{+}b_{k}+\sigma_{-}b^{\dagger}_{k})$ (3)

for the interaction between system and calorimeter. Finally, we write the
noise bath Hamiltonian as

$H_{N}=\sum_{E}E\operatorname{P}^{N}_{E},$ (4)

where orthogonal projectors $\operatorname{P}^{N}_{E}$ project on states of
energy $E$ of the noise bath. This way, together with the corresponding
operator for the boson reservoir $\operatorname{P}_{E}$, we can now introduce
the projector $\mathcal{P}_{E}$ on the joint reservoir and noise bath energy
$E$, i.e.,

$\mathcal{P}_{E}=\sum_{E^{\prime}}\operatorname{P}_{E^{\prime}}\otimes\operatorname{P}^{N}_{E-E^{\prime}}$
(5)

In our derivation of the energy-qubit state master equation below, we follow
the usual weak coupling treatment and assume that the qubit-calorimeter-noise
bath initial state is of the form

$\rho_{0}=\sum_{E}\rho(E,0)\otimes\mathcal{P}_{E}$ (6)

with the partially reduced density $\rho(E,0)$ of the qubit at fixed energy
$E$ of the total environment. Accordingly, taking the trace
$\operatorname{tr}(\mathcal{P}_{E}\rho_{0})=\operatorname{tr}_{Q}{\rho(E,0)}$,
we observe that the trace of the unnormalised qubit state $\rho(E,0)$ is the
probability to measure the environmental energy $E$ at the initial time.
Eliminating the noise reservoir, leads us to the qubit-calorimeter state

$\operatorname{tr}_{N}\rho_{0}=\sum_{E}\rho(E,0)\otimes\sum_{E^{\prime}}\operatorname{P}_{E^{\prime}}\operatorname{tr}_{N}\\{\operatorname{P}^{N}_{E-E^{\prime}}\\}\,.$
(7)

The latter distribution captures the energy partition between calorimeter and
the noise bath which for the sake of simplicity is taken to be of the form

$\operatorname{tr}_{N}\\{\operatorname{P}^{N}_{E-E^{\prime}}\\}\propto
e^{-(E-E^{\prime})^{2}/k}$ (8)

with a tunable parameter $k$. However, any other model originating, for
example, from a more microscopic description for a specific experimental set-
up is applicable as well.

##  III Derivation of the Hybrid Master Equation

We are now in a position to derive a qubit-energy hybrid master equation for
the partially reduced density $\rho(E,t)$ by generalizing the usual steps of a
weak coupling Born-Markov treatment Breuer and Petruccione (2002). An
alternative route in terms of the Nakajima-Zwanzig projection operator
technique Nakajima (1958); Zwanzig (1960) is presented in Appendix A, see also
Steinigeweg et al. (2007); Mallayya et al. (2019) for similar applications.

We start with the interaction-picture dynamics of the qubit-calorimeter-noise
bath as described by the Liouville-von Neumann equation

$\frac{d}{dt}\tilde{\rho}(t)=-i[\tilde{H}_{I}(t),\tilde{\rho}(t)]$ (9)

with the interaction picture Hamiltonian defined as

$\displaystyle\tilde{H}_{I}(t)=$ $\displaystyle
e^{-i(H_{Q}+H_{B})t}H_{I}e^{i(H_{Q}+H_{B})t}$ $\displaystyle=$
$\displaystyle\sum_{k}g_{k}\left[e^{i(\omega-\omega_{k})t}\sigma_{+}b_{k}+e^{-i(\omega-\omega_{k})t}\sigma_{-}b^{\dagger}_{k}\right]$
(10)

and $\tilde{\rho}$ the corresponding interaction picture state operator.

### Born-Markov approximation.

Integrating equation (9) and plugging it back into itself gives

$\frac{d}{dt}\tilde{\rho}(t)=-i[\tilde{H}_{I}(t),\rho_{0}]-\int_{0}^{t}ds\,[\tilde{H}_{I}(t),[\tilde{H}_{I}(s),\tilde{\rho}(s)]].$
(11)

To trace out the calorimeter and noise bath degrees of freedom, we proceed by
applying a modified version of the Born approximation. Since the calorimeter-
noise bath energy is continuously measured, we assume that at any time $t$ the
qubit-reservoir-noise bath state $\tilde{\rho}(t)$ has the same structure as
the initial state (6), i.e.

$\tilde{\rho}(t)\approx\sum_{E}\tilde{\rho}(E,t)\otimes\mathcal{P}_{E}.$ (12)

Additionally, we perform the Markov approximation by replacing
$\tilde{\rho}(s)$ in the integral in equation (11) by $\tilde{\rho}(t)$.
Taking the partial trace $\operatorname{tr}_{C+N}(\mathcal{P}_{E}\,\,.\,\,)$
on both sides of (11) gives

$\displaystyle\frac{d}{dt}\tilde{\rho}(E,t)=-i\operatorname{tr}_{C+N}(\mathcal{P}_{E}[\tilde{H}_{I}(t),\rho_{0}])$
$\displaystyle\quad-\int_{0}^{t}ds\,\operatorname{tr}_{C+N}(\mathcal{P}_{E}[\tilde{H}_{I}(t),[\tilde{H}_{I}(s),\sum_{E^{\prime}}\tilde{\rho}(E^{\prime},t)\otimes\mathcal{P}_{E^{\prime}}]].$
(13)

We observe that traces
$\operatorname{tr}(\mathcal{P}_{E}b_{k}\mathcal{P}_{E^{\prime}})=\operatorname{tr}(\mathcal{P}_{E}b^{\dagger}_{k}\mathcal{P}_{E^{\prime}})=0$,
which allows us to conclude that the first term on the right hand side of the
first line of (III) is zero. Next, the change of variables $s\rightarrow t-s$
is performed in the remaining integral and it is assumed that the calorimeter-
noise correlation functions decay over a time scale $\tau_{B}$ much smaller
than the qubit relaxation time $\tau_{R}$. Under said assumption we are
permitted to let the upper limit of the integral in (III) go to infinity to
arrive at the master equation

$\displaystyle\frac{d}{dt}\tilde{\rho}(E,t)=$
$\displaystyle\quad-\int_{0}^{\infty}ds\,\operatorname{tr}_{C+N}(\mathcal{P}_{E}\tilde{H}_{I}(t),[\tilde{H}_{I}(t-s),\sum_{E^{\prime}}\tilde{\rho}(E^{\prime},t)\otimes\mathcal{P}_{E^{\prime}}]].$
(14)

### Master Equation.

From equation (III) we proceed similar to the usual derivation of the Lindblad
equation Breuer and Petruccione (2002). Evaluating the time integral,
reordering terms and dropping the Lamb shift, we arrive in the Schrödinger
picture at the hybrid master equation

$\displaystyle\frac{d}{dt}\rho(E,t)=-i[H_{Q},\rho(t)]$
$\displaystyle+\Gamma_{\uparrow}(E+\omega)\sigma_{+}\rho(E+\omega,t)\sigma_{-}-\frac{\Gamma_{\uparrow}(E)}{2}\\{\sigma_{-}\sigma_{+},\rho(E,t)\\}$
$\displaystyle+\Gamma_{\downarrow}(E-\omega)\sigma_{-}\rho(E-\omega,t)\sigma_{+}-\frac{\Gamma_{\downarrow}(E)}{2}\\{\sigma_{+}\sigma_{-},\rho(E,t)\\}.$
(15)

The transition rates at fixed environmental energy $E$ are given by

$\displaystyle\Gamma_{\uparrow}(E)=$
$\displaystyle\kappa(\omega)\frac{\operatorname{tr}_{C+N}\\{b^{\dagger}_{\omega}b_{\omega}\mathcal{P}_{E}\\}}{\operatorname{tr}_{C+N}\\{\mathcal{P}_{E}\\}}$
(16a) $\displaystyle\Gamma_{\downarrow}(E)=$
$\displaystyle\kappa(\omega)\frac{\operatorname{tr}_{C+N}\\{b_{\omega}b^{\dagger}_{\omega}\mathcal{P}_{E}\\}}{\operatorname{tr}_{C+N}\\{\mathcal{P}_{E}\\}},$
(16b)

with $b_{\omega}$ denoting the ladder operator of the reservoir mode resonant
to the qubit frequency $\omega$ with coupling strength $\kappa(\omega)\propto
g_{\omega}^{2}$ [see (3)]. Exploiting our assumption on the noise bath traces
(8), we obtain the more explicit expressions

$\displaystyle\Gamma_{\uparrow}(E)=$
$\displaystyle\kappa(\omega)\frac{\sum_{E^{\prime}}\operatorname{tr}_{C}\\{b^{\dagger}_{\omega}b_{\omega}\operatorname{P}_{E^{\prime}}\\}e^{-(E-E^{\prime})^{2}/k}}{\sum_{E^{\prime}}\operatorname{tr}_{C}\\{\operatorname{P}_{E^{\prime}}\\}e^{-(E-E^{\prime})^{2}/k}}$
(17a) $\displaystyle\Gamma_{\downarrow}(E)=$
$\displaystyle\kappa(\omega)\frac{\sum_{E^{\prime}}\operatorname{tr}_{C}\\{b_{\omega}b^{\dagger}_{\omega}\operatorname{P}_{E^{\prime}}\\}e^{-(E-E^{\prime})^{2}/k}}{\sum_{E^{\prime}}\operatorname{tr}_{C}\\{\operatorname{P}_{E^{\prime}}\\}e^{-(E-E^{\prime})^{2}/k}}.$
(17b)

These results reduce to their perfect measurement counterparts by considering
$k\downarrow 0$ so that in (17) $\mathcal{P}_{E}$ is replaced by the
calorimeter energy projector $\operatorname{P}_{E}$ and in the sums over
energies only the contribution with $E=E^{\prime}$ survives Suomela et al.
(2016). Furthermore, for any finite $k$ and sufficiently large reservoirs we
expect the size of the energy eigenspaces to increase with the energy. This
implies
$\operatorname{tr}_{C}\\{\operatorname{P}_{E^{\prime}}\\},\,\operatorname{tr}_{C}\\{b^{\dagger}_{\omega}b_{\omega}\operatorname{P}_{E^{\prime}}\\}$
and
$\operatorname{tr}_{C}\\{b_{\omega}b^{\dagger}_{\omega}\operatorname{P}_{E^{\prime}}\\}$
to grow with $E^{\prime}$ and thus, as $k$ increases, the rates primarily feel
increased contributions from higher energies. This in turn may appear as an
effective heating up of the environment as, by way of example, will be
qualitatively confirmed in Section IV.

### Stochastic Evolution.

The hybrid master equation (III) can be mapped onto an equivalent stochastic
state vector dynamics according to

$\rho(E,t)=\mathsf{E}(|\psi(t)\rangle\langle\psi(t)|\delta(E-E(t)))\,,$ (18)

where $\mathsf{E}(\cdot)$ denotes a proper average over noise realizations.
Accordingly, one finds the following set of coupled stochastic differential
equations for the state of the qubit $\psi$ and the measured energy $E$ of the
calorimeter-noise bath

$\begin{cases}d\psi(t)=&-iG(\psi(t))\,dt+\left(\frac{\sigma_{-}\psi(t)}{\|\sigma_{-}\psi(t)\|}-\psi(t)\right)dN_{\downarrow}\\\
&+\left(\frac{\sigma_{+}\psi(t)}{\|\sigma_{+}\psi(t)\|}-\psi(t)\right)dN_{\uparrow}\\\
dE(t)=&\omega(dN_{\downarrow}-dN_{\uparrow})\,.\end{cases}$ (19)

Here, the continuous evolution of the state vector is given by

$\displaystyle-iG(\psi)=$
$\displaystyle-i\omega\sigma_{+}\sigma_{-}-\frac{1}{2}\Gamma_{\downarrow}(E(t))(\sigma_{+}\sigma_{-}-\|\sigma_{-}\psi\|^{2})\psi$
$\displaystyle-\frac{1}{2}\Gamma_{\uparrow}(E(t))(\sigma_{-}\sigma_{+}-\|\sigma_{+}\psi\|^{2})\psi$

and $N_{\uparrow}$, $N_{\downarrow}$ are Poisson processes with increments
obeying

$\displaystyle\mathsf{E}(dN_{\downarrow}|\psi,E)=$
$\displaystyle\Gamma_{\downarrow}(E)\|\sigma_{-}\psi\|^{2}dt$
$\displaystyle\mathsf{E}(dN_{\uparrow}|\psi,E)=$
$\displaystyle\Gamma_{\uparrow}(E)\|\sigma_{+}\psi\|^{2}dt\,.$

##  IV Example: Calorimeter of Resonant Oscillators

The central ingredients for the hybrid master equation (III) are the rates
(17). By way of example and drawing on Suomela et al. (2016), we study a
system of $N$ resonant harmonic oscillators coupled to a qubit. In this case
we are able to explicitly compute all traces required to compute the rates
(17). The number of micro-states of the calorimeter at a given energy
$E=n\omega$, i.e. the size of the calorimeter energy eigenspaces, is

$\operatorname{tr}_{C}\\{\operatorname{P}_{n\omega}\\}={n+N-1\choose N-1}$
(20)

so that

$\displaystyle\operatorname{tr}_{C}\\{a^{\dagger}_{\omega}a_{\omega}\operatorname{P}_{n\omega}\\}=n{n+N-1\choose
N-1}$ (21a)
$\displaystyle\operatorname{tr}_{C}\\{a_{\omega}a^{\dagger}_{\omega}\operatorname{P}_{n\omega}\\}=(n+N){n+N-1\choose
N-1}\,.$ (21b)

Note that the above traces indeed grow with the value of $n$. For the noise
bath, we take

$H_{N}=\sum_{n=-N_{C}}^{N_{C}}P^{N}_{n\omega},\quad\operatorname{tr}_{N}\\{P^{N}_{n\omega}\\}\propto
e^{-n^{2}\omega^{2}/k}.$ (22)

The noise bath thus has two free parameters: $k$ and the cut-off $N_{C}$.
Figure 2 shows the dependence of the rates on these parameters. As expected,
we observe that the rate increase with $k$ and eventually plateau at a value
which depends on $N_{C}$. The inset displays the energy dependence of the
rates in the case of perfect measurements (i.e. $k\rightarrow 0$) which
reveals that increasing $k$ has a similar effect as increasing $E$. Hence, in
an imperfect detection considered here, the temperature of the calorimeter
appears to be elevated compared to the measured value of the energy.

Figure 2: Dependency of the rates (17) on $k$ for different values of the cut-
off $N_{C}$ of the noise bath (22): $N_{C}=100$ (solid), $N_{C}=500$ (dashed),
$N_{C}=1000$ (dotted) for $E=0$. The inset shows the energy dependence of the
rates in the case of perfect measurements (i.e. $k\rightarrow 0$).

### Energy flow to Calorimeter.

In a next step, we explore how an imperfect measurement influences the
detection of the energy flow towards the calorimeter when the qubit is driven
by an external periodic signal, a generic situation for quantum
thermodynamical set-ups. For this purpose, a driving term is added to the
qubit Hamiltonian (1), i.e.

$H_{Q}(t)=\omega\sigma_{+}\sigma_{-}+\lambda(e^{i\omega
t}\sigma_{+}e^{-i\omega t}\sigma_{-})\,.$ (23)

The coupled jump equations (19) are simulated (time step $dt=0.03$) and the
energy flow to the calorimeter is extracted as a function of the energy
uncertainty parameter $k$. Figure 3 reports the average change in the measured
energy $\Delta E=E_{f}-E_{i}$ over 5 periods while the inset figure depicts
the same quantity in the perfect measurement case as a function of the initial
energy $E_{i}$ of the calorimeter. Obviously, in the latter case for
increasing initial energy the average energy flowing to the calorimeter
decreases. Imperfect measurements exhibit a similar behaviour for increasing
$k$ which implies that effectively a less perfect measurement shows the same
behavior as a hotter calorimeter.

Figure 3: Average change in measured energy of the calorimeter after 5 periods
of driving, the initial measured energy is $E_{i}=0$ and the qubit is
initially in the ground state. The inset plot shows the measured power in the
case of perfect measurements as a function the initial energy $E$. The
parameters are $\kappa(\omega)=0.001\omega$, $\lambda=0.05\omega$ and
$N_{C}=100$.

### Power at Steady State.

Finally, we address the power flowing through the qubit in steady state which
in calorimetric measurements is a viable tool to retrieve qubit properties
without ‘touching’ the qubit directly Ronzani et al. (2018); Senior et al.
(2020). To study the steady state properties, we add the loss term $-\omega
dN_{loss}$ to the energy jump process (19), where $N_{loss}$ is a Poisson
process with rate

$\mathsf{E}(dN_{loss})=\gamma\langle E\rangle dt\,$ (24)

and $\langle E\rangle$ is the expected energy of the calorimeter when the
energy $E$ was measured

$\langle
E\rangle=\frac{\sum_{E^{\prime}}\operatorname{tr}_{C}E^{\prime}\operatorname{tr}_{C}(\operatorname{P}_{E^{\prime}})e^{-(E-E^{\prime})^{2}/k}}{\sum_{E^{\prime}}\operatorname{tr}_{C}(\operatorname{P}_{E^{\prime}})e^{-(E-E^{\prime})^{2}/k}}\,.$
(25)

When the power from the qubit to the calorimeter and the loss term balance
each other out, the calorimeter-noise bath reaches a steady state. We call the
average energy at steady state $E_{s}$ and approximate the power flowing
through the calorimeter by

$P_{s}=\gamma\langle E_{s}\rangle.$ (26)

Figure 4 shows the power at steady state $P_{s}$ as a function of the measured
average energy $E_{s}$ for different values of $k$. The inset shows the
measured steady state energies $E_{s}$ as a function of $k$. As a comparison
the (blue) crosses display the power at steady state when not taking the
measurement error into account. In this case our estimate is then
$P^{\prime}_{s}=\gamma E_{s}$. We observe that in the imperfect situation the
actual power flowing through the calorimeter is nearly constant. This is not
surprising as the qubit driving remains constant when varying $k$. However,
when not taking the measurement errors into account, a significant
underestimation of the power can occur.

Figure 4: Power at steady state as a function of the measured energy at the
different values of $k$. The inset shows the measured energies as a function
of $k$. The (black) dots show the estimate for the power at steady state (26)
taking the value of $k$ into account, the (blue) crosses show the estimate
without taking the imperfect measurement into account. The parameters are
given in the caption of Figure 3 and $\gamma=0.0005\omega$.

##  V Experimental Realisation

To put the above theoretical results into the context of current experimental
activities, we discuss two different experimental realizations, by way of
example.

First, we think about the class of set-ups recently developed by Pekola and
co-workersRonzani et al. (2018); Senior et al. (2020); Karimi et al. (2020)
also discussed in the introduction. They consider a dedicated system, for
example a transmon qubit, which is in thermal contact with a small metallic
island (resistor). The Fermi gas of the latter acts as a calorimeter, the
energy/temperature variations of which are monitored via rf-thermometry
Schmidt et al. (2003); Gasparinetti et al. (2015). The Fermi gas contains a
sufficiently large number of electrons (typically on the order of about
$10^{8}$) with sufficiently fast internal relaxation time compared to the
electron-phonon coupling to qualify for a heat bath. Any residual on-chip
noise source may feed energy in the detection lines and cannot be as clearly
be identified as, for example, amplifier noise.

A second example, refers to a Josephson Photonics set-up, where a dc-voltage
biased Josephson junction is placed in series with a resonant mode of
frequency $\omega_{0}$ (LC-oscillator, cavity) Hofheinz et al. (2011); Westig
et al. (2017); Rolland et al. (2019). For voltages below the superconducting
gap, Cooper pairs can only be transferred if the excess energy $2eV$ can be
deposited as electromagnetic excitation $\hbar\omega_{0}$ in the cavity. Due
to the finite photon lifetime, photon radiation from the cavity can be then
detected. These and related set-ups have recently be shown to operate as
relatively bright sources for quantum microwaves. It turns out that the main
source of ‘noise’ is voltage noise at the junction, typically in the low
frequency regime. Even spurious additional ‘resonances’ may appear due to low
frequency modes that are often unavoidable by circuit design. In fact, by
slightly de-tuning the voltage from the resonance condition allows to
effectively cool these resonant modes and thus to reduce the impact of voltage
noise. A proper description requires to extend master equations applied so far
Gramich et al. (2013) to include also this additional harmonic low frequency
(classical) ‘heat bath’, for example, to explore quantum thermodynamics. The
results presented here, may provide a proper powerful theoretical framework.

##  VI Discussion

We analysed the role noise in calorimetric measurements by introducing a
simple model for imperfect calorimetric measurements. We introduced a noise
bath in addition to the system and the calorimeter. Only the last two
interact, the role of the noise bath is that its energy is measured
simultaneously as the calorimeter and as such disturbs the measurement.

Under weak coupling assumptions we derived a hybrid master equation for the
system state and the measured energy (III) and the corresponding stochastic
evolution (19). Remarkably, the only change compared to the perfect
measurement case is the values of the jump rates. The rates average
contributions from different energies which leads to an apparent heating of
the calorimeter. We study a simple example of a driven qubit in contact with a
reservoir of resonant bosons to study the qualitative behaviour of our model
for imperfect measurements. As the measurement imperfection increases, the
observed energy transfer from the qubit to the calorimeter decreases.

To study the effect of noise in steady state measurements such as in Ronzani
et al. (2018) to extract the power flowing from the qubit to the calorimeter,
we add a loss term to the calorimeter energy process. We find that for growing
imperfection of the measurement, not taking the imperfection into account
leads to a significant underestimation of the power.

The model for imperfect measurements we presented can be applied to general
systems and reservoirs. Further studies could include more complex systems,
e.g. multiple qubits, and fermion reservoirs. Another potential avenue is to
derive the noise bath from a microscopic description of the noise in an
experimental setup. Changes in the noise bath distribution (8) could have a
significant impact on the measurement outcomes.

##  VII Acknowledgements

We thank Paolo Muratore-Ginanneschi, Bayan Karimi, Jukka Pekola and Kalle
Koskinen for useful discussions. B.D. acknowledges support from the AtMath
collaboration at the University of Helsinki. This work has been supported by
IQST, the Zeiss Foundation, and the German Science Foundation (DFG) under
AN336/12-1 (For2724).

##  Appendix A Nakajima-Zwanzig projection operator technique

We define a projector $\operatorname{P}$, which acts on a qubit-calorimeter-
noise bath state $\rho$ as

$\operatorname{P}\rho=\sum_{E}\operatorname{tr}_{C+N}(\mathcal{P}_{E}\rho)\otimes\frac{\mathcal{P}_{E}}{\operatorname{tr}_{C+N}(\mathcal{P}_{E})}.$
(27)

It is straightforward to check that $\operatorname{P}$ is indeed a projector,
i.e. $\operatorname{P}^{2}\rho=\operatorname{P}\rho$.

We introduce the orthogonal projector
$\operatorname{Q}=\operatorname{1}-\operatorname{P}$, such that
$\operatorname{Q}\operatorname{P}=\operatorname{P}\operatorname{Q}=0$ and
define the hybrid state

$\rho(E,t)\equiv\operatorname{tr}(\mathcal{P}_{E}\operatorname{P}\rho)=\operatorname{tr}_{C+N}(\mathcal{P}_{E}\rho).$
(28)

In order to apply the Nakajima-Zwanzig projection operator technique, see for
example Rivas and Huelga (2012), we require three hypotheses to be satisfied.

### Hypothesis 1:

The initial state (6) satisfies $\operatorname{P}\rho_{0}=\rho_{0}$.

### Hypothesis 2:

For any qubit-calorimeter-noise bath state $\rho$ we have that

$[H_{C}+H_{N},\operatorname{P}\rho]=0.$ (29)

### Hypothesis 3:

For any qubit-calorimeter-noise bath state $\rho$ we have that

$\operatorname{tr}_{C+N}([H_{I},\operatorname{P}\rho])=0.$ (30)

A direct computation shows that the above three hypotheses are satisfied for
our model and choice of $\operatorname{P}$. Going though the Nakajima-Zwanzig
projection operator technique, we find that
$\rho(E,t)=\operatorname{tr}(\mathcal{P}_{E}\operatorname{P}\rho)$ satisfies a
closed differential equation

$\frac{d}{dt}\rho(E,t)=-\int_{0}^{t}ds\operatorname{tr}\left(\mathcal{P}_{E}[H_{I}(t),\operatorname{G}(t,s)[H_{I}(s),\operatorname{P}\rho]]\right)$
(31)

where $\operatorname{G}$ satisfies the differential equation

$\frac{d}{dt}\operatorname{G}(t,s)=-i\operatorname{Q}[H_{I}(t),\operatorname{G}(t,s)].$
(32)

Up to lowest order in the coupling strength
$\operatorname{G}(t,s)\approx\operatorname{1}$, such that (31) equals (III) up
to second order in the coupling.

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# Undecidability Results and Their Relevance in Modern Music Making

Halley Young<EMAIL_ADDRESS>
(September 2023)

###### Abstract

This paper delves into the intersection of computational theory and music,
examining the concept of undecidability and its significant, yet overlooked,
implications within the realm of modern music composition and production. It
posits that undecidability, a principle traditionally associated with
theoretical computer science, extends its relevance to the music industry. The
study adopts a multidimensional approach, focusing on five key areas: (1) the
Turing completeness of Ableton, a widely used digital audio workstation, (2)
the undecidability of satisfiability in sound creation utilizing an array of
effects, (3) the undecidability of constraints on polymeters in musical
compositions, (4) the undecidability of satisfiability in just intonation
harmony constraints, and (5) the undecidability of ”new ordering systems”. In
addition to providing theoretical proof for these assertions, the paper
elucidates the practical relevance of these concepts for practitioners outside
the field of theoretical computer science. The ultimate aim is to foster a new
understanding of undecidability in music, highlighting its broader
applicability and potential to influence contemporary computer-assisted (and
traditional) music making.

##  1 Introduction - Brief overview of the paper’s goals

The primary objective of this paper is to explore the concept of
undecidability within the context of modern music making. By examining various
aspects of music production and theory, we aim to demonstrate that
undecidability is not just an abstract notion limited to theoretical computer
science, but a relevant and thought-provoking concept that has tangible
implications for contemporary music composition and production.

###  1.1 Undecidability results

We will achieve this by investigating four distinct areas of music making:

  1. 1.

Proving that Ableton, a popular digital audio workstation, is Turing complete.

  2. 2.

Establishing the undecidability of satisfiability in sound creation using
audio effects.

  3. 3.

Demonstrating the undecidability of constraints on polymeters in musical
compositions.

  4. 4.

Proving that the satisfiability of just intonation harmony constraints is
undecidable.

  5. 5.

Proving that the satisfiability of ”new ordering systems” is undecidable.

These will all rely on reductions to problems already known to be undecidable
in the computer science literature.

###  1.2 Relevance of these results

We will also provide detailed explanations of why someone who is not a
theoretical computer scientist should care about each of these properties. In
particular, we will discuss why Rice’s theorem argues for a real limitation in
the possibility of analyzing output of a Turing complete system (which Ableton
is), and that undecidability of satisfiability constraints provides a
theoretical limit on to what extent an (automated or human) composer can
”think abstractly, then compose concretely” within different domains.

##  2 Decidability, Turing Completeness, and Their Implications for Music
Production

This section aims to introduce the fundamental concepts that underpin our
exploration of Ableton Live’s computational capabilities and the implications
of these for the undecidability of general properties of Ableton projects and
the satisfiability of a composer’s constraints or vision.

###  2.1 Decidability: An Overview

Decidability, a concept from theoretical computer science, refers to the
solvability of a problem through algorithmic means. A problem is considered
decidable if an algorithm exists that can provide a definitive solution to
every instance of the problem in a finite timeframe. Conversely, a problem is
undecidable if no such algorithm can be found. Understanding decidability is
crucial as it delineates the boundary between problems that can be addressed
using computational techniques and those that remain fundamentally unsolvable.

###  2.2 Turing Machines and Turing Completeness

The concept of a Turing machine, a theoretical model of computation, is
central to understanding the abilities and limitations of computational
systems. A system or programming language is considered Turing complete if it
can simulate a Turing machine’s behavior, meaning that it can execute any
computation that a Turing machine can, given adequate time and resources.

Ableton Live, a digital audio workstation, is such a Turing complete system.
Its computational power, coupled with its capability to manipulate audio and
MIDI data, makes it an immensely flexible tool for musical creation. However,
this Turing completeness also implies that certain questions about the
behavior of Ableton Live projects are undecidable, leading to intriguing
implications for music production.

###  2.3 Rice’s Theorem and Its Consequences

Rice’s theorem is a result in computability theory that states any non-trivial
property concerning a Turing machine’s computed function is undecidable [2].
This theorem suggests that many questions about the behavior of computational
systems and programs are fundamentally unanswerable, including those related
to the properties of Ableton Live projects.

Thus, the power and flexibility of Ableton Live, as embodied in its Turing
completeness, bring with them a fascinating paradox. While they allow for
virtually limitless musical creativity, they also introduce elements of
undecidability that challenge our ability to fully understand or predict the
outcomes of complex musical projects.

###  2.4 Undecidability of Satisfiability Problems

Satisfiability problems, which involve determining whether there exists a
solution that satisfies a set of conditions or constraints [3], also encounter
undecidability issues. These problems are pervasive in computer science and
mathematics, often manifesting in various domains, including logic [14],
number theory [7] , and algebra [9].

In the context of music composition, satisfiability problems can occur when a
composer defines a specific ”vision” for their piece, expressed in terms of
rhythmic, harmonic, or structural constraints. This vision might involve
intricate polyrhythms, complex harmonic progressions, or specific structural
properties.

However, the satisfiability of these constraints—whether there exists a
musical piece that fulfills all of them—is often undecidable. This means there
is no algorithm that can determine, for every possible set of constraints,
whether a satisfying musical piece exists. Therefore, a composer, whether
human or machine, might have a vision for a piece without any guarantee that
it can be realized.

In the following sections, we will explore how these undecidability issues can
limit a composer’s ability to know a priori if their vision can ever be
realized, highlighting the inherent tension between the vast expressive power
of a Turing complete system like Ableton Live and the fundamental limits
imposed by undecidability.

##  3 Ableton Live’s Turing Completeness

###  3.1 An Introduction to Ableton Live

Ableton Live is a comprehensive digital audio workstation (DAW) that offers a
diverse set of tools for computer musicians, catering to various aspects of
music production and performance. A key feature of Ableton Live is its
flexible audio and MIDI routing capabilities. You can freely route audio and
MIDI between tracks, enabling complex signal flows, layering of sounds, and
intricate cross-processing. This flexibility opens up vast creative
possibilities, from creating intricate soundscapes to designing complex
rhythmic patterns.

Additionally, Ableton Live provides a vast array of MIDI and audio effect
devices. MIDI effects transform MIDI notes and control signals, influencing
parameters such as pitch, velocity, and timing. Audio effects manipulate the
sound, offering control over parameters such as frequency, amplitude, and
time-based effects. What’s even more compelling is that these effects can
interact with each other, allowing the parameters of one effect to influence
another. This interoperability allows for the crafting of unique sonic
textures and innovative musical ideas. The combination of these features makes
Ableton Live a versatile and powerful tool, enabling musicians to push
boundaries and expand their creative potential in music production and
performance.

###  3.2 Proof of Turing Completeness

In this section, we present our proof of Ableton Live’s Turing completeness by
constructing a Turing machine simulation using its built-in audio and MIDI
devices.

####  3.2.1 Infinite Tape

Create an audio track representing the infinite tape with an unbounded audio
recording. Each audio sample (or short sequence of samples) represents a
symbol, mapped to different frequency ranges.

####  3.2.2 Read/Write Head

  * •

Reading: Use Granulator II to read the audio track in real-time. Automate the
FilePos parameter to control the read head’s position. Use EQ Eight to isolate
the frequency range representing a specific symbol, and use Envelope Follower
to analyze the output, determining the symbol at the current position.

  * •

Writing: Use Utility to control the amplitude of the audio track. Add an
Expression Control device and map its output to the Gain parameter of the
Utility device, effectively overwriting the symbol at the current position
with the new symbol according to the Turing machine’s rules.

####  3.2.3 States

Place an Audio Effect Rack on the reading track, containing multiple audio
effect chains, each representing a different state of the Turing machine. Each
chain contains a combination of audio and MIDI effects that determine the next
state and tape action based on the symbol read by Granulator II and analyzed
by the Envelope Follower.

####  3.2.4 Rules

Implement the Turing machine’s rules using MIDI Effect Racks, Chord devices,
and MIDI routing in this manner

  * •

Create an audio track for the infinite tape and read/write head, referred to
as the ”reading track.” This track will contain the Granulator II device for
reading symbols, the EQ Eight and Envelope Follower for symbol analysis, and
the Utility and Expression Control devices for overwriting symbols.

  * •

Create a separate MIDI track for each state in the Turing machine. Label these
tracks as ”State 1,” ”State 2,” and so on.

  * •

Route the MIDI output of each ”State” track to the Chain Selector parameter of
the Audio Effect Rack on the reading track. To do this, set the ”MIDI To”
option in the I/O section of each ”State” track to the reading track, and then
select ”Chain Selector” as the target parameter.

  * •

On each ”State” MIDI track, place a MIDI Effect Rack with multiple chains,
each chain representing a different rule for the current state based on the
input symbol. For example, in ”State 1” track’s MIDI Effect Rack, create
chains for each possible symbol the read/write head may encounter while in
State 1.

  * •

Configure the Chain Selector in the MIDI Effect Rack on each ”State” track to
choose the appropriate chain based on the symbol read by the read/write head.
Use the Envelope Follower’s output value (from the reading track) to control
the Chain Selector on the corresponding ”State” track. To do this, you can use
MIDI mapping or automation.

###  3.3 The Validity of the Proof

The method we employed to prove Ableton Live’s Turing completeness is a widely
accepted technique within the realm of theoretical computer science [6]. It’s
based on the concept of simulation, where the key components of a Turing
machine - an infinite tape, a read/write head, states, and transition rules -
are mapped to corresponding functionalities within Ableton Live’s music
production environment. By demonstrating that Ableton Live can effectively
simulate a Turing machine, we establish its Turing completeness. This is
because any system that can simulate a Turing machine, which is the
theoretical model for all computation, is itself Turing complete. Therefore,
our mapping approach not only demonstrates the richness and flexibility of
Ableton Live as a music production tool but also highlights its computational
universality.

###  3.4 Implications of Turing completeness in music production

One implication of Ableton’s Turing completeness is that it is, in theory,
possible to build any program in Ableton (a neural network, a compiler of a
different equation, etc.) However, I argue that the most important implication
of Ableton’s Turing completeness for music analysts and creators (particularly
those interested in automating musical tasks) comes from Rice’s theorem.
Imagine that we are designing an audio installation that is supposed to go on
indefinitely, and we want to avoid ever having the perception of dissonance
(defined according to an existing metric like having lots of energy at two
beating frequencies). According to Rice’s theorem, there does not exist a
program in any language which, regardless of the contents of the installation,
could check that it observes that property. Similarly, if an automated system
wanted to design procedural music to a videogame which continued as long as
the game was being played and responded to the game’s inputs, there could
never be a program that could check every such procedural music and determine
whether it responds in a logical way. This undecidability result means that by
using Ableton’s complexity, we must sacrifice some degree of certainty about
its outputs.

##  4 The Undecidability of Understanding Audio Effects

Consider a scenario where a sound engineer aspires to generate a specific
audio output using a particular audio effect. This effect can be modeled as a
real-valued function. The question that might arise in such a scenario is:
”Given any possible input audio signals, will applying this specific audio
effect consistently produce the desired output audio signal?”

To figure out whether we can answer this, we must turn to work in
computability theory. The undecidability of the universal theory of the reals,
a result that stems from the seminal work of mathematicians like Alonzo
Church, Alan Turing, and others, is a profound and far-reaching theorem in
mathematical logic and computability theory [4] [15] [16]. It states that
there does not exist a universal algorithm that can decide whether an
arbitrary mathematical statement concerning real numbers is true or false.
This result is a consequence of Gödel’s incompleteness theorem and the
negative solution to Hilbert’s Entscheidungsproblem, the decision problem.
Note that certain statements about reals, e.g., first order theories of real
numbers, are decidable [8]; however, they become undecidable when the sine
function (which is fundamental to audio processing) is included in the theory.

Now, let’s translate this abstract mathematical concept into our audio
engineering context. The real-valued function representing an audio effect can
be thought of as an equation involving real numbers. Just like the equations
in the universal theory of the reals, this audio effect function might have
many possible inputs, which correspond to the different potential input audio
signals. The output of this function, meanwhile, corresponds to the resulting
sound.

When the sound engineer seeks to find an audio effect that can consistently
produce a desired output from any possible input, they are essentially trying
to solve a problem similar to deciding whether a specific equation holds for
all real numbers. In mathematical terms, they are trying to find a function
(audio effect) that satisfies a particular condition (produces the desired
sound) for all possible inputs (audio signals). Furthermore, they are likely
implicitly including functions such as the sine function in their equation.

Given the undecidability result of the universal theory of the reals, however,
we know that there cannot exist a universal algorithm capable of deciding
whether an arbitrary equation holds for all real numbers. By analogy, then,
there cannot exist a universal algorithm capable of deciding whether an
arbitrary audio effect can produce a desired output from any possible input.

This has far-reaching implications for sound engineering and, more broadly,
any field that involves transformations of real-valued signals or data. It
suggests that there may be no systematic method for determining whether a
particular transformation (such as an audio effect) can achieve a specific
desired result from any possible input.

For a human music producer, the undecidability of understanding audio effects
can have significant implications. On the one hand, it emphasizes the limits
of formal, algorithmic approaches and the potential unpredictability of audio
production processes. This can encourage producers to embrace a more
exploratory, creative, and intuitive approach to using audio effects,
leveraging their personal experiences, aesthetic judgments, and experimental
practices. On the other hand, it also highlights the inherent complexity and
open-ended nature of audio production, which can be both challenging and
exciting. It suggests that there are no definitive answers or universal
recipes for achieving specific sounds, and that the creative possibilities are
vast and potentially unbounded.

In the case of automated systems for music production, the undecidability of
understanding audio effects can also have important consequences. It implies
that there are inherent limitations to the capabilities of these systems,
particularly in terms of their ability to predict and control the outcomes of
applying audio effects. This can impact the design and development of such
systems, emphasizing the need for robustness, adaptability, and flexibility.
It might also necessitate the use of probabilistic and heuristic methods, as
well as machine learning techniques, which can cope with uncertainty and make
educated guesses in the face of undecidability. However, it also opens up
opportunities for creative applications of AI in music production, where the
unpredictability of audio effects is not a bug, but a feature that can be used
to generate new and unexpected musical ideas.

##  5 Undecidability of Satisfiability with Polyrhythms or Just Intonation

###  5.1 Undecidability of Satisfiability with Multiplication of Unknown
Integers

The theory of undecidability of integer variable multiplication traces its
origins to the field of mathematical logic and theoretical computer science.
Here, it is associated with the principle of the universal theory of the
reals, which focuses on the properties of real numbers and the mathematical
structures derived from that set. This theory consists of several profound
results, including the undecidability of multiplication involving unknown
integer variables.

The undecidability of the satisfiability of multiplication by unknown integer
variables is not an obvious concept, and it took revolutionary work by
computer scientists and logicians to bring it to light. Essentially, the claim
of this theory is that there is no universal method or computational process
(termed ’algorithm’) that can definitively determine whether any general
mathematical statement, particularly one involving the multiplication of real
numbers or unknown variables, is always true or false under all circumstances.

This proposition was explicitly proved through the work of Yuri Matiyasevich,
building on contributions by three leading logicians of the 20th century:
Julia Robinson, Martin Davis, and Hilary Putnam [5] [13] [12]. Their quest to
resolve a significant mathematical problem of the time, known as Hilbert’s
Tenth Problem, inadvertently led to the discovery of the undecidability.
Hilbert’s Tenth Problem asked for an algorithm that could determine whether
any given Diophantine equation had integer solutions. A Diophantine equation
is a polynomial equation that seeks integer solutions.

Matiyasevich built upon the work of the aforementioned logicians, and in 1970,
he provided the final piece of the puzzle, proving that no such algorithm
exists for Hilbert’s Tenth Problem, thus confirming the resolution of the
problem in the negative [10]. This negative resolution subsequently implied
that certain specific sets, in this case, the sets that represent the
satisfiability of certain Diophantine equations or equivalently, the
multiplication of integer unknown variables, are undecidable.

###  5.2 Relationship to Polymeter and Polyrhythm

In metric music, beats can be visually represented using pairs of integers,
labeled as (n, m). Here, ’n’ signifies the beat number or the position of the
beat within a rhythmic cycle and ’m’ represents the time signature. This
notation system enables the mapping of complex rhythms onto a numeric grid and
permits us to vislauize and grapple with abstract musical concepts.

Taking a hypothetical scenario in which we want two beats from different
rhythms to coincide at a definite point in time, we can denote this intention
using an equation: (n1, k1) * (n2, k2) = (a, b). Here, (n1, k1) and (n2, k2)
correspond to the starting beats of the two rhythms, while (a, b) symbolizes
the point of conjunction. The variables k1 and k2 in this equation are
unknown, thereby making the equation a case of multiplication involving
unknown variables.

The theory of the undecidability of the satisfiability of multiplication by
unknown variables becomes highly relevant here. There are no definitive
computational procedures that can yield actual values for k1 and k2, making
the equation true. This implies that it’s not systematically determinable, on
solely mathematical grounds, whether we can make two beats from differing
rhythms coincide at a specific time point—a given polyrhythm feasible.

This undecidability offers profound implications for music theory and
composition, particularly when it comes to polyrhythms. Composers often have
to deal with the complexity and unpredictability that comes with writing
polyrhythms. The challenge lies in systematically specifying and reasoning
about whether it is possible for certain beats within a complex polyrhythm to
align. Because the algorithmic and procedural methods cannot provide
definitive answers due to the undecidability at play, composers must often
resort to heuristic or empirical methods. These methods, based on
experimentation, trial and error, or intuitive judgment, highlight the often
underappreciated complexity and creative challenge in music composition.

###  5.3 Just Intonation

Just intonation is a system of musical tuning in which the frequencies of
notes are related by ratios of small whole numbers. Theoretically, it results
in a pure and consonant sound that is often more pleasing to the ear compared
to other tuning systems. However, the process of creating just intonation
involves the multiplication of ratios, which inherently corresponds to the
multiplication of unknowns in the context of mathematical equations.

The undecidability of the satisfiability of multiplication by unknowns
directly impacts our capacity to devise a universally applicable algorithm for
creating just intonation automatically. In other words, there is no general
algorithm or decision procedure that can definitively determine whether it’s
possible to tune a piece of music in just intonation by multiplying specific
frequency ratios. This undecidability makes it inherently complex to
systematize the process of creating just intonation.

For instance, let’s consider a situation where we want to tune a piece of
music that uses a seven-note diatonic scale in just intonation. We may want to
determine the frequency ratios that will provide a perfect fifth (a frequency
ratio of 3:2) between each pair of successive notes. We could express this
requirement using equations involving multiplication of unknowns, such as a/b
* c/d = 3/2, where a/b and c/d represent the frequency ratios of the two
notes. However, due to the undecidability of the satisfiability of
multiplication by unknowns, we can’t conclusively determine algorithmically
whether there exist specific values of a, b, c, and d that will satisfy these
equations for all pairs of notes. .

###  5.4 Implications

For human composers, the undecidability of these harmonic and rhythmic
constraints has profound implications. It means that there might be times when
they conceive of a musical idea, a particular harmonic or rhythmic language,
but they might not be able to determine if it is even plausible to realize
that idea given the constraints of the harmonic space. The inability to
formally or algorithmically reason about these constraints can introduce an
element of profound uncertainty into the creative process. Rather than seeing
this as a limitation, it can be interpreted as an invitation to the composer
to delve deeper into the rich and unexplored territories of the harmonic
space. This uncertainty, then, serves not as a boundary, but as a stimulus for
innovation and experimentation in music composition.

When it comes to machine models used for automatic composition or music
analysis, the undecidability of reasoning about just intonation harmony or
polyrhythms presents substantial challenges as well. It suggests that AI
systems cannot be guaranteed to always find a solution to satisfy certain
harmonic constraints, or even determine whether such a solution exists. This
necessitates the development of AI models that are capable of handling these
inherent uncertainties, perhaps by leveraging probabilistic models or machine
learning techniques. At the same time, the undecidability could also be seen
as a source of creative potential, enabling AI models to generate novel and
unexpected musical ideas within the uncharted territories of the harmonic
space.

###  5.5 The Undecidability of Horn Clauses and Composer-Generated Systems of
Organization

###  5.6 Proof of undecidability of Horn Clauses

The proof for the undecidability of Horn clauses is rooted in the reduction
from the Halting Problem, which is a well-known undecidable problem within
computer science. One begins by establishing that the set of all true ground
(i.e., variable-free) Horn clauses is recursively enumerable [1]. This means
that there exists a Turing machine which will list all true Horn clauses.
Then, a reduction is crafted from the Halting Problem to the problem of
deciding if a given ground Horn clause is true. The subsistence of this
reduction implies that if there were an algorithm to decide the truthfulness
or groundness of any given Horn clause, such an algorithm could be used to
solve the Halting Problem, contradicting its established status as
undecidable.

David A. Plaisted, in his comprehensive study ”The Undecidability of Ground
Term Rewrite Systems” [11], was able to prove that establishing satisfiability
for sets of Horn clauses is undecidable when clauses could contain function
symbols of arity $>0$. His work is considered significant in demonstrating the
undecidability of Horn clauses satisfiability. Plaisted obtained this result
by reducing the word problem for semi-Thue systems (which is a known
undecidable problem) to the satisfiability problem of Horn clauses.

###  5.7 Implications for Composers

The notion of equating a musical system to general Horn clauses suggests that
each musical rule in a system, whether it governs melody, harmony, or rhythm,
can be formulated as a Horn clause.

In order to achieve this transmutation, one must assign a literal or
proposition to every possible musical event or condition. For instance, we can
designate ’note X is followed by note Y’ as a positive literal, and ’chord
progression A leads to chord progression B’ as a negative literal or a
combination of both. As such, each literal serves as a concrete representation
of a musical condition. These literals are the building blocks with which the
Horn clauses are formulated, effectively creating a musical language with
syntactic rules and structures mirroring those in propositional logic.

Collectively, these translated musical rules defined by a composer form a
system that resembles the characteristics of a set of general Horn clauses in
propositional logic. Each rule represented as a Horn clause becomes a
stipulation that demands satisfaction. In this context, satisfaction implies a
musical sequence that corresponds and adheres to the designed compositional
system or the logical constraints impounding it.

In propositional logic, the notion of whether a model exists that satisfies a
given set of clauses is described as the satisfiability problem. Analogously
applied to the domain of music, this translates to the existential question:
can a sequence of musical events, a composition, be constructed satisfying all
the rules of a given system simultaneously?

However, drawing this parallel presents an inherent challenge grounded in the
conclusively proven undecidability of general Horn clause satisfiability. This
paradigm transforms from an intellectual exercise into a practical dilemma for
composers who wish to implement such systems in their work. Without guaranteed
certifying mechanisms, composers following this approach walk a delicate path
between logical structure and potential unsatisfiability. The pursuit of an
aesthetically pleasing composition that simultaneously adheres to intricate
logical constraints places an extraordinary demand on both the creative and
logical faculties of the composer.

##  6 Conclusion

In this paper, we have presented several key findings that have profound
implications for both the practice of music composition and the development of
automated music generation systems. Our results highlight the inherent
complexities and limitations encountered when attempting to computationally
represent and generate musical pieces, whether the context is traditional
musical forms like string quartets or more experimental formats such as real-
time, non-time-bounded media installations.

Our first major result was establishing the Turing completeness of Ableton
Live, a popular digital audio workstation. This result not only underscores
Ableton Live’s vast expressive power as a musical tool but also sets the stage
for our subsequent investigations into the inherent computational limits of
music creation within such a powerful system.

We then demonstrated the undecidability of satisfiability for polyrhythmic and
just intonation constraints, which are common elements in contemporary music
composition. This result underscores the fundamental limits of formalizing and
algorithmically processing certain aspects of musical composition. We showed
that even with a Turing complete system, there is no guarantee that a
composer’s specific vision for a piece can always be realized, whether the
constraints involve intricate rhythmic structures, complex harmonic
progressions, or other specific musical properties.

Moreover, we extended this result to general composer-generated structural
constraints, further highlighting the challenges composers face when
attempting to realize their musical ideas, particularly when those ideas
involve intricate or complex structural properties.

These findings have significant implications for both human and automated
composers. For human composers, our results underscore the importance of
intuition, experience, and exploratory techniques in the composition process,
particularly when dealing with complex musical constraints that cannot be
fully formalized or algorithmically processed. For automated systems, our
results highlight the limitations of relying solely on algorithmic methods for
generating music that satisfies specific constraints. They also underscore the
need for incorporating heuristic or learning-based methods to navigate the
vast and complex landscape of musical possibilities.

In conclusion, our findings serve as a bridge between the worlds of
computational theory and music composition, shedding light on the fascinating
interplay between computation and creativity, and ultimately, between machines
and the art of music.

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