# Measuring relational information between quantum states, and applications

Michał Oszmaniec<EMAIL_ADDRESS>Center for Theoretical Physics, Polish
Academy of Sciences, Al. Lotników 32/46, 02-668 Warsaw, Poland  Daniel J. Brod
<EMAIL_ADDRESS>Instituto de Física, Universidade Federal Fluminense,
Niterói, RJ, 24210-340, Brazil  Ernesto F. Galvão<EMAIL_ADDRESS>International Iberian Nanotechnology Laboratory (INL), Av. Mestre Jose Veiga,
4715-330, Braga, Portugal  Instituto de Física, Universidade Federal
Fluminense, Niterói, RJ, 24210-340, Brazil

###### Abstract

The geometrical arrangement of a set of quantum states can be completely
characterized using relational information only. This information is encoded
in the pairwise state overlaps, as well as in Bargmann invariants of higher
degree written as traces of products of density matrices. We describe how to
measure Bargmann invariants using suitable generalizations of the SWAP test.
This allows for a complete and robust characterization of the projective-
unitary invariant properties of any set of pure or mixed states. As
applications, we describe basis-independent tests for linear independence,
coherence, and imaginarity. We also show that Bargmann invariants can be used
to characterize multi-photon indistinguishability.

Introduction. The relative geometrical arrangement of a set of quantum states
determines properties such as the Hilbert space dimension they span or whether
they can be simultaneously diagonalized. These properties are physically
meaningful, being independent of gauge choices for the global phase of each
state’s wave-function, and invariant under application of the same unitary
transformation on all quantum states in the set. More generally, we use the
terms _relational_ or _Projective-Unitary_ (PU) invariants to refer to
properties of tuples of quantum states that are invariant under application of
the same unitary transformation to each state in the tuple. The simplest
example of a relational invariant is the two-state overlap
$\Delta_{12}=\mathrm{tr}(\rho_{1}\rho_{2})$, which for pure states reduces to
$\Delta_{12}=\left\lvert\left\langle\psi_{1}|\psi_{2}\right\rangle\right\lvert^{2}$,
with a clear operational physical interpretation.

One way to fully characterize relational information in a tuple of (generally
mixed) quantum states, $\mathcal{S}=(\rho_{i})_{i=1}^{N}$, is to perform
individual tomographic reconstruction of each state. This is both
experimentally costly and unnecessary, providing much more information than
strictly needed. A more economical approach is to directly measure a set of
invariant properties of a tuple of quantum states. We call such a set
_complete_ if it allows us to decide whether two tuples of quantum states
$\mathcal{S}=(\rho_{i})_{i=1}^{N},\mathcal{S^{\prime}}=(\sigma_{i})_{i=1}^{N}$
are unitarily equivalent, in the sense that there exists a unitary operator
$U$ taking states from $\mathcal{S}$ onto their counterparts in
$\mathcal{S}^{\prime}$: $\sigma_{i}=U\rho_{i}U^{\dagger}$. In the above, we
identified the relational properties of quantum states with equivalence
classes (orbits) of the unitary group acting on tuples of quantum states. This
approach follows the spirit of the Erlangen program in Geometry Klein (1900)
(i.e. studying geometry via the lens of group actions and their invariants)
and is justified by the fact that unitary channels constitute the most general
inevitable operations allowed in quantum theory. Note that the same paradigm
motivated the usage of invariant polynomials in the context of classification
of entanglement classes subject to local unitary transformations Leifer  _et
al._ (2004).

In this Letter we introduce quantum circuits we call cycle tests, which enable
the direct measurement of complete sets of Bargmann invariants for both mixed
and pure quantum states. We review previous mathematical literature
identifying complete sets of invariants for different scenarios. For the case
of pure states, we show all necessary invariants can be incorporated in a
single Gram matrix $G_{ij}=\left\langle\psi_{i}|\psi_{j}\right\rangle$, with
suitable gauge choices so that all parameters in $G$ are PU-invariants
measurable by our proposed circuits. This operational approach to measuring
and using Bargmann invariants results in applications such as multi-photon
indistinguishability certification, and basis-independent tests of linear
independence, imaginarity, and coherence.

We start by reviewing the definition of Bargmann invariants. Consider a tuple
of $m$ pure quantum states (understood as rank one normalized projectors)
$\Psi=(\psi_{1},\psi_{2},\ldots\psi_{m})$, where we used the shorthand
$\psi_{j}=\left|{\psi_{j}}\right\rangle\\!\left\langle{\psi_{j}}\right|$. We
later consider the generalization to mixed states. The Bargmann invariant of
this tuple of states is defined as Bargmann (1964); Chien and Waldron (2016):

$\Delta_{12\cdots
m}=\left\langle\psi_{1}|\psi_{2}\right\rangle\left\langle\psi_{2}|\psi_{3}\right\rangle\cdots\left\langle\psi_{m}|\psi_{1}\right\rangle.$
(1)

We say that this invariant has _degree_ $m$. Due to the suitable arrangement
of bras and kets, the above expression is a well-defined PU-invariant function
of states in $\Psi$. Note that, for $m=2$, this invariant reduces to the
overlap
$\Delta_{12}=\left|\left\langle\psi_{1}|{\psi_{2}}\right\rangle\right|^{2}$.
Bargmann invariants have been studied in the context of characterization of
photonic indistinguishability Menssen  _et al._ (2017); Jones  _et al._
(2020); Minke  _et al._ (2021) and geometric phases Simon and Mukunda (1993);
Dariusz Chruscinski (2004). The phase of $\Delta_{12\cdots m}$ is precisely
the Pancharatnam geometric phase Pancharatnam (1956) acquired by a
wavefunction $|{\psi_{1}}\rangle$ subjected to a sequence of projective
measurements onto states $\psi_{m},...,\psi_{2},\psi_{1}$. Equivalently, it is
the Berry phase acquired by a cyclic trajectory along the shortest (geodesic)
cyclic path visiting those states Samuel and Bhandari (1988) – for a review of
these ideas, see Dariusz Chruscinski (2004).

Measuring Bargmann invariants. Let us now describe a quantum circuit family
that measures any Bargmann invariant $\Delta_{12\cdots m}$. We call these
circuits cycle tests, and they are generalizations of the well-known SWAP test
used to measure the two-state overlap $\Delta_{12}$ (see Fig. 1-a). The key
gate in the cycle test circuit of Fig. 1-b is a controlled cycle permutation,
which implements the cyclic permutation $(12\cdots m)$ of the input states
when the control qubit is in state $|{1}\rangle$, and identity otherwise. It
is easy to check that the final measurement of the auxiliary qubit gives an
estimate of $\Re(\Delta_{12\cdots m})$. The circuit of Fig. 1-a was proposed
in Ekert  _et al._ (2002) as a way to measure non-linear functionals of a
single density matrix. Here, we use it with different inputs to estimate
$\Re(\Delta_{12\cdots m})$, and propose a modification with an added phase
gate $P=diag(1,i)$, to enable the measurement of $\Im(\Delta_{12\cdots m})$
(see Fig. 1-b). Note that the cycle test reduces to the SWAP test for $m=2$
states. Overlaps can also be evaluated as a particular case of degree-3
invariants, as
$\Delta_{iij}=\Delta_{ij}=\left|\left\langle\psi_{i}|\psi_{j}\right\rangle\right|^{2}$
— lower-degree invariants can always be recovered from higher-degree
invariants with repeated indices. The cycle test circuit can be implemented in
linear depth using nearest-neighbor gates only, or in a depth that scales like
$O(\log m)$ if we assume all-to-all qubit connectivity, as we describe in
Appendix A.

Figure 1: Circuits for measuring projective-unitary invariants of a tuple of
states. (a) SWAP test to measure the two-state overlap
$\Delta_{12}=\left|\left\langle\psi_{1}|{\psi_{2}}\right\rangle\right|^{2}$.
The probability of outcome 0 is $p(0)=(1+\Delta_{12})/2$. (b) Cycle test to
measure the $m$-state invariant $\Delta_{12\cdots m}$. If the white square is
the identity, we have $p(0)=(1+\Re(\Delta_{12\cdots m}))/2$. If the white
square is the $P=\textrm{diag}(1,i)$ gate, we have
$p(0)=(1+\Im(\Delta_{12\cdots m}))/2$. The controlled gate implements the
cyclic permutation $(123\cdots m)$ of the inputs if the control is in state
$|{1}\rangle$, and does nothing otherwise.

Complete characterization of relational information between states. The
following Theorem gives a complete characterization of PU-invariant properties
of a collection of $N$ pure sates in terms of Bargmann invariants.

######  Theorem 1 (Characterization of relational properties of pure states).

Let $\Psi=(\psi_{i})_{i=1}^{N}$ be a tuple of pure quantum states. Then, the
PU-equivalence class of $\Psi$ is uniquely specified by values of at most
$(N-1)^{2}$ Bargmann invariants. The invariants are of degree $m\leq N$ and
their choice depends on $\Psi$.

The above Theorem follows from results obtained by Chien and Waldron in Chien
and Waldron (2016). In what follows we present a simplified and shortened
proof. Our strategy is based on encoding complete PU-invariants in a single
Gram matrix in a way that depends on orthogonality relations of states in
$\Psi$. This allows us to upper bound the number of necessary invariants by
$(N-1)^{2}$. and employ the results to the problem of approximate PU
equivalence of quantum states.

###### Proof.

We start with the connection between PU equivalence of two tuples of pure
states $\Psi=(\psi_{i})_{i=1}^{N}$, $\Phi=(\phi_{i})_{i=1}^{N}$ and unitary
equivalence between the associated tuples of wave-functions. Namely, $\Psi$ is
PU equivalent to $\Phi$ if and only if it is possible to find representing
wave functions $\mathcal{A}=\left(|{\psi_{i}}\rangle\right)_{i=1}^{N}$,
$\mathcal{B}=\left(|{\phi_{i}}\rangle\right)_{i=1}^{N}$ that are unitarily
equivalent. That is, there exists an unitary operator $U$ such that
$|{\phi_{i}}\rangle=U|{\psi_{i}}\rangle$, for $i=1,\ldots,N$. The problem of
unitary equivalence of tuples of vectors is equivalent to equality of the
corresponding Gram matrices i.e.,
$G^{\mathcal{A}}_{ij}=\left\langle{\psi_{i}}\middle|{\psi_{j}}\right\rangle=\left\langle{\phi_{i}}\middle|{\phi_{j}}\right\rangle=G^{\mathcal{B}}_{ij}$
(this follows e.g. from the uniqueness the Cholesky decomposition of $G$
Johnston (2021)). The phase of individual wave functions $|{\psi_{i}}\rangle$
is an unphysical, gauge degree of freedom, and therefore the Gram matrix of a
collection of pure states $\Psi$ is uniquely defined only up to conjugation
via a diagonal unitary matrix. Assume now that for every tuple of quantum
states $\Psi=(\psi_{i})_{i=1}^{N}$ we have a construction (presented in the
next paragraph) of a valid Gram matrix $\tilde{G}^{\Psi}$ whose entries can be
expressed solely in terms of PU-invariants of states from $\Psi$. It then
follows from the above considerations that $\Psi$ is PU equivalent to $\Phi$
if and only if $\tilde{G}^{\Psi}=\tilde{G}^{\Phi}$.

The construction of $\tilde{G}^{\Psi}$ proceeds as follows. We first introduce
the frame graph $\Gamma^{\Psi}$ as an (undirected) graph whose vertices are
labeled by $i=1,\ldots,N$ with edges connecting $i$ and $j$ if and only if
$\mathrm{tr}(\psi_{i}\psi_{j})\neq 0$. Without loss of generality we assume
that $\Gamma^{\Psi}$ is connected 111If $G^{\Psi}$ is not connected then the
problem of PU equivalence of states from $\Psi$ reduces to independent
problems concerning states associated with connected components in
$\Gamma^{\Psi}$. i.e., that every pair of vertices in $\Gamma^{\Psi}$ can be
connected via a path in $\Gamma^{\Psi}$. We can now choose a subgraph of
$\Gamma^{\Psi}$, denoted by $\mathcal{T}^{\Psi}$, which is connected and
contains the same vertices as $\Gamma^{\Psi}$, but no cycles. This is known as
a _spanning tree_ of $\Gamma^{\Psi}$, and there exists at least one spanning
tree for any connected graph Gross and Yellen (1998) – see Fig. 2 for an
illustration. We now choose vector representatives $|{\psi_{i}}\rangle$ of
states in $\Psi$ in such a way that, for $\\{i,j\\}$ an edge in
$\mathcal{T}^{\Psi}$,
$\left\langle{\psi_{i}}\middle|{\psi_{j}}\right\rangle=\left|\left\langle{\psi_{i}}\middle|{\psi_{j}}\right\rangle\right|$.
Every other inner product
$\left\langle{\psi_{i}}\middle|{\psi_{j}}\right\rangle$ will be either 0, or
its phase will be fixed as follows. We first find a path from $j$ to $i$
within $\mathcal{T}^{\Psi}$. Since $\mathcal{T}^{\Psi}$ is a spanning tree,
such a path is guaranteed to exist and to be unique. Suppose this path has $k$
vertices $(\alpha_{1}=j,\alpha_{2},\dots,\alpha_{k-1},\alpha_{k}=i)$. Consider
now the $k$-cycle that would be formed by adding the vertex $j$ at the end of
this path (note that vertices $i$ and $j$ are connected by an edge in
$\Gamma^{\Psi}$) and denote it by $C_{ij}$. By construction, every edge in
$C_{ij}$ except for $\\{i,j\\}$ is in $\mathcal{T}^{\Psi}$, and therefore all
the inner products
$\left\langle{\psi_{\alpha_{j}}}\middle|{\psi_{\alpha_{j+1}}}\right\rangle$
for $j\in\\{1,2,\dots k\\}$ have been chosen as positive. Hence, if we denote
the degree-$k$ Bargmann invariant associated to $C_{ij}$ as
$\Delta(C_{ij}):=\Delta_{\alpha_{1}\alpha_{2}\ldots\alpha_{k-1}\alpha_{k}}$ we
can write

$\displaystyle\Delta(C_{ij})=\left\langle{\psi_{i}}\middle|{\psi_{j}}\right\rangle\prod_{l=2}^{k-2}\left\langle{\psi_{\alpha_{l}}}\middle|{\psi_{\alpha_{l+1}}}\right\rangle.$
(2)

Therefore, we can fix the phase of every nonzero inner product that is not in
$\mathcal{T}^{\Psi}$ as

$\left\langle{\psi_{i}}\middle|{\psi_{j}}\right\rangle=\frac{\Delta(C_{ij})}{\prod_{l=2}^{k-2}\left\langle{\psi_{\alpha_{l}}}\middle|{\psi_{\alpha_{l+1}}}\right\rangle}.$
(3)

Thus, all matrix elements of the so-constructed Gram matrix
$\tilde{G}^{\Psi}_{ij}=\left\langle{\psi_{i}}\middle|{\psi_{j}}\right\rangle$
are expressed via Bargmann invariants of degree at most $N$. The total number
of invariants required to construct $\tilde{G}^{\Psi}$ in this way is upper-
bounded by $(N-1)^{2}$, as we first measure $N(N-1)/2$ degree-2 invariants to
obtain the frame graph, and then at most $(N-1)(N/2-1)$ additional invariants
to fix the phases of the non-null elements of $\tilde{G}^{\Psi}$ corresponding
to edges $\\{i,j\\}\notin\Gamma^{\Psi}$. ∎

Figure 2: Frame graph for the example described in the text. Edges in black
correspond to the choice of spanning tree, in this case a star graph.

We remark that in general we cannot characterize the relational properties of
even single-qubit states without using Bargmann invariants of degree $m>2$. As
an example, take two triples of single-qubit states
$\Psi=(\left|{X_{+}}\right\rangle\\!\left\langle{X_{+}}\right|,\left|{Y_{+}}\right\rangle\\!\left\langle{Y_{+}}\right|,\left|{Z_{+}}\right\rangle\\!\left\langle{Z_{+}}\right|)$,
$\Phi=(\left|{X_{+}}\right\rangle\\!\left\langle{X_{+}}\right|,\left|{Y_{-}}\right\rangle\\!\left\langle{Y_{-}}\right|,\left|{Z_{+}}\right\rangle\\!\left\langle{Z_{+}}\right|)$,
where $\left|{P_{\pm}}\right\rangle\\!\left\langle{P_{\pm}}\right|$ are
projectors on Pauli eigenvectors with $\pm 1$ eigenvalues. All overlaps within
each tuple have the same value $\Delta_{ij}=1/2$. Therefore, the tuples only
differ in that
$\Delta_{123}^{\Psi}=\frac{1+i}{4}\neq\frac{1-i}{4}=\Delta_{123}^{\Phi}$.

Let us now consider an application of 1 to a $N=9$ state tuple
$(\psi_{i})_{i=1\ldots 9}$. The first step is to construct the frame graph.
This is done by measuring all $N(N-1)/2=36$ overlaps
$\left|\left\langle\psi_{i}|\psi_{j}\right\rangle\right|$. Suppose the frame
graph is the one represented in fig. 2 (recall that missing edges correspond
to null overlaps). One choice of spanning tree is a star graph, where vertex 1
is connected to every other. We can fix the phases so that every degree-2
invariant $\left\langle\psi_{1}|\psi_{i}\right\rangle>0$. Finally, we consider
all cycles that can be built by adding edges from the original frame graph to
the spanning tree (in fig. 2, these correspond to gray edges). In each such
cycle, there is only one new edge whose phase has yet to be fixed; in this
example only 3-cycles appear. We then measure the degree-3 Bargmann invariants
$\Delta_{1ij}$ for all distinct pairs $\\{i,j\\}$, and attribute its phase to
edge $\\{i,j\\}$. There are 28 of these for this example size, though in fig.
2 only 10 of them are nonzero. This procedure leads to the Gram matrix:

$\tilde{G}^{\Psi}_{ij}=\begin{cases}\sqrt{\Delta_{ii}}=\left|\left\langle\psi_{i}|\psi_{i}\right\rangle\right|=1&\text{for
}i=j\\\
\sqrt{\Delta_{1j}}=\left|\left\langle\psi_{1}|\psi_{j}\right\rangle\right|,&\text{for
}i=1,\\\ \frac{\Delta_{ij1}}{\sqrt{|\Delta_{1j}||\Delta_{1i}|}}&\text{for
}j>i.\end{cases}$ (4)

The gauge choices involved in defining the Gram matrix given in Eq. (4) mean
that all parameters in $\tilde{G}^{\Psi}$ are physically meaningful
quantities, measurable using the cycle test circuits and fully characterizing
relative information among states in $\Psi$.

The choice of spanning tree is not unique. In the above example, we could have
chosen instead e.g. a simple path starting in vertex 1 and visiting all
vertices sequentially up to vertex 9. Building the Gram matrix
$\tilde{G}^{\Psi}$ using this path requires the same number of invariants, but
they are more experimentally involved to measure, as some will have degree
$>3$. An interesting open question is to find alternative characterization
procedures for general scenarios that minimize the experimental effort
required.

Robustness, and mixed states. Any implementation of cycle tests will have to
deal with experimental imperfections, and the fact that the prepared states
will, at best, be mixed-state approximations of the ideal pure states. In
Appendix B we give an argument that, as long as the prepared density matrices
have high purity, then the Bargmann invariants measured will be close to the
Bargmann invariants of the target pure states (understood to be the projector
onto the eigenvector corresponding to the largest eigenvalue of a density
matrix). Furthermore, in Appendix C we show that if Bargmann invariants
evaluated for two tuples of pure states $\Psi=(\psi_{i})_{i=1}^{N}$,
$\Phi=(\phi_{i})_{i=1}^{N}$ differ from each other only slightly, then states
comprising them can be _approximately transformed onto each other_.
Specifically, using the encoding of Bargmann invariants into Gram matrices we
show that there exists a unitary transformation $U$ that _approximately_
transforms $\Psi$ onto $\Phi$ in the sense that the average infidelity is
upper bounded by

$\frac{1}{N}\sum_{i=1}^{N}\left(1-\mathrm{Tr}(U\psi_{i}U^{\dagger}\phi_{i})\right)\leq\frac{C_{\Psi}}{N}\|\tilde{G}^{\Psi}-\tilde{G}^{\Phi}\|^{2}_{2}\
,$ (5)

where $C_{\Psi}$ is a constant (depending on $\Psi$),
$\tilde{G}^{\Psi},\tilde{G}^{\Phi}$ are Gram matrices from the proof of
Theorem 1, and $\|X\|_{2}=\sqrt{\mathrm{Tr}(X^{2})}$ is the Hilbert-Schmidt
norm. This shows that Bargmann invariants offer a robust characterization of
PU equivalence of tuples of pure states.

Identifying complete sets of PU-invariants for tuples of mixed states is a
much more challenging problem. If we use mixed states $\rho_{i}\
(i=1,2,\dots,m)$ as inputs to the cycle test circuits of Fig. 1, the output
gives an operational meaning to mixed-state Bargmann invariants:
$\Delta_{12...m}=\mathrm{Tr}\left(\rho_{1}\rho_{2}\ldots\rho_{m}\right)$. In
Appendix D we show that for a tuple of $N$ mixed states
$\mathcal{S}=(\rho_{i})_{i=1}^{N}$ in dimension $d$ Bargmann invariants of
degree $m\leq d^{2}$ form a complete set of invariants characterizing PU
invariant properties of $\mathcal{S}$. Moreover, the number of independent
invariants can be chosen to be $Nd^{2}+1$. These results follow from known
results from invariant theory Procesi (1976); Formanek ; Wigderson (2019),
which we (partially) reprove with the help of Schur-Weyl duality.

Application: Testing for linear independence. Whether a set of states is
linearly independent (LI) is clearly a unitary-invariant property. LI states
can be probabilistically cloned, and can be unambiguously discriminated.
Recognizing linear dependence of a set of states enables us to find a more
compact description of the set; in machine learning, this corresponds to a
procedure for dimensionality reduction. Using a complete set of Bargmann
invariants, we can compute the hypervolume generated by the set of states to
determine whether they are linearly independent. For the case of $m$ non-
orthogonal vectors, we can use the Gram matrix encoding of the complete set of
invariants we have described to write the linear independence condition simply
as: $\det(G)>0$. In Appendix E we calculate $\det(G)$ explicitly as a function
of the invariants for the cases of 3 and 4 non-orthogonal states.

Application: Basis-independent imaginarity witnesses. Measuring Bargmann
invariants of degree $3$ and above can serve as a basis-independent witness
that the input quantum states have complex-valued amplitudes. The simple
reason is that $\Delta_{12\dots m}$ is a gauge-invariant quantity, so if we
measure $\Im(\Delta_{12\dots m})\neq 0$, some of the density matrices at the
input must necessarily have complex-valued elements, independently of the
basis chosen. As an example, it is easy to check that the 3 Pauli operator
eigenstates
$|{A}\rangle=|{0}\rangle,|{B}\rangle=\frac{1}{\sqrt{2}}(|{0}\rangle+|{1}\rangle),|{C}\rangle=\frac{1}{\sqrt{2}}(|{0}\rangle+i|{1}\rangle)$
achieves the maximum value of $\Im(\Delta_{ABC})=1/4$ for this 3-state
scenario. In Appendix F we describe a tuple of $n$ single-qubit states that
are vertices of a regular spherical $n$-gon on the Bloch sphere, for which
$\Im(\Delta)\to 1$ as $n\to\infty$.

Experimental tests have been recently proposed to discriminate whether complex
amplitudes (or “imaginarity”) are an essential feature of quantum theory Renou
_et al._ (2021); Wu  _et al._ (2021). The application of our proposal as an
imaginarity witness differs fundamentally from the proposals of Renou  _et
al._ (2021); Wu  _et al._ (2021). In particular, though basis-independent, our
approach is not device-independent. Concretely, we can only interpret the
outcome of the circuit of fig. 1(b) as measuring a set of Bargmann invariants
if the input is promised to be a product state. Therefore, we cannot rule out
an adversarial preparation of the states that might fool our imaginarity
witness by using an entangled real-amplitude state, similar to techniques of
McKague  _et al._ (2009).

Application: Basis-independent coherence witnesses. If the $m$ states
$\rho_{i}$ are simultaneously diagonalizable in a single reference basis, the
degree-$m$ Bargmann invariant $\Delta_{12\dots
m}=\mathrm{Tr}(\prod_{i}\rho_{i})$ is simply the probability that independent
measurements of the reference basis on all states give the same outcome Galvão
and Brod (2020). Hence, for diagonal states these invariants must be real and
in the range $\Delta_{12\cdots m}\in[0,1]$. Measuring $\Im(\Delta_{12\dots
m})\neq 0$, or real values outside of the allowed range for diagonal states,
serves as a basis-independent witness of coherence, generalizing the overlap-
based witnesses of Galvão and Brod (2020). Note that witnesses of imaginarity,
as discussed in the previous section, are also witnesses of coherence, as
complex-valued entries of a density matrix must be off-diagonal. In Appendix G
we look at the simplest scenarios where coherence can be witnessed by Bargmann
invariant measurements when we have sets with either 2 or 3 states.

Application: Characterization of photonic indistinguishability. The formalism
of Bargmann invariants and projective unitary invariance is particularly well-
tailored to experiments with single photons in linear optics Simon and Mukunda
(1993). Each photon in an $m$-photon experiment is described by a state over
its internal degrees of freedom - such as time-of-arrival, polarization, and
orbital angular momentum Tichy (2015). Suppose we apply some linear-optical
transformation to these photons that is insensitive to their internal degrees
of freedom, and measure the occupation number at the output of the device. The
outcome statistics will not depend on exactly which internal state a
particular photons is in, as the apparatus is blind to this information, but
it will depend on the relational information, such as imperfect overlaps,
which can cause the photons to be effectively partially distinguishable. In
other words, the output probabilities of such an experiment can only depend on
the Bargmann invariants of the set of states (as well as the linear-optical
transformation). This can be seen in several recent experimental and
theoretical works concerning genuine 3- and 4-photon indistinguishability
Giordani  _et al._ (2021); Galvão and Brod (2020); Menssen  _et al._ (2017);
Jones  _et al._ (2020); Minke  _et al._ (2021).

In Galvão and Brod (2020), it was shown how measurements of some overlaps
among a set of $m$ states can result in nontrivial lower bounds for all
overlaps. Bargmann invariants of higher degree can be useful in the same way,
as by definition $|\Delta_{12\dots m}|^{2}$ is a lower bound to overlaps of
all neighboring states in the associated cycle. This suggests improved designs
for photonic measurements of Bargmann invariants may help in the
characterization of multi-photon sources.

Conclusion. We reviewed how Bargmann invariants encode the projective-unitary
invariant properties of a set of states. For the case of pure states, we have
shown how to represent the complete information in a Gram matrix of inner
products, written in terms of invariant quantities only, representing
relational information about the states in the set. We have proposed a way of
measuring real and imaginary parts of arbitrary Bargmann invariants using
cycle tests, a natural generalization of the SWAP test. We discussed several
applications: coherence and imaginarity witnesses, testing for linear
independence, and characterization of multi-photon indistinguishability. Open
problems include finding efficient NISQ methods for measuring and using
Bargmann invariants in algorithmic applications, studying possible
applications to self-testing in a semi-device independent context Tavakoli
_et al._ (2018); Miklin and Oszmaniec (2021), and developing further the
resource theories of imaginarity and coherence based on Bargmann invariants,
whose first steps we have described.

###### Acknowledgements.

We thank Adam Sawicki for his comments regarding the role of Schur-Weyl
duality for the problem of PU equivalence of mixed states. MO acknowledges
support by the Foundation for Polish Science through the TEAM-NET project
(contract no. POIR.04.04.00-00-17C1/18-00). DJB acknowledges support from
Instituto Nacional de Ciência e Tecnologia de Informação Quântica (INCT-IQ,
CNPq) and FAPERJ. EFG acknowledges funding of the Portuguese institution FCT –
Fundação para Ciência e Tecnologia via project CEECINST/00062/2018. This work
was supported by the H2020-FETOPEN Grant PHOQUSING (GA no.: 899544).

##  Appendix A Efficient constructions for cycle test circuits

In the main text we described how to use an $m$-state cycle test (cf. Fig. 1
of main text) to measure the $m$th order Bargmann invariant of the
corresponding set of states. The main ingredient of the cycle test is a
controlled permutation gate which applies a cyclic permutation when the
control qubit is in the $|{1}\rangle$ state (and does nothing otherwise). We
now describe two efficient ways of implementing this controlled cycle
permutation. For simplicity, we consider first how to decompose a cyclic
permutation in terms of pairwise permutations (i.e. transpositions), without
accounting for the control qubit, to which we return at the end of this
Section.

Figure 3: Decomposition of a 16-qubit cyclic permutation in terms of nearest-
neighbor SWAP gates.

Let us denote $C_{k}=(1,2,3,\ldots,k)$ as the cyclic permutation on $k$
elements, and $(i,k)$ as a permutation of two elements (or transposition) $i$
and $k$. A well-known decomposition of a cyclic permutation in terms of
transpositions is given by

$(1,2,3,\ldots,k)=(1,2)(2,3)\ldots(k-1,k)$ (6)

This decomposition uses only adjacent transpositions, which is equivalent to
swapping only nearest-neighbor qubits at a time. It leads to a circuit as
exemplified in Fig. 3. This circuit uses the optimal number of $(k-1)$
nearest-neighbor transpositions and has depth $(k-1)$. That $(k-1)$ is the
optimal number follows from the need to move the first element to the last
position, which requires exactly $(k-1)$ nearest-neighbor transpositions.

It is possible to give an alternative decomposition that has much smaller
depth, at the cost of using longer-range transpositions. To see it, consider
first the following Lemma:

######  Lemma 1.

Consider the following two disjoint cycles $(1,2,\ldots k)$ and
$(k+1,k+2,\ldots,n)$. We have the following decomposition:

$C_{n}(1,\ldots,n)=(1,k+1)(1,2,\ldots k)(k+1,k+2,\ldots,n)$ (7)

The proof of this Lemma is straightforward, it simply requires checking the
images of numbers $1,k,k+1$, and $n$ under the sequence of permutations, as
all other numbers get trivially mapped to the correct places.

Figure 4: Decomposition of a 16-qubit cyclic permutation in terms of SWAP
gates. This circuit is much more parallelized compared to that of fig. 3,
although the trade-off is the requirement of long-range swaps.

A very efficient decomposition of $C_{m}$ can now be obtained by applying
lemma 1 repeatedly. Suppose, for simplicity, that $m$ is a power of 2. Then we
can first decompose $C_{m}$ into two cycles acting on $m/2$ elements followed
by one transposition, i.e. $C_{m}=(1,m/2+1)(1,2,\ldots,m/2)(m/2+1,m/2+2\ldots
m)$. We then break each cycle of size $m/2$ again in two cycles, acting each
on $m/4$ elements. This divide-and-conquer algorithm is iterated until all
that is left are transpositions. It is easy to see that this procedure
terminates after O$(\log m)$ steps, and hence that the final circuit has depth
O$(\log m)$ as well. An example of this for $m=16$ is shown in fig. 4.

These are not the only decompositions of the cycle permutation in terms of
transpositions, but they represent two extreme regimes. In the first, we have
only nearest-neighbor SWAP gates, but the depth of the circuit is large. In
the second, the depth is logarithmic, but longer-range gates are required.
Which is the most efficient depends on the constraints of the particular
architecture being used—the latter has the benefit of mitigating accumulation
of errors in the circuit, but in particular architectures long-range gates
might not be an option.

Let us now return to the issue of decomposing a _controlled_ cycle
permutation. At first glance, neither decomposition we described has the
promised properties. The circuit in fig. 3 is no longer composed only of
nearest-neighboring controlled-SWAP (or Fredkin) gates, since the first qubit
must control the swapping of states that are distant to it. At the same time,
the circuit of fig. 4 does not have logarithmic depth since every gate in the
circuit must be controlled by the same control qubit, which presumably means
that all gates must be done sequentially. Nonetheless, both circuits can be
adapted to restore the desired properties, as follows.

Figure 5: An adaptation of the circuit of fig. 3 that obtains a controlled
cycle permutation with SWAP and Fredkin gates acting only on neighboring
state. The first qubit is swapped along the 1D chain as each Fredkin gate is
applied. The star at the output indicates where the control qubit is
effectively left at the end of the circuit.

In fig. 5 we show how the controlled version of circuit in fig. 3 can be
implemented using only gates (SWAP and Fredkin) that act on neighboring
qubits.

To obtain a controlled version of fig. 4 that has depth O$(\log m)$, as in the
original, we need to include more ancilla qubits to parallelize the control
operation. Instead of controlling the permutations with a single
$\tfrac{1}{\sqrt{2}}(|{0}\rangle+|{1}\rangle)$, we need to build an
$m/2$-qubit GHZ state $\tfrac{1}{\sqrt{2}}(|{0}\rangle^{\otimes
m/2}+|{1}\rangle^{\otimes m/2})$, such that each qubit in this state can
control a different SWAP gate in parallel. The whole construction still has
logarithmic depth due to the follow Lemma (also cf. Figure 2 in Cruz  _et al._
(2019)):

######  Lemma 2.

Cruz  _et al._ (2019) The $m$-qubit GHZ state:

$\frac{1}{\sqrt{2}}(|{0}\rangle^{\otimes m}+|{1}\rangle^{\otimes m})$ (8)

can be generated by a circuit that acts on the $|{0}\rangle^{\otimes m}$
state, is composed only of a single Hadamard and a sequence of CNOT gates, and
has depth O$(\log m)$.

###### Proof.

We initialize $m$ qubits in the $|{0}\rangle^{\otimes m}$ state, and apply a
Hadamard gate to the first one. Our goal is now to apply a sequence of CNOT
gates that will flip every qubit conditioned on the first qubit being
$|{1}\rangle$ and does nothing otherwise. This can be done as follows. In the
first round, we apply a CNOT gate from qubit 1 to qubit 2. In the second
round, we apply two CNOTs, from qubits 1 and 2 to qubits 3 and 4,
respectively. We repeat this procedure by reusing previously flipped qubits as
controls for the next layer. The number of qubits flipped in this manner
doubles with each layer. Hence, it follows that this will generate an
$m$-qubit GHZ state in O$(\log m)$ layers of two-qubit gates. ∎

For the controlled version of fig. 4 to work properly when using the GHZ state
as the ancilla, we need to perform the operation described in the proof of
lemma 1 in reverse at the end of the cycle test protocol. This is required to
disentangle the ancillas with the control qubit, such that interference can
occur properly at the last $H$ gate in fig. 1. In any case, it follows that
the entire controlled-cycle protocol works in depth O$(\log m)$.

##  Appendix B Bargmann invariants for almost pure states

In experiments to measure Bargmann invariants, the preparations will result in
mixed states that only approximate the target pure states. Here we prove two
lemmas that show that Bargman invariants computed for mixed states with high
purity will have values close to those of the ideal, target pure states.

######  Lemma 3.

Let $\rho$ be a mixed state on $\mathbb{C}^{d}$ (with arbitrary dimensionality
$d$), with $\mathrm{Tr}(\rho^{2})\geq 1-\epsilon$ for some $\epsilon\leq 1/2$.
Let $\psi$ be the normalized projector onto the eigenspace with largest
eigenvalue of $\rho$. Then $||\rho-\psi||_{1}\leq 2\epsilon$, where
$\|X\|_{1}=\mathrm{Tr}(\sqrt{XX^{\dagger}})$ denotes the trace norm.

###### Proof.

: Let $\lambda$ be the maximal eigenvalue of $\rho$. Write
$\rho=\lambda\psi+(1-\lambda)G$ (with $G\perp\psi$). Then

$\displaystyle\mathrm{Tr}(\rho^{2})$
$\displaystyle=\lambda^{2}+(1-\lambda)^{2}\mathrm{Tr}(G^{2})$ (9)
$\displaystyle\leq\lambda^{2}+(1-\lambda)^{2}$ (10)
$\displaystyle=2\lambda^{2}-2\lambda+1.$ (11)

(as the maximal value for $\mathrm{Tr}(G^{2})=1$). Then we have:

$\displaystyle\mathrm{Tr}(\rho^{2})\geq 1-\epsilon$ $\displaystyle\Rightarrow
2\lambda(\lambda-1)\geq-\epsilon.$ (12)
$\displaystyle\Rightarrow\lambda\geq\frac{1}{2}\left(1+\sqrt{1-2\epsilon}\right).$
(13)

As a result we obtain

$\displaystyle\|\rho-\psi\|_{1}$
$\displaystyle=\|(\lambda-1)\psi+(1-\lambda)G\|_{1}$ (14)
$\displaystyle=2(1-\lambda)$ (15) $\displaystyle\leq 1-\sqrt{1-2\epsilon}$
(16) $\displaystyle=\frac{2\epsilon}{1+\sqrt{1-2\epsilon}}\leq 2\epsilon,$
(17)

as we wanted to demonstrate. ∎

######  Lemma 4.

Assume we have two $m$-tuples of (generally mixed) quantum states
$(\rho_{1},\ldots,\rho_{m})$, $(\sigma_{1},\ldots,\sigma_{m})$ which satisfy
$||\rho_{i}-\sigma_{i}||_{1}\leq\delta$ for $i=1,2,\ldots,m$. Then,

$\left|\mathrm{Tr}\left(\prod_{i=1}^{m}\rho_{i}\right)-\mathrm{Tr}\left(\prod_{i=1}^{m}\sigma_{i}\right)\right|\leq
m\delta\ .$ (18)

###### Proof.

We use the formula expressing Bargmann invariant in terms of expectation value
of cycle permutation $C_{m}$ on tensor product of the relevant quantum states

$\mathrm{Tr}\left(\prod_{i=1}^{m}\sigma_{i}\right)=\mathrm{Tr}\left(C_{m}\otimes_{i=1}^{m}\sigma_{i}\right)\
.$ (19)

It follows that

$\displaystyle\left|\mathrm{Tr}\left(\prod_{i=1}^{m}\rho_{i}\right)-\mathrm{Tr}\left(\prod_{i=1}^{m}\sigma_{i}\right)\right|$
(20)
$\displaystyle=\left|\mathrm{Tr}\left[C_{m}\left(\bigotimes_{i=1}^{m}\rho_{i}-\bigotimes_{i=1}^{m}\sigma_{i}\right)\right]\right|$
(21)
$\displaystyle\leq\|C_{m}\|_{\infty}\|\otimes_{i=1}^{m}\rho_{i}-\otimes_{i=1}^{m}\sigma_{i}\|_{1}$
(22) $\displaystyle\leq m\delta\ ,$ (23)

where in (22) we employed a well-known inequality
$|\mathrm{Tr}(XY)|\leq\|X\|_{\infty}\|Y\|_{1}$, while in (23) we used
$\|C_{m}\|_{\infty}=1$ ($C_{m}$ is a unitary operator) and subaddtitivity of
1-norm:

$\|\otimes_{i=1}^{m}\rho_{i}-\otimes_{i=1}^{m}\sigma_{i}\|_{1}\leq\sum_{i=1}^{m}\|\rho_{i}-\sigma_{i}\|_{1}\leq
m\delta\ .$ (24)

∎

Combining the above two lemmas we get that Bargmann invariants computed for
mixed states with high purity approximate Bargmann invariants computed for
collection of pure states corresponding to projectors onto eigenvectors
corresponding to the highest eigenvalue.

######  Corollary 1.

Let $\mathcal{S}=\\{\rho_{1},\ldots,\rho_{N}\\}$ be collection of mixed states
that satisfy $\mathrm{Tr}(\rho_{i}^{2})\geq 1-\epsilon$ for
$\epsilon\in[0,1/2]$ and let $\Psi=\\{\psi_{1},\ldots,\psi_{N}\\}$, where
$\psi_{i}$ is a projector onto an eigenvector corresponding to the largest
eigenvalue of $\rho_{i}$. Then for for arbitrary sequence
$i_{k}\in\\{{1,\ldots,N\\}}$ of length $m$ we have

$|\mathrm{Tr}(\rho_{i_{1}}\rho_{i_{2}}\ldots\rho_{i_{m}})-\mathrm{Tr}(\psi_{i_{1}}\psi_{i_{2}}\ldots\psi_{i_{m}})|\leq
2m\epsilon.$ (25)

##  Appendix C Stability of PU equivalence from Bargmann invariants

The following result is a quantitative variant of Theorem 1—it shows two
collections of pure states $\Psi$,$\Phi$ can be transformed onto each other
approximately provided the relevant Bargmann invariants approximately match.

######  Lemma 5 (Robust Characterization of PU equivalence via Bargmann
invariants).

Let $\Psi,\Phi$ be two tuples of pure states with $m$ elements each:

$\Psi=(\psi_{1},\psi_{2},\ldots,\psi_{N}),\Phi=(\phi_{1},\phi_{2},\ldots,\phi_{N})\
.$ (26)

Moreover, assume that Bargmann invariants used to construct
$\tilde{G}^{\Psi},\tilde{G}^{\Phi}$ are such that

$\mathrm{rank}(\tilde{G}^{\Psi})\geq\mathrm{rank}(\tilde{G}^{\Phi}),\
\|\tilde{G}^{\Psi}-\tilde{G}^{\Phi}\|_{2}\|(\tilde{G}^{\Psi})^{\ddagger}\|_{\infty}^{\frac{1}{2}}\leq\frac{1}{2},$
(27)

where, $\|\cdot\|_{\infty},\|\cdot\|_{2}$ denote operator and Frobenius norm
respectively and $A^{\ddagger}$ is a Moore-Penrose pseudo inverse Arias-Castro
_et al._ (2018). Then, there exist a unitary $U$ such that

$\frac{1}{N}\sum_{i=1}^{N}\left(1-\mathrm{Tr}(U\psi_{i}U^{\dagger}\phi_{i})\right)\leq\frac{C_{\Psi}\|\tilde{G}^{\Psi}-\tilde{G}^{\Phi}\|^{2}_{2}}{N}\,$
(28)

where $C_{\Psi}=(1+\sqrt{2})^{2}\|(\tilde{G}^{\Psi})^{\ddagger}\|_{\infty}$.

In other words, closeness of Bargmann invariants defining Gram matrices
$\tilde{G}^{\Psi},\tilde{G}^{\Phi}$ implies that under a suitable unitary
transformation $U$ the average-state infidelity between states $\Psi$ and
$\Phi$ can be upper bounded as in (28).

######  Remark 1.

We note that in order to construct Gram matrices
$\tilde{G}^{\Psi},\tilde{G}^{\Phi}$ we need to assume that
$\mathrm{tr}(\psi_{i}\psi_{j})\neq 0$ if and only if
$\mathrm{tr}(\phi_{i}\phi_{j})\neq 0$ (in other words that frame graphs
defined in the proof of Theorem 1 are identical,
$\Gamma^{\Psi}=\Gamma^{\Phi}$). Furthermore, we assume that both matrices
$\tilde{G}^{\Psi},\tilde{G}^{\Phi}$ are constructed using the same spanning
tree of the corresponding frame graph.

###### Proof.

The proof follows from the stability of the celebrated Procrustes problem
discussed in Arias-Castro  _et al._ (2018). In Theorem 1 from that work it was
shown that if Gram matrices of two tuples of vectors

$\displaystyle\mathcal{A}$
$\displaystyle=\left(|{\psi_{1}}\rangle,|{\psi_{2}}\rangle,\dots|{\psi_{N}}\rangle\right),$
(29) $\displaystyle\mathcal{B}$
$\displaystyle=\left(|{\phi_{1}}\rangle,|{\phi_{2}}\rangle,\dots|{\phi_{N}}\rangle\right),$
(30)

satisfy

$\mathrm{rank}(G^{\mathcal{A}})\geq\mathrm{rank}(G^{\mathcal{B}}),\
\|G^{\mathcal{A}}-G^{\mathcal{B}}\|_{2}\|(G^{\mathcal{A}})^{\ddagger}\|_{\infty}^{\frac{1}{2}}\leq\frac{1}{2}\
,$ (31)

then there exist a unitary transformation $U$ such that

$\frac{1}{N}\sum_{i=1}^{N}\|U|{\psi}\rangle-|{\phi}\rangle\|^{2}\leq\frac{(1+\sqrt{2})^{2}\|(G^{\mathcal{A}})^{\ddagger}\|}{N}\|G^{\mathcal{A}}-G^{\mathcal{B}}\|^{2}_{2}\
.$ (32)

We can now apply this result directly to Gram matrices $\tilde{G}^{\Psi}$,
$\tilde{G}^{\Phi}$ constructed form Bargmann invariants (and vector
representatives of states form $\Psi,\Phi$ compatible with these Graam
matrices). To pass from (32) to (28) we note that for normalized wavefunctions
$|{\psi}\rangle,|{\phi}\rangle$ we have
$1-\mathrm{Tr}(U\psi_{i}U^{\dagger}\phi_{i})\leq\|U|{\psi_{i}}\rangle-|{\phi_{i}}\rangle\|^{2}$.
∎

##  Appendix D Complete PU invariants for mixed states

In this Appendix we show that Bargmann invariants can be used to decide
unitary equivalence of two tuples of (generically mixed) quantum states
$\mathcal{S}=({\rho_{i})}_{i=1}^{N},\mathcal{S^{\prime}}=({\sigma_{i})}_{i=1}^{N}$.
This result was previously established in Procesi (1976). Here, we present a
self-consistent proof based on Schur-Weyl duality. After presenting this
general result, we remark on the connection of the unitary equivalence problem
with the question of simultaneous conjugation of tuples of matrices. This
connection allows us to state bounds on the degrees and number of independent
Bargmann invariants.

We begin by introducing polynomial invariants of linear group actions of
compact groups in real vector spaces (see e.g. Fulton and Harris (1991);
Goodman and Wallach (2009) for an introduction to representation theory and
invariant theory). Let $G$ be a compact group and let

$\Pi:G\ni g\mapsto\Pi_{g}\in\mathrm{End}(V)$ (33)

be a representation of $G$ in a finite dimensional real vector space $V$
222Typically, in the context of invariant theory one considers _complex_
vector spaces. In that setting the derivation of invariant polynomials is
slightly more complicated - this is a consequence of the fact that even for
one dimensional complex space $\mathbb{C}$, general polynomials can be
understood as two variate polynomials in $\mathrm{Re}z,\mathrm{Im}z$. We
decided to limit ourselves to _real_ vector spaces to avoid unnecessary
technical complication. . Since $G$ is compact we can assume that $\Pi_{g}$ is
an orthogonal transformation. Homogeneous polynomials of degree $m$ in $V$ are
mappings of the form

$p(v)=\langle\tilde{p},v^{\otimes m}\rangle\ ,$ (34)

where $\langle\cdot,\cdot\rangle$ is the invariant inner product in
$V^{\otimes k}$ (induced by the inner product on $V$), and $\tilde{p}\in
V^{\otimes m}$ is a tensor encoding the polynomial $p$. We will denote by
$\mathbb{P}_{k}(V)$ the space of real homogeneous polynomials on $V$ of degree
$m$. By $\mathbb{P}(V)$ we will denote the space off all polynomials on $V$.

Invariant homogeneous polynomials of degree $m$ are polynomials that
additionally satisfy

$p(\Pi_{g}v)=p(v)\ \text{for every }v\in V,g\in G\ .$ (35)

Using orthogonality of $\Pi_{g}$ it can be shown that (35) is equivalent to
requiring

$\Pi_{g}^{\otimes m}\tilde{p}=\tilde{p}\ \text{for every }g\in G\ .$ (36)

We will denote the set of all invariant polynomials by $\mathbb{P}(V)^{G}$. It
is a well known result in invariant theory that in the case of compact groups
we can use invariant polynomials $\mathbb{P}(V)^{G}$ to decide about
equivalence of elements of $V$ under the action of $G$. In other words for
$v,w\in V$ we have $w=\Pi_{g}v$, for some $g\in G$ if and only if for every
invariant polynomial $p\in\mathbb{P}(V)^{G}$ we have $p(w)=p(v)$ (see e.g.
Theorem 2.4.1 in Vrana (2012) for the elementary exposition in the context of
entanglement theory).

The problem of unitary equivalence of $N$ tuples of density matrices in
$\mathbb{C}^{d}$ can be presented using the formalism outlined above. Let

$V=\overbrace{\mathrm{Herm}(\mathbb{C}^{d})\oplus\ldots\oplus\mathrm{Herm}(\mathbb{C}^{d})}^{N\
\textrm{times }}\ ,$ (37)

where $\mathrm{Herm}(\mathbb{C}^{d})$ denotes a (real) vector space of
Hermitian matrices on $\mathbb{C}^{d}$. Every $\mathbf{X}\in V$ can be
identified with a tuple of linear operators:

$\mathbf{X}=(X_{1},\ldots,X_{N})\ .$ (38)

Let $\mathrm{U}(d)$ denote the group of unitary matrices in $\mathbb{C}^{d}$.
This group can be represented on $V$ via simultaneous conjugation:

$\Pi_{U}\left[(X_{1},\ldots,X_{N})\right]=\left(\mathrm{Ad}_{U}[X_{1}],\ldots,\mathrm{Ad}_{U}[X_{N}]\right)\
,$ (39)

where $\mathrm{Ad}_{U}[X]=UXU^{\dagger}$ is the adjoint representation of
$\mathrm{U}(d)$ on $\mathrm{Herm}(\mathbb{C}^{d})$. We remark that the problem
of orbit classification with respect to this so-defined representation of
$\mathrm{U}(d)$ corresponds _exactly_ to the problem of unitary equivalence of
tuples of mixed states
$\mathcal{S}=(\rho_{1},\ldots,\rho_{N}),\mathcal{S}^{\prime}=(\sigma_{1},\ldots,\sigma_{N})$
discussed in the main text. Using the unitary equivalence of vector spaces

$\overbrace{\mathrm{Herm}(\mathbb{C}^{d})\oplus\ldots\oplus\mathrm{Herm}(\mathbb{C}^{d})}^{N\
\textrm{times }}\simeq\mathrm{Herm}(\mathbb{C}^{d})\otimes\mathbb{R}^{N}\ ,$
(40)

we can represent $\mathbf{X}\in V$ via

$\mathbf{X}=\sum_{i=1}^{N}X_{i}\otimes|{i}\rangle\ ,$ (41)

where $\\{{|{i}\rangle\\}}_{i=1}^{N}$ is an orthogonal basis of
$\mathbb{R}^{N}$. In this language the representation $\Pi_{U}$ defined in
(39) is equivalent to

$\Pi_{U}\simeq\mathrm{Ad}_{U}\otimes\mathbb{I}_{N}\ ,$ (42)

where $\mathbb{I}_{N}$ is the identity operator on $\mathbb{R}^{N}$. We note
that an invariant inner product with respect to representation $\Pi_{U}$ can
be chosen as

$\langle\mathbf{X},\mathbf{Y}\rangle=\sum_{i=1}^{N}\mathrm{Tr}(X_{i}Y_{i})\ .$
(43)

The equivalence (42) allows us to conveniently describe conditions that
$\tilde{p}\in V^{\otimes m}$ has to satisfy in order for it to define an
invariant polynomial of degree $m$ (cf. Eq. (36)). Indeed, upon reordering the
factors in the tensor product we have

$V^{\otimes
m}\simeq\left(\mathrm{Herm}(\mathbb{C}^{d})\otimes\mathbb{R}^{N}\right)^{\otimes
m}\simeq\mathrm{Herm}(\mathbb{C}^{d})^{\otimes
m}\otimes(\mathbb{R}^{N})^{\otimes m}\ ,$ (44) $\Pi_{U}^{\otimes
m}\simeq\left(\mathrm{Ad}_{U}\otimes\mathbb{I}_{N}\right)^{\otimes
m}\simeq\mathrm{Ad}_{U}^{\otimes N}\otimes\mathbb{I}^{\otimes m}_{N}\ .$ (45)

Using (44) we get that an arbitrary element of $\tilde{p}\in V^{\otimes m}$
can be written as

$\tilde{p}=\sum_{i_{1}=1}^{N}\sum_{i_{2}=1}^{N}\ldots\sum_{i_{m}=1}^{N}P_{i_{1}i_{2}\ldots
i_{m}}\otimes|{i_{1}}\rangle|{i_{2}}\rangle\ldots|{i_{m}}\rangle\ ,$ (46)

where $P_{i_{1}i_{2}\ldots i_{m}}\in\mathrm{Herm}((\mathbb{C}^{d})^{\otimes
m})$ is a Hermitian operator acting on $(\mathbb{C}^{d})^{\otimes m}$. Note
that we implicitly used the equivalence
$\mathrm{Herm}((\mathbb{C}^{d})^{\otimes
m})\simeq(\mathrm{Herm}(\mathbb{C}^{d}))^{\otimes m}$]. Recall that
$\tilde{p}$ defines an invariant polynomial
$p\in\mathbb{P}_{m}(V)^{\mathrm{U}(d)}$ if and only if $\Pi_{U}^{\otimes
m}\tilde{p}=\tilde{p}$ for all $Y\in\mathrm{U}(d)$. By virtue of the above
this is equivalent to demanding that for all $U$ $\mathrm{Ad}_{U}^{\otimes
m}\otimes\mathbb{I}^{\otimes m}_{N}\tilde{p}=\tilde{p}$. Using (46) and the
fact that vectors $|{i_{1}}\rangle|{i_{2}}\rangle\ldots|{i_{m}}\rangle$ are
linearly independent we get that this amounts to requiring

$[U^{\otimes m},P_{i_{1}i_{2}\ldots i_{m}}]=0\ ,$ (47)

for all $U\in\mathrm{U}(d)$ and all sequences $i_{1},i_{2},\ldots,i_{m}$. From
the celebrated Schur-Weyl duality Goodman and Wallach (2009) it follow that
(47) is equivalent to $P_{i_{1}i_{2}\ldots i_{m}}$ being a linear combination
of permutation operators $\tau$ that permute factors of the space
$(\mathbb{C}^{d})^{\otimes m}$:

$\tau|{\psi_{1}}\rangle|{\psi_{2}}\rangle\ldots|{\psi_{m}}\rangle\mathrel{:=}|{\psi_{\tau^{-1}(1)}}\rangle|{\psi_{\tau^{-1}(2)}}\rangle\ldots|{\psi_{\tau^{-1}(m)}}\rangle\
.$ (48)

We note that the expansion of $P_{i_{1}i_{2}\ldots i_{m}}$ onto permutations
$\tau$ must be such that the resulting operator is Hermitian (generically it
will be a real linear combination of operators of the form
$\tau+\tau^{\dagger}$, $i(\tau-\tau^{\dagger})$, which are themselves
Hermitian).

Let us evaluate a (possibly complex valued) polynomial corresponding to
$\tilde{p}_{\tau}=\tau\otimes|{i_{1}}\rangle|{i_{2}}\rangle\ldots|{i_{m}}\rangle$
on a tuple of operators $\textbf{X}=(X_{1},\ldots,X_{N})$ which we represent
by $\textbf{X}=\sum_{j=1}^{N}X_{j}\otimes|{j}\rangle$. We get

$p_{\tau}(\mathbf{X})=\langle\tilde{p}_{\tau},\mathbf{X}^{\otimes
m}\rangle=\left\langle\tau\otimes|{i_{1}}\rangle|{i_{2}}\rangle\ldots|{i_{m}}\rangle,\mathbf{X}^{\otimes
m}\right\rangle\ .$ (49)

By expanding $\mathbf{X}^{\otimes m}$ into individual tensors we obtain

$\mathbf{X}^{\otimes
m}=\sum_{j_{1}=1}^{N}\sum_{j_{2}=1}^{N}\ldots\sum_{j_{m}=1}^{N}\left(\bigotimes_{k=1}^{m}X_{j_{k}}\right)\otimes|{j_{1}}\rangle|{j_{2}}\rangle\ldots|{j_{m}}\rangle\
.$ (50)

Inserting this into (49) and using the inner product structure (43) (promoted
to $V^{\otimes m}$) and orthonormality of vectors
$|{j_{1}}\rangle|{j_{2}}\rangle\ldots|{j_{p}}\rangle$ gives

$p_{\tau}(\mathbf{X})=\mathrm{Tr}\left(\tau\bigotimes_{k=1}^{m}X_{i_{k}}\right)\
,$ (51)

where $i_{1},i_{2},\ldots,i_{m}$ is an _arbitrary_ sequence of numbers form
$\\{1,\ldots,N\\}$. Using the decomposition of $\tau$ into disjoint cycles
$\tau=C_{k_{1}}C_{k_{2}}\ldots C_{k_{m}}$ (indices $k_{i}$ denote lengths of
disjoint cycles composing $\tau$), we get that

$p_{\tau}(\mathbf{X})=\mathrm{Tr}\left(\prod_{l_{1}=1}^{k_{1}}X_{l_{1}}\right)\mathrm{Tr}\left(\prod_{l_{2}=1}^{k_{2}}X_{l_{2}}\right)\ldots\mathrm{Tr}\left(\prod_{l_{p}=1}^{k_{p}}X_{l_{p}}\right)\
,$ (52)

for suitably chosen sequences $\\{l_{1}\\}_{l_{1}}^{k_{1}}$,…,
$\\{l_{p}\\}_{l_{p}}^{k_{p}}$, that depend on $\tau$ and the sequence
$i_{1},i_{2},\ldots,i_{m}$. This shows that $p_{\tau}(\mathbf{X})$ can be
expressed as a product of Bargmann invariants of degree at most $m$.

Since an _arbitrary_ real polynomial invariant
$p\in\mathbb{P}(V)^{\mathrm{U}(d)}$ can be expressed via a suitable linear
combination of $p_{\tau}$, we finally conclude that Bargmann invariants decide
simultaneous unitary equivalence of two tuples of Hermitian operators
$\mathbf{X}=(X_{1},\ldots,X_{N}),\mathbf{Y}=(Y_{1},\ldots,Y_{N})$ operating in
$\mathbb{C}^{d}$. Restricting to density matrices we see that the above
considerations proved the desired result for equivalence of tuples of mixed
states.

######  Theorem 2.

Let
$\mathcal{S}=(\rho_{1},\ldots,\rho_{N}),\mathcal{S}^{\prime}=(\sigma_{1},\ldots,\sigma_{N})$
be two tuples of mixed states on $\mathbb{C}^{d}$. Then, there exist
$U\in\mathrm{U}(d)$ such that for every $i=1,\ldots,N$

$\sigma_{i}=U\rho_{i}U^{\dagger}\ $ (53)

if and only if for every $m$ and for every sequence $i_{1},i_{2},\ldots,i_{m}$
of numbers from $\\{1,\ldots,N\\}$ the corresponding Bargman invariants of
degree $m$ agree

$\mathrm{Tr}\left(\rho_{i_{1}}\rho_{i_{2}}\ldots\rho_{i_{m}}\right)=\mathrm{Tr}\left(\sigma_{i_{1}}\sigma_{i_{2}}\ldots\sigma_{i_{m}}\right).$
(54)

We conclude this part by remarking about the degree and number of generating
polynomials that suffice to decide unitary equivalence of tuples of density
matrices in $\mathbb{C}^{d}$. Bargmann invariants
$\mathrm{Tr}(X_{i_{1}}X_{i_{2}}\ldots X_{i_{m}})$ are relevant also for the
problem of simultaneous matrix conjugation Procesi (1976); Formanek ;
Wigderson (2019). This is a related equivalence problem in which the unitary
group $\mathrm{U}(d)$ is replaced by the special linear group $\mathrm{SL}(d)$
(a group of $d\times d$ matrices with unit determinant) with a representation
on $N$-tuples of matrices defined by

$\Pi_{Z}\left[(X_{1},\ldots,X_{N})\right]=\left(ZX_{1}Z^{-1},\ldots,ZX_{N}Z^{-1}\right)\
,$ (55)

where $X_{i}\in\mathrm{End}(\mathbb{C}^{d})$ and $Z\in\mathrm{SL}(d)$. It
turns out that _the same_ Bargmann invariants decide the problem of
equivalence of (closures of) orbits of group action in this scenario. Because
of this the ring of invariant polynomials generated by
$\mathrm{Tr}(X_{i_{1}}X_{i_{2}}\ldots X_{i_{m}})$ has been extensively studied
in the context of invariant theory - see Section 13.9.2 in Wigderson (2019)
for an accessible exposition. It follows from the discussion contained there
that the degrees of Bargmann invariants generating the whole invariant ring
$\mathbb{P}(V)^{\mathrm{U}(d)}$ are at most $m\leq d^{2}$. Furthermore, the
number of independent invariants can be chosen to be $Nd^{2}+1$ and can be
found efficiently in time polynomial in both $d$ and $N$.

##  Appendix E Conditions for linear independence

As described in the main text, we can parameterize the Gram matrix $G$ so that
a complete set of Bargmann invariants are represented in the matrix elements.
The condition for linear independence is simply:

$\det(G)>0.$ (56)

Having measured all the necessary invariants, the determinant of the resulting
numerical Gram matrix $G$ can be computed efficiently.

To illustrate the type of dependence on the Bargmann invariants that results,
below we will work out the expression for $\det(G)$ for $m=3,4$ normalized
pure states (which we assume to be non-orthogonal). We will use the
construction of the Gram matrix presented in Theorem 1 of the main text,
choosing for the spanning tree the star graph with state $|{\psi_{1}}\rangle$
as the central node. For 3 pure states
$|{\psi_{1}}\rangle,|{\psi_{2}}\rangle,|{\psi_{3}}\rangle$, we write the Gram
matrix:

$\displaystyle G=$
$\displaystyle\begin{pmatrix}1&\left|\left\langle\psi_{1}|\psi_{2}\right\rangle\right|&\left|\left\langle\psi_{1}|\psi_{3}\right\rangle\right|\\\
\left|\left\langle\psi_{1}|\psi_{2}\right\rangle\right|&1&\left\langle\psi_{2}|\psi_{3}\right\rangle\\\
\left|\left\langle\psi_{1}|\psi_{3}\right\rangle\right|&\left\langle\psi_{2}|\psi_{3}\right\rangle^{*}&1\end{pmatrix}$
$\displaystyle=$
$\displaystyle\begin{pmatrix}1&\sqrt{\Delta_{12}}&\sqrt{\Delta_{13}}\\\
\sqrt{\Delta_{12}}&1&\sqrt{\Delta_{23}}e^{i\phi_{23}}\\\
\sqrt{\Delta_{13}}&\sqrt{\Delta_{23}}e^{-i\phi_{23}}&1\end{pmatrix},$ (57)

where we have parameterized $G$ in terms of overlaps, i.e. 2-vertex invariants
$\Delta_{ij}=\left|\left\langle\psi_{i}|\psi_{j}\right\rangle\right|^{2}=\frac{|\Delta_{ijj}|^{2}}{\Delta_{1ii}\Delta_{1jj}}$
and the single 3-state invariant phase
$e^{i\phi_{23}}=\Delta_{123}/|\Delta_{123}|$. For the case we consider here
(no null overlaps), the linear independence condition is then:

$\displaystyle\det(G)$ $\displaystyle>0\Leftrightarrow$ $\displaystyle
1-(\Delta_{12}+\Delta_{13}+\Delta_{23})$
$\displaystyle+2\sqrt{\Delta_{12}\Delta_{13}\Delta_{23}}\cos(\phi_{23})>0.$

This condition has appeared previously in the literature, in the context of
unambiguous state discrimination Bergou  _et al._ (2012). Note that
measurement of overlaps only may suffice to guarantee linear independence, as
$1-(\Delta_{12}+\Delta_{13}+\Delta_{23})-2\sqrt{\Delta_{12}\Delta_{13}\Delta_{23}}>0$
implies $\det(G)>0$ independently of the value of $\phi_{23}$. This was the
approach to dimension witnesses based on overlaps proposed in Galvão and Brod
(2020), and measured experimentally in Giordani  _et al._ (2021).

The case of 4 states with no null overlaps can be analyzed similarly, using
overlaps and phases of degree-3 invariants. The Gram matrix in this case can
be written thus:

$G=\begin{pmatrix}1&\sqrt{\Delta_{12}}&\sqrt{\Delta_{13}}&\sqrt{\Delta_{14}}\\\
\sqrt{\Delta_{12}}&1&\sqrt{\Delta_{23}}e^{i\phi_{23}}&\sqrt{\Delta_{24}}e^{i\phi_{24}}\\\
\sqrt{\Delta_{13}}&\sqrt{\Delta_{23}}e^{-i\phi_{23}}&1&\sqrt{\Delta_{34}}e^{i\phi_{34}}\\\
\sqrt{\Delta_{14}}&\sqrt{\Delta_{24}}e^{-i\phi_{24}}&\sqrt{\Delta_{34}}e^{-i\phi_{34}}&1\end{pmatrix},$

where again we have the overlaps
$\Delta_{ij}=\left|\left\langle\psi_{i}|\psi_{j}\right\rangle\right|^{2}$ and
the 3-state Bargmann invariant phases
$e^{i\phi_{ij}}=\Delta_{1ij}/|\Delta_{1ij}|$.

The linear independence condition can then be stated:

$\displaystyle\det(G)>0\Leftrightarrow
1-(\Delta_{12}+\Delta_{13}+\Delta_{14}+\Delta_{23}+\Delta_{24}+\Delta_{34})+\left(\Delta_{12}\Delta_{34}+\Delta_{13}\Delta_{24}+\Delta_{14}\Delta_{23}\right)+$
$\displaystyle+2\left[\sqrt{\Delta_{12}\Delta_{13}\Delta_{23}}\cos(\phi_{23})+\sqrt{\Delta_{12}\Delta_{14}\Delta_{24}}\cos(\phi_{24})+\sqrt{\Delta_{13}\Delta_{14}\Delta_{34}}\cos(\phi_{34})+\sqrt{\Delta_{23}\Delta_{24}\Delta_{34}}\cos(\phi_{34}-\phi_{24})\right]$
$\displaystyle-2\left[\sqrt{\Delta_{12}\Delta_{13}\Delta_{14}\Delta_{34}}\cos(\phi_{34})+\sqrt{\Delta_{12}\Delta_{14}\Delta_{23}\Delta_{34}}\cos(\phi_{34}-\phi_{23})+\sqrt{\Delta_{13}\Delta_{14}\Delta_{23}\Delta_{24}}\cos(\phi_{24}-\phi_{23})\right]>0.$

It is clear that any set of $m$ states without null overlaps can be analyzed
similarly, with the linear independence condition expressed in terms of
invariants of degree 3. As discussed in the main text, more complex
orthogonality conditions results in frame graphs that require higher-order
invariants for a complete characterization of the PU-equivalence class.

##  Appendix F Maximizing $\Im(\Delta)$

In this Appendix we describe the maximum values we found for $\Im(\Delta)$ for
$n$-state Bargmann invariants, and show that for a suitable choice of a tuple
of $n$ states $\Im(\Delta)\to 1$ as $n\to\infty$. Take a tuple of $n$ pure,
single-qubit states forming successive vertices of a regular spherical $n$-gon
on the surface of the Bloch sphere. They can be chosen as

$|{\psi_{k}}\rangle=\cos(\theta/2)|{0}\rangle+\sin(\theta/2)e^{2\pi
ik/n}|{1}\rangle,$ (58)

for $k\in\\{0,1,2,\dots,(n-1)\\}$, for some choice of $\theta$. For $n=3$, it
is easy to show that $\Im(\Delta)$ reaches its maximal value of
$\Im(\Delta)=1/4$ for the spherical triangle vertices given by
$\theta=\arccos{(1/\sqrt{3})}$. More generally, for the regular spherical
$n$-gon vertices we have:

$\Delta=[1+\sin^{2}(\theta/2)(e^{2\pi i/n}-1)]^{n}$ (59)

For $n\geq 3$, the tuples of states described by Eq. (58) have the highest
value of $\Im(\Delta)$ that we have found. For example, for $n=10$ we obtain
$\Im(\Delta)\approx 0.6949$ for $\theta\approx 0.9972$. Using Eq. (59) it is
easy to show that $\Im(\Delta)\to 1$ as $n\to\infty$, if we choose
$\theta=\pi/3$.

##  Appendix G Basis-independent coherence witnesses

Here we look at the simplest scenarios for which coherence witnesses can be
given in terms of Bargmann invariants. For $m=2$ states $\rho_{1},\rho_{2}$,
it is possible to guarantee coherence if we measure the overlap $\Delta_{12}$
together with each state’s purity $\Delta_{ii}=Tr(\rho_{i}^{2})$ \- for
example, $\Delta_{11}=\Delta_{22}=1$ means both states are pure, so that any
$\Delta_{12}\neq 0,1$ will serve as a witness of coherence. The situation is
more interesting when we only measure relational information between different
states.

Let us now analyze the scenario with $m=3$ states, where we measure 2- and
3-state Bargmann invariants $\Delta_{12},\Delta_{13},\Delta_{23}$, and
$\Delta_{123}$. As discussed before, for simultaneously diagonal (i.e.,
coherence-free) states $\Delta_{123}$ is the probability that independent
measurements of the reference observable on the 3 states give the same outcome
(and similarly for $\Delta_{ij}$ and state pair $\rho_{i},\rho_{j}$). So the
first type of witness would result if the 3-state Bargmann invariant was
complex, or real negative. Following Galvão and Brod (2020), we find the
polytope of allowed 4-tuples
$(\Delta_{12},\Delta_{13},\Delta_{23},\Delta_{123})$ by listing the allowed
deterministic states in this space, and calculating their convex hull. For
this scenario, the allowed deterministic states are given in Table I.

$\Delta_{12}$ | $\Delta_{13}$ | $\Delta_{23}$ | $\Delta_{123}$
---|---|---|---
0 | 0 | 0 | 0
0 | 0 | 1 | 0
0 | 1 | 0 | 0
1 | 0 | 0 | 0
1 | 1 | 1 | 1
Table 1: Each row is one of the possible deterministic assignments to the
Bargmann invariants describing the 3-state scenario. The convex hull of the
rows characterizes all allowed 4-tuples of invariants for simultaneously
diagonal, i.e. coherence-free states.

The corresponding polytope facets, characterizing all 4-tuples of invariants
consistent with coherence-free states, are given by inequalities:

$\displaystyle\Delta_{123}$ $\displaystyle\geq 0,$ (60)
$\displaystyle\Delta_{123}$ $\displaystyle\leq\Delta_{ij},\text{for
}ij=12,13,23,$ (61) $\displaystyle\Delta_{123}$
$\displaystyle\geq\frac{1}{2}\left(\Delta_{12}+\Delta_{13}+\Delta_{23}-1\right).$
(62)

Any violation of ineqs. (60-62) is a witness of coherence, and these
inequalities are strictly stronger to detect coherence than the imaginarity
witnesses we have introduced, or previous basis-independent coherence
witnesses based on overlaps Galvão and Brod (2020), measured recently in
Giordani  _et al._ (2021). We can see this with a simple example. Consider 3
single-qubit states equally spaced on the $xz$ plane great circle of the Bloch
sphere. These states have real amplitudes, so their coherence cannot be
witnessed via imaginarity witnesses; it can be easily checked also that these
equal-valued overlaps $r_{12}=r_{13}=r_{23}=1/4$ are compatible with
coherence-free states Galvão and Brod (2020). This set, however, results in
$\Delta_{123}=-1/8$, hence witnesssing coherence due to violation of ineq.
(60).

## References

  * Klein (1900) Felix Klein, “A comparative review of recent researches in geometry,” Bulletin of the New York Mathematical Society 2, 215 – 249 (1900).
  * Leifer  _et al._ (2004) M. S. Leifer, N. Linden, and A. Winter, “Measuring polynomial invariants of multiparty quantum states,” Phys. Rev. A 69, 052304 (2004).
  * Bargmann (1964) V. Bargmann, “Note on wigner’s theorem on symmetry operations,” Journal of Mathematical Physics 5, 862 (1964).
  * Chien and Waldron (2016) T.-Y. Chien and S. Waldron, “A characterization of projective unitary equivalence of finite frames and applications,” SIAM J. Discrete Math., 30(2), 976–994. (19 pages) 30, 976–994 (2016).
  * Menssen  _et al._ (2017) A. J. Menssen, A. E. Jones, B. J. Metcalf, M. C. Tichy, S. Barz, W. S. Kolthammer, and I. A. Walmsley, “Distinguishability and Many-Particle Interference,” Phys. Rev. Lett. 118, 153603 (2017).
  * Jones  _et al._ (2020) Alex E. Jones, Adrian J. Menssen, Helen M. Chrzanowski, Tom A. W. Wolterink, Valery S. Shchesnovich, and Ian A. Walmsley, “Multiparticle interference of pairwise distinguishable photons,” Phys. Rev. Lett. 125, 123603 (2020).
  * Minke  _et al._ (2021) Alexander M. Minke, Andreas Buchleitner, and Christoph Dittel, “Characterizing four-body indistinguishability via symmetries,” (2021), arXiv:2103.04600 [quant-ph] .
  * Simon and Mukunda (1993) R. Simon and N. Mukunda, “Bargmann invariant and the geometry of the güoy effect,” Phys. Rev. Lett. 70, 880–883 (1993).
  * Dariusz Chruscinski (2004) Andrzej Jamiolkowski Dariusz Chruscinski,  _Geometric phases in classical and quantum mechanics_, 1st ed., Progress in Mathematical Physics (Birkhäuser Boston, 2004).
  * Pancharatnam (1956) S. Pancharatnam, “Generalized theory of interference, and its applications,” Proceedings of the Indian Academy of Sciences - Section A 44, 247–262 (1956).
  * Samuel and Bhandari (1988) Joseph Samuel and Rajendra Bhandari, “General setting for berry’s phase,” Phys. Rev. Lett. 60, 2339–2342 (1988).
  * Ekert  _et al._ (2002) Artur K. Ekert, Carolina Moura Alves, Daniel K. L. Oi, Michał Horodecki, Paweł Horodecki, and L. C. Kwek, “Direct estimations of linear and nonlinear functionals of a quantum state,” Phys. Rev. Lett. 88, 217901 (2002).
  * Johnston (2021) N. Johnston,  _Advanced Linear and Matrix Algebra_ (Springer International Publishing, 2021).
  * Note (1) If $G^{\Psi}$ is not connected then the problem of PU equivalence of states from $\Psi$ reduces to independent problems concerning states associated with connected components in $\Gamma^{\Psi}$.
  * Gross and Yellen (1998) J.L. Gross and J. Yellen,  _Graph Theory and Its Applications, Second Edition_, Discrete Mathematics and Its Applications (Taylor & Francis, 1998).
  * Procesi (1976) C Procesi, “The invariant theory of n Ã— n matrices,” Advances in Mathematics 19, 306–381 (1976).
  * (17) E. Formanek,  _The Polynomial Identities and Invariants of N X N Matrices_, CBMS Regional Conference Series (Conference Board of the Mathematical Sciences).
  * Wigderson (2019) A. Wigderson,  _Mathematics and Computation: A Theory Revolutionizing Technology and Science_ (Princeton University Press, 2019).
  * Renou  _et al._ (2021) Marc-Olivier Renou, David Trillo, Mirjam Weilenmann, Le Phuc Thinh, Armin Tavakoli, Nicolas Gisin, Antonio Acin, and Miguel Navascues, “Quantum physics needs complex numbers,” (2021), arXiv:2101.10873 [quant-ph] .
  * Wu  _et al._ (2021) Kang-Da Wu, Tulja Varun Kondra, Swapan Rana, Carlo Maria Scandolo, Guo-Yong Xiang, Chuan-Feng Li, Guang-Can Guo, and Alexander Streltsov, “Operational resource theory of imaginarity,” Phys. Rev. Lett. 126, 090401 (2021).
  * McKague  _et al._ (2009) Matthew McKague, Michele Mosca, and Nicolas Gisin, “Simulating quantum systems using real hilbert spaces,” Phys. Rev. Lett. 102, 020505 (2009).
  * Galvão and Brod (2020) Ernesto F. Galvão and Daniel J. Brod, “Quantum and classical bounds for two-state overlaps,” Phys. Rev. A 101, 062110 (2020).
  * Tichy (2015) M. C. Tichy, “Sampling of partially distinguishable bosons and the relation to the multidimensional permanent,” Phys. Rev. A 91, 022316 (2015).
  * Giordani  _et al._ (2021) Taira Giordani, Chiara Esposito, Francesco Hoch, Gonzalo Carvacho, Daniel J. Brod, Ernesto F. Galvão, Nicolò Spagnolo, and Fabio Sciarrino, “Witnesses of coherence and dimension from multiphoton indistinguishability tests,” Physical Review Research (in press) (2021).
  * Tavakoli  _et al._ (2018) A. Tavakoli, J. Kaniewski, T. Vértesi, D. Rosset, and N. Brunner, “Self-testing quantum states and measurements in the prepare-and-measure scenario,” Phys. Rev. A 98, 062307 (2018).
  * Miklin and Oszmaniec (2021) Nikolai Miklin and Michał Oszmaniec, “A universal scheme for robust self-testing in the prepare-and-measure scenario,” Quantum 5, 424 (2021).
  * Cruz  _et al._ (2019) Diogo Cruz, Romain Fournier, Fabien Gremion, Alix Jeannerot, Kenichi Komagata, Tara Tosic, Jarla Thiesbrummel, Chun Lam Chan, Nicolas Macris, Marc-André Dupertuis, and Clément Javerzac-Galy, “Efficient quantum algorithms for ghz and w states, and implementation on the ibm quantum computer,” Advanced Quantum Technologies 2, 1900015 (2019).
  * Arias-Castro  _et al._ (2018) Ery Arias-Castro, Adel Javanmard, and Bruno Pelletier, “Perturbation Bounds for Procrustes, Classical Scaling, and Trilateration, with Applications to Manifold Learning,” arXiv e-prints , arXiv:1810.09569 (2018), arXiv:1810.09569 [cs.LG] .
  * Fulton and Harris (1991) W. Fulton and J. Harris,  _Representation Theory: A First Course_, Graduate Texts in Mathematics (Springer New York, 1991).
  * Goodman and Wallach (2009) R. Goodman and N.R. Wallach,  _Symmetry, Representations, and Invariants_, Graduate Texts in Mathematics (Springer New York, 2009).
  * Note (2) Typically, in the context of invariant theory one considers _complex_ vector spaces. In that setting the derivation of invariant polynomials is slightly more complicated - this is a consequence of the fact that even for one dimensional complex space $\mathbb{C}$, general polynomials can be understood as two variate polynomials in $\mathrm{Re}z,\mathrm{Im}z$. We decided to limit ourselves to _real_ vector spaces to avoid unnecessary technical complication.
  * Vrana (2012) Peter Vrana, “Group representations in entanglement theory,” Phd Thesis (2012).
  * Bergou  _et al._ (2012) János A. Bergou, Ulrike Futschik, and Edgar Feldman, “Optimal unambiguous discrimination of pure quantum states,” Phys. Rev. Lett. 108, 250502 (2012).