# Kirkwood-Dirac nonclassicality, support uncertainty and complete
incompatibility

Stephan De Bièvre1 1Univ. Lille, CNRS, UMR 8524, Inria - Laboratoire Paul
Painlevé, F-59000 Lille, France

###### Abstract

Given two orthonormal bases ${\mathcal{A}}=\\{|a_{i}\rangle\\}$ and
${\mathcal{B}}=\\{|b_{j}\rangle\\}$ in a $d$-dimensional Hilbert space
${\mathcal{H}}$, one associates to each state its Kirkwood-Dirac (KD) quasi-
probability distribution. KD-nonclassical states – for which the KD-
distribution takes on negative and/or nonreal values – have been shown to
provide a quantum advantage in quantum metrology and information, raising the
question of their identification. Under suitable conditions of incompatibility
between the two bases, we provide sharp lower bounds on the support
uncertainty of states that guarantee their KD-nonclassicality. In particular,
when the bases are _completely incompatible_ , a notion we introduce, states
whose support uncertainty is not equal to its minimal value $d+1$ are
necessarily KD-nonclassical. The implications of these general results for
various commonly used bases, including the mutually unbiased ones, and their
perturbations, are detailed.

###### pacs:

vvv

##  I Introduction

The nonclassical features of quantum mechanical states can be of a very
diverse nature. Incompatible, noncommuting and complementary observables,
(de)coherence, interference, uncertainty principles, negativity or non-reality
of quasi-probability distributions, entanglement, noncontextuality and
nonlocality constitute a non-exhaustive list of concepts used to evaluate the
degree to which quantum states of a variety of physical systems may or may not
exhibit manifestly nonclassical behaviour in various experimental situations.
Partially in order to obtain a better understanding of quantum mechanics and
partially because such nonclassical behaviour has proven essential for a
number of tasks in quantum information theory and metrology, the study of
their properties and of the relationship between them attracts continued
attention Johansen (2007); Spekkens (2008); Heinosaari and Wolf (2010); Ferrie
(2011); Lundeen and Bamber (2012); Bamber and Lundeen (2014); Pusey (2014);
Dressel (2015); Thekkadath  _et al._ (2016); Heinosaari  _et al._ (2016);
Yunger Halpern  _et al._ (2018); Lostaglio (2018); Designolle  _et al._
(2019); Arvidsson-Shukur  _et al._ (2020a, b); Carmeli  _et al._ (2020); Uola
_et al._ (2021). This is in particular so when the Hilbert space of states
${\mathcal{H}}$ is finite dimensional, as for systems of qudits or qubits,
which is our focus here.

Given orthonormal bases ${\mathcal{A}}=\\{|a_{i}\rangle\\}$ and
${\mathcal{B}}=\\{|b_{j}\rangle\\}$, the Kirkwood-Dirac (KD) distribution of a
state $\psi$ Kirkwood (1933); Dirac (1945) is the quasi-probability
distribution

$Q(\psi)_{ij}=\langle a_{i}|\psi\rangle\langle\psi|b_{j}\rangle\langle
b_{j}|a_{i}\rangle,1\leq i,j\leq d,$ (1)

similar in spirit to the Wigner distribution Cahill and Glauber (1969a, b) in
continuous variable quantum mechanics. It is complex-valued and satisfies
$\sum_{ij}Q(\psi)_{ij}=1,$ with marginals $\sum_{j}Q(\psi)_{ij}=|\langle
a_{i}|\psi\rangle|^{2},\sum_{i}Q(\psi)_{ij}=|\langle b_{j}|\psi\rangle|^{2}.$
A state $\psi\in{\mathcal{H}}$ is _KD-classical_ if its KD-distribution is
real nonnegative everywhere so that its KD-distribution is a probability
distribution. If not, it is _KD-nonclassical_. KD-nonclassicality is used in
quantum tomography Lundeen and Bamber (2012); Bamber and Lundeen (2014);
Thekkadath  _et al._ (2016) as well as in the theory and applications of weak
measurements, (non)contextuality, and their relation to nonclassical effects
in quantum mechanics Spekkens (2008); Dressel (2015); Pusey (2014); Lostaglio
(2018). Also, KD-nonclassicality provides an operational quantum advantage in
postselected metrology Arvidsson-Shukur  _et al._ (2020a).

This poses the question how to ensure the prevalence of KD-nonclassical
states? The KD-distribution and hence the KD-nonclassicality of $\psi$ depend
not only on $\psi$, but also on ${\mathcal{A}}$ and ${\mathcal{B}}$. In
applications their choice is linked to the identification of two observables
$A$ and $B$ of which they are eigenbases. The questions are therefore: under
what conditions on ${\mathcal{A}}$ and ${\mathcal{B}}$ are most states KD-
nonclassical and where in ${\mathcal{H}}$ are those states located?

A form of incompatibility or of noncommutativity is needed between the
projectors $|a_{i}\rangle\langle a_{i}|$ and $|b_{j}\rangle\langle b_{j}|$ for
KD-nonclassical states to exist. Indeed, if for example $A$ and $B$ have
nondegenerate spectra, and if they are compatible in the usual sense that
$[A,B]=0$, then all those projectors commute, and therefore each
$|a_{i}\rangle$ is up to a phase equal to some $|b_{j}\rangle$. It is then
immediate from (1) that $Q_{ij}(\psi)\geq 0$ for all $\psi$ so that there are
no KD-nonclassical states in ${\mathcal{H}}$. Under the assumption that $A$
and $B$ are incompatible in the sense that they do not commute, a sufficient
but non-optimal condition for a state to be KD-nonclassical was given in
Arvidsson-Shukur  _et al._ (2020b). In the present paper, we sharpen and
optimize that result in several ways. We start our analysis from the
observation that the above notion of incompatibility, and others close to it
Heinosaari and Wolf (2010); Heinosaari  _et al._ (2016); Designolle  _et al._
(2019); Carmeli  _et al._ (2020); Uola  _et al._ (2021), is weak, since it is
based on the negation of a very strong notion of compatibility, namely that
$A$ and $B$ commute. We introduce two increasingly restrictive notions of
incompatibility, _strong incompatibility_ and _complete incompatibility_ , not
obtained by negating a form of commutativity, but emanating directly from an
analysis of incompatible successive measurements.

We show that under these hypotheses, the KD-(non)classical states can be
identified through their _support uncertainty_ , a measure of their
uncertainty with respect to their ${\mathcal{A}}$\- and
${\mathcal{B}}$-representations [See Eq. (2)]. We illustrate the power of
these general results by analyzing the KD-nonclassicality for spin bases, for
mutually unbiased bases (MUBs), including the discrete Fourier transform
(DFT), and their perturbations.

##  II Strongly incompatible bases

Let $U$ be the unitary transition operator between ${\mathcal{A}}$ and
${\mathcal{B}}$, defined as $U|a_{j}\rangle=|b_{j}\rangle$, with matrix
elements $U_{ij}=\langle a_{i}|b_{j}\rangle$ in ${\mathcal{A}}$. Let
$m_{{\mathcal{A}},{\mathcal{B}}}^{2}=\min_{i,j}|\langle
a_{i}|b_{j}\rangle|^{2}$,
$M_{{\mathcal{A}},{\mathcal{B}}}^{2}=\max_{i,j}|\langle
a_{i}|b_{j}\rangle|^{2}$, then

$0\leq m_{{\mathcal{A}},{\mathcal{B}}}\leq d^{-1/2}\leq
M_{{\mathcal{A}},{\mathcal{B}}}\leq 1.$

It is easily seen that ${\mathcal{A}}$ and ${\mathcal{B}}$ are incompatible in
the sense of “noncommuting” iff there exist $1\leq i,j\leq d$ so that
$0<|\langle a_{i}|b_{j}\rangle|^{2}<1$. Hence, if
$M_{{\mathcal{A}},{\mathcal{B}}}<1$, the bases are incompatible. The converse
is not true; however, we note that $M_{{\mathcal{A}},{\mathcal{B}}}=1$ if and
only if there exist $k\geq 1$ states in ${\mathcal{A}}$ that are equal, up to
a phase, to states in ${\mathcal{B}}$. Reordering the bases, we have
$|a_{i}\rangle=|b_{i}\rangle$ for $i\in\llbracket 1,k\rrbracket$. The KD-
distribution then satisfies $Q_{ij}=|\langle
a_{i}|\psi\rangle|^{2}\delta_{i,j}$ for $i,j\in\llbracket 1,k\rrbracket$. It
is moreover block-diagonal and one can therefore restrict the KD-
nonclassicality analysis to the $(d-k)$-dimensional Hilbert subspace
orthogonal to the common basis vectors. Without loss of generality, we
therefore assume that $M_{{\mathcal{A}},{\mathcal{B}}}<1$.

######  Definition 1.

We say ${\mathcal{A}}$ and ${\mathcal{B}}$ are _strongly incompatible
(STROINC) bases_ if, for all $1\leq i,j\leq d$, $\langle
a_{i}|b_{j}\rangle\not=0$ or, equivalently, if
$0<m_{{\mathcal{A}},{\mathcal{B}}}^{2}\leq|\langle a_{i}|b_{j}\rangle|^{2}\leq
M_{{\mathcal{A}},{\mathcal{B}}}^{2}<1.$

Strong incompatibility is easily checked to be equivalent to the requirement
that _none_ of the projectors $|a_{i}\rangle\langle a_{i}|$ commutes with
_any_ of the $|b_{j}\rangle\langle b_{j}|$. It is therefore stronger than the
usual notion of incompatibility, which only requires that at least one pair of
those projectors does not commute. For example, consider an integer spin $s$.
Then it is easily checked using the explicit expression for the Wigner
rotation matrices J.J. (1994) that

$m_{J_{x}J_{z}}:=\min_{m,m^{\prime}}|\langle
m_{x}=m|m_{z}=m^{\prime}\rangle|=0,$

where $J_{x},J_{y},J_{z}$ are the spin operators and, for
$m\in\llbracket-s,s\rrbracket$,
$J_{\ell}|m_{\ell}=m\rangle=m|m_{\ell}=m\rangle$. (See Appendix D.) For
example, when $s=1$, $\langle m_{z}=0|m_{x}=0\rangle=0$. Hence, $J_{x}$ and
$J_{z}$ do not commute, but they are not strongly incompatible.

The condition $m_{{\mathcal{A}},{\mathcal{B}}}>0$ means that, if the system is
in the state $|a_{i}\rangle$, and a measurement in the ${\mathcal{B}}$-basis
is made, then any of the post-measurement states $|b_{j}\rangle$ occurs with a
nonvanishing probability $|\langle a_{i}|b_{j}\rangle|^{2}\geq
m_{{\mathcal{A}},{\mathcal{B}}}^{2}$. The same holds with the roles of
${\mathcal{A}}$ and ${\mathcal{B}}$ reversed. Consequently, the larger
$m_{{\mathcal{A}},{\mathcal{B}}}$, the greater the uncertainty in the
measurement outcomes. Hence a larger value of
$m_{{\mathcal{A}},{\mathcal{B}}}$ points to a stronger incompatibility. The
uncertainty is maximal when $m_{{\mathcal{A}},{\mathcal{B}}}^{2}=d^{-1}$, in
which case one easily sees $|\langle a_{i}|b_{j}\rangle|^{2}=d^{-1}$ for all
$i,j$ so that all the aforementioned measurement outcomes are equally
probable. In view of this observation such bases can be considered maximally
incompatible. They are known as mutually unbiased bases (MUBs) and have found
numerous applications in various quantum information protocols Ivonovic
(1981); Cerf  _et al._ (2002); Lundeen  _et al._ (2011); Lundeen and Bamber
(2012); Aguilar  _et al._ (2018); Farkas and Kaniewski (2019). We refer to
Planat  _et al._ (2006); Durt  _et al._ (2010) for reviews on MUBs. We can
conclude that strong incompatibility naturally interpolates between mere
incompatibility and mutual unbiasedness with the parameter
$0<m_{{\mathcal{A}},{\mathcal{B}}}\leq d^{-1/2}$ providing a measure of the
strength of the incompatibility.

##  III Support uncertainty

To decide if $\psi$ is KD-nonclassical with respect to two STROINC bases, we
use its support uncertainty $n_{{\mathcal{A}},{\mathcal{B}}}(\psi)$, a notion
of uncertainty that has proven useful in various contexts previously Donoho
and Stark (1989); Matusiak  _et al._ (2004); Tao (2005); Ghobber and Jaming
(2011); Wigderson and Wigderson (2021):

$n_{{\mathcal{A}},{\mathcal{B}}}(\psi):=n_{{\mathcal{A}}}(\psi)+n_{{\mathcal{B}}}(\psi).$
(2)

Here $n_{\mathcal{A}}(\psi)$ (respectively $n_{\mathcal{B}}(\psi)$) is the
number of nonvanishing $\langle a_{i}|\psi\rangle$ (respectively $\langle
b_{j}|\psi\rangle$). One should think of $n_{\mathcal{A}}(\psi)$ as the size
of the support or the “spread” of the probability distributions $|\langle
a_{i}|\psi\rangle|^{2}$ and $\langle b_{j}|\psi\rangle|^{2}$ of the state
$\psi$ in the ${\mathcal{A}}$\- and ${\mathcal{B}}$-representations, which is
one possible measure of their uncertainty. Many other such measures exist,
notably the entropic ones Maassen and Uffink (1988); Coles  _et al._ (2017).
More precisely, we introduce the ${\mathcal{A}}$-support and
${\mathcal{B}}$-support of $\psi$: $S_{\psi}=\\{i\in\llbracket
1,d\rrbracket\mid\langle a_{i}|\psi\rangle\not=0\\}$ and
$T_{\psi}=\\{j\in\llbracket 1,d\rrbracket\mid\langle
b_{j}|\psi\rangle\not=0\\}.$ Then

$n_{\mathcal{A}}(\psi)=|S_{\psi}|\not=0,\
n_{\mathcal{B}}(\psi)=|T_{\psi}|\not=0;$ (3)

here $|S|$ denotes the number of elements in $S$. Clearly,
$n_{{\mathcal{A}},{\mathcal{B}}}(\psi)$ is a global measure of the uncertainty
inherent in $\psi$, with respect to these two representations. We also
introduce the _minimal support uncertainty_
$n_{{\mathcal{A}},{\mathcal{B}}}^{\min}$ of the bases
${\mathcal{A}},{\mathcal{B}}$ as
$n_{{\mathcal{A}},{\mathcal{B}}}^{\min}=\min_{\psi\not=0}n_{{\mathcal{A}},{\mathcal{B}}}(\psi).$
We call the collection of all points $(n_{\mathcal{A}},n_{\mathcal{B}})$ in
the $n_{\mathcal{A}}$-$n_{\mathcal{B}}$ plane for which there is a $\psi$ so
that $n_{\mathcal{A}}(\psi)=n_{\mathcal{A}}$ and
$n_{\mathcal{B}}(\psi)=n_{\mathcal{B}}$ the uncertainty diagram of the bases;
see Fig. 1. To delimit the uncertainty diagram from below, we use an
uncertainty principle originally shown for the Fourier transform on finite
groups Donoho and Stark (1989), but which has much larger validity Ghobber and
Jaming (2011); Wigderson and Wigderson (2021). It reads

$n_{\mathcal{A}}(\psi)n_{\mathcal{B}}(\psi)\geq
M_{{\mathcal{A}},{\mathcal{B}}}^{-2}.$ (4)

This follows immediately from

$\displaystyle 1$ $\displaystyle=$
$\displaystyle|\sum_{ij}Q_{ij}|\leq\sum_{ij}|Q_{ij}|\leq
M_{{\mathcal{A}},{\mathcal{B}}}\sum_{ij}|\langle a_{i}|\psi\rangle||\langle
b_{j}|\psi\rangle|$ (5) $\displaystyle\leq$ $\displaystyle
M_{{\mathcal{A}},{\mathcal{B}}}(\sum_{{}_{i\in S_{\psi}}}|\langle
a_{i}|\psi\rangle|)(\sum_{j\in T_{\psi}}|\langle b_{j}|\psi\rangle|)$
$\displaystyle\leq$ $\displaystyle
M_{{\mathcal{A}},{\mathcal{B}}}\sqrt{n_{\mathcal{A}}(\psi)n_{\mathcal{B}}(\psi)}.$

Figure 1: Uncertainty diagrams. Dashed curve:
$n_{\mathcal{A}}(\psi)n_{\mathcal{B}}(\psi)=M_{{\mathcal{A}},{\mathcal{B}}}^{-2}$.
Dot-dashed line: $n_{\mathcal{A}}(\psi)+n_{\mathcal{B}}(\psi)=d+1$. Diamonds
(blue): KD-nonclassical states. Squares (red): KD-classical states. (a)
Complex MUBs; $d=4$, $M_{{\mathcal{A}},{\mathcal{B}}}^{-2}=4$,
$n_{{\mathcal{A}},{\mathcal{B}}}^{\min}=4<5$. (b) Perturbed MUB as in Eqn.
(7); $d=4$, $\epsilon=0.1$, $M_{{\mathcal{A}},{\mathcal{B}}}^{-2}=2.97<4$,
$n_{{\mathcal{A}},{\mathcal{B}}}^{\min}=5$. (c) DFT; $d=5$,
$M_{{\mathcal{A}},{\mathcal{B}}}^{-2}=5$,
$n_{{\mathcal{A}},{\mathcal{B}}}^{\min}=6$. (d) DFT, $d=6$,
$M_{{\mathcal{A}},{\mathcal{B}}}^{-2}=6$,
$n_{{\mathcal{A}},{\mathcal{B}}}^{\min}=5<7$. (e) Spin 2 transition matrix;
$d=5$, $M_{{\mathcal{A}},{\mathcal{B}}}^{-2}=8/3<5$,
$n_{{\mathcal{A}},{\mathcal{B}}}^{\min}=4<6$. Hexagons (magenta): KD-classical
and KD-nonclassical states.

It follows the entire uncertainty diagram lies above or on the hyperbola
$n_{\mathcal{A}}n_{\mathcal{B}}=M_{{\mathcal{A}},{\mathcal{B}}}^{-2}$. Eqn.
(4) implies that

$n_{{\mathcal{A}},{\mathcal{B}}}^{\min}\geq\frac{2}{M_{{\mathcal{A}},{\mathcal{B}}}}.$
(6)

Eqn. (4) yields more information when $M_{{\mathcal{A}},{\mathcal{B}}}$ is
small than when $M_{{\mathcal{A}},{\mathcal{B}}}$ is large. In the case of
MUBs, Eqns. (4) and (6) become $n_{\mathcal{A}}(\psi)n_{\mathcal{B}}(\psi)\geq
d$, $n_{{\mathcal{A}},{\mathcal{B}}}^{\min}\geq 2\sqrt{d}$. Fig. 1 (a)-(d)
shows representative examples of uncertainty diagrams of STROINC bases, on
which the KD-(non)classical nature of the states has also been indicated. [For
the details of the computations, see below and the Appendices.] In dimension
$4$, all MUBs are known Banica (2019). They all display the same uncertainty
diagram, shown in Fig. 1-(a). Fig. 1-(b) shows the uncertainty diagram for a
perturbed MUB matrix $U$ of the form

$U(\epsilon)=\exp(-i\epsilon L)U,$ (7)

where $L$ is self-adjoint. In the figure, $L_{jk}=-L_{kj}=i$, $1\leq j<k\leq
d$. Fig 1-(c)&(d) concern bases with transition matrix
$U_{\textrm{DFT},i,j}:=\langle
a_{i}|b_{j}\rangle=\frac{1}{\sqrt{d}}\exp(i\frac{2\pi}{d}{ij}),$ the discrete
Fourier transform (DFT) in dimensions $d=5$ and $d=6$. Fig. 1-(e) shows the
uncertainty diagram for the eigenbases of $J_{z}$ and $J_{x}$ for a spin $2$.
As indicated above, these bases are not STROINC. One notes the presence of
numerous classical states in the uncertainty diagram, a phenomenon we shall
explain.

Our results below show that the observed location of the KD-(non)classical
states in these and other uncertainty diagrams can be predicted from the
incompatibility properties of the bases used and the uncertainty properties of
the states considered.

##  IV Characterizing KD-nonclassicality

One observes in all panels of Fig. 1 that there are no KD-classical states
above the _nonclassicality edge_ , by which we mean the line segment in the
first quadrant defined by $n_{{\mathcal{A}}}+n_{{\mathcal{B}}}=d+1$. This is
explained by the following theorem.

######  Theorem 2.

Let ${\mathcal{A}},{\mathcal{B}}$ be STROINCs on a $d$-dimensional Hilbert
space ${\mathcal{H}}$. Then, if $\psi\in{\mathcal{H}}$ satisfies

$n_{{\mathcal{A}},{\mathcal{B}}}(\psi)>d+1,$ (8)

then $\psi$ is KD-nonclassical. Equivalently, if $\psi$ is KD-classical then
$n_{{\mathcal{A}},{\mathcal{B}}}(\psi)\leq d+1$.

The proof, which sharpens an argument in Arvidsson-Shukur  _et al._ (2020b),
is given in Appendix A. The lower bound in (8) is optimal in the sense that
the support uncertainty of the basis vectors of STROINCs, which are all KD-
classical, equals $d+1$. Nevertheless, while both optimal and sufficient for
KD-nonclassicality, (8) is not necessary: as can be observed in Fig. 1
(a)-(d), KD-nonclassical states may occur below and on the nonclassicality
edge. While the transition matrix between the $J_{z}$ and $J_{x}$ bases for a
spin $2$ is not STROINC since it contains zeroes, it will be shown elsewhere
De Bièvre (2021) that the conclusion of the theorem holds under weaker
conditions than strong incompatibility that are satisfied in this case. This
explains the absence of KD-classical states above the nonclassicality edge in
Fig. 1-(e).

In Arvidsson-Shukur  _et al._ (2020b), it was shown that, if none of the
$|a_{i}\rangle$ are equal (up to a phase) to any of the $|b_{j}\rangle$, then
the condition

$n_{\mathcal{A}}(\psi)+n_{\mathcal{B}}(\psi)>\lfloor{3d/2}\rfloor$ (9)

implies $\psi$ is KD-nonclassical; here $\lfloor x\rfloor$ is the integer part
of $x$. While this estimate holds under weaker conditions on the overlaps
$\langle a_{i}|b_{j}\rangle$ than $m_{{\mathcal{A}},{\mathcal{B}}}>0$, it is
not optimal as for example when the bases are STROINC and if $d\geq 4$, since
then $\lfloor 3d/2\rfloor>d+1$; the difference between the lower bounds is
increasingly pronounced for larger $d$. For example, for $d=6$ (see Fig. 1
(d)), our bound shows all states with $n_{{\mathcal{A}},{\mathcal{B}}}\geq 8$
are KD-nonclassical, while the bound of Arvidsson-Shukur  _et al._ (2020b)
only guarantees this if $n_{{\mathcal{A}},{\mathcal{B}}}\geq 10$.

The theorem provides an upper bound to the support uncertainty of KD-classical
states. As such, it is interesting to compare the situation with the one
familiar from quantum mechanics and quantum optics where the “classical”
states are the coherent states. They can be characterized as those that
minimize the total noise $(\Delta Q)^{2}+(\Delta P)^{2}$, a known measure of
uncertainty for pure states Hillery (1989); Yadin  _et al._ (2018); De Bièvre
_et al._ (2019); Kwon  _et al._ (2019). Here $Q$ and $P$ are two conjugate
observables that satisfy the canonical commutation relation $[Q,P]=i$ which
expresses the very strong sense in which they fail to commute, and – in this
sense – their very strong incompatibility. The question then naturally arises
in the discrete setting as well under which condition on the bases
${\mathcal{A}}$ and ${\mathcal{B}}$ the support uncertainty of all KD-
classical states equals the minimal support uncertainty
$n_{{\mathcal{A}},{\mathcal{B}}}^{\min}$? Note that this is the case in Fig.
1-(b)&(c), but _not_ in Fig. 1-(a), (d)&(e). We show below that a sufficient
condition for this to happen is that the two bases are _completely
incompatible_ , a notion we now introduce.

##  V Complete incompatibility

###  V.1 Completely incompatible bases: definition

Let ${\mathcal{A}}$ be as above and define, for any index set
$S\subset\llbracket 1,d\rrbracket$, the orthogonal projector

${\Pi_{{\mathcal{A}}}}(S)=\sum_{i\in S}|a_{i}\rangle\langle a_{i}|.$

We write $\Pi_{\mathcal{A}}(S){\mathcal{H}}$ for the $|S|$-dimensional
subspace of ${\mathcal{H}}$ onto which it projects. One should think of
$\Pi_{\mathcal{A}}(S){\mathcal{H}}$ as the set of all states $\psi$ whose
${\mathcal{A}}$-support $S_{\psi}$ lies in $S$.

######  Definition 3.

We say that two bases ${\mathcal{A}}$ and ${\mathcal{B}}$ are completely
incompatible (COINC) if and only if all index sets $S,T$ in $\llbracket
1,d\rrbracket$ for which $|S|+|T|\leq d$ have the property that
${\Pi_{{\mathcal{A}}}}(S){\mathcal{H}}\cap{\Pi_{{\mathcal{B}}}}(T){\mathcal{H}}=\\{0\\}$.

As a first motivation for this definition, we point out that, while it is only
formulated in finite dimension, conjugate operators $Q$ and $P$ satisfy an
analogous property. Indeed, it is well known that there do not exist states
$\psi$ for which both the $Q$-representation $\psi(x)$ vanishes outside some
bounded set $S\subset\mathbb{R}$ and the $P$-representation $\hat{\psi}(p)$
vanishes outside some bounded set $T\subset\mathbb{R}$ Folland and Sitaram
(1997). The above definition naturally transcribes this crucial property of
conjugate operators to the finite-dimensional case. The restriction
$|S|+|T|\leq d$ is then unavoidable since, for dimensional reasons, whenever
$|S|+|T|>d$, the intersection
${\Pi_{{\mathcal{A}}}}(S){\mathcal{H}}\cap{\Pi_{{\mathcal{B}}}}(T){\mathcal{H}}$
must be nontrivial.

While the definition is purely algebraic, its physical interpretation is
readily given in terms of the quantum theory of selective projective
measurements Johansen (2007); Nielsen and Chuang (2010); Claude  _et al._
(2015). Given a basis ${\mathcal{A}}$, to every partition $S_{1},\dots,S_{L}$
of $\llbracket 1,d\rrbracket$ we associate a projective partition of unity
${\Pi_{{\mathcal{A}}}}(S_{1}),\dots\Pi_{{\mathcal{A}}}(S_{L})$. If initially
the system is in the state $\psi\in{\mathcal{H}}$ and the outcome $S_{\ell}$
is realized in a selective projective measurement then the post-measurement
state is
${\Pi_{{\mathcal{A}}}}(S_{\ell})\psi/\|{\Pi_{{\mathcal{A}}}}(S_{\ell})\psi\|$.
The probability of this outcome is
$\|{\Pi_{{\mathcal{A}}}}(S_{\ell})\psi\|^{2}$. This is referred to as a
measurement in the basis ${\mathcal{A}}$. When $L=d$ and
$S_{\ell}=\\{\ell\\}$, for $\ell=1,\dots,d$, the measurement is said to be
fine-grained. Otherwise it is coarse-grained. When $L=2$, the corresponding
measurement has only two possible outcomes and one has $S_{1}=S,S_{2}=S^{c}$,
for some $S\subset\llbracket 1,d\rrbracket$. Here $S^{c}=\llbracket
1,d\rrbracket\setminus S$, the complementary set of $S$. Note that projectors
are observables with eigenvalues $1$ and $0$. Repeated selective measurements
of $\Pi_{\mathcal{A}}(S)$ and of $\Pi_{\mathcal{B}}(T)$ on a system initially
prepared in $\psi$ systematically yield the outcome $1$ if and only if
$\Pi_{\mathcal{B}}(T)\Pi_{\mathcal{A}}(S)\psi$ belongs to
$\Pi_{\mathcal{A}}(S){\mathcal{H}}\setminus\\{0\\}$. Hence, if this occurs,
${\Pi_{{\mathcal{A}}}}(S){\mathcal{H}}\cap{\Pi_{{\mathcal{B}}}}(T){\mathcal{H}}\not=\\{0\\}$.
In other words, the definition of COINC bases (COINCs) is equivalent to the
statement that such repeated _compatible selective measurements_ cannot occur
for any $S,T$ for which $|S|+|T|\leq d$; they cannot occur for any
insufficiently coarse-grained measurements. Since, whenever $|S|+|T|>d$,
necessarily
${\Pi_{{\mathcal{A}}}}(S){\mathcal{H}}\cap{\Pi_{{\mathcal{B}}}}(T){\mathcal{H}}\not=\\{0\\}$,
sufficiently coarse-grained measurements can be compatible.

When ${\mathcal{A}},{\mathcal{B}}$ are COINC, $\langle
a_{i}|b_{j}\rangle\not=0$ for all $i,j$ and hence they are STROINC. Indeed, if
for example $\langle a_{1}|b_{1}\rangle=0$, then $|a_{1}\rangle$ belongs to
${\Pi_{{\mathcal{A}}}}(S){\mathcal{H}}\cap{\Pi_{{\mathcal{B}}}}(T){\mathcal{H}}$
for $S=\\{1\\}$ and $T=\\{2,\dots,d\\}$. But since $|T|+|S|=d$, this
contradicts the definition. Hence each basis vector $|a_{i}\rangle$ of
${\mathcal{A}}$ has full ${\mathcal{B}}$-support and
$m_{{\mathcal{A}},{\mathcal{B}}}>0$. Suppose one now introduces some
uncertainty in the pre-measurement state $\psi$ by considering a coherent
superposition of two basis vectors
$\psi=c_{1}|a_{1}\rangle+c_{2}|a_{2}\rangle$ with $c_{1}\not=0\not=c_{2}$.
Then, by choosing $c_{1},c_{2}$ to ensure $\langle b_{1}|\psi\rangle=0$ one
reduces the uncertainty on the measurement outcomes of a measurement in the
basis ${\mathcal{B}}$ in the sense that the post-measurement state can no
longer be $|b_{1}\rangle$. In other words, we can give up some of the
precision on a measurement in ${\mathcal{A}}$ in order to reduce the
uncertainty on a measurement in ${\mathcal{B}}$. Provided ${\mathcal{A}}$ and
${\mathcal{B}}$ are COINC, it then follows that $\langle
b_{j}|\psi\rangle\not=0$, for all $j\not=1$. Indeed, if for example $\langle
b_{2}|\psi\rangle=0$ as well, then
$\psi\in{\Pi_{{\mathcal{A}}}}(S){\mathcal{H}}\cap{\Pi_{{\mathcal{B}}}}(T){\mathcal{H}}$
with $S=\\{1,2\\}$ and $T=\\{3,4,\dots,d\\}$ so that
$n_{{\mathcal{A}},{\mathcal{B}}}(\psi)\leq d$, which is a contradiction. In
this sense, when the bases are COINC, the increase of information on a
measurement in ${\mathcal{B}}$ is constrained optimally by the loss of
information on a measurement in ${\mathcal{A}}$.

###  V.2 Complete incompatibility: a criterion and examples

A useful criterion for complete incompatibility is:

######  Lemma 4.

${\mathcal{A}}$ and ${\mathcal{B}}$ are COINC if and only if none of the
minors of the matrix $U$ vanishes.

Recall that a $k$-minor of $U$ is the determinant of a $k$ by $k$ submatrix of
$U$ obtained by removing $d-k$ rows and $d-k$ columns from $U$. The statement
and proof are implicit in Tao (2005); we give a straightforward argument using
linear algebra in Appendix E.

As an immediate application, one sees that, in dimensions $d=2$ and $d=3$, two
bases ${\mathcal{A}}$ and ${\mathcal{B}}$ are COINC iff they are STROINC,
_i.e._ iff $1\leq i,j\leq d$, $\langle a_{i}|b_{j}\rangle\not=0$. In dimension
$2$ this is obvious. In dimension $3$, note that each column of $U$ is a
multiple of the complex conjugate of the vector product of the two other
columns. Since the components of the vector product are minors of order $2$,
their nonvanishing follows from the nonvanishing of all matrix elements of
$U$. Hence the bases are COINC. In dimension more than three, the above is no
longer true. Indeed, it is proven in Tao (2005) that none of the minors of the
DFT transition matrix vanish if and only if the dimension $d$ is a prime
number. Lemma 4 then implies the DFT is COINC iff $d$ is a prime number. This
implies MUBs – although maximally incompatible in the sense explained above –
are not necessarily COINC. In fact, when $d=4$, no MUBs are COINC, as easily
seen from their explicit expression in Eqn. (11), and using Lemma 4.

###  V.3 Complete incompatibility, support uncertainty, and KD-
nonclassicality.

The link between complete incompatibility and support uncertainty is given by:

######  Theorem 5.

${\mathcal{A}}$ and ${\mathcal{B}}$ are COINC iff
$n_{{\mathcal{A}},{\mathcal{B}}}^{\min}=d+1$.

_Proof._ For all states $\psi$,
$\psi\in{\Pi_{{\mathcal{A}}}}(S_{\psi}){\mathcal{H}}\cap{\Pi_{{\mathcal{B}}}}(T_{\psi}){\mathcal{H}}\not=\\{0\\}$.
If the bases are COINC, this implies $|S_{\psi}|+|T_{\psi}|>d$. so
$n_{{\mathcal{A}},{\mathcal{B}}}^{\min}>d$. Since for the basis vectors we
know $n_{{\mathcal{A}},{\mathcal{B}}}(|a_{i}\rangle)=d+1$, we conclude
$n_{{\mathcal{A}},{\mathcal{B}}}^{\min}=d+1$. We prove the converse by proving
its contraposition. Suppose ${\mathcal{A}}$ and ${\mathcal{B}}$ are not COINC.
Then there exist $S,T$, with $|S|+|T|\leq d$ and
$\Pi_{\mathcal{A}}(S){\mathcal{H}}\cap\Pi_{\mathcal{B}}(T){\mathcal{H}}\not=\\{0\\}$.
Let
$0\not=\psi\in\Pi_{\mathcal{A}}(S){\mathcal{H}}\cap\Pi_{\mathcal{B}}(T){\mathcal{H}}$.
For this state
$n_{{\mathcal{A}}}(\psi)\leq|S|,n_{{\mathcal{B}}}(\psi)\leq|T|$. Hence
$n_{{\mathcal{A}},{\mathcal{B}}}(\psi)\leq d$ and
$n_{{\mathcal{A}},{\mathcal{B}}}^{\min}\leq d$. ∎
The theorem asserts that the uncertainty diagram of
${\mathcal{A}},{\mathcal{B}}$ lies above the KD-nonclassicality edge iff the
bases are COINC. This is illustrated in Fig. 1 where panels (a), (d) and (e)
show the uncertainty diagram of bases that are not COINC as follows from our
previous analysis: one does indeed observe states below the nonclassicality
edge. Panels (b) and (c) on the other hand concern COINCs.

When two bases are COINC, Theorems 2 and 5 imply that all KD-classical states
$\psi$ have minimal support uncertainty:
$n_{{\mathcal{A}},{\mathcal{B}}}(\psi)=n_{{\mathcal{A}},{\mathcal{B}}}^{\min}=d+1$
(Fig. 1-(b)&(c)). Note however that it is not true that all states with
minimal support uncertainty are KD-classical contrary to what happens with
conjugate continuous variables $Q$ and $P$. In other words, when the bases are
COINC all states, except possibly some of those with minimal support
uncertainty are KD-nonclassical.

##  VI Conclusion

It has been observed recently that KD-nonclassical states can furnish a
quantum advantage. This raises the question under what conditions on the
observables used to define the KD-distribution such KD-nonclassicality
prevails in Hilbert space? We have established that complete incompatibility –
a notion we introduce – implies only states with minimal support uncertainty
can be KD-classical, all others being KD-nonclassical. This is therefore the
optimal situation when the goal is to reduce to a minimum the presence of KD-
classical states. Our findings further imply that if two bases are mutually
unbiased as well as completely incompatible, their incompatibility mimicks
most closely the incompatibility of two conjugate observables in continuous
variable quantum mechanics. When the bases are strongly but not completely
incompatible, the support uncertainty of states still can serve as a witness
of KD-nonclassicality. A number of open questions will be explored elsewhere
De Bièvre (2021), in particular the link between strong incompatiblity and
noncommutativity. Our findings provide an improved understanding of the
general structural properties of KD-nonclassicality which we expect to be
important for the conception of experiments and protocols capable of
harnessing KD-nonclassicality for quantum information tasks.

###### Acknowledgements.

This work was supported in part by the Agence Nationale de la Recherche under
grant ANR-11-LABX-0007-01 (Labex CEMPI) and by the Nord-Pas de Calais Regional
Council and the European Regional Development Fund through the Contrat de
Projets État-Région (CPER). The author thanks David Arvidsson-Shukur for
stimulating discussions on the subject matter of this work.

APPENDICES

##  Appendix A Proof of Theorem 2

The proof follows from a refinement of the arguments in Arvidsson-Shukur  _et
al._ (2020b). We proceed by contradiction and suppose $\psi$ is KD-classical.
Since the KD-distribution is insensitive to global phase rotations
$|a_{i}\rangle\to\exp(i\phi_{i})|a_{i}\rangle,|b_{j}\rangle\to\exp(i\phi_{j}^{\prime})|b_{j}\rangle$,
we can suppose that all $\langle a_{i}|\psi\rangle$ and
$\langle\psi|b_{j}\rangle$ are nonnegative (hence real) for $1\leq i,j\leq d$.
Possibly relabeling the basis vectors, we can suppose that $\langle
a_{i}|\psi\rangle\not=0\not=\langle b_{j}|\psi\rangle$ for $1\leq i\leq
n_{\mathcal{A}}(\psi),1\leq j\leq n_{\mathcal{B}}(\psi)$ whereas all other
$\langle a_{i}|\psi\rangle$, $\langle b_{j}|\psi\rangle$ vanish. By
hypothesis, the KD-distribution of $\psi$ is real and nonnegative. Hence, for
the same range of $i$ and $j$, we can conclude $\langle a_{i}|b_{j}\rangle$ is
real and nonnegative. Since, by hypothesis, $\langle
a_{i}|b_{j}\rangle\not=0$, this implies that $\langle a_{i}|b_{j}\rangle>0$
for these values of $i$ and $j$.

Suppose now first that $n_{\mathcal{A}}(\psi)=d=n_{\mathcal{B}}(\psi)$. Then
all matrix elements $U_{ij}$ are real and nonnegative. This is in
contradiction with the fact that the columns of $U$ are orthogonal. Let us now
assume that $n_{\mathcal{B}}(\psi)\leq n_{\mathcal{A}}(\psi)<d$. Then, for
$1\leq j<j^{\prime}\leq n_{\mathcal{B}}(\psi)$, we have

$0=\langle
b_{j}|b_{j^{\prime}}\rangle=\sum_{i=1}^{n_{\mathcal{A}}(\psi)}\langle
b_{j}|a_{i}\rangle\langle
a_{i}|b_{j^{\prime}}\rangle+\sum_{i=n_{\mathcal{A}}(\psi)+1}^{d}\langle
b_{j}|a_{i}\rangle\langle a_{i}|b_{j^{\prime}}\rangle.$

From the above, we know that

$\sum_{i=1}^{n_{\mathcal{A}}(\psi)}\langle b_{j}|a_{i}\rangle\langle
a_{i}|b_{j^{\prime}}\rangle>0.$ (10)

It then follows that, for all $1\leq j<j^{\prime}\leq n_{\mathcal{B}}(\psi)$,
one has

$\sum_{i=n_{\mathcal{A}}(\psi)+1}^{d}\langle b_{j}|a_{i}\rangle\langle
a_{i}|b_{j^{\prime}}\rangle<0.$

Defining, for each $1\leq j\leq n_{\mathcal{B}}(\psi)$ the vector
$d_{j}=(\langle a_{n_{\mathcal{A}}(\psi)+1}|b_{j}\rangle,\dots,\langle
a_{d}|b_{j}\rangle)\in\mathbb{C}^{d-n_{\mathcal{A}}}$ we see from the above
that $\langle d_{j}|d_{j^{\prime}}\rangle<0$. It then follows from Lemma 6
below that

$n_{\mathcal{B}}(\psi)\leq d-n_{\mathcal{A}}(\psi)+1.$

This proves the result. The case where $n_{\mathcal{A}}(\psi)\leq
n_{\mathcal{B}}(\psi)<d$ is treated similarly, inverting the roles of the
columns and the rows. ∎

It remains to prove the following lemma, which puts an upper bound on the
number of vectors in $\mathbb{C}^{n}$ that can have an obtuse angle between
them, two by two. It is a refinement of an argument in Arvidsson-Shukur  _et
al._ (2020b).

######  Lemma 6.

Let $n,k\in\mathbb{N}_{*}$ and $v_{1},v_{2},\dots,v_{k}\in\mathbb{C}^{n}$.
Then the following holds: if $\langle v_{i}|v_{j}\rangle<0$ for all $1\leq
i<j\leq k$, then $k\leq n+1$.

###### Proof.

The proof goes by induction. For $n=1$, one may note that one can always take
$v_{1}>0$, by applying a common phase rotation to all $v_{i}\in\mathbb{C}$,
which does not change the inner products $\overline{v}_{i}v_{j}$ between them.
Hence $v_{j}<0$ for all $j\not=1$. But if $k>2$, then this contradicts the
requirement that $v_{2}v_{3}>0$. So $k\leq 2$ when $n=1$. Suppose now the
result holds for some $n\in\mathbb{N}_{*}$. We show it holds for $n+1$. Let
$v_{1},\dots,v_{k}\in\mathbb{C}^{n+1}$. As above, we can suppose
$v_{1}=a_{1}e_{1}$, $a_{1}>0$. Write $v_{j}=a_{j}e_{1}+w_{j}$, with $\langle
w_{j},e_{1}\rangle=0$, for all $j=2,\dots,k$. By hypothesis, $\langle
v_{1}|v_{j}\rangle<0$, so that all $a_{j}<0$. As a result, for all $2\leq
i<j\leq k$,

$0>\langle v_{i}|v_{j}\rangle=a_{i}a_{j}+\langle w_{i}|w_{j}\rangle.$

Hence, for all $2\leq i<j\leq k$, $\langle w_{i}|w_{j}\rangle<0$. They
therefore constitute $(k-1)$ nonvanishing vectors in $\mathbb{C}^{n}$, and
since their mutual inner products are all negative, the induction hypothesis
allows to conclude that $k-1\leq n+1$ so that $k\leq(n+1)+1$ which is the
desired result. ∎

##  Appendix B Uncertainty diagrams for the mutually unbiased bases in
dimension $4$

We will, in this section, draw up the uncertainty diagrams for all mutually
unbiased bases (MUBs) in dimension $d=4$. Before doing so, we remark that the
definitions of strong and complete incompatibility as well as of mutual
unbiasedness depend on the two bases ${\mathcal{A}}$ and ${\mathcal{B}}$ only
through the unitary transition operator between them, defined as
$U|a_{j}\rangle=|b_{j}\rangle$, with matrix elements $U_{ij}=\langle
a_{i}|b_{j}\rangle$ in the ${\mathcal{A}}$-basis. Indeed, if
${\mathcal{A}}^{\prime}$ and ${\mathcal{B}}^{\prime}$ are two other bases,
where, for some unitary operator $V$, $|a^{\prime}_{i}\rangle=V|a_{i}\rangle$
and $|b^{\prime}_{j}\rangle=V|b_{j}\rangle$, then ${\mathcal{A}}$ and
${\mathcal{B}}$ are strongly/completely incompatible or mutually unbiased if
and only if ${\mathcal{A}}^{\prime}$ and ${\mathcal{B}}^{\prime}$ are, as is
readily checked; the unitary transition matrix is the same in both cases. One
could in fact identify ${\mathcal{H}}$ with $\mathbb{C}^{d}$ and
systematically use the canonical basis of $\mathbb{C}^{d}$ as the
${\mathcal{A}}$-basis. The choice of the ${\mathcal{B}}$ basis is then
completely determined by the choice of a unitary transition matrix $U$. We
will say $U$ is strongly/completely incompatible or mutually unbiased whenever
the bases ${\mathcal{A}}$ and ${\mathcal{B}}$ are. These properties are also
not affected by a renumbering of the basis vectors, nor by global phase
changes of the basis vectors as is also readily checked.

Up to permutations of rows and columns, and global phase rotations, the
transition matrices of mutually unbiased bases (MUBs) in dimension $4$ are
known to be all of the form Banica (2019)

$U(s)=\frac{1}{2}\begin{pmatrix}1&1&1&1\\\ 1&1&-1&-1\\\ 1&-1&s&-s\\\
1&-1&-s&s\end{pmatrix},\quad|s|=1;U^{\dagger}(s)=U(\overline{s}).$ (11)

The case $s=i$ corresponds to the discrete Fourier transform (DFT) in
dimension $4$. It is clear $U$ is not completely incompatible (COINC) since it
has vanishing $2$-minors (Lemma 4). We now draw up the uncertainty diagram of
$U(s)$, for $s\not=\pm 1$. The results are shown in Fig. 1-(a).

It is readily checked that for all points
$(n_{{\mathcal{A}}},n_{{\mathcal{B}}})$ above the nonclassicality edge
$n_{{\mathcal{A}}}+n_{{\mathcal{B}}}=5$ there exists $\psi\in{\mathcal{H}}$ so
that $n_{{\mathcal{A}}}(\psi)=n_{{\mathcal{A}}}$ and
$n_{{\mathcal{B}}}(\psi)=n_{{\mathcal{B}}}$. Since Theorem 2 applies we can
conclude that all corresponding states are KD-nonclassical.

Consider then states along the nonclassicality edge
$n_{\mathcal{A}}(\psi)+n_{\mathcal{B}}(\psi)=5$. The basis vectors, with
$n_{\mathcal{A}}(\psi)=1,n_{\mathcal{B}}(\psi)=4$ or _vice versa_ , are in
this case; they are KD-classical.

Next, if $\psi$ is such that $T_{\psi}=\\{1,2\\}$, then $|S_{\psi}|\not=3$ as
is readily checked. The same is true when $T_{\psi}=\\{3,4\\}$.

Consider therefore $\psi$ so that $T_{\psi}=\\{1,3\\}$. To obtain
$|S_{\psi}|=3$, one needs $\psi=\psi_{\pm}$, where

$\displaystyle\psi_{+}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(|b_{1}\rangle+|b_{3}\rangle)$
$\displaystyle=$
$\displaystyle\frac{1}{2\sqrt{2}}\left(2|a_{1}\rangle+(1+s)|a_{3}\rangle+(1-s)|a_{4}\rangle\right)$
$\displaystyle\psi_{-}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(|b_{1}\rangle-|b_{3}\rangle)$
$\displaystyle=$
$\displaystyle\frac{1}{2\sqrt{2}}\left(2|a_{2}\rangle+(1-s)|a_{3}\rangle+(1+s)|a_{4}\rangle\right).$

Then $S_{\psi_{+}}=\\{1,3,4\\}$ and $S_{\psi_{-}}=\\{2,3,4\\}$ since
$s\not=\pm 1$. The corresponding KD-distributions are

$\displaystyle Q_{+}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\begin{pmatrix}1&0&1&0\\\ 0&0&0&0\\\
1/2(1+\overline{s})&0&1/2(1+s)&0\\\
1/2(1-\overline{s})&0&1/2(1-s)&0\end{pmatrix},$ $\displaystyle Q_{-}$
$\displaystyle=$ $\displaystyle\frac{1}{4}\begin{pmatrix}0&0&0&0\\\ 1&0&1&0\\\
1/2(1-\overline{s})&0&1/2(1-s)&0\\\
1/2(1+\overline{s})&0&1/2(1+s)&0\end{pmatrix}.$

Consequently, $\psi_{\pm}$ are both KD-nonclassical. The case where
$T_{\psi}=\\{1,4\\}$ is similar, with $s$ replaced by $-s$. The cases where
$T_{\psi}=\\{2,3\\}$ or $T=\\{2,4\\}$ and $|S_{\psi}|=3$ are obtained by
permuting the rows and/or the columns of the KD-distributions $Q_{\pm}$
obtained above. One therefore sees in this example that, along the
nonclassicality edge, there co-exist both KD-classical states and KD-
nonclassical states. The only KD-classical states are the basis vectors. Since
$U^{\dagger}(s)=U(\overline{s})$, the same results are obtained when one
switches the roles of ${\mathcal{A}}$ and ${\mathcal{B}}$.

As a result of Eq. (4) it only remains to investigate the states for which
$n_{{\mathcal{A}},{\mathcal{B}}}(\psi)=4$. This can only happen when
$n_{\mathcal{A}}(\psi)=2=n_{\mathcal{B}}(\psi)$ since $U$ has no zeros.
Straightforward calculations show such states exist and that they are all KD-
classical. The latter statement also follows from (5), which can be rewritten
as

$1\leq{\mathcal{N}}_{\mathrm{NC}}(\psi)\leq
M_{{\mathcal{A}},{\mathcal{B}}}\sqrt{n_{\mathcal{A}}(\psi)n_{\mathcal{B}}(\psi)},$
(12)

where in the present case $M_{{\mathcal{A}},{\mathcal{B}}}=\frac{1}{2}$

${\mathcal{N}}_{\mathrm{NC}}(\psi)=\sum_{i,j}|Q_{ij}|\geq 1,$ (13)

is the so-called nonclassicality of the state. Clearly, a state is KD-
nonclassical if and only if ${\mathcal{N}}_{\mathrm{NC}}(\psi)>1$. Hence Eqn.
(12) implies that if $\psi$ saturates the uncertainty principle (4), then it
is KD-classical.

When $s=\pm 1$, the situation is slightly different. There are then, for
example, no states with $|S_{\psi}|=3$ and $|T_{\psi}|=2$, as is readily
checked.

In Fig. 1-(b) we show the uncertainty diagram of the unitary $U(\epsilon)$ of
Eqn. (7), with $\epsilon=0.1$, which is a perturbation of a MUB. We checked
numerically that all its minors are nonvanishing, thereby proving it is COINC.
When $\epsilon\not=0$, it is no longer possible to obtain simple closed
analytical expressions for the various states along the nonclassicality edge.
We therefore proceeded to a numerical computation of those states and we
established they are KD-nonclassical, with the exception of the basis states
themselves.

##  Appendix C The discrete Fourier transform

Recall that the discrete Fourier transform (DFT) is

$U_{ij}=\langle a_{i}|b_{j}\rangle=\frac{1}{\sqrt{d}}\exp(i\frac{2\pi}{d}ij).$

It is convenient to choose $0\leq i,j\leq d-1$. When $d$ is prime, it follows
from Lemma 4 and Tao (2005) that the DFT is COINC. As a result,
$n_{\mathcal{A}}(\psi)+n_{\mathcal{B}}(\psi)\geq d+1$ for all $\psi$. This is
no longer true when $d$ is not prime: in that case, there do exist states for
which $n_{\mathcal{A}}(\psi)+n_{\mathcal{B}}(\psi)<d+1$ as we will see below.
This is illustrated in Fig. 1-(c)&(d).

For all dimensions, Theorem 2 implies that all states for which
$n_{\mathcal{A}}(\psi)+n_{\mathcal{B}}(\psi)>d+1$ are KD-nonclassical. It
remains therefore to investigate the KD-nonclassicality of the states with
$n_{\mathcal{A}}(\psi)+n_{\mathcal{B}}(\psi)\leq d+1$. Recall that the support
uncertainty principle guarantees that

$n_{\mathcal{A}}(\psi)n_{\mathcal{B}}(\psi)\geq d.$

When $d$ is not prime, there may therefore exist states in the region of the
$(n_{\mathcal{A}},n_{\mathcal{B}})$-plane on or above the hyperbola
$n_{\mathcal{A}}n_{\mathcal{B}}=d$ and on or below the straight line segment
$n_{{\mathcal{A}}}+n_{\mathcal{B}}=d+1$ that we refer to as the
_nonclassicality edge_. We know that all KD-classical states lie in this
region. But, as we shall illustrate, there may also be some KD-nonclassical
states there. We will investigate these issues now, one dimension at a time.

When $d=2$, and $U$ is the DFT, it follows from what precedes that all states
except the basis states are KD-nonclassical.

When $d=3$, the DFT is equivalent, after dephasing and permuting rows and
columns, to

$U=\frac{1}{\sqrt{3}}\begin{pmatrix}1&1&1\\\ 1&\omega&\omega^{2}\\\
1&\omega^{2}&\omega\end{pmatrix},\quad\textrm{where}\quad\omega=\exp(i\frac{2\pi}{3}).$
(14)

Using Lemma 4 it is easily checked to be COINC, a fact that also follows also
from the general result cited, since $3$ is prime. Straigthforward explicit
computations show that, when $|S_{\psi}|=2=|T_{\psi}|$, $\psi$ is KD-
nonclassical. As in dimension $2$, the only classical states are the basis
vectors.

The case $d=4$ was treated in the previous section ($s=i$). Note that the
classical states are now on the hyperbola $n_{\mathcal{A}}n_{\mathcal{B}}=d$
which contains one point $n_{\mathcal{A}}(\psi)=2=n_{\mathcal{B}}(\psi)$
strictly below the nonclassicality edge
$n_{\mathcal{A}}(\psi)+n_{\mathcal{B}}(\psi)=d+1=5$. So this is the lowest
dimension for which the DFT admits classical states other than the basis
states (Fig. 1-(a)).

When $d=5$ the DFT is COINC, since $5$ is a prime number. There are now no
states below the KD-nonclassicality edge $n_{\mathcal{A}}+n_{\mathcal{B}}=6$.
It remains to investigate the KD-nonclassicality for the states on the
nonclassicality edge. Apart from the basis states, which are classical, they
are of three types: $n_{\mathcal{A}}(\psi)=2$, $n_{\mathcal{B}}(\psi)=4$;
$n_{\mathcal{A}}(\psi)=3$, $n_{\mathcal{B}}(\psi)=3$;
$n_{\mathcal{A}}(\psi)=4$, $n_{\mathcal{B}}(\psi)=2$. We computed those states
numerically and showed they are nonclassical by observing numerically that
${\mathcal{N}}_{\mathrm{NC}}(\psi)>1$. (See Fig. 1-(c).)

Suppose now $d$ is not prime and $2<d=pq$, with $1<p,q<d$; the divisors $p$
and $q$ need not be prime. One can then show that that the DFT is not COINC by
noting that there exist states $\psi$ for which
$n_{\mathcal{A}}(\psi)+n_{\mathcal{B}}(\psi)\leq d$. In fact, it is easy to
construct states for which $n_{\mathcal{A}}(\psi)=q,n_{\mathcal{B}}(\psi)=p$,
so that the lower bound $n_{\mathcal{A}}(\psi)n_{\mathcal{B}}(\psi)=d$ is
reached in this case. For that purpose, consider, for $0\leq m<p,0\leq s<q$,

$|m,s\rangle=\frac{1}{\sqrt{q}}\sum_{k=0}^{q-1}\exp(i\frac{2\pi}{q}sk)|a_{kp+m}\rangle,$

which are readily checked to form an orthonormal basis. Clearly
$n_{\mathcal{A}}(|m,s\rangle)=q$. For $j=\ell q+r$, $0\leq\ell<p$, $0\leq
r<q$, one has

$\langle
b_{j}|m,s\rangle=\frac{1}{\sqrt{q}}\exp(-i\frac{2\pi}{d}mj)\delta_{sr}.$

Hence $n_{\mathcal{B}}(|m,s\rangle)=p$. It can be shown (see Matusiak  _et
al._ (2004) and references therein) that the $|m,s\rangle$ are the only states
with the property that $n_{\mathcal{A}}(\psi)n_{\mathcal{B}}(\psi)=d$. As a
result of (12), these states are all KD-classical. The smallest non-prime
dimension is $d=4$, which we already treated above. When $d=6$, the previous
development shows there are classical states with
$n_{\mathcal{A}}(\psi)=2,n_{\mathcal{B}}(\psi)=3$ and _vice versa_. We know
from Theorem 2 that all states with
$n_{\mathcal{A}}(\psi)+n_{\mathcal{B}}(\psi)>7$ are KD-nonclassical. However,
we are no longer ensured that, given a point
$(n_{\mathcal{A}},n_{\mathcal{B}})$ with $n_{\mathcal{A}}+n_{\mathcal{B}}>7$,
there exists a state with
$n_{\mathcal{A}}(\psi)=n_{\mathcal{A}},n_{\mathcal{B}}(\psi)=n_{\mathcal{B}}$
since the DFT is not COINC when $d=6$. In addition, there may now be points
$(n_{\mathcal{A}},n_{\mathcal{B}})$ below the nonclassicality edge and above
$n_{\mathcal{A}}n_{\mathcal{B}}=d$ for which such a state does exist.

We now explore these phenomena.

We first consider all states for which $n_{\mathcal{B}}(\psi)=2$. It is easy
to check that, for $0\leq k_{1}<k_{2}\leq 5$, $0\leq i_{1}<5$

$|k_{1},k_{2},i_{1}\rangle=\frac{1}{\sqrt{2}}\left(\omega^{i_{1}k_{2}}|b_{k_{1}}\rangle-\omega^{i_{1}k_{1}}|b_{k_{2}}\rangle\right)$

is the only state with the property that

$\langle a_{i_{1}}|k_{1},k_{2},i_{1}\rangle=0.$

Hence $n_{\mathcal{A}}(k_{1},k_{2},i_{1})\leq 5$. Now note that, with $0\leq
i_{1}<i_{2}\leq 5$,

$\langle a_{i_{2}}|k_{1},k_{2},i_{1}\rangle=0$

as well if and only if $(k_{2}-k_{1})(i_{2}-i_{1})=0$ modulo $6$. Hence, if
this condition is not satisfied, then
$n_{{\mathcal{A}},{\mathcal{B}}}(k_{1},k_{2},i_{1})=2+5=7$ and if it is,
$n_{{\mathcal{A}},{\mathcal{B}}}(k_{1},k_{2},i_{1})\leq 2+4=6$. We have
computed the nonclassicality of these states numerically and have observed the
states are KD-nonclassical except when
$n_{{\mathcal{A}},{\mathcal{B}}}(k_{1},k_{2},i_{1})=2+3=5$, as we saw above.
The same observations hold true when the roles of ${\mathcal{A}}$ and
${\mathcal{B}}$ are interchanged.

We have further established through a numerical computation that there are no
states for which $n_{\mathcal{A}}=3=n_{\mathcal{B}}$. For that purpose, we
constructed all states of the form

$|k_{1},k_{2},k_{3}\rangle=d_{k_{1}}|b_{k_{1}}\rangle+d_{k_{2}}|b_{k_{2}}\rangle+d_{k_{3}}|d_{k_{3}}\rangle,$

for which $\langle a_{i_{1}}|k_{1},k_{2},k_{3}\rangle=0=\langle
a_{i_{2}}|k_{1},k_{2},k_{3}\rangle=\langle
a_{i_{3}}|k_{1},k_{2},k_{3}\rangle$, with $0\leq k_{1}<k_{2}<k_{3}\leq 5$,
$0<i_{1}<i_{2}<i_{3}\leq 5$ and computed for each $n_{\mathcal{A}}$ and
$n_{\mathcal{B}}$. Clearly, for all such states, $n_{\mathcal{A}}\leq
3,n_{\mathcal{B}}\leq 3$. As it turns out, equality is achieved for none.

We similarly explored the pairs $(n_{\mathcal{A}},n_{\mathcal{B}})=(3,4)$ and
$(4,3)$. We found numerically all $\psi$ so that $n_{\mathcal{A}}(\psi)=3$,
$n_{\mathcal{B}}(\psi)=4$ and observed they are all KD-nonclassical. The
results of these computations are summarized in Fig. 1-(d).

It is of course impractical to treat the case of general $d$ in this manner.
An analytic treatment would probably involve a fair amount of number theory,
such as the techniques used to prove that for prime $d$, the DFT is COINC Tao
(2005). A question that comes to mind when looking at the uncertainty diagrams
for the dimensions $2\leq d\leq 6$ is whether it is true that the only KD-
classical states for the DFT are the ones on the hyperbola
$n_{{\mathcal{A}}}n_{{\mathcal{B}}}=d$? For prime dimensions, this would mean
the only KD-classical states are the basis states.

##  Appendix D Spin $s$

For general integral spin $s$ we show that the $J_{x}$ and $J_{z}$ bases are
not STROINC. For that purpose, we need to investigate the matrix elements

$\displaystyle\langle J_{z}=m^{\prime}|J_{x}=m\rangle$ $\displaystyle=$
$\displaystyle\langle J_{z}=m^{\prime}|\exp(-i\beta J_{y}|J_{z}=m\rangle$
$\displaystyle=$ $\displaystyle d_{m^{\prime},m}^{(s)}(\beta)$

with $\beta=\frac{\pi}{2}$ and where the matrix
$d_{m^{\prime},m}^{(s)}(\beta)$ is Wigner’s well-known little matrix J.J.
(1994). It is known that

$d_{m,m^{\prime}}^{(s)}(\beta)=(-1)^{m^{\prime}-m}d_{m^{\prime},m}^{(s)}(\beta).$

From Wigner’s formula for $d_{m^{\prime},m}^{(s)}(\pi/2)$ one readily
concludes that, for all integer $s$, and $m\geq 0$,

$d_{0,m}^{(s)}(\pi/2)=s!\sum_{k=m}^{s}(-1)^{k}\frac{\sqrt{(s+m)!(s-m)!}}{(s-k)k!(s-(k-m))!(k-m)!}.$

Making the change of variables $k^{\prime}=s+m-k$, one observes that
$d_{0,m}^{(s)}(\pi/2)=0$ for all odd $m$ when $s$ is even and for all even $m$
when $s$ is odd. Indeed, there are $s-m+1$ terms in the sum above and they
cancel two by two whenever this number is even, which means $s-m$ must be odd.

When $s=2$, the explicit form of the Wigner matrix is readily computed:

$U=\frac{1}{2}\begin{pmatrix}\frac{1}{2}&1&\sqrt{\frac{3}{2}}&1&\frac{1}{2}\\\
-1&-1&0&1&1\\\ \sqrt{\frac{3}{2}}&0&-1&0&\sqrt{\frac{3}{2}}\\\ -1&1&0&-1&1\\\
\frac{1}{2}&-1&\sqrt{\frac{3}{2}}&-1&\frac{1}{2}\end{pmatrix}.$

We constructed the uncertainty diagram of $U$ numerically, proceeding as for
the DFT in the previous section. The result is displayed in Fig. 1-(e). Note
that, along the nonclassicality edge, when
$n_{{\mathcal{A}}}=2,n_{{\mathcal{B}}}=4$ (or vice versa), there exist both
KD-classical and KD-nonclassical states in this case.

##  Appendix E Proof of Lemma 4

_Proof of_ $\Rightarrow$. We proceed by contraposition. Suppose a $k$-minor of
$U$ vanishes, for some $1\leq k\leq d-1$. Reordering the rows and columns of
$U$ (which amounts to relabeling the basis vectors), we can assume that the
matrix

$\begin{pmatrix}\langle a_{1}|b_{1}\rangle&\dots&\langle a_{1}|b_{k}\rangle\\\
\vdots&\vdots&\vdots\\\ \langle a_{k}|b_{1}\rangle&\dots&\langle
a_{k}|b_{k}\rangle\end{pmatrix}$

has vanishing determinant. Hence there exist
$(\beta_{1},\dots\beta_{k})\not=0$ so that
$|\psi\rangle=\sum_{j=1}^{k}\beta_{j}|b_{j}\rangle$ satisfies, for all
$i\in\llbracket 1,k\rrbracket$, $\langle a_{i}|\psi\rangle=0$. So
$\psi\in{\Pi_{{\mathcal{A}}}}(S){\mathcal{H}}\cap{\Pi_{{\mathcal{B}}}}(T)$
with $S=\llbracket k+1,d\rrbracket$ and $T=\llbracket 1,k\rrbracket$. Since
$|S|+|T|=d$, this implies ${\mathcal{A}}$ and ${\mathcal{B}}$ are not COINC.
_Proof of_ $\Leftarrow$. We proceed again by contraposition. Suppose
${\mathcal{A}}$ and ${\mathcal{B}}$ are not COINC. Then there exist
$S,T\subset\llbracket 1,d\rrbracket$ with $|S|+|T|\leq d$, and for which
${\Pi_{{\mathcal{A}}}}(S){\mathcal{H}}\cap{\Pi_{{\mathcal{B}}}}(T){\mathcal{H}}\not=\\{0\\}$.
Let
$0\not=\psi\in{\Pi_{{\mathcal{A}}}}(S){\mathcal{H}}\cap{\Pi_{{\mathcal{B}}}}(T){\mathcal{H}}$.
Possibly reordering the elements of ${\mathcal{B}}$, we can suppose
$T=\llbracket 1,k\rrbracket$ and write
$\psi=\sum_{j=1}^{k}\beta_{j}|b_{j}\rangle$. By hypothesis
${\Pi_{{\mathcal{A}}}}(S^{c})|\psi\rangle=0$ so that, for all $i\in S^{c}$,
$\langle a_{i}|\psi\rangle=0$. Since $|S|+|T|\leq d$, $|T|\leq|S^{c}|$.
Reordering the basis ${\mathcal{A}}$, we can assume $S^{c}=\llbracket
1,\ell\rrbracket$, with $k\leq\ell$. It follows that

$\begin{pmatrix}\langle a_{1}|b_{1}\rangle&\dots&\langle a_{1}|b_{k}\rangle\\\
\vdots&\vdots&\vdots\\\ \langle a_{\ell}|b_{1}\rangle&\dots&\langle
a_{\ell}|b_{k}\rangle\end{pmatrix}\begin{pmatrix}\beta_{1}\\\ \vdots\\\
\beta_{k}\end{pmatrix}=0.$

Since $(\beta_{1},\dots\beta_{k})\not=0$, one can extract from the $\ell$ by
$k$ matrix in the left-hand side a vanishing $k$-minor of $U$. ∎

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# The Effect of Robot Posture and Idle Motion on Spontaneous Emotional
Contagion during Robot-Human Interactions

Isabel Casso1, Bing Li1, Tatjana Nazir1, Yvonne N. Delevoye-Turrell1 1Univ.
Lille, CNRS, UMR 9193 - SCALab - Sciences Cognitives et Sciences Affectives,
F-59000 Lille, France

###### Abstract

In the next decade, social robots will be implemented in many public spaces to
provide services to humans. We question the properties of these social robots
to afford acceptance and spontaneous emotional interactions. More
specifically, in the present study, we report the effects of idle motion
frequency in a robot on emotional contagion, in a face-to-face interactive
task with a human participant.

The robotic system Buddy was programmed to adopt a sad posture and facial
expression while telling three sad stories and moving its head up/down at low,
medium, and high frequency. Each participant (N=15 total) was invited to sit
in front of Buddy and listen to the stories. Unconscious changes in posture in
the human participant were recorded using a 3D motion capture system
(Qualysis). Results show greater inclinations of the shoulder/torso towards
the ground in low-frequency trials and more rigid postures in high-frequency
trials. The quantity of spontaneous movement was also greater when Buddy moved
at slow frequencies. These findings echo results reported in experimental
psychology when two individuals are engaged in social interactions. The scores
obtained in the Godspeed questionnaire further suggest that emotional
contagion may occur when Buddy moves slowly because the robotic system is
perceived as more natural and knowledgeable, e.g., at a speed coherent with
the expressed emotion. Our work opens the question of the degree of importance
of body posture and frequency of idle motion in the conception of robotic
systems. Such additions could provide social robots that afford emotional
contagion in effortless robot-human collaborative tasks.

##  I INTRODUCTION

Humans tend to engage in spontaneous unconscious movements during social
human-human interactions. When we are on the receiver side of the interaction,
we tend to nod our head or mimic the changes in postures of the speaker. These
gross behaviors are social indicators that we are engaging attention in the
exchange. However, there are also micro changes in body sway, with slow shifts
in body postures at frequencies that echo the emotional ambiance of the social
exchange. Such idle motions have been defined as active states of
communication [1] and can offer effortless collaborative situations.

Idle motions can influence the perception of a person’s emotional state and
thus, modulate acceptance in social situations. They are related to the
intensity of energy of the speaker with frequencies that are greater in high-
energy emotions (anger and joy) than in low-energy emotions (sadness and
contentment). Applying such idle movements to artificial agents may influence
acceptance as human individuals perceive the robotic system as more animated
and alive [2][3][4].

In the present study, we wanted to test whether manipulating the frequency of
the idle motion would modify how human participants perceived a robot that was
telling a sad story about its past life experiences. We controlled the
frequency of the idle motion and measured the spontaneous body oscillations
and shifts in body posture of the human participant while she/he was attentive
to the negative content of the stories. We hypothesized that more emotion
contagion would be observed in human participants when the robot oscillated at
a low frequency as the expressed state of emotion was sad (low energy). We
also hypothesized that a frizzing state of emotion, i.e., inhibition of
spontaneous oscillations, should be observed in the human participants when
the robot moved at high idle frequencies because such situations should be
perceived as a threat from an angry robot (high energy). A total of three
stories were used to give time for the emotional contagion to take place.

Figure 1: Experimental paradigm and positioning of motion capture markers on a
participant

##  II METHODS

###  II-A Robotic system

Buddy is a social robot developed by Blue Fog robotics capable of displaying
facial expressions, generating speech, and moving independently across
environments.

###  II-B Experimental paradigm

As depicted in Figure 1, the participant was invited to sit on an ergonomic
ball and watch a fictional 1-minute video documentary about the life of Buddy,
the robot. Each participant was then instructed to listen to three negative
stories told by the robot in the first person, with sad music playing in the
background [5]. At the end of each story, participants were invited to reply
to a short series of questionnaires to self-evaluate their experience.

This study followed the ethical principles of the WMA Declaration of Helsinki
for research involving humans. Each participant signed a consent form before
inclusion and was debriefed about the aims of the study. The unique session
lasted a total of 40 minutes.

While narrating the stories, Buddy moved its head up/down in the sagittal
plane at three different frequencies: low (16s period), medium (12s period),
and high (4s period). The idle frequencies were counterbalanced across the
participants while the stories were told in the same pre-set order. A total of
15 participants were tested individually. They were college French-native
students with little experience with robot systems and were aged 22-26 years.

###  II-C Material and measures

The following measures were taken to assess the degree of emotional contagion:

  * •

Physiological measures Five infrared motion capture cameras were used to
collect the 3D coordinates of four reflective markers placed on each
participant’s shoulders, the base of the neck, and the forehead. We used the
Qualisys Track Manager (QTM) software to calibrate, record, and measure body
movements.

  * •

Questionnaires Before and after the experiment, the participant was invited to
report the valence and arousal level of their affective states using the
affect grid derived from the circumplex model of affect [6].

At the end of each trial, the participant indicated the sadness level of the
narration on a 0 to 5-point scale. The GODSPEED questionnaire was also filled
out [7], which offered the means to evaluate the perception of artificial
agents within five characteristics: anthropomorphism, likeability, animacy,
perceived intelligence, and perceived safety. Each category comprised five
semantic differential scales (e.g., Fake-Natural, Apathetic-Responsive,
Unfriendly-Friendly).

##  III Movement data analysis and results

The motion capture data was recorded at a sampling frequency of 200 Hz. Marker
velocity was extracted to compute 3D acceleration vectors. A filtering
procedure was applied to extract all absolute acceleration peaks above 1000
$mm/s^{2}$; the remaining data series were interpolated before further
analyses.

We computed the torso inclination as a function of idle frequency (high;
medium; low) for each trial and participant using the shoulder markers.

A time-series spectral analysis was then calculated by applying wavelet
transforms on the chest marker to extract body motion. As shown in Figure 2,
we grouped the graphs of the power spectrum of each story (presented in rows)
as a function of idle frequency (presented in columns), averaged across
participants. The y-axis represents the period of the center marker in seconds
as a function of the trial time, for the total story duration, in seconds. The
average power spectral density (PSD) is represented with a color bar with
brighter zones illustrating the highest power.

Figure 2: Power spectrum density of center marker averaged across participants
for each trial

Results showed that participants looked more to the ground at the end than at
the start of a trial (M=1.8 deg).

Participants did not engage in visible spontaneous torso oscillations with
Buddy (see the absence of frequencies at 4, 8, and 12s in Figure 2).
Nevertheless, the spectral analysis revealed a greater power density in the
low idle conditions and an absence of movement in the high idle condition.
This pattern of results was observed across all three stories, even if only
stories 1 and 3 are presented in Figure 2. These findings suggest that
participants tended to relax and engage in spontaneous oscillations when Buddy
moved slowly; when idle was at high frequency, participants tended to freeze
and inhibit spontaneous oscillations.

Figure 3: Mean power spectral density as a function of period interval for
high (a) and low (b) idle frequency of the robot

Figure 3 confirms that at high idle frequency, participants moved much less
than for trials at low idle frequency. Nevertheless, in the high idle
condition, movements at high frequency were more frequent than those at low
frequency, confirming the presence of emotional contagion, even if to a lesser
degree. This phenomenon resembles the posture changes reported when humans
experience sadness or angry states during human-human interactions [8].

##  IV Questionnaire analysis and results

The Perceived Safety category of the GODSPEED was taken out as Buddy is small
and harmless in appearance. For brevity, we only report the scores that
obtained significant statistical differences, i.e., the Anthropomorphism and
the Perceived intelligence categories (see Figure LABEL:fig:anth). Results are
shown as a function of idle frequency (y-axis) and the mean score (x-axis).

(a)

(b)

Figure 4: Average scores for anthropomorphism and perceived intelligence of
Buddy

An ANOVA on the GOODSPEED individual scores revealed a main idle effect for
Anthropomorphism $(p<0.001)$ and Perceived Intelligence $(p<0.001)$ [3].
Participants perceived Buddy to be more anthropomorphic and more intelligent
when it moved slowly (see Figure 4a). The analysis showed no effects of idle
frequency on Likeability and Animacy. Such findings corroborate results
reported previously in studies assessing emotional alignment during human-
robot interactions [4].

##  V DISCUSSION

In the not-so-far future, Robots will become more present in our environment.
However, we must scrutinize how the interaction with these systems will
influence us, humans [9]. In this study, we questioned whether the oscillatory
movement of a robot’s head could modulate the spontaneous engagement of a
human during face-to-face interaction. More specifically, we report how the
idle motion of Buddy’s head can modulate emotional contagion and thus, impact
a person’s affective state. In addition, our results suggest that Buddy is
perceived as more anthropomorphous and intelligent when moving slowly. Future
studies will need to test whether such findings are also found when using
positive induction methods during human-robot interactions.

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