Present address: ]Faculty of Physics, University of Warsaw, Pasteura 5, 02-093
Warsaw, Poland

# Many-body entanglement in fermion systems

N. Gigena  [  IFLP/CONICET and Departamento de Física, Universidad Nacional de
La Plata, C.C. 67, La Plata (1900), Argentina  M. Di Tullio  IFLP/CONICET and
Departamento de Física, Universidad Nacional de La Plata, C.C. 67, La Plata
(1900), Argentina  R. Rossignoli  IFLP/CONICET and Departamento de Física,
Universidad Nacional de La Plata, C.C. 67, La Plata (1900), Argentina
Comisión de Investigaciones Científicas (CIC), La Plata (1900), Argentina

###### Abstract

We discuss a general bipartite-like representation and Schmidt decomposition
of an arbitrary pure state of $N$ indistinguishable fermions, based on states
of $M<N$ and $(N$-$M)$ fermions. It is directly connected with the reduced
$M$\- and $(N$-$M)$-body density matrices (DMs), which have the same spectrum
in such states. The concept of $M$-body entanglement emerges naturally in this
scenario, generalizing that of one-body entanglement. Rigorous majorization
relations satisfied by the normalized $M$-body DM are then derived, which
imply that the associated entropy will not increase, on average, under a class
of operations which have these DMs as post-measurement states. Moreover, such
entropy is an upper bound to the bipartite entanglement entropy generated by a
class of operations which map the original state to a bipartite state of $M$
and $N-M$ effectively distinguishable fermions. Analytic evaluation of the
spectrum of $M$-body DMs in some strongly correlated fermionic states is also
provided.

##  I introduction

A remarkable feature of quantum mechanics is the existence of correlations
between quantum systems that cannot be emulated by their classical
counterpart. Entanglement is the most celebrated manifestation of such
correlations and it has been object of intense research in quantum physics,
particularly within the field of quantum information theory Nielsen and Chuang
(2000). Particle indistinguishability is another fundamental feature of
quantum mechanics, lying at the heart of condensed matter physics and quantum
field theories. An interesting problem combining these two fundamental
concepts is that of the study and quantification of correlations between
indistinguishable particles, a topic that has received increasing attention in
the last years Benatti  _et al._ (2020). Indistinguishability poses a
nontrivial difficulty in the study of quantum correlations, because the notion
of entanglement is intimately connected with that of local operations, and the
latter are possible only if the constituents of the system can be
distinguished. Different approaches to this problem have been considered, like
mode entanglement Zanardi (2002); Shi (2003); Friis  _et al._ (2013),
extensions based on correlations between observables Barnum  _et al._ (2004);
Zanardi  _et al._ (2004); Sasaki  _et al._ (2011); Balachandran  _et al._
(2013); Benatti  _et al._ (2014) and entanglement beyond symmetrization
Schliemann  _et al._ (2001a, b); Eckert  _et al._ (2002); Ghirardi  _et al._
(2002); Paškauskas and You (2001); Wiseman and Vaccaro (2003); Iemini and
Vianna (2013); Oszmaniec and Kuś (2013); Sárosi and Lévay (2014); Gigena and
Rossignoli (2015); Majtey  _et al._ (2016), which is independent of the choice
of single particle (sp) basis. The relation between these forms of
entanglement has been analysed by different authors Benatti  _et al._ (2020);
Friis  _et al._ (2013); Wiseman and Vaccaro (2003); Gigena and Rossignoli
(2015); Dasenbrook  _et al._ (2016); Gigena and Rossignoli (2017); Benatti
_et al._ (2017); Di Tullio  _et al._ (2018, 2019); da Silva Souza  _et al._
(2018); Ding and Schilling (2020); Debarba  _et al._ (2020) as well as the
question regarding whether symmetrization correlations should be addressed as
entanglement Cavalcanti  _et al._ (2007); Killoran  _et al._ (2014); Compagno
_et al._ (2018); Lo Franco and Compagno (2018); Morris  _et al._ (2020).

In this paper we explore the generalization of the bipartite formulation
devised in Gigena  _et al._ (2020) for the notion of one-body entanglement, a
measure of correlations beyond (anti)symmetrization introduced in Gigena and
Rossignoli (2015). We start by considering operators on the fermion Fock space
of the system creating general $M$-fermion states, which are used to show that
a general $N$-fermion state can be always written as a bipartite-like state of
$M<N$ and $N-M$ fermions. This bipartite representation is connected with the
definition of the $M$ and $N-M$-body reduced density matrices (DMs) Carlson
and Keller (1961); Ando (1963); Amosov and Filippov (2017) in a way that
closely resembles the case of bipartite states of distinguishable
constituents. The ensuing $(M,N-M)$ Schmidt-like decomposition of the state
determines the diagonal form of these DMs, entailing they share the same non-
zero eigenvalues. Pushing forward this analogy we propose to link the
correlations between $M$ and $N-M$-body observables, which we call $M$-body
entanglement, to the mixedness of the $M$-body DM, as formalized by eigenvalue
majorization. We introduce a family of entropic measures of such correlations
and show that there exists a class of operations not increasing the amount of
these correlations in any $N$-fermion state. Finally, we prove the existence
of a family of quantum maps converting $N$-fermion states into bipartite
states of effectively distinguishable $M$ and $N-M$-fermions, in the sense of
occupying orthogonal sp subspaces. Conversion by means of any of these maps is
such that the entanglement entropy of the bipartite target state is bounded
from above by the $M$-body entropy, assigning the latter a clear operational
meaning. Explicit examples of $M$-body DMs and their eigenvalues in some
physical states are also provided.

##  II Formalism

We consider a sp space $\cal H$ of finite dimension $D$, spanned by fermion
operators $c_{i}$, $c^{\dagger}_{i}$, $i=1,\ldots,D$ satisfying the
anticommutation relations

$\\{c_{i},c^{\dagger}_{j}\\}=\delta_{ij},\;\\{c_{i},c_{j}\\}=\\{c^{\dagger}_{i},c^{\dagger}_{j}\\}=0\,.$
(1)

We also define the $M$-fermion creation operators

$C^{(M){\dagger}}_{\bm{\alpha}}=c^{\dagger}_{i_{1}}\ldots
c^{\dagger}_{i_{M}}\,,$ (2)

where $i_{1}<i_{2}<\ldots<i_{M}$ and $\bm{\alpha}=(i_{1},\ldots,i_{M})$ labels
all $\binom{D}{M}=\frac{D!}{M!(D-M)!}$ distinct sets of $M$ sp states sorted
in increasing order. These operators satisfy

$\displaystyle\langle
0|C^{(M)}_{\bm{\alpha}}C^{(M^{\prime}){\dagger}}_{\bm{\alpha^{\prime}}}|0\rangle$
$\displaystyle=$
$\displaystyle\delta^{MM^{\prime}}\delta_{\bm{\alpha\alpha^{\prime}}}\,,$ (3)
$\displaystyle\sum_{\bm{\alpha}}C^{(M){\dagger}}_{\bm{\alpha}}C^{(M)}_{\bm{\alpha}}$
$\displaystyle=$ $\displaystyle\binom{\hat{N}}{M}\,,$ (4)

where $\hat{N}=\sum_{i}c^{\dagger}_{i}c_{i}$ is the fermion number operator
and $\binom{\hat{N}}{M}$ is the operator taking the value $\binom{N}{M}$ in a
state of $N$ fermions
($\binom{\hat{N}}{M}|\Psi\rangle=\binom{N}{M}|\Psi\rangle$ for
$\hat{N}|\Psi\rangle=N|\Psi\rangle$, with $\binom{\hat{N}}{1}=\hat{N}$,
$\binom{\hat{N}}{2}=\frac{\hat{N}^{2}-\hat{N}}{2}$, etc.). Eq. (4) is a
generalization of the number operator, representing the number of “$M$-fermion
composites”. The states $C^{M{\dagger}}_{\bm{\alpha}}|0\rangle$ are Slater
Determinants (SDs) and form, for all $\bm{\alpha}$ and $0\leq M\leq D$, an
orthonormal basis of the $2^{D}$-dimensional Fock space associated to ${\cal
H}$.

A normalized pure state $|\Psi\rangle$ of $N$ fermions
($\hat{N}|\Psi\rangle=N|\Psi\rangle$) can then be expanded in this basis as

$\displaystyle|\Psi\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{N!}\sum_{i_{1},\ldots,i_{N}}\\!\\!\Gamma_{i_{1}\ldots
i_{N}}\,c^{\dagger}_{i_{1}}\ldots c^{\dagger}_{i_{N}}|0\rangle$ (5)
$\displaystyle=$
$\displaystyle\sum_{\bm{\alpha}}\Gamma^{(N)}_{\bm{\alpha}}C^{(N){\dagger}}_{\bm{\alpha}}|0\rangle\,,$
(6)

where $\Gamma_{i_{1}\ldots i_{N}}$ are the elements of a fully antisymmetric
tensor and $\Gamma^{(N)}_{\bm{\alpha}}=\langle
0|C^{(N)}_{\bm{\alpha}}|\Psi\rangle=\Gamma_{i_{1}\ldots i_{N}}$ (for
$\bm{\alpha}=(i_{1},\ldots,i_{n})$, $i_{1}<i_{2}<\ldots<i_{N}$), with

$\sum_{\bm{\alpha}}|\Gamma^{(N)}_{\bm{\alpha}}|^{2}=\frac{1}{N!}\sum_{i_{1},\ldots,i_{N}}|\Gamma_{i_{1}\ldots
i_{N}}|^{2}=1\,.$ (7)

Thus,
$|\Gamma^{(N)}_{\bm{\alpha}}|^{2}=\langle\Psi|C_{\bm{\alpha}}^{(M){\dagger}}C_{\bm{\alpha}}^{(M)}|\Psi\rangle$
is the probability of finding the $N$ sp states ${\bm{\alpha}}$ occupied in
$|\Psi\rangle$.

###  II.1 The $(M,N-M)$ representation and the
$M$-body DM

For $0\leq M\leq N$ we can also rewrite the state (5) as

$|\Psi\rangle={\textstyle\binom{N}{M}^{-1}}\sum_{\bm{\alpha},\bm{\beta}}\Gamma^{(M)}_{\bm{\alpha}\bm{\beta}}C^{(M){\dagger}}_{\bm{\alpha}}C^{(N-M){\dagger}}_{\bm{\beta}}|0\rangle\,,$
(8)

where
$\Gamma^{(M)}_{\bm{\alpha\beta}}\equiv\Gamma^{(M,N-M)}_{\bm{\alpha\beta}}$ is
given by

$\Gamma^{(M)}_{\bm{\alpha}\bm{\beta}}=\langle
0|C^{(N-M)}_{\bm{\beta}}C^{(M)}_{\bm{\alpha}}|\Psi\rangle=\Gamma_{i_{1}\ldots
i_{M}j_{1}\ldots j_{N-M}}\,,$ (9)

for $\bm{\alpha}=(i_{1},\ldots,i_{M})$, $\bm{\beta}=(j_{1},\ldots,j_{N-M})$,
and the sum in (8) is over all $\binom{D}{M}$ and $\binom{D}{N-M}$ distinct
sets of $M$ and $N-M$ sp states respectively, with (7) implying

$\sum_{\bm{\alpha},\bm{\beta}}\left|\Gamma^{(M)}_{\bm{\alpha\beta}}\right|^{2}=\binom{N}{M}\,.$
(10)

We will denote the expression (8) as the $(M,N-M)$-body representation of the
$N$-fermion state $\eqref{psi1}$. It is a bipartite-like expansion of
$|\Psi\rangle$ in orthogonal $M$\- and $(N-M)$-fermion states, leading to a
$\binom{D}{M}\times\binom{D}{N-M}$ matrix representation $\Gamma^{(M)}$ of the
original tensor $\Gamma$ in (5). Of course, decompositions $(M,N-M)$ and
$(N-M,M)$ are equivalent, with

$\Gamma^{(N-M)}=(-1)^{M(N-M)}(\Gamma^{(M)})^{T}\,,$ (11)

due the antisymmetry of $\Gamma$ ($T$ denotes transpose). Eq. (6) is the
trivial $(N,0)$ representation.

From the antisymmetry of $\Gamma$ it also follows that

$\displaystyle C^{(M)}_{\bm{\alpha}}|\Psi\rangle$ $\displaystyle=$
$\displaystyle\sum_{\bm{\beta}}\Gamma^{(M)}_{\bm{\alpha\beta}}C^{(N-M){\dagger}}_{\bm{\beta}}|0\rangle\,,$
(12)

which represents the (unnormalized) state of the remaining $N-M$ fermions when
the $M$ sp states labelled by $\bm{\alpha}$ are occupied in $|\Psi\rangle$.
Eqs. (12) and (3) imply that the $M$-body density matrix Carlson and Keller
(1961); Amosov and Filippov (2017), whose elements are defined as

$\displaystyle\rho^{(M)}_{\bm{\alpha\alpha^{\prime}}}$ $\displaystyle:=$
$\displaystyle\langle\Psi|C^{(M){\dagger}}_{\bm{\alpha^{\prime}}}C^{(M)}_{\bm{\alpha}}|\Psi\rangle\,,$
(13)

can be expressed in terms of $\Gamma^{(M)}$ as

$\rho^{(M)}=\Gamma^{(M)}\Gamma^{(M){\dagger}}\,,$ (14)

i.e.
$\rho^{(M)}_{\bm{\alpha\alpha^{\prime}}}=\sum_{\beta}\Gamma^{(M)}_{\bm{\alpha\beta}}\Gamma^{(M)\,*}_{\bm{\alpha^{\prime}\beta}}$,
in the same way as the reduced density matrix $\rho^{A}={\rm
Tr}_{B}|\Psi_{AB}\rangle\langle\Psi_{AB}|$ is obtained from a general state
$|\Psi_{AB}\rangle=\sum_{i,j}G_{ij}|i_{A},j_{B}\rangle$ of a system of two
distinguishable components
($\rho^{A}_{ii^{\prime}}=(GG^{\dagger})_{ii^{\prime}}$ for
$\rho^{A}_{ii^{\prime}}=\langle\Psi_{AB}||i^{\prime}_{A}\rangle\langle
i_{A}|\otimes\mathbbm{1}_{B}|\Psi_{AB}\rangle$). In particular,
$\rho^{(M)}_{\bm{\alpha\alpha}}$ is the probability of finding the $M$ sp
states specified by $\bm{\alpha}$ occupied in $|\Psi\rangle$. Eq. (14) is a
positive semidefinite $\binom{D}{M}\times\binom{D}{M}$ matrix which determines
the average of any $M$-body operator
$\hat{O}^{(M)}=\sum_{\bm{\alpha},\bm{\alpha^{\prime}}}O^{(M)}_{\bm{\alpha\alpha}^{\prime}}C^{(M){\dagger}}_{\bm{\alpha}}C^{(M)}_{\bm{\alpha}^{\prime}}$
through

$\langle\Psi|\hat{O}^{(M)}|\Psi\rangle={\rm Tr}\,[\rho^{(M)}O^{(M)}]\,.$ (15)

Its trace is given by

${\rm Tr}\,[\rho^{(M)}]=\binom{N}{M}\,,$ (16)

as implied by (4) or (10). We also notice that

$C^{(N-M)}_{\beta}|\Psi\rangle=\sum_{\alpha}\Gamma^{(N-M)}_{\beta\alpha}C_{\alpha}^{(M){\dagger}}|0\rangle\,.$
(17)

Hence, using (11) the partner $(N-M)$-body DM, of elements
$\rho^{(N-M)}_{\beta\beta^{\prime}}=\langle\Psi|C^{(N-M){\dagger}}_{\beta^{\prime}}C^{(N-M)}_{\beta}|\Psi\rangle$,
is

$\rho^{(N-M)}=\Gamma^{(N-M)}\Gamma^{(N-M){\dagger}}=(\Gamma^{(M)})^{T}\Gamma^{(M)*}\,,$
(18)

which shows it has the same non-zero eigenvalues (and hence the same trace) as
the $M$-body DM (14) Carlson and Keller (1961); Amosov and Filippov (2017), as
discussed in II.2 in more detail.

In particular, for $M=1$, $C^{(1){\dagger}}_{\bm{\alpha}}=c^{\dagger}_{i}$ and
we recover from (8) the $(1,N-1)$ representation Gigena  _et al._ (2020)

$|\Psi\rangle=\frac{1}{N}\sum_{i,\bm{\alpha}}\Gamma^{(1)}_{i\bm{\alpha}}c^{\dagger}_{i}C^{(N-1){\dagger}}_{\bm{\alpha}}|0\rangle\,,$
(19)

where the $D\times\binom{D}{N-1}$ matrix $\Gamma^{(1)}$ determines the one-
body DM (also denoted as SPDM)
$\rho^{(1)}_{ii^{\prime}}=\langle\Psi|c^{\dagger}_{i^{\prime}}c_{i}|\Psi\rangle$
through

$\rho^{(1)}=\Gamma^{(1)}\Gamma^{(1)^{\dagger}}\,.$ (20)

We finally remark that Eq. (8) is a particular case of the more general
$k$-partite representation of the state (5),

$\displaystyle|\Psi\rangle$ $\displaystyle=$
$\displaystyle{\textstyle\frac{M_{1}!\ldots
M_{k}!}{N!}}\\!\\!\sum_{\bm{\alpha}_{1},\ldots,\bm{\alpha}_{k}}\\!\\!\\!\\!\Gamma^{M_{1}\ldots
M_{k}}_{\bm{\alpha}_{1}\ldots\bm{\alpha}_{k}}C^{(M_{1}){\dagger}}_{\bm{\alpha}_{1}}\ldots
C^{(M_{k}){\dagger}}_{\bm{\alpha}_{k}}|0\rangle\,,\;\;\;$ (21)

where $\Gamma^{M_{1}\ldots
M_{k}}_{\bm{\alpha}_{1}\ldots\bm{\alpha}_{k}}=\langle
0|C^{(M_{k})}_{\bm{\alpha}_{k}}\ldots
C^{(M_{1})}_{\bm{\alpha}_{1}}|\Psi\rangle=\Gamma_{i_{1}\ldots i_{N}}$ for
$\bm{\alpha}_{1}\equiv(i_{1},\ldots,i_{M_{1}}),\ldots,\bm{\alpha}_{k}\equiv(i_{N-M_{k}+1},\ldots,i_{N})$
and $\sum_{j=1}^{k}M_{j}=N$, with $k\leq N$ and sums running over all
$\binom{D}{M_{j}}$ sets of $M_{j}$ sp states. The basic expansion (5)
corresponds to $k=N$ and $M_{j}=1$ for $j=1,\ldots,N$.

###  II.2 The $(M,N-M)$ Schmidt representation

We can now employ the singular value decomposition of the matrix
$\Gamma^{(M)}$,

$\displaystyle\Gamma^{(M)}$ $\displaystyle=$ $\displaystyle
U^{(M)}D^{(M)}V^{(N-M){\dagger}}\,,$ (22a) $\displaystyle\
D^{(M)}_{\nu\nu^{\prime}}$ $\displaystyle=$
$\displaystyle\sqrt{\lambda^{(M)}_{\nu}}\delta_{\nu\nu^{\prime}}\,,$ (22b)

where $U^{(M)},V^{(N-M)}$ are $\binom{D}{M}\times\binom{D}{M}$ and
$\binom{D}{N-M}\times\binom{D}{N-M}$ unitary matrices and $D^{(M)}\equiv
D^{(M,N-M)}$ a $\binom{D}{M}\times\binom{D}{N-M}$ diagonal matrix with non-
negative elements. Here $\lambda^{(M)}_{\nu}$ denote the square of the
singular values of $\Gamma^{(M)}$, i.e. the eigenvalues of
$\Gamma^{(M)}\Gamma^{(M){\dagger}}=\rho^{(M)}$ or equivalently
$\Gamma^{(M)T}\Gamma^{(M)*}=\rho^{(N-M)}$, which have the same spectrum
(except for the number of zero eigenvalues). It then becomes possible to
rewrite Eq. (8) in the Schmidt-like diagonal form

$|\Psi\rangle={\textstyle\binom{N}{M}^{-1}}\sum_{\nu=1}^{n_{M}}\sqrt{\lambda^{(M)}_{\nu}}A^{(M){\dagger}}_{\nu}B^{(N-M){\dagger}}_{\nu}|0\rangle\,,$
(23)

where $n_{M}$ is the rank of $\Gamma^{(M)}$ and

$\displaystyle A^{(M){\dagger}}_{\nu}$ $\displaystyle=$
$\displaystyle\sum_{\bm{\alpha}}U^{(M)}_{\bm{\alpha}\nu}C^{(M){\dagger}}_{\bm{\alpha}}\,,$
(24) $\displaystyle B^{(N-M)\,{\dagger}}_{\nu}$ $\displaystyle=$
$\displaystyle\sum_{\bm{\beta}}V_{\bm{\beta}\nu}^{(N-M)*}C^{(N-M)\,{\dagger}}_{\bm{\beta}}\,,$
(25)

are “collective” operators creating, respectively, $M$ and $N-M$ fermions in
generally entangled (i.e., non-SD’s for $M\geq 2$) states. Nevertheless, since
they are unitarily related to the original operators
$C^{(M){\dagger}}_{\bm{\alpha}}$ and $C^{(N-M){\dagger}}_{\bm{\beta}}$, they
still satisfy, for $1\leq M\leq N-1$,

$\displaystyle\langle 0|A^{(M)}_{\nu}A^{(M){\dagger}}_{\nu^{\prime}}|0\rangle$
$\displaystyle=$ $\displaystyle\delta_{\nu\nu^{\prime}}=\langle
0|B^{(N-M)}_{\nu}B^{(N-M){\dagger}}_{\nu^{\prime}}|0\rangle,\;\;\;\;\;\;$ (26)
$\displaystyle\sum_{\nu}A^{(M){\dagger}}_{\nu}A^{(M)}_{\nu}$ $\displaystyle=$
$\displaystyle\binom{\hat{N}}{M}=\sum_{\nu}B^{(N-M){\dagger}}_{\nu}B^{(N-M)}_{\nu},\;\;\;\;\;\;\;$
(27) $\displaystyle\langle
0|B^{(N-M)}_{\nu^{\prime}}A^{(M)}_{\nu}|\Psi\rangle$ $\displaystyle=$
$\displaystyle\sqrt{\lambda^{(M)}_{\nu}}\delta_{\nu\nu^{\prime}}\,.$ (28)

Moreover, Eqs. (12), (22) and (24) lead to

$A^{(M)}_{\nu}|\Psi\rangle=\sqrt{\lambda_{\nu}^{(M)}}B^{(N-M)\,{\dagger}}_{\nu}|0\rangle\,,$
(29)

such that $B^{(N-M){\dagger}}_{\nu}|0\rangle$ is the state of remaining $N-M$
fermions when $M$ fermions are measured to be in the “normal” or “natural”
state $A^{(M){\dagger}}_{\nu}|0\rangle$. Eqs. (26)–(29) also imply that the
normal operators $A^{(M)}_{\nu}$, $B^{(N-M)}_{\nu}$ diagonalize the $M$\- and
$(N-M)$-body DM’s:

$\displaystyle\langle\Psi|A^{(M){\dagger}}_{\nu^{\prime}}A^{(M)}_{\nu}|\Psi\rangle$
$\displaystyle=$
$\displaystyle(U^{(M){\dagger}}\rho^{(M)}U^{(M)})_{\nu\nu^{\prime}}$
$\displaystyle=$ $\displaystyle\lambda_{\nu}^{(M)}\delta_{\nu\nu^{\prime}}$
$\displaystyle=$
$\displaystyle\langle\Psi|B^{(N-M)\,{\dagger}}_{\nu^{\prime}}B^{(N-M)}_{\nu}|\Psi\rangle\,.$

For $M=1$ we recover from (24)–(25) the diagonal representation of the one-
body DM $\rho^{(1)}$ Gigena  _et al._ (2020), with
$A^{(1){\dagger}}_{\nu}=\sum_{i}U_{i\nu}c^{\dagger}_{i}=c^{\dagger}_{\nu}$ the
operators creating a fermion in the ensuing natural sp orbitals.

In the trivial case $M=N$, $\rho^{(N)}$ has a single non-zero eigenvalue
$\lambda^{(N)}_{1}=1$ associated with the operator
$A^{(N)\,{\dagger}}_{1}=\sum_{\bm{\alpha}}\Gamma^{(N)}_{\bm{\alpha}}C^{(N){\dagger}}_{\bm{\alpha}}$
creating the state. On the other hand, in an $N$-fermion SD, which can be
always written as $|\Psi\rangle=c^{\dagger}_{1}\ldots
c^{\dagger}_{N}|0\rangle$ by a suitable choice of the operators
$c^{\dagger}_{i}$, $\rho^{(M)}$ has just $\binom{N}{M}$ non-zero eigenvalues
$\lambda^{(M)}_{\nu}=1$, associated with the $\binom{N}{M}$ operators
$A^{(M){\dagger}}_{\nu}=c^{\dagger}_{i_{1}}\ldots c^{\dagger}_{i_{M}}$, $1\leq
i_{1}<\ldots<i_{M}\leq N$, with support on the $N$ occupied sp states,
satisfying
$\langle\Psi|A^{(M){\dagger}}_{\nu}A^{(M)}_{\nu^{\prime}}|\Psi\rangle=\delta_{\nu\nu^{\prime}}$.
For instance, the decomposition (23) of an $N=3$ SD
$c^{\dagger}_{1}c^{\dagger}_{2}c^{\dagger}_{3}|0\rangle$ for $M=1$ is just
$|\Psi\rangle=\frac{1}{3}\sum_{i=1}^{3}c^{\dagger}_{i}B^{(2){\dagger}}_{i}|0\rangle$,
with $B^{(2){\dagger}}_{1}=c^{\dagger}_{2}c^{\dagger}_{3}$,
$B^{(2){\dagger}}_{2}=-c^{\dagger}_{1}c^{\dagger}_{3}$,
$B^{(2){\dagger}}_{3}=c^{\dagger}_{1}c^{\dagger}_{2}$ and $\langle
c^{\dagger}_{i}c_{j}\rangle=\delta_{ij}=\langle
B^{(2){\dagger}}_{i}B^{(2)}_{j}\rangle$ for $i,j\leq 3$. Similarly for general
$N$. Thus, in a SD $\rho^{(M)}$ is idempotent: $(\rho^{(M)})^{2}=\rho^{(M)}$
$\forall M\leq N$.

And for a general pure two-fermion state
$|\Psi_{2}\rangle=\frac{1}{2}\sum_{i<j}\Gamma_{ij}c^{\dagger}_{i}c^{\dagger}_{j}|0\rangle$,
with $\Gamma_{ij}=-\Gamma_{ji}$, the (non-zero) singular values of
$\Gamma^{(1)}=\Gamma$ for the $M=1$ decomposition $(1,1)$, and hence the
eigenvalues of $\rho^{(1)}=\Gamma\Gamma^{\dagger}$, are always two-fold
degenerate Schliemann  _et al._ (2001a), such that the natural operators can
be paired as $A^{(1){\dagger}}_{\nu}=c^{\dagger}_{\nu}$,
$A^{(1){\dagger}}_{\bar{\nu}}=c^{\dagger}_{\bar{\nu}}$ for
$\lambda^{(1)}_{\nu}=\lambda^{(1)}_{\bar{\nu}}$, with
$B^{(1)}_{\nu}=c^{\dagger}_{\bar{\nu}}$,
$B^{(1)}_{\bar{\nu}}=-c^{\dagger}_{\nu}$ and
$c^{\dagger}_{\nu(\bar{\nu})}=\sum_{j}U_{j\nu(\bar{\nu})}c^{\dagger}_{j}$.
Then Eq. (23) leads to

$\displaystyle|\Psi_{2}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{\nu}\sqrt{\lambda_{\nu}^{(1)}}\,(A^{(1){\dagger}}_{\nu}B^{(1){\dagger}}_{\nu}+A^{(1){\dagger}}_{\bar{\nu}}B^{(1){\dagger}}_{\bar{\nu}})|0\rangle\;\;\;\;$
(31a) $\displaystyle=$
$\displaystyle\sum_{\nu}\sqrt{\lambda_{\nu}^{(1)}}\,c^{\dagger}_{\nu}c^{\dagger}_{\bar{\nu}}|0\rangle\,,$
(31b)

where (31b) is the well-known Slater decomposition of a two-fermion state
Schliemann  _et al._ (2001a); Eckert  _et al._ (2002), with $\langle
c^{\dagger}_{\nu}c_{\nu^{\prime}}\rangle=\langle
c^{\dagger}_{\bar{\nu}}c_{\bar{\nu}^{\prime}}\rangle=\lambda^{(1)}_{\nu}\delta_{\nu\nu^{\prime}}$,
$\langle c^{\dagger}_{\nu}c_{\bar{\nu}^{\prime}}\rangle=0$ and
$\sum_{\nu}\lambda^{(1)}_{\nu}=1$. For $D=4$ (two fermions in 4 sp levels),
there are just two distinct eigenvalues $\lambda_{\nu}^{(1)}$, given by
$\lambda_{\pm}=\frac{1\pm\sqrt{1-C^{2}}}{2}$, with
$C=2|\Gamma_{12}\Gamma_{34}-\Gamma_{13}\Gamma_{24}+\Gamma_{14}\Gamma_{23}|=2\sqrt{\lambda_{+}\lambda_{-}}$
the fermionic concurrence Schliemann  _et al._ (2001a); Eckert  _et al._
(2002); Gigena and Rossignoli (2015), and (31b) can be rewritten as
$(\sqrt{\lambda_{+}}c^{\dagger}_{1}c^{\dagger}_{\bar{1}}+\sqrt{\lambda_{-}}c^{\dagger}_{2}c^{\dagger}_{\bar{2}})|0\rangle$.

###  II.3 The eigenvalues of the $M$-body DM

While the eigenvalues
$\lambda^{(1)}_{\nu}=\langle\Psi|c^{\dagger}_{\nu}c_{\nu}|\Psi\rangle$ of the
SPDM $\rho^{(1)}$ always lie in the interval $[0,1]$ (as $c^{\dagger}_{\nu}$
are standard fermion operators), for $M\geq 2$ those of $\rho^{(M)}$ can be
greater than 1 when $|\Psi\rangle$ is not a SD, since the normal operators
$A^{(M){\dagger}}_{\nu}$ will in general no longer be a single product of $M$
fermion creation operators. These operators may exhibit boson-like features
for even $M\geq 2$, or in general features of boson + fermion creation
operators for odd $M\geq 3$.

We first note that any operator
$A^{(M){\dagger}}=\sum_{\bm{\alpha}}\gamma_{\bm{\alpha}}C^{(M){\dagger}}_{\bm{\alpha}}$
creating $M<N$ fermions can be expanded in the normal operators (24) as
$A^{(M){\dagger}}=\sum_{\nu}\gamma_{\nu}A^{(M){\dagger}}_{\nu}$, with
$\gamma_{\nu}=\sum_{\bm{\alpha}}U_{{\bm{\alpha}}\nu}^{*}\gamma_{\bm{\alpha}}$.
Hence, using (II.2),

$\langle\Psi|A^{(M){\dagger}}A^{(M)}|\Psi\rangle=\sum_{\nu}\lambda^{(M)}_{\nu}|\gamma_{\nu}|^{2}\,,$
(32)

with $\langle
0|A^{(M)}A^{(M){\dagger}}|0\rangle=\sum_{\nu}|\gamma_{\nu}|^{2}$. Thus for
normalized operators satisfying $\langle
0|A^{(M)}A^{(M){\dagger}}|0\rangle=1$, the largest eigenvalue
$\lambda^{(M)}_{\rm max}$ of $\rho^{(M)}$ bounds any such average:

$\langle\Psi|A^{(M){\dagger}}A^{(M)}|\Psi\rangle\leq\lambda^{(M)}_{\rm
max}\,.$ (33)

In a SD, $\lambda^{(M)}_{\rm max}=1$ and
$\langle\Psi|A^{(M){\dagger}}A^{(M)}|\Psi\rangle\leq 1$ $\forall$ $M$ and
normalized $A^{(M){\dagger}}$. This bound can of course be broken in more
general fermion states.

For instance, let us define, assuming even sp dimension $D$, the collective
pair creation operator

$A^{{\dagger}}=\frac{1}{\sqrt{D/2}}\sum_{i=1}^{D/2}c^{{\dagger}}_{2i-1}c^{\dagger}_{2i}\,,$
(34)

which satisfies $[A^{\dagger},\hat{N}]=-2A^{\dagger}$ and

$[A,A^{\dagger}]=1-\tfrac{2}{D}\hat{N}\,,$ (35)

implying $\langle 0|AA^{\dagger}|0\rangle=1$. We then consider the normalized
states ($0\leq k\leq D/2$)

$\displaystyle|\Psi_{2k}\rangle$ $\displaystyle=$
$\displaystyle\frac{(A^{\dagger})^{k}}{k!\sqrt{(\frac{2}{D})^{k}\binom{D/2}{k}}}|0\rangle=\frac{\sum_{\mu}C^{(2k){\dagger}}_{\mu}}{\sqrt{\binom{D/2}{k}}}|0\rangle\,,$
(36)

which contain $k$ of such pairs and hence $N=2k$ fermions, and where
$C^{(2k){\dagger}}_{\mu}=\prod_{i}(c^{\dagger}_{2i-1}c^{\dagger}_{2i})^{n_{i\mu}}$
with $n_{i\mu}=0,1$, $\sum_{\mu}n_{i\mu}=k$ and $1\leq\mu\leq\binom{D/2}{k}$.
They satisfy

$\displaystyle A^{\dagger}|\Psi_{2k}\rangle$ $\displaystyle=$
$\displaystyle{\textstyle\sqrt{(k+1)(1-2k/D)}}|\Psi_{2(k+1)}\rangle$ (37a)
$\displaystyle A|\Psi_{2k}\rangle$ $\displaystyle=$
$\displaystyle{\textstyle\sqrt{k(1-2\frac{k-1}{D})}}|\Psi_{2(k-1)}\rangle\,.$
(37b)

so that for these states $A^{\dagger}$ behaves as a perfect ladder operator
and can therefore be considered an ideal coboson according to the definition
given in Law (2005).

Since in $|\Psi_{2k}\rangle$ all sp states have the same probability of being
occupied, and fermions are created in pairs, it is apparent that the SPDM
$\rho^{(1)}$ will have just a single degenerate eigenvalue
$\lambda^{(1)}_{\nu}=N/D\leq 1$ (see Appendix A). Then it will be uniformly
mixed, i.e., proportional to the identity and hence diagonal in any sp basis:

$\rho^{(1)}=\frac{2k}{D}\mathbbm{1}\,,$ (38)

Hence the states (36) lead to maximum one-body entanglement compatible with a
given value of $N$, with (38) showing explicitly that they are not SDs for
$2k<D$.

In contrast, from Eqs. (37) it follows that
$A^{\dagger}A|\Psi_{2k}\rangle=k(1-2\frac{k-1}{D})|\Psi_{2k}\rangle$, implying
that the two-body DM $\rho^{(2)}$ has one large nondegenerate eigenvalue (see
Appendix A)

$\displaystyle\lambda_{\rm max}^{(2)}$ $\displaystyle=$
$\displaystyle\langle\Psi_{2k}|A^{\dagger}A|\Psi_{2k}\rangle=k\left(1-2\frac{k-1}{D}\right)\geq
1\,,$ (39)

associated with the normal operator $A^{\dagger}$, which satisfies
$\lambda_{\rm max}^{(2)}>1$ for $2\leq k\leq\frac{D}{2}-1$
[$\lambda^{(2)}_{\rm max}=1+(k-1)(1-\tfrac{2k}{D})$]. All remaining
$\binom{D}{2}-1$ eigenvalues of $\rho^{(2)}$ are small and identical to (see
Appendix A)

$\lambda_{\rm rest}^{(2)}=\frac{4k(k-1)}{D(D-2)}\leq 1\,,$ (40)

such that $\lambda^{(2)}_{\rm max}+(\binom{D}{2}-1)\lambda_{\rm
rest}^{(2)}=\binom{N}{2}$ (Eq. (16)). Since
$\langle\Psi|A^{\dagger}A|\Psi\rangle\leq 1$ in any SD (Eq. (33)), (39)
clearly signals as well that $|\Psi_{2k}\rangle$ is not a SD.

The dominant eigenvalue (39) is maximum in the half-filled case
$k=[\frac{D+2}{4}]$, where $\lambda^{(2)}_{{\rm max}}\approx D(1+2/D)^{2}/8$
increases linearly with $D$ for large $D$, and can then become arbitrarily
large. For $D\gg k$, $\lambda^{(2)}_{\rm max}\approx k=N/2$ is just the number
of pairs, whereas $\lambda^{(2)}_{\rm min}\approx(2k/D)^{2}$ becomes very
small. As seen from (35)–(36), for $D\rightarrow\infty$ at fixed $N$,
$A^{\dagger}$ becomes a “true” boson ($[A,A^{\dagger}]\rightarrow 1$), with
$|\Psi_{2k}\rangle\rightarrow\frac{(A^{\dagger})^{k}}{\sqrt{k!}}|0\rangle$ a
condensate of $k$-bosons.

Eigenvalues of odd $M$ DM’s can also exceed $1$. For instance, in an odd state
$|\Psi_{2k+1}\rangle=c^{\dagger}_{D+1}|\Psi_{2k}\rangle$, where we have
enlarged the sp space with one additional state, the largest eigenvalue of the
three-body DM $\rho^{(3)}$ is again given by (39):
$\langle\Psi_{2k+1}|c^{\dagger}_{D+1}A^{\dagger}A\,c_{D+1}|\Psi_{2k+1}\rangle=\lambda^{(2)}_{\rm
max}\geq 1$. It corresponds to the number of pairs times the number of “single
fermions” (1). The eigenvalues of $\rho^{(3)}$ in the even state (36) can also
be analytically determined (see Appendix A). Its largest eigenvalue,

${\textstyle\lambda^{(3)}_{\rm
max}=\frac{2k(k-1)(1-\frac{2}{D}(k-1))}{D-2}\,,}$ (41)

while smaller than (39), still satisfies $\lambda^{(3)}_{\rm max}>1$ for
$1+\sqrt{D/2}<k<D/2$, reaching $\approx\frac{2D}{27}$ at $k\approx\frac{D}{3}$
for $D\gg 1$. Fig. 1 depicts $\lambda^{(M)}_{\rm max}$ vs. $k$ for $M\leq 4$
in the states (36).

From (37) it also follows that
$(A^{\dagger})^{m}A^{m}|\Psi_{2k}\rangle\propto|\Psi_{2k}\rangle$ for $m\leq
k$. Thus, the largest eigenvalue of the $2m$-body DM in the state (36) is
associated to the normalized operator
$A^{(2m){\dagger}}=\frac{(A^{\dagger})^{m}}{m!\sqrt{(\frac{2}{D})^{m}\binom{D/2}{m}}}$
(see also Appendix A):

$\lambda^{(2m)}_{\rm
max}=\langle\Psi_{2k}|\frac{(A^{\dagger})^{m}A^{m}}{m!^{2}(\frac{2}{D})^{m}\binom{D/2}{m}}|\Psi_{2k}\rangle=\frac{\binom{k}{m}\binom{D/2-k+m}{m}}{\binom{D/2}{m}}\,,$
(42)

which generalizes Eq. (39) ($m=1$ case). Thus, $\lambda^{(2m)}_{\rm max}>1$
for $m<k<D/2$, with $\lambda^{(2m)}_{\rm max}\approx\binom{k}{m}$ if $D\gg k$.
Similarly, in the odd state
$|\Psi_{2k+1}\rangle=c^{{\dagger}}_{D+1}|\Psi_{2k}\rangle$, $\lambda_{\rm
max}^{(2m+1)}=\lambda_{\rm max}^{(2m)}$.

The eigenvalues of $\rho^{(M)}$ can also be all smaller than $1$. For
instance, in a $N=D/2$ GHZ-like fermion state

$|\Psi_{D/2}\rangle=\frac{1}{\sqrt{2}}(c^{\dagger}_{1}\ldots
c^{\dagger}_{D/2}+c^{\dagger}_{D/2+1}\ldots c^{\dagger}_{D})|0\rangle\,,$ (43)

all nonzero eigenvalues of $\rho^{(M)}$ are easily seen to be

$\lambda^{(M)}_{\nu}=1/2\,,$ (44)

for $1\leq M\leq N-1$, $2\binom{N}{M}$ degenerate. This example shows that
distinct $N$-fermion states which appear identical at the one-body level, like
(36) and (43) for $N=D/2$ ($\rho^{(1)}=\frac{1}{2}\mathbbm{1}$), can differ
significantly at higher $M$-body levels ($\lambda^{(2)}_{1}=1/2$ in (43) while
$\lambda^{(2)}_{1}=\frac{D}{8}(1+\frac{4}{D})>1$ in (36) for $k=D/4$ and
$D\geq 8$).

Figure 1: Maximum eigenvalue $\lambda_{\rm max}=\lambda^{(M)}_{\rm max}$ of
the $M$-body density matrix $\rho^{(M)}$ for $M\leq 4$ in the state (36), as a
function of the number of pairs $k$, for sp space dimension $D=30$ (all labels
and quantities plotted are dimensionless).

##  III $M$-body Entanglement

###  III.1 Mixedness of the $M$-body DM

The Schmidt-like decomposition (23) of an $N$-fermion pure state, and the fact
that the positive numbers $\lambda^{(M)}_{\nu}$ represent the non-zero
eigenvalues of both the $M$ and $(N-M)$-body DM’s, naturally lead to a notion
of $M$-body entanglement based on the spread of these eigenvalues. It
characterizes the correlations between $M$ and $(N-M)$-body observables in an
$N$-fermion state. More precisely, given two pure states $|\Psi\rangle$,
$|\Phi\rangle$ of $N$ fermions, we will say that $|\Psi\rangle$ is not less
$(M,N-M)$ entangled, or simply not less $M$-body entangled than
$|\Phi\rangle$, if $\rho^{(M)}_{\Psi}$ is more mixed than (or equally mixed
as) $\rho^{(M)}_{\Phi}$, i.e., if their eigenvalues satisfy the majorization
relation

$\bm{\lambda}(\rho_{\Psi}^{(M)})\prec\bm{\lambda}({\rho}_{\Phi}^{(M)})\,,$
(45)

where $\bm{\lambda}(\rho^{(M)})$ denotes the spectrum of $\rho^{(M)}$, sorted
in decreasing order. Explicitly, Eq. (45) means that all inequalities Bhatia
(1997); Marshall  _et al._ (2011)

$\sum_{\nu=1}^{j}\lambda_{\nu}(\rho^{(M)}_{\Psi})\leq\sum_{\nu=1}^{j}\lambda_{\nu}(\rho^{(M)}_{\Phi})\,,\;\;j=1,\ldots,{\textstyle\binom{D}{M}}-1\,,$
(46)

are to be satisfied by the sorted eigenvalues $\lambda_{\nu}$ of
$\rho^{(M)}_{\Psi}$ and $\rho^{(M)}_{\Phi}$, with equality for
$j=\binom{D}{M}$, implying that those of $\rho_{\Psi}^{(M)}$ are more spread
out than those of ${\rho}_{\Phi}^{(M)}$. Of course, one may likewise employ
the partner DM $\rho^{(N-M)}$ in (45)–(46) since they share the same non-zero
eigenvalues. For $M=1$ we recover the concept of one-body entanglement,
determined by the SPDM $\rho^{(1)}$ Gigena  _et al._ (2020); Gigena and
Rossignoli (2015).

Eq. (45) is analogous to that satisfied by local reduced states in systems of
distinguishable components (where it warrants that $|\Psi\rangle$ can be
converted to $|\Phi\rangle$ by LOCC operations Nielsen (1999); Nielsen and
Chuang (2000); Nielsen and Vidal (2001)). Notice, however, that majorization
provides a partial order, entailing that two states may be uncomparable
according to previous criterion.

For example, in an $N$-fermion SD, $\rho^{(M)}$ presents just $\binom{N}{M}$
non-zero eigenvalues equal to $1$, while in the GHZ-like state (43), all
$2\binom{N}{M}$ non-zero eigenvalues of $\rho^{(M)}$ are equal to $1/2$ for
$M<N$, implying

$\bm{\lambda}(\rho^{(M)}_{\Psi_{D/2}})\prec\bm{\lambda}(\rho^{(M)}_{\rm SD})$
(47)

for $1\leq M\leq N-1$. Then the state (43) is more entangled than a SD at any
$M$-body level ($1\leq M\leq N-1$).

However, in the pair condensate $|\Psi_{2k}\rangle$ of Eq. (36) with $2\leq
k\leq D/2-1$, while clearly
$\bm{\lambda}(\rho^{(1)}_{\Psi_{2k}})\prec\bm{\lambda}(\rho^{(1)}_{\rm SD})$
(Eq. (38)), neither
$\bm{\lambda}(\rho^{(2)}_{\Psi_{2k}})\prec\bm{\lambda}(\rho^{(2)}_{\rm SD})$
nor $\bm{\lambda}(\rho^{(2)}_{\rm
SD})\prec\bm{\lambda}(\rho^{(2)}_{\Psi_{2k}})$ are fulfilled, since the
largest eigenvalue of $\rho^{(2)}_{\Psi_{2k}}$ is greater than $1$ whereas
remaining $\binom{D}{2}-1$ eigenvalues are non-zero and lower than $1$ (Eqs.
(39)–(40)), with $\binom{D}{2}>\binom{N}{2}$ for $D>N$. Hence, SDs no longer
provide the least mixed two-body DM. The same occurs with the three-body DM
when its largest eigenvalue in the state (36) exceeds $1$ (Eq. (41)), in which
case $\bm{\lambda}(\rho^{(3)}_{\Psi_{2k}})\nprec\bm{\lambda}(\rho^{(3)}_{\rm
SD})$ and $\bm{\lambda}(\rho^{(3)}_{\rm
SD})\nprec\bm{\lambda}(\rho^{(3)}_{\Psi_{2k}})$.

###  III.2 $M$-body entropies

Associated with previous definition (45), we may first consider the $M$-body
entropies

$\displaystyle E_{f}^{(M)}(|\Psi\rangle)$ $\displaystyle=$ $\displaystyle
S_{f}(\rho_{\Psi}^{(M)})=S_{f}(\rho_{\Psi}^{(N-M)})\,,$ (48a) $\displaystyle=$
$\displaystyle\sum_{\nu}f[\lambda_{\nu}(\rho^{(M)}_{\Psi})]\,,$ (48b)

where $S_{f}(\rho)={\rm Tr}f(\rho)$ is a trace-form entropy Canosa and
Rossignoli (2002); Rossignoli and Canosa (1999), with $f:\mathbb{R}_{\geq
0}\rightarrow\mathbb{R}_{\geq 0}$ a strictly concave non-negative function
complying with $f(0)=0$. These entropies will satisfy

$E_{f}^{(M)}(|\Psi\rangle)\geq E_{f}^{(M)}(|\Phi\rangle)$ (49)

whenever Eq. (45) is fulfilled Bhatia (1997); Marshall  _et al._ (2011);
Rossignoli and Canosa (2002).

Eq. (48) is particularly suitable for defining a one-body entanglement entropy
Gigena and Rossignoli (2015); Gigena  _et al._ (2020) since
$\lambda^{(1)}_{\nu}\in[0,1]$ and standard entropic measures can be employed.
For $M\geq 2$ it is possible to employ measures such as the bosonic-like
entropy, obtained for
$f(\lambda)=-\lambda\log\lambda+(1+\lambda)\log(1+\lambda)$ (such that
$\sum_{\nu}f(\lambda_{\nu})$ represents the von Neumann entropy of independent
bosons in the grand canonical ensemble with average occupation numbers
$\lambda_{\nu}$ Rossignoli and Canosa (1999)), which is just an example of an
increasing concave function of $\lambda$ satisfying $f(0)=0$.

A second possibility, strongly motivated by the majorization relations derived
in the next subsections, is to consider the entropy of the normalized
densities

$\rho_{n}^{(M)}=\rho^{(M)}/{\textstyle\binom{N}{M}}\,,$ (50)

which have eigenvalues $\lambda^{(M)}_{\nu}/\binom{N}{M}\in[0,1]$ and satisfy
${\rm Tr}[\rho_{n}^{(M)}]=1$. In this case we may employ any entropic measure
$S_{f}(\rho)={\rm Tr}\,f(\rho)$ intended for standard probabilities and define
an associated $M$-body entropy as

$\displaystyle E_{n_{f}}^{(M)}(|\Psi\rangle)$ $\displaystyle=$ $\displaystyle
S_{f}(\rho_{n\Psi}^{(M)})=S_{f}(\rho_{n\Psi}^{(N-M)})\,.$ (51)

In particular, the von Neumann entropy $S(\rho)=-{\rm Tr}\rho\log_{2}\rho$
leads to $S(\rho^{(M)}_{n})=S(\rho^{(M)})/\binom{N}{M}+\log_{2}\binom{N}{M}$.
Other Schur-concave functions Bhatia (1997) of $\rho_{n}^{(M)}$ can also be
used. Since at fixed $N$ Eq. (45) is fully equivalent to

$\bm{\lambda}(\rho^{(M)}_{n\Psi})\prec\bm{\lambda}(\rho^{(M)}_{n\Phi})\,,$
(52)

as all eigenvalues are just rescaled by the same factor, it will also imply
$E_{n_{f}}^{(M)}(|\Psi\rangle)\geq E_{n_{f}}^{(M)}(|\Phi\rangle)$. And any
pair of $N$-fermion states states uncomparable with (45) will remain
uncomparable with (52), and viceversa.

On the other hand, Eq. (52) can also be used to compare the mixedness of
reduced DMs $\rho^{(M)}_{n}$ for states with distinct $N$, as $\rho^{(M)}_{n}$
has fixed trace, implying

$\bm{\lambda}(\rho_{n\Psi}^{(M)})\prec\bm{\lambda}(\rho_{n\Phi}^{(M)})\Rightarrow
E_{n_{f}}^{(M)}(|\Psi\rangle)\geq E_{n_{f}}^{(M)}(|\Phi\rangle)\,.$ (53)

We finally remark that the converse of Eq. (53) does not hold in general: Only
if the entropic inequality $E_{n_{f}}^{(M)}(|\Psi\rangle)\geq
E_{n_{f}}^{(M)}(|\Phi\rangle)$ holds for all concave functions $f$ (and not
just a particular choice) it can be ensured that
$\bm{\lambda}(\rho_{n\Psi}^{(M)})\prec\bm{\lambda}(\rho_{n\Phi}^{(M)})$
Rossignoli and Canosa (2002). And this implies
$\bm{\lambda}(\rho_{\Psi}^{(M)})\prec\bm{\lambda}(\rho_{\Phi}^{(M)})$ only
when $\rho_{\Psi}^{(M)}$ and $\rho_{\Phi}^{(M)}$ have the same trace, i.e.
$|\Psi\rangle$ and $|\Phi\rangle$ the same fermion number. For states with
distinct $N$ Eq. (45) is to be replaced by (52).

###  III.3 Operations not increasing the $M$-body entropy

Let us now determine the basic operations which do not increase the $M$-body
entropy (51). We first note that one-body unitary transformations

$|\Psi\rangle\rightarrow{\cal U}|\Psi\rangle\,,\;\;{\cal
U}=\exp[-i\sum_{i,j}H_{ij}c^{\dagger}_{i}c_{j}]\,,$ (54)

where $H^{\dagger}=H$, lead to unitary transformations of all $M$-body DMs,
thus leaving their eigenvalues and hence $M$-body entanglement unchanged:
Since ${\cal U}^{\dagger}c_{i}{\cal U}=\sum_{j}U_{ij}c_{j}$ with
$U=\exp[-iH]$, then $\rho^{(1)}\rightarrow U\rho^{(1)}U^{\dagger}$ and
$\rho^{(M)}\rightarrow U^{(M)}\rho^{(M)}U^{(M){\dagger}}$, with
$U^{(M)}_{\bm{\alpha\alpha}^{\prime}}=\epsilon_{i_{1}\ldots
i_{m}}U_{\alpha_{1}\alpha^{\prime}_{i_{1}}}\ldots
U_{\alpha_{m}\alpha^{\prime}_{i_{m}}}$ and $\epsilon$ the fully antisymmetric
tensor.

We now show, for both pure states $|\Psi\rangle$ with fixed fermion number $N$
and also mixed states

$\rho=\sum_{i}q_{i}|\Psi_{i}\rangle\langle\Psi_{i}|\,,$ (55)

with definite $N$ ($q_{i}\geq 0$, $\sum_{i}q_{i}=1$ and
$\hat{N}|\Psi_{i}\rangle=N|\Psi_{i}\rangle$ $\forall$ $i$), the following
theorem:
Theorem 1. The quantum operation described by the Kraus operators

${\cal M}^{(1)}_{j}=\frac{c_{j}}{\sqrt{N}}\,,\;\;j=1,\ldots,D$ (56)

which satisfy $\sum_{j=1}^{D}{{\cal M}^{(1)}_{j}}^{\dagger}{\cal
M}^{(1)}_{j}=\hat{N}/N=\mathbbm{1}_{N}$ within the subspace of states with
definite fermion number $N$, does not increase, on average, the mixedness of
the normalized $M$-body DMs $\rho^{(M)}_{n}=\rho^{(M)}/\binom{N}{M}$ for
$1\leq M\leq N-1$, and implies the majorization relation

$\bm{\lambda}(\rho^{(M)}_{n})\prec\sum_{j}p_{j}\bm{\lambda}(\rho^{(M)}_{jn})\,,$
(57)

between the spectrum of the initial and post-measurement normalized $M$-body
DM’s.

In (57) $\rho^{(M)}_{jn}=\rho^{(M)}_{j}/\binom{N-1}{M}$ are the normalized
$M$-body DMs determined by the post-selected states

$\displaystyle\rho_{j}$ $\displaystyle=$ $\displaystyle p_{j}^{-1}\,{\cal
M}_{j}^{(1)}\rho{{\cal M}^{(1)}_{j}}^{\dagger}=\frac{c_{j}\rho
c^{\dagger}_{j}}{\langle c^{\dagger}_{j}c_{j}\rangle}\,,$ (58)

while $p_{j}$ is the probability of outcome $j$:

$\displaystyle p_{j}$ $\displaystyle=$ $\displaystyle{\rm Tr}\,[\rho\,{{\cal
M}^{(1)}_{j}}^{\dagger}{\cal M}_{j}^{(1)}]=\langle
c^{\dagger}_{j}c_{j}\rangle/N\,,$ (59)

with $\sum_{j=1}^{D}p_{j}=1$. This measurement, with $D$ distinct outcomes,
corresponds for instance to the detection of a single fermion through its
momentum or sp energy (labelled by $j$), with $\rho_{j}$ the ensuing state of
remaining fermions. Notice that Eq. (57) implies

$S_{f}(\rho^{(M)}_{n})\geq
S_{f}[\sum_{j}p_{j}\bm{\lambda}(\rho^{(M)}_{jn})]\geq\sum_{j}p_{j}S_{f}(\rho^{(M)}_{jn})\,,$
(60)

for any Schur-concave function of $\rho^{(M)}_{n}$, including in particular
the entropies (51). Thus, Theorem 1 implies that the entropy
$S_{f}(\rho^{(M)}_{n})$ of the normalized $M$-body DMs will not increase, on
average, after such operation. For pure states this means that the $M$-body
entropy (51) will on average not increase, and will in general decrease, after
such measurement:

$E_{n_{f}}^{(M)}(|\Psi\rangle)\geq\sum_{j}p_{j}E_{n_{f}}^{(M)}(|\Psi_{j}\rangle)\,,$
(61)

where $|\Psi_{j}\rangle={\cal
M}_{j}|\Psi\rangle/\sqrt{p_{j}}=c_{j}|\Psi\rangle/\sqrt{\langle
c^{\dagger}_{j}c_{j}\rangle}$ is the state after outcome $j$. For mixed states
$\rho$ with definite $N$, Eq. (61) implies a similar inequality

$E^{(M)}_{n_{f}}(\rho)\geq\sum_{j}p_{j}E^{(M)}_{n_{f}}(\rho_{j})\,,$ (62)

for the convex-roof extension of (51), the $M$-body entanglement of formation
$E^{(M)}_{n_{f}}(\rho)={\rm
Min}\sum_{i}q_{i}E^{(M)}_{n_{f}}(|\Psi_{i}\rangle)$, where the minimum is over
all representations (55) of $\rho$ as convex mixture of pure states with
definite $N$. Eq. (62) follows from (61) by using the representation of $\rho$
minimizing $E_{n_{f}}^{(M)}(\rho)$, such that $E_{n_{f}}^{(M)}(\rho)$
$=\sum_{i}q_{i}E_{n_{f}}^{(M)}(|\Psi_{i}\rangle)\geq\sum_{i,j}q_{i}p_{ij}E_{n_{f}}^{(M)}(|\Psi_{ij}\rangle)\geq\sum_{j}p_{j}E_{n_{f}}^{(M)}(\rho_{j})$,
where $p_{j}=\sum_{i}q_{i}p_{ij}$ and
$p_{ij}=\langle\Psi_{i}|c^{\dagger}_{j}c_{j}|\Psi_{i}\rangle/N$.

Proof of Eq. (57): Let $\rho$ be the state of an $N$ fermion system upon which
the operation defined by the operators (56) is performed. After outcome $j$ is
obtained, the elements of the $M$-body DM $\rho^{(M)}_{j}$ determined by the
ensuing state (58) are, for $M\leq N-1$,

$\displaystyle({\rho^{(M)}_{j}})_{\bm{\alpha\alpha}^{\prime}}$
$\displaystyle=$ $\displaystyle p_{j}^{-1}\,{\rm Tr}\,{\cal
M}_{j}^{(1)}\rho{\cal
M}^{(1){\dagger}}_{j}C^{(M){\dagger}}_{\bm{\alpha}^{\prime}}C^{(M)}_{\bm{\alpha}}$
(63) $\displaystyle=$ $\displaystyle p_{j}^{-1}\,{\rm
Tr}\,\rho\,C^{(M){\dagger}}_{\bm{\alpha}^{\prime}}{{\cal
M}^{(1)}_{j}}^{\dagger}{\cal M}_{j}^{(1)}C^{(M)}_{\bm{\alpha}},$ (64)

where the last line holds because operators $C^{(M)}_{\bm{\alpha}}$ and ${\cal
M}_{j}^{(1)}$ either commute or anticommute. Then

$\sum_{j}p_{j}(\rho^{(M)}_{j})_{\bm{\alpha}\bm{\alpha}^{\prime}}={\rm
Tr}\,[\rho\,C^{(M){\dagger}}_{\bm{\alpha}^{\prime}}\frac{\hat{N}}{N}C_{\bm{\alpha}}^{(M)}]=\frac{N-M}{N}\rho^{(M)}_{\bm{\alpha}\bm{\alpha}^{\prime}},$
(65)

$\forall$ $\bm{\alpha},\bm{\alpha}^{\prime}$, implying
$\sum_{j}p_{j}\rho^{(M)}_{j}=\frac{N-M}{N}\rho^{(M)}$ and hence

$\rho^{(M)}/{\textstyle\binom{N}{M}}=\sum_{j}p_{j}\,\rho^{(M)}_{j}/{\textstyle\binom{N-1}{M}}\,,$
(66)

for the normalized $M$-body densities, i.e.,

$\rho^{(M)}_{n}=\sum_{j}p_{j}\rho^{(M)}_{jn}\,.$ (67)

This equation implies the majorization relation (57) since

$\bm{\lambda}(\sum_{i}A_{i})\prec\sum_{i}\bm{\lambda}(A_{i})$ (68)

for any set of hermitian matrices $A_{i}$ of the same dimension Nielsen and
Vidal (2001); Nielsen (2001); Bhatia (1997); Marshall  _et al._ (2011). ∎

In Gigena  _et al._ (2020) we have shown that one-body entanglement, i.e., the
one quantified by the mixedness of the SPDM $\rho^{(1)}$, is also non
increasing under measurements of the occupancy of a fixed sp state. Such
measurement, with just two-possible outcomes ($1$ or $0$), is described by the
number conserving projection operators ${\cal
P}_{j}=c^{\dagger}_{j}c_{j}\,,\;\;\;{\cal
P}_{\bar{j}}=c_{j}c^{\dagger}_{j}=\mathbbm{1}-{\cal P}_{j}$ and leads to the
majorization relation Gigena  _et al._ (2020)

$\bm{\lambda}(\rho^{(1)})\prec
n_{j}\bm{\lambda}(\rho^{(1)}_{j})+(1-n_{j})\bm{\lambda}(\rho^{(1)}_{\bar{j}})\,,$
(69)

between the spectra of the initial and post-selected SPDMs, where
$n_{j}=\langle\Psi|c^{\dagger}_{j}c_{j}|\Psi\rangle$ and $\rho^{(1)}_{j}$,
$\rho^{(1)}_{\bar{j}}$ are the SPDM’s determined by the post-selected states
$|\Psi_{j}\rangle=c^{\dagger}_{j}c_{j}|\Psi\rangle/\sqrt{n_{j}}$ and
$|\Psi_{\bar{j}}\rangle=c_{j}c^{\dagger}_{j}|\Psi\rangle/\sqrt{1-n_{j}}$. For
an $N$-fermion state $|\Psi\rangle$ an identical relation obviously holds for
the normalized DMs $\rho^{(1)}_{n}=\rho^{(1)}/N$,
$\rho^{(1)}_{jn}=\rho^{(1)}_{j}/N$.

However, Eq. (69) does not hold, in general, for $\rho^{(M)}$ with $M\geq 2$,
implying that $M$-body entanglement will not necessarily decrease after such
measurement. A simple analytic example is provided in Appendix B. Essentially,
measurement of the occupancy of a sp state reduces the available sp space for
“collective pairs” in a state like (36), implying a lower maximum eigenvalue
$\lambda^{(M)}_{1}$ in the post-measurement states and hence violation of the
inequality (69) by $\rho^{(M)}$ with $M\geq 2$. This result is expected as
these measurements do not necessarily increase our knowledge of the $M$-body
DM for $M\geq 2$.

###  III.4 $M$-body density operators and generalized majorization relations

To each $M$-body DM we can associate an $M$-body density operator (DO)

$\hat{\rho}^{(M)}=\sum_{\bm{\alpha},\bm{\alpha}^{\prime}}\rho_{\bm{\alpha}\bm{\alpha}^{\prime}}^{(M)}C_{\bm{\alpha}}^{(M){\dagger}}|0\rangle\langle
0|C_{\bm{\alpha}^{\prime}}^{(M)}\,,$ (70)

which is the unique mixed state of $M$ fermions satisfying

${\rm
Tr}\,[\hat{\rho}^{(M)}C^{(M){\dagger}}_{\bm{\alpha}^{\prime}}C_{\bm{\alpha}}^{(M)}]=\rho^{(M)}_{\bm{\alpha}\bm{\alpha}^{\prime}}$
(71)

$\forall$ $\bm{\alpha},\bm{\alpha}^{\prime}$, due to Eq. (3). The normal
decomposition (23)–(II.2) provides its diagonal representation:

$\hat{\rho}^{(M)}=\sum_{\nu}\lambda_{\nu}^{(M)}A_{\nu}^{(M){\dagger}}|0\rangle\langle
0|A_{\nu}^{(M)}\,.$ (72)

Now consider again the measurement (56) applied on a general mixed state
$\rho$ of $N$ fermions. If the result is unknown, the post-measurement state
(with no post-selection) $\rho^{\prime}=\sum_{j}p_{j}\rho_{j}$, is the $N-1$
fermion state

$\rho^{\prime}=\sum_{j}{\cal M}^{(1)}_{j}\,\rho\,{\cal
M}_{j}^{(1)^{\dagger}}=\frac{1}{N}\sum_{j}c_{j}\,\rho\,c_{j}^{\dagger}\,.$
(73)

Using Eq. (65) for $M=N-1$ we obtain

${\rm
Tr}\,[\rho^{\prime}\,C^{(N-1){\dagger}}_{\bm{\alpha}^{\prime}}C^{(N-1)}_{\bm{\alpha}}]=\frac{1}{N}\rho^{(N-1)}_{\bm{\alpha}\bm{\alpha}^{\prime}}$
(74)

$\forall$ $\bm{\alpha},\bm{\alpha}^{\prime}$, implying
$\rho^{\prime}=\hat{\rho}_{n}^{(N-1)}:=\hat{\rho}^{(N-1)}/N$, the normalized
$(N-1)$-body DO (${\rm Tr}\,\hat{\rho}^{(N-1)}_{n}=1$). In summary,

$\hat{\rho}^{(N-1)}=\sum_{j}c_{j}\rho c_{j}^{\dagger}\,,$ (75)

The $(N-1)$-body DO is then just proportional to the post-measurement state
(73), with the operation in (75) playing the role of a partial trace.
Expressions similar or equivalent to (75) have been previously derived in
refs. Carlson and Keller (1961); Amosov and Filippov (2017); da Silva Souza
_et al._ (2018), with Amosov and Filippov (2017) discussing its extension to
states with no fixed fermion number.

These results, together with Eqs. (57)–(62), can be extended to $L$-body
measurements, in which $L$ fermions are annihilated. Such measurement can be
obtained by applying previous measurement $L$ times, i.e., by composing it
with itself $L$ times, and is described by the Kraus operators

${\cal
M}^{(L)}_{\bm{\beta}}=\frac{C^{(L)}_{\bm{\beta}}}{\sqrt{\binom{N}{L}}}=\frac{c_{\beta_{1}}\ldots
c_{\beta_{L}}}{\sqrt{\binom{N}{L}}}\,,$ (76)

which, due to Eq. (4) satisfy

$\sum_{\bm{\beta}}{\cal M}^{(L){\dagger}}_{\bm{\beta}}{\cal
M}^{(L)}_{\bm{\beta}}={\textstyle\frac{1}{\binom{N}{L}}}\sum_{\bm{\beta}}C^{(L){\dagger}}_{\bm{\beta}}C_{\bm{\beta}}^{(L)}=\mathbbm{1}_{N}$
(77)

within the subspace of $N$-fermion states. Then,

$\displaystyle{\rm
Tr}\sum_{\bm{\beta}}C_{\bm{\beta}}^{(L)}\rho\,C_{\bm{\beta}}^{(L){\dagger}}C^{(M){\dagger}}_{\bm{\alpha}^{\prime}}C^{(M)}_{\bm{\alpha}}$
$\displaystyle=$
$\displaystyle{\textstyle\binom{N-M}{L}}\rho^{(M)}_{\bm{\alpha}\bm{\alpha}^{\prime}}$
(78)

$\forall$ $\bm{\alpha},\bm{\alpha}^{\prime}$ for $M\leq N-L$. Hence, for
$L=N-M$ it implies

$\displaystyle\hat{\rho}^{(M)}$ $\displaystyle=$
$\displaystyle\sum_{\bm{\beta}}C_{\bm{\beta}}^{(N-M)}\rho\,C_{\bm{\beta}}^{(N-M){\dagger}}$
$\displaystyle=$
$\displaystyle{\textstyle\frac{1}{(N-M)!}}\\!\\!\\!\sum_{j_{1},\ldots,j_{N-M}}\\!\\!\\!c_{j_{1}}\ldots
c_{j_{N-M}}\rho\,c^{\dagger}_{j_{N-M}}\ldots c^{\dagger}_{j_{1}}$
$\displaystyle=$
$\displaystyle{\textstyle\binom{N}{N-M}}\sum_{\bm{\beta}}{\cal
M}_{\bm{\beta}}^{(N-M)}\,\rho\,{\cal M}_{\bm{\beta}}^{(N-M){\dagger}}\,,$ (80)

which generalizes Eq. (75). The sum in (80) is the post-measurement state
(without post-selection) of the $L=(N-M)$-fermion measurement (76).

For general $L\leq N$, this measurement has $\binom{D}{L}$ distinct outcomes
$\bm{\beta}$, with probabilities

$p_{\bm{\beta}}={\rm Tr}\,\rho\,{\cal M}_{\bm{\beta}}^{(L){\dagger}}{\cal
M}_{\bm{\beta}}^{(L)}=\rho^{(L)}_{\bm{\beta}\bm{\beta}}/{\textstyle\binom{N}{L}}\,,$
(81)

satisfying $\sum_{\bm{\beta}}p_{\bm{\beta}}=1$, and post-selected states

$\rho_{\bm{\beta}}=p_{\bm{\beta}}^{-1}\,{\cal M}_{\bm{\beta}}^{(L)}\rho{\cal
M}^{(L){\dagger}}_{\bm{\beta}}=C_{\bm{\beta}}^{(L)}\rho\,C_{\bm{\beta}}^{(L){\dagger}}/\rho^{(L)}_{\bm{\beta}\bm{\beta}}\,,$
(82)

which generalize Eqs. (58)–(59). From Eq. (78) we then obtain, for the ensuing
conditional $M$-body DMs of elements
$(\rho^{(M)}_{\bm{\beta}})_{\bm{\alpha}\bm{\alpha}^{\prime}}={\rm
Tr}\,\rho_{\bm{\beta}}\,C^{(M){\dagger}}_{\bm{\alpha}^{\prime}}C^{(M)}_{\bm{\alpha}}$,

$\sum_{\bm{\beta}}p_{\bm{\beta}}\rho^{(M)}_{\bm{\beta}}=\frac{\binom{N-M}{L}}{\binom{N}{L}}\rho^{(M)}\,,$
(83)

for $M\leq N-L$. Eq. (83) is equivalent to

$\sum_{\bm{\beta}}p_{\bm{\beta}}\rho^{(M)}_{\bm{\beta}}/{\textstyle\binom{N-L}{M}}=\rho^{(M)}/{\textstyle\binom{N}{M}}\,,$
(84)

and hence to

$\rho^{(M)}_{n}=\sum_{\beta}p_{\bm{\beta}}\rho^{(M)}_{\bm{\beta}n}\,,$ (85)

for the normalized $M$-body DMs. Eq. (85) then implies the general
majorization relation

$\bm{\lambda}(\rho^{(M)}_{n})\prec\sum_{\beta}p_{\bm{\beta}}\bm{\lambda}(\rho^{(M)}_{\bm{\beta}n})\,,$
(86)

between the initial and post-measurement conditional normalized $M$-body DMs,
which generalizes Eq. (57) to the measurement described by the operators (76).
The associated entropies satisfy

$S_{f}(\rho^{(M)}_{n})\geq\sum_{\bm{\beta}}p_{\bm{\beta}}S_{f}(\rho^{(M)}_{\bm{\beta}n})\,.$
(87)

For pure states this implies that the entanglement entropies (51) will not
increase, on average, after these measurements. The same occurs with the
associated entanglement of formation for initial mixed states.

###  III.5 Mapping to bipartite systems

In the previous sections it was proposed to link the mixedness of either
$\rho^{(M)}$ or $\rho^{(N-M)}$, as reduced DM’s of a given $N$-fermion state,
to the amount of correlations between $M$-body and $N-M$-body observables on
such state. In this section we will show this relation is operationally
justified by the existence of a class of quantum maps converting states of
indistinguishable fermions into states of effectively distinguishable
fermions, in such a way that entanglement in the target state is bounded by
the entropy (51) of the normalized $M$-body DM.

Let $|\Psi\rangle$ be an $N$-fermion state with support in a sp subspace
${\cal H}$ of dimension $D\geq N$ and let ${\cal H}_{A}$ be a sp subspace of
dimension $D_{A}\geq M$ orthogonal to ${\cal H}$, such that
$\\{c_{i_{A}},c_{j}\\}=\\{c^{\dagger}_{i_{A}},c^{\dagger}_{j}\\}=\\{c_{i_{A}},c^{{\dagger}}_{j}\\}=0$
for $i_{A}$ ($j$) labelling sp states in ${\cal H}_{A}$ (${\cal H}$)
(entailing $\langle i_{A}|j\rangle=\langle
0|c_{i_{A}}c^{\dagger}_{j}|0\rangle=0$ $\forall\,i_{A},j$). They are of course
subspaces of a complete sp space ${\cal H}_{T}$ such that ${\cal H}\oplus{\cal
H}_{A}\subset{\cal H}_{T}$. Consider now a CPTP map ${\cal T}_{M}$ described
by Kraus operators

$\displaystyle{\cal T}^{r}$ $\displaystyle=$
$\displaystyle{\textstyle\frac{1}{\sqrt{\binom{N}{M}}}}\sum_{\bm{\mu},\bm{\alpha}}T^{r}_{\bm{\mu}\bm{\alpha}}\,C^{(M){\dagger}}_{\bm{\mu}}C^{(M)}_{\bm{\alpha}}\,,$
(88)

where
$C^{(M){\dagger}}_{\bm{\mu}}=(c^{\dagger}_{i_{1_{A}}},\ldots,c^{\dagger}_{i_{M_{A}}})$
creates $M$ fermions in ${\cal H}_{A}$ while
$C^{(M)}_{\bm{\alpha}}=(c_{j_{M}},\ldots,c_{j_{1}})$ annihilates $M$ fermions
in ${\cal H}$, and $T^{r}$ is a $\binom{D_{A}}{M}\times\binom{D}{M}$ matrix.
Assuming ${\cal H}_{A}$ initially “empty” and the condition
$\sum_{r}T^{r{\dagger}}T^{r}=\mathbbm{1}$, they will satisfy

$\sum_{r}{\cal T}^{r{\dagger}}{\cal
T}^{r}={\textstyle\frac{1}{\binom{N}{M}}}\sum_{\bm{\alpha},\bm{\alpha}^{\prime},r}(T^{r{\dagger}}T^{r})_{\bm{\alpha}^{\prime}\bm{\alpha}}C^{(M){\dagger}}_{\bm{\alpha}^{\prime}}C^{(M)}_{\bm{\alpha}}=\mathbbm{1}_{N}$

by virtue of Eq. (4), within the Fock space ${\cal F}_{N}({\cal H})$ of $N$
fermion states with support in ${\cal H}$. Its action on an $N$-fermion state
$|\Psi\rangle\in{\cal F}_{N}({\cal H})$ is, using Eqs. (8) and (12),

$\displaystyle{\cal T}^{r}|\Psi\rangle$ $\displaystyle=$
$\displaystyle{\textstyle\frac{1}{\sqrt{\binom{N}{M}}}}\sum_{\bm{\mu},\bm{\alpha},\bm{\beta}}T^{r}_{\bm{\mu}\bm{\alpha}}\Gamma^{(M)}_{\bm{\alpha}\bm{\beta}}C_{\bm{\mu}}^{(M){\dagger}}C_{\bm{\beta}}^{(N-M){\dagger}}|0\rangle$
(89) $\displaystyle=$
$\displaystyle\sum_{\bm{\mu},\bm{\beta}}\Gamma^{r}_{\bm{\mu}\bm{\beta}}C_{\bm{\mu}}^{(M){\dagger}}C_{\bm{\beta}}^{(N-M){\dagger}}|0\rangle\,,$

where $\Gamma^{r}=T^{r}\Gamma^{(M)}/\sqrt{\binom{N}{M}}$. It is verified that
the probability of outcome $r$,

$p_{r}=\langle\Psi|{\cal T}^{r{\dagger}}{\cal T}^{r}|\Psi\rangle={\rm
Tr}(\Gamma^{r{\dagger}}\Gamma^{r})\,,$ (90)

satisfies $\sum_{r}p_{r}={\frac{1}{\binom{N}{M}}}{\rm
Tr}(\Gamma^{(M){\dagger}}\Gamma^{(M)})=1$ due to Eq. (16).

Operators (88) can therefore be regarded as Kraus operators describing a CPTP
map on the $N$-fermion Fock space that takes, with probability $p_{r}$, the
original state $|\Psi\rangle$ into the $(M,N-M)$ “bipartite” state
$|\Phi_{AB}^{r}\rangle=(\sqrt{p_{r}})^{-1}{\cal T}^{r}|\Psi\rangle$ containing
$N-M$ fermions in ${\cal H}$ and $M$ fermions in the orthogonal sp space
${\cal H}_{A}$.

After the map is implemented with outcome $r$, we can look at the reduced
state of the fermionic modes in ${\cal H}_{A}$ in the state
$|\Phi_{AB}^{r}\rangle$ (which will contain $M$ fermions),
${\rho}_{A}^{r}={\rm Tr}_{B}|\Phi^{r}_{AB}\rangle\langle\Phi^{r}_{AB}|$, such
that $\langle\Phi_{AB}^{r}|O_{A}|\Phi_{AB}^{r}\rangle={\rm
Tr}\,[\rho_{A}^{r}O_{A}]$ for any number conserving operator $O_{A}$ depending
just on operators $c^{\dagger}_{i_{A}}$, $c_{i_{A}}$ with support in ${\cal
H}_{A}$. Using (89) we obtain
${\rho}_{A}^{r}=\sum_{\bm{\mu},\bm{\mu}^{\prime}}(\rho_{A}^{r})_{\bm{\mu}\bm{\mu}^{\prime}}C^{(M){\dagger}}_{\bm{\mu}}|0\rangle\langle
0|C^{(M)}_{\bm{\mu}^{\prime}}$ with

$\rho_{A}^{r}=\Gamma^{r}\Gamma^{r{\dagger}}/p_{r}\,.$ (91)

Similarly $\rho_{B}^{r}=\Gamma^{rT}\Gamma^{r*}/p_{r}$.

As a direct consequence of expression (91) we can derive the following
important result: If $\bm{\lambda}(\rho^{r}_{A})$ and
$\bm{\lambda}(\rho^{(M)}_{n})$ denote again the eigenvalue vectors (sorted in
decreasing order) of $\rho_{A}^{r}$ and
$\rho^{(M)}_{n}=\rho^{(M)}/\binom{N}{M}$ respectively, then the following
majorization relation holds:

$\bm{\lambda}(\rho^{(M)}_{n})\prec\sum_{r}p_{r}\bm{\lambda}(\rho_{A}^{r}),$
(92)

where the vector of smaller dimension is assumed to be completed with zeros to
match the dimensions.

###### Proof.

From Eqs. (91) and (14) we obtain

$p_{r}\rho_{A}^{r}={\textstyle\frac{1}{\binom{N}{M}}}T^{r}\Gamma^{(M)}\Gamma^{(M){\dagger}}T^{r{\dagger}}=T^{r}\rho^{(M)}_{n}T^{r{\dagger}}\,.$

We can also write

$\rho^{(M)}_{n}=\sum_{r}\sqrt{\rho^{(M)}_{n}}T^{r{\dagger}}T^{r}\sqrt{\rho^{(M)}_{n}}\,,$

which implies, using again (68),

$\bm{\lambda}(\rho^{(M)}_{n})\prec\sum_{r}\bm{\lambda}\left(\sqrt{\rho^{(M)}_{n}}T^{r{\dagger}}T^{r}\sqrt{\rho^{(M)}_{n}}\right)=\sum_{r}p_{r}\bm{\lambda}(\rho_{A}^{r})$
(93)

since the nonzero eigenvalues of
$(\sqrt{\rho^{(M)}_{n}}T^{r{\dagger}})(T^{r}\sqrt{\rho^{(M)}_{n}})$ are the
same as those of
$T^{r}\sqrt{\rho^{(M)}_{n}}\sqrt{\rho^{(M)}_{n}}T^{r{\dagger}}=p_{r}\rho_{A}^{r}$.
∎

In particular, Eq. (92) implies that the average bipartite entanglement
entropy between $A$ and $B$ of the final states, defined by
$\sum_{r}p_{r}E_{f}(|\Phi^{r}_{AB}\rangle)$ with
$E_{f}(|\Phi^{r}_{AB}\rangle=S_{f}(\rho_{A}^{r})=S_{f}(\rho_{B}^{r})$, will
never exceed the $M$-body entropy (51), which provides, therefore, an upper
bound to the generated bipartite entanglement:

$E_{f_{n}}^{(M)}(|\Psi\rangle)=S_{f}(\rho_{n}^{(M)})\geq\sum_{r}p_{r}S_{f}(\rho_{A}^{r})\,.$
(94)

A second immediate corollary of (92) is that we can derive a condition for the
conversion of a pure $N$-fermion state $|\Psi\rangle$ into a bipartite state
$|\Phi_{AB}\rangle$ of $M$ and $N-M$ fermions in orthogonal sp spaces, i.e.,
with $M$ fermions occupying sp states in ${\cal H}_{A}$, and $N-M$ in ${\cal
H}$, by means of a CPTP map described by the operators (88): for such
conversion to be possible the relation

$\bm{\lambda}(\rho^{(M)}_{n})\prec\bm{\lambda}(\rho_{A}),$ (95)

must hold, where $\rho_{A}$ is the reduced state of $A$ in
$|\Phi_{AB}\rangle$.

###### Proof.

If the map takes $|\Psi\rangle$ into $|\Phi_{AB}\rangle$ and is described by
the operators (88), then for all $r$ we have

${\cal T}^{r}|\Psi\rangle=\sqrt{p_{r}}|\Phi_{AB}\rangle,$ (96)

with $p_{r}$ given by (90). It follows that $\rho^{r}_{A}=\rho_{A}$ $\forall$
$r$, and by virtue of (92), Eq. (95) follows. ∎

We have therefore identified a class of quantum maps that could be used to
transform a pure state $|\Psi\rangle$ of $N$ indistinguishable fermions into
another state $|\Phi_{AB}\rangle$ which contains a definite number of fermions
in two orthogonal sp subspaces, such that $M$ fermions can be effectively
“distinguished” from the remaining $N-M$ due the orthogonality (for instance,
$M$ fermions confined in a spatial region well separated from that of the
remaining $N-M$ fermions, or in orbitals orthogonal to those occupied in
$|\Psi\rangle$). The conversion is such that the entanglement between these
two sets of particles in the target state is bounded from above, on average,
by the entropy of the normalized $M$-body DM. This provides a clear
operational meaning to the suggested link between the mixedness of the
eigenvalues of this matrix and the amount of correlations between the
indistinguishable particles in $|\Psi\rangle$.

As a trivial example, consider a two fermion state

$|\Psi\rangle=(\alpha c^{\dagger}_{1}c^{\dagger}_{2}+\beta
c^{\dagger}_{3}c^{\dagger}_{4})|0\rangle\,.$ (97)

As mentioned in II.2, any two-fermion state with support in a sp subspace of
dimension $4$ (i.e. occupying just 4 sp levels) can be written in this way
Schliemann  _et al._ (2001b, a); Gigena and Rossignoli (2015). Its SPDM
$\rho^{(1)}$ is diagonal in this basis, with $\langle
c^{\dagger}_{j}c_{i}\rangle=\delta_{ij}f_{i}$ and $f_{i}=|\alpha|^{2}$ for
$i=1,2$, $f_{i}=|\beta^{2}|$ for $i=3,4$ and $f_{i}=0$ otherwise, such that
its nonzero eigenvalues are
$\bm{\lambda}(\rho^{(1)})=(|\alpha|^{2},|\alpha|^{2},|\beta|^{2},|\beta|^{2})$,
with $|\alpha|^{2}+|\beta|^{2}=1$. Consider now the simple map ${\cal
T}=\frac{1}{\sqrt{2}}\sum_{i}c^{\dagger}_{i_{A}}c_{i}$, where $c_{i}$
annihilates a fermion in ${\cal H}$ ($i=1,\ldots,D$) while
$c^{\dagger}_{i_{A}}$ creates a fermion in ${\cal H}_{A}$, initially empty
(and of dimension $D_{A}\geq D$). Then ${\cal T}^{\dagger}{\cal
T}=\mathbbm{1}_{2}$ in ${\cal F}_{2}({\cal H})$ and

${\cal
T}|\Psi\rangle=\frac{1}{\sqrt{2}}[\alpha(c^{\dagger}_{1_{A}}c^{\dagger}_{2}-c^{\dagger}_{2_{A}}c^{\dagger}_{1})+\beta(c^{\dagger}_{3_{A}}c^{\dagger}_{4}-c^{\dagger}_{4_{A}}c^{\dagger}_{3})]|0\rangle$
(98)

is a normalized two-fermion state with one fermion in ${\cal H}_{A}$ and one
in ${\cal H}$. It leads to reduced states $\rho_{A(B)}$ with spectrum
$\bm{\lambda}(\rho_{A(B)})=(|\alpha|^{2},|\alpha|^{2},|\beta|^{2},|\beta|^{2})/2$
identical to that of $\rho^{(1)}/2$, thus saturating the inequalities of the
majorization relation (92). We note that even if $\beta=0$, in which case
$|\Psi\rangle$ is a SD, ${\cal T}|\Psi\rangle$ is no longer a SD and has
therefore non-zero one-body entanglement, in agreement with the result that
bipartite entanglement with fixed fermion number or number parity at each side
requires one-body entanglement Gigena  _et al._ (2020).

In contrast, for ${\cal T}^{r}=\frac{1}{\sqrt{2}}c^{\dagger}_{1_{A}}c_{r}$,
$r=1,\ldots D$, still $\sum_{r}{\cal T}^{r{\dagger}}{\cal
T}^{r}=\mathbbm{1}_{2}$ in ${\cal F}_{2}({\cal H})$ but ${\cal
T}^{r}|\Psi\rangle$ is either $0$ or proportional to a state
$c^{\dagger}_{1_{A}}c^{\dagger}_{j}|0\rangle$ with
$c^{\dagger}_{j}|0\rangle\in{\cal H}$, and no direct $A-B$ entanglement is
generated, thus trivially satisfying Eq. (92).

##  IV Conclusions

We have examined a general bipartite-like representation and Schmidt
decomposition of arbitrary $N$-fermion states, which expresses it in terms of
general $M$\- and ($N$-$M$)-fermion creation operators. It is directly
connected to the reduced $M$ and ($N$-$M$)-body DMs, which share the same non-
zero eigenvalues, formally resembling the standard case of distinguishable
components. It naturally leads to the concept of $M$-body entanglement,
determined by the mixedness of the $M$ or ($N$-$M$)-body DMs, which
generalizes one-body entanglement and characterizes the correlations between
$M$\- and $(N$-$M)$-body operators. Such entanglement is of course independent
of the choice of sp basis (it is not a mode-entanglement) and in fact also of
the choice of $M$-fermion creation operators, as long as they satisfy Eqs.
(26)–(27).

The full set of $M$-body DM’s $\rho^{(M)}$ ($1\leq M\leq N/2$) provides a
detailed characterization of the structure of correlated $N$-fermion states.
We have explicitly seen that correlated states which look similar at the level
of the one-body DM, can lead to very distinct $M$-body DMs, as in the case of
the states (43) and (36). In the latter, emergence of bosonic-like features is
signaled by the appearance of an eigenvalue larger than $1$ in the $M$-body DM
for $M\geq 2$. On the other hand, all SDs (i.e. all free- or independent-
fermion states) lead to idempotent $M$-body DMs (i.e. with eigenvalues $1$ or
$0$) $\forall$ $M$. The eigenvalues of $\rho^{(M)}$ determine the average of
$M$-body operators and its largest eigenvalue provides an upper bound to all
averages $\langle A^{(M){\dagger}}A^{(M)}\rangle$ (Eq. (33)) determined by
“collective” operators $A^{(M){\dagger}}$ creating a normalized state of $M$
fermions.

Finally, we have investigated some operational implications of $M$-body
entanglement. By demonstrating the majorization relation (57), we have shown
that the entropy (51) of the normalized $M$-body DM will not increase (and
will in general decrease) under single fermion measurements determined by the
Kraus operators (56). This result can be extended to general $L$-fermion
measurements based on the operators (76) (Eqs. (86)–(87)), which have the
reduced $M$-body DMs as post-measurement states. Moreover, by proving the
majorization relation (92), we have shown that that such $M$-body entropy also
provides an upper bound to the average bipartite entanglement entropy between
$M$ and $N-M$ effectively distinguishable fermions generated by the mapping
determined by the operators (88). Present results then provide the basis for a
general theory of many-body entanglement beyond anti-symmetrization in fermion
systems.

###### Acknowledgements.

The authors acknowledge support from CONICET (NG, MDT) and CIC (RR) of
Argentina. Work supported by CONICET PIP Grant No. 11220150100732.

##  Appendix A Eigenvalues of $\rho^{(M)}$ in the states $|\Psi_{2k}\rangle$

We first derive here the eigenvalues of the first three $M$-body DM’s in the
states (36). Since fermions are created in pairs
$c^{\dagger}_{2i-1}c^{\dagger}_{2i}$ with equal probability, the elements of
the one-body DM in these states are easily seen to be

$\langle\Psi_{2k}|c^{\dagger}_{i}c_{j}|\Psi_{2k}\rangle=\delta_{ij}N/D\,,$
(99)

implying $\rho^{(1)}=\frac{N}{D}\mathbbm{1}$. Thus, it is the maximally mixed
SPDM compatible with the total fermion number $N$, being hence diagonal in any
sp basis with a single $D$-fold degenerate eigenvalue $N/D$.

On the other hand, the elements of the two-body DM are blocked in two
submatrices. The first one, comprising the contiguous pair creation operators
$c^{\dagger}_{2i-1}c^{\dagger}_{2i}$ that form the operator $A^{\dagger}$ of
Eq. (34), has elements

$\langle\Psi_{2k}|c^{\dagger}_{2i-1}c^{\dagger}_{2i}c_{2j}c_{2j-1}|\Psi_{2k}\rangle=\alpha\delta_{ij}+\beta(1-\delta_{ij})\,,$
(100)

where, using $k=N/2$ and assuming $D$ even,

$\alpha={\textstyle\frac{\binom{D/2-1}{k-1}}{\binom{D/2}{k}}=\frac{2k}{D}\,,\;\;\beta=\frac{\binom{D/2-2}{k-1}}{\binom{D/2}{k}}=\frac{2k(D-2k)}{D(D-2)}\,.}$

Hence, since this $\frac{D}{2}\times\frac{D}{2}$ block is just
$(\alpha-\beta)\mathbbm{1}+M$, with $M$ a rank one matrix with all elements
equal to $\beta$, it has just two distinct eigenvalues: a nondegenerate
eigenvalue

$\lambda^{(2)}_{\rm max}=\alpha+(\frac{D}{2}-1)\beta=k(1-2(k-1)/D)\,,$ (101)

which is precisely that associated with the collective uniformly weighted pair
creation operator $A^{\dagger}$:

$\langle\Psi_{2k}|A^{\dagger}A|\Psi_{2k}\rangle=\lambda^{(2)}_{\rm max}\,,$
(102)

and a $(D/2-1)$-fold degenerate smaller eigenvalue

$\lambda^{(2)}_{\rm rest}=\alpha-\beta=\frac{4k(k-1)}{D(D-2)}\,.$ (103)

The other block comprises the remaining $\binom{D}{2}-\frac{D}{2}$ pair
creation operators $c^{{\dagger}}_{i}c^{\dagger}_{j}$ involving two distinct
pairs and is directly diagonal, with elements
$\frac{\binom{D/2-2}{k-2}}{\binom{D/2}{k}}=\lambda^{(2)}_{\rm rest}$. Thus,
the final result is one large nondegenerate eigenvalue $\lambda^{(2)}_{\rm
max}\geq 1$, plus $\binom{D}{2}-1$ smaller identical eigenvalues
$\lambda^{(2)}_{\rm rest}<1$, satisfying

${\textstyle\lambda^{(2)}_{\rm
max}+\left(\binom{D}{2}-1\right)\lambda^{(2)}_{\rm rest}=\binom{N}{2}\,.}$
(104)

The same procedure can be applied to determine the eigenvalues of the three-
body DM $\rho^{(3)}$. For creation of three fermions with two of them in one
of the contiguous pairs ($2i-1,2i$), we obtain

$\langle\Psi_{2k}|c^{\dagger}_{2i-1}c^{\dagger}_{2i}c^{\dagger}_{j}c_{k}c_{2l}c_{2l-1}|\Psi_{2k}\rangle=\delta_{jk}[\gamma\delta_{il}+\eta(1-\delta_{il})]$
(105)

where $k\neq 2l$, $k\neq 2l-1$, $j\neq 2i$, $j\neq 2i-1$, and

$\gamma=\frac{\binom{D/2-2}{k-2}}{\binom{D/2}{k}}\,,\;\;\;\;\eta=\frac{\binom{D/2-3}{k-2}}{\binom{D/2}{k}}\,.$

Hence, we obtain $D$ identical $(\frac{D}{2}-1)\times(\frac{D}{2}-1)$ blocks,
each of which has a large non-degenerate eigenvalue

$\displaystyle\lambda^{(3)}_{\rm max}$ $\displaystyle=$
$\displaystyle{\textstyle\gamma+(\frac{D}{2}-2)\eta=\frac{2k(k-1)(1-2(k-1)/D)}{D-2}}\;\;\;\;\;\;\;$
(106) $\displaystyle=$
$\displaystyle\langle\Psi_{2k}|A^{(3){\dagger}}_{j}A^{(3)}_{j}|\Psi_{2k}\rangle,$
(107)

where
$A^{(3){\dagger}}_{j}=\frac{1}{\sqrt{D/2-1}}\sum_{i}c^{\dagger}_{2i-1}c^{\dagger}_{2i}c^{\dagger}_{j}$,
and $D/2-2$ smaller identical eigenvalues

$\lambda^{(3)}_{\rm
rest}=\gamma-\eta={\textstyle\frac{8k(k-1)(k-2)}{D(D-2)(D-4)}}\,,$ (108)

associated to orthogonal operators $A^{(3){\dagger}}_{\nu}$. On the other
hand, remaining $\binom{D}{3}-D(\frac{D}{2}-1)$ triplets
$c^{\dagger}_{i}c^{\dagger}_{j}c^{\dagger}_{k}$ belonging to distinct pairs
lead to a diagonal block in $\rho^{(3)}$ with identical diagonal elements
$\frac{\binom{D/2-3}{k-3}}{\binom{D/2}{k}}=\lambda^{(3)}_{\rm rest}$.
Therefore, there are $D$ eigenvalues equal to $\lambda^{(3)}_{\rm max}$ plus
$\binom{D}{3}-D$ eigenvalues equal to $\lambda^{(3)}_{\rm rest}$, satisfying

${\textstyle D\lambda^{(3)}_{\rm
max}+\left(\binom{D}{3}-D\right)\lambda^{(2)}_{\rm rest}=\binom{N}{3}\,.}$
(109)

It should be noticed that while $\lambda_{\rm rest}^{(3)}\leq 1$,
$\lambda_{\rm max}^{(3)}\geq 1$ for $1+\sqrt{D/2}\leq k\leq D/2$, reaching its
maximum for $k\approx D/3$ for $D\geq 6$, where $\lambda^{(3)}_{\rm
max}\approx 2D/27$.

For obtaining the largest eigenvalue $\lambda^{(2m)}_{\rm max}$ of the
$M=2m$-body DM algebraically, we may directly note that it will arise from a
$\binom{D/2}{m}\times\binom{D/2}{m}$ block containing products of $m$ distinct
pairs $c^{\dagger}_{2i-1}c^{\dagger}_{2i}$ and $c_{2j}c_{2j-1}$. All elements
$\langle\Psi_{2k}|c^{\dagger}_{2i_{1}-1}c^{\dagger}_{2i_{1}}\ldots
c_{2j_{1}}c_{2j_{1}-1}|\Psi_{2k}\rangle$ will be positive with all rows of
this block having the same sum owing to symmetry. Its largest eigenvalue will
then be equal to this sum (as verified in (101) for $m=1$) and is associated
to the uniform eigenvector $\propto(1,1,\ldots,1)$, i.e. to the collective
operator $A^{(2m){\dagger}}\propto(A^{\dagger})^{m}$, of Eq. (36), implying
Eq. (42).

##  Appendix B Behavior of $\rho^{(2)}$ under single mode occupancy
measurement

We will now prove that in the states (36), measurement of the occupancy of one
sp mode through the operators ${\cal P}_{k}=c^{\dagger}_{k}c_{k}$ and ${\cal
P}_{\bar{k}}=c_{k}c^{\dagger}_{k}$, will break the analogous of Eq. (69) for
$\rho^{(2)}$. We will prove in fact that the largest eigenvalue (101) of
$\rho^{(2)}$ is greater than that of $\rho^{(2)}_{k}$ and
$\rho^{(2)}_{\bar{k}}$, implying

$\lambda^{(2)}_{1}>p_{k}\lambda^{(2)}_{1k}+(1-p_{k})\lambda^{(2)}_{1\bar{k}}\,,$
(110)

which breaks the first majorization inequality.
Proof: If sp state $k$ is measured to be occupied, which will occur with
probability $p_{k}=N/D$, the associated contiguous pair $(k,k+1)$ ($k$ odd) or
$(k-1,k)$ ($k$ even) becomes “freezed” and the maximum eigenvalue
$\lambda^{(2)}_{k\,{\rm max}}$ of the ensuing DM $\rho^{(2)}_{k}$ will arise
from the remaining $N-2$ fermions occupying the other $D-2$ sp states.
Consequently, using Eq. (101) for $D\rightarrow D-2$ and $k=N/2\rightarrow
N/2-1$,

${\textstyle\lambda^{(2)}_{k\,{\rm
max}}=\frac{N-2}{2(D-2)}(D+2-N)<\frac{N}{2D}(D+2-N)=\lambda^{(2)}_{\rm max}}$

where the inequality holds for $N<D$. Similarly, if state $k$ is found to be
empty, a similar reasoning leads to

$\lambda^{(2)}_{\bar{k}\,{\rm
max}}=\frac{N}{2(D-2)}(D-N)<\frac{N}{2D}(D+2-N)=\lambda^{(2)}_{\rm max}$

where the inequality holds for $N>2$. These two results imply Eq. (110) and
hence violation of the majorization relation similar to (69) for $M=2$.
Analogous results can be obtained for $M=3$ in the same states (36).

For completeness, we also verify that for the measurement (56) in the same
states (36), the first inequality

$\frac{\lambda_{\rm
max}^{(2)}}{\binom{N}{2}}\leq\sum_{k}p_{k}\frac{\lambda^{(2)}_{k\,{\rm
max}}}{\binom{N-1}{2}}\,,$ (111)

in (57) for the largest eigenvalues of the initial and post-measurement
normalized two-body DMs, does hold: Using again Eq. (101) for $\lambda_{\rm
max}^{(2)}$ (with $k=N/2$) and $\lambda_{k\,{\rm max}}^{(2)}$ (with $k=N/2-1$
and $D\rightarrow D-2$), with $p_{k}=\frac{\langle
c^{\dagger}_{k}c_{k}\rangle}{N}=\frac{1}{D}$, we obtain, in agreement with
(111),

$\tfrac{1}{D}\sum_{k}\tfrac{(N-2)(D+2-N)}{2(D-2)\binom{N-1}{2}}=\tfrac{D+2-N}{(D-2)(N-1)}\geq\tfrac{(D+2-N)}{D(N-1)}=\tfrac{\lambda_{\rm
max}^{(2)}}{\binom{N}{2}}.$

## References

  * Nielsen and Chuang (2000) M. A. Nielsen and I. L. Chuang,  _Quantum Computation and Quantum Information_ (Camb. Univ. Press, 2000).
  * Benatti  _et al._ (2020) F. Benatti, R. Floreanini, F. Franchini, and U. Marzolino, “Entanglement in indistinguishable particle systems,” Phys. Rep. 878, 1 (2020).
  * Zanardi (2002) P. Zanardi, “Quantum entanglement in fermionic lattices,” Phys. Rev. A 65, 042101 (2002).
  * Shi (2003) Y. Shi, “Quantum entanglement of identical particles,” Phys. Rev. A 67, 024301 (2003).
  * Friis  _et al._ (2013) N. Friis, A.R. Lee, and D.E. Bruschi, “Fermionic-mode entanglement in quantum information,” Phys. Rev. A 87, 022338 (2013).
  * Barnum  _et al._ (2004) H. Barnum, E. Knill, G. Ortiz, R. Somma, and L. Viola, “A subsystem-independent generalization of entanglement,” Phys. Rev. Lett. 92, 107902 (2004).
  * Zanardi  _et al._ (2004) P. Zanardi, D.A. Lidar, and S. Lloyd, “Quantum tensor product structures are observable induced,” Phys. Rev. Lett. 92, 060402 (2004).
  * Sasaki  _et al._ (2011) T. Sasaki, T. Ichikawa, and I. Tsutsui, “Entanglement of indistinguishable particles,” Phys. Rev. A 83, 012113 (2011).
  * Balachandran  _et al._ (2013) A.P. Balachandran, T.R. Govindarajan, A. R. de Queiroz, and A.F. Reyes-Lega, “Entanglement and particle identity: a unifying approach,” Phys. Rev. Lett. 110, 080503 (2013).
  * Benatti  _et al._ (2014) F. Benatti, R. Floreanini, and U. Marzolino, “Entanglement in fermion systems and quantum metrology,” Phys. Rev. A 89, 032326 (2014).
  * Schliemann  _et al._ (2001a) J. Schliemann, J. I. Cirac, M. Kuś, M. Lewenstein, and D. Loss, “Quantum correlations in two-fermion systems,” Phys. Rev. A 64, 022303 (2001a).
  * Schliemann  _et al._ (2001b) J. Schliemann, D. Loss, and A. H. MacDonald, “Double-occupancy errors, adiabaticity, and entanglement of spin qubits in quantum dots,” Phys. Rev. B 63, 085311 (2001b).
  * Eckert  _et al._ (2002) K Eckert, J. Schliemann, D. Bruß, and M. Lewenstein, “Quantum correlations in systems of indistinguishable particles,” Ann. Phys. 299, 88–127 (2002).
  * Ghirardi  _et al._ (2002) G. Ghirardi, L. Marinatto, and T. Weber, “Entanglement and properties of composite quantum systems: a conceptual and mathematical analysis,” J. Stat. Phys. 108, 49–122 (2002).
  * Paškauskas and You (2001) R. Paškauskas and L. You, “Quantum correlations in two-boson wave functions,” Phys. Rev. A 64, 042310 (2001).
  * Wiseman and Vaccaro (2003) H. M. Wiseman and J. A. Vaccaro, “Entanglement of indistinguishable particles shared between two parties,” Phys. Rev. Lett. 91, 097902 (2003).
  * Iemini and Vianna (2013) F. Iemini and R. O. Vianna, “Computable measures for the entanglement of indistinguishable particles,” Phys. Rev. A 87, 022327 (2013), F. Iemini, T. Debarba and R.O. Vianna, Phys. Rev. A 89, 032324 (2014).
  * Oszmaniec and Kuś (2013) M. Oszmaniec and M. Kuś, “Universal framework for entanglement detection,” Phys. Rev. A 88, 052328 (2013), M. Oszmaniec, J. Gutt and M. Kuś, Phys. Rev. A 90, 020302 (2014).
  * Sárosi and Lévay (2014) G. Sárosi and P. Lévay, “Entanglement in fermionic fock space,” J. Phys. A: Math. Theor. 47, 115304 (2014).
  * Gigena and Rossignoli (2015) N. Gigena and R. Rossignoli, “Entanglement in fermion systems,” Phys. Rev. A 92, 042326 (2015).
  * Majtey  _et al._ (2016) A.P. Majtey, P.A. Bouvrie, A. Valdés-Hernández, and A.R. Plastino, “Multipartite concurrence for identical-fermion systems,” Phys. Rev. A 93, 032335 (2016), A.R. Plastino, D. Manzano and J.S. Dehesa, EPL 86 20005 (2009).
  * Dasenbrook  _et al._ (2016) D. Dasenbrook, J. Bowles, J. Bohr Brask, P.P. Hofer, C. Flindt, and N. Brunner, “Single-electron entanglement and nonlocality,” New J. Phys. 18, 043036 (2016).
  * Gigena and Rossignoli (2017) N. Gigena and R. Rossignoli, “Bipartite entanglement in fermion systems,” Phys. Rev. A 95, 062320 (2017).
  * Benatti  _et al._ (2017) F. Benatti, R. Floreanini, F. Franchini, and U. Marzolino, “Remarks on entanglement and identical particles,” Open Syst. Inf. Dyn. 24, 1740004 (2017).
  * Di Tullio  _et al._ (2018) M. Di Tullio, N. Gigena, and R. Rossignoli, “Fermionic entanglement in superconducting systems,” Phys. Rev. A 97, 062109 (2018).
  * Di Tullio  _et al._ (2019) M. Di Tullio, R. Rossignoli, M. Cerezo, and N. Gigena, “Fermionic entanglement in the lipkin model,” Phys. Rev. A 100, 062104 (2019).
  * da Silva Souza  _et al._ (2018) L. da Silva Souza, T. Debarba, D. L. Braga-Ferreira, F. Iemini, and R. O. Vianna, “Completely positive maps for reduced states of indistinguishable particles,” Phys. Rev. A 98, 052135 (2018).
  * Ding and Schilling (2020) K. Ding and C. Schilling, “Correlation paradox of the dissociation limit: A quantum information perspective,” J. Chem. Theory Comput. 16, 4159 (2020).
  * Debarba  _et al._ (2020) T. Debarba, F. Iemini, G. Giedke, and N. Friis, “Teleporting quantum information encoded in fermionic modes,” Phys. Rev. A 101, 052326 (2020).
  * Cavalcanti  _et al._ (2007) D. Cavalcanti, L.M. Malard, F.M. Matinaga, M.O. Terra Cunha, and M. F. Santos, “Useful entanglement from the Pauli principle,” Phys. Rev. B 76, 113304 (2007).
  * Killoran  _et al._ (2014) N. Killoran, M. Cramer, and M. B. Plenio, “Extracting entanglement from identical particles,” Phys. Rev. Lett. 112, 150501 (2014).
  * Compagno  _et al._ (2018) G. Compagno, A. Castellini, and R. Lo Franco, “Dealing with indistinguishable particles and their entanglement,” Phil. Trans. R. Soc. A 376, 20170317 (2018).
  * Lo Franco and Compagno (2018) R. Lo Franco and G. Compagno, “Indistinguishability of elementary systems as a resource for quantum information processing,” Phys. Rev. Lett. 120, 240403 (2018).
  * Morris  _et al._ (2020) B. Morris, B. Yadin, M. Fadel, T. Zibold, P. Treutlein, and G. Adesso, “Entanglement between identical particles is a useful and consistent resource,” Phys. Rev. X 10, 041012 (2020).
  * Gigena  _et al._ (2020) N. Gigena, M. Di Tullio, and R. Rossignoli, “One-body entanglement as a quantum resource in fermionic systems,” Phys. Rev. A 102, 042410 (2020).
  * Carlson and Keller (1961) B. C. Carlson and J. M. Keller, “Eigenvalues of density matrices,” Phys. Rev. 121, 659 (1961).
  * Ando (1963) T. Ando, “Properties of fermion density matrices,” Rev. Mod. Phys. 35, 690 (1963).
  * Amosov and Filippov (2017) G.G. Amosov and S.N. Filippov, “Spectral properties of reduced fermionic density operators and parity superselection rule,” Quantum Inf. Process. 16, 2 (2017).
  * Law (2005) C. K. Law, “Quantum entanglement as an interpretation of bosonic character in composite two-particle systems,” Phys. Rev. A 71, 034306 (2005).
  * Bhatia (1997) R. Bhatia,  _Matrix Analysis_ (Springer, New York, 1997).
  * Marshall  _et al._ (2011) A. W. Marshall, I. Olkin, and B. C. Arnold,  _Inequalities: Theory of Majorization and its Applications_ (Springer, New York, 2011).
  * Nielsen (1999) M. A. Nielsen, “Conditions for a class of entanglement transformations,” Phys. Rev. Lett. 83, 436 (1999).
  * Nielsen and Vidal (2001) M. A. Nielsen and G. Vidal, “Majorization and the interconversion of bipartite states,” Quantum Inf. Comput. 1, 76 (2001).
  * Canosa and Rossignoli (2002) N. Canosa and R. Rossignoli, “Generalized nonadditive entropies and quantum entanglement,” Phys. Rev. Lett. 88, 170401 (2002).
  * Rossignoli and Canosa (1999) R. Rossignoli and N. Canosa, “Non-additive entropies and quantum statistics,” Phys. Lett. A 264, 148 (1999).
  * Rossignoli and Canosa (2002) R. Rossignoli and N. Canosa, “Generalized entropic criterion for separability,” Phys. Rev. A 66, 042306 (2002), R. Rossignoli, N. Canosa, Violation of majorization relations in entangled states and its detection by means of generalized entropic forms, Phys. Rev. A 67, 042302 (2003).
  * Nielsen (2001) M. A. Nielsen, “Characterizing mixing and measurement in quantum mechanics,” Phys. Rev. A 63, 022114 (2001).