# Quantum properties in the four-node network

Yanwen Liang1 College of Physics, Hebei Key Laboratory of Photophysics
Research and Application, Hebei Normal University, Shijiazhuang 050024, China
Fengli Yan1<EMAIL_ADDRESS>College of Physics, Hebei Key Laboratory of
Photophysics Research and Application, Hebei Normal University, Shijiazhuang
050024, China  Ting Gao2<EMAIL_ADDRESS>School of Mathematical Sciences,
Hebei Normal University, Shijiazhuang 050024, China

###### Abstract

There are different preparable quantum states in different network structures.
The four nodes as a whole has two situations: one is the four nodes in a
plane, the other is the four nodes in the space. In this paper, we obtain some
properties of the quantum states that can be prepared in four-node network
structures. These include the properties of entropy, entanglement measure,
rank and multipartite entangled states. These properties also mean that the
network structures impose some constraints on the states that can be prepared
in a four-node quantum network. In order to obtain these properties we also
define $n$-partite mutual information of the quantum system, which satisfies
symmetry requirement.

###### pacs:

03.67.Mn, 03.65.Ud, 03.67.-a

##  I Introduction

By using quantum network one can exchange corresponding quantum information on
different nodes through quantum channels, such as quantum entanglement 1 ; 2 ;
3 ; 4 ; 5 ; 6 ; 7 ; 8  and quantum key distribution 15 ; 115 . Therefore,
quantum network plays an important role in the study of quantum correlation 9
; 10 ; 11 . The great progress has been made in the research and application
of quantum network structure and quantum entangled states prepared in the
quantum network in both theory and experiment 12 ; 13 ; 14 ; 16 ; 17 ; 18 . Of
course, the practical application of quantum network technologies may also
affect the real life in the future.

In the network system, quantum states are distributed to different nodes 12 ;
13 . It is assumed that sources are independent in the quantum network. It
means different sources are separable. However, different particles in the
same source can be entangled 14 ; 15 , so there are a lot of situations. For
example, one can has the situation in which a source can only generate two
particle entangled states. Of course, there also exists the situation in which
three particle entangled states can be produced by the source.

Apparently, the quantum system consisting of different number particles have
many quantum network structures and a lots of different preparable states 16 ;
17 . This paper only studies several different quantum network structures of
four nodes. The structure of four-node quantum network has two different
situations: one is the network structure of four nodes in a plane, the other
is the network structure of four nodes in the space. We assume that the
sources do not intersect. Under the assumption of no intersection of sources,
the four nodes in a plane have only one quadrangular network structure, while
the four nodes in the space have two kinds of network structures. Different
quantum states can be prepared in different network structures. Based on the
pioneer work in three nodes 18 , we will mainly discuss the network structures
in the above three special situations, and deduce some properties and
constraints of the quantum states that can be prepared.

The paper is organized as follows. In section II we define $n$-partite mutual
information in quantum systems and obtain some properties of states prepared
in a quadrangular network structure. The properties of quantum states prepared
in two kinds of network structures with four nodes in the space are mainly
investigated in section III. Among them, the two network structures of four
nodes in the space with six sources, where each source generates the two-
partite entangled state, and four sources, where every source produces three-
partite entangled states are discussed respectively. A summary is given in
section IV.

##  II Four nodes in a plane

Let us discuss the situation in which four nodes are in a plane. This network
is called independent quadrangular network (IQN). As shown in Fig.1(a), it has
four nodes, namely _A_ , _B_ , _C_ , and _D_ , which are formed by pairing of
two particles in each of the four sources $\rho_{\alpha}$, $\rho_{\beta}$,
$\rho_{\gamma}$, and $\rho_{\delta}$ 19 . The states of two particles in each
source are the two-partite entangled states 20 ; 21 ; 22 . $\rho_{\alpha}$,
$\rho_{\beta}$, $\rho_{\gamma}$, and $\rho_{\delta}$ is shared respectively by
two nodes of _A_ , _B_ , _C_ , and _D_. It means that each of the four nodes
_A_ , _B_ , _C_ , and _D_ receives two particles. Exactly, _A_ receives two
particles respectively from the source $\rho_{\alpha}$ and $\rho_{\beta}$; _B_
receives the two particles respectively from the source $\rho_{\beta}$ and
$\rho_{\gamma}$; _C_ receives two particles respectively from the source
$\rho_{\gamma}$ and $\rho_{\delta}$; _D_ receives two particles respectively
from the source $\rho_{\delta}$ and $\rho_{\alpha}$. The two particles at the
each node can be applied by a local unitary matrix. The four local unitary
matrices are denoted by $U_{A}$, $U_{B}$, $U_{C}$, and $U_{D}$ respectively.

Figure 1: Quadrangular network. (a) An independent quadrangular network (IQN)
in which each source is a two-partite entangled state. (b) A classically
correlated quadrangular network (CQN) in which every source and every node are
classically correlated by sharing the random variable $\lambda$. (c) The four-
partite quantum states which can not be generated in IQN and CQN.

We use $\Delta_{\mathrm{IQN}}$ to represent the set of quantum states that can
be prepared in an independent quadrangular network structure shown in
Fig.1(a). Obviously, the quantum state $\rho$ in the set
$\Delta_{\mathrm{IQN}}$ can be written as

$\displaystyle\rho=(U_{A}\otimes U_{B}\otimes U_{C}\otimes
U_{D})(\rho_{\alpha}\otimes\rho_{\beta}\otimes\rho_{\gamma}\otimes\rho_{\delta})(U_{A}^{{\dagger}}\otimes
U_{B}^{{\dagger}}\otimes U_{C}^{{\dagger}}\otimes U_{D}^{{\dagger}}).$ (1)

Here, $U_{A}$ only acts on the system consisting of a particle of
$\rho_{\alpha}$ and a particle of $\rho_{\delta}$, it does not act on the
other particles which are not at the node _A_ at same time. Similarly, other
unitary matrices cannot act on the states of $\rho_{\alpha}$, $\rho_{\beta}$,
$\rho_{\gamma}$, and $\rho_{\delta}$ at same time.

Next we will discuss Von Neumann entropy of the quantum states in the set
$\Delta_{\mathrm{IQN}}$. In this paper, entropy refers to Von Neumann entropy.

Because entropy $S(\rho)$ of quantum state $\rho$ is invariant under unitary
transformation, and additive on tensor products, for the quantum states shown
by Eq.(1), we have

$S(\rho)=S(\rho_{\alpha})+S(\rho_{\beta})+S(\rho_{\gamma})+S(\rho_{\delta}).$

The three-partite entropy is extended to

$S(ABC)=S(\mathrm{tr}_{D}\rho_{\alpha})+S(\rho_{\beta})+S(\rho_{\gamma})+S(\mathrm{tr}_{D}\rho_{\delta}),$

the two-partite entropy is extended to

$S(AB)=S(\mathrm{tr}_{D}\rho_{\alpha})+S(\rho_{\beta})+S(\mathrm{tr}_{C}\rho_{\gamma}),$

the one-partite entropy is extended to

$S(A)=S(\mathrm{tr}_{D}\rho_{\alpha})+S(\mathrm{tr}_{B}\rho_{\beta}),$

and similarly for others.

Based on entropy, one can study the mutual information of the quantum system.
The two-partite quantum mutual information of a quantum system composed of _X_
and _Y_ is defined as

$I_{2}(X:Y)=S(X)-S(X|Y)=S(X)+S(Y)-S(XY),$ (2)

where $S(X|Y)$ stands for conditional entropy and $S(XY)$ is the joint
entropy. Three-partite quantum mutual information measuring the common
information of subsystems _X_ , _Y_ , and _Z_ , was defined as 18

$I_{3}(X:Y:Z)=I_{2}(X:Y)+I_{2}(X:Z)-I_{2}(X:YZ)=S(X)+S(Y)+S(Z)+S(XYZ)-S(XY)-S(XZ)-S(YZ).$
(3)

We define four-partite quantum mutual information of the quantum system
composed of _T_ , _X_ , _Y_ , and _Z_ as

$\displaystyle I_{4}(T:X:Y:Z)=I_{3}(T:X:Y)+I_{3}(T:X:Z)-I_{3}(T:X:YZ).$ (4)

By substituting the formula of the three-partite quantum mutual information
into the four-partite quantum mutual information, one can obtain

$\displaystyle I_{4}(A:B:C:D)=$ $\displaystyle
S(A)+S(B)+S(C)+S(D)+S(ABC)+S(ABD)+S(BCD)+S(ACD)$ (5)
$\displaystyle-S(AB)-S(AC)-S(AD)-S(BC)-S(BD)-S(CD)-S(ABCD)$ $\displaystyle=$
$\displaystyle x_{1}-x_{2},$

where

$x_{1}=S(A)+S(B)+S(C)+S(D)+S(ABC)+S(ABD)+S(BCD)+S(ACD),$

and

$x_{2}=S(AB)+S(AC)+S(AD)+S(BC)+S(BD)+S(CD)+S(ABCD).$

Obviously, the four-partite quantum mutual information is invariant under
arbitrary permutation of the subsystems.

Furthermore, we define the $n$-partite quantum mutual information of the
system consisting of subsystems $X_{1},X_{2},\cdots,X_{n}$. We use
$\sigma_{X_{1}X_{2}\cdots X_{n}}$ to express a quantum state of the system. We
define the $n$-partite quantum mutual information of the system as

$\begin{array}[]{ll}I_{n}(X_{1}:X_{2}:\cdots:X_{n})=&(-1)^{1-1}\sum_{i=1}^{n}S(\sigma_{X_{i}})+(-1)^{2-1}\sum_{1\leq
i<j\leq n}S(\sigma_{X_{i}X_{j}})+(-1)^{3-1}\sum_{1\leq i<j<k\leq
n}S(\sigma_{X_{i}X_{j}X_{k}})\\\ &+(-1)^{4-1}\sum_{1\leq i<j<k<l\leq
n}S(\sigma_{X_{i}X_{j}X_{k}X_{l}})+\cdots+(-1)^{n-1}S(\sigma_{X_{1}X_{2}\cdots
X_{n}}),\\\ \end{array}$ (6)

where $\sigma_{X_{i}X_{j}\cdots X_{l}}=\mathrm{tr}_{X_{q}X_{r}\cdots
X_{t}}\sigma_{X_{1}X_{2}\cdots X_{n}}$ is the reduced density matrix for the
subsystems $X_{i},X_{j},\cdots,X_{l}$ and the set
$\\{q,r,\cdots,t\\}=\\{1,2,\cdots,n\\}\setminus\\{i,j,\cdots,l\\}.$ Clearly,
$I_{n}(X_{1}:X_{2}:\cdots:X_{n})$ is symmetry about the subsystems.

For the quantum states in the set $\triangle_{\mathrm{IQN}}$, it is easy to
obtain the following result.

_Conclusion 1-1._ $I_{4}(A:B:C:D)=0$ for any
$\rho\in\triangle_{\mathrm{IQN}}$.

Now we discuss the effect of the local channels on the four-partite quantum
mutual information. Suppose that there are three local channels $\Lambda_{A}$,
$\Lambda_{B}$, and $\Lambda_{C}$ acting on nodes _A_ , _B_ , and _C_
respectively. The four-partite quantum mutual information of the quantum
system is $I_{4}[(\Lambda_{A}\otimes\Lambda_{B}\otimes\Lambda_{C}\otimes
I_{D})\rho]$. Here $I_{D}$ is the identity operator on the node $D$, and
$\rho\in\triangle_{\mathrm{IQN}}$.

Clearly, in this case for two particles in _D_ we have
$S(D)=S(D_{\alpha})+S(D_{\delta})$. By considering the actions of the local
quantum channels we can get

$\displaystyle x_{1}=$ $\displaystyle
S(A_{\alpha}A_{\beta})+S(B_{\beta}B_{\gamma})+S(C_{\gamma}C_{\delta})+S(D_{\delta})+S(D_{\alpha})+S(A_{\alpha}A_{\beta}B_{\beta}B_{\gamma}C_{\gamma}C_{\delta})$
$\displaystyle+S(A_{\alpha}A_{\beta}D_{\alpha})+S(C_{\gamma}C_{\delta}D_{\delta})+S(B_{\beta}B_{\gamma}C_{\gamma}C_{\delta}D_{\delta})+S(D_{\alpha})$
$\displaystyle+S(A_{\alpha}A_{\beta}B_{\beta}B_{\gamma}D_{\alpha})+S(D_{\delta}),$
$\displaystyle x_{2}=$ $\displaystyle
S(A_{\alpha}A_{\beta}B_{\beta}B_{\gamma})+S(A_{\alpha}A_{\beta})+S(C_{\gamma}C_{\delta})+S(A_{\alpha}A_{\beta}D_{\alpha})+S(D_{\delta})$
$\displaystyle+S(B_{\beta}B_{\gamma}C_{\gamma}C_{\delta})+S(B_{\beta}B_{\gamma})+S(D_{\alpha})+S(D_{\delta})+S(C_{\gamma}C_{\delta}D_{\delta})+S(D_{\alpha})$
$\displaystyle+S(D_{\alpha}A_{\alpha}A_{\beta}B_{\beta}B_{\gamma}C_{\gamma}C_{\delta}D_{\delta}).$

Therefore, we derive that

$\displaystyle I_{4}[(\Lambda_{A}\otimes\Lambda_{B}\otimes\Lambda_{C}\otimes
I_{D})\rho]$ $\displaystyle=$ $\displaystyle x_{1}-x_{2}$ $\displaystyle=$
$\displaystyle
S(A_{\alpha}A_{\beta}B_{\beta}B_{\gamma}C_{\gamma}C_{\delta})+S(A_{\alpha}A_{\beta}B_{\beta}B_{\gamma}D_{\alpha})+S(B_{\beta}B_{\gamma}C_{\gamma}C_{\delta}D_{\delta})$
$\displaystyle-S(A_{\alpha}A_{\beta}B_{\beta}B_{\gamma})-S(B_{\beta}B_{\gamma}C_{\gamma}C_{\delta})-S(A_{\alpha}A_{\beta}B_{\beta}B_{\gamma}C_{\gamma}C_{\delta}D_{\delta}D_{\alpha}).$
$\displaystyle=$ $\displaystyle
S(ABC)+S(ABD_{\alpha})+S(BCD_{\delta})-S(AB)-S(BC)-S(ABCD_{\delta}D_{\alpha}).$

Hence, generally speaking, one can not determine whether $I_{4}$ is larger
than zero or not for the quantum state
$(\Lambda_{A}\otimes\Lambda_{B}\otimes\Lambda_{C}\otimes I_{D})\rho$.

However, for two adjacent local channels 24 ; 25 ; 26 , the four-partite
quantum mutual information of the quantum system becomes
$I_{4}[(\Lambda_{A}\otimes\Lambda_{B}\otimes I_{C}\otimes I_{D})\rho]$. For
this case, one has

$\displaystyle I_{4}[(\Lambda_{A}\otimes\Lambda_{B}\otimes I_{C}\otimes
I_{D})\rho]$ $\displaystyle=$ $\displaystyle
S(A_{\alpha}A_{\beta}B_{\beta}B_{\gamma}C_{\gamma})+S(A_{\alpha}A_{\beta}B_{\beta}B_{\gamma}D_{\alpha})-S(A_{\alpha}A_{\beta}B_{\beta}B_{\gamma})-S(A_{\alpha}A_{\beta}B_{\beta}B_{\gamma}C_{\gamma}D_{\alpha})$
$\displaystyle=$ $\displaystyle
S(ABC_{\gamma})+S(ABD_{\alpha})-S(AB)-S(ABC_{\gamma}D_{\alpha}).$

By using the strong subadditivity inequality

$S(XY)+S(XZ)\geq S(X)+S(XYZ)$ (7)

for three quantum systems $X,Y,Z,$ it is easy to obtain that the four-partite
quantum mutual information of quantum system satisfies
$I_{4}[(\Lambda_{A}\otimes\Lambda_{B}\otimes I_{C}\otimes I_{D})\rho]\geq 0$.

For two non-adjacent local channels, the four-partite quantum mutual
information of the quantum system turns out to be $I_{4}[(\Lambda_{A}\otimes
I_{B}\otimes\Lambda_{C}\otimes I_{D})\rho]$. In this case one can deduce

$I_{4}[(\Lambda_{A}\otimes I_{B}\otimes\Lambda_{C}\otimes
I_{D})\rho]=S(AB_{\beta}D_{\alpha})+S(B_{\gamma}CD_{\delta})-S(AB_{\beta}D_{\alpha}B_{\gamma}CD_{\delta}).$
(8)

By the subadditivity inequality

$S(X)+S(Y)\geq S(XY)$ (9)

for two quantum systems $X,Y$, we arrive at that the four-partite quantum
mutual information of the quantum system $I_{4}[(\Lambda_{A}\otimes
I_{B}\otimes\Lambda_{C}\otimes I_{D})\rho]$ is still large than zero.
Combining the above two cases we have the following result.

_Conclusion 1-2._ For any pair of local channels one finds that the four-
partite quantum mutual information for the quantum states obtained by acting
the pair of the local quantum channels on the quantum state in
$\triangle_{\mathrm{IQN}}$ cannot be rendered negative.

Although in general we can not decide
$I_{4}[(\Lambda_{A}\otimes\Lambda_{B}\otimes\Lambda_{C}\otimes I_{D})\rho]$ is
large or not than zero, however, we have the following conclusion.

_Conclusion 1-3._ The four-partite quantum mutual information
$I_{4}[(\Lambda_{A}\otimes\Lambda_{B}\otimes\Lambda_{C}\otimes I_{D})\rho]$
are bounded as

$\displaystyle
2I_{2}(A:D)+2I_{2}(B:D)+2I_{2}(C:D)-[I_{2}(AB:D)+I_{2}(BC:D)+I_{2}(AC:D)]$
$\displaystyle\leq$ $\displaystyle
I_{4}[(\Lambda_{A}\otimes\Lambda_{B}\otimes\Lambda_{C}\otimes I_{D})\rho]$
$\displaystyle\leq$ $\displaystyle
I_{2}(ABC:D)-[I_{2}(A:D)+I_{2}(B:D)+I_{2}(C:D)]$

and

$\displaystyle
2I_{2}(A:D)+2I_{2}(B:D)+2I_{2}(C:D)-[I_{2}(AB:D)+I_{2}(BC:D)+I_{2}(AC:D)]\leq
0,$ (11) $\displaystyle I_{2}(ABC:D)-[I_{2}(A:D)+I_{2}(B:D)+I_{2}(C:D)]\geq
0.$ (12)

Here $\rho$ is a quantum state in the set $\Delta_{\mathrm{IQN}}$.

For detailed proof of above conclusion please refer to Appendix A.

As there exists quantum entanglement in Fig. 1, so the entanglement measure
$\varepsilon[\sigma]$ should be introduced. The entanglement measure should
satisfy three requirements: (1) if the quantum state is separable, the
entanglement measure will disappear; (2) under local operation and classical
communication, the average entanglement measure does not increase; (3) under
local unitary transformation, the entanglement measure does not change. For
present situation, we also require that the entanglement measure is additive
on tensor products, that is

$\displaystyle\varepsilon[\sigma_{1}\otimes\sigma_{2}\otimes\sigma_{3}\otimes\sigma_{4}]=\varepsilon[\sigma_{1}]+\varepsilon[\sigma_{2}]+\varepsilon[\sigma_{3}]+\varepsilon[\sigma_{4}].$
(13)

Furthermore, we require the entanglement measure satisfies the monogamy
relation 23 ,

$\displaystyle\varepsilon_{T|X}[\sigma_{TX}]+\varepsilon_{T|Y}[\sigma_{TY}]+\varepsilon_{T|Z}[\sigma_{TZ}]\leq\varepsilon_{T|XYZ}[\sigma_{TXYZ}],$
(14)

where $\sigma_{TX}=\mathrm{tr}_{YZ}\sigma_{TXYZ}$,
$\sigma_{TY}=\mathrm{tr}_{XZ}\sigma_{TXYZ}$. Clearly, not arbitrary
entanglement measure meets the above requirements, however the squashed
entanglement does 35 ; 36 .

For the quantum state (1), as the local unitary matrices do not change the
entanglement between the parties, we have

$\varepsilon_{A|BCD}=\varepsilon_{A_{\alpha}A_{\beta}|B_{\beta}B_{\gamma}C_{\gamma}C_{\delta}D_{\delta}D_{\alpha}}=\varepsilon_{A_{\alpha}|D_{\alpha}}+\varepsilon_{A_{\beta}|B_{\beta}}=\varepsilon_{A|D}+\varepsilon_{A|B},$
(15)

where $A_{\alpha}$ stands for a particle that $A$ receives from
$\rho_{\alpha}$, similarly for others. Therefore for the entanglement measure
$\varepsilon[\sigma]$ satisfying the above requirements, we have the following
conclusion.

_Conclusion 1-4._ For any $\rho\in\triangle_{\mathrm{IQN}}$,
$\varepsilon_{T|XYZ}[\rho]=\varepsilon_{T|X}[\mathrm{tr}_{YZ}\rho]+\varepsilon_{T|Z}[\mathrm{tr}_{XY}\rho]$
holds for all bipartitions, such as $A|BCD$, $B|ACD$, $C|ABD$, and $D|ABC$.

After that we discuss the ranks of the global state (1) and its marginals.
Because the unitary matrices $U_{A}$, $U_{B}$, $U_{C}$, and $U_{D}$ don’t
change the rank of the global state, for all $\rho\in\Delta_{\mathrm{IQN}}$,
we have the rank of quantum state $\rho$

$\mathrm{rk}(\rho)=\mathrm{rk}(\rho_{\alpha})\mathrm{rk}(\rho_{\beta})\mathrm{rk}(\rho_{\gamma})\mathrm{rk}(\rho_{\delta})=r_{\alpha}r_{\beta}r_{\gamma}r_{\delta},$

where $r_{\alpha}=\mathrm{rk}(\rho_{\alpha})$ is the rank of quantum state
$\rho_{\alpha}$, similarly for others. The rank of the local reduced state is

$\mathrm{rk}(\mathrm{tr}_{BCD}\rho)=\mathrm{rk}(\mathrm{tr}_{D}\rho_{\alpha})\mathrm{rk}(\mathrm{tr}_{B}\rho_{\beta})=r_{\alpha}^{A}r_{\beta}^{A},$

the rank of the two-partite reduced state is

$\mathrm{rk}(\mathrm{tr}_{CD}\rho)=\mathrm{rk}(\mathrm{tr}_{D}\rho_{\alpha})\mathrm{rk}(\mathrm{tr}_{CD}\rho_{\beta})\mathrm{rk}(\mathrm{tr}_{C}\rho_{\gamma})=r_{\alpha}^{A}r_{\beta}r_{\gamma}^{B},$

the rank of the three-partite reduced state is

$\mathrm{rk}(\mathrm{tr}_{D}\rho)=\mathrm{rk}(\mathrm{tr}_{D}\rho_{\alpha})\mathrm{rk}(\rho_{\beta})\mathrm{rk}(\rho_{\gamma})\mathrm{rk}(\mathrm{tr}_{D}\rho_{\delta})=r_{\alpha}^{A}r_{\beta}r_{\gamma}r_{\delta}^{C}.$

Here $r_{\alpha}^{A}=\mathrm{rk}(\mathrm{tr}_{D}\rho_{\alpha})$, similarly for
others.

Then we arrive at

$\begin{array}[]{llllll}\mathrm{rk}(\rho)=r_{\alpha}r_{\beta}r_{\gamma}r_{\delta},&&&\\\
\mathrm{rk}(\mathrm{tr}_{BCD}\rho)=r_{\alpha}^{A}r_{\beta}^{A},&\quad\mathrm{rk}(\mathrm{tr}_{CDA}\rho)=r_{\beta}^{B}r_{\gamma}^{B},&\quad\mathrm{rk}(\mathrm{tr}_{DAB}\rho)=r_{\gamma}^{C}r_{\delta}^{C},&\quad\mathrm{rk}(\mathrm{tr}_{ABC}\rho)=r_{\alpha}^{D}r_{\delta}^{D},\\\
\mathrm{rk}(\mathrm{tr}_{CD}\rho)=r_{\alpha}^{A}r_{\beta}r_{\gamma}^{B},&\quad\mathrm{rk}(\mathrm{tr}_{CA}\rho)=r_{\alpha}^{D}r_{\beta}^{B}r_{\gamma}^{B}r_{\delta}^{D},&\quad\mathrm{rk}(\mathrm{tr}_{CB}\rho)=r_{\alpha}r_{\beta}^{A}r_{\delta}^{D},&\\\
\mathrm{rk}(\mathrm{tr}_{DA}\rho)=r_{\beta}^{B}r_{\gamma}r_{\delta}^{C},&\quad\mathrm{rk}(\mathrm{tr}_{DB}\rho)=r_{\alpha}^{A}r_{\beta}^{A}r_{\gamma}^{C}r_{\delta}^{C},&\quad\mathrm{rk}(\mathrm{tr}_{AB}\rho)=r_{\alpha}^{D}r_{\gamma}^{C}r_{\delta},&\\\
\mathrm{rk}(\mathrm{tr}_{A}\rho)=r_{\alpha}^{D}r_{\beta}^{B}r_{\gamma}r_{\delta},&\quad\mathrm{rk}(\mathrm{tr}_{B}\rho)=r_{\alpha}r_{\beta}^{A}r_{\gamma}^{C}r_{\delta},&\quad\mathrm{rk}(\mathrm{tr}_{C}\rho)=r_{\alpha}r_{\beta}r_{\gamma}^{B}r_{\delta}^{D},&\quad\mathrm{rk}(\mathrm{tr}_{D}\rho)=r_{\alpha}^{A}r_{\beta}r_{\gamma}r_{\delta}^{C}.\end{array}$
(16)

If the Hilbert space of every particle is d-dimensional, we have $r_{\alpha}$,
$r_{\beta}$, $r_{\gamma}$, $r_{\delta}$ $\in[1,d^{2}]$, $r_{\alpha}^{A}$,
$r_{\alpha}^{D}$, $r_{\beta}^{A}$, $r_{\beta}^{B}$, $r_{\gamma}^{B}$,
$r_{\gamma}^{C}$, $r_{\delta}^{C}$, $r_{\delta}^{D}$ $\in[1,d]$. Hence one has
the conclusion as follows.

_Conclusion 1-5._ For $\rho\in\triangle_{\mathrm{IQN}}$ there exist integers
$r_{\alpha}$, $r_{\beta}$, $r_{\gamma}$, $r_{\delta}$$\in[1,d^{2}]$,
$r_{\alpha}^{A}$, $r_{\alpha}^{D}$, $r_{\beta}^{A}$, $r_{\beta}^{B}$,
$r_{\gamma}^{B}$, $r_{\gamma}^{C}$, $r_{\delta}^{C}$,
$r_{\delta}^{D}$$\in[1,d]$, which satisfy rank relationship Eq. (16).

Next we discuss the scenario where the four nodes and the four sources are
classically correlated 18 . As shown in Fig. 1(b), we assume that every source
and every node are correlated by sharing the random variable $\lambda$ 18 . In
this case, the quantum states are called classically correlated states, which
read

$\displaystyle\rho=$ $\displaystyle\int\texttt{d}\lambda
p(\lambda)(U_{A}(\lambda)\otimes U_{B}(\lambda)\otimes U_{C}(\lambda)\otimes
U_{D}(\lambda))(\rho_{\alpha}(\lambda)\otimes\rho_{\beta}(\lambda)\otimes\rho_{\gamma}(\lambda)\otimes\rho_{\delta}(\lambda)$
(17) $\displaystyle(U_{A}^{{\dagger}}(\lambda)\otimes
U_{B}^{{\dagger}}(\lambda)\otimes U_{C}^{{\dagger}}(\lambda)\otimes
U_{D}^{{\dagger}}(\lambda)).$

Here $p(\lambda)$ is a probability distribution function of the random
variable $\lambda$. We use CQN to denote this classically correlated
quadrangular network, $\Delta_{\mathrm{CQN}}$ to represent the set of quantum
states that can be prepared in CQN. In other words, a quantum state
$\rho\in\Delta_{\mathrm{CQN}}$ can be written as
$\rho=\sum_{\lambda}p_{\lambda}\rho_{\lambda}$, where
$\rho_{\lambda}\in\Delta_{\mathrm{IQN}}$, $p_{\lambda}$ is a probability
distribution.

Based on the properties of the ranks of quantum states, we will verify that
there are no four-qubit genuine multipartite entangled states in the set
$\Delta_{\mathrm{CQN}}$. Here four-qubit genuine multipartite entangled state
refers to it can be embedded in larger dimensional systems. Evidently, if a
four-qubit genuine multipartite entangled state is pure, then the rank of the
global state $\rho$ is equal to 1 27 ; 28 ; 29  and it is entangled along
every bipartition. Considering the Schmidt decomposition of the four-qubit
genuine multipartite entangled state, one has that all single node reduced
state has rank 2. However, as a matter of fact, when each source in the IQN is
entangled among two partitions. According to the Schmidt decomposition, the
rank of the reduced state of a particle in each source is 2. Therefore, in IQN
as shown in Fig. 1(a), with sources which prepare 2-qubit entangled state, the
local rank of the reduced state at each node is 4. So it is impossible to
prepare four-qubit genuine multipartite entangled states in IQN. Similarly, we
can show that the statement holds for other cases of resources. Furthermore,
because four-qubit genuine entangled states need pure four-qubit genuine
multipartite entangled states 30 , so it is also impossible to prepare four-
qubit genuine multipartite entangled states in CQN. Hence we arrive at the
following conclusion.

_Conclusion 1-6._ No four-qubit genuine multipartite entangled state can be
prepared in IQN and CQN.

Fig. 1(c) shows the four-partite quantum states which can not be generated in
IQN and CQN.

##  III Four nodes in the space

Clearly, four nodes in the space form a tetrahedron. By the assumption of no
intersection of sources, in this case there are two different quantum network
structures: one is the triangular cone network structure 1 with six two-
partite sources as shown in Fig. 2(a); the other is the triangular cone
network structure 2 with four three-partite sources as illustrated in Fig.
3(a).

We first study the triangular cone network structure 1. It has four nodes,
namely _A_ , _B_ , _C_ , and _D_. These four nodes are formed by pairs of two
particles in each of the six sources $\rho_{\alpha}$, $\rho_{\beta}$,
$\rho_{\gamma}$, $\rho_{\delta}$, $\rho_{\theta}$, and $\rho_{\tau}$. Two
particles in each source are in two-partite entangled states. $\rho_{\alpha}$,
$\rho_{\beta}$, $\rho_{\gamma}$, $\rho_{\delta}$, $\rho_{\theta}$ and
$\rho_{\tau}$ are shared by $\\{A,B\\}$, $\\{B,D\\}$, $\\{A,D\\}$,
$\\{A,C\\}$, $\\{C,D\\}$, $\\{B,C\\}$, respectively. Four nodes do not share
common information. Each of four nodes _A_ , _B_ , _C_ , and _D_ receives
three particles, for example, ${A}$ receives three particles from sources
$\rho_{\alpha}$, $\rho_{\gamma}$, and $\rho_{\delta}$, similarly for others.
One can apply a local unitary matrix to the three received particles. Four
local unitary matrices are denoted by $U_{A}$, $U_{B}$, $U_{C}$, and $U_{D}$
respectively. The global quantum state in an independent triangular cone
network structure 1 (ITCN1) is denoted by $\rho$. We use
$\Delta_{\mathrm{ITCN1}}$ to represent the set of quantum states that can be
prepared in ITCN1.

Figure 2: Triangular cone network structure 1. (a) Independent triangular cone
network structure 1 (ITCN1), where each of these sources is a two-partite
entangled state. (b) Classically correlated triangular cone network structure
1 (CTCN1), where each source and each node are classically correlated by
sharing the random variable $\lambda$. (c) The four-partite quantum states
which can not be prepared in ITCN1 and CTCN1.

Apparently, the quantum states prepared in ITCN1 read

$\rho=(U_{A}\otimes U_{B}\otimes U_{C}\otimes
U_{D})(\rho_{\alpha}\otimes\rho_{\beta}\otimes\rho_{\gamma}\otimes\rho_{\delta}\otimes\rho_{\theta}\otimes\rho_{\tau})(U_{A}^{{\dagger}}\otimes
U_{B}^{{\dagger}}\otimes U_{C}^{{\dagger}}\otimes U_{D}^{{\dagger}}).$ (18)

By using that entropy is invariant under unitary transformation, and for the
tensor product state, entropy has additivity, we have the entropy of the
quantum state (18)

$S(\rho)=S(\rho_{\alpha})+S(\rho_{\beta})+S(\rho_{\gamma})+S(\rho_{\delta})+S(\rho_{\theta})+S(\rho_{\tau}).$

The three-partite entropy, the two-partite entropy, the one-partite entropy
are extended to

$S(ABC)=S(\rho_{\alpha})+S(\mathrm{tr}_{D}\rho_{\beta})+S(\mathrm{tr}_{D}\rho_{\gamma})+S(\rho_{\delta})+S(\mathrm{tr}_{D}\rho_{\theta})+S(\rho_{\tau}),$
$S(AB)=S(\rho_{\alpha})+S(\mathrm{tr}_{D}\rho_{\beta})+S(\mathrm{tr}_{D}\rho_{\gamma})+S(\mathrm{tr}_{C}\rho_{\delta})+S(\mathrm{tr}_{C}\rho_{\tau}),$
$S(A)=S(\mathrm{tr}_{B}\rho_{\alpha})+S(\mathrm{tr}_{D}\rho_{\gamma})+S(\mathrm{tr}_{C}\rho_{\delta}),$

respectively, and similarly for the other cases. It is easy to obtain that
$I_{4}(A:B:C:D)=0$. Hence we have the following result.

_Conclusion 2-1-1._ For any $\rho\in\triangle_{\mathrm{ITCN1}}$, the four-
partite quantum mutual information $I_{4}(A:B:C:D)$ of quantum state $\rho$ is
zero.

Now we discuss entanglement measure. We require the entanglement measure
$\varepsilon$ satisfies the requirements stated in section II. It is easy to
deduce

$\begin{array}[]{ll}\varepsilon_{A|BCD}&=\varepsilon_{A_{\alpha}A_{\gamma}A_{\delta}|B_{\alpha}B_{\beta}B_{\tau}C_{\delta}C_{\theta}C_{\tau}D_{\beta}D_{\gamma}D_{\theta}}=\varepsilon_{A_{\alpha}|B_{\alpha}}+\varepsilon_{A_{\gamma}|D_{\gamma}}+\varepsilon_{A_{\delta}|C_{\delta}}=\varepsilon_{A|B}+\varepsilon_{A|D}+\varepsilon_{A|C}.\\\
\end{array}$ (19)

So the following result holds.

_Conclusion 2-1-2._ For any $\rho\in\triangle_{\mathrm{ITCN1}}$, we have that
$\varepsilon_{T|XYZ}[\rho]=\varepsilon_{T|X}[\mathrm{tr}_{YZ}\rho]+\varepsilon_{T|Y}[\mathrm{tr}_{XZ}\rho]+\varepsilon_{T|Z}[\mathrm{tr}_{XY}\rho]$,
for all the bipartitions, such as $A|BCD$, $B|ACD$, $C|ABD$, and $D|ABC$.

Later on we turn our attention to the rank of the quantum states. Because the
unitary matrices $U_{A}$, $U_{B}$, $U_{C}$, and $U_{D}$ don’t change the rank
of the whole quantum system, for all $\rho\in\Delta_{\mathrm{ITCN1}}$, we have
the rank of the quantum state $\rho$,

$\mathrm{rk}(\rho)=\mathrm{rk}(\rho_{\alpha})\mathrm{rk}(\rho_{\beta})\mathrm{rk}(\rho_{\gamma})\mathrm{rk}(\rho_{\delta})\mathrm{rk}(\rho_{\theta})\mathrm{rk}(\rho_{\tau})=r_{\alpha}r_{\beta}r_{\gamma}r_{\delta}r_{\theta}r_{\tau},$

the rank of the local reduced state $\mathrm{tr}_{BCD}\rho$ is

$\mathrm{rk}(\mathrm{tr}_{BCD}\rho)=\mathrm{rk}(\mathrm{tr}_{B}\rho_{\alpha})\mathrm{rk}(\mathrm{tr}_{D}\rho_{\gamma})\mathrm{rk}(\mathrm{tr}_{C}\rho_{\delta})=r_{\alpha}^{A}r_{\gamma}^{A}r_{\delta}^{A},\\\
$

the rank of the two-partite reduced state $\mathrm{tr}_{CD}\rho$ is

$\mathrm{rk}(\mathrm{tr}_{CD}\rho)=\mathrm{rk}(\rho_{\alpha})\mathrm{rk}(\mathrm{tr}_{D}\rho_{\beta})\mathrm{rk}(\mathrm{tr}_{D}\rho_{\gamma})\mathrm{rk}(\mathrm{tr}_{C}\rho_{\delta})\mathrm{rk}(\mathrm{tr}_{C}\rho_{\tau})=r_{\alpha}r_{\beta}^{B}r_{\gamma}^{A}r_{\delta}^{A}r_{\tau}^{B},\\\
$

the rank of the three-partite reduced state $\mathrm{tr}_{D}\rho$ is

$\mathrm{rk}(\mathrm{tr}_{D}\rho)=\mathrm{rk}(\rho_{\alpha})\mathrm{rk}(\mathrm{tr}_{D}\rho_{\beta})\mathrm{rk}(\mathrm{tr}_{D}\rho_{\gamma})\mathrm{rk}(\rho_{\delta})\mathrm{rk}(\mathrm{tr}_{D}\rho_{\theta})\mathrm{rk}(\rho_{\tau})=r_{\alpha}r_{\beta}^{B}r_{\gamma}^{A}r_{\delta}r_{\theta}^{C}r_{\tau}.\\\
$

Here $r_{\alpha}^{A}=\mathrm{rk}(\mathrm{tr}_{B}\rho_{\alpha})$ stands for the
rank of the quantum state $\mathrm{tr}_{B}\rho_{\alpha}$, similarly for the
others.

Therefore we have

$\begin{array}[]{llll}\mathrm{rk}(\rho)=r_{\alpha}r_{\beta}r_{\gamma}r_{\delta}r_{\theta}r_{\tau},&&&\\\
\mathrm{rk}(\mathrm{tr}_{BCD}\rho)=r_{\alpha}^{A}r_{\gamma}^{A}r_{\delta}^{A},&\quad\mathrm{rk}(\mathrm{tr}_{CDA}\rho)=r_{\alpha}^{B}r_{\beta}^{B}r_{\tau}^{B},&\quad\mathrm{rk}(\mathrm{tr}_{DAB}\rho)=r_{\delta}^{C}r_{\theta}^{C}r_{\tau}^{C},&\quad\mathrm{rk}(\mathrm{tr}_{ABC}\rho)=r_{\beta}^{D}r_{\gamma}^{D}r_{\theta}^{D},\\\
\mathrm{rk}(\mathrm{tr}_{CD}\rho)=r_{\alpha}r_{\beta}^{B}r_{\gamma}^{A}r_{\delta}^{A}r_{\tau}^{B},&\quad\mathrm{rk}(\mathrm{tr}_{CA}\rho)=r_{\alpha}^{B}r_{\beta}r_{\gamma}^{D}r_{\theta}^{D}r_{\tau}^{B},&\quad\mathrm{rk}(\mathrm{tr}_{CB}\rho)=r_{\alpha}^{A}r_{\beta}^{D}r_{\gamma}r_{\delta}^{A}r_{\theta}^{D},&\\\
\mathrm{rk}(\mathrm{tr}_{DA}\rho)=r_{\alpha}^{B}r_{\beta}^{B}r_{\delta}^{C}r_{\theta}^{C}r_{\tau},&\quad\mathrm{rk}(\mathrm{tr}_{DB}\rho)=r_{\alpha}^{A}r_{\gamma}^{A}r_{\delta}r_{\theta}^{C}r_{\tau}^{C},&\quad\mathrm{rk}(\mathrm{tr}_{AB}\rho)=r_{\beta}^{D}r_{\gamma}^{D}r_{\delta}^{C}r_{\theta}r_{\tau}^{C},&\\\
\mathrm{rk}(\mathrm{tr}_{A}\rho)=r_{\alpha}^{B}r_{\beta}r_{\gamma}^{D}r_{\delta}^{C}r_{\theta}r_{\tau},&\quad\mathrm{rk}(\mathrm{tr}_{B}\rho)=r_{\alpha}^{A}r_{\beta}^{D}r_{\gamma}r_{\delta}r_{\theta}r_{\tau}^{C},&\quad\mathrm{rk}(\mathrm{tr}_{C}\rho)=r_{\alpha}r_{\beta}r_{\gamma}r_{\delta}^{A}r_{\theta}^{D}r_{\tau}^{B},&\quad\mathrm{rk}(\mathrm{tr}_{D}\rho)=r_{\alpha}r_{\beta}^{B}r_{\gamma}^{A}r_{\delta}r_{\theta}^{C}r_{\tau}.\end{array}$
(20)

Suppose that the Hilbert space of single particle is d-dimensional. So
$r_{\alpha}$, $r_{\beta}$, $r_{\gamma}$, $r_{\delta}$, $r_{\theta}$,
$r_{\tau}$$\in[1,d^{2}]$, $r_{\alpha}^{A}$, $r_{\alpha}^{B}$, $r_{\beta}^{B}$,
$r_{\beta}^{D}$, $r_{\gamma}^{A}$, $r_{\gamma}^{D}$, $r_{\delta}^{A}$,
$r_{\delta}^{C}$, $r_{\theta}^{C}$, $r_{\theta}^{D}$, $r_{\tau}^{B}$,
$r_{\tau}^{C}$$\in[1,d]$. Thus we arrive at the following conclusion.

_Conclusion 2-1-3._ For $\rho\in\triangle_{\mathrm{ITCN1}}$ there exist
integers $r_{\alpha}$, $r_{\beta}$, $r_{\gamma}$, $r_{\delta}$, $r_{\theta}$,
$r_{\tau}$$\in[1,d^{2}]$, $r_{\alpha}^{A}$, $r_{\alpha}^{B}$, $r_{\beta}^{B}$,
$r_{\beta}^{D}$, $r_{\gamma}^{A}$, $r_{\gamma}^{D}$, $r_{\delta}^{A}$,
$r_{\delta}^{C}$, $r_{\theta}^{C}$, $r_{\theta}^{D}$, $r_{\tau}^{B}$,
$r_{\tau}^{C}$$\in[1,d]$, which satisfy Eq. (20).

After that we invistegate the situation where the four nodes and the six
sources are classically correlated 18 . The classically correlated refers to
that every source and every node are correlated by sharing the random variable
$\lambda$. The classically correlated triangular cone network structure 1 is
shown in Fig. 2(b). We use CTCN1 to denote this classically correlated
triangular cone network, $\Delta_{\mathrm{CTCN1}}$ to represent the set of
quantum states that can be prepared in CTCN1. A quantum state
$\rho\in\Delta_{\mathrm{CTCN1}}$ can be written as
$\rho=\sum_{\lambda}p_{\lambda}\rho_{\lambda}$, where
$\rho_{\lambda}\in\Delta_{\mathrm{ITCN1}}$, $p_{\lambda}$ is a probability
distribution.

Next we will use the rank relationship to show that there exists no four-qubit
genuine multipartite entangled state in the set $\Delta_{\mathrm{CTCN1}}$.
First we consider the case that the four-qubit genuine multipartite entangled
state is a pure state. Clearly, in this case, the rank of the four-qubit
genuine multipartite entangled state is equal to 1 and it is entangled along
every bipartition. By the Schmidt decomposition of the four-qubit genuine
entangled state, we know that all single node reduced state has rank 2.
However, in fact, when each source in the set
$\\{\rho_{\alpha},\rho_{\beta},\rho_{\gamma},\rho_{\theta},\rho_{\tau},\rho_{\delta}\\}$
is entangled among two particles. According to the Schmidt decomposition, the
rank of the reduced state of a particle in each source is 2, if source is in
2-qubit entangled state. For the network structure ITCN1 with six sources as
shown in Fig. 2(a), the local rank of the reduced states of each node is 8.
Therefore, it is impossible to prepare pure four-qubit genuine multipartite
entangled state in ITCN1. Similarly, one can verify that the claim is true for
the other cases of resources. Moreover, as four-qubit genuine multipartite
entangled states need pure four-qubit genuine multipartite entangled states.
So one can not prepare four-qubit genuine multipartite entangled states in
CTCN1 also. Hence, we obtain the following result.

_Conclusion 2-1-4._ No four-qubit genuine multipartite entangled state can be
prepared in ITCN1 and CTCN1.

The four-partite e quantum states which can not be generated in ITCN1 and
CTCN1 is illustrated in Fig. 2(c).

Now let us consider the independent triangular cone network structure 2
(ITCN2). As shown in Fig. 3(a), it has four nodes, namely _A_ , _B_ , _C_ ,
and _D_. These four nodes are formed by pairs of three particles in each of
the four sources $\rho_{\alpha}$, $\rho_{\beta}$, $\rho_{\gamma}$, and
$\rho_{\delta}$. Each source produces entangled three-partite quantum states.
$\rho_{\alpha}$ is shared by _A_ , _B_ , and _D_ , $\rho_{\beta}$ is shared by
_A_ , _B_ , and _C_ , $\rho_{\gamma}$ is shared by _A_ , _C_ , and _D_ ,
$\rho_{\delta}$ is shared by _B_ , _C_ , and _D_. We assume that four nodes do
not share common information. _A_ receives three particles from source
$\rho_{\alpha}$, $\rho_{\gamma}$, and $\rho_{\gamma}$, _B_ receives three
particles from source $\rho_{\alpha}$, $\rho_{\beta}$, and $\rho_{\delta}$,
_C_ receives three particles from source $\rho_{\beta}$, $\rho_{\gamma}$, and
$\rho_{\delta}$, _D_ receives three particles from source $\rho_{\alpha}$,
$\rho_{\gamma}$, and $\rho_{\delta}$. Each node can be applied a local unitary
matrix to the three received particles. Four local unitary matrices are
denoted by $U_{A}$, $U_{B}$, $U_{C}$, and $U_{D}$. We use $\rho$ to express
the quantum state in ITCN2 and utilize $\Delta_{\mathrm{ITCN2}}$ to denote the
set of quantum states that can be produced in ITCN2.

Figure 3: Triangular cone network structure 2. (a) Independent triangular cone
network structure 2 (ITCN2), where each of these sources is a three-partite
entangled state. (b) Classically correlated triangular cone network structure
2 (CTCN2), where each source and each node are classically correlated by
sharing the random variable $\lambda$. (c) Four-partite quantum states which
can not be produced in ITCN2 and CTCN2.

Obviously, the quantum states $\rho$ can be expressed as

$\rho=(U_{A}\otimes U_{B}\otimes U_{C}\otimes
U_{D})(\rho_{\alpha}\otimes\rho_{\beta}\otimes\rho_{\gamma}\otimes\rho_{\delta})(U_{A}^{{\dagger}}\otimes
U_{B}^{{\dagger}}\otimes U_{C}^{{\dagger}}\otimes U_{D}^{{\dagger}}).$ (21)

Because at each node the three particles are coming from different sources, so
the unitary matrix at one node cannot act on the global quantum states
$\rho_{\alpha}$, $\rho_{\beta}$, $\rho_{\gamma}$, and $\rho_{\delta}$ at same
time.

Evidently, we have that the entropy of quantum state (21)

$S(\rho)=S(\rho_{\alpha})+S(\rho_{\beta})+S(\rho_{\gamma})+S(\rho_{\delta}).$

The three-partite entropy is extended to

$S(ABC)=S(\mathrm{tr}_{D}\rho_{\alpha})+S(\rho_{\beta})+S(\mathrm{tr}_{D}\rho_{\gamma})+S(\mathrm{tr}_{D}\rho_{\delta}),$

the two-partite entropy is extended to

$S(AB)=S(\mathrm{tr}_{D}\rho_{\alpha})+S(\mathrm{tr}_{C}\rho_{\beta})+S(\mathrm{tr}_{CD}\rho_{\gamma})+S(\mathrm{tr}_{CD}\rho_{\delta}),$

the one-partite entropy is extended to

$S(A)=S(\mathrm{tr}_{BD}\rho_{\alpha})+S(\mathrm{tr}_{BC}\rho_{\beta})+S(\mathrm{tr}_{CD}\rho_{\gamma}).$

By calculating, one finds that for quantum state
$\rho\in\triangle_{\mathrm{ITCN2}}$, $I_{4}(A:B:C:D)=0$. This can be restated
as the following conclusion.

_Conclusion 2-2-1._ The four-partite quantum mutual information
$I_{4}(A:B:C:D)=0$ for any $\rho\in\triangle_{\mathrm{ITCN2}}$.

Then we discuss the entanglement measure of the quantum states. For the
entanglement measure $\varepsilon$ which satisfies the requirements stated in
section II, we can obtain

$\varepsilon_{A|BCD}=\varepsilon_{A_{\alpha}A_{\beta}A_{\gamma}|B_{\alpha}B_{\beta}B_{\delta}C_{\beta}C_{\gamma}C_{\delta}D_{\alpha}D_{\gamma}D_{\delta}}=\varepsilon_{A_{\alpha}|B_{\alpha}D_{\alpha}}+\varepsilon_{A_{\beta}|B_{\beta}C_{\beta}}+\varepsilon_{A_{\gamma}|C_{\gamma}D_{\gamma}}\\\
=\varepsilon_{A|BD}+\varepsilon_{A|BC}+\varepsilon_{A|CD}.$ (22)

So we have the following result.

_Conclusion 2-2-2._ For any $\rho\in\triangle_{\mathrm{ITCN2}}$, we have that
$\varepsilon_{T|XYZ}[\rho]=\varepsilon_{T|XY}[\mathrm{tr}_{Z}\rho]+\varepsilon_{T|XZ}[\mathrm{tr}_{Y}\rho]+\varepsilon_{T|YZ}[\mathrm{tr}_{X}\rho]$,
which holds for all the bipartitions, such as $A|BCD$, $B|ACD$, $C|ABD$, and
$D|ABC$.

As the unitary matrices $U_{A}$, $U_{B}$, $U_{C}$, and $U_{D}$ don’t change
the rank of the whole quantum state, so for $\rho\in\Delta_{\mathrm{ITCN2}}$,
we have the rank of quantum state $\rho$

$\mathrm{rk}(\rho)=\mathrm{rk}(\rho_{\alpha})\mathrm{rk}(\rho_{\beta})\mathrm{rk}(\rho_{\gamma})\mathrm{rk}(\rho_{\delta}).$

The rank of the local reduced state is

$\mathrm{rk}(\mathrm{tr}_{BCD}\rho)=\mathrm{rk}(\mathrm{tr}_{BD}\rho_{\alpha})\mathrm{rk}(\mathrm{tr}_{BC}\rho_{\beta})\mathrm{rk}(\mathrm{tr}_{CD}\rho_{\gamma})=r_{\alpha}^{A}r_{\beta}^{A}r_{\gamma}^{A},\\\
$

the rank of the two-partite reduced state is

$\mathrm{rk}(\mathrm{tr}_{CD}\rho)=\mathrm{rk}(\mathrm{tr}_{D}\rho_{\alpha})\mathrm{rk}(\mathrm{tr}_{C}\rho_{\beta})\mathrm{rk}(\mathrm{tr}_{CD}\rho_{\gamma})\mathrm{rk}(\mathrm{tr}_{CD}\rho_{\delta})=r_{\alpha}^{AB}r_{\beta}^{AB}r_{\gamma}^{A}r_{\delta}^{B},\\\
$

the rank of the three-partite reduced state is

$\begin{split}\mathrm{rk}(\mathrm{tr}_{D}\rho)=\mathrm{rk}(\mathrm{tr}_{D}\rho_{\alpha})\mathrm{rk}(\rho_{\beta})\mathrm{rk}(\mathrm{tr}_{D}\rho_{\gamma})\mathrm{rk}(\mathrm{tr}_{D}\rho_{\delta})=r_{\alpha}^{AB}r_{\beta}r_{\gamma}^{AC}r_{\delta}^{BC}.\end{split}$

Here $r_{\alpha}^{AB}=\mathrm{rk}(\mathrm{tr}_{D}\rho_{\alpha})$ is the rank
of reduced quantum state $\mathrm{tr}_{D}\rho_{\alpha}$, similarly for the
others. Therefore we have

$\begin{array}[]{llll}\mathrm{rk}(\rho)=r_{\alpha}r_{\beta}r_{\gamma}r_{\delta},&&&\\\
\mathrm{rk}(\mathrm{tr}_{BCD}\rho)=r_{\alpha}^{A}r_{\beta}^{A}r_{\gamma}^{A},&\quad\mathrm{rk}(\mathrm{tr}_{CDA}\rho)=r_{\alpha}^{B}r_{\beta}^{B}r_{\delta}^{B},&\quad\mathrm{rk}(\mathrm{tr}_{DAB}\rho)=r_{\beta}^{C}r_{\gamma}^{C}r_{\delta}^{C},&\quad\mathrm{rk}(\mathrm{tr}_{ABC}\rho)=r_{\alpha}^{D}r_{\gamma}^{D}r_{\delta}^{D},\\\
\mathrm{rk}(\mathrm{tr}_{CD}\rho)=r_{\alpha}^{AB}r_{\beta}^{AB}r_{\gamma}^{A}r_{\delta}^{B},&\quad\mathrm{rk}(\mathrm{tr}_{CA}\rho)=r_{\alpha}^{BD}r_{\beta}^{B}r_{\gamma}^{D}r_{\delta}^{BD},&\quad\mathrm{rk}(\mathrm{tr}_{CB}\rho)=r_{\alpha}^{AD}r_{\beta}^{A}r_{\gamma}^{AD}r_{\delta}^{D},&\\\
\mathrm{rk}(\mathrm{tr}_{DA}\rho)=r_{\alpha}^{B}r_{\beta}^{BC}r_{\gamma}^{C}r_{\delta}^{BC},&\quad\mathrm{rk}(\mathrm{tr}_{DB}\rho)=r_{\alpha}^{A}r_{\beta}^{AC}r_{\gamma}^{AC}r_{\delta}^{C},&\quad\mathrm{rk}(\mathrm{tr}_{AB}\rho)=r_{\alpha}^{D}r_{\beta}^{C}r_{\gamma}^{CD}r_{\delta}^{CD},&\\\
\mathrm{rk}(\mathrm{tr}_{A}\rho)=r_{\alpha}^{BD}r_{\beta}^{BC}r_{\gamma}^{CD}r_{\delta},&\quad\mathrm{rk}(\mathrm{tr}_{B}\rho)=r_{\alpha}^{AD}r_{\beta}^{AC}r_{\gamma}r_{\delta}^{CD},&\quad\mathrm{rk}(\mathrm{tr}_{C}\rho)=r_{\alpha}r_{\beta}^{AB}r_{\gamma}^{AD}r_{\delta}^{BD},&\quad\mathrm{rk}(\mathrm{tr}_{D}\rho)=r_{\alpha}^{AB}r_{\beta}r_{\gamma}^{AC}r_{\delta}^{BC}.\end{array}$
(23)

Assume that the Hilbert space of single particle is _d_ -dimensional. So
$r_{\alpha}^{A}$, $r_{\alpha}^{B}$, $r_{\alpha}^{D}$, $r_{\beta}^{A}$,
$r_{\beta}^{B}$, $r_{\beta}^{C}$, $r_{\gamma}^{A}$, $r_{\gamma}^{C}$,
$r_{\gamma}^{D}$, $r_{\delta}^{B}$, $r_{\delta}^{C}$,
$r_{\delta}^{D}$$\in[1,d]$, $r_{\alpha}^{AB}$, $r_{\alpha}^{AD}$,
$r_{\alpha}^{BD}$, $r_{\beta}^{AB}$, $r_{\beta}^{AC}$, $r_{\beta}^{BC}$,
$r_{\gamma}^{AC}$, $r_{\gamma}^{AD}$, $r_{\gamma}^{CD}$, $r_{\delta}^{BC}$,
$r_{\delta}^{BD}$, $r_{\delta}^{CD}$$\in[1,d^{2}]$, and $r_{\alpha}$,
$r_{\beta}$, $r_{\gamma}$, $r_{\delta}$$\in[1,d^{3}]$.

_Conclusion 2-2-3._ For $\rho\in\triangle_{\mathrm{ITCN2}}$ there exist
integers $r_{\alpha}$, $r_{\beta}$, $r_{\gamma}$, $r_{\delta}$$\in[1,d^{3}]$,
$r_{\alpha}^{AB}$, $r_{\alpha}^{AD}$, $r_{\alpha}^{BD}$, $r_{\beta}^{AB}$,
$r_{\beta}^{AC}$, $r_{\beta}^{BC}$, $r_{\gamma}^{AC}$, $r_{\gamma}^{AD}$,
$r_{\gamma}^{CD}$, $r_{\delta}^{BC}$, $r_{\delta}^{BD}$,
$r_{\delta}^{CD}$$\in[1,d^{2}]$, $r_{\alpha}^{A}$, $r_{\alpha}^{B}$,
$r_{\alpha}^{D}$, $r_{\beta}^{A}$, $r_{\beta}^{B}$, $r_{\beta}^{C}$,
$r_{\gamma}^{A}$, $r_{\gamma}^{C}$, $r_{\gamma}^{D}$, $r_{\delta}^{B}$,
$r_{\delta}^{C}$, $r_{\delta}^{D}$$\in[1,d]$, which satisfy Eq. (23).

Similar to the case investigated in ITCN1 we can study the scenario where the
four nodes and the four sources are classically correlated in ITCN2. If every
source and every node are correlated by sharing the random variable $\lambda$,
we call the quantum state prepared the classically correlated state as shown
in Fig. 3(b). Let CTCN2 denote this classically correlated triangular cone
network strcture 2, $\Delta_{\texttt{CTCN2}}$ express the set of quantum
states that can be produced in CTCN2. The classically correlated state
$\rho\in\Delta_{\texttt{CTCN2}}$ reads
$\rho=\sum_{\lambda}p_{\lambda}\rho_{\lambda}$, where
$\rho_{\lambda}\in\Delta_{\mathrm{ITCN2}}$, $p_{\lambda}$ is a probability
distribution.

Based on rank relationship we can demonstrate that there exists no four-qubit
genuine multipartite entangled state in the set $\Delta_{\mathrm{CTCN2}}$
also. First we assume four-qubit genuine multipartite entangled state is a
pure state. It is easy to show that all single node reduced state has rank 2.
However, when each source in the set
$\\{\rho_{\alpha},\rho_{\beta},\rho_{\gamma},\rho_{\delta}\\}$ is entangled
among three particles and it is entangled along every bipartition. Based on
the Schmidt decomposition, the rank of the reduced state of one particle in
each source is 2, if source is in 3-qubit entangled state. So the local rank
of the reduced states of each node is 8. This shows that it is impossible to
generate pure four-qubit genuine multipartite entangled states in ITCN2.
Similarly, we can verify that the claim holds for other cases of resources.
Moreover, as mixed four-qubit genuine multipartite entangled states require
pure four-qubit genuine multipartite entangled states. So there exists no
four-qubit genuine multipartite entangled state in the set
$\Delta_{\mathrm{CTCN2}}$. That can be restated as the following result.

_Conclusion 2-2-4._ No four-qubit genuine multipartite entangled state can be
prepared in ITCN2 and CTCN2.

Fig.3(c) shows the four-partite quantum states which can not be produced in
ITCN2 and CTCN2.

##  IV Summary

We investigate the quantum states that can be prepared and the quantum states
that cannot be prepared in three kinds of four-node network structures,
including a four-node network structure in a plane and two four-node network
structures in the space. The $n$-partite mutual information of quantum system,
which satisfies the symmetry requirement, is defined. We analyzed the
properties of four-partite quantum mutual information of quantum states
prepared in different network structures and the effect of the local channels
on the four-partite quantum mutual information. The entanglement properties of
the quantum states prepared in the four-node quantum network structures are
discussed. The constraints on the rank of global state and the local ranks of
the reduced states are deduced. We also prove that the four-qubit genuine
multipartite entangled states can not be prepared classically in the four-node
network structures. We hope that these results may be useful for the further
study of quantum network.

###### Acknowledgements.

This work was supported by the National Natural Science Foundation of China
under Grant No. 12071110, the Hebei Natural Science Foundation of China under
Grant No. A2020205014, and funded by Science and Technology Project of Hebei
Education Department under Grant Nos. ZD2020167, ZD2021066.

##  Appendix A  The proof of conclusion 1-3

By Eq.(3), and according to Observation 4 of Ref.18 , for quantum state
$(\Lambda_{A}\otimes\Lambda_{B}\otimes\Lambda_{C}\otimes I_{D})\rho$, one gets

$I_{3}(A:B:D)=-I_{2}(AB:D)+I_{2}(A:D)+I_{2}(B:D)\leq 0,$ (24)

where $\rho\in\Delta_{\mathrm{IQN}}$, $\Lambda_{A}$, $\Lambda_{B}$,
$\Lambda_{C}$ are three local quantum channels, $I_{D}$ is the identity
operator. Similarly, we have

$I_{3}(B:C:D)=-I_{2}(BC:D)+I_{2}(B:D)+I_{2}(C:D)\leq 0,$ (25)
$I_{3}(A:C:D)=-I_{2}(AC:D)+I_{2}(A:D)+I_{2}(C:D)\leq 0,$ (26) $\displaystyle
I_{3}(AB:C:D)=-I_{2}(ABC:D)+I_{2}(AB:D)+I_{2}(C:D)\leq 0,$ (27) $\displaystyle
I_{3}(BC:A:D)=-I_{2}(ABC:D)+I_{2}(BC:D)+I_{2}(A:D)\leq 0.$ (28)

By summing the Eqs.(A1-A3), we have

$-I_{2}(AB:D)-I_{2}(AC:D)-I_{2}(BC:D)+2I_{2}(A:D)+2I_{2}(B:D)+2I_{2}(C:D)\leq
0.$ (29)

By summing the Eqs.(A1,A2, A4,A5), we have

$-I_{2}(ABC:D)+I_{2}(A:D)+I_{2}(B:D)+I_{2}(C:D)\leq 0.$ (30)

By the definition of four-partite quantum mutual information for an arbitrary
quantum state sharing by nodes $A,B,C,D$ one can rewrite Eq.(5) as

$\begin{array}[]{lll}&&I_{4}(A:B:C:D)\\\
&=&[S(ABC)+S(D)-S(ABCD)]+[S(A)+S(D)-S(AD)]\\\
&&+[S(B)+S(D)-S(BD)]+[S(C)+S(D)-S(CD)]\\\
&&-[S(AB)+S(D)-S(ABD)]-[S(AC)+S(D)-S(ACD)]\\\ &&-[S(BC)+S(D)-S(BCD)]\\\
&=&I_{2}(ABC:D)+I_{2}(A:D)+I_{2}(B:D)+I_{2}(C:D)\\\
&&-I_{2}(AB:D)-I_{2}(AC:D)-I_{2}(BC:D).\\\ \end{array}$ (31)

So, for the four-partite quantum mutual information of the quantum state
$(\Lambda_{A}\otimes\Lambda_{B}\otimes\Lambda_{C}\otimes I_{D})\rho$, by using
Eqs. (29,30,31), we have

$\begin{array}[]{lll}&&I_{4}[(\Lambda_{A}\otimes\Lambda_{B}\otimes\Lambda_{C}\otimes
I_{D})\rho]\\\ &=&I_{2}(ABC:D)+I_{2}(A:D)+I_{2}(B:D)+I_{2}(C:D)\\\
&&-I_{2}(AB:D)-I_{2}(AC:D)-I_{2}(BC:D)\\\
&\leq&I_{2}(ABC:D)+I_{2}(A:D)+I_{2}(B:D)+I_{2}(C:D)\\\
&&-[2I_{2}(A:D)+2I_{2}(B:D)+2I_{2}(C:D)]\\\
&=&I_{2}(ABC:D)-[I_{2}(A:D)+I_{2}(B:D)+I_{2}(C:D)]\\\ \\\ \end{array}$ (32)

and

$\begin{array}[]{lll}&&I_{4}[(\Lambda_{A}\otimes\Lambda_{B}\otimes\Lambda_{C}\otimes
I_{D})\rho]\\\ &=&I_{2}(ABC:D)+I_{2}(A:D)+I_{2}(B:D)+I_{2}(C:D)\\\
&&-I_{2}(AB:D)-I_{2}(AC:D)-I_{2}(BC:D)\\\
&\geq&2I_{2}(A:D)+2I_{2}(B:D)+2I_{2}(C:D)\\\
&&-[I_{2}(AB:D)+I_{2}(BC:D)+I_{2}(AC:D)].\\\ \end{array}$ (33)

Therefore, $I_{4}[(\Lambda_{A}\otimes\Lambda_{B}\otimes\Lambda_{C}\otimes
I_{D})\rho]$ are bounded as follows

$\displaystyle
2I_{2}(A:D)+2I_{2}(B:D)+2I_{2}(C:D)-[I_{2}(AB:D)+I_{2}(BC:D)+I_{2}(AC:D)]$
$\displaystyle\leq$ $\displaystyle
I_{4}[(\Lambda_{A}\otimes\Lambda_{B}\otimes\Lambda_{C}\otimes I_{D})\rho]$
$\displaystyle\leq$ $\displaystyle
I_{2}(ABC:D)-[I_{2}(A:D)+I_{2}(B:D)+I_{2}(C:D)].$

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# Hebbian learning with gradients: Hebbian convolutional neural networks with
modern deep learning frameworks

Thomas Miconi
ML Collective
<EMAIL_ADDRESS>

###### Abstract

Deep learning networks generally use non-biological learning methods. By
contrast, networks based on more biologically plausible learning, such as
Hebbian learning, show comparatively poor performance and difficulties of
implementation. Here we show that Hebbian learning in hierarchical,
convolutional neural networks can be implemented almost trivially with modern
deep learning frameworks, by using specific losses whose gradients produce
exactly the desired Hebbian updates. We provide expressions whose gradients
exactly implement a plain Hebbian rule ($\Delta{w}\propto xy$), Grossberg’s
instar rule ($\Delta{w}\propto y(x-w)$), and Oja’s rule ($\Delta{w}\propto
y(x-yw)$). As an application, we build Hebbian convolutional multi-layer
networks for object recognition. We observe that higher layers of such
networks tend to learn large, simple features (Gabor-like filters and blobs),
explaining the previously reported decrease in decoding performance over
successive layers. To combat this tendency, we introduce interventions (denser
activations with sparse plasticity, pruning of connections between layers)
which result in sparser learned features, massively increase performance, and
allow information to increase over successive layers. We hypothesize that more
advanced techniques (dynamic stimuli, trace learning, feedback connections,
etc.), together with the massive computational boost offered by modern deep
learning frameworks, could greatly improve the performance and biological
relevance of multi-layer Hebbian networks.

##  1 Introduction

Recent advances in deep learning have greatly improved the state of the art in
visual object recognition. Deep learning networks take inspiration from
features of the ventral visual cortex (including a hierarchical organization
in successive layers and a topological organization of connections). Their
internal activity shows similarities, as well as differences, with neural
responses to natural images (Yamins and DiCarlo, 2016). One important
difference is that deep learning networks are usually trained by gradient
descent and backpropagation. By contrast, learning in the visual cortex is
generally thought to occur primarily by local learning (with some influences
from both downstream and upstream activity), presumably according to the
Hebbian principle of reinforcing connections that cause a cell to fire.
Investigating Hebbian learning in large hierarchical networks could therefore
provide a more relevant model of the visual system, as well as offering new
methods for unsupervised and self-supervised (label-free) learning.

There is a long history of research in hierarchical Hebbian networks. The
Neocognitron (an early hierarchical model for translation- and scaling-
tolerant object recognition) used an effectively Hebbian learning rule
(Fukushima and Miyake, 1982). The HMAX model used an “imprinting” learning
method to approximate Hebbian learning in a large network in a computationally
tractable manner (Serre et al., 2007). The VISNet model also studied various
forms of Hebbian learning with temporal dynamics to learn invariant
representations (Wallis and Rolls, 1997; Rolls and Milward, 2000). While these
models and others provided rich insight, their performance remained far from
current deep learning techniques. Some of this difference may result from the
massive computational advances that supported (and were prompted by) the deep
learning revolution, including both specialized hardware (Pinto et al., 2009;
Krizhevsky et al., 2012) and software frameworks to interact with this
hardware in an intuitive manner (Paszke et al., 2019; Abadi et al., 2015).

Another motivation for investigating Hebbian networks is that Hebbian learning
can be generalized to implement many forms of learning, including sparse
coding (Brito and Gerstner, 2016), K-means (Hu et al., 2014) and similarity-
matching (Bahroun and Soltoggio, 2017). However, pure Hebbian multi-layer
networks suffer from poor performance. Notably, a consistent result is that
decoding performance _decreases_ over successive layers, suggesting loss of
information (Amato et al., 2019; Bahroun et al., 2017). This is a particularly
disconcerting finding, since the purpose of hierarchical networks is precisely
to aggregate information and produce more exploitable representations over
successive layers DiCarlo and Cox (2007); Serre et al. (2007).

Recently several authors have proposed using modern deep learning framework
for Hebbian learning in multi-layer convolutional networks (Amato et al.,
2019; Talloen et al., 2021). These projects rely on intricate hand-crafted
machinery to implement Hebbian learning. While Talloen et al. (2021) expose
this machinery as a remarkable reusable framework, this necessarily constrains
possible usage to whatever abilities the framework provides, in addition to
burdening users with the need to learn the framework itself.

Here we show that Hebbian learning can be implemented very easily with modern
deep learning frameworks, by using surrogate losses whose gradients turn out
to produce exactly the desired Hebbian updates. As a result, we use the
frameworks as they were intended to be used, with little additional burden,
freeing us to concentrate on actual network design. Importantly, the actual
outputs of the network are arbitrary and need not be differentiable, since the
surrogate losses are only used to set up the computational graph. We provide
expressions for losses whose gradients exactly implement several Hebbian
rules, namely: plain Hebbian ($\Delta{w}\propto xy$), Grossberg’s instar rule
($\Delta{w}\propto y(x-w)$), and Oja’s rule ($\Delta{w}\propto y(x-yw)$).

As an application, we build simple convolutional multi-layer networks for
object recognition, which we test on the CIFAR10 dataset. We observe that the
higher layers of such networks do not spontaneously learn complex shapes.
Rather, they simply learn large, simple features (Gabor-like oriented edge
detectors and blobs), by combining lower-level features appropriately. This
explains the reduction in decoding performance over successive layers, despite
a higher number of filters, as was observed in previous work (Amato et al.,
2019; Bahroun et al., 2017; Bahroun and Soltoggio, 2017). To combat this
tendency, we introduce several interventions (“triangle” method for computing
activations (Coates et al., 2011) and massive pruning of connections between
layers) which both prevent the formation of high-level Gabors and massively
increase higher-level performance, allowing higher layers to produce more
informative representations than the first layer. We hypothesize that
incorporating more advanced features of previous models (dynamic stimuli with
trace learning, feedback connections, etc.), together with the massive
computational boost offered by modern deep learning frameworks, could greatly
improve the performance and biological relevance of multi-layer Hebbian
networks.

##  2 Methods

###  2.1 Hebbian learning with deep learning frameworks

Modern deep learning frameworks implement automatic differentiation with
transparent access to GPU computation, batched processing, and convolutional
filtering. As such, they are ideally suited to compute batch gradients over
multi-layer convolutional networks (Paszke et al., 2019; Abadi et al., 2015).

Implementing Hebbian learning poses no difficulty with fully-connected layers,
or fully recurrent networks, because the input and output are simple vectors
(or matrices when considering the batch dimension). As a result, computing the
Hebbian weight update (based on a product of inputs and outputs) requires
little more than an outer vector product. No special machinery is needed
(Miconi et al., 2018).

Things are not so simple with convolutional networks, because in this case the
inputs to the weights are composed of _overlapping_ patches within the
previous layer’s output. As such, we can not simply grab the inputs and
outputs and perform a straightforward outer product. We propose two methods,
not necessarily exclusive, to implement such computations with deep learning
frameworks.

####  2.1.1 Method 1: Hebbian updates as gradients of surrogate losses

By far the simplest method is to compute a _surrogate loss_ , the gradient of
which just happens to be the desired Hebbian weight update.

We first compute the “real” output of that layer, which is arbitrary and can
involve any computations, differentiable or not (in this work we mostly use a
winner-take-all competition, which is not straightforwardly differentiable).
Then, we compute a _surrogate loss_ for this layer, designed specifically to
produce the desired update as a gradient (see Appendix A for equations and
code samples). After this surrogate loss is computed, setting in place the
appropriate computational graph, we overwrite the values of this surrogate
output with the “real” output values previously computed, without impacting
the computational graph (because the gradient of the surrogate loss must be
evaluated at the “real” value of the output). Then we simply invoke the
framework’s backward pass function to compute and apply the desired weight
updates (see Appendix A for details).

At first sight this seems to require two separate forward passes: one to
compute the real output, and one to compute the surrogate loss. However, this
is not the case, because the computationally intensive element (namely, the
linear convolution by the weight filters) is common to both real and surrogate
output, and thus only needs to be computed once. The only additional
computation is the surrogate loss function, which is generally negligible in
comparison to the convolution itself.

Equations and code samples for plain Hebb, Instar and Oja’s rule are provided
in Appendix A. All code for the experiments described in this paper is
available at https://github.com/ThomasMiconi/HebbianCNNPyTorch.

####  2.1.2 Method 2: Unfolding the input with fixed convolutions

Alternatively, we can eliminate overlapping input patches by separating the
weight convolution into two successive convolutions: a fixed, binary
convolution that rearranges every input patch into a single column vector,
followed by a $1\times 1$ convolution that contains the actual weights. More
precisely, suppose our original convolution has input size $h\times w\times
n_{i}$ (where $h$ and $w$ are the height and width of the convolutional
filter, and $n_{i}$ is the number of channels in the input), with $n_{o}$
output channels. Then, we can first pass the input through a fixed convolution
of input size $h\times w\times n_{i}$ with $hwn_{i}$ output channels, with a
fixed weight vector set to 1 for the weights that links input $x,y,i$ to
output $xyi$ (where $x$, $y$ and $i$ run from 1 to $h$, $w$ and $n_{i}$
respectively) and 0 everywhere else. This rearranges (and duplicates) the
values of each input patch of the original convolution into single, non-
overlapping column vectors. Afterwards we can apply the actual weights of the
original convolution with a simple $1\times 1$ convolution, which can be
performed by a simple tensor product with appropriate broadcasting if
necessary.

This somewhat clunkier method does require two convolutional steps, as well as
additional memory usage. However, it also provides finer-grained control. For
example, it makes it easy to separately normalize and whiten each input patch
of the convolution separately, which is commonly done in computer vision
models (Coates et al., 2011; Olshausen and Field, 1997), but not easy to do
with a normal convolution. In addition, it makes it possible to compute the
Hebbian weight update by hand, since the inputs of each weight application are
neatly separated into non-overlapping vectors. This may be of interest if one
seeks more complex updates that are not easily reducible to a gradient.
Conversely, note that this method and the previous one are not mutually
incompatible: it is possible to use Method 2 to provide individual patch
normalization (and whitening, see below), and then Method 1 to compute the
actual weight updates.

All results reported in this work use Method 1 exclusively. In our
experiments, individual patch normalization did not seem to improve
performance (though individual patch whitening in the first layer did improve
performance somewhat over the whole-image whitening method described below),
and we are not studying any Hebbian rule beyond the ones mentioned above,
making the added complexity of Method 2 superfluous.

###  2.2 Whitening

Natural images exhibit considerable correlations between pixels. It is
generally desirable to remove this correlation in order to make the actual
signal easier to extract (Olshausen and Field, 1997; Hyvärinen and Oja, 2000;
Coates et al., 2011; Bahroun and Soltoggio, 2017). In grayscale images, this
correlation mostly takes the form of spatial autocorrelation, resulting in
much higher energy at low frequencies; as a result, it is usually eliminated
by a simple band-pass filter (in the spatial domain, such filters tend to
resemble the center-surround structure of retinal receptive fields) (Olshausen
and Field, 1997). In color images, however, the correlations are more complex
and involve inter-channel interactions. As a result, it is customary to use a
more complex form of whitening, such as the so-called ZCA method which makes
the covariance matrix of the transformed data equal to the identity matrix
(Bell and Sejnowski, 1997; Krizhevsky and Hinton, 2009; Coates et al., 2011;
Bahroun et al., 2017; Hyvärinen and Oja, 2000). We first normalize each image
vector $\mathbf{x}$ (i.e. subtract the mean and divide by the standard
deviation111In theory, the mean and standard deviations are to be taken over
the dataset. In practice, we found that taking them over each individual image
works well, and indeed better than taking them over the batch dimension.),
then multiply it by the matrix $\mathbf{ED}^{-1/2}\mathbf{E}^{\text{T}}$,
where $\mathbf{E}$ is the matrix of the eigenvectors of the covariance matrix
of the $\mathbf{x}$ vectors and $\mathbf{D}$ the diagonal matrix of its
eigenvalues (to which we add a regularization constant of $10^{-3}$ to avoid
instabilities). Note that the spatial domain visualization of this filter also
has a center-surround structure, though one which depends on the preferred
color of the center and surround neurons (see e.g. Figure 1.3 in (Krizhevsky
and Hinton, 2009)), suggesting a plausible biological implementation. In this
work we first compute the ZCA matrix over the entire training set, then apply
it to each input image during training and testing.

###  2.3 Network architecture

All experiments use the CIFAR10 dataset Krizhevsky and Hinton (2009), which
contains 60000 color images of size $32\times 32$ pixels (50000 for training
and 10000 for testing). Each image is centered and whitened as explained above
before being fed to the network.

Our network uses three convolutional layers, with kernel size $5\times 5$ in
the first layer and $3\times 3$ in the others. Before each layer, we normalize
each element in the batch by subtracting its mean and dividing it by its
standard deviation (over all positions and channels).

Our basic network applies a winner-take-all selection (WTA) to each layer’s
outputs: for each position, we only preserve the channel with the highest
activation and set all other ones to 0 (the winning channel output is set to
1; while not strictly necessary, this binarizing makes learning more stable).
This choice parallels previous work (Fukushima and Miyake, 1982; Masquelier et
al., 2007) and is influenced by Hu et al. (2014)’s proof that Hebbian
plasticity with WTA in the outputs is essentially equivalent to K-means
clustering, which is known to produce highly efficient and biologically
plausible receptive fields (Coates et al., 2011; Hu et al., 2014). Later we
describe different forms of selection that improve performance considerably
over this baseline.

After each layer a $2\times 2$ average pooling and downsampling is applied
(note that plasticity occurs before pooling, so pooling at a layer has no
effect on the learning in the same layer). We use relatively small networks
with 100, 196 and 400 filters at layers 1, 2 and 3 respectively (196 is the
closest perfect square to 200; we prefer square numbers of filters for
presentation purposes). With these settings, receptive fields at layer 1, 2
and 3 have size $5\times 5$, $10\times 10$ and $20\times 20$ pixels,
respectively. The final output of the network has dimensions $400\times
2\times 2$. We train the network over 20 epochs, each containing the whole
training set; we verified that this duration was sufficient for stabilizing
the learned weights.

In addition, we use adaptive thresholds to ensure similar firing rates across
neurons: after computing the total input to each neuron, but before applying
WTA, we add a bias term to each neuron’s activation that is adaptively
modulated over time to ensure the firing rate of the neuron remains close to
$1/N$, where N is the number of neurons in the layer (when using $k$-WTA, the
target rate for each neuron is instead $k$/N). These thresholds are not
strictly necessary, but help prevent the appearance of “dead” cells that never
fire at all (though we note that such cells are in themselves an object of
study (Shoham et al., 2006; Thorpe, 2011)).

We also keep the weight vectors of all neurons normalized to Euclidean norm 1.
While Oja’s rule can theoretically constrain the norms of weight vector to one
under some conditions, in practice we found that it did not fully stabilize
weights in our settings (presumably due to interference from nonlinearities
and adaptive thresholds).

In all experiments, we use Grossberg’s Instar rule (Vasilkoski et al., 2011),
also used in previous work (Hu et al., 2014; Amato et al., 2019). This is
implemented by the surrogate gradient method described above. In practice, we
found that with weights constrained to norm 1, the choice of plain Hebbian,
Instar or Oja’s rule made little difference in the results.

###  2.4 Decoding and evaluation

We assess network performance with a simple linear decoder: we freeze weights,
then feed the entire training set to the network and regress the one-hot label
vectors over the network outputs. We then apply these regression weights to
the network’s response to test set images, and pick the category with the
largest output for each image. This simple regression classifier is orders of
magnitude faster than the linear SVM used in other work (Coates et al., 2011;
Bahroun et al., 2017), at only a minor performance cost.

To evaluate the information contained in each layer below the top one, we use
the “quadrants” method introduced by Coates et al. (2011). We divide the
layer’s output spatially in four equal quadrants, average the value of each
channel across all positions within each quadrants, and concatenate the
results. For layer 1, this results in a vector of size 400 (100 filters times
4 quadrants), to which we apply the linear decoding procedure described above.
We note that the quadrants method is an extremely powerful baseline, reaching
a test accuracy of $43\%$ on CIFAR10 from the output of the first layer of an
_untrained_ network, i.e. with randomly initialized weights (see Table 1).
Note that for the last layer, because the network output has spatial
dimensions $2\times 2$, the quadrants method is identical to simply using the
output of the network as it is.

The receptive fields of layer 1, which take input directly from the RGB
representation of input images, can be visualized by simply using the weights
as image pixels (since weights can be negative, we normalize to the $[0,1]$
range, ensuring that all receptive fields use a common value for weight 0).
The receptive fields of higher levels are reconstructed as the weighted sums
of previous-layer receptive fields at the appropriate location. This
reconstruction is necessarily approximate, due to the pooling operation after
each layer. Figures 1 and 2 show example receptive fields after training (full
sets of learned receptive fields are shown in Figures B.1 and B.2).

##  3 Results

###  3.1 Unconstrained hierarchical Hebbian learning builds simple higher-
level features

Figure 1: Example receptive fields from layers 1 (left), 2 (center) and 3
(right) in a standard Hebbian network with WTA competition. Receptive fields
have size $5\times 5$ (layer 1), $10\times 10$ (layer 2) and $20\times 20$
(layer 3). The full sets of receptive fields for each layer are shown in
Figure B.1.

We observe poor performance on CIFAR10, reaching a test accuracy of $34.5\%\pm
0.4$ (mean and standard deviation over 10 runs). As previously reported
(Bahroun et al., 2017; Bahroun and Soltoggio, 2017; Amato et al., 2019), we
observe that object information actually _decreases_ across successive layers,
despite the increasing number of channels. In particular, the quadrant methods
applied to layers 1 and 2 result in test accuracies of $49.4\%\pm 0.4$ and
$45.3\%\pm 0.6$, respectively (see Table 1 for a summary of results).

Inspection of learned receptive fields provides an explanation for this poor
performance. As expected from previous work (Krizhevsky and Hinton, 2009;
Bahroun et al., 2017), layer 1 receptive fields include a mixture of grayscale
Gabor-like features, color-opponent cells (cyan-red, blue-orange and green-
purple), and flat fields of white, black, red, green and blue (this outcome,
including the specific colors, was extremely reproducible across many
parameter and design choices). However, unlike surmised in traditional models
of the visual system (Serre et al., 2007; Wallis and Rolls, 1997), the higher
layers do not learn increasingly complex, shape-selective combinations of
lower-level features. Instead, they simply learn larger Gabor-like features
and poorly localized blobs. Notably, the layers do not simply perform an
identity mapping, merely transmitting specific individual lower-level channels
(as observed in the single-column case by Olshausen and Field (1997));
instead, they precisely combine lower-level features to form higher-level,
larger oriented edge detectors and blobs. As a result, the information in
higher layers is essentially a less spatially precise version of that provided
by layer 1, making the loss of decoding performance unsurprising.

Varying the number of winners in each layer (that is, using $k$-WTA instead of
WTA, for various values of $k$ ranging from 1 to hundreds) had only minimal
impact on performance, and simply resulted in larger, more blurry receptive
fields.

###  3.2 Dense responses, pruned networks and dispersed weights improve
performance

We introduce two manipulations which, put together, considerably improve
performance and produce very different receptive fields.

  1. 1.

We relax the competition between neurons for the purpose of computing
activations, but _not_ for the purpose of applying plasticity. Activations are
now computed by the “triangle” method introduced by Coates et al. (2011): at
each position, we subtract the mean activation (across all channels at this
position) from all channels, and rectify negative values to 0 (before pooling
and downsampling). This results in less sparse, but still competitive
responses: for each stimulus, on average, about half the cells at any given
column have non-zero response. Importantly, this determines the responses
transmitted to the next layer, but _not_ the plasticity: the output used for
plasticity at each position is still determined by binary WTA, as before (that
is, only the most active neuron in any column undergoes plasticity). This
differentiation between selection for activation and for plasticity
considerably improved performance over alternative choices (e.g. WTA or
$k$-WTA for both responses and plasticity, or triangle method for both).

  2. 2.

We massively prune the connectivity of the network, such that only 1% of all
possible connections can be nonzero. This is done by generating a random
binary mask for each weight vector at the beginning of the experiment,
containing 99% of zeros and 1% of ones, and multiplying the weight vectors by
their mask after every weight modification (we still normalize the pruned
weights to norm 1).

Note that these methods are only applied to layers 2 and above, since layer 1
is supposed to take inputs directly from the “retinal” image input.

These manipulations, put together, greatly improve performance. With these
alterations, the model reaches a test accuracy of $64.55\%\pm 0.5$ (mean and
standard deviation over 10 runs). This is well above the decoding accuracy
from the quadrants method on layer 1 ($58.56\%\pm 0.3$), and similar to the
same from layer 2 ($64.2\pm 0.3$), showing that information now increases or
persists over successive layers (see Table 1 for a summary of results).

Figure 2: Example receptive fields from layers 1 (left), 2 (center) and 3
(right) in a Hebbian network with dense activations and pruned connections
(plasticity still occurs through WTA competition). The full sets of receptive
fields for each layer are shown in Figure B.2.

##  4 Discussion

Table 1: Test accuracy on CIFAR10, for the default network (unpruned weights, WTA competition) and the pruned network with triangle method, before training (i.e. at initialization) and after Hebbian training on 20 epochs over the training dataset. | L1 quadrants | L2 quadrants | Final output
---|---|---|---
Default network, untrained $w$ | $43.3\%\pm 0.7$ | $21.05\%\pm 0.5$ | $10.56\%\pm 0.4$
Default network, trained $w$ | $49.94\%\pm 0.4$ | $45.3\%\pm 0.6$ | $34.5\%\pm 0.4$
Network with triangle method & pruning, untrained $w$ | $52.38\%\pm 0.4$ | $57.17\%\pm 0.3$ | $56.3\%\pm 0.3$
Network with triangle method & pruning, trained $w$ | $58.56\%\pm 0.3$ | $64.2\%\pm 0.4$ | $64.55\%\pm 0.4$

Transparent use of deep learning frameworks for multi-layer Hebbian learning
opens the door to many avenues of exploration. In this work, we showed that
unconstrained Hebbian learning tends to settle on simple, information-poor
features, explaining the paradoxical reduction in information and decoding
performance over successive layers that had been previously reported (Bahroun
et al., 2017; Amato et al., 2019). We countered this with a more permissive
selection in activations (but not for plasticity), and a pruning of synaptic
trees. However, these only scratch the surface of possibilities. Further
possible directions of research include:

  1. 1.

_Dynamic stimuli and temporal learning rules:_ There is a long history of
using the temporal structure of visual stimuli to extract transformation-
invariant representations (Földiák, 1991; Wallis and Rolls, 1997; Masquelier
et al., 2007). In this work we have only considered static stimuli, but it
would be highly desirable to exploit the wealth of temporal information
available in natural stimuli.

  2. 2.

_Decorrelation in higher layers:_ Here we use ZCA decorrelation in the first
layer, and rely on WTA to prevent highly correlated firing. More aggressive
forms of decorrelation in the upper layers might prove beneficial. We
experimented with channel-wise lateral inhibition and online ZCA, but neither
improved performance. More elaborate (or more biologically inspired) scheme
might produce different results.

  3. 3.

_Feedback connections:_ in the visual system, feedback connections carry
information from upper to lower layers. The computational role of these
connections has been debated (Lee and Mumford, 2003; Hupé et al., 1998; Rao
and Ballard, 1999). Recent computational work suggests that appropriately
configured feedback can greatly improve learning (Payeur et al., 2021; Lindsey
and Litwin-Kumar, 2020). How to incorporate feedback connections into large
networks remains an area of active research, which could greatly benefit from
the computational power of deep learning frameworks.

We note that, although our interventions greatly improved the performance of
the network, the learned features (see Figure 2) are difficult to interpret.
They are also quite different from the organized, shape-selective features
posited by hierarchical models of the visual cortex. In these models, lower
layers are expected to respond to simple conjunctions of edges and angles,
while intermediate layers respond to middle-scale components such as
curvature, and the higher layers develop selectivity for individual objects
and faces across multiple poses. This raises the question: how can
biologically plausible hierarchical networks develop selectivities to
specific, increasingly complex stimuli, elaborating their selectivities across
successive layers in a way that matches the increasing specificity of visual
cortex? We suggest that the computational power of modern deep learning
frameworks is likely to prove extremely useful for such investigations.

In another direction, several meta-learning approaches seek to discover new
learning rules for particular domains Bengio et al. (1997); Risi and Stanley
(2010); Najarro and Risi (2020). The method described above could conceivably
be harnessed to search over possible loss expressions (as functions of $x,y,w$
or any other variable accessible to the framework), seeking losses whose
gradients would produce optimal learning, whether in unsupervised, self-
supervised, or supervised learning.

In conclusion, we showed that modern deep learning frameworks are a powerful
tool for exploring Hebbian learning in multi-layer convolutional networks. We
believe these frameworks offer exciting opportunities for further research,
both in terms of performance for unsupervised and self-supervised learning,
and of biological relevance and insight for models of the visual cortex.

## Appendix

##  Appendix A Hebbian updates with gradients

All code for the experiments described above is available at
https://github.com/ThomasMiconi/HebbianCNNPyTorch.

We want to build a surrogate loss function, such that the gradient of this
loss is equal to the desired Hebbian update. To do this, we note that the loss
$\mathcal{L}$ is a function of the network output $y$, and thus its derivative
over the weight $w$ can be written as follows (by the chain rule):

$\frac{\partial\mathcal{L}}{\partial w}=\frac{\partial\mathcal{L}}{\partial
y}\Bigr{|}_{\hat{y}}\frac{\partial y}{\partial w},$ (1)

where $\hat{y}$ is the (“real”) numerical value of the output $y$. We can thus
obtain our desired Hebbian updates as the gradients of the following
expressions for $\mathcal{L}$ and $y$:

$\displaystyle\mathcal{L}=\frac{1}{2}y^{2}$ (2) $\displaystyle
y=\begin{cases}wx&\mbox{(Plain Hebb)}\\\ wx-w^{2}/2&\mbox{(Instar rule)}\\\
wx-\hat{y}w^{2}/2&\mbox{(Oja's rule)}\end{cases}$ (3)

By using Equation 1, it is easily verified that the derivatives of the above
expressions over $w$ are $xy$, $y(x-w)$ and $y(x-\hat{y}w)$, respectively. We
have also verified that the resulting gradients computed by the framework are
equal to hand-computed Hebbian updates in the code.

The following code implements the method in PyTorch:

⬇

optimizer.zero_grad()

# Perform the weight convolution, setting up the first part

# of the computational graph.

prelimy = F.conv2d(x, w)

# We now compute the "real" output realy, with a k-WTA

realy = prelimy.clone().detach() # We don’t want to affect the graph

tk = torch.topk(realy.data, K, dim=1, largest=True)[0]

realy.data[realy.data < tk.data[:,-1,:,:][:, None, :, :]] = 0

realy.data = (realy.data > 0).float() # Binarizing

# We now compute the surrogate output y, used only to set up the

# proper computational graph.

yforgrad = prelimy # Plain Hebb, dw ~= xy

# The following expressions implement the Instar rule (dw ~= y(x-w)) and Oja’s

# rule (dw ~= y(x-wy)), respectively. Note the dimensional rearrangements and

# broadcasting.

# yforgrad = prelimy - 1/2 * torch.sum(w * w, dim=(1,2,3))[None,:, None, None]

# yforgrad = prelimy - 1/2 * torch.sum(w * w, dim=(1,2,3))[None,:, None, None]
* realy.data

# We overwrite the values of yforgrad with the "real" y.

yforgrad.data = realy.data

# Compute the loss and perform the backward pass, which applies the

# desired Hebbian updates.

loss = torch.sum( -1/2 * yforgrad * yforgrad)

loss.backward()

optimizer.step()

##  Appendix B Full sets of learned receptive fields

Figure B.1 shows the full set of learned receptive field in the original,
unconstrained Hebbian network with WTA competition. Figure B.2 shows the full
set of learned receptive field in a Hebbian network with “triangle”
activations and pruned connectivity (with plasticity still determined by WTA
competition).

Figure B.1: Full set of learned receptive fields from layers 1, 2 and 3 in a
standard Hebbian network with WTA competition.

Figure B.2: Full set of learned receptive fields from layers 1, 2 and 3 in a
Hebbian network with dense activations and pruned connections (plasticity
still occurs through WTA competition).

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