# Shannon theory for quantum systems and beyond: information compression for
fermions

Paolo Perinotti<EMAIL_ADDRESS>QUIT group, Physics Dept., Pavia
University, and INFN Sezione di Pavia, via Bassi 6, 27100 Pavia, Italy
Alessandro Tosini<EMAIL_ADDRESS>QUIT group, Physics Dept., Pavia
University, and INFN Sezione di Pavia, via Bassi 6, 27100 Pavia, Italy
Leonardo Vaglini<EMAIL_ADDRESS>QUIT group, Physics
Dept., Pavia University, and INFN Sezione di Pavia, via Bassi 6, 27100 Pavia,
Italy

(27th August 2024)

###### Abstract

We address the task of compression of fermionic quantum information. Due to
the parity superselection rule, differently from the case of encoding of
quantum information in qubit states, part of the information carried by
fermionic systems is encoded in their delocalised correlations. As a
consequence, reliability of a compression protocol must be assessed in a way
that necessarily accounts also for the preservation of correlations. This
implies that input/output fidelity is not a satisfactory figure of merit for
fermionic compression schemes. We then discuss various aspects regarding the
assessment of reliability of an encoding scheme, and show that entanglement
fidelity in the fermionic case is capable of evaluating the preservation of
correlations, thus revealing itself strictly stronger than input/output
fidelity, unlike the qubit case. We then introduce a fermionic version of the
source coding theorem showing that, as in the quantum case, the von Neumann
entropy is the minimal rate for which a fermionic compression scheme exists,
that is reliable according to the entanglement fidelity criterion.

##  I Introduction

The task of _data compression_ addresses the primary question in information
theory as to how redundant is the information contained in a message and to
what extent the message can then be compressed.

In classical information theory this question is answered by the source coding
theorem 6773024 , which establishes the fundamental role of Shannon entropy in
information theory and its operational interpretation. The coding theorem
recognizes the Shannon entropy as the fundamental limit for the compression
rate in the i.i.d. setting. This means that if one compresses at a rate above
the Shannon entropy, then the compressed data can be recovered perfectly in
the asymptotic limit of infinitely long messages, while this is not possible
for compression rate below the Shannon entropy. As a result the Shannon
entropy, which can be intuitively thought of as the uncertainty about the
outcome of an experiment that we are going to perform on a classical system,
quantifies in a stringent way the amount of “non-redundant” information that
is encoded in the state of the classical system, what one would definitely
call _information content_.

In quantum information theory the Shannon entropy is replaced by the von
Neumann entropy. In particular, the quantum source coding theorem
PhysRevA.51.2738  identifies von Neumann entropy as the rate at which quantum
compression can be reliably achieved. Consider a quantum information source
described by a system $\mathrm{A}$ and density operator
$\rho\in\mathsf{St}{(\mathrm{A})}$, with $\mathsf{St}{(\mathrm{A})}$ the set
of states of system $\mathrm{A}$. The density operator describes the
preparation of a state $\sigma_{i}$ from any ensemble $\\{p_{i}\sigma_{i}\\}$,
with probabilities $p_{i}$, such that $\sum_{i}p_{i}\sigma_{i}=\rho$. A
quantum message of $N$ letters can now be understood in terms of $N$ uses of
the quantum source, that output a sequence of $N$ states
$p_{i_{j}}\sigma_{i_{j}}$, with $1\leq j\leq N$, drawn independently. One
instance of this preparation protocol thus produces
$\sigma_{\mathrm{\mathbf{i}}}\coloneqq\bigotimes_{j=1}^{N}\sigma_{i_{j}}$,
with probability $p_{\mathrm{\mathbf{i}}}\coloneqq\prod_{j}p_{i_{j}}$. Each of
the $N$ systems has density operator $\rho$, and the density operator of the
entire message is then given by $\rho^{\otimes N}$. A compression scheme for
messages from the above described source consists of two steps. _Encoding:_
Alice encodes the system $\mathrm{A}^{\otimes N}$ according to a compression
map given by a channel $\mathscr{E}:\mathsf{St}(\mathrm{A}^{\otimes
N})\rightarrow\mathsf{St}(B)$, where $\mathrm{B}$ is generally a system with
dimension $d_{\mathrm{B}}(N)$ smaller then $\mathrm{A}^{\otimes N}$. The
compression rate is defined as the asymptotic quantity
$R=\lim_{N\rightarrow\infty}\log_{2}d_{\mathrm{B}}(N)/N$. Typically, one
estimates the “size” of the compressed message through the capacity of system
$\mathrm{B}$ given in terms of $\log_{2}d_{\mathrm{B}}(N)$, namely the number
of qubits that are needed to simulate $\mathrm{B}$. Alice then sends the
system $\mathrm{B}$ to Bob using $NR$ noiseless qubit channels. _Decoding:_
Finally, Bob applies a decompression map
$\mathscr{D}:\mathsf{St}(B)\rightarrow\mathsf{St}(\mathrm{A}^{\otimes N})$ to
the message encoded in system $\mathrm{B}$, with the purpose of recovering the
original message as reliably as possible.

As one might expect, the above scheme generally introduces an error in the
decoding: we now discuss the figure of merit by which we estimate the error
introduced by the compression scheme. In order to understand the operational
meaning of the figure of merit, think of a referee (Charlie) who prepares the
states $\sigma_{\mathrm{\mathbf{i}}}$ with probability
$p_{\mathrm{\mathbf{i}}}$, and receives the final states
$\mathscr{D}\mathscr{E}(\sigma_{\mathrm{\mathbf{i}}})$. The figure of merit
that we use corresponds to the probability that, after receiving Bob’s final
state, Charlie is able to distinguish it from the input one. For a single
instance, this probability is a linear function of the trace-norm distance
$\|\sigma_{\mathrm{\mathbf{i}}}-\mathscr{D}\mathscr{E}(\sigma_{\mathrm{\mathbf{i}}})\|_{1}$.
The probability of successful discrimination is thus evaluated to
$\sum_{\mathrm{\mathbf{i}}}p_{\mathrm{\mathbf{i}}}\|\sigma_{\mathrm{\mathbf{i}}}-\mathscr{D}\mathscr{E}(\sigma_{\mathrm{\mathbf{i}}})\|_{1}=\|\rho^{\otimes
N}-\mathscr{D}\mathscr{E}(\rho^{\otimes N})\|_{1}$. The protocol has then
error $\epsilon$ if the compressed and decompressed states
$\mathscr{D}\mathscr{E}(\sigma_{\mathrm{\mathbf{i}}})$ are $\epsilon$-close to
the original states $\sigma_{\mathrm{\mathbf{i}}}$, in the trace-norm
distance. In the case of qubits the above quantity is equivalent to fidelity,
thanks to the Fuchs-van der Graaf inequalities 761271 . The optimal quantum
encoding will then make the error arbitrarily small for $N$ large enough, with
rate $R$ as small as possible. Schumacher’s quantum source coding theorem
shows that the optimal rate is equal to the von Neumann entropy $S(\rho)$ of
the state $\rho$.

Another way to evaluate the error for a compression scheme is the following:
Charlie prepares a purification of the density operator $\rho^{\otimes N}$ and
sends the $N$ copies of system $\mathrm{A}$ to Alice. Alice then sends her
share of the pure state to Bob, sending as few qubits to Bob as possible.
After decompressing the received qubits, Bob shares an entangled state with
Charlie. The quality of the compression scheme can then be evaluated
considering how well Charlie can distinguish the initial state from the final
one, after receiving Bob’s $N$ systems. The probability that Charlie detects a
compression error can be evaluated through the input/output fidelity. Again,
Schumacher’s theorem states that Alice can transfer her share of the pure
state to Bob by sending $NS(\rho)$ qubits and achieving arbitrarily good
fidelity, increasing the length $N$ of the message. This second perspective
answers the question whether the compression protocol preserves the
correlations that system $\mathrm{A}^{\otimes N}$ has with a remote system
$\mathrm{C}$.

Equivalence of the two approaches in assessing the quality of a compression
scheme shows that the ability to send quantum superpositions is equivalent to
the ability to send entanglement. In other terms, the amount of quantum
information preserved by a compression scheme represents the dimension of the
largest Hilbert space whose superpositions can be reliably compressed, or
equivalently the amount of entanglement that can be reliably compressed.

According to the above discussion a crucial point in the compression protocol
is to quantify the reliability of the compression map
$\mathscr{C}:=\mathscr{D}\mathscr{E}$, which in the asymptotic limit of
$N\to\infty$ must coincide with the identity map. In quantum theory checking
the reliability of the compression map looking only at its local action,
namely via the fidelity between states $\mathscr{C}(\rho^{\otimes N})$ and
$\rho^{\otimes N}$, or at the effects on correlations, namely via entanglement
fidelity, is equivalent. This follows from _local process tomography_ of
quantum theory where, given a map $\mathscr{C}$ on system $\mathrm{A}$ one has

$\displaystyle(\mathscr{C}\otimes\mathscr{I}_{\mathrm{C}})(\Psi)=$
$\displaystyle\Psi\qquad\forall\Psi\in\mathsf{St}(\mathrm{A}\mathrm{C})$ (1)
$\displaystyle\Leftrightarrow$ $\displaystyle\mathscr{C}(\rho)=$
$\displaystyle\rho\qquad\forall\rho\in\mathsf{St}{(\mathrm{A})}.$

This equivalence is due to _local discriminability_ PhysRevA.81.062348 ;
PhysRevA.84.012311 ; DAriano:2017up  of quantum theory, where the
discrimination of bipartite quantum states can always be performed using local
measurements only (this property is equivalent to the one known in the
literature as local tomography, or tomographic locality Araki:1980tr ;
dakic_brukner_2011 ; Masanes_2011 ; Barnum:2014vt ). However, in the absence
of local discriminability, a map preserving local states still can affect
correlations with remote systems DAriano2020information . This raises a
crucial issue if one aims at studying the compression task beyond quantum
theory, where the reliability of a protocol generally needs to be verified on
extended systems. Indeed, in general, testing a compression scheme using
ancillary systems is strictly stronger than testing them with local schemes.

As a first step in the direction of generalizing the compression protocol to
an arbitrary information theory, in this paper we consider the case of
fermionic systems as carriers of information. Fermionic computation has been
proposed in Ref. Bravyi2002210  and later studied in several works Wolf2006 ;
Banuls2007 ; Friis2013 ; DAriano2014a ; PhysRevA.101.052326 . Differently from
quantum systems, fermions obey the _parity superselection rule_. As a
consequence, fermionic information theory does not satisfy local
discriminability, thus providing a physically relevant example of a theory
where the task of compression is not straightforward. Indeed, in the case of
study, a map $\mathscr{C}$ that acts as the identity on local states
$\rho^{\otimes N}$ could still destroy the correlations with remote systems,
and then be mistakenly considered as a reliable compression map.

After reviewing the structure of fermionic quantum information, we prove that
the entanglement fidelity is a valid criterion for the reliability of a
fermionic compression map. We then show an analogous of the quantum source
coding theorem in the fermionic scenario, showing that the minimal compression
rate for which a reliable compression scheme exists is the von Neumann entropy
of the fermionic state. We conclude therefore that the von Neumann entropy
provides the informational content of the state also in the case of fermionic
theory, namely in the presence of parity superselection. The above result,
however, is not a straightforward consequence of Schumacher’s source coding
theorem.

##  II Fermionic information theory

We now briefly review fermionic information theory. The systems of the theory
are made by local fermionic modes (LFMs). A LFM is the counterpart of the
qubit in quantum theory, and can be thought of as a cell that can be either
empty or occupied by a fermionic excitation. An $L$-LFMs system, denoted
$\mathrm{L}_{\mathrm{F}}$, is described by $L$ fermionic fields $\varphi_{i}$,
satisfying the canonical anticommutation rule (CAR)
$\\{\varphi_{i},\varphi_{j}^{\dagger}\\}=\delta_{ij}I$,
$\\{\varphi_{i},\varphi_{j}\\}=0$ where $i,j=1,\dots,L$. With these fields one
constructs the occupation number operators $\varphi_{i}^{\dagger}\varphi_{i}$,
which can be easily proved to have only eigenvalues 0 and 1. The common
eigenvector $\left|{\Omega}\right\rangle$ of the operators
$\varphi_{i}^{\dagger}\varphi_{i}$, $i=1,\ldots,L$ with eigenvalue 0 defines
the vacuum state $\left|{\Omega}\right\rangle\left\langle{\Omega}\right|$ of
$\mathrm{L}_{\mathrm{F}}$, representing the state in which all the modes are
not excited. The fermionic vacuum state in terms of the field operators is
given by
$\left|{\Omega}\right\rangle\left\langle{\Omega}\right|=\prod_{i=1}^{L}\varphi_{i}\varphi_{i}^{\dagger}$.
By applying the operators $\varphi_{i}^{\dagger}$ to
$\left|{\Omega}\right\rangle$ the corresponding $i$-th mode is excited and, by
raising $\left|{\Omega}\right\rangle$ in all possible ways, we get the $2^{L}$
orthonormal vectors forming the Fock basis in the occupation number
representation: a generic element of this basis is

$\displaystyle\left|{n_{1},\dots,n_{L}}\right\rangle:=(\varphi_{1}^{\dagger})^{n_{1}}\dots(\varphi_{L}^{\dagger})^{n_{L}}\left|{\Omega}\right\rangle,$
(2)

with $n_{i}=\\{0,1\\}$ corresponding to the occupation number at the $i$-th
site. The linear span of these vectors corresponds to the antisymmetric Fock
space $\mathcal{F}_{L}$ of dimension $d_{\mathcal{F}_{L}}=2^{L}$. Notice that
the Fock space $\mathcal{F}_{L}$ is isomorphic to the Hilbert space
$\mathcal{H}_{L}$ of $L$ qubits, by the trivial identification of the
occupation number basis with the qubit computational basis. This
correspondence lies at the basis of the Jordan-Wigner isomorphism Jordan1928 ;
Verstraete2005 ; Pineda2010  typically used in the literature to map fermionic
systems to qubits systems and vice-versa. We recall here the definition of the
Jordan-Wigner map

$\displaystyle
J_{L}(\varphi_{i})=\left(\bigotimes_{l=1}^{i-1}\sigma^{z}_{l}\right)\otimes\sigma^{-}_{i}\otimes\left(\bigotimes_{k=i+1}^{L}I_{k}\right),$
(3) $\displaystyle J_{L}(\varphi^{\dagger}_{i})=J_{L}(\varphi_{i})^{\dagger},$
$\displaystyle J_{L}(XY)=J_{L}(X)J_{L}(Y),$ $\displaystyle
J_{L}(aX+bY)=aJ_{L}(X)+bJ_{L}(Y),$

with $X,Y$ linear combinations of products of field operators on the $L$-LFMs,
and where we used the standard notation for Pauli sigma operators. In the
following we will drop the dependence on the number of LFMs in the Jordan-
Wigner map, namely we will write $J(X)$ in place of $J_{L}(X)$, when it will
be clear from the context. Notice that the Jordan-Wigner isomorphism is
implicitly defined in Eq. (2), and, as such, it depends on the arbitrary
ordering of the modes. All such representations are unitarily equivalent.

Differently from standard qubits, fermionic systems satisfy the parity
superselection rule Schuch2004 ; Kitaev2004 ; Schuch2004a ; DAriano2014a ;
fermionic_theory . One can decompose the Fock space $\mathcal{F}_{L}$ of
system $\mathrm{L}_{\mathrm{F}}$ in the direct sum
$\mathcal{F}_{L}=\mathcal{F}_{L}^{e}\oplus\mathcal{F}_{L}^{o}$, with
$\mathcal{F}^{e}_{L}$ and $\mathcal{F}^{o}_{L}$ the spaces generated by
vectors with even and odd total occupation number, respectively. The convex
set of states $\mathsf{St}{(\mathrm{L}_{\mathrm{F}})}$ corresponds, in the
Jordan-Wigner representation, to the set of density matrices on
$\mathcal{F}_{L}$ of the form $\rho=\rho_{e}+\rho_{o}$, with
$\rho_{e},\rho_{o}\geq 0$,
$\operatorname{Tr}[\rho_{o}]+\operatorname{Tr}[\rho_{e}]\leq 1$ and with
$\rho_{e}$ and $\rho_{o}$ having support on $\mathcal{F}_{L}^{e}$ and
$\mathcal{F}_{L}^{o}$, respectively, and pure states are represented by rank
one density operators. Moreover, the density matrices representing the states
represent linear combinations of products of an even number of field operators
(see appendix A and fermionic_theory  for further details). Viceversa, every
linear combination of products of an even number of field operators that is
represented by a density matrix is an admissible state. Analogously, effects
in the set $\mathsf{Eff}{(\mathrm{L}_{\mathrm{F}})}$ are represented by
positive operators on $\mathrm{L}_{\mathrm{F}}$ of the form $a=a_{e}+a_{o}$,
with $a_{e}$ and $a_{o}$ having support on $\mathcal{F}_{L}^{e}$ and
$\mathcal{F}_{L}^{o}$, respectively. Notice that set of states and effects of
system $\mathrm{L}_{\mathrm{F}}$ have dimension
$d^{2}_{\mathcal{F}_{L}}/2=2^{2L-1}$, compared to the quantum case where the
set of states and effects associated to the Hilbert space $\mathcal{H}_{L}$ of
$L$ qubits has dimension $d^{2}_{\mathcal{H}_{L}}=2^{2L}$.

Given a state $\rho\in\mathsf{St}(\mathrm{L}_{\mathrm{F}})$ we define the
refinement set of $\rho$ as
$\mathsf{Ref}(\rho):=\\{\sigma\in\mathsf{St}(\mathrm{L}_{\mathrm{F}})|\exists\tau\in\mathsf{St}(\mathrm{L}_{\mathrm{F}}):\rho=\sigma+\tau\\}$,
and a state is pure when all the elements in the refinement are proportional
to the state itself. In the following we will denote by
$\mathsf{PurSt}(\mathrm{L}_{\mathrm{F}})$ and
$\mathsf{St}_{1}(\mathrm{L}_{\mathrm{F}})$ the set of pure states and the set
of normalized states (of trace one) of system $\mathrm{L}_{\mathrm{F}}$,
respectively.

Given two fermionic systems $L_{\mathrm{F}}$ and $M_{\mathrm{F}}$, we
introduce the composition of the two as the system made of $K\equiv L+M$ LFMs,
denoted with the symbol
$\mathrm{K}_{\mathrm{F}}\coloneqq\mathrm{L}_{\mathrm{F}}\boxtimes\mathrm{M}_{\mathrm{F}}$,
or simply
$\mathrm{K}_{\mathrm{F}}\coloneqq\mathrm{L}_{\mathrm{F}}\mathrm{M}_{\mathrm{F}}$.
We use the symbol $\boxtimes$ to distinguish the fermionic parallel
composition rule from the quantum one, corresponding to the tensor product
$\otimes$.

Given a state
$\Psi\in\mathsf{St}(\mathrm{L}_{\mathrm{F}}\mathrm{M}_{\mathrm{F}})$, one can
discard the subsystem $\mathrm{M}_{\mathrm{F}}$ and consider the marginal
state, which we denote by
$\sigma:=\operatorname{Tr}^{f}_{\mathrm{M}_{\mathrm{F}}}(\Psi)$. We use the
symbol $\operatorname{Tr}^{f}_{\mathrm{M}_{\mathrm{F}}}$ to denote the
fermionic partial trace on the subsystem $\mathrm{M}_{\mathrm{F}}$. This is
computed by performing the following steps (see ref. fermionic_theory  for
further details): (i) drop all those terms in $\Psi$ containing an odd number
of field operators in any of the LFMs in $\mathrm{M}_{\mathrm{F}}$; (ii)
remove all the field operators corresponding to the LFMs in
$\mathrm{M}_{\mathrm{F}}$ from the remaining terms. The fermionic trace
$\operatorname{Tr}^{f}(\rho)$ of a state
$\rho\in\mathsf{St}(\mathrm{M}_{\mathrm{F}})$ is then defined as a special
case of the partial one, corresponding to the case in which $L=0$.

Finally, the set of transformations from $\mathrm{L}_{\mathrm{F}}$ to
$\mathrm{M}_{\mathrm{F}}$, denoted by
$\mathsf{Tr}(\mathrm{L}_{\mathrm{F}}\rightarrow\mathrm{M}_{\mathrm{F}})$, is
given by completely positive maps from $\mathsf{St}(\mathrm{L}_{\mathrm{F}})$
to $\mathsf{St}(\mathrm{M}_{\mathrm{F}})$ in the Jordan-Wigner representation.
Moreover, we denote by
$\mathsf{Tr}_{1}(\mathrm{L}_{\mathrm{F}}\rightarrow\mathrm{M}_{\mathrm{F}})$
the set of deterministic transformations, also called _channels_ , from
$\mathrm{L}_{\mathrm{F}}$ to $\mathrm{M}_{\mathrm{F}}$, corresponding to
trace-preserving completely positive maps. Like in quantum theory, any
fermionic transformation
$\mathscr{C}\in\mathsf{Tr}(\mathrm{L}_{\mathrm{F}}\rightarrow\mathrm{M}_{\mathrm{F}})$
can be expressed in Kraus form $\mathscr{C}(\rho)=\sum_{i}C_{i}\rho
C^{\dagger}_{i}$, with deterministic transformations having Kraus operators
$\\{C_{i}\\}$ such that $J(\sum_{i}C_{i}^{\dagger}C_{i})=I_{\mathcal{H}_{L}}$,
$I_{\mathcal{H}_{L}}$ denoting the identity operator on $\mathcal{H}_{L}$. For
a map
$\mathscr{C}\in\mathsf{Tr}(\mathrm{L}_{\mathrm{F}}\rightarrow\mathrm{M}_{\mathrm{F}})$
with Kraus operators $\\{C_{i}\\}$, we define its Jordan-Wigner representative
$J(\mathscr{C})$ as the quantum map with Kraus operators $\\{J(C_{i})\\}$.
Now, given two transformations
$\mathscr{C}\in\mathsf{Tr}(\mathrm{L}_{\mathrm{F}}\rightarrow\mathrm{M}_{\mathrm{F}})$
and
$\mathscr{D}\in\mathsf{Tr}(\mathrm{K}_{\mathrm{F}}\rightarrow\mathrm{N}_{\mathrm{F}})$,
we denote by
$\mathscr{C}\boxtimes\mathscr{D}\in\mathsf{Tr}(\mathrm{L}_{\mathrm{F}}\mathrm{K}_{\mathrm{F}}\to\mathrm{M}_{\mathrm{F}}\mathrm{N}_{\mathrm{F}})$
the _parallel composition_ of $\mathscr{C}$ and $\mathscr{D}$, with Kraus
operators $\\{C_{i}D_{j}\\}$, where $\\{C_{i}\\}$ are Kraus operators for
$\mathscr{C}$ and $\\{D_{j}\\}$ for $\mathscr{D}$. We observe that in the
Jordan-Wigner representation one generally has $J_{L+K}(C_{i}D_{j})\neq
J_{L}(C_{i})\otimes J_{K}(D_{j})$, and
$J_{L+K}(\mathscr{C}\boxtimes\mathscr{D})\neq J_{L}(\mathscr{C})\otimes
J_{K}(\mathscr{D})$. If $\mathscr{C}$ is a transformation in
$\mathsf{Tr}(\mathrm{L}_{\mathrm{F}}\rightarrow\mathrm{M}_{\mathrm{F}})$, its
extension to a composite system
$\mathrm{L}_{\mathrm{F}}\mathrm{N}_{\mathrm{F}}$, is given by
$\mathscr{C}\boxtimes\mathscr{I}$, with $\mathscr{I}$ the identity map of
system $\mathrm{N}_{\mathrm{F}}$—whose Jordan-Wigner representative is the
quantum identity map—and its Kraus operators involve field operators on the
$\mathrm{L}_{\mathrm{F}}$ modes only. It is worth noticing that, despite the
Jordan-Wigner representative of this map is not necessarily of the form
$J_{L}(\mathscr{C})\otimes\mathscr{I}$, upon suitable choice of the ordering
of the LFMs that defines the representation, one can always reduce to the case
where, actually,
$J_{L+N}(\mathscr{C}\boxtimes\mathscr{I})=J_{L}(\mathscr{C})\otimes\mathscr{I}$.

As a special case of the above composition rule, one can define
$\rho\boxtimes\sigma\coloneqq\rho\sigma\in\mathsf{St}(\mathrm{L}_{\mathrm{F}}\mathrm{M}_{\mathrm{F}})$
for the parallel composition of states
$\rho\in\mathsf{St}(\mathrm{L}_{\mathrm{F}})$ and
$\sigma\in\mathsf{St}(\mathrm{M}_{\mathrm{F}})$, and similarly $a\boxtimes
b\coloneqq ab\in\mathsf{Eff}(\mathrm{L}_{\mathrm{F}}\mathrm{M}_{\mathrm{F}})$
for the parallel composition of effects
$a\in\mathsf{Eff}(\mathrm{L}_{\mathrm{F}})$ and
$b\in\mathsf{Eff}(\mathrm{M}_{\mathrm{F}})$.

A useful characterization of fermionic maps in
$\mathsf{Tr}(\mathrm{L}_{\mathrm{F}}\rightarrow\mathrm{L}_{\mathrm{F}})$,
proved in Ref. DAriano2014 , is the following:

######  Proposition II.1 (Fermionic transformations).

All the transformations in
$\mathsf{Tr}(\mathrm{L}_{\mathrm{F}}\rightarrow\mathrm{L}_{\mathrm{F}})$ with
Kraus operators being linear combinations of products of either an even number
or an odd number of field operators are admissible fermionic transformations.
Viceversa, each admissible fermionic transformation in
$\mathsf{Tr}(\mathrm{L}_{\mathrm{F}}\rightarrow\mathrm{L}_{\mathrm{F}})$ has
Kraus operators being superpositions of products of either an even number or
an odd number of field operators.

######  Corollary II.1 (Fermionic effects).

Fermionic effects are positive operators bounded by the identity operator that
are linear combinations of products of an even number of field operators.
Viceversa, every linear combination of products of an even number of field
operators that is represented by a positive operator bounded by the identity
is a fermionic effect.

The corollary follows immediately from Proposition II.1, since an effect $A$
is obtained as a fermionic transformation $\mathscr{A}$ followed by the
discard map, i.e. the trace. Thus

$\displaystyle\operatorname{Tr}[\rho
A]=\operatorname{Tr}[\mathscr{A}(\rho)]=\sum_{i}\operatorname{Tr}[K_{i}\rho
K_{i}^{\dagger}]=\operatorname{Tr}[\rho\sum_{i}K_{i}^{\dagger}K_{i}],$

namely $A=\sum_{i}K_{i}^{\dagger}K_{i}$. Having the polynomial $K_{i}$ a
definite parity (though not necessarily the same for every $i$), $A$ is an
even polynomial.

In the following we denote by $\mathcal{L}(\mathcal{H}_{L})$ the set of linear
operators on the Hilbert space $\mathcal{H}_{L}$ of $L$-qubits and by
$\mathcal{L}(\mathcal{H}_{L},\mathcal{H}_{M})$ the set of linear operators
from $\mathcal{H}_{L}$ to $\mathcal{H}_{M}$. It is useful to introduce the
isomorphism between operators $X$ in
$\mathcal{L}(\mathcal{H}_{L},\mathcal{H}_{M})$ and vectors
$|X\rangle\\!\rangle$ in $\mathcal{H}_{M}\otimes\mathcal{H}_{L}$ given by

$|X\rangle\\!\rangle=(X\otimes
I_{\mathcal{H}_{L}})|I_{\mathcal{H}_{L}}\rangle\\!\rangle=(I_{\mathcal{H}_{M}}\otimes
X^{T})|I_{\mathcal{H}_{M}}\rangle\\!\rangle,$ (4)

where $I_{\mathcal{H}_{L}}$ is the identity operator in $\mathcal{H}_{L}$,
$|I_{\mathcal{H}_{L}}\rangle\\!\rangle\in\mathcal{H}_{L}^{\otimes 2}$ is the
maximally entangled vector
$|I_{\mathcal{H}_{L}}\rangle\\!\rangle=\sum_{l}|l\rangle|l\rangle$ (with
$\\{|l\rangle\\}$ a fixed orthonormal basis for $\mathcal{H}_{L}$), and
$X^{T}\in\mathcal{L}(\mathcal{H}_{M},\mathcal{H}_{L})$ is the transpose of $X$
with respect to the two fixed bases chosen in $\mathcal{H}_{L}$ and
$\mathcal{H}_{M}$. Notice also the useful identity

$\displaystyle Y\otimes Z|X\rangle\\!\rangle=|YXZ^{T}\rangle\\!\rangle,$ (5)

where $X\in\mathcal{L}(\mathcal{H}_{L},\mathcal{H}_{M})$,
$Y\in\mathcal{L}(\mathcal{H}_{M},\mathcal{H}_{N})$ and
$Z\in\mathcal{L}(\mathcal{H}_{L},\mathcal{H}_{K})$. Moreover, for
$X,Y\in\mathcal{L}(\mathcal{H}_{L},\mathcal{H}_{M})$, one has
$\operatorname{Tr}_{\mathcal{H}_{L}}[|X\rangle\\!\rangle\langle\\!\langle
Y|]=XY^{\dagger}$, and
$\operatorname{Tr}_{\mathcal{H}_{M}}[|X\rangle\\!\rangle\langle\\!\langle
Y|]=X^{T}Y^{*}$. We remark that, in the above paragraph, we are dealing with
abstract linear operators on an Hilbert space, disregarding their possible
interpretation as Jordan-Wigner representatives of some fermionic operator.

A notion that will be used in the following is that of states dilation.

######  Definition II.1 (Dilation set of a state $\rho$).

For any $\rho\in\mathsf{St}{(\mathrm{L}_{\mathrm{F}})}$, we say that
$\Psi_{\rho}\in\mathsf{St}(\mathrm{L}_{\mathrm{F}}\mathrm{M}_{\mathrm{F}})$
for some system $\mathrm{M}_{\mathrm{F}}$, is a dilation of $\rho$ if
$\rho=\operatorname{Tr}^{f}_{\mathrm{M}_{\mathrm{F}}}[\Psi_{\rho}]$. We denote
by $D_{\rho}$ the set of all possible dilations of $\rho$. A pure dilation
$\Psi_{\rho}\in\mathsf{PurSt}(\mathrm{L}_{\mathrm{F}}\mathrm{M}_{\mathrm{F}})$
of $\rho$ is called a purification.

Naturally, any purification of $\rho$ belongs to $D_{\rho}$, more precisely
the set of purifications of $\rho$ is the subset of $D_{\rho}$ containing pure
states. A main feature of quantum theory that is valid also for fermionic
systems is the existence of a purification of any state, that is unique modulo
channels on the purifying system.

######  Proposition II.2 (Purification of states).

For every $\rho\in\mathsf{St}{(\mathrm{L}_{\mathrm{F}})}$, there exists a
purification
$\Psi_{\rho}\in\mathsf{PurSt}(\mathrm{L}_{\mathrm{F}}\mathrm{M}_{\mathrm{F}})$
of $\rho$ for some system $\mathrm{M}_{\mathrm{F}}$. Moreover, the
purification is unique up to channels on the purifying system: if
$\Psi_{\rho}\in\mathsf{PurSt}(\mathrm{L}_{\mathrm{F}}\mathrm{M}_{\mathrm{F}})$
and
$\Phi_{\rho}\in\mathsf{PurSt}(\mathrm{L}_{\mathrm{F}}\mathrm{K}_{\mathrm{F}})$
are two purifications of $\rho$ then there exists a channel
$\mathscr{V}\in\mathsf{Tr}_{1}(\mathrm{M}_{\mathrm{F}}\rightarrow\mathrm{K}_{\mathrm{F}})$
such that
$(\mathscr{I}_{\mathrm{L}_{\mathrm{F}}}\boxtimes\mathscr{V})(\Psi_{\rho})=\Phi_{\rho}$.

###### Proof.

It can be easily verified that every purification of
$\rho\in\mathsf{St}{(\mathrm{L}_{\mathrm{F}})}$, having even part $\rho_{e}$
and odd part $\rho_{o}$, can be obtained in terms of the minimal one
$J^{-1}(|F\rangle\\!\rangle\langle\\!\langle
F|)\in\mathsf{PurSt}(\mathrm{L}_{\mathrm{F}}\mathrm{M}_{\mathrm{F}})$, with
$F=J(\rho)^{\frac{1}{2}}$, $M=\lceil\log_{2}{2r}\rceil$ and
$r=\max(\operatorname{rank}(\rho_{e}),\operatorname{rank}(\rho_{o}))$. Now,
let
$\Psi_{\rho}\in\mathsf{PurSt}(\mathrm{L}_{\mathrm{F}}\mathrm{M}_{\mathrm{F}})$
and
$\Phi_{\rho}\in\mathsf{PurSt}(\mathrm{L}_{\mathrm{F}}\mathrm{K}_{\mathrm{F}})$
be two purifications of $\rho$. If $M=K$, let us choose the ordering defining
the Jordan-Wigner isomorphism of Eq. (2) in such a way that the modes in the
purifying systems $\mathrm{M}_{\mathrm{F}}$ precede the modes of
$\mathrm{L}_{\mathrm{F}}$. Then, using the quantum purification theorem, we
know that there exists a reversible map $\mathscr{U}$ with unitary Kraus $U$
such that $|F_{\rho}\rangle\\!\rangle=(U\otimes I)|P_{\rho}\rangle\\!\rangle$,
where

$\displaystyle|F_{\rho}\rangle\\!\rangle\langle\\!\langle
F_{\rho}|=J(\Phi_{\rho}),\quad|P_{\rho}\rangle\\!\rangle\langle\\!\langle
P_{\rho}|=J(\Psi_{\rho}).$

The unitary $U$ can be chosen in such a way that $J^{-1}(\mathscr{U})$ is an
admissible fermionic map, namely in such a way that it respects the parity
superselection rule (see Lemma B.1 in Appendix B). Moreover, due to Lemma B.2
in Appendix B, $J^{-1}(U\otimes I)$ cannot contain field operators on the
modes in $\mathrm{L}_{\mathrm{F}}$, and is then local on the purifying system
$\mathrm{K}_{\mathrm{F}}$. Now, let $K>M$. Then, we can consider a pure state
$\omega$ on the $K-M$ modes and take the parallel composition
$\Psi_{\rho}\boxtimes\omega$. This is still a purification of $\rho$, and by
the previous argument, there exists a reversible channel
$\mathscr{U}\in\mathsf{Tr}_{1}(\mathrm{K}_{\mathrm{F}}\rightarrow\mathrm{K}_{\mathrm{F}})$
such that
$\Phi_{\rho}=(\mathscr{I}_{\mathrm{L}_{\mathrm{F}}}\boxtimes\mathscr{U})(\Psi_{\rho}\boxtimes\omega)=(\mathscr{I}_{\mathrm{L}_{\mathrm{F}}}\boxtimes\mathscr{V})(\Psi_{\rho})$
where $\mathscr{V}$ is the channel defined by
$\mathscr{V}=\mathscr{U}(\mathscr{I}\boxtimes\omega)$. If $K<M$, we consider
$\Phi_{\rho}\boxtimes\omega$, where $\omega$ is any pure state on $N=M-K$
modes system, and we have
$\Phi_{\rho}\boxtimes\omega=(\mathscr{I}_{\mathrm{L}_{\mathrm{F}}}\boxtimes\mathscr{U})(\Psi_{\rho})$.
Now we discard the additional modes, and the channel connecting the
purifications is the sequential composition of $\mathscr{U}$ and the
discarding map:
$\mathscr{V}:=(\mathscr{I}_{\mathrm{K}_{\mathrm{F}}}\boxtimes\operatorname{Tr}^{f}_{\mathrm{N}_{\mathrm{F}}})\mathscr{U}$.
∎

The main difference between fermionic and quantum information lies in the
notion of what Kraus operators correspond to local maps. While in the case of
qubit systems local maps acting on the $i$-th qubit of a composite system have
Kraus operators that can be factorized as a non trivial operator on the $i$-th
tensor factor $\mathbb{C}^{2}$ of the total Hilbert space, in the case of the
fermionic Fock space $\mathcal{F}_{L}$ a local transformation on the $i$-th
mode can be represented in the Jordan-Wigner isomorphism by operators that act
non trivially on factors $\mathbb{C}^{2}$ different from the $i$-th one. This
fact is the source of all the differences between the theory of qubits and
fermionic theory, including superselection and features that it affects, such
as the notion of entanglement DAriano2014  and local states discrimination
protocols fermLOCC1 ; fermLOCC2 . Due to parity superselection, fermionic
theory does not satisfy _local process tomography_ , namely the property
stating that two transformations
$\mathscr{C}_{1},\mathscr{C}_{2}\in\mathsf{Tr}(\mathrm{L}_{\mathrm{F}}\rightarrow\mathrm{M}_{\mathrm{F}})$
are equal iff they act in the same way on the local states in
$\mathsf{St}(\mathrm{L}_{\mathrm{F}})$, namely
$\mathscr{C}_{1}(\rho)=\mathscr{C}_{2}(\rho)$ for every
$\rho\in\mathsf{St}(\mathrm{L}_{\mathrm{F}})$ (see for example Eq. (1) in the
introduction on the equality between the compression map $\mathscr{C}$ and the
identity map). As a consequence, fermionic theory also violates _local
tomography_. A typical example of a transformation that is locally equivalent
to the identity but differs from it when extended to multipartite systems is
the parity transformation, as shown in the following. Let us consider a single
fermionic mode system $\mathrm{1}_{\mathrm{F}}$, whose possible states are
constrained to be of the form
$J(\rho)=q_{0}\left|{0}\right\rangle\left\langle{0}\right|+q_{1}\left|{1}\right\rangle\left\langle{1}\right|$
by the parity superselection rule. Let $P_{0}$ and $P_{1}$ be the projectors
on $\left|{0}\right\rangle$ and $\left|{1}\right\rangle$ respectively, namely
on the even and odd sector of the Fock space. The parity transformation
$\mathscr{P}$, that in the Jordan-Wigner representation $J(\mathscr{P})$ has
Kraus operators $P_{0}$ and $P_{1}$, acts as the identity
$\mathscr{I}_{\mathrm{1}_{\mathrm{F}}}$ when applied to states in
$\mathsf{St}{(\mathrm{1}_{\mathrm{F}})}$. However, taking the system
$\mathrm{2}_{\mathrm{F}}$ and considering the extended transformation
$\mathscr{P}\boxtimes\mathscr{I}_{\mathrm{1}_{\mathrm{F}}}$ on
$\mathsf{St}{(\mathrm{2}_{\mathrm{F}})}$ one notices that $\mathscr{P}$
differs from the identity map $\mathscr{I}_{\mathrm{1}_{\mathrm{F}}}$. Indeed,
the state $J^{-1}(\left|{\Psi}\right\rangle\left\langle{\Psi}\right|)$, with
$\left|{\Psi}\right\rangle=\frac{1}{\sqrt{2}}(\left|{00}\right\rangle+\left|{11}\right\rangle)$
is a legitimate fermionic state in $\mathsf{St}{(\mathrm{2}_{\mathrm{F}})}$,
and one can straightforwardly verify that

$\displaystyle(\mathscr{P}\boxtimes\mathscr{I}_{\mathrm{1}_{\mathrm{F}}})[J^{-1}(\left|{\Psi}\right\rangle\left\langle{\Psi}\right|)]$
$\displaystyle=\frac{1}{2}J^{-1}(\left|{00}\right\rangle\left\langle{00}\right|+\left|{11}\right\rangle\left\langle{11}\right|)$
$\displaystyle\neq
J^{-1}(\left|{\Psi}\right\rangle\left\langle{\Psi}\right|).$

###  II.1 Identical channels upon-input of $\rho$

In the following we will be interested in quantitatively assessing how closely
a channel (the compression map) resembles another one (the identity map),
provided that we know that the input state corresponds to a given $\rho$. To
this end we introduce the notion of identical fermionic channels upon-input of
$\rho$.

Given two fermionic channels
$\mathscr{C}_{1},\mathscr{C}_{2}\in\mathsf{Tr}_{1}{(\mathrm{L}_{\mathrm{F}}\to\mathrm{M}_{\mathrm{F}})}$
and a state $\rho\in\mathsf{St}(\mathrm{L}_{\mathrm{F}})$, we say that
$\mathscr{C}_{1}$ and $\mathscr{C}_{2}$ are equal upon-input of $\rho$ if

$(\mathscr{C}_{1}\boxtimes\mathscr{I})(\Sigma)=(\mathscr{C}_{2}\boxtimes\mathscr{I})(\Sigma)\qquad\forall\Sigma\in\mathsf{Ref}(D_{\rho}).$
(6)

Operationally, this means that one cannot discriminate between
$\mathscr{C}_{1}$ and $\mathscr{C}_{2}$ when applied to any dilation
$\Psi_{\rho}$ of the state $\rho$, independently of how $\Psi_{\rho}$ has been
prepared. Suppose that $\Psi_{\rho}\in D_{\rho}$ was prepared as
$\Psi_{\rho}=\sum_{i}\Sigma_{i}$, for some refinement of $\Psi_{\rho}$. Even
using the knowledge of the preparation, one cannot distinguish between
$\mathscr{C}_{1}$ and $\mathscr{C}_{2}$. Notice that, differently from the
quantum case here it is necessary to check the identity between channels on
bipartite systems.

Following the above definition, one can quantify how close two channels are.
One has that $\mathscr{C}_{1}$ and $\mathscr{C}_{2}$ are $\varepsilon$-close
upon-input of $\rho$ if

$\displaystyle\sum_{i}\lVert[(\mathscr{C}_{1}-\mathscr{C}_{2})\boxtimes\mathscr{I}](\Sigma_{i})\rVert_{1}\leq\varepsilon\quad\forall\\{\Sigma_{i}\\}:\
\sum_{i}\Sigma_{i}\in D_{\rho},$ (7)

where $\lVert X\rVert_{1}$ is the $1$-norm of $J(X)$ . One can
straightforwardly prove that the trace distance
$d(\rho,\sigma):=\frac{1}{2}\lVert\rho-\sigma\rVert_{1}$ has a clear
operational interpretation in terms of the maximum success probability of
discrimination between the two states $\rho$ and $\sigma$. Eq. (7) provides
then an upper bound for the probability of discriminating between
$\mathscr{C}_{1}$ and $\mathscr{C}_{2}$ when applied to the dilations of
$\rho$, including their refinements: $\mathscr{C}_{1}$ and $\mathscr{C}_{2}$
cannot be distinguished with a succes probability bigger than
$\frac{1}{2}+\frac{1}{4}\varepsilon$. Accordingly, a sequence of channels
$\mathscr{C}_{N}\in\mathsf{Tr}_{1}{(\mathrm{L}_{\mathrm{F}}\to\mathrm{M}_{\mathrm{F}})}$
converges to the channel
$\mathscr{C}\in\mathsf{Tr}_{1}{(\mathrm{L}_{\mathrm{F}}\to\mathrm{M}_{\mathrm{F}})}$
upon-input of $\rho$ if

$\lim_{N\to\infty}\lVert[(\mathscr{C}_{N}-\mathscr{C})\boxtimes\mathscr{I}](\Sigma)\rVert_{1}=0\quad\forall\Sigma\in\mathsf{Ref}(D_{\rho}).$

##  III Fermionic compression

Consider now a system $\mathrm{L}_{\mathrm{F}}$ and let
$\rho\in\mathsf{St}{(\mathrm{L}_{\mathrm{F}})}$ be the generic state of the
system. As usual the source of fermionic information is supposed to emit $N$
independent copies of the state $\rho$ . A fermionic compression scheme
$(\mathscr{E}_{N},\mathscr{D}_{N})$ consists of the following two steps:

  1. 1.

Encoding: Alice encodes the system $\mathrm{L}_{\mathrm{F}}^{\boxtimes N}$ via
a channel $\mathscr{E}_{N}:\mathsf{St}(\mathrm{L}_{\mathrm{F}}^{\boxtimes
N})\rightarrow\mathsf{St}(\mathrm{M}_{\mathrm{F}})$, where the target system
is generally a system of $M$-LFMs. The map $\mathscr{E}_{N}$ produces a
fermionic state $\mathscr{E}(\rho^{\boxtimes N})$ with support
$\mathsf{Supp}(\mathscr{E}(\rho^{\boxtimes N}))$ on a Fock space
$\mathcal{F}_{M}$ of dimension $d_{\mathcal{F}_{M}}(N)$ smaller than the one
of the original state $\rho^{\boxtimes N}$. The compression rate is defined as
the quantity

$\displaystyle R=\log_{2}d_{\mathcal{F}_{M}}(N)/N.$

Alice sends the system $\mathrm{M}_{\mathrm{F}}$ to Bob using $N\lceil
R\rceil$ noiseless fermionic channels.

  2. 2.

Decoding: Finally Bob sends the system $\mathrm{M}_{\mathrm{F}}$ through a
decompression channel
$\mathscr{D}_{N}:\mathsf{St}(\mathrm{M}_{\mathrm{F}})\rightarrow\mathsf{St}(\mathrm{L}_{\mathrm{F}}^{\boxtimes
N})$.

The scheme $(\mathscr{E}_{N},\mathscr{D}_{N})$ overall transforms the
$L^{\boxtimes N}$ LFMs, with a compression map
$\mathscr{C}_{N}:=\mathscr{D}_{N}\mathscr{E}_{N}$. The latter can be more or
less “good” (in a sense that will be precisely defined) in preserving the
information which is contained in the system, depending on $\rho$ itself. The
goal now is to define the notion of reliable compression scheme once we are
provided with an information source $\rho$.

###  III.1 Reliable compression scheme

The aim of a compression scheme, besides reducing the amount of information
carriers used, is to preserve all the information that is possibly encoded in
a given state $\rho$. What we actually mean is not only to preserve the input
state and keep track of the correlations of our system with an arbitrary
ancilla, but also to preserve these informations for any procedure by which
the input system and its correlations have been prepared. In other words, even
the agent that prepared the system along with possible ancillary systems, must
have a small success probability in detecting the effects of compression on
the original preparation. This amounts to require that the compression channel
$\mathscr{C}_{N}$ must be approximately equal to the identity channel upon-
input of $\rho$, and more precisely that in the limit of $N\to\infty$ the two
channels must coincide upon-input of $\rho$.

In Section II.1 we introduced the notion of $\varepsilon$-close channels upon-
input of $\rho$. This notion can now be used to quantify the error, say
$\varepsilon$, introduced by the map $\mathscr{C}_{N}$ in a compression
protocol given the source $\rho$. According to Eq. (7) we have indeed the
following definition of a reliable compression scheme

######  Definition III.1 (Reliable compression scheme).

Given a state $\rho\in\mathsf{St}(\mathrm{L}_{\mathrm{F}})$, a compression
scheme $(\mathscr{E}_{N},\mathscr{D}_{N})$ is $\varepsilon$-reliable if
$\sum_{i}\left\lVert(\mathscr{C}_{N}\boxtimes\mathscr{I})(\Sigma_{i})-\Sigma_{i}\right\rVert<\varepsilon$
for every $\\{\Sigma_{i}\\}$ such that $\sum_{i}\Sigma_{i}\in
D_{\rho^{\boxtimes N}}$, where
$\mathscr{C}_{N}:=\mathscr{D}_{N}\mathscr{E}_{N}$.

It is clear from the definition that in order to check the reliability of a
fermionic compression map one should test it on states of an arbitrary large
system, since the dilation set $D_{\rho^{\boxtimes N}}$ includes dilations on
any possible ancillary system. It is then necessary to find a simpler
criterion to characterize the reliability of a compression scheme. Let us
start with a preliminary definition.

######  Definition III.2.

Let $\rho\in\mathsf{St}(\mathrm{L}_{\mathrm{F}})$. We define its _square root_
$\rho^{\frac{1}{2}}$ as follows

$\displaystyle\rho^{\frac{1}{2}}\coloneqq J^{-1}[J(\rho)^{\frac{1}{2}}].$ (8)

One can easily prove that the square root of a fermionic state is well
defined, i.e. it does not depend on the particular Jordan-Wigner
representation $J$ chosen (see Appendix C). In the following we show that a
useful criterion for reliability can be expressed via _entanglement fidelity_
:

######  Definition III.3 (Entanglement fidelity).

Let $\rho\in\mathsf{St}_{1}(\mathrm{L}_{\mathrm{F}})$,
$\mathscr{C}\in\mathsf{Tr}_{1}(\mathrm{L}_{\mathrm{F}}\rightarrow\mathrm{M}_{\mathrm{F}})$
and
$\Phi_{\rho}\in\mathsf{PurSt}(\mathrm{L}_{\mathrm{F}}\mathrm{K}_{\mathrm{F}})$
be any purification of $\rho$. The entanglement fidelity is defined as and
$F(\rho,\mathscr{C})=F(\Phi_{\rho},(\mathscr{C}\boxtimes\mathscr{I})(\Phi_{\rho}))^{2}$,
where $F(\rho,\sigma):=\operatorname{Tr}[J(\rho^{1/2}\sigma\rho^{1/2})^{1/2}]$
denotes the Uhlmann’s fidelity between states
$\rho,\sigma\in\mathsf{St}_{1}(\mathrm{L}_{\mathrm{F}})$.

We notice that the Uhlmann fidelity of fermionic states is well defined,
namely it is independent of the ordering of the fermionic modes (see Appendix
C). As a consequence also the Entanglement fidelity, given in terms of the
Uhlmann one, must be well defined.

Since by definition the Uhlmann fidelity of fermionic states coincides with
the one of their Jordan-Wigner representatives and the same for their trace-
norm distance, given $\rho,\sigma\in\mathsf{St}(\mathrm{L}_{\mathrm{F}})$, the
Fuchs-van der Graaf inequalities 761271  hold as a trivial consequence of
their quantum counterparts

$1-F(\rho,\sigma)\leq\frac{1}{2}\lVert\rho-\sigma\rVert_{1}\leq\sqrt{1-F(\rho,\sigma)^{2}}.$
(9)

The following proposition summarizes the main properties of fermionic
entanglement fidelity that will be used in the remainder.

######  Proposition III.1.

Let $\rho\in\mathsf{St}_{1}(\mathrm{L}_{\mathrm{F}})$,
$\mathscr{C}\in\mathsf{Tr}_{1}(\mathrm{L}_{\mathrm{F}}\rightarrow\mathrm{L}_{\mathrm{F}})$
and
$\Phi_{\rho}\in\mathsf{PurSt}(\mathrm{L}_{\mathrm{F}}\mathrm{K}_{\mathrm{F}})$
be any purification of $\rho$. Entanglement fidelity has the following
properties.

  1. 1.

$F(\rho,\mathscr{C})$ is independent of the particular choice for the
purification $\Phi_{\rho}$.

  2. 2.

If the ordering is chosen in such a way that the $L$ modes are all before the
purifying ones, the following identity holds:

$F(\rho,\mathscr{C})=\sum_{i}|\operatorname{Tr}[J(\rho)C_{i}]|^{2}$ (10)

for arbitrary Kraus decomposition $J(\mathscr{C})=\sum_{i}C_{i}\cdot
C_{i}^{\dagger}$ of the Jordan-Wigner representative $J(\mathscr{C})$. From
the second inequality in (9) it follows that, if $F(\rho,\mathscr{C})\geq
1-\delta$, one has

$\lVert(\mathscr{C}\boxtimes\mathscr{I}_{\mathrm{C}})(\Phi_{\rho})-\Phi_{\rho}\rVert_{1}\leq
2\sqrt{\delta}$ (11)

for every purification $\Phi_{\rho}$ of $\rho$.

###### Proof.

Let
$\Phi_{\rho}\in\mathsf{PurSt}(\mathrm{L}_{\mathrm{F}}\mathrm{M}_{\mathrm{F}})$
be a purification of $\rho$. If we choose the trivial ordering for the LFMs,
the Kraus of $J(\mathscr{C}\boxtimes\mathscr{I})$ are of the form
$C_{i}\otimes I$. Moreover, since the minimal purification
$|F\rangle\\!\rangle\langle\\!\langle F|$ (introduced in the proof of
proposition II.2) and $J(\Phi_{\rho})$ both purify the same quantum state,
they are connected through an isometry $V$. Recalling that for quantum states
$\left|{\psi}\right\rangle\left\langle{\psi}\right|$ and $\sigma$ the quantum
Uhlmann fidelity is given by
$F(\left|{\psi}\right\rangle\left\langle{\psi}\right|,\sigma)=\left\langle{\psi}\right|\sigma\left|{\psi}\right\rangle^{1/2}$,
we find

$\displaystyle F(\rho,\mathscr{C})=$
$\displaystyle\sum_{i}\operatorname{Tr}(|FV^{T}\rangle\\!\rangle\langle\\!\langle{FV^{T}}|{C_{i}FV^{T}}\rangle\\!\rangle\langle\\!\langle
C_{i}FV^{T}|)$ $\displaystyle=$
$\displaystyle\sum_{i}|\operatorname{Tr}(|C_{i}FV^{T}\rangle\\!\rangle\langle\\!\langle
FV^{T}|)|^{2}=$ $\displaystyle=$
$\displaystyle\sum_{i}|\operatorname{Tr}[J(\rho)C_{i}]|^{2},$

namely, the claimed formula in (10). Since $\Phi_{\rho}$ is arbitrary, this
also implies independence from the choice of the purification. ∎

In quantum theory a compression scheme $(\mathscr{E}_{N},\mathscr{D}_{N})$ is
reliable when the entanglement fidelity $F(\rho^{\boxtimes
N},\mathscr{C}_{N})$, with $\mathscr{C}_{N}:=\mathscr{D}_{N}\mathscr{E}_{N}$,
approaches $1$ as $N\rightarrow\infty$. Here we prove an analogous reliability
criterion for the fermionic case.

We can now prove the following proposition and the subsequent corollary
providing a simple reliability criterion for fermionic compression.

######  Proposition III.2.

Given a state $\rho\in\mathsf{St}_{1}(\mathrm{L}_{\mathrm{F}})$ and a channel
$\mathscr{C}\in\mathsf{Tr}_{1}(\mathrm{L}_{\mathrm{F}}\to\mathrm{L}_{\mathrm{F}})$,
$\forall\varepsilon>0$ there exists $\delta>0$ such that if
$F(\rho,\mathscr{C})>1-\delta$ then
$\sum_{i}\left\lVert[(\mathscr{C}-\mathscr{I})\boxtimes\mathscr{I}](\Sigma_{i})\right\rVert_{1}\leq\varepsilon$
for every $\\{\Sigma_{i}\\}$ such that $\sum_{i}\Sigma_{i}\in D_{\rho}$.

###### Proof.

Firstly we observe that, given a set of states $\\{\Sigma_{i}\\}$ such that
$\sum_{i}\Sigma_{i}\in D_{\rho}$, considering any purification
$\Psi_{\rho}\in\mathsf{PurSt}(\mathrm{L}_{\mathrm{F}}\mathrm{K}_{\mathrm{F}}\mathrm{N}_{\mathrm{F}})$
of $\Sigma\coloneqq\sum_{i}\Sigma_{i}$, one can find a POVM
$\\{b_{i}\\}\in\mathsf{Eff}(\mathrm{N}_{\mathrm{F}})$ such that
$\Sigma_{i}=\operatorname{Tr}_{\mathrm{N}_{\mathrm{F}}}[(I_{\mathrm{L}_{\mathrm{F}}\mathrm{K}_{\mathrm{F}}}\boxtimes
b_{i})\Psi_{\rho}]$. As a consequence, we have

$\displaystyle\sum_{i}$
$\displaystyle\lVert(\mathscr{C}-\mathscr{I})\Sigma_{i}\rVert_{1}$
$\displaystyle=\sum_{i}\lVert\operatorname{Tr}_{\mathrm{N}_{\mathrm{F}}}[\\{(\mathscr{C}-\mathscr{I})\boxtimes\mathscr{I}_{\mathrm{N}_{\mathrm{F}}}\\}(\Psi_{\rho})(I_{\mathrm{L}_{\mathrm{F}}\mathrm{K}_{\mathrm{F}}}\boxtimes
b_{i})]\rVert_{1}$
$\displaystyle\leq\lVert\\{(\mathscr{C}-\mathscr{I})\boxtimes\mathscr{I}_{\mathrm{N}_{\mathrm{F}}}\\}(\Psi_{\rho})\rVert_{1},$

where the last inequality follows from the equivalent definition of the 1-norm
for $X\in\mathsf{St}_{\mathbb{R}}(\mathrm{L}_{\mathrm{F}})$

$\displaystyle\|X\|_{1}=\max_{b\in\mathsf{Eff}(\mathrm{L}_{\mathrm{F}})}\operatorname{Tr}[Xb],$

and from the fact that, for
$\\{a_{i}\\}\subseteq\mathsf{Eff}(\mathrm{L}_{\mathrm{F}}\mathrm{K}_{\mathrm{F}})$
such that

$\displaystyle\lVert\operatorname{Tr}_{\mathrm{N}_{\mathrm{F}}}[$
$\displaystyle\\{(\mathscr{C}-\mathscr{I})\boxtimes\mathscr{I}_{\mathrm{N}_{\mathrm{F}}}\\}(\Psi_{\rho})(I_{\mathrm{L}_{\mathrm{F}}\mathrm{K}_{\mathrm{F}}}\boxtimes
b_{i})]\rVert_{1}$
$\displaystyle=\operatorname{Tr}_{\mathrm{N}_{\mathrm{F}}}[\\{(\mathscr{C}-\mathscr{I})\boxtimes\mathscr{I}_{\mathrm{N}_{\mathrm{F}}}\\}(\Psi_{\rho})(a_{i}\boxtimes
b_{i})],$

one can write

$\displaystyle\sum_{i}\lVert($
$\displaystyle\mathscr{C}-\mathscr{I})\Sigma_{i}\rVert_{1}$
$\displaystyle=\sum_{i}\operatorname{Tr}_{\mathrm{N}_{\mathrm{F}}}[\\{(\mathscr{C}-\mathscr{I})\boxtimes\mathscr{I}_{\mathrm{N}_{\mathrm{F}}}\\}(\Psi_{\rho})(a_{i}\boxtimes
b_{i})]$
$\displaystyle=\operatorname{Tr}_{\mathrm{N}_{\mathrm{F}}}[\\{(\mathscr{C}-\mathscr{I})\boxtimes\mathscr{I}_{\mathrm{N}_{\mathrm{F}}}\\}(\Psi_{\rho})A],$

where $A\coloneqq\sum_{i}(a_{i}\boxtimes b_{i})$. Now, by the Fuchs-van der
Graaf inequalities, if $F(\rho,\mathscr{C})\geq 1-\delta$, then

$\displaystyle\lVert\\{(\mathscr{C}-\mathscr{I})\boxtimes\mathscr{I}_{\mathrm{N}_{\mathrm{F}}}\\}(\Psi_{\rho})\rVert_{1}\leq
2\sqrt{1-F(\rho,\mathscr{C})}\leq 2\sqrt{\delta}.$

The thesis is then obtained just taking $\delta\leq\varepsilon^{2}/4$. ∎

######  Corollary III.1 (Reliable compression scheme).

Given a state $\rho\in\mathsf{St}(\mathrm{L}_{\mathrm{F}})$, a compression
scheme $(\mathscr{E}^{N},\mathscr{D}^{N})$ is $\epsilon$-reliable if one has
$F(\rho^{\boxtimes N},\mathscr{C}_{N})>1-\delta$, where
$\delta=\epsilon^{2}/4$, and
$\mathscr{C}_{N}:=\mathscr{D}^{N}\mathscr{E}^{N}$.

###  III.2 Fermionic typical subspace

At the basis of the quantum source coding theorem lies the notion of typical
subspace, that in turn generalizes to the quantum case that of typical
sequences and typical sets of classical information. We now introduce the
notion of typical subspace also for fermionic systems and use it to show that,
like in quantum theory, the von Neumann entropy of a fermionic state is the
rate that separates the region of rates for which a reliable compression
scheme exists from that of unachievble rates. In order to do this we have to
verify that the compression map given in terms of the projection on the
typical subspace represents an admissible fermionic map.

We start by defining the notion of logarithm of a fermionic state

######  Definition III.4.

Let $\rho$ be a fermionic state. We define its logarithm as

$\displaystyle\log_{2}\rho=J^{-1}[\log_{2}J(\rho)].$ (12)

Then we define the von Neumann entropy of a fermionic state via its Jordan-
Wigner representative.

######  Definition III.5.

Given a fermionic state $\rho$, its von-Neumann entropy is defined as

$\displaystyle
S_{f}(\rho):=S(J(\rho))=-\operatorname{Tr}(J(\rho)\log_{2}J(\rho)).$ (13)

These definitions are independent of the particular Jordan-Wigner transform
corresponding to a given ordering of the modes (see Appendix C).

When we use the orthonormal decomposition for
$J(\rho)=\sum_{x_{i}}p_{i}|{x_{i}}\rangle\langle{x_{i}}|$, this reduces to the
Shannon entropy of the classical random variable $X$ that takes values in
$\mathsf{Rng}(X)=\\{x_{1},x_{2},\ldots x_{n}\\}$, called range of $X$, with
probability distribution $(p_{1},p_{2},\ldots,p_{n})$:
$S_{f}(\rho)=H(X)=-\sum_{i}p_{i}\log_{2}p_{i}$. We remind that $N$ i.i.d.
copies of the state $\rho$ are represented as $J(\rho^{\boxtimes
N})=J(\rho)^{\otimes
N}=\sum_{x_{\mathrm{\mathbf{i}}}\in\mathsf{Rng}(X)^{N}}p_{\mathrm{\mathbf{i}}}|{x_{\mathrm{\mathbf{i}}}}\rangle\langle{x_{\mathrm{\mathbf{i}}}}|,$.
With $\mathsf{T}_{N,\varepsilon}(\rho)$ we will denote the typical set of the
random variable $X$.

######  Definition III.6 (Typical subspace).

Let $\rho\in\mathsf{St}(\mathrm{L}_{\mathrm{F}})$ with orthonormal
decomposition
$J(\rho)=\sum_{x_{i}\in\mathsf{Rng}(X)}p_{i}|{x_{i}}\rangle\langle{x_{i}}|$.
The $\varepsilon$-typical subspace $\mathsf{F}_{N,\varepsilon}(\rho)$ of
$\mathcal{H}^{\otimes N}_{L}$ is defined as

$\mathsf{F}_{N,\varepsilon}(\rho):=\mathsf{Span}\\{\left|{x_{\mathrm{\mathbf{i}}}}\right\rangle\
|\ x_{\mathrm{\mathbf{i}}}\in\mathsf{T}_{N,\varepsilon}(X)\\},$ (14)

where
$\left|{x_{\mathrm{\mathbf{i}}}}\right\rangle:=\left|{x_{i_{1}}}\right\rangle\left|{x_{i_{2}}}\right\rangle\ldots\left|{x_{i_{N}}}\right\rangle$,
and $X$ is the random variable with $\mathsf{Rng}(X)=\\{x_{i}\\}$ and
$\mathbb{P}_{X}(x_{i}):=p_{i}$.

It is an immediate consequence of the definition of typical subspace that

$\mathsf{F}_{N,\varepsilon}(\rho):=\mathsf{Span}\left\\{\left|{x_{\mathrm{\mathbf{i}}}}\right\rangle\
|\
\left|\frac{1}{N}\log_{2}\frac{1}{\mathbb{P}_{X^{N}}(x_{\mathrm{\mathbf{i}}})}-S_{f}(\rho)\right|\leq\varepsilon\right\\}.$

We will denote the projector on the typical subspace as

$\displaystyle P_{N,\varepsilon}(\rho):=$
$\displaystyle\sum_{x_{\mathrm{\mathbf{i}}}\in\mathsf{T}_{N,\varepsilon}(X)}|{x_{\mathrm{\mathbf{i}}}}\rangle\langle{x_{\mathrm{\mathbf{i}}}}|$
(15) $\displaystyle=$
$\displaystyle\sum_{x_{\mathrm{\mathbf{i}}}\in\mathsf{T}_{N,\varepsilon}(X)}|{x_{i_{1}}}\rangle\langle{x_{i_{1}}}|\otimes\cdots\otimes|{x_{i_{N}}}\rangle\langle{x_{i_{N}}}|,$

and we have that
$\dim(\mathsf{F}_{N,\varepsilon}(\rho))=\operatorname{Tr}[P_{N,\varepsilon}(\rho)]=|\mathsf{T}_{N,\varepsilon}(X)|$.

Notice that some of the superpositions of vectors in the typical subspace
might not be legitimate fermionic pure states, as their parity might be
different. However, up to now, we only defined the typical subspace as a
mathematical tool, and it does not need a consistent physical interpretation.
We will come back to this point later (see Lemma III.1), when we will discuss
the physical meaning of the projection $P_{N,\varepsilon}(\rho)$. Now, it is
immediate to see that

$\displaystyle\operatorname{Tr}[P_{N,\varepsilon}(\rho)J(\rho)^{\otimes N}]$
$\displaystyle\qquad=\sum_{x_{\mathrm{\mathbf{i}}}\in\mathsf{T}_{N,\varepsilon}(\mathrm{X})}\mathbb{P}_{X^{N}}(x_{\mathrm{\mathbf{i}}})=\mathbb{P}_{\mathrm{X}^{N}}[x_{\mathrm{\mathbf{i}}}\in\mathsf{T}_{N,\varepsilon}(X)].$
(16)

As in quantum theory, also the fermionic typical subspace has the following
features:

######  Proposition III.3 (Typical subspace).

Let $\rho\in\mathsf{St}(\mathrm{L}_{\mathrm{F}})$. The following statements
hold:

  1. 1.

For every $\varepsilon>0$ and $\delta>0$ there exists $N_{0}$ such that for
every $N\geq N_{0}$

$\operatorname{Tr}[P_{N,\varepsilon}(\rho)J(\rho)^{\otimes N}]\geq 1-\delta.$
(17)

  2. 2.

For every $\epsilon>0$ and $\delta>0$ there exists $N_{0}$ such that for every
$N\geq N_{0}$ the dimension of the typical subspace
$\mathsf{F}_{N,\varepsilon}(\rho)$ is bounded as

$(1-\delta)2^{N(S_{f}(\rho)-\varepsilon)}\leq\dim(\mathsf{F}_{N,\varepsilon}(\rho))\leq
2^{N(S_{f}(\rho)+\varepsilon)}$ (18)

  3. 3.

For given $N$, let $S_{N}$ denote an arbitrary orthogonal projection on a
subspace of $\mathcal{F}_{L}^{\otimes N}$ with dimension
$\operatorname{Tr}(S_{N})<2^{NR}$, with $R<S_{f}(\rho)$ fixed. Then for every
$\delta>0$ there exists $N_{0}$ such that for every $N\geq N_{0}$ and every
choice of $S_{N}$

$\operatorname{Tr}[S_{N}J(\rho)^{\otimes N}]\leq\delta.$ (19)

The proof of the above properties is exactly the same as the one of quantum
theory (see for instance QInielsenchuang ). However, order to exploit the same
scheme proposed by Schumacher for the quantum case, one has to check that the
encoding and decoding channels given in the constructive part of the proof are
admissible fermionic maps. In particular, the encoding channel makes use of
the projector $P_{N,\varepsilon}(\rho)$ as a Kraus operator, therefore, we
have to show that it is a legitimate Kraus for a fermionic map. This is proved
in the following lemma based on characterization of fermionic transformations
of Proposition II.1.

######  Lemma III.1.

Let $\rho$ be a fermionic state. The projector $P_{N,\varepsilon}(\rho)$ of eq
15 is the Kraus operator of an admissible fermionic transformation.

###### Proof.

By proposition II.1 the projector on the typical subspace
$P_{N,\varepsilon}(\rho)$ is a legitimate fermionic Kraus if it is the sum of
products of either an even or an odd number of fermionic fields. Let us
consider the single projection
$\left|{x_{\mathrm{\mathbf{i}}}}\right\rangle\left\langle{x_{\mathrm{\mathbf{i}}}}\right|$.
This is given by the tensor product
$\left|{x_{i_{1}}}\right\rangle\left\langle{x_{i_{1}}}\right|\otimes\dots\otimes\left|{x_{i_{N}}}\right\rangle\left\langle{x_{i_{N}}}\right|$,
where each $\left|{x_{i_{k}}}\right\rangle$ is an eigenvector of the density
matrix $J(\rho)$ representing the fermionic state $\rho$, and, as such, it has
a definite parity. Thus, each factor in the above expression of
$\left|{x_{\mathrm{\mathbf{i}}}}\right\rangle\left\langle{x_{\mathrm{\mathbf{i}}}}\right|$
is the Jordan-Wigner representative of an even polynomial, and also the
projection
$\left|{x_{\mathrm{\mathbf{i}}}}\right\rangle\left\langle{x_{\mathrm{\mathbf{i}}}}\right|$
is thus the representative of an even polynomial for every
$\mathrm{\mathbf{i}}$, which is given, in detail, by the product
$J^{-1}(\left|{x_{\mathrm{\mathbf{i}}}}\right\rangle\left\langle{x_{\mathrm{\mathbf{i}}}}\right|)=\prod_{j=1}^{N}J^{-1}(\left|{x_{i_{j}}}\right\rangle\left\langle{x_{i_{j}}}\right|)$.
Now, by Proposition II.1, $P_{N,\varepsilon}(\rho)$ is the Jordan-Wigner
representative of a legitimate fermionic Kraus operator. ∎

###  III.3 Fermionic source coding theorem

We can now prove the source coding theorem for fermionic information theory.

######  Theorem III.1 (Fermionic source coding).

Let $\rho\in\mathsf{St}_{1}(\mathrm{L}_{\mathrm{F}})$ be a state of system
$\mathrm{L}_{\mathrm{F}}$. Then for every $\delta>0$ and $R>S_{f}(\rho)$ there
exists $N_{0}$ such that for every $N\geq N_{0}$ one has a compression scheme
$\\{\mathscr{E}_{N},\mathscr{D}_{N}\\}$ with rate $R$, and $F(\rho^{\boxtimes
N},\mathscr{D}_{N}\mathscr{E}_{N})\geq 1-\delta$. Conversely, for every
$R<S_{f}(\rho)$ there is $\delta\geq 0$ such that for every compression scheme
$\\{\mathscr{E}_{N},\mathscr{D}_{N}\\}$ with rate $R$ one has
$F(\rho^{\boxtimes N},\mathscr{D}_{N}\mathscr{E}_{N})\leq\delta$.

The proof follows exactly the lines of the original proof for standard quantum
compression, that can be found e.g. in Ref. QInielsenchuang . As the direct
proof is constructive, we only need to take care of the legitimacy of the
compression protocol as a fermionic map. To this end, we recapitulate the
construction here.

  1. 1.

Encoding: Perform the measurement
$\\{P_{N,\varepsilon}(\rho),I-P_{N,\varepsilon}(\rho)\\}$. If the outcome
corresponding to $P_{N,\varepsilon}(\rho)$ occurs, then leave the state
unchanged. Otherwise, if the outcome corresponding to
$I-P_{N,\varepsilon}(\rho)$ occurs, replace the state by a standard state
$|{S}\rangle\langle{S}|$, with
$\left|{S}\right\rangle\in\mathsf{F}_{N,\varepsilon}(\rho)$. Such an map is
described by the channel $\mathscr{M}_{N}:\mathrm{L}_{\mathrm{F}}^{\boxtimes
N}\to\mathrm{L}^{\boxtimes N}_{\mathrm{F}}$ given by

$\displaystyle J(\mathscr{M}_{N})(\sigma)\coloneqq$ $\displaystyle\quad
P_{N,\varepsilon}(\rho)\sigma
P_{N,\varepsilon}(\rho)+\operatorname{Tr}[(I-P_{N,\varepsilon}(\rho))\sigma]\left|{S}\right\rangle\left\langle{S}\right|$

Notice that this is a well defined transformation since by Lemma III.1 the
projector on the typical subspace is a legitimate fermionic Kraus operator.
The second term is a measure and prepare channel, which is also a legitimate
fermionic transformation. Then consider a system $\mathrm{M}_{\mathrm{F}}$
made of $M:=N\lceil R\rceil$ LFMs and the (partial) isometric embedding
$V:\mathsf{F}_{N,\varepsilon}(\rho)\rightarrow\mathcal{H}_{N\lceil R\rceil}$
such that $V^{\dagger}V=I_{\mathsf{F}_{N,\varepsilon}(\rho)}$. Since the first
stage of the protocol never produces states in the complement of
$\mathsf{F}_{N,\varepsilon}(\rho)$, we can complete the map $V\cdot
V^{\dagger}$ to a fermionic channel $\mathscr{V}_{N}$. The encoding is then
given by the composite map $\mathscr{E}_{N}:=\mathscr{V}_{N}\mathscr{M}_{N}$.

  2. 2.

Decoding: For the decoding channel, we simply choose the co-isometry
$V^{\dagger}$, which inverts $V$ on $\mathsf{F}_{N,\varepsilon}(\rho)$.

As for the converse statement, the proof for quantum compression is based on
item 3, which we proved for fermionic theory as well. Thus, the quantum proof
applies to the fermionic case.

##  IV Discussion

We have studied information compression for fermionic systems, showing the
fermionic counterpart of the quantum source coding theorem. In spite of parity
superselection rule and the non locality of the Jordan-Wigner representation
of fermionic operators, the von Neumann entropy of fermionic states can still
be interpreted as their information content, providing the minimal rate for
which a reliable compression is achievable.

The novelty in this paper is the analysis of compression in the absence of
local tomography. Here, the properties of a map, and in the specific case of
study of the compression map, cannot be accessed locally. This poses stronger
constraints on the set of reliable compression maps.

Despite the significant differences between fermionic and quantum information
DAriano2014 , the source coding theorem holds also for fermions. We can now
wonder which are the minimal features of a theory that lie behind the coding
theorem. As we learn from classical, quantum, and now also fermionic
information theory, the task of information compression is intimately related
to the notion of entropy. However, it is known that information theories
beyond quantum exhibit inequivalent notions of entropy KIMURA2010175 ;
Barnum_2010 ; Short_2010 . This is the main issue one has to face in order to
introduce the notion of information content in the general case. On one side
one has to provide a definition of information content including a broad class
of probabilistic theories. On the other side one can compare such a notion
with the different notions of entropy, identifying the one that plays the same
role of Shannon entropy in the compression task.

###### Acknowledgements.

A.T. acknowledges financial support from the Elvia and Federico Faggin
Foundation through the Silicon Valley Community Foundation, Grant No.
2020-214365. This work was supported by MIUR Dipartimenti di Eccellenza
2018-2022 project F11I18000680001.

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##  Appendix A Fermionic States

In a $L$-LFM system, fermionic states in
$\mathsf{St}(\mathrm{L}_{\mathrm{F}})$ are represented by density matrices on
the antisymmetric Fock space $\mathcal{F}_{L}$ satisfying the parity
superselection rule. As such they can be written as combinations of products
of field operators. Indeed, a fermionic state $\rho$ can be split in its even
and odd part as follows

$\displaystyle\rho=\sum_{e}E_{e}\left|{\Omega}\right\rangle\left\langle{\Omega}\right|E^{\dagger}_{e}+\sum_{o}O_{o}\left|{\Omega}\right\rangle\left\langle{\Omega}\right|O^{\dagger}_{o},$

where $E_{e}$ and $O_{o}$ are linear combinations of products of even and odd
number of field operators respectively. By recalling that
$\left|{\Omega}\right\rangle\left\langle{\Omega}\right|=\prod_{i=1}^{L}\varphi_{i}\varphi_{i}^{\dagger}$
one can easily realize that $\rho$ can be written as combination of products
of even number of field operators. Moreover, by using the CAR, the generic
state can be written as follows

$\rho=\sum_{\underline{s},\underline{t}}\rho_{\underline{s}\underline{t}}\prod_{i=1}^{L}\varphi_{i}^{\dagger
s_{i}}\varphi_{i}\varphi_{i}^{\dagger}\varphi_{i}^{t_{i}},$

where $\underline{s},\underline{t}\in\\{0,1\\}^{L}$ and
$\rho_{\underline{s}\underline{t}}\in\mathbb{C}$.

##  Appendix B Technical Lemmas

Here we show two lemmas that are used in the proof of Proposition II.2 in the
main text.

As a preliminary notion we define quantum states with definite parity. Let
$\mathcal{H}_{L}$ be an Hilbert space of $L$-qubits and let
$\mathsf{St}{(\mathcal{H}_{L})}$ be the corresponding set of states. The
vectors of the computational basis

$\displaystyle\left|{s_{1},s_{2},\ldots,s_{L}}\right\rangle,\quad
s_{i}=\\{0,1\\},\quad i=1,\ldots L,$ (20)

can be divided into even $p=0$ and odd $p=1$ vectors according to their parity
$p:=\oplus_{i=1}^{L}s_{i}$. Denoting by $\mathcal{H}_{L}^{0}$ and
$\mathcal{H}_{L}^{1}$, with
$\mathcal{H}_{L}=\mathcal{H}_{L}^{0}\oplus\mathcal{H}_{L}^{1}$, the spaces
generated by even and odd vectors respectively, one says that a state
$\rho\in\mathsf{St}{(\mathcal{H}_{L})}$ has definite parity if it is of the
form $\rho=\rho_{0}+\rho_{1}$, with $\rho_{0}$ and $\rho_{1}$ having support
on $\mathcal{H}_{L}^{0}$ and $\mathcal{H}_{L}^{1}$ respectively. As a special
case, a pure state of definite parity $p$ must have support only on
$\mathcal{H}_{L}^{p}$. We can now prove the following lemma.

######  Lemma B.1.

Consider a quantum state $\rho\in\mathsf{St}{(\mathcal{H}_{L})}$ and two
purifications $\Psi,\Phi\in\mathsf{St}{(\mathcal{H}_{L}\mathcal{H}_{M})}$ with
definite parity. Then it is alway possible to find a unitary channel
$\mathscr{U}$ that maps states of definite parity into states of definite
parity and such that $(\mathscr{I}\otimes\mathscr{U})(\Psi)=\Phi$.

###### Proof.

Let $\left|{\Psi}\right\rangle\in\mathcal{H}_{LM}^{p}$ and
$\left|{\Phi}\right\rangle\in\mathcal{H}_{LM}^{q}$, for $p,q\in\\{0,1\\}$.
Since the two states are purification of the same state
$\rho\in\mathsf{St}{(\mathcal{H}_{L})}$ their Schmidt decomposition can always
be taken as follows

$\displaystyle\left|{\Psi}\right\rangle=\sum_{i}\lambda_{i}\left|{i}\right\rangle\left|{\Psi_{i}}\right\rangle,\qquad\left|{\Phi}\right\rangle=\sum_{i}\lambda_{i}\left|{i}\right\rangle\left|{\Phi_{i}}\right\rangle,$

where $\\{\left|{i}\right\rangle\\}\in\mathcal{H}_{L}$ is the same orthonormal
set for the two states, while
$\\{\left|{\Psi_{i}}\right\rangle\\},\\{\left|{\Phi_{i}}\right\rangle\\}\in\mathcal{H}_{M}$
are two generally different orthonormal sets. Notice that, since $\Psi$ and
$\Phi$ are pure states of definte parity, any element in the above orthonormal
sets must be a vector of definite parity. Within the set
$\\{\left|{i}\right\rangle\\}=\\{\\{\left|{i_{0}}\right\rangle\\},\\{\left|{i_{1}}\right\rangle\\}\\}$
one can separate even $\\{\left|{i_{0}}\right\rangle\\}$ and odd
$\\{\left|{i_{0}}\right\rangle\\}$ parity vectors, and then write $\Psi$ and
$\Phi$ (respectively of parity $p$ and $q$) as

$\displaystyle\left|{\Psi}\right\rangle=\sum_{i_{0}}\lambda_{i_{0}}\left|{i_{0}}\right\rangle\left|{\Psi^{p}_{i_{0}}}\right\rangle+\sum_{i_{1}}\lambda_{i_{1}}\left|{i_{1}}\right\rangle\left|{\Psi^{\bar{p}}_{i_{1}}}\right\rangle,$
$\displaystyle\left|{\Phi}\right\rangle=\sum_{i_{0}}\lambda_{i_{0}}\left|{i_{0}}\right\rangle\left|{\Phi^{q}_{i_{0}}}\right\rangle+\sum_{i_{1}}\lambda_{i_{1}}\left|{i_{1}}\right\rangle\left|{\Phi^{\bar{q}}_{i_{1}}}\right\rangle,$

where $\bar{r}=r\oplus 1$, and in the orthonormal sets
$\\{\left|{\Psi^{p}_{i_{0}}}\right\rangle,\left|{\Psi^{\bar{p}}_{i_{1}}}\right\rangle\\}$
and
$\\{\left|{\Phi^{q}_{i_{0}}}\right\rangle,\left|{\Phi^{\bar{q}}_{i_{1}}}\right\rangle\\}$
we separated vectors according to their parity. We can now complete the above
two sets to orthonormal bases in such a way that all vectors in both bases
have definite parity. Let us take for example the basis
$\\{\left|{\Psi^{p}_{i_{0}}}\right\rangle,\left|{\Psi^{\bar{p}}_{i_{1}}}\right\rangle\\},|\Psi_{k}^{r(k)}\rangle\\}$
and
$\\{\left|{\Phi^{1}_{i_{0}}}\right\rangle,\left|{\Phi^{\bar{q}}_{i_{1}}}\right\rangle,|\Phi_{k}^{t(k)}\rangle\\}$
with $r(k),t(k)\in\\{0,1\\}$. It is now straightforward to see that the
unitary map $\mathscr{U}$ having Kraus operator

$\displaystyle
U=\sum_{i_{0}}\left|{\Psi^{p}_{i_{0}}}\right\rangle\left\langle{\Phi^{q}_{i_{0}}}\right|+\sum_{i_{1}}\left|{\Psi^{\bar{p}}_{i_{1}}}\right\rangle\left\langle{\Phi^{\bar{q}}_{i_{1}}}\right|+\sum_{k}|\Psi^{r(k)}_{k}\rangle\langle\Phi^{t(k)}_{k}|$

is such that $(I\otimes
U)\left|{\Psi}\right\rangle=\left|{\Phi}\right\rangle$. Moreover $\mathscr{U}$
maps states of definite parity into states of definite parity. ∎

######  Lemma B.2.

Let $\mathrm{N}_{\mathrm{F}}:=\mathrm{L}_{\mathrm{F}}\mathrm{K}_{\mathrm{F}}$
and
$\mathscr{C}\in\mathsf{Tr}(\mathrm{N}_{\mathrm{F}}\rightarrow\mathrm{N}_{\mathrm{F}})$
be a single Kraus transformation with Kraus $C$ having Jordan-Wigner
rapresentative $J(C)=U\otimes I_{\mathrm{K}_{F}}$, $U$ acting on the first $L$
qubits. Then $\mathscr{C}$ is local on the first $L$ modes.

###### Proof.

Due to Proposition II.1, the Kraus operator of $\mathscr{C}$ can be written as
$C=\sum_{i}C_{i}$, where either each $C_{i}$ is a product of an even number of
field operators, or each $C_{i}$ is a product of an odd one. The set
$\\{C_{i}\\}$ can be taken to be linearly independent without loss of
generality. Let us assume by contradiction that $\mathscr{C}$ is not local on
the first $L$ modes. Therefore, since a set of independent operators
generating the algebra of the $j$-th mode is
$\\{\varphi_{j},\varphi_{j}^{\dagger},\varphi_{j}^{\dagger}\varphi_{j},\varphi_{j}^{\dagger}\varphi_{j}+\varphi_{j}\varphi_{j}^{\dagger}\\}$,
there exists at least one product $C_{i}$ that contains one of the factors
$\varphi_{j}$, $\varphi_{j}^{\dagger}$, or $\varphi_{j}\varphi_{j}^{\dagger}$,
for some mode $j$ of the system $\mathrm{K}_{\mathrm{F}}$. Let $j(i)$ be the
mode with largest label in the chosen ordering of the $N=L+K$ modes, such that
the corresponding factor in the product $C_{i}$ is not the identity (i.e.
$\varphi_{j}^{\dagger}\varphi_{j}+\varphi_{j}\varphi_{j}^{\dagger}$).
Accordingly, one has that the Jordan-Wigner representative of $C_{i}$ is of
the form

$\displaystyle J(C_{i})=K\otimes
O_{j(i)}\otimes\left(\bigotimes_{l=j(i)+1}^{N}I_{l}\right),$

where $K$ is an operator on the first $1,\ldots,{j(i)-1}$ qubits, and
$O_{j(i)}$ is one of the factors
$\sigma_{j(i)}^{+},\sigma_{j(i)}^{-},\sigma_{j(i)}^{+}\sigma_{j(i)}^{-}$ on
the $j$-th qubit. This contradicts the hypothesis on the form of $J(C)$. ∎

##  Appendix C Jordan-Wigner independence

In this appendix we show the consistency of definitions III.2, III.3, III.4
and III.5 given in text. In particular, we check that they are independent of
the particular choice of the order of the fermionic modes, which defines the
Jordan-Wigner transform. We remember that all Jordan-Wigner representations
are unitarily equivalent.

######  Lemma C.1.

Let $\rho$ be a fermionic state. The square root and the logarithm of $\rho$
are well defined.

###### Proof.

Once we have fixed the ordering of the modes, the square root of a fermionic
state $\rho$ is defined via its Jordan-Wigner representative as follows

$\rho^{\frac{1}{2}}:=J^{-1}[J(\rho)^{\frac{1}{2}}]$

If $\tilde{J}$ is the Jordan-Wigner isomorphism associated to a different
ordering, then consider $X:=\tilde{J}^{-1}[\tilde{J}(\rho)^{\frac{1}{2}}]$. We
can now prove that $X=\rho^{\frac{1}{2}}$ and then independence of the square
root from the ordering. Indeed, one has

$\tilde{J}(X)^{2}=\tilde{J}(\rho)=UJ(\rho)U^{\dagger},$

with $U$ unitary. It follows that

$J(\rho)=U^{\dagger}\tilde{J}(X)UU^{\dagger}\tilde{J}(X)U=J(X)^{2}\implies
J(X)=J(\rho)^{\frac{1}{2}}.$

Since $J$ is an isomorphism, by taking $J^{-1}$ we finally get

$X=J^{-1}[J(\rho^{\frac{1}{2}})]=\rho^{\frac{1}{2}}.$

Analogously, the logarithm of a fermionic state is defined thorugh its Jordan-
Wigner representative

$\log_{2}(\rho):=J^{-1}[\log_{2}(J(\rho))]$

Again, let $\tilde{J}$ be the Jordan-Wigner isomorphism corresponding to a
different ordering, and let $X=\tilde{J}^{-1}[\log_{2}(\tilde{J}(\rho))]$.
Firstly we notice that

$\log_{2}[\tilde{J}(\rho)]=\log_{2}[UJ(\rho)U^{\dagger}]=U\log_{2}[J(\rho)]U^{\dagger}$

since $U$ is unitary (we remind that the logarithm of a positive operator is
defined via its spectral decomposition, and a unitary map preserves the
spectrum). Therefore, we find

$J(X)=\log_{2}[J(\rho)]\implies X=J^{-1}[\log_{2}(J(\rho))]=\log_{2}(\rho).$

that concludes the proof. ∎

Based on the above lemma we have the following proposition.

######  Proposition C.1.

Let $\rho$ and $\sigma$ be two fermionic states. The Uhlmann fidelity
$F(\rho,\sigma)$ and the von Neumann entropy $S_{f}(\rho)$ of definitions
III.3 and III.5 are well defined.

###### Proof.

These two quantities are given by a trace of two well defined operators, as
proved in the previous lemma. Moreover, since a reordering of the modes
corresponds to a unitarily change of basis, the trace is Jordan-Wigner
independent, and so are $F(\rho,\sigma)$ and $S_{f}(\rho)$.

∎


# dm2gal: Mapping Dark Matter to Galaxies with Neural Networks

Noah Kasmanoff
Center for Data Science
New York University
New York, NY 10011
<EMAIL_ADDRESS>
Francisco Villaescusa-Navarro
Department of Astrophysical Sciences
Princeton University
Princeton NJ 08544
<EMAIL_ADDRESS>
Jeremy Tinker
Center for Cosmology and Particle Physics
New York University
New York, NY 10011
<EMAIL_ADDRESS>
Shirley Ho
Center for Computational Astrophysics
Flatiron Institute
New York, NY 10010
<EMAIL_ADDRESS>

###### Abstract

Maps of cosmic structure produced by galaxy surveys are one of the key tools
for answering fundamental questions about the Universe. Accurate theoretical
predictions for these quantities are needed to maximize the scientific return
of these programs. Simulating the Universe by including gravity and
hydrodynamics is one of the most powerful techniques to accomplish this;
unfortunately, these simulations are very expensive computationally.
Alternatively, gravity-only simulations are cheaper, but do not predict the
locations and properties of galaxies in the cosmic web. In this work, we use
convolutional neural networks to paint galaxy stellar masses on top of the
dark matter field generated by gravity-only simulations. Stellar mass of
galaxies are important for galaxy selection in surveys and thus an important
quantity that needs to be predicted. Our model outperforms the state-of-the-
art benchmark model and allows the generation of fast and accurate models of
the observed galaxy distribution.††Code available at
https://github.com/nkasmanoff/dm2gal

##  1 Introduction

Galaxies are not randomly distributed in the sky, but follow a particular
pattern known as the cosmic web. Galaxies concentrate in high-density regions
composed of dark matter halos, and galaxy clusters usually lie within these
dark matter halos and they are connected via thin and long filaments. Those
filaments are surrounded by very large low-density regions with almost no
galaxies in them: cosmic voids. Cosmologists use the cosmic web as a
laboratory to learn about the fundamental laws and constituents of our
Universe. The scientific community has invested billions of dollars in
missions, both from ground and space, to survey the cosmic web as accurately
as possible. In order to maximize the scientific return of those missions,
accurate theoretical predictions are needed to extract the relevant
information from observational data. Since these surveys observe galaxies and
their properties such as stellar masses (the galaxy mass in stars), we need
theoretical predictions for those quantities.

Cosmological hydrodynamic simulations are probably the best way to obtain
these predictions; however, due to their large computational cost (millions of
CPU hours), they only allow predictions of very small volumes. On the other
hand, gravity-only simulations are much cheaper, but do not model galaxies nor
their properties. In this work we try to bridge the gap between these two
approaches using convolutional neural networks. Our purpose is to show that
neural networks can learn to paint galaxy properties on top of gravity-only
simulations. This will speed up the process of creating predicted galaxy
distributions used to analyze data from astronomical missions. In this work,
we focus our attention in one of the most important galaxy properties, the
stellar mass, i.e. the mass in stars a galaxy contains.

The mapping we want to perform is

$M^{h}_{*}(\vec{x})=f(M^{g}_{\rm dm}(\vec{x}),M^{g}_{\rm dm}(\vec{y}))~{},$
(1)

where $M_{*}^{h}(\vec{x})$ represents the stellar mass at position $\vec{x}$
according to the hydrodynamic simulation, $M^{g}_{\rm dm}(\vec{x})$
corresponds to the dark matter mass from the gravity-only simulation at
position $\vec{x}$. We emphasize that the stellar mass of a galaxy will likely
depend on its environment in a very complicated way. Although the underlying
structure of the simulation pairs are the same, baryonic effects give rise to
minor variations. That is the reason why we included the term $M^{g}_{\rm
dm}(\vec{y})$ in the above equation, where $\vec{y}\neq\vec{x}$.Our purpose in
this paper is to show that convolutional neural networks can approximate the
function $f$.

Some of the studies that inspired this work are [1],[2] and [3].

##  2 Methods

###  2.1 Data

We use data from the state-of-the-art magneto-hydrodynamic simulation TNG100-1
[4, 5], and its gravity-only counterpart, TNG100-1-Dark, at present time.
Those simulations contain, among other things, the position and mass of all
particles in the simulations. Each simulation also contains a catalogue of
dark matter halos with their properties (e.g. mass and position). We construct
the stellar mass and dark matter mass density fields from the particle
positions and masses of the hydrodynamic and gravity-only simulations,
respectively. Since galaxies are expected to reside in dark matter subhalos,
we facilitate the training of the network by using also the mass-weighted
subhalo field, that we construct from the gravity-only simulation. The fields
span a volume of $(75~{}h^{-1}{\rm Mpc})^{3}$ ($1$ ${\rm Mpc}$ corresponds to
$3.26$ million light-years) and they contain $2048^{3}$ voxels.

One of the greatest challenges of working with this data is its sparsity: most
of the voxels in this simulations do not contain galaxies (i.e. stellar mass
is zero). We circumvent this problem by training the network only on regions
centered on a subhalo with a stellar mass larger than
$10^{8}~{}h^{-1}M_{\odot}$.

###  2.2 Model

Our network takes as input a two-channel 3D volume with $65^{3}$ voxels each:
the dark matter and subhalos fields. The output of the model is the value of
the stellar mass in the central voxel of the 3D fields. Our architecture
consists of a series of blocks composed of convolutional, batch normalization,
and ReLU activation layers that alternate between having a larger kernel size
of $k\geq 5$ and stride 1, to a smaller kernel size, $k=3$, with stride 2.
Both block types capture information on different scales, while efficiently
down-sampling this large input to a single value. After six blocks, the latent
representation is flattened into a vector that is passed through two fully
connected layers which produces a predicted stellar mass. We will refer to
this network as dm2gal, as its purpose is to map dark matter from gravity-only
simulations to galaxies in hydrodynamic simulations.

Weighted sampling. The abundance of voxels with different stellar masses (the
target) varies by many orders of magnitude. This poses a problem to our
network, that learns to predict the stellar masses of the voxels with lowest
stellar masses (the most abundant), but fails for the less frequent voxels
with large stellar masses. We overcome this problem with a weighted sampling
procedure. We first bin the voxels with stellar masses of the training data
into $100$ bins logarithmically spaced between the minimum
($10^{8}~{}h^{-1}M_{\odot}$) and maximum ($10^{11.5}~{}h^{-1}M_{\odot}$)
target values. We associate to each training sample a weight corresponding to
the inverse the number count of values within its assigned bin. We also made
use of data augmentation (3D random rotations) to increase our training data
set.

Training and validation. From the $2048^{3}$ voxel fields, we reserve two
cubes, one for validation and one for testing. The validation and testing
cubes have $840^{3}$ ($30.76~{}h^{-1}\rm Mpc$) and $868^{3}$
($31.78~{}h^{-1}\rm Mpc$) voxels, respectively. We save the model that best
matched the cosmological statistics first on the validation cube, and then
report performance on the testing cube. These regions were selected by
requiring that they were representative enough, i.e. avoiding they contain big
voids or very massive halos. The remaining voxels are used for training.

We train our model by minimizing the mean square error
$L_{\mathrm{MSE}}=(\frac{1}{n})\sum_{i=1}^{n}(M_{*}^{h}(i)-M_{*}^{\mathrm{NN}}(i))^{2}$,
where $M_{*}^{h}(i)$ and $M_{*}^{\mathrm{NN}}(i)$ are the stellar masses from
the simulation and the prediction of the neural network, respectively. The sum
runs over all samples in the training set. After trained to convergence, we
select for testing models that best match the stellar mass power spectrum of
the validation region. Because the validation MSE value may correspond to good
performance on only reconstructing low-mass galaxies, we avoid using it for
indicating performance after training.

Hyper-parameter search. We utilize PyTorch and PyTorch-Lightning [6] to
quickly train in parallel a broad range of hyper-parameter configurations,
with learning rates between $10^{-5}$ to $10^{-1}$, weight decay between $0$
to $10^{-1}$, and capacity (number of channels in each layer). We employ a
learning rate scheduler which decreases by a factor of $10$ for every 5 epochs
in which the validation loss does not improve. Each model’s best performing
validation score was achieved within 24 hours of training on a single NVIDIA
P100 GPU.

###  2.3 Benchmark model

We now describe the benchmark model we use to compare our results with. We
refer to this method as HOD, from halo occupation distribution [7, 8, 9, 10].
The most important assumption of this model is that it considers that all
galaxies reside within dark matter halos. The method works as follows. First,
the dark matter halos from the hydrodynamic simulation are assigned to
different bins according to their halo masses. Within each halo mass bin,
galaxies are split into centrals and satellites, and their stellar mass
distribution is calculated. Each halo mass bin will then have two stellar mass
distributions: one for the centrals and one for the satellites. The HOD works
as follows. We take a halo from the gravity-only simulation and its subhalos
are split into central and satellites; the subhalo stellar masses are assigned
by sampling the distribution obtained from the hydrodynamic simulation. We
also correct for the effects of baryons on the halo mass function and number
of satellites by multiplying the HOD prediction by the ratio of satellites,
and overall halo mass between the simulations. We expect our HOD to perform
better than the traditional one, where neither subhalo positions, nor halo
mass corrections are considered.

##  3 Results

We now investigate the performance of our network on the test set, and compare
the results against the HOD model. The first two panels of the upper row of
Fig. 1 show the spatial distribution of dark matter and subhalos from a
$(2.4~{}h^{-1}{\rm Mpc})^{3}$ ($65^{3}$ voxels) region of the test set. These
represent the input to the network, that outputs the value of the stellar mass
in the central voxel. With inputs to the network at different spatial
positions, the 3D stellar mass field can be predicted; we show it in the third
panel. The stellar mass fields from the hydrodynamic simulation and the HOD
model are shown in the fourth and fifth panels, respectively. From visual
inspection, we find that dm2gal performs better than the HOD, and closely
match the results of the hydrodynamic simulation.

Figure 1: The upper row shows the spatial distribution of dark matter and
subhalos from the fast gravity-only simulations. Those fields are the inputs
of the network, that outputs the stellar mass in the central voxel. By
choosing different input regions the 3D stellar mass field can be predicted;
we show it in the third panel. The fourth and fifth panels display the stellar
mass from the expensive hydrodynamic simulation and the benchmark HOD model.
The bottom panels compare different summary statistics (power spectrum-left,
bispectrum-center, PDF-right) from the simulations (blue), dm2gal (red), and
HOD (green). As can be seen, dm2gal outperforms the HOD in the clustering
statistics (power spectrum and bispectrum) while yielding similar performance
than the HOD for the relevant range of stellar masses.

We now quantify the agreement between the predicted, HOD, and simulation
stellar mass fields using three different summary statistics: 1) the power
spectrum, 2) the bispectrum, and 3) the probability distribution function
(PDF).

Power spectrum. Given a 3D field, $\delta(\vec{x})$, we can compute its
Fourier transform as $\delta(\vec{k})=\int
e^{-i\vec{k}\cdot\vec{x}}\delta(\vec{x})d^{3}\vec{x}$ (using the discrete
version for finite fields). The power spectrum can be computed as
$P(k_{i})=1/N_{k_{i}}\sum_{k\in[k,k+dk]}|\delta(\vec{k})|^{2}$, where
$N_{k_{i}}$ is the number of independent modes in the internal $[k,k+dk]$. The
power spectrum is one of the most important quantities in cosmology, as it
describes the cosmic web on large, linear, scales. The first panel on the
bottom row of Fig. 1 shows the results. We find strong agreement on large
scales (low values of $k$) between all fields for the power spectrum. This is
expected for the HOD, but is a prediction for dm2gal. On smaller scales,
dm2gal outperforms the HOD, with the exception of scales $k\geq 30~{}h{\rm
Mpc}^{-1}$. We emphasize that these are extremely small scales (in
cosmological terms), and most non-linear regime of physics. We believe that
with more training on a higher resolution input, this fit will improve.

Bispectrum. The bispectrum is a higher-order statistic that contains non-
Gaussian information from density fields [11]. It is calculated as
$B(k_{1},k_{2},\theta)=1/N_{k}\sum_{\vec{k}_{1},\vec{k}_{2}|\vec{k}_{1}+\vec{k}_{2}+\vec{k}_{3}=\vec{0}}[\delta(\vec{k}_{1})\delta(\vec{k}_{2})\delta(\vec{k}_{3})]$,
where $N_{k}$ is the number of independent modes in the considered interval in
$k_{1}$, $k_{2}$ and $\theta$, that is the angle between $\vec{k}_{1}$ and
$\vec{k}_{2}$. We have taken a configuration with $k_{1}=3~{}h{\rm Mpc}^{-1}$
and $k_{2}=4~{}h{\rm Mpc}^{-1}$ and show the results of the bispectrum, as a
function of $\theta$, in the middle panel of the bottom row of Fig. 1. In this
case, we find that dm2gal outperforms the HOD for all angles. We have repeated
the exercise for other triangle configurations, finding similar results.

Probability distribution function. Finally, we consider the probability
distribution function, that we compute as the number of voxels with a certain
stellar mass, as a function of the stellar mass value (for clarity, we do not
normalize the distribution to have an area equal to 1 under it). This quantity
contains additional information to the one embedded into the power spectrum
and bispectrum [12], and therefore, represents a different way to quantify the
agreement between the different methods. We show the results in the bottom
right panel of Fig. 1. We find that for stellar masses
$M_{*}>10^{8.5}~{}h^{-1}M_{\odot}$, both dm2gal and the HOD outputs a
distribution very similar to the one from the hydrodynamic simulation. On the
other hand, at the low mass end of the stellar mass PDF, the HOD outperforms
dm2gal. We note that it is expected that the HOD model works very well for the
PDF, as it is built to reproduce this statistic. We believe that with further
training and tuning of the hyperparameters we can improve the results of the
network in that regime. However, we emphasize that the low stellar mass regime
is not very important for cosmological analysis, as astronomical surveys will
have a hard time detecting such low mass objects.

##  4 Conclusion

We have shown, for the first time, that convolutional neural networks can be
used to paint stellar masses into the dark matter field of computationally
cheap gravity-only simulations. This method allows the production of stellar
mass fields over large cosmological volumes. Generating these fields using
hydrodynamic simulations will have a computational cost between 10x and 100x
higher than with our method, that only requires running a gravity-only
simulation. In terms of its performance, we have shown, that our model
outperforms the traditional HOD method, while being more computationally
efficient.

This work has made use of simulations where the cosmological and astrophysics
model is fixed. In the future, we plan to generalize our network to models
with different cosmologies and astrophysics. We have also neglected any
dependence on time that the mapping between dark matter to stellar mass may
have. We plan to quantify the importance of that term by training the network
using inputs at different times or by training using information from the
merger trees. We also plan to extend the network to be able to predict other
galactic properties such as metallicity, luminosity, and radius.

## Acknowledgments

We thank Jacky Yip, Carlos Fernandez-Granda, Gabriella Contardo, Yin Li, and
Sigurd Naess for insightful discussions. This work was conducted using the
computational resources at New York University and Princeton University

## Broader Impact

This work will benefit upcoming cosmological missions by speeding up the
computational time needed to generate mock galaxy catalogues, needed to
analyze the collected data. Before using this method for cosmological
analyses, it is important to perform blind tests with simulations to
corroborate that the network produce unbiased results for the required
precision of the data. No ethical aspects are relevant for this work.

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but not sensitive at all to $M$. This is encouraging since a large amount of
computational overhead comes from the ensemble. Note that while smaller
$\varphi$ performs the best here, we use a larger value ($\varphi=30$) in
combination with TEE.

In Figure 17(c), we show the effect of only training the feature extractor
using the gradient from one member of the ensemble at every iteration. The
results are computed on: ninja, plunder, jumper, caveflyer, bigfish, leaper,
climber for 1 seed. We observe that always training the feature extractor
leads to lower performance, corroborating our intuition that the feature
extractor should be trained at the same speed as the individual ensemble
members.

(a) $\varphi$ ablation.

(b) $M$ ablation.

(c) Feature extractor ablation.

Figure 17: Aggregated min-max normalized test scores for $\varphi$ (for fixed
$M=3$, and training feature extractor for all value heads), $M$ (for fixed
$\varphi=50$ and training feature extractor for all value heads), and whether
feature extractor is trained with all value head (for fixed $\varphi=50$ and
$M=3$).

In Figure 18, we study the performance under different hyperparameter values
of TEE. We use fixed $M=5$ and $\varphi=30$ and vary the values of either
$\alpha$ or $\lambda$ while holding the other one fixed. We observe no
significant difference across these, suggesting that the algorithm is robust
to the values of $\alpha$ and $\lambda$.

Figure 18: Aggregated min-max normalized test scores for $\lambda$ (for fixed
$\alpha=7$) and $\alpha$ (for fixed $\lambda=0.6$) on 4 games: bossfight,
climber, plunder, starpilot.

##  Appendix K Hardware

The experiments are conducted on 2080 and V100 and take approximately 250 GPU
days.