# Fully passive entanglement based quantum key distribution scheme

Bo Liu<EMAIL_ADDRESS>College of Advanced Interdisciplinary Studies,
National University of Defense Technology, Changsha, 410073, China  Institute
for Quantum Optics and Quantum Information (IQOQI), Austrian Academy of
Sciences, Boltzmanngasse 3, 1090 Vienna, Austria  Vienna Center for Quantum
Science and Technology (VCQ), Faculty of Physics, University of Vienna,
Boltzmanngasse 5, A-1090 Vienna, Austria  Matej Pivoluska  Institute of
Computer Science, Masaryk University, Czech Republic  Institute of Physics,
Slovak Academy of Sciences, Bratislava, Slovakia  Johannes Handsteiner
Institute for Quantum Optics and Quantum Information (IQOQI), Austrian Academy
of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria  Vienna Center for Quantum
Science and Technology (VCQ), Faculty of Physics, University of Vienna,
Boltzmanngasse 5, A-1090 Vienna, Austria  Dominik Rauch  Institute for Quantum
Optics and Quantum Information (IQOQI), Austrian Academy of Sciences,
Boltzmanngasse 3, 1090 Vienna, Austria  Vienna Center for Quantum Science and
Technology (VCQ), Faculty of Physics, University of Vienna, Boltzmanngasse 5,
A-1090 Vienna, Austria  Marcus Huber  Institute for Quantum Optics and Quantum
Information (IQOQI), Austrian Academy of Sciences, Boltzmanngasse 3, 1090
Vienna, Austria  Vienna Center for Quantum Science and Technology,
Atominstitut, Technische Universit$\ddot{a}$t Wien, 1020 Vienna, Austria
Fabian Steinlechner  Institute for Quantum Optics and Quantum Information
(IQOQI), Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria
Vienna Center for Quantum Science and Technology (VCQ), Faculty of Physics,
University of Vienna, Boltzmanngasse 5, A-1090 Vienna, Austria  Fraunhofer
Institute for Applied Optics and Precision Engineering IOF, Albert-Einstein-
Strasse 7, 07745 Jena, Germany  Rupert Ursin  Institute for Quantum Optics and
Quantum Information (IQOQI), Austrian Academy of Sciences, Boltzmanngasse 3,
1090 Vienna, Austria  Vienna Center for Quantum Science and Technology (VCQ),
Faculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090 Vienna,
Austria  Thomas Scheidl<EMAIL_ADDRESS>Institute for Quantum
Optics and Quantum Information (IQOQI), Austrian Academy of Sciences,
Boltzmanngasse 3, 1090 Vienna, Austria  Vienna Center for Quantum Science and
Technology (VCQ), Faculty of Physics, University of Vienna, Boltzmanngasse 5,
A-1090 Vienna, Austria

###### Abstract

In entanglement based quantum key distribution (QKD), generation of entangled
photon pairs and the random choice of measurement basis can be implemented
with passive optical devices. However, auxiliary random numbers are still
necessary for conducting the privacy amplification procedure. In this paper,
we solve this issue and propose a fully passive entanglement based QKD scheme
in which certified random bits for privacy amplification are extracted from
detection events of measurement-basis-mismatched entangled photon pairs. The
privacy of these random bits can be certified by estimating the phase and bit
error rate in the sifted key. The idea of extracting certified random numbers
from measurement-basis-mismatched photons, can be extended to all practical
QKD schemes.

###### pacs:

Valid PACS appear here

††preprint: APS/123-QED

##  I Introduction

Quantum Key Distribution (QKD) facilitates distributing unconditionally secure
random keys between distant parties. There are many different types of QKD
systems and one might distinguish between active and passive systems. Typical
active systems implement decoy state QKD protocols, employing sources of weak
coherent laser pulses (WCP) Yin  _et al._ (2016); Kiyoshi  _et al._ (2016);
Zhao  _et al._ (2006); Peng  _et al._ (2007); Rosenberg2007 ; Yuan  _et al._
(2007); Liu  _et al._ (2010); Wang  _et al._ (2013); Tang  _et al._ (2014a, b)
and require true random numbers for actively modulating certain parameters of
the pulses (e.g. intensity, polarization, phase, etc.) and the choice of
measurement settings, according to the needs of the specific QKD protocol.
Contrary, it has been emphasized Curty  _et al._ (2010a) that passive systems
might be advantageous in scenarios, where operation at a high transmission
rate is required. Moreover, passive systems do not require the use of
auxiliary random numbers and optical modulators, and are thus immune against
attacks through side-channels provided by imperfections of these devices.
Additionally, for the very same reason they provide the potential for
significantly reducing the overall costs and complexity for the implementation
of QKD systems. This can be advantageous for example when deploying QKD system
on a satellite. Many groups have proposed schemes to implement passive decoy
state QKD systems based on parametric down conversion (PDC) Rarity  _et al._
(1994); Adachi  _et al._ (2007); Mauerer and Silberhorn (2007); Xiongfeng and
Hoi-Kwong (2008); Wang  _et al._ (2007a); Yoritoshi  _et al._ (2009); TAN and
CAI (2011); Wang  _et al._ (2016) as well as on WCP Curty  _et al._ (2010a);
Wang  _et al._ (2007b); Curty  _et al._ (2010b); Li  _et al._ (2014); Shan
_et al._ (2014); Curty  _et al._ (2015) and both types of sources have
recently already been implemented Guan  _et al._ (2015); Sun  _et al._ (2016);
Qi-Chao  _et al._ (2014); Krapick  _et al._ (2014); Zhang  _et al._ (2012,
2010). Another scheme for passive QKD, which does not rely on the generation
of decoy states, is based on entanglement and has been implemented in a wide
range of systems with the most prominent utilizing the polarization or the
time degree of freedom Jennewein  _et al._ (2000); Tittel  _et al._ (2000);
Resch  _et al._ (2005); Peng  _et al._ (2005); Marcikic  _et al._ (2004);
Ursin  _et al._ (2007); Poppe  _et al._ (2004); Ma  _et al._ (2007); Marcikic
_et al._ (2006); Ali Khan and Howell (2006); Honjo  _et al._ (2008); Fasel
_et al._ (2004); Alexander  _et al._ (2009); Takesue  _et al._ (2010); Scheidl
_et al._ (2009). Even though passive QKD systems do not require auxiliary true
random numbers for the state generation and also the random choice of the
measurement basis can easily be implemented in a passive way using e.g. a
$50/50$ beam splitter, true random numbers are still necessary for conducting
the most important post-processing procedure – privacy amplification Bennett
_et al._ (1988, 1995); Watanabe (2007); Bo  _et al._ (2013). Privacy
amplification reduces the mutual information between the eavesdropper and the
communication parties Bennett  _et al._ (1988, 1995); Fung  _et al._ (2010);
Liu  _et al._ (2013); Bo  _et al._ (2013); Liu  _et al._ (2016); Hayashi
(2011); Watanabe (2007); Renner and König (2005) by applying a randomly chosen
universal hash function to the sifted and error-corrected key. Therefore, the
universal hash function is usually constructed with the help of a trusted
auxiliary random source Hayashi (2009, 2011); Hayashi and Tsurumaru (2016,
2015).

Since the generation of true random numbers must rely on an unpredictable
physical process Pironio  _et al._ (2010), great effort has been put into the
investigation and development of quantum random number generators (QRNGs),
based on the intrinsic randomness of the quantum measurement process Jennewein
_et al._ (2000); Fiorentino  _et al._ (2007); Wei and Guo (2009); Svozil
(2009); Fürst  _et al._ (2010); Jofre  _et al._ (2011); Gallego  _et al._
(2013); Abellán  _et al._ (2014); Dhara  _et al._ (2013); Pironio  _et al._
(2010); Marangon  _et al._ (2017); Acín and Masanes (2016); Acín  _et al._
(2016); Vallone  _et al._ (2014); Rarity  _et al._ (1994); Herrero-Collantes
and Garcia-Escartin (2017); Pivoluska  _et al._ (2020). Today, high quality
devices are already commercially available passing the statistical tests on
the randomness of the output bits. However, statistical tests alone can not
certify that the numbers are not known to an adversary Marangon  _et al._
(2017). A sequence of bits is thought to be “certified random”, if it is
perfectly random and unpredictable by any observer. In the device independent
frame work, it has been shown that random numbers can be generated utilizing
entangled states, while the privacy can simultaneously be certified via the
violation of a Bell inequality Acín and Masanes (2016). We want to point out
that the device independent generation of certified random numbers essentially
requires a loophole-free violation of a Bell inequality and is thus very hard
to achieve in any practical system. Meanwhile, under the assumption of trusted
or well-known and characterized measurement devices, the privacy of random
numbers can be certified by performing given positive-operator-valued measures
Vallone  _et al._ (2014); Acín  _et al._ (2016).

Here we propose a fully passive entanglement-based QKD scheme in which
certified random bits for privacy amplification are extracted from detection
events of measurement-basis-mismatched photons. The randomness is guaranteed
by the intrinsic randomness of the outcomes of measurements on entangled
states while, building on the very same assumptions on which the security of
any practical QKD system is based on, the privacy of these random bits can be
certified by estimating the phase and bit error rate in the sifted key. The
idea of extracting certified random numbers from measurement-basis-mismatched
photons is very attractive for satellite-based QKD systems, which provides the
potential for significantly simplify the implementation of QKD payload on
satellite.

##  II Preliminaries

The security of practical QKD systems is based on the following underlying
assumptions:

  1. 1.

The quantum formalism is valid and all devices as well as the adversary are
constrained by the laws of quantum mechanics.

  2. 2.

Alice and Bob share a certain amount of secure key prior to starting the QKD
session.

  3. 3.

Alice and Bob share independent and identically distributed (IID) quantum
systems and know the dimension of the Hilbert space describing these systems
Acín  _et al._ (2006).

  4. 4.

The detection system is compatible with the squashing model Beaudry  _et al._
(2008); Gittsovich  _et al._ (2014).

Usually, communication parties of practical QKD systems should also be
equipped with trusted random number generators and trusted key management. For
the practical entanglement based QKD scenario, Alice and Bob have to trust
their local measurement systems.

The standard implementation of the entanglement based QKD protocol mainly
consists of four phases: _quantum communication phase_ , _basis
reconciliation_ , _error correction_ and _privacy amplification_. In the
quantum communication phase, the communicating parties Alice and Bob share
pairs of polarization entangled photons, which are passively generated at a
separated source and distributed through quantum channels. Alice and Bob each
measure every impinging photon randomly in one of two complementary
polarization measurement bases and associate the two possible measurement
outcomes in any measurement basis with the binary values “0” and “1”,
respectively. The random basis choice is either implemented with active
polarization switches controlled by local random number generators or with
passive devices like a 50/50 beam splitter. Finally, Alice and Bob only keep
those results where both photons of a pair have been detected. This is called
the raw key $K_{r}$ with length $n_{r}$.

The quantum communication phase is followed by basis reconciliation, where
Alice and Bob publicly exchange information about the measurement bases they
actually implemented for their detected photons. Owing to the properties of
the maximally entangled state, they typically discard all bits for which their
measurement bases did not match, since in these cases their measurement
results are maximally uncorrelated. Contrary, their measurement results are
perfectly (anti-)correlated when they used the same bases. In doing so, Alice
and Bob end up with the sifted key $K_{sift}$ with length $n_{s}$.

In a realistic scenario, $K_{sift}$ always contains errors, which all have to
be attributed to a potential eavesdropping attack. During error correction,
these errors are corrected implementing classical algorithms Bennett  _et al._
(1992a); Brassard and Salvail (1994); Sugimoto and Yamazaki (2000); Yan  _et
al._ (2008); Pedersen and Toyran (2015); Buttler  _et al._ (2003); Cui  _et
al._ (2013); Elkouss  _et al._ (2009); Wilde and Guha (2013) via public
channels, such that Alice and Bob end up sharing an identical key $K_{EC}$.
Finally, all information an eavesdropper might have obtained during the
communication phase or error correction is removed with the help of privacy
amplification.

Privacy amplification is a procedure allowing two communication parties, which
share a weak secret key $K_{EC}$ to agree on a final secrete key $K_{f}$ of
length $n_{f}$ via communication over a authenticated public channel. Thereby,
the public channel might even be controlled by the adversary Eve, who is
granted unbounded computational power. Let us denote $E$ an arbitrarily string
that Eve may have learned during the QKD procedure. During privacy
amplification, Alice typically uses an auxiliary random number generator to
generate a random bit string and sends it to Bob through a public channel.
Then, Alice and Bob use this random string to construct a universal hash
function, which they apply to $K_{EC}$.

Even in QKD protocols with passive quantum communication phase, other sub-
procedures of the protocol often require local randomness and therefore QKD
implementations are usually equipped with proprietary random number
generators. Consider for example, squashing operation for multiple-click
events, which often require some local randomness, because double-click events
in one basis are typically assigned a random outcome value. Another example is
error correction where some protocols use local randomness as a free resource
(e.g. CASCADE protocol introduced in Brassard and Salvail (1994)).
Additionally, in determining the error rates of practical QKD protocols, Alice
and Bob need to exchange small randomly chosen subset of their raw key. Last
but not least, as mentioned above, privacy amplification step requires local
randomness in order to construct a universal hash function.

In order to present a fully passive protocol in this manuscript we consider
sub-procedures, which do not require local randomness wherever possible. For
example, one can count the number of double-click events and perform privacy
amplification based on the counted number of the double-click events
Gittsovich  _et al._ (2014); Takesue  _et al._ (2015) without resorting to
extra local random strings, or use error correction based on standardized LDPC
codes, which also does not utilize additional randomness Mink and Nakassis
(2014). Further, since error rates can be well estimated from logarithmically
small samples, local randomness needed to choose them can be neglected
compared to privacy amplification. This leaves privacy amplification as the
only sub-procedure which requires a significant amounts of local randomness.
In this paper, we show that enough local randomness for privacy amplification
can be obtained by utilising measurement outcomes in mismatched measurement
bases of Alice and Bob, which are discarded in a typical protocol.

Let us denote the length of the final key $K_{f}$ as $n_{f}$. Then, $n_{f}$ is
given by

$n_{f}=n_{f}^{x}+n_{f}^{z},$ (1)

where $n_{f}^{x}$ ($n_{f}^{z}$) is the final secure key length extracted in
$X$ ($Z$) basis, which can be obtained using Ma  _et al._ (2007):

$\displaystyle n_{f}^{x}$ $\displaystyle\geq
n_{s}^{x}\left[H_{\text{min}}^{x}(K_{sift}^{A}|E)-f\left(e_{bx}\right)H^{x}\left(K_{sift}^{A}|K_{sift}^{B}\right)\right]$
$\displaystyle\geq
n_{s}^{x}\left[1-H_{2}\left(e_{px}^{U}\right)-f\left(e_{bx}\right)H_{2}\left(e_{bx}\right)\right].$
(2)

Here $H_{\text{min}}^{x}(K_{sift}^{A}|E)$ is min-entropy of a single Alice’s
sifted key bit obtained by measurement in the $X$ basis conditioned on the
knowledge of the adversary and $H^{x}\left(K_{sift}^{A}|K_{sift}^{B}\right)$
is the error rate between sifted keys of Alice and Bob, obtained by
measurement in the $X$ basis. Further, $n_{s}^{x}$ is the number of bits in
the sifted key obatained by measurement in the $X$ basis, $e_{px}^{U}$ and
$e_{bx}$ are the estimated upper bound of phase error rate and the measured
quantum bit error rate (QBER) in $X$ basis, $f(\cdot)$ is the error correction
efficiency and $H_{2}(\cdot)$ is the binary entropy function. Analogously, the
length of the part of the final key, which Alice and Bob can obtain by
measurement in $Z$ basis is defined as Ma  _et al._ (2007):

$\displaystyle n_{f}^{z}$ $\displaystyle\geq
n_{s}^{z}\left[H_{\text{min}}^{z}(K_{sift}^{A}|E)-f\left(e_{bz}\right)H^{z}\left(K_{sift}^{A}|K_{sift}^{B}\right)\right]$
$\displaystyle\geq
n_{s}^{z}\left[1-H_{2}\left(e_{pz}^{U}\right)-f\left(e_{bz}\right)H_{2}\left(e_{bz}\right)\right].$
(3)

##  III Passive Entanglement Based QKD Scheme

The fully passive entanglement based QKD scheme is shown in Fig. 1. Entangled
photon pairs can be produced by parametric down conversion with nonlinear
crystals. With beam splitter, polarizing beam splitter and other passive
optical elements, Alice and Bob implement the passive receiver for detecting
the entangled photon pairs, in particular, the measurement basis choice is
implemented with a $50/50$ beam splitter. Thus, the quantum physical
communication between Alice and Bob is done in a fully passive way. Here, we
focus on how to conduct privacy amplification with the certified random bits
extracted from measurement-basis-mismatched photons, which mainly includes
four steps.

Figure 1: Fully passive entanglement based quantum key distribution scheme.
$K_{mis}^{A}$ is the reserved sifted key string, which is measured in
mismatched basis between Alice and Bob. Random seed $W^{\ast}$ is used to
publicly choose the universal hash function. $K_{sift}$ is the key string
after the basis reconciliation. $K_{EC}$ is the key string after the error
correction phase. $\widetilde{e}_{p}$ is the estimated phase error rate.
Entangled photons are transmitted to Alice and Bob through the quantum
channel, specified in red lines. Classical information exchange between Alice
and Bob is by the ITS authenticated classical channel, specified in green
lines.

Step 1. Generation of local random bit-strings. First, during the basis
reconciliation procedure, one has to reserve the bit string $K_{mis}$ at
Alice’s or Bob’s side, respectively, resulting from local measurement results,
which have been detected in measurement-basis-mismatched photons. In this
paper, we gather Alice’s mismatched basis measurement results into a string
labeled $W=K_{mis}^{A}$. Intuitively, this string contains as much randomness
and privacy as the $K_{sift}$, which can be used in the privacy amplification
part of the protocol.

Step 2. Randomness certification. The amount of information about $W$, which
is unknown to anybody else before the post-processing procedure started (under
the assumptions in Sec. II) is given by its min-entropy. Hence, the goal is to
find a lower bound for $H_{\min}\left(W|E\right)$, which allows to certify its
randomness and privacy.

As an adversary cannot influence the local measurement devices, her attack
strategy is the same for photon pairs measured in either matched or mismatched
bases. This means we have

$\displaystyle\\!H_{\min}\\!\left(W|E\right)\\!$
$\displaystyle=\\!m_{x}H_{\min}^{x}\\!\left(K_{sift}^{A}|E\right)\\!+\\!m_{z}H_{\min}^{z}\\!\left(K_{sift}^{A}|E\right)$
$\displaystyle=\\!m_{x}\\!\left[1\\!-\\!H_{2}\\!\left(e_{pz}^{U}\right)\right]\\!+\\!m_{z}\\!\left[1\\!-\\!H_{2}\\!\left(e_{px}^{U}\right)\right]\\!,$
(4)

where $m_{x}$ and $m_{z}$ are the number of measurement results obtained in
$X$ and $Z$ basis in the mismatched rounds. The second equation was obtained
by using (2) and (3) and omitting the error correction cost
$f\left(e_{bz}\right)H_{2}\left(e_{bz}\right)$ and
$f\left(e_{bx}\right)H_{2}\left(e_{bx}\right)$, since the local randomness
does not need error correction.

After estimating the min-entropy of $W$ from the phase error rates
${e_{px}^{U}}$ and ${e_{pz}^{U}}$, an adequate randomness extractor has to be
implemented to distill perfect and uniformly distributed random string
$W^{\ast}$ Herrero-Collantes and Garcia-Escartin (2017). This can be done with
the use of the seeded extractors Trevisan (2001); Ma  _et al._ (2013); Konig
and Renner (2011); Skorski (2015) with the help of some additional randomness.
This additional randomness can either be taken from the pre-shared secure key,
or the protocol can start with a short local random string. The main idea is
that the seed used to extract uniform local randomness from $W$ is not made
public, therefore it can be reused multiple times with help of leftover
hashing lemma Ma  _et al._ (2013). This is important, since it shows that the
amount of local randomness this extraction step consumes is negligible.

Step 3. Public Hash Function Discussion. Alice sends the extracted random seed
$W^{\ast}$ to Bob via a authenticated public channel.

Step 4. Key Extraction. Alice and Bob use $W^{\ast}$ to construct a universal
hash function for privacy amplification, which they apply to their error
corrected key $K_{EC}$ resulting the final secure key $K_{f}$.

In order to evaluate the key rate of our protocol, we need to specify the
family of hash functions used in the privacy amplification. This choice
crucially influences the amount of local uniform randomness $W^{*}$ needed to
extract the final key. The main idea is that if the mismatched measurements do
not produce enough local randomness in order to extract $K_{f}$ from $K_{EC}$,
Alice will reassign $\epsilon$ bits from $K_{EC}$ to $W$. By doing so she
reduces the amount of local randomness needed for privacy amplification, which
usually depends on the length of $K_{EC}$ and/or its min-entropy, both of
which decrease by reassigning key bits to local randomness, while
simultaneously increasing the amount of local randomness in $W$. Let us define
a hash-family-specific function $h(n_{s},n_{f})$, which returns the size of
the local seed needed to extract the final key of length $n_{f}$ (which is
equivalent to the total min-entropy of $K_{EC}$) from the error corrected key
of length $n_{s}$. Then the length of $K^{A}_{m}is$ is $(n_{r}-n_{s})$.
Further, for simplicity let us denote
$\widetilde{e}_{p}:=\max(e_{pz}^{U},e_{px}^{U})$ and
$\widetilde{e}_{b}:=\max(e_{bz},e_{bx})$. Then with $K_{raw}$ of length
$n_{r}$ and $K_{EC}$ of length $n_{s}$, we have

$\displaystyle H_{\text{min}}(W|E)=(n_{r}-n_{s})[1-H_{2}(\widetilde{e}_{p})]$
(5)

and the total entropy of Alice’s error corrected key is

$\displaystyle
H_{\text{min}}(K_{EC}^{A}|E)=n_{s}[1-H_{2}(\widetilde{e}_{p})-f(\widetilde{e}_{b})H_{2}(\widetilde{e}_{b})].$
(6)

Note that we use $\widetilde{e}_{p}$ and $\widetilde{e}_{b}$ only for
convenience of notation, since it allows us to disregard the information about
the measurement basis. The result can be generalized in a straightforward way
to take into account also the information in which basis the measurement
outcomes forming both $W$ and $K_{EC}^{A}$ were obtained. Taking into account
the procedure to enlarge $W$ by $\epsilon$ bits, together with equations (5)
and (6) the final secure key length of fully passive scheme is defined as

$n_{f}^{\ast}\geq(n_{s}-\epsilon)[1-H_{2}(\widetilde{e}_{p})-f(\widetilde{e}_{b})H_{2}(\widetilde{e}_{b})],$
(7)

where $\epsilon$ is the smallest integer such that

$\displaystyle{[n_{r}\\!-\\!n_{s}+\epsilon]}$
$\displaystyle{[1\\!-\\!H_{2}(\widetilde{e}_{p})]}\\!\geq$ (8)
$\displaystyle{h\left[n_{s}\\!-\\!\epsilon,\\!(n_{s}\\!-\\!\epsilon)[1\\!-\\!H_{2}(\widetilde{e}_{p})\\!-\\!f(\widetilde{e}_{b})H_{2}(\widetilde{e}_{b})]\right],}$

i.e. the smallest $\epsilon$ that leads to a sufficient length of the local
random string $W^{*}$ usable as a seed for hash function with requirements
$h(n_{s},n_{f})$. In order for this paper to be self sufficient, in table 1 we
list a table of functions $h(n_{s},n_{f})$ for different families of hash
functions studied in Hayashi and Tsurumaru (2016).

Table 1: Seed requirements of different families of hash functions. Hash function family | $h(n_{s},n_{f})$
---|---
$f_{F1,R}$ and $f_{F2,R}$ Hayashi and Tsurumaru (2016) | $n_{s}-n_{f}$
$f_{F3,R}$ and $f_{F4,R}$ Hayashi and Tsurumaru (2016) | $n_{f}$
Hash functions using Toeplitz matrix | $n_{s}$
Trevisian’s extractor De  _et al._ (2012) | $\log^{3}(n_{s})$
Hash functions in the TSSR paper Tomamichel  _et al._ (2011) | $2n_{f}$
$\epsilon$-almost pairwise independent hash functions Miller and Shi (2016) | $4n_{f}$

##  IV Simulation Results

We simulate the performance of the fully passive entanglement based QKD with
the parameters shown in Table. 2. We compare fully passive and standard
entanglement (BBM92) QKD schemes Bennett  _et al._ (1992b). In order to
establish optimal secure key rate, we take the optimal photon pair number per
coincidence window ($\mu$). Finite-size-effect is considered with post-
processing block size equals to $10^{6}$ and the failure probability
$\varepsilon^{ph}=10^{-7}$ for estimating the phase error rates.

Table 2: Parameters used for simulation. Parameters | Values
---|---
Dark Count Rate $p_{d}$ | $10^{-6}$
Detector Efficiency $\eta_{d}$ | 0.40
Misalignment error rate $e_{d}$ | 0.015
Error Correction Efficiency $f$ | 1.15
Photon Pair Number per Coincidence Window $\mu$ | Optimal
Basis Reconciliation Factor $q$ | 0.50
Phase Error Estimation Failure Probability $\varepsilon^{ph}$ | $10^{-7}$
Post-processing Block Size | $10^{6}$
Figure 2: a) Secure key rate per pulse versus channel loss (source in
between). For each secure key rate plot, we numerically optimize the mean
photon pair number $\mu$. The hash function $h$ performed in this paper is
shown in Table 1.

The simulation results are shown in Fig. 2. The secure key rate per pulse of
fully passive entanglement based QKD scheme is varying a lot depending on the
performed hash function in the privacy amplification procedure. When
$f_{F3,R}$, $f_{F4,R}$, TSSR hash functions or Trevisan’s extractor are used,
the fully passive entanglement based QKD scheme achieves the same secure key
extracting rate as the BBM92 scheme. Considering the computational complexity
of the privacy amplification procedure, we suggest the implementation scheme
with hash functions in family $f_{F3,R}$ and $f_{F4,R}$, which achieve
computational complexity of $O(n\log n)$.

##  V Discussion

We described a fully passive entanglement based QKD protocol, which does not
require any auxiliary random number generators. We achieve this by producing
certified local random bits extracted from mismatched basis measurements
during a run of the protocol. These measurement-basis-mismatched events are
typically discarded during the basis reconciliation procedure of QKD
protocols. However, as we show above, these events can be used to produce a
local random string, which can afterwards, be used in privacy amplification
procedure. To our knowledge, this simple but powerful observation has never
been mentioned in literature before. The advantages of a fully passive
protocol are twofold. Firstly, since auxiliary random number generators and
active optical modulators are not utilized, our protocol is robust against
attacks through side-channels caused by imperfections of these devices.
Secondly, the fact that our protocol does not require auxiliary random numbers
provides the potential for significantly reducing the overall costs and
complexity for the implementation of QKD systems, which can be beneficial for
example in reducing payload on a satellite. Last but not least, the idea of
utilizing mismatched measurement results is very simple and can be used in
conjunction with plethora of existing practical QKD protocols.

###### Acknowledgements.

We thank Anton Zeilinger for for fruitful discussions. We acknowledge funding
from the FWF START project (Y879-N27) and the FWF-GACR joint international
project (I3053-N27 and GF17-33780L), the Austrian Research Promotion Agency
(FFG) via the Austrian Science and Applications Programme (ASAP12, Contract
6238191), the Austrian Federal Ministry of Education, Science and Research
(BMBWF) and the University of Vienna via the project QUESS, the National
Natural Science Foundation of China via project No. 61972410 as well as from
the research plan of National University of Defense Technology under Grant No.
ZK19-13 and No. 19-QNCXJ-107.

## References

  * Yin  _et al._ (2016) H.-L. Yin, T.-Y. Chen, Z.-W. Yu, H. Liu, L.-X. You, Y.-H. Zhou, S.-J. Chen, Y. Mao, M.-Q. Huang, W.-J. Zhang, H. Chen, M. J. Li, D. Nolan, F. Zhou, X. Jiang, Z. Wang, Q. Zhang, X.-B. Wang, and J.-W. Pan, Physical Review Letters 117, 190501 (2016), pRL.
  * Kiyoshi  _et al._ (2016) T. Kiyoshi, C. Marcos, and L. Marco, New Journal of Physics 18, 065008 (2016).
  * Zhao  _et al._ (2006) Y. Zhao, B. Qi, X. Ma, H.-K. Lo, and L. Qian, Physical Review Letters 96, 070502 (2006), pRL.
  * Peng  _et al._ (2007) C.-Z. Peng, J. Zhang, D. Yang, W.-B. Gao, H.-X. Ma, H. Yin, H.-P. Zeng, T. Yang, X.-B. Wang, and J.-W. Pan, Physical Review Letters 98, 010505 (2007), pRL.
  * (5) [Rosenberg  _et al._(2007)Rosenberg, Harrington, Rice, Hiskett, Peterson, Hughes, Lita, Nam, and Nordholt]D. Rosenberg, J. Harrington, P. Rice, P. Hiskett, C. Peterson, R. Hughes, A. Lita, S. Nam, and J. Nordholt, Physical Review Letters 98, 010503 (2007), pRL.
  * Yuan  _et al._ (2007) Z. Yuan, A. Sharpe, and A. Shields, Applied physics letters 90, 011118 (2007).
  * Liu  _et al._ (2010) Y. Liu, T.-Y. Chen, J. Wang, W.-Q. Cai, X. Wan, L.-K. Chen, J.-H. Wang, S.-B. Liu, H. Liang, L. Yang, C.-Z. Peng, K. Chen, Z.-B. Chen, and J.-W. Pan, Optics Express 18, 8587 (2010).
  * Wang  _et al._ (2013) J. Wang, B. Yang, S. Liao, L. ZHang, Q. Shen, X. Xu, J. Wu, S. Yang, H. Jiang, Y. Tang, B. Zhong, H. Liang, W. Liu, Y. Hu, Y. Huang, B. Qi, J. Ren, G. Pan, J. Yin, J. Jia, Y. Chen, K. Chen, C.-Z. Peng, and J.-W. Pan, Nature Photonics 7 (2013), 10.1038/nphoton.2013.89 10.1038/NPHOTON.2013.89.
  * Tang  _et al._ (2014a) Z. Tang, Z. Liao, F. Xu, B. Qi, L. Qian, and H.-K. Lo, Physical Review Letters 112, 190503 (2014a), pRL.
  * Tang  _et al._ (2014b) Y. Tang, H. Yin, S. Chen, Y. Liu, W. Zhang, X. Jiang, L. Zhang, J. Wang, L. You, J. Guan, D. Yang, Z. Wang, H. Liang, Z. Zhang, N. Zhou, X. Ma, T.-Y. Chen, Q. Zhang, and J.-W. Pan, Physical Review Letters 113 (2014b), 10.1103/PhysRevLett.113.190501.
  * Curty  _et al._ (2010a) M. Curty, X. Ma, B. Qi, and T. Moroder, Physical Review A 81, 022310 (2010a).
  * Rarity  _et al._ (1994) J. G. Rarity, P. C. M. Owens, and P. R. Tapster, Journal of Modern Optics 41, 2435 (1994).
  * Adachi  _et al._ (2007) Y. Adachi, T. Yamamoto, M. Koashi, and N. Imoto, Physical Review Letters 99, 180503 (2007), pRL.
  * Mauerer and Silberhorn (2007) W. Mauerer and C. Silberhorn, Physical Review A 75, 050305 (2007), pRA.
  * Xiongfeng and Hoi-Kwong (2008) M. Xiongfeng and L. Hoi-Kwong, New Journal of Physics 10, 073018 (2008).
  * Wang  _et al._ (2007a) Q. Wang, X. B. Wang, G. Björk, and A. Karlsson, EPL (Europhysics Letters) 79, 40001 (2007a).
  * Yoritoshi  _et al._ (2009) A. Yoritoshi, Y. Takashi, K. Masato, and I. Nobuyuki, New Journal of Physics 11, 113033 (2009).
  * TAN and CAI (2011) Y.-G. TAN and Q.-Y. CAI, International Journal of Quantum Information 09, 903 (2011).
  * Wang  _et al._ (2016) Q. Wang, C.-H. Zhang, and X.-B. Wang, Physical Review A 93, 032312 (2016), pRA.
  * Wang  _et al._ (2007b) Q. Wang, X.-B. Wang, and G.-C. Guo, Physical Review A 75, 012312 (2007b), pRA.
  * Curty  _et al._ (2010b) M. Curty, X. Ma, H.-K. Lo, and N. Lütkenhaus, Physical Review A 82, 052325 (2010b).
  * Li  _et al._ (2014) Y. Li, W.-s. Bao, H.-w. Li, C. Zhou, and Y. Wang, Physical Review A 89, 032329 (2014), pRA.
  * Shan  _et al._ (2014) Y.-Z. Shan, S.-H. Sun, X.-C. Ma, M.-S. Jiang, Y.-L. Zhou, and L.-M. Liang, Physical Review A 90, 042334 (2014), pRA.
  * Curty  _et al._ (2015) M. Curty, M. Jofre, V. Pruneri, and M. W. Mitchell, Entropy 17, 4064 (2015).
  * Guan  _et al._ (2015) J.-Y. Guan, Z. Cao, Y. Liu, G.-L. Shen-Tu, J. S. Pelc, M. M. Fejer, C.-Z. Peng, X. Ma, Q. Zhang, and J.-W. Pan, Physical Review Letters 114, 180502 (2015), pRL.
  * Sun  _et al._ (2016) S.-H. Sun, G.-Z. Tang, C.-Y. Li, and L.-M. Liang, Physical Review A 94, 032324 (2016).
  * Qi-Chao  _et al._ (2014) S. Qi-Chao, W. Wei-Long, L. Yang, Z. Fei, S. P. Jason, M. M. Fejer, P. Cheng-Zhi, C. Xianfeng, M. Xiongfeng, Z. Qiang, and P. Jian-Wei, Laser Physics Letters 11, 085202 (2014).
  * Krapick  _et al._ (2014) S. Krapick, M. S. Stefszky, M. Jachura, B. Brecht, M. Avenhaus, and C. Silberhorn, Physical Review A 89, 012329 (2014), pRA.
  * Zhang  _et al._ (2012) Y. Zhang, S. Wang, Z.-Q. Yin, W. Chen, W.-Y. Liang, H.-W. Li, G.-C. Guo, and Z.-F. Han, Chinese Physics B 21, 100307 (2012).
  * Zhang  _et al._ (2010) Y. Zhang, W. Chen, S. Wang, Z.-Q. Yin, F.-X. Xu, X.-W. Wu, C.-H. Dong, H.-W. Li, G.-C. Guo, and Z.-F. Han, Optics Letters 35, 3393 (2010).
  * Jennewein  _et al._ (2000) T. Jennewein, C. Simon, G. Weihs, H. Weinfurter, and A. Zeilinger, Physical Review Letters 84, 4729 (2000), pRL.
  * Tittel  _et al._ (2000) W. Tittel, J. Brendel, H. Zbinden, and N. Gisin, Physical Review Letters 84, 4737 (2000), pRL.
  * Resch  _et al._ (2005) K. J. Resch, M. Lindenthal, B. Blauensteiner, H. R. Böhm, A. Fedrizzi, C. Kurtsiefer, A. Poppe, T. Schmitt-Manderbach, M. Taraba, R. Ursin, P. Walther, H. Weier, H. Weinfurter, and A. Zeilinger, Optics Express 13, 202 (2005).
  * Peng  _et al._ (2005) C.-Z. Peng, T. Yang, X.-H. Bao, J. Zhang, X.-M. Jin, F.-Y. Feng, B. Yang, J. Yang, J. Yin, and Q. Zhang, Physical review letters 94, 150501 (2005).
  * Marcikic  _et al._ (2004) I. Marcikic, H. de Riedmatten, W. Tittel, H. Zbinden, M. Legré, and N. Gisin, Physical Review Letters 93, 180502 (2004), pRL.
  * Ursin  _et al._ (2007) R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, and P. Trojek, Nature physics 3, 481 (2007).
  * Poppe  _et al._ (2004) A. Poppe, A. Fedrizzi, R. Ursin, H. R. Böhm, T. Lorünser, O. Maurhardt, M. Peev, M. Suda, C. Kurtsiefer, H. Weinfurter, T. Jennewein, and A. Zeilinger, Optics Express 12, 3865 (2004).
  * Ma  _et al._ (2007) X. Ma, C.-H. Fung, and H.-K. Lo, Physical Review A 76, 012307 (2007), pRA.
  * Marcikic  _et al._ (2006) I. Marcikic, A. Lamas-Linares, and C. Kurtsiefer, Applied Physics Letters 89, 101122 (2006).
  * Ali Khan and Howell (2006) I. Ali Khan and J. C. Howell, Physical Review A 73, 031801 (2006), pRA.
  * Honjo  _et al._ (2008) T. Honjo, S. W. Nam, H. Takesue, Q. Zhang, H. Kamada, Y. Nishida, O. Tadanaga, M. Asobe, B. Baek, R. Hadfield, S. Miki, M. Fujiwara, M. Sasaki, Z. Wang, K. Inoue, and Y. Yamamoto, Optics Express 16, 19118 (2008).
  * Fasel  _et al._ (2004) S. Fasel, N. Gisin, G. Ribordy, and H. Zbinden, The European Physical Journal D-Atomic, Molecular, Optical and Plasma Physics 30, 143 (2004).
  * Alexander  _et al._ (2009) T. Alexander, P. Andreas, H. Michael, F. Daniele, L. Thomas, Q. Edwin, M. Thomas, H. Hannes, and Z. Anton, New Journal of Physics 11, 045013 (2009).
  * Takesue  _et al._ (2010) H. Takesue, K.-i. Harada, K. Tamaki, H. Fukuda, T. Tsuchizawa, T. Watanabe, K. Yamada, and S.-i. Itabashi, Optics Express 18, 16777 (2010).
  * Scheidl  _et al._ (2009) T. Scheidl, R. Ursin, A. Fedrizzi, S. Ramelow, X.-S. Ma, T. Herbst, R. Prevedel, L. Ratschbacher, J. Kofler, and T. Jennewein, New Journal of Physics 11, 085002 (2009).
  * Bennett  _et al._ (1988) C. H. Bennett, G. Brassard, and J.-M. Robert, SIAM journal on Computing 17, 210 (1988).
  * Bennett  _et al._ (1995) C. H. Bennett, G. Brassard, C. Crépeau, and U. M. Maurer, Information Theory, IEEE Transactions on 41, 1915 (1995).
  * Watanabe (2007) Y. Watanabe, Journal of physics A: Mathematical and theoretical 40 (2007), 10.1088/1751-8113/40/3/F03.
  * Bo  _et al._ (2013) L. Bo, Z. Baokang, L. Bo, and W. Chunqing, J. UCS 19, 2420 (2013).
  * Fung  _et al._ (2010) C.-H. F. Fung, X. Ma, and H. F. Chau, Physical Review A 81, 012318 (2010), pRA.
  * Liu  _et al._ (2013) B. Liu, B. Liu, B. Zhao, D. Zou, C. Wu, W. Yu, and I. You, “A real-time privacy amplification scheme in quantum key distribution,” in  _Information and Communication Technology:
International Conference, ICT-EurAsia 2013,
Yogyakarta, Indonesia, March 25-29, 2013. Proceedings_ (Springer Berlin
Heidelberg, Berlin, Heidelberg, 2013) pp. 453–458.

  * Liu  _et al._ (2016) B. Liu, B.-K. Zhao, W.-R. Yu, and C.-Q. Wu, Journal of Internet Technology 17, 309 (2016).
  * Hayashi (2011) M. Hayashi, IEEE Transactions on Information Theory 57, 3989 (2011).
  * Renner and König (2005) R. Renner and R. König, “Universally composable privacy amplification against quantum adversaries,” in  _Theory of Cryptography_ (Springer, 2005) pp. 407–425.
  * Hayashi (2009) M. Hayashi, Physical Review A 79, 020303 (2009), pRA.
  * Hayashi and Tsurumaru (2016) M. Hayashi and T. Tsurumaru, IEEE Transactions on Information Theory 62, 2213 (2016).
  * Hayashi and Tsurumaru (2015) M. Hayashi and T. Tsurumaru, in  _2015 IEEE International Symposium on Information Theory (ISIT)_ (2015) pp. 1786–1790.
  * Pironio  _et al._ (2010) S. Pironio, A. Acín, S. Massar, A. B. de la Giroday, D. N. Matsukevich, P. Maunz, S. Olmschenk, D. Hayes, L. Luo, T. A. Manning, and C. Monroe, Nature 464, 1021 (2010), 10.1038/nature09008.
  * Fiorentino  _et al._ (2007) M. Fiorentino, C. Santori, S. M. Spillane, R. G. Beausoleil, and W. J. Munro, Physical Review A 75, 032334 (2007), pRA.
  * Wei and Guo (2009) W. Wei and H. Guo, Optics Letters 34, 1876 (2009).
  * Svozil (2009) K. Svozil, Physical Review A 79, 054306 (2009), pRA.
  * Fürst  _et al._ (2010) H. Fürst, H. Weier, S. Nauerth, D. G. Marangon, C. Kurtsiefer, and H. Weinfurter, Optics Express 18, 13029 (2010).
  * Jofre  _et al._ (2011) M. Jofre, M. Curty, F. Steinlechner, G. Anzolin, J. P. Torres, M. W. Mitchell, and V. Pruneri, Optics Express 19, 20665 (2011).
  * Gallego  _et al._ (2013) R. Gallego, L. Masanes, G. De La Torre, C. Dhara, L. Aolita, and A. Acín, Nature Communications 4, 2654 (2013).
  * Abellán  _et al._ (2014) C. Abellán, W. Amaya, M. Jofre, M. Curty, A. Acín, J. Capmany, V. Pruneri, and M. W. Mitchell, Optics Express 22, 1645 (2014).
  * Dhara  _et al._ (2013) C. Dhara, G. Prettico, and A. Acín, Physical Review A 88, 052116 (2013), pRA.
  * Marangon  _et al._ (2017) D. G. Marangon, G. Vallone, and P. Villoresi, Physical Review Letters 118, 060503 (2017), pRL.
  * Acín and Masanes (2016) A. Acín and L. Masanes, Nature 540, 213 (2016).
  * Acín  _et al._ (2016) A. Acín, S. Pironio, T. Vértesi, and P. Wittek, Physical Review A 93, 040102 (2016), pRA.
  * Vallone  _et al._ (2014) G. Vallone, D. G. Marangon, M. Tomasin, and P. Villoresi, Physical Review A 90, 052327 (2014), pRA.
  * Herrero-Collantes and Garcia-Escartin (2017) M. Herrero-Collantes and J. C. Garcia-Escartin, Reviews of Modern Physics 89, 015004 (2017), rMP.
  * Pivoluska  _et al._ (2020) M. Pivoluska, M. Plesch, M. Farkas, N. Ružičková, C. Flegel, N. Herrera Valencia, W. McCutcheon, M. Malik, and E. A. Aguilar, arXiv e-prints , arXiv:2002.12295 (2020), arXiv:2002.12295 [quant-ph] .
  * Acín  _et al._ (2006) A. Acín, N. Gisin, and L. Masanes, Physical Review Letters 97, 120405 (2006), pRL.
  * Beaudry  _et al._ (2008) N. J. Beaudry, T. Moroder, and N. Lütkenhaus, Physical Review Letters 101, 093601 (2008), pRL.
  * Gittsovich  _et al._ (2014) O. Gittsovich, N. J. Beaudry, V. Narasimhachar, R. R. Alvarez, T. Moroder, and N. Lütkenhaus, Physical Review A 89, 012325 (2014), pRA.
  * Bennett  _et al._ (1992a) C. H. Bennett, F. Bessette, G. Brassard, L. Salvail, and J. Smolin, Journal of cryptology 5, 3 (1992a).
  * Brassard and Salvail (1994) G. Brassard and L. Salvail, in  _advances in Cryptology—EUROCRYPT’93_ (Springer, 1994) pp. 410–423.
  * Sugimoto and Yamazaki (2000) T. Sugimoto and K. Yamazaki, IEICE Trans Fundamentals E83, 1987 (2000).
  * Yan  _et al._ (2008) H. Yan, T. Ren, X. Peng, X. Lin, and H. Guo, “Information reconciliation protocol in quantum key distribution system,”  (2008).
  * Pedersen and Toyran (2015) T. B. Pedersen and M. Toyran, Quantum Info. Comput. 15, 419 (2015).
  * Buttler  _et al._ (2003) W. T. Buttler, S. K. Lamoreaux, J. R. Torgerson, G. H. Nickel, C. H. Donahue, and C. G. Peterson, Physics Review A 67, 1 (2003).
  * Cui  _et al._ (2013) K. Cui, J. Wang, H. F. Zhang, C. L. Luo, G. Jin, and T. Y. Chen, IEEE Transactions on Information Forensics and Security 8, 184 (2013).
  * Elkouss  _et al._ (2009) D. Elkouss, A. Leverrier, R. Aeaume, and e. al., EEE international symposium on information theory , 1879 (2009).
  * Wilde and Guha (2013) M. M. Wilde and S. Guha, IEEE Transactions on Information Theory 59, 1175 (2013).
  * Takesue  _et al._ (2015) H. Takesue, T. Sasaki, K. Tamaki, and M. Koashi, Nature Photonics 9, 827 (2015).
  * Mink and Nakassis (2014) A. Mink and A. Nakassis, in  _Quantum Information and Computation XII_, Vol. 9123, edited by E. Donkor, A. R. Pirich, H. E. Brandt, M. R. Frey, S. J. L. Jr., and J. M. Myers, International Society for Optics and Photonics (SPIE, 2014) pp. 19 – 31.
  * Trevisan (2001) L. Trevisan, Journal of the ACM 48, 860 (2001).
  * Ma  _et al._ (2013) X. Ma, F. Xu, H. Xu, X. Tan, B. Qi, and H.-K. Lo, Physical Review A 87 (2013), 10.1103/PhysRevA.87.062327.
  * Konig and Renner (2011) R. Konig and R. Renner, Information Theory, IEEE Transactions on 57, 4760 (2011).
  * Skorski (2015) M. Skorski, “True random number generators secure in a changing environment: Improved security bounds,” in  _SOFSEM 2015: Theory and Practice of Computer Science: 41st International Conference on Current Trends in Theory and Practice of Computer Science, Pec pod Sněžkou, Czech Republic, January 24-29, 2015. Proceedings_, edited by G. F. Italiano, T. Margaria-Steffen, J. Pokorný, J.-J. Quisquater, and R. Wattenhofer (Springer Berlin Heidelberg, Berlin, Heidelberg, 2015) pp. 590–602.
  * De  _et al._ (2012) A. De, C. Portmann, T. Vidick, and R. Renner, SIAM Journal on Computing 41, 915 (2012).
  * Tomamichel  _et al._ (2011) M. Tomamichel, C. Schaffner, A. Smith, and R. Renner, IEEE Trans. Inform. Theory 57, 5524 (2011).
  * Miller and Shi (2016) C. A. Miller and Y. Shi, J. ACM 63, 1 (2016).
  * Bennett  _et al._ (1992b) C. Bennett, G. Brassard, and N. Mermin, Physical Review Letters 68, 557 (1992b), pRL.


# A Rational Analysis of the Speech-to-Song Illusion

Raja Marjieh1, Pol van Rijn2, Ilia Sucholutsky3, Harin Lee2,4, Thomas L.
Griffiths1,3,*, Nori Jacoby2,*
1Department of Psychology, Princeton University
2Computational Auditory Perception Group, Max Planck Institute for Empirical
Aesthetics
3Department of Computer Science, Princeton University
4Max Planck Institute for Human Cognitive and Brain Sciences
*Equal Contribution
{raja.marjieh, is2961<EMAIL_ADDRESS>{pol.van-rijn, harin.lee,
<EMAIL_ADDRESS>

###### Abstract

The speech-to-song illusion is a robust psychological phenomenon whereby a
spoken sentence sounds increasingly more musical as it is repeated. Despite
decades of research, a complete formal account of this transformation is still
lacking, and some of its nuanced characteristics, namely, that certain phrases
appear to transform while others do not, is not well understood. Here we
provide a formal account of this phenomenon, by recasting it as a statistical
inference whereby a rational agent attempts to decide whether a sequence of
utterances is more likely to have been produced in a song or speech. Using
this approach and analyzing song and speech corpora, we further introduce a
novel prose-to-lyrics illusion that is purely text-based. In this illusion,
simply duplicating written sentences makes them appear more like song lyrics.
We provide robust evidence for this new illusion in both human participants
and large language models.

Keywords: speech-to-song, music cognition, rational analysis, Bayesian
modeling

## Introduction

First published as an audio demonstration just over two decades ago (?, ?, ?),
the speech-to-song illusion concerns a curious phenomenon whereby spoken
phrases can be made to sound more song-like simply by repetition. This
observation has been replicated and elaborated on in numerous studies (?, ?,
?, ?, ?, ?, ?, ?, ?, ?), and it taps into the deep relationship between speech
and music which itself has been the center of considerable inquiry (?, ?, ?,
?, ?, ?) tracing as far back as the works of eighteenth century philosophers
such as Rousseu and Herder (?, ?, ?).

Figure 1: The speech-to-song paradigm and the newly proposed prose-to-lyrics
paradigm.

Despite this rich research tradition, there is still no unifying theoretical
framework to account for the full phenomenology of the illusion, namely, (i)
what causes repetition on average to transform spoken phrases to song? (?, ?)
(ii) why does the strength of the effect increase with the number of
repetitions? (?, ?) (iii) why is it that certain phrases transform whereas
others do not? (?, ?) and (iv) how does this connect to the statistical
properties of speech and music learned throughout life and culture? (?, ?, ?)

In the present work, we propose one such framework by analyzing the abstract
computational problem underlying the speech-to-song illusion and recasting it
as rational statistical inference (?, ?, ?). Specifically, we argue that the
phenomenon can be understood in terms of the extent to which a given stimulus
(or its repetitions) is more likely to originate from a generative model of
speech versus song. Using Bayesian inference, the problem of determining
whether a stimulus is more speech-like or song-like reduces to analyzing the
probability of that stimulus under suitable speech and song corpora. We show
that textual transcription datasets of speech and songs, both naturalistic and
synthetically generated by large language models (LLMs), are indeed sufficient
to reproduce some of the key features of the illusion, and motivate a
hypothesis regarding a new “prose-to-lyrics” illusion that is based purely on
text (i.e., no acoustic signal is provided) which we evaluate in both humans
and LLMs (GPT-4; ?, ?).

Our framework provides a principled account of the speech-to-song illusion
that explicitly roots it in the learned statistics of speech and music, and
allows for generalization across multiple modalities given suitable generative
models. The paper proceeds as follows. We first review the empirical
literature on the speech-to-song illusion, and the fundamentals of rational
analysis in the context of a two-hypothesis decision problem. We then
explicitly apply the latter to the speech-to-song problem and use it to
motivate the textual version of the illusion. We then summarize the technical
details of the corpus analysis and the human and LLM experiments, then present
our results and distill their implications.

## Background

### Speech-to-Song Illusion

In its original form, the speech-to-song paradigm involves presenting
participants with a spoken phrase (famously, “sometimes behave so strangely”)
and repeating it several times (?, ?, ?). In the pause between each
repetition, participants report how the last utterance sounds to them from
“exactly like speech” to “exactly like song” using a numeric Likert scale
(Figure 1; Speech-to-Song). ? (?) showed that this simple manipulation is
sufficient to drive participants to perceive the speech excerpt as more like
song than speech relative to the original (and identical) presentation. In a
subsequent experiment, the authors asked participants to repeat the utterance
after one and ten presentations, and found that in the former case the
acoustic characteristics of the participants’ responses (their fundamental
frequency $f_{0}$ contour) resembled those of speech utterances, whereas in
the latter case those of a tonal melody. The fact that stimuli in all
repetitions are identical is significant as it allows for studying the
internal processing of speech and music without relying on different stimuli.

Empirical research into the speech-to-song phenomenon has expanded
substantially since the original study of ? (?). First, ? (?) provided some
evidence suggesting that the phenomenon may be affected by linguistic
proficiency. Specifically, the authors showed that speech excerpts from
languages that are hard to pronounce for native English speakers tended to
transform more readily than those from languages that are easier to pronounce.
Second, the increase of musicality of repeated sounds has been shown to
generalize to non-speech sounds such as random tone sequences (?, ?),
environmental sounds like dripping water (?, ?), and animal sounds (?, ?).
Third, while repetition tends to make stimuli sound more musical on average,
it has been shown that the effect at the level of individual stimuli is more
nuanced, with some transforming to song effectively while others resisting
transformation altogether (?, ?). Specifically, ? (?) meticulously scanned
audiobooks in English to find “song-like” sentences that show enhanced effect
of repetition. They then contrasted these stimuli with similar utterances
that, without repetition, have comparable speech vs. song ratings, but do not
exhibit significant transformation when repeated.

Further research (?, ?, ?) revealed that this interaction also occurs when
participants are presented with only the fundamental frequency ($f_{0}$)
contour of a spoken utterance. However, simple acoustic manipulations of the
contour (such as changing the beat consistency, pitch slope, and melodic
structure) that were supposed to enhance the effect did not influence
musicality ratings as expected. These findings highlight the complexity of
extracting acoustic correlates to predict whether a given phrase should
transform under repetition (?, ?, ?).

Figure 2: Aggregate repetition analysis. A. Log-likelihood curves for the
naturalistic dataset as a function of the number of repetitions (aggregated
across all words). B. Log-odds ratio for the naturalistic dataset (positive
values favor song). C. and D. Similar to A and B for the synthetic (LLM-
generated) dataset. Dashed red line indicates equal-likelihood threshold.
Shaded area corresponds to 95% confidence intervals bootstrapped over
documents.

### Rational Analysis

Rational analysis is a cognitive modeling strategy that has been effectively
applied across a wide array of topics including categorization (?, ?), causal
induction (?, ?), perception (?, ?), and semantic memory (?, ?). The core idea
behind rational analysis is to analyze human behavior from the perspective of
the abstract computational problem that it attempts to solve and the optimal
solution to that problem (?, ?). This is usually implemented as Bayesian
inference whereby the posterior probabilities of different hypotheses $p(h|d)$
are estimated by inverting a generative model of the data $d$ specified by
some prior $p(h)$ and likelihood $p(d|h)$ using Bayes rule

$p(h|d)=\frac{p(d|h)p(h)}{\sum_{h}p(d|h)p(h)}$ (1)

The prior captures inductive bias towards certain hypotheses independent of
the data, the likelihood captures how well hypotheses are supported by the
data, and the posterior integrates the two. The main challenge in applying
Bayes rule is computing the denominator as it often requires summing over a
large number of hypotheses. In the context of binary decision problems,
however, in which an agent needs to decide between two hypotheses $h_{0}$ and
$h_{1}$, a simpler formula can be derived by taking the ratio between the
posteriors of the two hypotheses. This is known as the log-odds form

$\log\frac{p(h_{1}|d)}{p(h_{0}|d)}=\log\frac{p(d|h_{1})}{p(d|h_{0})}+\log\frac{p(h_{1})}{p(h_{0})}$
(2)

From here we see that the extent to which a given hypothesis is preferred over
the other ultimately depends on the likelihood ratio of the data as well as
the prior odds.

## A Generative Account of Speech-to-Song

We now leverage the two-hypothesis Bayesian log-odds formula to formulate a
rational model of the speech-to-song task. The data in this case is the
presented utterance $s$, or its $n$-repetitions, which we will denote by
$s^{n}$. The log-odds formula for the speech-to-song task then becomes

$\log\frac{p(\text{song}|s^{n})}{p(\text{speech}|s^{n})}=\log\frac{p(s^{n}|\text{song})}{p(s^{n}|\text{speech})}+\log\frac{p(\text{song})}{p(\text{speech})}$
(3)

Note that the log prior odds is just a constant independent of $n$, and by
design we can set it to zero (a priori speech and song stimuli are equally
likely). Thus, determining whether a sample is more speech-like or song-like
becomes a matter of estimating the probability of the repeated stimulus
$s^{n}$ under generative models of speech and song. We hypothesize that
differences in the strength of transformation for different sentences will
follow from the statistics of those sentences under the different likelihood
models. Equation (3) can also be used to predict the general effect of
repetition by estimating $p(n|\text{speech})$ and $p(n|\text{song})$, i.e.,
the probability of observing an $n$-repetition in speech and song,
respectively.

Finding generative audio models to estimate the probability of $s^{n}$ is
traditionally a hard problem (though recent advances in machine learning have
changed this status; see Discussion). As an approximation, we can consider
text corpora of speech and song transcripts. The idea here being that textual
repetition in lyrics is indicative to some degree of repetition in melody
(e.g., through refrains). By estimating the probability of sentences and their
repetitions under such textual corpora we can evaluate whether their log-odds
will exhibit the phenomenology of the illusion. Moreover, this treatment
motivates the hypothesis that repetition should have a parallel effect in a
purely textual variant of the illusion (Figure 1; Prose-to-Lyrics), as humans
may also internalize similar statistical expectations in prose and lyrics. We
will next present the modeling analysis and then proceed to the behavioral
studies.

## Modeling Methods

Figure 3: Log-odds profiles for common sentences constructed from the
naturalistic speech and song corpora under a naïve Bayes approximation. A.
Absolute log-odds profiles, dashed red line indicates equal-likelihood
threshold, positive values favor song. B. Relative log-odds profiles, with the
value at zero subtracted (indicated by the dashed red line).

### Text Preprocessing and Analysis

To construct repetition probabilities from text corpora we applied a bag-of-
words approach. Specifically, we measured the repetition of words within each
text document irrespective of their location within the document. This is a
simplifying assumption that increases statistical power and is common to other
textual analysis methods (e.g. topic modeling; ?, ?). The processing steps
were as follows. First, documents were tokenized, lemmatized, and cleaned from
stop words and non-alphabetic entries using the `nltk` Python package (?, ?).
Next, to create aggregate repetition probabilities per corpus, we counted the
number of times each token was repeated within a given document, then created
a histogram over the number of repetitions (i.e., the number of times a word
was repeated $n$ times within a document), normalized, and finally averaged
over all documents. A similar procedure was applied for word-level repetition
probabilities (including pronouns), by counting the number of documents in
which a given word was repeated $n$ times. Sentence-level repetition
probabilities were constructed using a naïve Bayes approach (by multiplying
the repetition probabilities of individual words). This is of course an
approximation and can be generalized (see Discussion), but for the purpose of
our theoretical analysis of short sentences constructed of common words it is
sufficient. Specifically, the short sentences were constructed from the
jointly most common words found in the naturalistic speech and song corpora
(i.e., words that maximize the product of the number of documents from speech
and song in which they appeared). These were useful as they allow for the
construction of smooth word-level and sentence-level log-odds curves.
Confidence intervals were constructed by bootstrapping over documents with
replacement and 1,000 replicas.

Figure 4: Behavioral results of the prose-to-lyrics experiments. A. Mean human
ratings as a function of repetitions (rescaled to a 0-1 range via division by
a factor of 6; dashed red line indicates speech-song equality threshold).
Inset shows the data grouped based on a median split of the GPT4 scores at 4
repetitions. Shaded area indicates 95% confidence intervals (CIs) bootstrapped
over participants. B. Mean rating as a function of repetition for the GPT4
control experiment. C. Mean sentence-level human ratings. D. Mean sentence-
level human ratings relative to the value at 0 repetitions.

### Text Corpora for Speech and Song

For our prose and lyrics corpora we relied on two complementary approaches:
one where we mine text from an ecologically valid (“naturalistic”) source, but
accept the fact that different documents will have varied lengths, and another
where we synthesize text artificially using a large language model (GPT4) but
have full control over the document length. For the naturalistic song corpus,
we used a collection of 2,994 song lyrics extracted from songs on Shazam, a
mobile app for song identification. The subset was extracted from a larger
collection at random, whereby the lyrical language was detected using the
FastText algorithm (?, ?). For the speech corpus, we used text documents from
the Corpus of Contemporary American English (COCA; ?, ?), a large dataset
constructed from a variety of sources such as magazines, blogs, and academic
articles. Specifically, we focused on a set of 948 randomly sampled documents
from the magazine subset of COCA. Note that while there were more song
documents, these were naturally shorter, with a given document comprising 98
words on average as opposed to an average of 437 words for speech. To control
for this variability, we used GPT4 to generate a synthetic set of 1,400 songs
and 1,400 conversation transcripts, both constrained to approximately 20 lines
per document, by repeatedly querying the model using the following prompt
(with a temperature of 0.7 to encourage diversity in output): “Please write 10
[song lyrics|conversations] of length 20 lines each. Before each
[song|conversation], write [Song|Conversation] and the number of the
[song|conversation]”. The average number of tokens was indeed more aligned in
this case and amounted to 54 (95% CI: $[33,76]$) for song and 49 (95% CI:
$[32,66]$) for speech.

## Modeling Results

We begin with an empirical evaluation of the properties of the Bayes log-odds
formula as applied to the textual speech and song corpora. Our goal here is to
see if, despite its simplicity and pure reliance on text, the model is able to
capture the qualitative phenomenology of the speech-to-song illusion. Figure 2
shows the log-likelihoods and log-odds for both speech and song as a function
of general word repetition (naturalistic: Figure 2A-B, synthetic: Figure
2C-D). We see that for both cases, the log-likelihoods of repetition decrease
with the number of repetitions. Crucially, while these initially overlap for
song and speech, the tails of the former are much heavier, driving the log
odds formula to favor song within two repetitions. We also see that this
difference in tails leads to a monotonic increase in the log-odds formula
towards favoring song as a function of repetition. These results are
consistent with the expectations for the speech-to-song phenomenology, namely,
that repetition on average renders utterances more song-like (?, ?), and
increasingly so as a function of repetition (?, ?, ?, ?). Note that a richer
model could also capture ceiling effects by mapping the unbounded log-odds to
a finite interval (e.g., with a sigmoid).

Next, we computed repetition log-odds for short sentences constructed from
common words in the naturalistic corpus for which smooth profiles can be
reliably estimated (see Modeling Methods). The resulting profiles are shown in
Figure 3A. As is the case with Figure 2, we see a general trend of
transformation from speech to song under repetition. Indeed, plotting the
profiles relative to their value without repetition (i.e., collapsing the
origin to zero; Figure 3B) reveals that seven out of eight sentences increased
over their original value, and that the rate of change is different in
different sentences, similar to findings from the literature (?, ?, ?, ?). For
example, we see that intuitive sentences like “I love you” and “I want you”
transform at a faster rate than more generic ones like “I get it” and “go
away”. Overall, this provides a principled explanation for the observed
variation in phrase transformation rate based on natural statistics.

Having evaluated the qualitative predictions of the textual model, we are now
ready to test whether these are realized quantitatively in the hypothesized
prose-to-lyrics paradigm.

## Behavioral Methods

### Behavioral Paradigm

Humans For the prose-to-lyrics experiment, we recruited 120 UK participants
from the online recruitment platform Prolific (https://www.prolific.com/). All
participants provided informed consent prior to participation in accordance
with the Max Planck Ethics Council (#2021_42). Upon providing informed
consent, participants received the following instructions: ‘In this study you
will be presented with text transcripts of different audio recordings and your
task is to decide based on those transcripts whether the original audio is
more likely to have been a song or a speech excerpt. For each transcript you
will have 7 response options ranging from 0 (definitely speech) to 6
(definitely song).’ Participants then rated up to 20 randomly chosen stimuli
and responded to the prompt: ‘The following text transcript was extracted from
an audio recording: $\langle\text{transcript}\rangle$. Based on this
transcript alone, is the audio recording more likely to have been a speech or
a song excerpt?’ (Figure 1; Prose-to-Lyrics). Overall, there were 18 sentences
(see Stimuli) each repeated 0 to 4 times (overall 90 items), and each rating
item received an average of 63 ratings (range: 61 – 66). All items were
shuffled and participants were randomly assigned items from this set.

GPT4 The LLM control experiment was constructed in a similar fashion. We used
Azure’s OpenAI API to query GPT4 (with temperature set to zero; version 0613)
with the following prompt: “In what follows you will see a transcript of an
audio recording and your task is to decide whether this transcript came from a
song or a conversation. Please provide a numerical score between 0 and 1 where
0 is definitely conversation and 1 is definitely song. Do not provide any
explanation or any other text. Transcript: $\langle\text{transcript}\rangle$.
Answer:”.

### Stimuli

We used a set of 18 sentences based on the following sources. First, short
sentences constructed from common words in the naturalistic corpora (see
Modeling Methods). This set included, ‘I love you’, ‘I get it’, ‘I want you’,
‘stay back’, ‘stay still’, ‘go away’, ‘take it back’, and ‘I need time’.
Second, medium-length sentences that were randomly extracted from the speech
corpus. These were ‘how fast does a Zamboni go?’, ‘but once in a while you
step on it from one end to the other’, ‘if it takes an extra minute to do it,
that’s fine’, ‘and wait 10 seconds or so between each bounce’, ‘I drilled a
hole through the top of my rack’, ‘what kind of car would it be?’, ‘I grew up
watching them’, and ‘it has really kept us growing’. Third, two additional
sentences: ‘sometimes behave so strangely’ for its historical significance in
the original paradigm, and an additional technical sentence ‘I will head to
the department store tomorrow’. Repeated versions were constructed by simply
concatenating the sentences.

## Behavioral Results

Figure 4 shows the behavioral results for the set of 18 sentences where each
sentence was presented with 0-4 repetitions (see Behavioral Methods). Overall,
the human average ratings over all sentences (Figure 4A) exhibited an
excellent inter-rater reliability of $r=.96$ (95% CI: $[.94,.98]$) computed
via a split-half method and bootstrapped over participants. Inspecting Figure
4A, we see that repetition steadily transformed the sentences from initially
being significantly speech-like with mean rating of $.31$ (95% CI:
$[.28,.33]$; with 0 being definitely speech and 1 being definitely song) to
significantly being song-like with mean rating $.58$ (95% CI: $[.55,.61]$). A
similar pattern was observed in the case of GPT4 ratings which performed an
equivalent task (Figure 4B). We also found that the human and LLM data were
significantly correlated with a Pearson correlation of $r=.64$ (95% CI:
$[.60,.68]$). Moreover, splitting the human data into two groups based on a
median split of the GPT4 scores at four repetitions (Figure 4A, inset) nicely
separated the original curve into one that surpasses the 0.5 threshold and one
that does not. These results show that we can replicate the effect of
repetition in the speech-to-song illusion with our prose-to-lyrics illusion
without directly playing an audio recording.

Next, inspecting the sentence-level human mean ratings (Figure 4C), we see
that the aggregate trend of monotonic increase in song ratings as a function
of repetitions is also reflected at the level of individual sentences. This
becomes particularly clear when the value at zero repetitions is subtracted
(Figure 4D). We see that not only song ratings increase for all sentences, but
they also do so at different rates as found for the speech-to-song illusion
(?, ?, ?, ?). To further highlight the analogy to the speech-to-song illusion
findings (?, ?), we focus on a demonstrative subset of four sentences in
Figure 5. We see that while the pair ‘I love you’ and ‘sometimes behave so
strangely’ are not significantly separated at zero repetitions ($.46$, CI:
$[.38,.54]$, and $.46$, CI: $[.40,.53]$, respectively), the ratings of the
former increase at a significantly faster rate (at 4 repetitions: $.77$, CI:
$[.71,.83]$, and $.64$, CI: $[.59,.70]$, respectively).

Figure 5: Prose-to-lyrics illusion for demonstrative sentences. Lines
represent averages, shaded area represents 95% CIs.

Likewise, while the pair ‘I drilled a hole through the top of my rack’ and
‘how fast does a Zamboni go?’ were significantly speech-like at zero
repetitions and not significantly separated ($.22$, CI: $[.16,.27]$, and
$.23$, CI: $[.17,.29]$, respectively), only the latter transformed
significantly to being song-like ($.43$, CI: $[.37,.50]$, and $.59$, CI:
$[.52,.66]$, respectively). We confirmed this sentence-repetition interaction
using a non-parametric test ($p<0.001$ for both pairs). This provides clear
evidence analogous to that found in the speech-to-song illusion (?, ?, ?, ?).

## Discussion

By recasting the problem of deciding whether a stimulus is more like speech or
song as rational statistical inference, we constructed a theoretical account
of the speech-to-song illusion that not only provides principled explanations
of its phenomenology based on the statistics of corpora, but also successfully
predicts a new illusion that is purely textual.

Our framework opens up multiple avenues for future investigation. First, a
complete empirical treatment of the problem must ultimately use richer
generative models of text and audio to estimate the probabilities of
naturalistic stimuli and derive quantitative predictions. For text, we could
train or refine LLMs on lyrics and prose, and then use that to estimate the
probability of an entire (repeated) text conditioned on each subset. This
should be possible with an open access language model like Llama 2 (?, ?). For
audio, we believe that we can also provide direct probability estimates of
(repeated) utterances using modern audio transformer architectures such as the
Music Transformer (?, ?). This task can be made easier by referring back to
the observation of ? (?) that the speech-to-song effect can be replicated by
providing only the $f_{0}$ contours of a sentence. This is significant because
training a model only on $f_{0}$ contours of speech and sung utterances would
require significantly less data than a full audio model. Second, finding
balanced transforming and non-transforming illusion pairs is a hard problem
(?, ?). Our theoretical account could potentially allow for the development of
automated search algorithms for such pairs given suitable generative models.
Third, being able to switch between generative models that are trained on
music and speech corpora from different cultures could help develop principled
hypotheses about the cross-cultural variation of this phenomenon (?, ?, ?).
For example, would it be possible to find phrases that transform in one
language but not in another? We hope to engage with these ideas in future
work.

The speech-to-song illusion will continue to fascinate listeners for decades
to come. We believe that formalizing it as rational statistical inference
whereby a perceived object provides sufficient evidence for an alternative,
unconventional hypothesis serves as an exciting new step towards a
computational theory of perceptual illusions.

## Acknowledgments

This work was supported by Microsoft Azure credits supplied to Princeton and
by a Microsoft Foundation Models grant to TLG. The authors declare no
competing interests.

## References

  * Achiam et al.Achiam et al. Achiam, J., Adler, S., Agarwal, S., Ahmad, L., Akkaya, I., Aleman, F. L., … others  (2023).  GPT-4 technical report.  _arXiv preprint arXiv:2303.08774_.
  * Albouy, Benjamin, Morillon, ZatorreAlbouy et al. Albouy, P., Benjamin, L., Morillon, B., Zatorre, R. J.  (2020).  Distinct sensitivity to spectrotemporal modulation supports brain asymmetry for speech and melody.  _Science_ , _367_(6481), 1043–1047.
  * Albouy, Mehr, Hoyer, Ginzburg, ZatorreAlbouy et al. Albouy, P., Mehr, S. A., Hoyer, R. S., Ginzburg, J., Zatorre, R. J.  (2023).  Spectro-temporal acoustical markers differentiate speech from song across cultures.  _bioRxiv_ , 2023–01.
  * AndersonAnderson Anderson, J. R.  (1990).  _The adaptive character of thought_.  Psychology Press.
  * Bird, Klein, LoperBird et al. Bird, S., Klein, E., Loper, E.  (2009).  _Natural language processing with python: analyzing text with the natural language toolkit_.  O’Reilly.
  * DaviesDavies Davies, M.  (2010).  The corpus of contemporary american english as the first reliable monitor corpus of English.  _Literary and linguistic computing_ , _25_(4), 447–464.
  * DeutschDeutsch Deutsch, D.  (2003).  _Phantom words and other curiosities_.  Philomel Records.
  * DeutschDeutsch Deutsch, D.  (2019).  _Musical illusions and phantom words: How music and speech unlock mysteries of the brain_.  Oxford University Press.
  * Deutsch, Henthorn, LapidisDeutsch et al. Deutsch, D., Henthorn, T., Lapidis, R.  (2011).  Illusory transformation from speech to song.  _The Journal of the Acoustical Society of America_ , _129_(4), 2245–2252.
  * Ding et al.Ding et al. Ding, N., Patel, A. D., Chen, L., Butler, H., Luo, C., Poeppel, D.  (2017).  Temporal modulations in speech and music.  _Neuroscience & Biobehavioral Reviews_, _81_ , 181–187.
  * Falk, Rathcke, Dalla BellaFalk et al. Falk, S., Rathcke, T., Dalla Bella, S.  (2014).  When speech sounds like music.  _Journal of Experimental Psychology: Human Perception and Performance_ , _40_(4), 1491.
  * Griffiths, Steyvers, TenenbaumGriffiths et al. Griffiths, T. L., Steyvers, M., Tenenbaum, J. B.  (2007).  Topics in semantic representation.  _Psychological Review_ , _114_(2), 211.
  * Griffiths TenenbaumGriffiths Tenenbaum Griffiths, T. L., Tenenbaum, J. B.  (2007).  From mere coincidences to meaningful discoveries.  _Cognition_ , _103_(2), 180–226.
  * HerderHerder Herder, J. G. v.  (2002).  Treatise on the origin of language (1772).  In M. N. Forster (Ed.), _Herder: Philosophical writings_ (p. 65–164).  Cambridge University Press.
  * Huang et al.Huang et al. Huang, C.-Z. A., Vaswani, A., Uszkoreit, J., Shazeer, N., Hawthorne, C., Dai, A. M., … Eck, D.  (2018).  Music transformer: Generating music with long-term structure.  _arXiv preprint arXiv:1809.04281_.
  * Jaisin, Suphanchaimat, Figueroa Candia, WarrenJaisin et al. Jaisin, K., Suphanchaimat, R., Figueroa Candia, M. A., Warren, J. D.  (2016).  The speech-to-song illusion is reduced in speakers of tonal (vs. non-tonal) languages.  _Frontiers in Psychology_ , _7_ , 662.
  * Joulin et al.Joulin et al. Joulin, A., Grave, E., Bojanowski, P., Douze, M., Jégou, H., Mikolov, T.  (2016).  Fasttext. zip: Compressing text classification models.  _arXiv preprint arXiv:1612.03651_.
  * Kersten YuilleKersten Yuille Kersten, D., Yuille, A.  (2003).  Bayesian models of object perception.  _Current opinion in neurobiology_ , _13_(2), 150–158.
  * Kherwa BansalKherwa Bansal Kherwa, P., Bansal, P.  (2019).  Topic modeling: a comprehensive review.  _EAI Endorsed transactions on scalable information systems_ , _7_(24).
  * Margulis Simchy-GrossMargulis Simchy-Gross Margulis, E. H., Simchy-Gross, R.  (2016).  Repetition enhances the musicality of randomly generated tone sequences.  _Music Perception: An Interdisciplinary Journal_ , _33_(4), 509–514.
  * Margulis, Simchy-Gross, BlackMargulis et al. Margulis, E. H., Simchy-Gross, R., Black, J. L.  (2015).  Pronunciation difficulty, temporal regularity, and the speech-to-song illusion.  _Frontiers in Psychology_ , _6_ , 48.
  * Ozaki et al.Ozaki et al. Ozaki, Y., Tierney, A., Pfordresher, P., Mcbride, J., Benetos, E., Proutskova, P., … others  (2022).  Globally, songs and instrumental melodies are slower, higher, and use more stable pitches than speech [stage 2 registered report].
  * RousseauRousseau Rousseau, J.-J.  (1782).  _Collection complète des oeuvres de jj rousseau, citoyen de genève_ (Vol. 3).
  * Rowland, Kasdan, PoeppelRowland et al. Rowland, J., Kasdan, A., Poeppel, D.  (2019).  There is music in repetition: Looped segments of speech and nonspeech induce the perception of music in a time-dependent manner.  _Psychonomic Bulletin & Review_, _26_ , 583–590.
  * Simchy-Gross MargulisSimchy-Gross Margulis Simchy-Gross, R., Margulis, E. H.  (2018).  The sound-to-music illusion: Repetition can musicalize nonspeech sounds.  _Music & Science_, _1_ , 2059204317731992.
  * Tierney, Dick, Deutsch, SerenoTierney et al. Tierney, A., Dick, F., Deutsch, D., Sereno, M.  (2013).  Speech versus song: multiple pitch-sensitive areas revealed by a naturally occurring musical illusion.  _Cerebral Cortex_ , _23_(2), 249–254.
  * Tierney, Patel, BreenTierney et al. Tierney, A., Patel, A. D., Breen, M.  (2018a).  Acoustic foundations of the speech-to-song illusion.  _Journal of Experimental Psychology: General_ , _147_(6), 888.
  * Tierney, Patel, BreenTierney et al. Tierney, A., Patel, A. D., Breen, M.  (2018b).  Repetition enhances the musicality of speech and tone stimuli to similar degrees.  _Music Perception: An Interdisciplinary Journal_ , _35_(5), 573–578.
  * Touvron et al.Touvron et al. Touvron, H., Martin, L., Stone, K., Albert, P., Almahairi, A., Babaei, Y., … others  (2023).  Llama 2: Open foundation and fine-tuned chat models.  _arXiv preprint arXiv:2307.09288_.
  * Vanden Bosch der Nederlanden, Hannon, SnyderVanden Bosch der Nederlanden et al. Vanden Bosch der Nederlanden, C. M., Hannon, E. E., Snyder, J. S.  (2015).  Everyday musical experience is sufficient to perceive the speech-to-song illusion.  _Journal of experimental psychology: General_ , _144_(2), e43.
  * Zatorre, Belin, PenhuneZatorre et al. Zatorre, R. J., Belin, P., Penhune, V. B.  (2002).  Structure and function of auditory cortex: music and speech.  _Trends in Cognitive Sciences_ , _6_(1), 37–46.