Controlled remote implementation of operations via graph states Xinyu Qiu1and Lin Chen1 2y 1LMIBBeihang University Ministry of education

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Controlled remote implementation of operations via graph states

Xinyu Qiu1, ∗and Lin Chen1, 2, †

1LMIB(Beihang University), Ministry of education,

and School of Mathematical Sciences, Beihang University, Beijing 100191, China

2International Research Institute for Multidisciplinary Science, Beihang University, Beijing 100191, China

We propose protocols for controlled remote implementation of operations with convincing control

power. Sharing a (2N+1)-partite graph state, 2Nparticipants collaborate to prepare the stator and

realize the operation ⊗N

j=1 exp [iαjσnOj] on Nunknown states for distributed systems Oj, with the

permission of a controller. All the implementation requirements of our protocol can be satisﬁed by

means of local operations and classical communications, and the experimental feasibility is presented

according to current techniques. We characterize the entanglement requirement of our protocol in

terms of geometric measure of entanglement. It turns out to be economic to realize the control

function from the perspective of entanglement cost. Further we show that the control power of our

protocol is reliable by positive operator valued measurement.

I. INTRODUCTION

As the unique resource, entanglement allows the emergence and development of quantum informa-

tion processing, such as teleportation [1], dense coding [2] and cryptography [3,4]. Being expected to

oﬀer substantial speed-ups over classical counterparts, quantum computation has been paid a lot of

attention [5,6]. Several challenges have surfaced in its actual construction, such as decoherence and

dissipation, suﬃciently manipulating a large number of qubits, and undesirable interactions [7]. To

counter such challenges, distributed quantum computation has been proposed [8–10]. It requires to

transfer states from one place to the other and implement the operations on a remote state faithfully.

The ﬁrst requirement has been met by quantum teleportation [1,11–13]. The second one has been

tackled by remote implementation of operation (RIO) and controlled RIO (CRIO). RIO means that

the quantum operation performed on the sender’s local system is able to act on an unknown state

of a remote system that belongs to the receiver [14,15]. Then CRIO was proposed by extending

RIO to multipartite case with controller [16]. The idea of that is to implement remote operations,

but only with the permission of controller. It can deﬁnitely enhance the security of RIO. Both of

RIO and CRIO play an important role not only in distributed quantum computation, but also other

tasks in remote quantum information processing such as programming [17], operation sharing [18]

and remote state preparation [19]. They are realized by local operation and classical communication

(LOCC) and consume entanglement resource. Some works concerning RIO have been presented and

interesting progress has been made both theoretically [20–23] and experimentally [24–26]. As for

CRIO, it has been realized in terms of partially unknown quantum operations [16,27], various classes

of bipartite unitary operations [28], arbitrary dimensional controlled phase gate [29], and operators

on diﬀerent remote photon states [30]. Entangled states including Bell, Greenberger-Horne-Zeilinger

(GHZ), and ﬁve-qubit cluster states are employed as channels in these protocols. Theoretically,

many of these protocols can hardly be scalable, or technically complicated. What’s more, the highly

entangled states they employed are susceptible to noise, so it is a great challenge to realize them

under experimental techniques.

In this paper, we propose CRIO protocols for the operations exp [iασn] with convincing control

∗xinyuqiu@buaa.edu.cn

†linchen@buaa.edu.cn (corresponding author)

arXiv:2210.14674v2 [quant-ph] 18 Dec 2022

power. The diagram of our protocol via a graph state |h3iis shown in FIG. 1. We generalize

the protocol realizing RIO by stators [20] into CRIO. The stators in our protocol are constructed

from shared graph states by LOCC, only with the permission of the controller. The protocol via

a (2N+ 1)-partite graph state can be obtained by generalization, and that is used to implement

remote operations on Nunknown states. Graph states are employed as the channel in our protocol

for three reasons. Firstly, they make it possible to realize control function and enable our protocol

to be scalable. Secondly, they are a natural resource for much of quantum information, and known

to be most readily available multipartite resource in the laboratory [31,32]. Thirdly, many of the

graph states show advantages for being robust to noise [33]. All the implementation requirements

of our protocol can be satisﬁed by means of LOCC. The experimental feasibility is presented in

terms of current techniques. The local operations and measurements can be implemented by a

diamond nanophotonic resonator containing SiV quantum memory with an integrated microwave

stripline [26]. The entanglement requirement of our protocol is characterized in terms of geometric

measure of entanglement. It is presented in Proposition 2. Compared with the former protocol in

[20], ours is endowed with control function, while it only requires the same entanglement resource.

Hence it is economic to realize the control function from the perspective of entanglement cost.

Besides, we analyze the control power in our protocol, and obtain that without the permission of

the controller, other participants can hardly realize the remote implementation of operation. It is

shown in Proposition 3. Thus the control power of our protocol is convincing. Our protocol shows

advantage in stronger security, extensive applications, and advanced eﬃciency. It can contribute

to improving the ability of distributed quantum computing and stimulate more research work on

quantum information processing.

FIG. 1: Diagram showing CRIO via a graph state |h3ia,b,c in (13). Qubits a, b, c belong to Alice, Bob and Charlie,

respectively. These participants locate in distributed places and share the entangled state |h3i. Only with the

permission of controller Alice, then Bob and Charlie can construct the stator by local measurements and operation

Uc,C etc. Then with the help of stator S3, Bob can implement the remote operation eiασnCon an unknown state for

Charlie’s system Cby locally implementing eiασxb.

Graph states have a strong connection with quantum computation. Naturally, they can be char-

acterized by geometric measure of entanglement (GM), a well-known entanglement measure for

multipartite systems. GM not only provides a simple geometric picture, but also has signiﬁcant op-

erational meanings. It has connections with optimal entanglement witnesses [34], and multipartite

state discrimination under LOCC [35]. As one of the most widely used entanglement measures for

the multipartite states, GM fulﬁlls all the desired properties of an entanglement monotone [34]. It

has been utilized to determine the universality of resource states for one-way quantum computation

[36]. It also has been employed to show that most entangled states are too entangled to be useful

as computational resources [37].

The rest of this paper is organized as follows. In Sec. II, we introduce some basic concepts of

graph states, GM and stator. Then we simply recall the deduction of eigenoperator equation of the

stator. In Sec. III, we propose our CRIO protocol. We show the implementation of remote operation

on an unknown state for a single system via a tripartite graph state in Sec. III A. The protocol via

a ﬁve-partite graph state is presented in Sec. III B, which is slightly diﬀerent from the former one

and used to implement operations on two remote systems. Then we generalize it into the one via a

(2N+ 1)-partite graph state in Sec. III C. We show GM of the graph states used in our protocols in

Sec. IV. We do the control power analysis in Sec. V, and exhibit the experimental feasibility of our

protocol in Sec. VI. Finally, we conclude in Sec. VII.

II. PRELIMINARIES

In this section, we recall the deﬁnitions and some properties of graph states, GM and stator. In

Sec. II A, we recall the deﬁnition of graph states, and demonstrate a graph state with the help

of quantum gates. In Sec. II B, we recall the deﬁnition of GM and show a lemma we employ to

characterize the GM of graph states. In Sec. II C, we introduce the concept of stator, and present

the eigenoperator equation of the stator used in this paper.

A. Graph states

A graph is a pair G= (V, E), where Vis the set of vertices and E⊂[V]2is the set of edges. With

each graph, a graph state is associated. An axiomatic framework for mapping graphs to quantum

states is proposed in [38]. A graph state is a certain pure state on a Hilbert space H⊗V

2. Each vertex

of the graph labels a qubit. Each vertex a∈Vof the graph G= (V, E) is attached to a Hermitian

operator

K(a)

G=σ(a)

b∈Na

σ(b)

z.(1)

Here σ(a)

xand σ(a)

zare the Pauli matrices and the upper index speciﬁes the Hilbert space on which the

operator acts. K(a)

Gis an observable of the qubits related to vertex aand all of its neighbors b∈Na.

There are N=|V|operators in the set {K(a)

G}a∈V. The operators in this set are all commute.

We demonstrate a graph state with the help of Hadamard gate Hand two-qubit controlled-Z gate

CZ(j1,j2), where

H=1

√21 1

1−1,(2)

and the gate

CZ(j1,j2)= diag{1,1,1,−1}(3)

denotes the gate with control qubit j1and controlled qubit j2. By the two gates, we can show the

preparation of graph states in the quantum circuit conveniently.

A graph state |Hniis created from a graph G={V, E}of nvertices by assigning a qubit to each

vertex and initializing them by applying the Hadamard gate on each qubit. Let |+i= (|0i+|1i)/√2.

If two vertices j1, j2∈Vare connected by an edge e∈V, then we perform CZ(j1,j2)over the initialized

n-qubit state |+i⊗n. By implementing all the controlled-Z gates corresponding the edges e∈E, we

obtain the graph state

|Hni=Y

e∈E

CZe|+i⊗n.(4)

Graph states are useful resources with applications spanning many aspects of quantum information

processing, such as computation [39], cryptography [40], quantum error correction [41] and networks

[42]. Experimentally, diﬀerent techniques have been studied to implement graph states including ion

traps [43], superconducting qubits [44], and continuous variable optics [45].

B. Geometric measure of entanglement

Geometric measure of entanglement (GM) is a well-known entanglement measure for multipartite

systems [34]. It measures the closest distance in terms of overlap between a given state and the

set of separable states, or the set of pure product states. Originally introduced for pure bipartite

states, GM was subsequently generalized to multipartite and to mixed states. Several inequivalent

deﬁnitions of GM has surfaced by now. In this paper, we shall follow the deﬁnition given in (5) and

(6).

Λ2(ρ) := max

σ∈SEP Tr(ρσ) = max

|ϕi∈P ROhϕ|ρ|ϕi,(5)

G(ρ) := −2 log Λ(ρ).(6)

Here SEP denotes the separable states and PRO denotes the fully pure product states in the Hilbert

space ⊗N

j=1Hj. GM is known only for a few examples, such as bipartite pure states, GHZ-type states,

antisymmetric basis states, pure symmetric three-qubit states and some graph states [46–48].

We show the following fact given in Ref. [49]. It is a useful lemma concerning the closest prod-

uct states of non-negative states. Here the closest product state denotes any pure product state

maximizing (5). The non-negative state means that all its entries in the computational basis are

non-negative.

Lemma 1 The closest product state to a non-negative state ρcan be chosen to be non-negative.

The proof is presented in Lemma 8 of Ref. [49]. This lemma can be used to characterize GM of the

states that are non-negative or locally equivalent to non-negative states. In addition, it contributes to

prove the strong additivity of GM of the states including Bell diagonal states, maximally correlated

generalized Bell diagonal states, isotropic states, generalized Dicke states, mixture of Dicke states,

the Smolin state and D¨ur’s multipartite entangled states.

Since the graph states we employed in this paper are all locally equivalent to non-negative states,

we use Lemma 1to investigate GM of them. The calculation is presented in Sec. IV.

C. Stator and eigenoperator equation

The stator, a hybrid state operator, is an object that expresses quantum correlations between

states of one participant and operators of the other participant. It is ﬁrstly proposed in Ref. [20]

to implement a class of operations on remote systems. Given a well-prepared stator, the operation

on Bob’s system is remotely brought about by Alice’s local operations. The desired operation is

determined by Alice and unknown to Bob. Hence it demonstrates advantage in terms of security.

A stator SAB shared by remote observers Alice and Bob is in the space

SAB ∈ {HA⊗O(HB)},(7)

where HAand HBare the Hilbert spaces of Alice and Bob respectively, and O(HB) denotes the

operators acting on an arbitrary state in HB. A stator has the general form

SAB =

s=1

t=1

cst|si ⊗ Bt,(8)

where NA= Dim(HA), NB= Dim(HB), |si∈HA,Btacts on states in HBand cs,t are complex

numbers. For each stator, an eigenoperator equation can be constructed. We consider the following

stator used in this paper,

S=|0iA⊗IB+|1iA⊗σnB.(9)

Here σnB=nB·σ∈O(HB), where nB= [x, y, z] is the axis vector and σ= [σx, σy, σz] is the Pauli

matrix vector. The states |0Ai,|1Ai ∈ HAare the eigenstates of σzA. One can verify that σ2

nB=I.

Obviously, Ssatisﬁes the eigenoperator equation

σxAS=σnBS. (10)

Thus for any analytic function f, it also satisﬁes that

f(σxA)S=f(σnB)S, (11)

and particularly,

eiασxAS=eiασnBS, (12)

where αis any real number determined by Alice. Using the stator, a unitary operation on Alice’s

qubit gives rise to a similar unitary operation acting on Bob’s system, which is remote to Alice. The

construction of stator and the implementation of remote operations are both realized by LOCC.

III. CONTROLLED REMOTE IMPLEMENTATION OF OPERATIONS

In this section, we show the implementation of controlled remote operations via a graph state. The

2N+ 1 participants A1, A2, ..., A2N+1 share the entangled graph state as resource. They cooperate

to realize the remote operations U=⊗N

j=1 exp [iαjσnOj] on Nunknown states for remote systems Oj

by applying LOCC. Here αis the angle of rotation only known by participants A2, A3, ..., AN+1 and

unknown to others including the controller. Besides, the controller takes the responsibility to decide

whether or not and when the implementation of remote operations should be done on each system.

Thus our protocol shows advantage in terms of strong security and extensive applications. In Sec.

III A, we introduce the implementation of operation UC= exp [iασnC] on one remote system Cvia a

tripartite graph state. In Sec. III B, we show the protocol via a ﬁve-partite graph state, where two

operations are implemented on two remote systems respectively. In this protocol, appropriate local

Hadamard operations are employed. So it is diﬀerent from the protocol via tripartite state. Then

we generalize this protocol into the one via a (2N+ 1)-partite graph state (N≥2). It is presented

in Sec. III C.

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ControlledremoteimplementationofoperationsviagraphstatesXinyuQiu1,andLinChen1,2,y1LMIB(BeihangUniversity),Ministryofeducation,andSchoolofMathematicalSciences,BeihangUniversity,Beijing100191,China2InternationalResearchInstituteforMultidisciplinaryScience,BeihangUniversity,Beijing100191,ChinaWepropos...

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Controlled remote implementation of operations via graph states Xinyu Qiu1and Lin Chen1 2y 1LMIBBeihang University Ministry of education

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