# (Causal)-Activation of Complex Entanglement Structures in Quantum Networks

Seid Koudia  Department of Physics “Ettore Pancini”, University of Naples
Federico II, Naples, 80125 Italy.  Angela Sara Cacciapuoti  Quantum Internet
research group @ FLY: Future Communications Laboratory, Department of
Electrical Engineering and Information Technology (DIETI), University of
Naples Federico II, Naples, 80125 Italy. www.quantuminternet.it Marcello
Caleffi  Quantum Internet research group @ FLY: Future Communications
Laboratory, Department of Electrical Engineering and Information Technology
(DIETI), University of Naples Federico II, Naples, 80125 Italy.
www.quantuminternet.it Laboratorio Nazionale di Comunicazioni Multimediali,
CNIT: National Inter-University Consortium for Telecommunications, Naples,
80126 Italy  Corresponding author: Marcello Caleffi
(marcello.caleffi@unina.it).

###### Abstract

Entanglement represents “the” key resource for several applications of quantum
information processing, ranging from quantum communications to distributed
quantum computing. Despite its fundamental importance, deterministic
generation of maximally entangled qubits represents an on-going open problem.
Here, we design a novel generation scheme exhibiting two attractive features,
namely, i) deterministically generating genuinely multipartite entangled
states, ii) without requiring any direct interaction between the qubits.
Indeed, the only necessary condition is the possibility of coherently
controlling – according to the indefinite causal order framework – the causal
order among some unitaries acting on the qubits. Through the paper, we analyze
and derive the conditions on the unitaries for deterministic generation, and
we provide examples for unitaries practical implementation. We conclude the
paper by discussing the scalability of the proposed scheme to higher
dimensional GME states and by introducing some possible applications of the
proposal for quantum networks.

## Introduction

One of the most fundamental concepts within the quantum realm is the notion of
quantum entanglement. It is well established that entangled states – even in
the simplest form of two-qubit entangled states – are essential to enable the
marvels of quantum information processing [1, 2, 3, 4, 5] within the Quantum
Internet. And, as a matter of fact, both the theory of entanglement and its
experimental generation have been a topic of intensive research.
Indeed, several applications of quantum information processing – ranging from
quantum communications through distributed quantum key distribution to
distributed quantum computing – rely on the generation and the remote
distribution of entangled flying qubits [6, 7, 8, 9], with a wide consensus
within the research community on light being the ideal substrate for quantum
information carriers. Nevertheless, given the limitations of current schemes
for photonic entanglement generation, the research is still ongoing. In fact,
some of the available schemes are probabilistic [10, 11, 12], relying as
instance on some form of parametric down conversion. Other schemes require a
tight matter-flying interaction [13, 14]. Clearly, when it comes to multi-
partite entanglement, both the approaches hardly scale to large systems. This
has driven a recent interest in designing all-photonic deterministic sources
of entanglement [15, 16].
In this work, we contribute toward this research direction by resorting to a
recently proposed framework for quantum information processing, namely, the
superposition of causal orders [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27,
28, 29, 25, 30, 31, 32, 33, 34, 35, 36]. Specifically, we design an
entanglement generation scheme where a superposition of causal orders between
local unitaries, acting on qubits in pure product states, deterministically
generates genuinely multipartite entangled (GME) states. Interestingly, the
proposed scheme efficiently scales to higher dimensional GME states, due to
the simplicity and the modularity of the protocol architecture. Furthermore,
the scheme does not require any direct interaction among the input qubits or
between the input qubit and the qubit governing the quantum control of the
causal order between the unitaries. Indeed, the only requirement is the
possibility of coherently controlling the causal order among the unitaries.
It is worthwhile to note that – by exploiting the super-map formalism – the
design of the proposed scheme has been conducted without any specific
assumption on the particulars of the underlying qubit technology. However,
when it comes to practical implementation, we can recognize that the proposed
scheme for deterministic entanglement generation is achievable in near-term
quantum networks, as coherent control of causal orders is affordable by
current technology level and it has been successfully implemented for flying
qubits [27, 28, 35]. In this light, through the paper, we discuss some
possible applications of the proposal for quantum networks. Specifically, we
recognize that the coherent control of the causal order of collective single
qubit unitaries – either locally on particular nodes or globally on some given
cluster of nodes – constitutes a novel paradigm to generate and distribute
multipartite entangled states among remote nodes and in a scalable way.

## Results

### Background and Notation

(a) Graphical representation of the quantum switch placing the unitaries $U$
and $\tilde{U}$ in an even superposition of causal orders. The control qubit,
viewed as a fixed parameter of the communication network between the sender
and the receiver and set to $\ket{+}$, becomes part of the supermap
responsible for the coherent control of the causal orders and it is shown in
the bottom part of the graph. By measuring the control qubit in the coherent
basis $\\{\ket{-},\ket{+}\\}$, one of the two alternative states
$\ket{\psi_{\pm}}$ emerges as output.

(b) Pictorial representation of the output state $\ket{\psi}_{\pm}$, once the
control qubit is measured. The output $\ket{\psi_{\pm}}$ is given by the
coherent superposition of two terms. Within the first one, the unitaries
operate on the input according to the causal order where $\tilde{U}$ is
applied before $U$, whereas within the second one the unitaries operates
following the alternative order where $U$ is applied before $\tilde{U}$.

Figure 1: Quantum Switch implementing an even superposition of the two
alternative causal orders between two unitaries $U$ and $\tilde{U}$ operating
on the input state $\ket{\varphi}$.

With a series of recent works, researchers have shown that quantum placement
of quantum channels – namely, placing quantum channels in a coherent
superposition of alternative configurations – can provides significant
advantages for a number of problems, ranging from quantum computation [37, 26,
38] and quantum information processing [39, 40] through non-local games [41]
to communication complexity [42, 43, 44, 45, 46, 47, 48]. Instances of this
quantum placement range from superposition of alternative quantum channels
traversed by the information carrier to superposition of alternative causal
orders between the quantum channels. With reference to the superposition of
causal orders between quantum channels, the placement is realized through an
higher-order map known as quantum switch [26].
Mathematically, the quantum switch is described by a supermap $\mathcal{S}$
taking two channels $\mathcal{U}(\cdot)$ and $\tilde{\mathcal{U}}(\cdot)$ as
inputs, and giving as output a channel resulting from the combination of
$\mathcal{U}(\cdot)$ and $\tilde{\mathcal{U}}(\cdot)$ in a superposition of
causal orders, controlled by a quantum degree of freedom $\ket{\varphi_{c}}$.
Its action on quantum states is defined by the Kraus operators[26, 17]
$S_{ij}=U_{i}\tilde{U}_{j}\otimes|0\rangle\langle
0|_{c}+\tilde{U}_{j}U_{i}\otimes|1\rangle\langle 1|_{c}$, where $\\{U_{i}\\}$
and $\\{\tilde{U}_{j}\\}$ denote the Kraus operators of the primitive channels
$\mathcal{U}(\cdot)$ and $\tilde{\mathcal{U}}(\cdot)$, and
$\\{\ket{0}_{c},\ket{1}_{c}\\}$ denotes the orthogonal states of the control
system. Accordingly, the resultant channel implemented by the quantum switch
is given by:

$\mathcal{S}_{\rho_{c}}(\mathcal{U},\tilde{\mathcal{U}})(\rho)=\sum_{ij}S_{ij}(\rho\otimes\rho_{c})S_{ij}^{\dagger},$
(1)

with $\rho$ and $\rho_{c}$ denoting the density matrix of the input and the
control, respectively.
In the following, giving that we aim at generating maximally entangled states,
we focus on pure input states and unitary channels. Furthermore, we set the
control $\ket{\varphi_{c}}$ to $\ket{+}$, i.e., we place the primitive
channels in an even superposition of causal orders, accordingly to [44, 35,
26, 19, 20, 21]. Accordingly, the quantum switch supermap in (1) exhibits a
single unitary Kraus operator $S=U\tilde{U}\otimes|0\rangle\langle
0|_{c}+\tilde{U}U\otimes|1\rangle\langle 1|_{c}$, with $U$ and $\tilde{U}$
denoting the (single) Kraus operators of the primitive channels, leading to
the overall operation on the input state $\ket{\varphi}$ given by111The
notation has been simplified with respect to the one in (1) to highlight the
focus on pure input states and unitary channels.:

$S(\ket{\varphi}\otimes\ket{\varphi_{c}})=\frac{1}{2}\big{(}U\tilde{U}+\tilde{U}U\big{)}\ket{\varphi}\otimes\ket{+}_{c}+\frac{1}{2}\big{(}U\tilde{U}-\tilde{U}U\big{)}\ket{\varphi}\otimes\ket{-}_{c}$
(2)

After performing a measurement on the control qubit in the coherent basis, the
following outcome states – highlighting the superposition of causal orders
between the unitaries – emerge:

$\ket{\psi_{\pm}}=\frac{1}{\sqrt{L}_{\pm}}\big{(}U\tilde{U}\pm\tilde{U}U\big{)}\ket{\varphi}=\frac{1}{\sqrt{L}_{\pm}}\big{(}\overrightarrow{U}\pm\overleftarrow{U}\big{)}\ket{\varphi}$
(3)

where $L_{\pm}$ is a normalization constant, depending on both the unitaries
$U,\tilde{U}$ and on the postselected state. In (3), we introduced
$\overrightarrow{U},\overleftarrow{U}$ as a shorthand notation, with
$\overrightarrow{U}\stackrel{{\scriptstyle\triangle}}{{=}}U\tilde{U}$ denoting
the order where $\tilde{U}$ is applied before $U$ and
$\overleftarrow{U}\stackrel{{\scriptstyle\triangle}}{{=}}\tilde{U}U$ denoting
the alternative order. This is schematized in Fig. 1 where we omitted – as
extensively done whenever possible through the rest of the paper – the
normalization constant for the sake of simplicity. Indeed, from Fig. 1, it is
intuitive to grasp that, once the control qubit is measured, the output is a
coherent superposition of two contributes, where in each contribute the
unitaries process the input according to one of the two alternative causal
orders, namely, either $\overleftarrow{U}$ or $\overrightarrow{U}$.

### Bell states generation

(a) Graphical representation of the quantum switch generating a Bell state
through an even superposition of the causal order between the 2-qubit local
unitaries $V^{(2)}\stackrel{{\scriptstyle\triangle}}{{=}}U_{0}\otimes U_{1}$
and
$\tilde{V}^{(2)}\stackrel{{\scriptstyle\triangle}}{{=}}\tilde{U_{0}}\otimes\tilde{U_{1}}$.
By measuring the control qubit in the coherent basis, one of the two
alternative states $\ket{\psi^{(2)}_{\pm}}$ emerges as output.

(b) Pictorial representation of the outcome state $\ket{\psi^{(2)}_{\pm}}$,
once the control qubit is measured. The output is given by the coherent
superposition of two terms. Within the first one, the unitaries operate on the
input according to the causal order where $\tilde{U_{0}}$ and $\tilde{U_{1}}$
are applied before $U_{0}$ and $U_{1}$, whereas within the second one the
unitaries operates following the alternative order where $U_{0}$ and $U_{1}$
are applied before $\tilde{U_{0}}$ and $\tilde{U_{1}}$.

Figure 2: Bell state through an even superposition of two alternative causal
orders between two 2-qubit local unitaries $V^{(2)}$ and $\tilde{V}^{(2)}$
operating on a separable input state
$\ket{\varphi_{0}}\otimes\ket{\varphi_{1}}$.

(a) Concurrence $C(\rho_{+})$ of the state $\ket{\psi^{(2)}_{+}}$ given in
(4), obtained as output when the control qubit is measured as $\ket{+}$ with
$\rho_{+}\stackrel{{\scriptstyle\triangle}}{{=}}\ket{\psi^{(2)}_{+}}\bra{\psi^{(2)}_{+}}$.

(b) Concurrence $C(\rho_{-})$ of the state $\ket{\psi^{(2)}_{-}}$ given in
(4), obtained as output when the control qubit is measured as $\ket{-}$ with
$\rho_{-}\stackrel{{\scriptstyle\triangle}}{{=}}\ket{\psi^{(2)}_{-}}\bra{\psi^{(2)}_{-}}$.

Figure 3: 3D plot for the concurrence of the output states
$\ket{\psi^{(2)}_{\pm}}$ given in (4) as function of: i) the y-rotation
parameter $\lambda$ controlling the unitaries $\tilde{U_{0}}$ and
$\tilde{U_{1}}$, and ii) the superposition parameter $\alpha$ controlling the
input state $\ket{\varphi_{0}\varphi_{1}}$, assumed real. Unitaries $U_{0}$
and $U_{1}$ both set to a Pauli-z gate. Maximally entangled states exhibit
unitary concurrence.

Let us now consider two local unitary operators $V^{(2)}=U_{0}\otimes U_{1}$
and $\tilde{V}^{(2)}=\tilde{U}_{0}\otimes\tilde{U}_{1}$ and a 2-qubit input
system in the separable state $\ket{\varphi_{0}}\otimes\ket{\varphi_{1}}$.
Being the input in a product state and given the assumption of local
unitaries, the resulting outcome, for any causal order between the local
unitaries such as $V^{(2)}\tilde{V}^{(2)}$ or $\tilde{V}^{(2)}V^{(2)}$, will
be a product state as well. Furthermore, no entanglement can be distilled from
such a state in the asymptotic limit with two-way LOCC assistance [49].
Conversely, if we process the separable input through an even superposition of
causal orders between the two unitaries – as shown in Fig. 2 – by measuring
the control in the coherent basis, the following output emerges:

$\ket{\psi^{(2)}_{\pm}}=\frac{1}{\sqrt{L}_{\pm}}\big{(}V^{(2)}\tilde{V}^{(2)}\pm\tilde{V}^{(2)}V^{(2)}\big{)}\ket{\varphi_{0}\varphi_{1}}$
(4)

From (4) we note that, once the control qubit is measured, the output is a
coherent superposition of two contributes, where in each contribute the
unitaries process the separable input according to one of the two alternative
– i.e., either $V^{(2)}\tilde{V}^{(2)}$ or $\tilde{V}^{(2)}V^{(2)}$ – causal
orders. This is schematized in Figure 2.
Now, the main question arises: is there any entanglement within the quantum
state $\ket{\psi^{(2)}_{\pm}}$ emerging out of the controlled superposition of
causal orders? The answer to this question is definitely yes. Indeed, the
output state is maximally entangled if and only if the following condition on
the local unitaries $\\{U_{i}\,,\tilde{U}_{i}\\}_{i=0,1}$ holds (see Theorem 1
in Methods):

$\bra{\varphi_{0}}\overleftarrow{U_{0}}^{\dagger}\overrightarrow{U_{0}}\ket{\varphi_{0}}=0\quad\wedge\quad\bra{\varphi_{1}}\overleftarrow{U_{1}}^{\dagger}\overrightarrow{U_{1}}\ket{\varphi_{1}}=0$
(5)

with $\wedge$ denoting the Boolean operator AND, and with
$\overleftarrow{\cdot},\overrightarrow{\cdot}$ being the introduced shorthand
notations for the alternative causal orders among the unitaries, i.e.,
$\overrightarrow{U_{i}}\stackrel{{\scriptstyle\triangle}}{{=}}U_{i}\tilde{U_{i}}$
and
$\overleftarrow{U_{i}}\stackrel{{\scriptstyle\triangle}}{{=}}\tilde{U_{i}}U_{i}$
for $i\in\\{0,1\\}$.
Stemming from this result, we derive a lighter condition assuring the
separability of the output state (see Proposition 1 in Methods). Specifically,
if there exists at least one $i\in\\{0,1\\}$ so that
$\bra{\varphi_{i}}\overleftarrow{U_{i}}^{\dagger}\overrightarrow{U_{i}}\ket{\varphi_{i}}=1$,
the output state in (4) is separable.
Clearly, one could wonder which are the requirements in terms of unitaries and
input state so that condition (5) can be satisfied. Namely, how “easily”
entanglement can be obtained out of a superposition of causal orders.
To address this crucial aspect, we note that the sufficient and necessary
condition in (5) consists of two separate constraints, with the former
constraint operating only on $U_{0},\tilde{U_{0}}$ and $\ket{\varphi_{0}}$
whereas the latter one depends only on $U_{1},\tilde{U_{1}}$ and
$\ket{\varphi_{1}}$. This separability feature allows us to design unitaries
$U_{0}$ and $\tilde{U_{0}}$ independently from $U_{1},\tilde{U_{1}}$.
Furthermore, the condition in (5) can be satisfied with practical unitaries,
as shown in the following.
To this aim, let us assume as input state
$\ket{\varphi_{0}\varphi_{1}}=\ket{\eta\eta}$, with
$\ket{\eta}=\sqrt{\alpha}\ket{0}+\sqrt{1-\alpha}\ket{1}$ being an arbitrary
superposition of basis states. Furthermore, let us assume both $U_{0}$ and
$U_{1}$ representing the popular Pauli-z gate, i.e., $U_{i}=\sigma_{z}$.
Finally, let us assume both $\tilde{U_{0}}$ and $\tilde{U_{1}}$ being the
y-rotation gate $R_{y}(2\lambda)=e^{-i\sigma_{y}\lambda}$, with $\sigma_{y}$
denoting the Pauli-y gate. Let’s now consider the two possible events, namely,
control qubit measured either as $\ket{+}$ or as $\ket{-}$. In the former
case, the condition for entangled output state given in (4) translates to
$\lambda\neq 0,\frac{\pi}{2}$. In fact, only when the unitary parameter
$\lambda$ is either equal to $0$ or $\frac{\pi}{2}$, the output state
generated by the quantum switch shown in Figure 2 is separable. Furthermore,
the condition for maximally entangled output given in (5) is translated to
$\lambda=\frac{\pi}{4}$. This is shown in Figure 3 by plotting the concurrence
(see Methods) of the output state $\ket{\psi^{(2)}_{+}}$ as a function of the
parameters. As regards to the latter case – namely, whenever the control qubit
is measured as $\ket{-}$ – the output is either separable or maximally
entangled. And the state is separable only if $\lambda$ is either equal to $0$
or $\frac{\pi}{2}$, whereas it is maximally entangled for any $\lambda$ in
$(0,\frac{\pi}{2})$, as shown in Figure 3.
We have shown that the scheme can be implemented through a coherent control of
straightforward unitaries: the Pauli-z gate and the y-rotation gate
$R_{y}(\frac{\pi}{2})$.

### GHZ-like states generation

(a) Extension of the quantum switch represented in Figure 2 for generating a
GHZ-like state through an even superposition of the causal order between the
3-qubit local unitaries
$V^{(3)}\stackrel{{\scriptstyle\triangle}}{{=}}U_{0}\otimes U_{1}\otimes
U_{2}$ and
$\tilde{V}^{(3)}\stackrel{{\scriptstyle\triangle}}{{=}}\tilde{U_{0}}\otimes\tilde{U_{1}}\otimes\tilde{U_{2}}$.

(b) Pictorial representation of the output state $\ket{\psi^{(3)}_{\pm}}$,
once the control qubit is measured. Similarly to Figure 2, the output is given
by the coherent superposition of two terms, differing for the order between
the unitaries.

Figure 4: GHZ-like state through an even superposition of two alternative
causal orders between two 3-qubit local unitaries $V^{(3)}$ and
$\tilde{V}^{(3)}$ operating on a separable input state
$\ket{\varphi_{0}}\otimes\ket{\varphi_{1}}\otimes\ket{\varphi_{2}}$.

(a) GME concurrence $C_{\text{GME}}(\rho_{+})$ of the state
$\ket{\psi^{(3)}_{+}}$ given in (6), obtained as output when the control qubit
is measured as $\ket{+}$ with
$\rho_{+}\stackrel{{\scriptstyle\triangle}}{{=}}\ket{\psi^{(3)}_{+}}\bra{\psi^{(3)}_{+}}$.

(b) GME concurrence $C_{\text{GME}}(\rho_{-})$ of the state
$\ket{\psi^{(3)}_{-}}$ given in (6), obtained as output when the control qubit
is measured as $\ket{-}$ with
$\rho_{-}\stackrel{{\scriptstyle\triangle}}{{=}}\ket{\psi^{(3)}_{-}}\bra{\psi^{(3)}_{-}}$.

Figure 5: 3D plot for the GME concurrence of the output state
$\ket{\psi^{(3)}_{\pm}}$ given in (6) as function of: i) the y-rotation
parameter $\lambda$ controlling the unitaries $\\{\tilde{U_{i}}\\}$, and ii)
the superposition parameter $\alpha$ controlling the input state
$\ket{\varphi_{0}\varphi_{1}\varphi_{2}}$, assumed real. Unitaries
$\\{U_{i}\\}$ set to Pauli-z gate. GHZ-like states exhibit unitary GME
concurrence.

Similarly to the bipartite case, we consider an even superposition of the two
alternative causal orders between two 3-qubit local unitaries
$V^{(3)}=U_{0}\otimes U_{1}\otimes U_{2}$ and
$\tilde{V}^{(3)}=\tilde{U}_{0}\otimes\tilde{U_{1}}\otimes\tilde{U_{2}}$ acting
on an initially pure product tripartite state
$\ket{\varphi_{0}\varphi_{1}\varphi_{2}}$, as shown in Figure 4. By measuring
the control qubit in the coherent basis and according to eq. (3), we obtain
the following output state:

$\ket{\psi^{(3)}_{\pm}}=\frac{1}{\sqrt{L}_{\pm}}\big{(}V^{(3)}\tilde{V}^{(3)}\pm\tilde{V}^{(3)}V^{(3)}\big{)}\ket{\varphi_{0}\varphi_{1}\varphi_{2}}$
(6)

Indeed, as in the bipartite case, the output is a superposition of two
different input processing, with the two processing differing for the causal
order between the unitaries. This similarity maps as well into the necessary
and sufficient condition for the output in (6) being a GHZ-like state, which
is given by (see Theorem 2 in Methods):

$\bra{\varphi_{i}}\overleftarrow{U_{i}}^{\dagger}\overrightarrow{U_{i}}\ket{\varphi_{i}}=0\quad\forall\,i=0,1,2$
(7)

and $\overleftarrow{\cdot},\overrightarrow{\cdot}$ being the usual shorthand
notations for the alternative causal orders among the unitaries. Hence, the
output in (6) is a legitimate GHZ-like state if and only if
$\overleftarrow{U_{i}}\ket{\varphi_{i}}$ is orthonormal to
$\overrightarrow{U_{i}}\ket{\varphi_{i}}$.
Indeed, there exists a lighter condition (see Proposition 2 in Methods
assuring the separability of the output state (6), given by:

$\exists\,i\in\\{0,1,2\\}\;:\;\overleftarrow{U_{i}}\ket{\varphi_{i}}=\overrightarrow{U_{i}}\ket{\varphi_{i}}$
(8)

It is crucial to note that the scheme shown in Figure 4 straightforwardly
extends to $n$-partite GHZ-like states by considering two $n$-qubit local
unitaries $V^{(n)}$ and $\tilde{V}^{(n)}$ acting on a $n$-partite separable
state. In such a case, the output is a legitimate GHZ-like state as long as
(see Remark following Theorem 2 in Methods):

$\bra{\varphi_{i}}\overleftarrow{U_{i}}^{\dagger}\overrightarrow{U_{i}}\ket{\varphi_{i}}=0\quad\forall\,i=0,\ldots,n-1$
(9)

Clearly, the higher is the dimension of the GHZ-like state to be generated,
the higher is the number of constraints in (9) that must be simultaneously
satisfied. However, this is not an issue, given that the set of constraints
are separable, namely, the design of the $i$-th unitaries
$U_{i},\tilde{U_{i}}$ depends only from the $i$-th input $\ket{\varphi_{i}}$
and it is completely independent from the other inputs as well as the other
unitaries. Indeed, as long as all the separable input qubits are all set to
the same state $\ket{\eta}$ (which is reasonable), the condition for
deterministically generating GHZ states222It worthwhile to note that this
consideration holds also for Bell and W-like states generation. reduces to a
single constraint regardless of the dimension of the state to be generated.
Namely, the unitaries acting on the different qubits can be the same. This key
feature makes the protocol highly scalable.
This pivotal separability feature of the necessary and sufficient conditions
derived in (9) (as well as in (7)) allows us to easily address the issue of
designing unitaries and input state for generating a GHZ-like state. Indeed,
similarly to the bipartite case, by assuming $\ket{\eta\eta\eta}$ as input
state, with $\ket{\eta}=\sqrt{\alpha}\ket{0}+\sqrt{1-\alpha}\ket{1}$, as well
as $U_{i}=\sigma_{z}$ and $\tilde{U_{i}}=R_{y}(2\lambda)$ for any $i$, we have
that the condition for GHZ-like output state given in (7) translates to
$\lambda=\frac{\pi}{4}$. This is shown in Figure 5 by plotting the GME
concurrence (see Methods) of the output state $\ket{\psi^{(3)}_{\pm}}$ as a
function of the parameters, where we can appreciate that the GME concurrence
vanishes for $\lambda=0$, and reaches its maximum when
$\lambda=\frac{\pi}{4}$.

### W-like states generation

(a) Scheme for generating a W-like state through a superposition of the causal
order between 1-qubit local unitaries $U_{i},\tilde{U_{i}}$ acting on the
$i$-th qubit $\ket{\varphi_{i}}$.

(b) Pictorial representation of the output state $\ket{\psi^{(3)}_{\pm\pm}}$,
once the control qubit is measured. The output is given by the coherent
superposition of three terms, differing for the order between the unitaries
acting on each qubit.

Figure 6: W-like state through a superposition of two alternative causal
orders between two 1-qubit local unitaries $U_{i}$ and $\tilde{U_{i}}$
operating on the $i$-th qubit of a separable input state
$\ket{\varphi_{0}}\otimes\ket{\varphi_{1}}\otimes\ket{\varphi_{2}}$.

(a) GME concurrence $C_{GME}(\rho_{++})$ of the state $\ket{\psi^{(3)}_{++}}$
given in (11), obtained as output when the control qubit is measured as
$\ket{++}$ with
$\rho_{++}\stackrel{{\scriptstyle\triangle}}{{=}}\ket{\psi^{(3)}_{++}}\bra{\psi^{(3)}_{++}}$.

(b) GME concurrence $C_{GME}$ for the states $\ket{\psi^{(3)}_{+-}}$,
$\ket{\psi^{(3)}_{--}}$ and $\ket{\psi^{(3)}_{-+}}$ given in (11).

Figure 7: 3D plot for the GME concurrence of the output state
$\ket{\psi^{(3)}_{\pm\pm}}$ given in (11) as function of: i) the y-rotation
parameter $\lambda$ controlling the unitaries $\\{\tilde{U_{i}}\\}$, and ii)
the superpostion parameter $\alpha$ controlling the input state
$\ket{\varphi_{0}\varphi_{1}\varphi_{2}}$, assumed real. Unitaries
$\\{U_{i}\\}$ set to Pauli-z gate. W-like states exhibit GME concurrence equal
to $0.9$.

To generate W-like states, a coherent control of two alternative evolutions –
as previously done for both Bell and GHZ-like states – is not sufficient due
to distinguishing peculiarities of W-like states.
Basically, the (minimum, for a proper basis choice) number of superpositions
grows linearly with the number of parties in the state, in contrast to GHZ
states where the (minimum) number of superpositions is two regardless of
parties. This is expected, as the W-like states are from a nonequivalent class
with respect to GHZ-like states, also by means of stochastic LOCC. For this,
to generate W-like states from a coherent superposition of causal orders we
need to adjust our coherent control strategy.
Let us consider the scheme shown in Figure 6, where the $i$-th individual
qubit of an initially pure product $3$-partite state
$\ket{\varphi_{0}\varphi_{1}\varphi_{2}}$ evolves through a superposition of
two alternative causal orders between local unitaries $U_{i}$ and
$\tilde{U_{i}}$ controlled by a degree of freedom $\ket{\varphi_{c}}$ so that
the quantum switch supermap exhibits the unitary Kraus operator
$S=\tilde{U_{0}}U_{0}\otimes U_{1}\tilde{U_{1}}\otimes
U_{2}\tilde{U_{2}}\otimes|0\rangle\langle
0|_{c}+U_{0}\tilde{U_{0}}\otimes\tilde{U_{1}}U_{1}\otimes
U_{2}\tilde{U_{2}}\otimes|1\rangle\langle 1|_{c}+U_{0}\tilde{U_{0}}\otimes
U_{1}\tilde{U_{1}}\otimes\tilde{U_{2}}U_{2}\otimes|2\rangle\langle 2|_{c}$.
Accordingly, the overall unitary operation on the input state is given by:

$\displaystyle S$
$\displaystyle(\ket{\varphi_{0}\varphi_{1}\varphi_{2}}\otimes\ket{\varphi_{c}})=\frac{1}{\sqrt{3}}\left[\left(\tilde{U_{0}}\otimes
U_{1}\otimes U_{2}\cdot
U_{0}\otimes\tilde{U_{1}}\otimes\tilde{U_{2}}\right)\ket{\varphi_{0}\varphi_{1}\varphi_{2}}\otimes\ket{0}_{c}+\right.$
$\displaystyle\left.+\left(U_{0}\otimes\tilde{U_{1}}\otimes
U_{2}\cdot\tilde{U_{0}}\otimes
U_{1}\otimes\tilde{U_{2}}\right)\ket{\varphi_{0}\varphi_{1}\varphi_{1}}\otimes\ket{1}_{c}+\left(U_{0}\otimes
U_{1}\otimes\tilde{U_{2}}\cdot\tilde{U_{0}}\otimes\tilde{U_{1}}\otimes
U_{2}\right)\ket{\varphi_{1}\varphi_{1}\varphi_{2}}\otimes\ket{2}_{c}\right]$
(10)

Upon measuring the control degrees of freedom according to an appropriate
measurement setup (see Remark following Theorem 3 in Methods), the following
state emerges:

$\ket{\psi^{(3)}_{\pm\pm}}=\frac{1}{\sqrt{L_{\pm\pm}}}\left(\overleftarrow{U_{0}}\otimes\overrightarrow{U_{1}}\otimes\overrightarrow{U_{2}}\pm\overrightarrow{U_{0}}\otimes\overleftarrow{U_{1}}\otimes\overrightarrow{U_{2}}\pm\overrightarrow{U_{0}}\otimes\overrightarrow{U_{1}}\otimes\overleftarrow{U_{2}}\right)\ket{\varphi_{0}\varphi_{1}\varphi_{2}}$
(11)

with $L_{\pm\pm}$ being the appropriate normalization constant, with
$\overleftarrow{\cdot},\overrightarrow{\cdot}$ being the usual shorthand
notations for the alternative causal orders among the unitaries, and with
$\pm$ being equal to $+$ or $-$ depending on the measurement output of the
control qubit. Regardless of the particular expression of
$\ket{\psi^{(3)}_{\pm\pm}}$, the state is a superposition of three different
input processing, differing for the causal order between the unitaries acting
on each qubit. And, regardless of the particular expression, the necessary and
sufficient condition for the output in (11) being a W-like state is (see
Theorem 3 in Methods):

$\bra{\varphi_{i}}\overleftarrow{U_{i}}^{\dagger}\overrightarrow{U_{i}}\ket{\varphi_{i}}=0\quad\forall\,i=0,1,2$
(12)

Furthermore, similarly to the GHZ-like state, there exists a lighter condition
for the separability of the output state in (11), given by (see Proposition 3
in Methods)

$\exists\,i\in\\{0,1,2\\}:\overleftarrow{U_{i}}\ket{\varphi}=\overrightarrow{U_{i}}\ket{\varphi}$
(13)

It is worthwhile to note that the scheme in Figure 6 straightforwardly extends
to $n$-partite W-like states by simply extending condition (12) to any
$i=0,\ldots,n-1$, by reasoning as highlighted in the Remark following Theorem
2 in Methods. Furthermore, the same considerations in terms of unitaries
design made for the GHZ-like states continue to hold. This is confirmed by
assuming – as done for the GHZ-like states – as input state
$\ket{\eta\eta\eta}$, with
$\ket{\eta}=\sqrt{\alpha}\ket{0}+\sqrt{1-\alpha}\ket{1}$, as well as
$U_{i}=\sigma_{z}$ and $\tilde{U_{i}}=R_{y}(2\lambda)$ for any $i$. With this
setting, the necessary and sufficient condition for the output being a W-like
state translates to $\lambda=\frac{\pi}{4}$. This can be clearly seen from the
visualization of the GME concurrence of the states in (11) given in Fig. 7,
where the GME concurrence for all states reaches the maximum at the critical
value of $\lambda=\frac{\pi}{4}$, whereas it vanishes for
$\lambda=0,\frac{\pi}{2}$ when the output state becomes separable.

## Discussion

In the previous section, we have shown that the generation of GME states
belonging to different non-equivalent classes of states is deterministically
achievable through a proper superposition of causal order between local
unitaries. We discuss now the possible applications of the proposed scheme
under two complementary perspectives, namely, entanglement generation and
entanglement distribution, for the Quantum Internet [1, 2, 3].

### Local Entanglement Generation

We first consider the case where the entanglement generation is local rather
than distributed. Namely, the local unitaries $U_{i},\tilde{U}_{i}$ as well as
the controlling degree of freedom $\ket{\varphi_{c}}$ are all located within
the same quantum node, which locally implements the proper supermap – such as
(6) illustrated in Figure 4 or (10) illustrated in Figure 6 – for generating
GME states.
In this scenario, the entangled states are thus deterministically generated in
a (some) network node – acting as entanglement generator – and they are
subsequently distributed within the network through proper quantum
communication links.
Hence, the network node acting as entanglement generator implements a coherent
control strategy on the local unitaries processing some initial product state
– with both the unitaries and the input state considered as free resources –
for generating the GME states. Clearly, as discussed within the paper, the
coherent control strategy depends on the desired GME output state. Although it
is still technologically unclear whether a node can dynamically change the
coherent control strategy for generating entangled states belonging to
different non-equivalent classes, the proposed scheme for deterministic
entanglement generation is achievable in near-term quantum networks, as the
coherent control of orders of operations is affordable by current technology
level and it has been successfully implemented for single-qubit channels [28,
27, 18, 35, 36].
Regardless the control strategy being dynamic or fixed a-priori, three are the
crucial properties of the proposed scheme for deterministic GME state
generation. i) First, the individual input qubits don’t interact each others
or with the control in any way. Indeed, they only traverse their respective
local unitaries $U_{,}\tilde{U_{i}}$ in a coherent superposition of two
alternative causal orders. ii) Second, the (sufficient and necessary)
condition for deterministically generating GHZ- and W-like states consists of
separable constraints. Namely, the design of the $i$-th unitaries is
completely independent from the other unitaries as well as any input qubit
different from $\ket{\varphi_{i}}$. iii) Third, as long as all the separable
input qubits are all set to the same state $\ket{\eta}$ (which is reasonable),
the condition for deterministically generating GHZ- and W-like states reduces
to a single constraint regardless of the dimension of the state to be
generated. Namely, the unitaries acting on the different qubits can be the
same. These crucial features make the protocol highly scalable.

### Distributed Entanglement Generation

(a) Scheme for distributing multipartite entanglement states in a quantum
network. The network is composed by nine clients, organized in three sets.
each served by an intermediate entangler whose control qubit is controlled by
a entanglement coordinator.

(b) A magnification of the intermediate entangler $e_{i}$, which exploits a
superposition of causal orders to distribute multi-partite entanglement to
remote nodes.

Figure 8: Distributed multi-partite generation.

When it comes to multipartite entanglement distribution in a quantum network,
several issues arises. First, as the number of parties to be entangled
increases, the number of required multi-qubit gates increases as well. This
not only implies severe error propagation effects, but it also hardly scales –
as instance, in W-like states – with the number of parties. Furthermore,
regardless of the number of parties, whenever the size of the quantum network
grows to moderate- or large-scale [50], direct entanglement distribution is
not feasible anymore due to photon noise and losses. In this context, quantum
repeaters[51, 3, 52] are commonly accepted as the strategy for increasing the
entanglement distribution range. Unfortunately, regardless of the repeater
particulars that roughly depend on the repeater generation[53], quantum
repeaters require some sort of Bell state measurements for Bell pairs
distribution or other projective measurements for multipartite entangled
states, which are usually hard to implement and very noisy in practice.
Interestingly, our scheme for entanglement generation could provide an
alternative strategy for overcoming such issues in distributing multi-partite
entanglement, without the need of multi-qubits gates or any other interaction
among the qubits. Let us better clarify this with an example. Specifically, as
shown in Figure 8, multiple quantum switches – referred to as edge entanglers
– are geographically distributed through the network, so that each switch is
closely located (from the entanglement distribution perspective) to a certain
group of nodes. Each edge entangler implements the proper supermap – such as
(6) illustrated in Figure 4 for GHZ-like states – for generating GME states.
But each edge entangler uses, as control degree of freedom, the output of
another quantum switch – which acts as entangler coordinator – in order to
collectively generate the desired multi-partite entangled state. Clearly, each
edge entangler generates the required $k$-partite GHZ-like state according to
the number $k$ of nodes that are physically linked to it. As instance, in
Figure 8, $e_{1}$, $e_{2}$ and $e_{3}$ generate a tri-partite entangled state
by relying on the coherent control of the causal order between unitaries
$V^{3}$ and $\tilde{V}^{3}$ as illustrated in Figure 8, where the coherent
control is provided as a $3$-GHZ state generated by the entanglement
coordinator $e_{0}$. The overall state distributed through the network is a
valid $\ket{GHZ}_{9}$ state. It is worthwhile to note that the proposed scheme
can scale to large networks through a proper hierarchical multi-tier
architecture, where additional intermediate entanglers are deployed between
the coordinator and the edge entanglers.
It is important to note that, in the above example, we have only discussed the
distributed multipartite entanglement generation when the coherent control of
remotely located edge entanglers is obtained through a proper multi-partite
entangled state, shared between the edge entanglers. However, the scheme in
Figure 8 requires only the availability of a coherent control of the unitaries
among remotely located edge entanglers, regardless of the specific
implementation of such a control. And such a coherent control is considered
the genuine quantum feature of a quantum network[54], where a genuine quantum
coherence is an intrinsic property of the communication network.

### Entanglement mapping

Figure 9: Scheme for entanglement mapping between different quantum degrees of
freedom. By implementing a coherent control of three different quantum
switches through a proper 3-partite GME state and by measuring each control
qubit in the coherent basis, an initially pure product tripartite state
$\ket{\varphi_{0}\varphi_{1}\varphi_{2}}$ is deterministically transformed
into a 3-partite GME state.

Here we consider a scenario where entanglement – rather then generated – must
be mapped between different quantum degrees of freedom. As an example, let us
consider deterministic generation of photonic GME states, which may benefits
from a matter-photonic interface with a quantum degree of freedom – such as
superconducting-circuit based qubits [13] – where entanglement can be
generated easier than in photonic-circuits. Another example is represented by
matter-flying interfaces per-se, which represent a critical component for
quantum networks [1, 55, 56, 57], where matter qubits for information
processing/storing – based on heterogeneous technologies ranging from
transmons through quantum dots to ion traps – must be interfaced to flying
qubits – generally implemented with photons – acting as information carriers.
Regardless of the specific applications for an entanglement mapper, the
proposed scheme based on superposition of causal orders provides an
interesting approach toward deterministic entanglement mapping, worthwhile of
further investigation. As an example, let us consider the scheme shown in
Figure 9. The initially-entangled quantum degree of freedom, say333Clearly,
the proposed scheme applies to the deterministic mapping of $W$-like states as
well. in a GHZ state, is used to implement a coherent control among different
quantum switches. Each quantum switch implements the superposition of causal
orders between two unitaries given in (3) and shown in Figure 1, by acting on
the individual qubit $\ket{\varphi_{i}}$ of a second quantum degree of
freedom, initially in a separable state. As long as the condition for
deterministic generation of GHZ-like state in (7) is satisfied, the GHZ state
is deterministically mapped from the control degree of freedom to the
initially separable second degree of freedom, which becomes maximally
entangled. In a nutshell, the scheme harnesses the quantum correlation
embedded within the control degree of freedom to generate – through multiple
switch instances – a coherent evolution of the input degree of freedom, which
eventually exhibits a correlation of the same nature (i.e., class) of the
original entanglement.
It is worthwhile to note that the proposed mapping scheme does not require any
interaction between the input qubits or between the input qubit and the
control qubit, which represents a key feature whenever the input qubits weakly
interact each others or with the environment, as in the mentioned case of
photonic qubits.

## Methods

Here we give an overview of the main techniques used to establish our results.

### Entanglement measures

Quantum correlations has always been considered as a resource to perform tasks
that are unachievable through classical resources, or to enhance other ones.
Hence, a characterization and quantification of quantum correlations, in
particular entanglement, has been widely studied [49, 58].
For a two-qubit state with density matrix $\rho$, quantum entanglement is
completely characterized by concurrence[5, 49]
$C(\rho))=max\\{0,\mu_{1}-\mu-\mu_{3}-\mu_{4}\\}$, where $\\{\mu_{i}\\}$
denotes the set eigenvalues – in decreasing order – of the operator
$\rho\tilde{\rho}$, with
$\tilde{\rho}=(\sigma_{y}\otimes\sigma_{y})\leavevmode\nobreak\
\rho\leavevmode\nobreak\ (\sigma_{y}\otimes\sigma_{y})$. Concurrence
$C(\cdot)$ is an entanglement monotone metric, with value equal to $1$ for
maximally entangled states and value equal to $0$ for separable states.
For tripartite systems, entanglement measures are more intricate, and are only
analytically found for special classes of states [58, 59, 60]. For genuinely
multipartite entangled (GME) states – i.e, states that are not separable for
any bipartition – a known entanglement measure is the GME concurrence[61, 62,
63]
$C_{\text{GME}}(\rho)=\sqrt{2\min\\{1-\mathrm{Tr}(\rho_{1}^{2}),1-\mathrm{Tr}(\rho_{2}^{2}),1-\mathrm{Tr}(\rho_{3}^{2})\\}}$,
with $\rho_{i}\stackrel{{\scriptstyle\triangle}}{{=}}Tr_{jk}(\rho)$ (with
$j,k\neq i$) denoting the reduced density matrix for the $i$-th subsystem.

### Bell states

######  Theorem 1.

The states in (4) are maximally entangled bipartite states if and only if the
conditions in (5) hold.

###### Proof.

The proof follows by reasoning as in Theorem 2. ∎

######  Proposition 1.

The states in (4) are bi-separable if and only if the conditions:
$\exists\,i\in\\{0,1\\}\;:\;\overleftarrow{U_{i}}\ket{\varphi_{i}}=\overrightarrow{U_{i}}\ket{\varphi_{i}}$
hold.

###### Proof.

The proof follows by reasoning as in Proposition 2. ∎

### GHZ-like states

Here we derive in Theorem 2 the necessary and sufficient condition for
generating a 3-partite GHZ-like state through superposition of causal orders.

######  Theorem 2.

The states in (6) are GHZ-like states if and only if the condition given in
(7) holds.

###### Proof.

We first observe that any tripartite GHZ-like state $\ket{\Psi^{(3)}}$ is
equivalent to the GHZ state
$\ket{\text{GHZ}}=\frac{1}{\sqrt{2}}(\ket{000}+\ket{111})$ through local
unitaries [58] – i.e., $\ket{\text{GHZ}}=(U_{0}^{\text{LU}}\otimes
U_{1}^{\text{LU}}\otimes U_{2}^{\text{LU}})\ket{\Psi^{(3)}}$.
Case $\Longleftarrow$ (sufficient condition). By hypothesis,
$\overleftarrow{U_{i}}\ket{\varphi_{i}}=\left(\overrightarrow{U_{i}}\ket{\varphi_{i}}\right)^{\perp}$
for any $i$. Hence, (6) is equivalent to:

$\ket{\psi^{(3)}_{\pm}}=\frac{1}{\sqrt{L}_{\pm}}\overrightarrow{U_{0}}\ket{\varphi_{0}}\overrightarrow{U_{1}}\ket{\varphi_{1}}\overrightarrow{U_{2}}\ket{\varphi_{2}}\pm\left(\overrightarrow{U_{0}}\ket{\varphi_{0}}\right)^{\perp}\left(\overrightarrow{U_{1}}\ket{\varphi_{1}}\right)^{\perp}\left(\overrightarrow{U_{2}}\ket{\varphi_{2}}\right)^{\perp}$
(14)

By defining local unitaries such that
$U_{i}^{\text{LU}}\overrightarrow{U_{i}}\ket{\varphi_{i}}=\ket{0}$ (hence,
$U_{i}^{\text{LU}}\left(\overrightarrow{U_{i}}\ket{\varphi_{i}}\right)^{\perp}=\ket{1}$),
the thesis follows.
Case $\Longrightarrow$ (necessary condition). By hypothesis, (6) is a GHZ-like
state. We prove the case with a reductio ad absurdum by supposing that there
exists at least one $i$, say $i=0$, so that
$\overleftarrow{U_{0}}\ket{\varphi_{0}}\neq\left(\overrightarrow{U_{0}}\ket{\varphi_{0}}\right)^{\perp}$.
From GHZ-like state definition, there exist
$U_{1}^{\text{LU}},U_{2}^{\text{LU}}$ such that $\ket{\psi^{(3)}_{\pm}}$ is
LU-equivalent to the following (by neglecting the normalization factor for the
sake of simplicity):

$\left(I\otimes U_{1}^{\text{LU}}\otimes
U_{2}^{\text{LU}}\right)\ket{\psi^{(3)}_{\pm}}=\overrightarrow{U_{0}}\ket{\varphi_{0}}\otimes\ket{00}\pm\overleftarrow{U_{0}}\ket{\varphi_{0}}\otimes\ket{11}$
(15)

Then, there must exist another unitary $U_{0}^{\text{LU}}$ acting on the first
qubit such that (15) is equivalent to the $\ket{GHZ}$ state, i.e.:

$\left(U_{0}^{\text{LU}}\otimes I\otimes
I\right)\left(\overrightarrow{U_{0}}\ket{\varphi_{0}}\otimes\ket{00}\pm\overleftarrow{U_{0}}\ket{\varphi_{0}}\otimes\ket{11}\right)=\frac{1}{\sqrt{2}}(\ket{000}+\ket{111})$
(16)

Since unitary matrices preserve orthogonality, from (16), it follows that
$\overrightarrow{U_{0}}\ket{\varphi_{0}}=\left(\overleftarrow{U_{0}}\ket{\varphi_{0}}\right)^{\perp}$,
which constitutes a reductio ad absurdum. Hence, the thesis follow. ∎

######  Remark.

Clearly, the above result can be straightforwardly extended to $n$-partite
GHZ-like states $\ket{\psi^{(n)}_{\pm}}$ generated through an even
superposition of the two alternative causal orders between two $n$-qubit local
unitaries $V^{(n)}=\bigotimes_{i=0}^{n-1}U_{i}$ and
$\tilde{V}^{(n)}=\bigotimes_{i=0}^{n-1}\tilde{U_{i}}$ acting on an initially
pure product $n$-partite state $\bigotimes_{i=0}^{n-1}\ket{\varphi_{i}}$ by
following the same reasoning and, in such a case, the necessary and sufficient
condition becomes:
$\bra{\varphi_{i}}\overleftarrow{U_{i}}^{\dagger}\overrightarrow{U_{i}}\ket{\varphi_{i}}=0\;\forall\,i=0,\ldots,n-1$.

######  Proposition 2.

The states in (6) are bi-separable if and only if the condition given in (8)
holds.

###### Proof.

In the following, we directly prove the proposition for the arbitrary states
$\ket{\psi^{(n)}_{\pm}}$ generated through an even superposition of the two
alternative causal orders between two $n$-qubit local unitaries
$V^{(n)}=\bigotimes_{i=0}^{n-1}U_{i}$ and
$\tilde{V}^{(n)}=\bigotimes_{i=0}^{n-1}\tilde{U_{i}}$ acting on an initially
pure product $n$-partite state $\bigotimes_{i=0}^{n-1}\ket{\varphi_{i}}$.
Case $\Longleftarrow$ (sufficient condition). It is straightforward to
recognize that, whenever there exists at least an $i$ such that
$\overleftarrow{U_{i}}\ket{\psi_{i}}=\overrightarrow{U_{i}}\ket{\psi_{i}}$,
the states $\ket{\psi^{(n)}_{\pm}}$ in Eq. 6 are separable.
Case $\Longrightarrow$ (necessary condition). By hypothesis
$\ket{\psi^{(n)}_{\pm}}$ are separable. Hence, there exist a partition so that
$\ket{\psi^{(n)}_{\pm}}=\ket{\psi_{A}}\otimes\ket{\psi_{B}}$ with
$\ket{\psi_{A}}$ being pure state of the first subsystem $A$. Let us assume,
without loss of generality, subsystem $A$ consisting of the first two qubits
of (6). We prove the case with a reductio ad absurdum by supposing that
$\overleftarrow{U_{i}}\ket{\psi_{i}}\neq\overrightarrow{U_{i}}\ket{\psi_{i}}$
for any $i=\\{0,1,\ldots,n-1\\}$. Since $\ket{\psi^{(n)}_{\pm}}$ are
separable, there exist two local unitaries acting on the first two
qubits444The same reasoning – as well as the same result – holds by
considering local unitaries acting on the remaining $n-2$ qubits. such that:

$\displaystyle\left(U_{0}^{\text{LU}}\otimes
U_{1}^{\text{LU}}\otimes\underbrace{I\ldots\otimes
I}_{n-2}\right)\ket{\psi^{(n)}_{\pm}}=$
$\displaystyle\quad=U_{0}^{\text{LU}}\overrightarrow{U_{0}}\ket{\varphi_{0}}\otimes
U_{1}^{\text{LU}}\overrightarrow{U_{1}}\ket{\varphi_{1}}\otimes\left(\overrightarrow{U_{3}}\otimes\ldots\otimes\overrightarrow{U}_{n-1}\right)\ket{\varphi_{3}\ldots\varphi_{n-1}}\pm
U_{0}^{\text{LU}}\overleftarrow{U_{0}}\ket{\varphi_{0}}\otimes
U_{1}^{\text{LU}}\overleftarrow{U_{1}}\ket{\varphi_{1}}\otimes\left(\overleftarrow{U_{3}}\otimes\ldots\otimes\overleftarrow{U}_{n-1}\right)\ket{\varphi_{3}\ldots\varphi_{n-1}}$
$\displaystyle\quad=\ket{\uparrow\uparrow}\otimes\left(\overrightarrow{U_{3}}\otimes\ldots\otimes\overrightarrow{U}_{n-1}\right)\ket{\varphi_{3}\ldots\varphi_{n-1}}\pm
c\ket{\uparrow\uparrow}\otimes\left(\overleftarrow{U_{3}}\otimes\ldots\otimes\overleftarrow{U}_{n-1}\right)\ket{\varphi_{3}\ldots\varphi_{n-1}}$
(17)

with $\ket{\uparrow\uparrow}$ denoting a certain state for subsystem $A$.
Since unitary matrices preserve inner product, from (2) it results that
$\overleftarrow{U_{0}}\otimes\overleftarrow{U_{1}}\ket{\varphi_{0}\varphi_{1}}=c\,\overrightarrow{U_{0}}\otimes\overrightarrow{U_{1}}\ket{\varphi_{0}\varphi_{1}}$.
But this last equality requires that
$\overleftarrow{U_{i}}\ket{\psi_{i}}=\overrightarrow{U_{i}}\ket{\psi_{i}}$ for
any $i=\\{0,1\\}$, which constitutes a reductio ad absurdum. Hence, the thesis
follow. ∎

### W-like states

######  Theorem 3.

The states in (11) are W-like states if and only if the condition given in
(12) holds.

###### Proof.

We first note that any tripartite W-like state $\ket{\Psi^{(3)}}$ is
equivalent to $\ket{W}=\frac{1}{\sqrt{3}}(\ket{100}+\ket{010}+\ket{001})$ by
local unitaries -i.e., $\ket{W}=\big{(}U_{0}^{LU}\otimes U_{1}^{LU}\otimes
U_{2}^{LU}\big{)}\ket{\Psi^{(3)}}$. Similarly to the proof of Theorem. 2, the
proof of the sufficiency of the condition in (12) is straightforward. In the
meanwhile, the necessity can be proved by reductio ad absurdum as in Theorem.
2, by supposing that there exists at least one $i$ – say, without loss of
generality, $i=0$ – so that
$\overleftarrow{U}_{0}\ket{\varphi}\neq\big{(}\overrightarrow{U}_{0}\ket{\varphi_{0}}\big{)}^{\perp}$.
From the W-like state equivalence, there exists a local unitary of the form
$I\otimes U_{1}^{LU}\otimes U_{2}^{LU}$ such that

$\left(I\otimes U_{1}^{\text{LU}}\otimes
U_{2}^{\text{LU}}\right)\ket{\psi^{(3)}_{\pm\pm}}=\overleftarrow{U_{0}}\ket{\varphi_{0}}\otimes\ket{00}\pm\overrightarrow{U_{0}}\ket{\varphi_{0}}\otimes\ket{10}\pm\overrightarrow{U_{0}}\ket{\varphi_{0}}\otimes\ket{01}$
(18)

Then, there must exist another unitary $U_{0}^{\text{LU}}$ acting on the first
qubit such that (18) is equivalent to W state, i.e.:

$\left(U_{0}^{\text{LU}}\otimes I\otimes
I\right)\overleftarrow{U_{0}}\ket{\varphi_{0}}\otimes\ket{00}\pm\overrightarrow{U_{0}}\ket{\varphi_{0}}\otimes\ket{10}\pm\overrightarrow{U_{0}}\ket{\varphi_{0}}\otimes\ket{01}=\frac{1}{\sqrt{3}}\big{(}\ket{100}\pm\ket{010}\pm\ket{001}\big{)}$
(19)

Since unitary matrices preserve orthogonality, from (19), it follows that
$\overrightarrow{U_{0}}\ket{\varphi_{0}}=\left(\overleftarrow{U_{0}}\ket{\varphi_{0}}\right)^{\perp}$,
which constitutes a reductio ad absurdum. Hence, the thesis follow. ∎

######  Remark.

Although the deterministic generation of any $n$-partite GHZ-like state
requires only a qubit degree of freedom controlling two different evolutions
coherently, the deterministic generation of W-like states requires a higher-
order control of the causal orders. Specifically, $n$ local unitaries – with
$n$ being a power of $2$ – must be arranged in a particular way. The rationale
for this requirement lays in the necessity of having a maximally coherent
basis of states that serves as a measurement setup on the controlling degrees
of freedom, allowing the deterministic generation of the superposition
required in the W states on all outputs. This requirement can only be met in
Hilbert spaces of dimension which is a power of two. In such space, an
orthonormal basis of maximally coherent states exists, and it can be used to
coherentely control the order of the local unitaries. Therefore, generating
$n$-partite W-like states deterministically, where $n=2^{d}$, encounters no
problem as we can always find a maximally coherent orthonormal basis achieving
this. Instead, a slight adjustment on the control strategy needs to be handled
in order to achieve the deterministic generation of $n$-partite W states when
$n$ is not a power of $2$. To overcome this issue, we embed the control
degrees of freedom in a larger Hilbert space of dimension $2^{d}$ where
$d=\lceil\log_{2}n\rceil$.

######  Proposition 3.

The state in (11) is bi-separable if and only if the condition given in (13)
holds.

###### Proof.

The proof follows the similar steps of Proposition 2. ∎

## Data availability

Data sharing not applicable to this article as no datasets were generated or
analyzed during the current study.

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## Acknowledgements

This work was supported by …

## Competing interests

The authors declare no competing financial interests.


CP3-21-67

$tWZ$ production at NLO in QCD in the SMEFT

Hesham El Faham

Centre for Cosmology, Particle Physics and Phenomenology (CP3), Université
catholique de Louvain, B-1348 Louvain-la-Neuve, Belgium and Vrije Universiteit
Brussel, Pleinlaan 2, 1050 Brussels, Belgium

> At NLO in QCD, the $tWZ$ production process interferes with resonant LO
> processes, $t\bar{t}Z$ and $t\bar{t}$, and a method to meaningfully
> disentangle these overlapping processes needs to be employed. I discuss the
> diagram-removal procedure needed to obtain an operative definition of $tWZ$
> at NLO accuracy. I also show the predictions we obtain in the context of the
> Standard Model and in the Standard Model Effective Field Theory including
> the relevant dimension-six SMEFT operators.

> PRESENTED AT
>
>
>
>
> $14^{\mathrm{th}}$ International Workshop on Top Quark Physics
> (videoconference), 13–17 September, 2021

##  1 Introduction

Precision measurements at the LHC can potentially reveal deviations from the
Standard Model (SM) expectations, and therefore hint to the existence of new
physics phenomena. To exploit future LHC measurements, we have to be able to
predict SM processes to our best possible precision. Moreover, a tool by which
we can interpret information coming from these measurements is a crucial
element to move forward. Assuming new physics is “hidden” at some scale
$\Lambda$ that is currently higher than our reach, the Bottom-Up Standard
Model EFT (SMEFT) is one powerful candidate that provides a consistent and
model-independent framework where deviations from the SM are encapsulated in
higher dimension gauge-invariant operators and can be systematically analysed.

Even though $tWZ$ is a rare EW process with a cross-section around 100
femtobarns, it does feature a unique unitarity-violating behaviour induced in
its sub-amplitudes through the modification of the top quark EW interactions
[1, 2]. This provides sensitivity to new physics deviations which, despite the
complexity of $tWZ$ at NLO accuracy, makes it still an interesting process to
study.

##  2 $tWZ$ in the SM

In the five-flavor scheme (5FS) and at LO, $tWZ$ is identified through the
process $gb\to tWZ$ shown in Fig. 1. At NLO and because of the real radiation,
a real emission $tWZb$ final state arise from $gg(b\bar{b})\to tWZb$. This
real emission final state overlaps with a similar, however resonant, final
states induced by an intermediate top quark going on-shell. These resonant
final states do not belong to the genuine $tWZ$ production but to $t\bar{t}$,
with $\bar{t}\to WZ\bar{b}$, or to $t\bar{t}Z$, with $\bar{t}\to W\bar{b}$, as
displayed in Fig. 2. For a properly-behaved perturbative expansion and to be
able to define the genuine $tWZ$ process at NLO in QCD, such resonant
contributions need to be subtracted.

Figure 1:  Representative diagrams for $tWZ$ production in the 5FS at leading
order. Figure 2:  Representative diagrams of the $gg$-initiated production of
$t\bar{t}$ (_left_) and $t\bar{t}Z$ (_right_) processes with a potentially
resonant (anti-)top quark leading to the $t\bar{\ell}\ell Wb$ final state.

To this end, we follow the diagram-removal procedures presented in Ref. [3,
4]. The amplitude, $\mathcal{M}$, can be decomposed into a non-resonant and a
resonant part,

$\mathcal{M}=\mathcal{M}_{\rm non-res}+\sum_{i}\mathcal{M}_{{\rm res},i},$
(2.1)

where the index $i$ indicates different resonant processes. The amplitude
squared reads

$|\mathcal{M}|^{2}=\left|\mathcal{M}_{\rm non-
res}\right|^{2}+2\Re\left(\mathcal{M}_{\rm non-res}\sum_{i}\mathcal{M}_{{\rm
res},i}\right)+\left|\sum_{i}\mathcal{M}_{{\rm res},i}\right|^{2}.$ (2.2)

In the case of standard DR procedure (DR1), the resonant terms are discarded
before squaring the amplitude, i.e.

$|\mathcal{M}|^{2}_{\rm DR1}\equiv\left|\mathcal{M}_{\rm non-res}\right|^{2}.$
(2.3)

In the case of diagram-removal with interference (DR2), the squared resonant
matrix elements in the squared amplitude are discarded, but their interference
with the non-resonant part is retained,

$|\mathcal{M}|^{2}_{\rm DR2}\equiv\left|\mathcal{M}_{\rm non-
res}\right|^{2}+2\Re\left(\mathcal{M}_{\rm non-res}\sum_{i}\mathcal{M}_{{\rm
res},i}\right).$ (2.4)

Since the aim is to define $tWZ$ as an independent process, one has to
suppress the interference of the non-resonant amplitude with the resonant one,
i.e. the second term on the R.H.S of Eq. 2.4. We found that vetoing the
bottom-tagged jet ($b$-jet) that would stem from the top decay strongly
suppresses this term and thus renders DR1 and DR2 predictions in good
agreement. The reason why vetoing hard $b$-jets cuts away regions in the
phase-space in which $tWZ$ is non-dominant is because $b$-quarks originating
from real emission diagrams are instead enhanced in the soft and collinear
limits. This is evident in left panel of Fig. 3 which shows the differential
cross section predictions in the $Wll$ invariant mass bins for the $tWll$
process at NLO in QCD in the DR1 and DR2 schemes, both at inclusive-level and
after applying a veto on $b$-quarks with $|\eta|<2.5$ or $p_{T}>30$ GeV.

We proceed by comparing the cross-section of the $tWll$ process in one case
where we consider only diagrams with a resonant $Z$ boson to another case
where all diagrams are retained. As shown in the right panel of Fig. 3, we see
the two cases in good agreement in the [70, 150] GeV range, and only differ in
regions where the cross-section is suppressed. This leads us to simplify our
SMEFT study that follows by keeping the $Z$ boson stable. Finally and before
moving to the SMEFT study, in Tab. 1 we show the SM inclusive cross-section
for $tWZ$ at LO and at NLO for both the DR1 and the DR2 schemes.

| LO | NLO DR1 | NLO DR2
---|---|---|---
SM | $103.36(4)^{+12.76\%}_{-12.82\%}$ | $106.70(15)^{+4.97\%}_{-6.28\%}$ | $106.80(9)^{+5.04\%}_{-5.62\%}$

Table 1: The SM contributions [fb] to $tWZ$ production at LO and NLO,
including QCD scale uncertainties, for the LHC at $\sqrt{s}=$ 13 TeV.

Figure 3:  The $tWll$ differential cross section in the invariant mass bins of
the $Wll$ system (left). The first two insets show the relative scale
uncertainties and the third shows the DR1/DR2 ratio. The right panel shows DR1
and DR2 predictions in the invariant mass bins of the lepton pair, $M_{ll}$.
The ‘Normal’ case is when we keep all diagrams as opposed to the ‘sch’ where
only diagrams with a resonant $Z$ boson are retained. The inset shows the
ratios of the distributions w.r.t to the ‘Normal’ DR1 prediction.

##  3 $tWZ$ in the SMEFT

As previously mentioned, the $bW\to tZ$ sub-amplitudes of the $tWZ$ process
shown in Fig. 4 feature a unique sensitivity to unitarity-violating behaviour
through modified EW interactions. Therefore, searching for an induced
characteristic energy growth can be an interesting path for a new physics
search.

Figure 4:  $bW\to tZ$ sub-amplitudes of the $tWZ$ process. The blob on the far
right diagram shows the embedding of the $bW\to tZ$ sub-amplitude into $gb\to
tWZ$ production.

Given our sub-amplitudes of interest, the EW top interactions can be modified
by the following dimension-six SMEFT operators

$\displaystyle\mathcal{O}_{\scriptscriptstyle\varphi
Q}^{\scriptscriptstyle(3)},\mathcal{O}_{\scriptscriptstyle\varphi
Q}^{\scriptscriptstyle(-)},\mathcal{O}_{\scriptscriptstyle\varphi
t},\mathcal{O}_{\scriptscriptstyle tW},\mathcal{O}_{\scriptscriptstyle
tZ},\mathcal{O}_{\scriptscriptstyle tG}.$ (3.1)

The first two current operators modify the interactions of two fermions fields
with the gauge bosons. The third current operator,
$\mathcal{O}_{\scriptscriptstyle\varphi t}$, is the right handed $t\bar{t}Z$
interaction. The following two dipole operators modify the interaction of the
top quark with the weak isospin and the weak hypercharge gauge fields.
Finally, and even though not included in the $bW\to tZ$ sub-amplitude,
$\mathcal{O}_{\scriptscriptstyle tG}$ is included in our analysis as it
modifies the gluon-top quark interaction.

It is worth noting that in the high-energy phase-space regions, and since in
the SMEFT study we use the on-shell $Z$ approximation as discussed earlier,
the contribution of the $t\bar{t}$ overlap via $\bar{t}\to WZ\bar{b}$ decay
becomes non-resonant, and thus those $t\bar{t}$ contributions should not be
removed. Therefore, in the SMEFT study, only the $t\bar{t}Z$ overlap is of
concern. Moving into the SMEFT differential predictions, as with our SM
results, the $tWZ$ process is simulated via MadGraph5_aMC@NLO [5, 6] at NLO-
QCD accuracy in the SMEFT using SMEFTatNLO [7]. The $b$-veto as discussed in
the SM section proves efficient for the SMEFT case in suppressing the resonant
contributions to the process and providing stable SMEFT predictions. In Fig. 5
we show the Fixed Order (FO) DR2 cross-section predictions in the $WZ$
invariant mass bins for four of the six operators presented in Eq.3.1. The
reason we omitted $\mathcal{O}_{\scriptscriptstyle\varphi
Q}^{\scriptscriptstyle(-)}$ and $\mathcal{O}_{\scriptscriptstyle\varphi t}$
from our analysis is because they showed a rather insignificant energy growth.

Figure 5:  The invariant mass of the $WZ$ pair, $M_{WZ}$, at DR2 for the four
relevant SMEFT operators when $t\bar{t}Z$ is removed and the $b$-veto is
imposed, at FO NLO accuracy. The legends correspond to different computations
at different SMEFT contributions. The first inset shows the scale variations
in the process. The middle and the last insets show (at LO and at NLO) the
interference and the full SMEFT contribution relative to the SM, respectively.

##  4 Summary and conclusions

I presented a study of the $tWZ$ production process at NLO in QCD where the
diagram-removal procedure were used to suppress resonant contributions from LO
overlapping processes. We found that vetoing additional hard or central
$b$-jet is sufficient to define a region in the phase-space where the genuine
$tWZ$ process dominates.

In the SMEFT study, we considered the EW modifications of the underlying
$bW\to tZ$ sub-amplitude through different dimension-six SMEFT operators. The
aim of this study is to identify potential energy-growing effects that could
be exploited at future collider searches.

The Fixed Order differential predictions at NLO accuracy for the SM and for
the SMEFT including the four relevant operators were presented. Similar to the
SM case, we found our SMEFT results to be stable under the diagram-removal
procedures.

ACKNOWLEDGEMENTS

H.F is supported by the F.R.S.-FNRS under the “Excellence of Science” EOS be.h
project no. 3082081.

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