quantum entanglement transformations via local operations … · 2016. 11. 17. · entanglement...

Quantum Entanglement

Transformations via Local Operations

and Classical Communication

Student’s name: Cheng Guo

Supervisor: Prof. Runyao Duan

Centre for Quantum Computation and Intelligent Systems

Faculty of Engineering and Information Technology

University of Technology, Sydney

CERTIFICATE OF ORIGINAL AUTHORSHIP

I certify that the work in this thesis has not previously been submitted for a

degree nor has it been submitted as part of requirements for a degree except as

fully acknowledged within the text.

I also certify that the thesis has been written by me. Any help that I have

received in my research work and the preparation of the thesis itself has been

acknowledged. In addition, I certify that all information sources and literature

used are indicated in the thesis.

Ian

铅笔

Acknowledgments

Firstly, I would like to thank my supervisor Prof. Runyao Duan. He gives

me a deep impression with his patience and wisdom. I met him at Tsinghua

University, when I was looking for an interesting topic for Bachelor graduation

thesis. He introduced Quantum Information Theory to me. Then, I came to

University of Technology, Sydney. He has been a tremendous mentor for me.

His advices both on my research as well as on my career have been priceless. He

offers his continuous advice and encouragement throughout this thesis. I thank

him for his systematic guidance and great effort that he put into training me in

the scientific field.

I would like to thank Prof. Mingsheng Ying, Prof. Chengqi Zhang, Prof.

Yuan Feng, Prof. Sanjiang Li, Prof. Min-Hsiu Hsieh, Prof. Enrico Carlini and

Prof. Eric Chitambar for all their help and encouragement in these years.

Many thanks to Dr. Nengkun Yu for his pragmatic advices. I learned many

original ideas and techniques in our discussions. Thank Dr. Youming Qiao, Dr.

Ching-Yi Lai, Dr. Yangjia Li and Dr. Shenggang Ying for their kind help and

support.

Finally, I would like to thank my parents for everything they have done for

me. Words cannot express how grateful I am to my parents for all of the sacrifices

that you have made on my behalf.

AbstractThe primary goals of this thesis are two-fold: i) calculating the optimal entan-

glement transformation rate between two multipartite pure states via stochas-

tic local operations and classical communication (SLOCC), and ii) showing the

properties of a common resource for a set of multi-partite pure state via local

operations and classical communication (LOCC) or SLOCC.

We introduce a notion of entanglement transformation rate to character-

ize the asymptotic comparability of two multi-partite pure entangled states

under SLOCC. For two well known SLOCC inequivalent three-qubit states:

Greenberger-Horne-Zeilinger (GHZ) state and W state, we show that the entan-

glement transformation rate from GHZ state to W state is exactly 1. We then

apply similar techniques to obtain a lower bound on the entanglement transfor-

mation rates from an N -partite GHZ state to a class of Dicke states.

Then, we discuss the common resource for a set of pure states. We have com-

pletely solved the bipartite pure states case by explicitly constructing a unique

optimal common resource state for any given set of states via LOCC. In the

multi-partite setting, the general problem becomes quite complicated, and we

focus on finding non-trivial common resources for the whole multi-partite state

space of given dimensions. We show that |GHZ3〉 = (1/√

3)(|000〉+|111〉+|222〉)is a nontrivial common resource for three-qubit systems via LOCC. We also ob-

tain a number of interesting properties of non-trivial common resource states for

two N -qubit pure states and multi-partite systems via SLOCC.

Key words: Entanglement Transformation, LOCC, GHZ state, W state,

Tensor Rank, Common Resource States.

Contents

I. Introduction 1

A. Main topics and motivation 2

B. Overview of the thesis 4

II. Preliminaries 7

A. Linear Algebra 7

B. Postulates of Quantum Mechanics 9

C. Tensor rank, Schmidt decomposition, majorization and their

applications 13

III. Asymptotic Rate of State Transformation 21

A. Previous results about R(|GHZ〉, |W 〉) 23

B. R(|GHZ〉, |W 〉)=1 27

C. Generalization to Dicke states 31

D. Alternative proof 34

E. Computational complexity of matrix permanent 36

IV. Common resource via local operations and classical

communication 39

A. Optimal common resource of bipartite pure states 41

B. Optimal common resource of a special kind of multi-partite pure states44

C. A non-trivial common resource of 3-qubit system: |GHZ3〉 46

V. Common resource via stochastic local operations and classical

communication 55

A. Tripartite entangled pure states 57

B. Multi-partite qubit pure states 63

VI. Conclusion 69

References 71

I INTRODUCTION

I. INTRODUCTION

Quantum Mechanics was established in early 20th century. Entanglement

correlation in Quantum Mechanics is one of the most significant differences com-

pared with classical correlation in Classical Mechanics. Such quantum entangle-

ment correlation may exist in multi-subsystems even if they are far away from

each other, and random events taking place on one subsystem can affect the

state’s other part. Historically, the concept of entanglement first appeared in

Schrodinger “cat state” papers [1].

Due to the distances among the different subsystems of quantum entangled

states, we have the following two reasonable restrictions: (1) (local operations)

each party can only perform operations on his/her own subsystem and global

operations on the composite system are not allowed, and (2) (classical commu-

nication) each party can tell other parties his/her measurement outcomes by

classical channels in order to communicate and coordinate. This class of oper-

ations is called “local operations and classical communication”, abbreviated as

LOCC. The fundamental problem of quantum state transformation is to ask, for

a given state, what kind of states can be obtained from this one via LOCC. For

an arbitrary mutli-partite entangled state, it has been proven this problem is

hard except for bipartite pure states [2]. If the success probability of the trans-

formation is positive (but usually strictly less than 1), the protocol is named

“stochastic local operations and classical communication” (SLOCC). There are

many interesting open problems about SLOCC transformations.

In this thesis, we are mainly interested in two topics. The first topic is to

calculate the asymptotic rate of a state transformation from Greenberger-Horne-

1

I INTRODUCTION

Zeilinger (GHZ) state to W state. The second one is to study the properties of

common resource states of a set of multi-partite pure state via LOCC or SLOCC.

A. Main topics and motivation

The problem of transforming one entangled state to another by LOCC is of

central importance in quantum entanglement theory. In order to make quantita-

tive comparison between different types of quantum information resources, the

following fundamental entanglement transformation problem arises: whether an

N -partite state can be transformed into another given N -partite state via LOCC.

In bipartite case, a necessary and sufficient condition for entanglement transfor-

mation was reported by Nielsen [2]. In tripartite case, this condition doesn’t hold.

Dur, Vidal, and Cirac observed that within three-qubit systems, there exist two

distinct equivalence classes of genuinely tripartite entangled states: GHZ-type

states and W-type states [4].

What is the asymptotic rate of state transformation from GHZ state to W

state? This has been a longstanding open problem. We improve the upper bound

of tensor rank of the n-copies W state these years [5] [6] [7]. Finally, we find the

answer: though we all know W and GHZ cannot be transformed to each other

via LOCC or even SLOCC, it is interesting that when we have sufficiently large

number of copies of GHZ state, we can obtain nearly the same number of W

state via SLOCC.

The problem of common resource state naturally arises when we need to pre-

pare a number of different pure states. Due to the practical limitation, one eco-

nomic strategy is to prepare minimal number of standard quantum pure states,

2

I INTRODUCTION

then transform these standard states into the desired pure state via LOCC or

SLOCC. In other words, it is of great importance to find one state which can be

transformed to all other pure states in a given set. We can imagine the following

scene: a quantum laboratory staff is responsible for the quantum state prepa-

ration of a company. Everyday he receives a number of different preparation

requests of multi-partite pure states. If the number of these requests are large

enough, it is impossible to keep adjusting the quantum states preparation equip-

ments. Fortunately, these produced multi-partite quantum states are shared by

his colleagues. These colleagues can communicate with each other in a classical

way and each person can make quantum operations on their own subsystems.

Now the question becomes the following: assume all required quantum states are

known, can that quantum laboratory staff only prepare one state which can be

transformed into any required quantum state via LOCC or SLOCC?

In fact, there exist a trivial kind of entangled pure states in some special state

space which can be transformed into any other pure state in the same state space

by LOCC. These kinds of states are called maximally entangled states, and they

exist in spaces if and only if the dimension of one subsystem is no less than

the product of dimensions of all other subsystems [8]. Bell states are one such

example, which can be transformed into any pure state in two-qubit systems.

For tripartite and more than three parties case, “the product of dimensions of

all other subsystems” grows fast. Based on this fact, we can immediately know:

the maximally entangled states do not exist in tripartite and more than three

parties usually. For any given multi-partite pure state set (possibly a space), can

we find a common state to prepare them via SLOCC? Such a state, if exists, will

3

I INTRODUCTION

be a common resource state for the given set of states. How much local rank of

its each subsystem should be? If these local ranks cannot be less, the state is

named an “optimal common resource ” (OCR) sate. We will present a number

of nontrivial properties of OCR states in Section V.

B. Overview of the thesis

In the next section, we review the necessary background knowledge including

Linear Algebra, Postulates of quantum mechanics, and some properties of the

local operations and classical communication protocols.

In Section III, we introduce research history about asymptotic rate of state

transformation from |GHZ〉 to |W 〉 and tensor rank of |W 〉⊗n. Then, we prove

the result that this rate is just 1. We also discuss some related problem: compu-

tational complexity of matrix permanent. Part of section III has been published

in Physical Review Letters [18].

Section IV is about how to find a common resource for a set of pure states via

local operations and classical communication. This is a new topic in quantum

information theory. We show that |GHZ3〉 = (1/√

3)(|000〉+ |111〉+ |222〉) is a

nontrivial common resource for 3-qubit systems via LOCC.

Section V is about common resource problem with stochastic local operations

and classical communication. We show some properties of the non-trivial com-

mon resource of two N -qubit pure states via SLOCC. We also study the common

resource in multi-partite space system via SLOCC.

Manuscripts about the results in Section IV and Section V are in preparation

4

I INTRODUCTION

for submitting to journals [46][47]. A brief conclusion is drawn in the last section.

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II PRELIMINARIES

II. PRELIMINARIES

In this section, we will review the necessary background knowledge in this

research area, include the postulates and definitions about quantum mechan-

ics, theorems in linear algebra and some properties of the local operations and

classical communication protocols. Most of definitions and postulates are from

Nielsen and Chuang’s influential textbook “Quantum Computation and Quan-

tum Information” [3].

A. Linear Algebra

The study of quantum mechanics is based on a solid grasp of elementary linear

algebra. The basic objects of linear algebra are vector space. The space of all

n−tuples of complex numbers (z1, · · · , zn) is denoted as Cn.We will use the standard notation (Dirac notations) of quantum mechanics

for concepts from linear algebra. The standard quantum mechanical notation for

a vector in a vector space is as following:

|ψ〉 =

u1

u2...

un

, |φ〉 =

v1

v2...

vn

.The conjugate transpose of a vector is written as

〈ψ| = [u∗1 u∗2 · · · u∗n ] ,

where u∗ denotes the complex conjugate of a complex number u.

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II PRELIMINARIES

Outer product of |φ〉 and |ψ〉 is |φ〉〈ψ| =

v1u

∗1 v1u

∗2 · · · v1u∗n

v2u∗1 v2u

∗2 · · · v2u∗n

......

. . ....

vnu∗1 vnu

∗2 · · · vnu∗n

,and inner product is 〈φ|ψ〉 =

∑nj=1 v

∗juj.

For a matrix A = (aj,k)m×n, T denotes the matrix transpose, say

AT = (ak,j)n×m.

The tensor product, denoted as ⊗, is a way of putting vector spaces together

to form larger vector spaces. For a matrix B = (bj,k)p×q,

A⊗B =

a11B · · · a1nBa21B · · · a2nB

.... . .

...

am1B · · · amnB

.By definition the tensor product satisfies the following basic properties, where

V and W are linear spaces:

(1) For an arbitrary scalar z and elements |v〉 of V and |w〉 of W ,

z(|v〉 ⊗ |w〉) = (z|v〉)⊗ |w〉 = |v〉 ⊗ (z|w〉).

(2) For arbitrary |v1〉 and|v2〉 in V and|w〉 in W ,

(|v1〉 ± |v2〉)⊗ |w〉 = |v1〉 ⊗ |w〉 ± |v2〉 ⊗ |w〉.

(3) For arbitrary |v〉 in V , |w1〉 and |w2〉 in W ,

|v〉 ⊗ (|w1〉 ± |w2〉) = |v〉 ⊗ |w1〉 ± |v〉 ⊗ |w2〉.

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II PRELIMINARIES

For vectors |ψ〉 in system V ⊗ W , |v〉 of V , and identity matrix IW of W ,

〈v|ψ〉 denotes (〈v| ⊗ IW )|ψ〉, and 〈ψ|v〉 denotes 〈ψ|(|v〉 ⊗ IW ).

Suppose IV and IW are identity matrices on system V and W , we have:

(IV ⊗ EW )(EV ⊗ IW ) = EV ⊗ EW = (EV ⊗ IW )(IV ⊗ EW ).

The trace of a matrix M is defined as:

Tr (M) =∑j

〈j|M |j〉,

where |j〉 is an orthonormal basis.

B. Postulates of Quantum Mechanics

There are four postulates of quantum mechanics in quantum information the-

ory. They are the basics of the whole theory.

Postulate 1 is about how to characterize a quantum state to in a certain

system. Mathematically, it shows that we can use a unit vector to represent a

pure state.

Postulate 1: Any physical system is associated to a complex Hilbert vector

space, so-called the state space of the system. The quantum state of an isolated

system is completely described by its state vector, which is a unit vector in the

system’s state space.

A density operator of a pure state |ψ〉 is defined by ρ = |ψ〉〈ψ|. We usually

use |i〉 to denote an orthonormal basis of a Hilbert space H. For instance,

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II PRELIMINARIES

in 2-dimensional Hilbert space H2, we usually use |0〉 and |1〉 to denote vectors

[1, 0]T and [0, 1]T . A density operator is just a positive semi-definite matrix with

trace 1.

Mathematically, Postulate 2 shows that the evolution, in the same system

after a certain time, is just a linear transformation over the state space, no

matter what the initial status of a quantum state is. The state after the

evolution is to apply this linear operator to the original state. Postulate 2

further indicates this linear operator should be unitary.

Postulate 2: The evolution of a closed quantum system is described by a

unitary transformation. That is, the state |ψ〉 of the system at time t1 is related

to the state |ψ′〉 of the system at time t2 by a unitary operator U which depends

only on the times t1 and t2,

|ψ′〉 = U |ψ〉.

In practice, we need to observe the quantum system at the right time in

order to obtain relevant information. This requires the introduction of quantum

measurement or quantum observation.

Postulate 3: Quantum measurements are represented as a set of measure-

ment operators Mm, where∑

mM†mMm = I. These operators act on the state

space of the system being measured. The index m is the ordinal number of the

measurement output after the experiment. If the state of the quantum system

is |ψ〉, then the probability of result m taking place is given by

p(m) = 〈ψ|M †mMm|ψ〉

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II PRELIMINARIES

and after the measurement, the finial state of the system should be

Mm|ψ〉√〈ψ|M †

mMm|ψ〉.

For example, in an H2 system, all following sets are quantum measurements:

M0 = |0〉〈0|, M1 = |1〉〈1|,M0 = |0〉〈0|+ cos θ|1〉〈1|, M1 = sin θ|1〉〈1| and

M0 = (1/2)|0〉〈1|+ (√

2/2)|0〉〈0|+ (1/2)|1〉〈1|,M1 = (1/2)|0〉〈1| − (

√2/2)|0〉〈0|+ (1/2)|1〉〈1|.

In fact, for any unitary U , U †U = I and 〈ψ|U †U |ψ〉 = 〈ψ|ψ〉. Generally, we

can treat a unitary evolution as a quantum measurement with one outcome.

All above discussions are only about one sone quantum system. When more

than one subsystems have been combined as a whole system, we name those

states in this system as multi-partite states. The following postulate is the char-

acterisation of a multi-partite quantum state.

Postulate 4: The state space of a composite physical system is the tensor

product of the state spaces of the subsystems.

The simplest example is a product state. If we have n subsystems and each

subsystem has a pure state |ψj〉, 1 ≤ j ≤ n, then the composite state in the

whole system is |ψ1〉 ⊗ |ψ2〉 ⊗ · · · ⊗ |ψn〉, denoted as ⊗nj=1|ψj〉.From Postulate 4, we notice that there are not only product states in the com-

posite systems but also some other pure states: entangled states. For example,

a Bell state (1/√

2)(|00〉+ |11〉) is not a product state.

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II PRELIMINARIES

All pure states can be denoted in this way. For example,

|GHZ〉 = (1/√

2)(|000〉+ |111〉) = [1/√

2 0 0 0 0 0 0 1/√

2]T

and

|W 〉 = (1/√

3)(|001〉+ |010〉+ |100〉) = [0 1/√

3 1/√

3 0 1/√

3 0 0 0]T .

Furthermore, we can define partial trace and reduced density operator.

Suppose |ψ〉 is a multi-partite pure state combining with subsystem A,B, · · · .Partial trace of |ψ〉 on A subsystem can be defined as

∑j〈jA||ψ〉〈ψ||jA〉, denoted

as Tr A(|ψ〉〈ψ|), where |jA〉 is an orthonormal basis for subsystem A. If |ψ〉 is a

composite state with two subsystem A and B, the reduced density operator for

subsystem A is ρB = Tr A(|ψ〉〈ψ|).The local rank of a pure state |φ〉 in some space is the rank of the result after

taking the partial trace on all the other spaces, i.e. the local rank of a 3-qubit

pure state |φ〉 in space A, denoted by rA, is the matrix rank of Tr BC(|φ〉〈φ|).Without loss of generality, we will denote the state with local rank m, n and p

as an m⊗ n⊗ p state.

For instance, |W 〉 and |GHZ〉 are both 2⊗ 2⊗ 2 states.

A Bell state (√

2/2)(|00〉+ |11〉) is a 2⊗ 2 state.

|W 〉⊗2 = (1/3)(|003〉 + |030〉 + |300〉) + (1/24)((|0〉 + |1〉 + |2〉)⊗3 + (−|0〉 −|1〉+ |2〉)⊗3 + (−|0〉+ |1〉 − |2〉)⊗3 + (|0〉 − |1〉 − |2〉)⊗3) is a 4⊗ 4⊗ 4 state.

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II PRELIMINARIES

C. Tensor rank, Schmidt decomposition, majorization and their

applications

In this subsection, we will introduce tensor rank, Schmidt decomposition, and

majorization.

Consider an N -partite quantum state space such that the i-th subsystem

is described by a di-dimensional Hilbert space Hi, where i = 1, . . . , N . The

tensor rank rk(|ψ〉) of a state |ψ〉 ∈N⊗i=1

Hi, is defined as the smallest number

r of product states N⊗i=1

|φiα〉α=1···r, |φiα〉 ∈ Hi, whose linear span contains |ψ〉.

If we ignore the coefficient, this can also be written as rk(|ψ〉) = min r such

that |ψ〉 =r∑

α=1

N⊗i=1

|φiα〉 [9]. Then a tensor Ai,j,k corresponds to a tripartite state

|A〉 =∑

ijk aijk|ijk〉.The tensor rank of a multi-partite pure state (or tensor) characterizes the

minimal number of product states (decomposable tensors) required to linearly

represent the given state. This quantity, as a natural generalization of matrix

rank, has been playing a significant role in mathematics and computer science.

It is the key notion in determining the communication complexity and algebraic

computational complexity. For bipartite states, their tensor ranks are also named

Schmidt rank. Recently it has been gradually realized that the tensor rank is

also an indispensable tool in many problems of quantum information theory.

Indeed, the tensor rank provides a useful criterion for deciding the feasibil-

ity of entanglement transformation: to decide whether one entangled state can

be transformed into another state by using SLOCC. In certain important cases,

the tensor rank can fully characterize the feasibility of SLOCC transformations.

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II PRELIMINARIES

This property has been investigated actively, and has led to a new entanglement

measure, namely Schmidt measure, to quantify the amount of entanglement re-

sources contained in a state. However, it was well known that the tensor ranks

of general multipartite pure states are computationally difficult to determine,

and in fact belong to so called NP-hard problems. On the other hand, it is still

possible and of great interest to determine or estimate the tensor ranks of several

classes of special quantum states that frequently appear in various applications

in quantum information theory.

For any bipartite pure state, tensor rank is also named as Schmidt rank and

is essentially reduced to matrix rank. Let us introduce Schmidt decomposition.

Theorem 1. (Schmidt decomposition) [3] Suppose |φ〉 is a pure state of compos-

ite system HA ⊗ HB. Then there exist orthonormal states |iA〉 for system A

and orthonormal states |iB〉 for system B, such that

|φ〉 =r∑i=1

√λi|iA〉|iB〉,

where λi > 0 andr∑i=1

λi = 1, these λj are so-called Schmidt values of |φ〉. Then

r is the Schmidt rank of |φ〉.

The existence of a Schmidt decomposition in this case is simply due to the

fact that any bipartite state |φ〉 can be isomorphically mapped to a matrix.

After taking the singular value decomposition of matrix, we will get a Schmidt

decomposition of the state . This result is very useful. Many important properties

of quantum systems are completely determined by the eigenvalues of the reduced

density operator of the system, so for a pure state of a composite system such

properties will be the same for both subsystems.

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II PRELIMINARIES

Let |ψ〉 ∈ HA⊗HB⊗HC . We use ρAB to denote the reduced density operator

of |ψ〉〈ψ|, which means we take partial trace on |ψ〉 in HC . As ρAB is a positive

operator, it has a spectral decomposition ρAB =m∑k=1

pk|ψk〉〈ψk| where 0 < pk ≤ 1.

The vector span of |ψk〉 : 1 ≤ k ≤ m is called the support of ρAB and denoted

by supp(ρAB). The following lemma is very useful for determining the tensor

rank of many multipartite pure states.

Lemma 2. [5] Suppose |ψ〉 ∈ HA⊗HB⊗HC. The tensor rank of |ψ〉 is equal to

the minimum number of product states in HA ⊗HB whose linear span contains

supp(ρAB) = supp(Tr C(|ψ〉〈ψ|)).

The SLOCC transformation is of great importance. Firstly, the solution to

this problem are helpful for us to understand the weird nature of quantum en-

tanglement. Secondly, this problem plays a crucial role in understanding the

structures of LOCC and SLOCC protocols. Actually, the results will tell us what

kinds of states are useful resources for quantum (long-distance) communications.

Theorem 3. |ψ〉 can be transformed to |φ〉 via SLOCC, if and only if |ψ〉 and

|φ〉 have the relationship:

|φ〉 = ⊗Ni=1Ei|ψ〉,

where the linear operator Ei acts on space Hi only.

Let us review this fact briefly. It is easy to prove:

(IA ⊗ EB)(EA ⊗ IB) = EA ⊗ EB = (EA ⊗ IB)(IA ⊗ EB). (1)

If |φ〉 = ⊗Ni=1Ei|ψ〉, because all quantum operators Ei denote the successful local

operations, we can construct the operators Ei,√I − E†iEi one by one, which

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II PRELIMINARIES

means, with some positive non-zero probability, we get |φ〉 from |ψ〉 by acting

Ei ⊗j 6=i Ij on |ψ〉. Also, if the SLOCC transform can be done from |ψ〉 to |φ〉,what we have to do in each space Hi is just some local operations on that space

only. We know if we let Ei denote the whole operations on space Hi, for Eq. (1),

the SLOCC transformation from |ψ〉 to |φ〉 can be denoted by |φ〉 = ⊗Ni=1Ei|ψ〉.It is easy to see that the tensor rank is an SLOCC monotone: if |ψ〉 can be

transformed into |φ〉 via SLOCC, then rk(|ψ〉) ≥ rk(|φ〉). In general the converse

is not true. However, a GHZ-equivalent state |ψGHZ〉 can be SLOCC transformed

into |φ〉, if and only if rk(ψGHZ) ≥ rk(φ).

Every LOCC protocol is just a special SLOCC protocol with success prob-

ability 1. It is a generally hard problem to determine whether a multi-partite

state can be transformed into another state. However, after proving a state |ψ〉cannot be transformed into a state |φ〉 via SLOCC, we always know that the

state |ψ〉 cannot be transformed into the state |φ〉 via LOCC.

Before jumping into the study of entanglement transformation, let us first

acquaint ourselves with a few relevant facts about majorization [2]. Majorization

is an ordering on d-dimensional real vectors intended to capture the notion that

one vector is more or less disordered than another. More precisely, suppose

x = (x1, · · · , xd) and y = (y1, · · · , yd) are two d-dimensional vectors. We use

the notation x↓ to mean x re-ordered in decreasing order, x↓1 ≥ x↓2 ≥ · · · ≥ x↓d,

for example, x↓1 is the largest component of x. x ≺ y denotes “y majorizes x”,

ifk∑j=1

x↓j ≤k∑j=1

y↓j , for k = 1, · · · , d, with equality instead of inequality when

k = d.

The connection between majorization and entanglement transformation is eas-

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II PRELIMINARIES

ily stated yet rather surprising. Given a bipartite pure state |ψ〉, λψ denotes a

probability vector whose entries are in descending order of the Schmidt coeffi-

cients of |ψ〉. For instance, if |ψ〉 = 1√2|0〉|0〉+ 1√

6|1〉|2〉+ 1√

3|2〉|2〉, λψ = (1

2, 13, 16).

Nielsen established the following fundamental result:

Theorem 4. [2] A bipartite pure state |ψ〉 can be transformed to another pure

state |φ〉 by LOCC if and only if λψ ≺ λφ.

It is interesting that the tensor ranks of some symmetric states are related

to the polynomial ranks of homogenous polynomials, Strassen’s algorithm, and

multiplications in complex number [10] [11]. These problems may help us in

many other areas including algebraic computational complexity and quantum

information theory.

For computing certain quantum states, we mainly focus on the state |W 〉⊗n.

Also, we know some properties of states |GHZ〉⊗n.

As direct corollaries of the definition, we can readily verify the following simple

properties of tensor rank.

1. The tensor rank is subadditive under tensor product: rk(|φ〉 ⊗ |ψ〉) ≤rk(|φ〉)× rk(|ψ〉).

2. The tensor rank is nondecreasing: rk(|φ〉 ⊗ |ψ〉) ≥ maxrk(|φ〉), rk(|ψ〉).

These properties are very weak, and only can give rough estimation of tensor

ranks. In order to obtain better bounds on the tensor rank, many powerful

techniques have been introduced.

Let us now consider a more general class of symmetric states. We will in-

troduce a bijection between symmetric states and homogenous polynomials, and

17

II PRELIMINARIES

establish an one-to-one correspondence between the symmetric tensor rank and

the polynomial rank. We say anN -partite state Φ is a symmetric state if the state

is invariant if we exchange arbitrary two subsystems. For instance, |00〉+ |11〉 is

a symmetric state over H2⊗H2. Clearly, both |GHZ〉⊗n and |W 〉 are symmetric.

Suppose Psym(⊗nk=1|k〉⊗ak) is the sum of all permutation symmetric prod-

uct states of ak |k〉s in space ⊗Nj=1Hj, where ak are natural numbers and

N =∑n

k=1 ak. The Dicke state is simply a symmetrization of a product vec-

tor from a computation basis:

|D(j1 . . . jd)〉 :=

(N

j1 · · · jd

)−1/2Psym

(|1〉⊗j1 ⊗ · · · ⊗ |d〉⊗jd

). (2)

The symmetric tensor rank of symmetric states |ψ〉 ∈ ⊗Ni=1Hi, denoted by

srk(|ψ〉), is the smallest number r, such that |ψ〉 =∑r

i=1 |ψi〉⊗N , where |ψi〉 is in

Hilbert space Hi. So, for all symmetric states |ψ〉,

rk(|ψ〉) ≤ srk|ψ〉.

Let us define the asymptotic tensor rank of |φ〉 via the following way:

rk∞(|φ〉) = infn≥1

rk(|φ〉⊗n)1/n,

which equals to the number x when we use xn to approximate rk(|φ〉⊗n). One can

easily prove that “inf” can be replaced by “lim” in the above definition. From

rk(|W 〉n) ≥ 2n+1 − 1 [6], we have limn→+∞

rk(|W 〉⊗n)1/n ≥ 2.

Border rank can be defined as follows: border rank of a quantum state |φ〉 is

the minimum r such that there exists a sequence of quantum states whose limit

is |φ〉 and each of the state in the sequence with rank no more than r.

18

III ASYMPTOTIC RATE OF STATE TRANSFORMATION

III. ASYMPTOTIC RATE OF STATE TRANSFORMATION

Multi-partite entanglement has been widely studied [12][13][14][15] since it

is a proven asset to information processing and computational tasks. In order

to quantitatively compare between different types of quantum information re-

sources, the following fundamental entanglement transformation problem arises:

whether a pure N -partite state |ψ〉 can be transformed into another given N -

partite state |φ〉 via LOCC. Nielsen solved all bipartite cases [2]. After that,

multiple-copy entanglement transformation was studied: Duan et al. proved that

entanglement-assisted transformation of bipartite case is asymptotically equiv-

alent to multiple-copy transformation [16]; in a multipartite setting, Ji et al

showed that the entanglement transformation rate between any two genuinely

entangled states are positive, that is, it is always feasible to exactly transform a

genuinely N -partite entangled pure state with sufficient many but a finite num-

ber of copies to any other N -partite state by LOCC [20], where a multipartite

pure entangled state is said to be genuine if it is not in a product form between

any bipartite partition of the parties.

One of the major difficulties in evaluating the entanglement transformation

rate of multipartite case is that the class of LOCC is still not satisfactorily

understood. Another one is the richness of multipartite entanglement. Gen-

erally, there exist incomparable states, even in three-qubit systems. It is still

unclear how to determine whether one multipartite state can be transformed

to another by LOCC. To partially remedy these obstacles, we relax the restric-

tion of LOCC and consider SLOCC [4, 9, 17, 19, 21]. The ability to trans-

form a state |ψ〉 to another state |φ〉 with SLOCC is symbolically expressed as

21


|ψ〉 SLOCC−→ |φ〉. The physical meaning of SLOCC operations is that they can be

implemented by LOCC operations with nonzero probability. In fact, SLOCC has

been used to study entanglement classification [4, 27] and entanglement trans-

formation [5–7]. The whole multi-partite state space can be divided into SLOCC

equivalence classes. For instance, Dur et al observed that within three qubit sys-

tems, there exist two distinct equivalence classes of genuinely tripartite entangled

states,|GHZ〉 = 1√2(|000〉+ |111〉) and |W 〉 = 1√

3(|100〉+ |010〉+ |001〉) [4].

Comparing with LOCC entanglement transformation, SLOCC entanglement

transformation of pure states has a much simpler mathematical structure that

can be directly characterized. In order to consider the asymptotic SLOCC entan-

glement transformation between pure states, we only need to deal with SLOCC

equivalence classes. By employing the concept of tensor rank, which is defined

as the smallest number of product states whose linear span contains the given

state, many interesting results are obtained. For three-qubit systems, it was

showed that 3 copies of GHZ-state can be transformed into 2 copies of W -state

[5]. In [6], we proved that |GHZ〉⊗m SLOCC−→ |W 〉⊗2n is valid if 2m ≥ 7n. Later,

it was demonstrated that SLOCC protocol can transform 4 copies of GHZ-state

to 3 copies of W -state [7]. These increasing lower bounds reflect both the rich-

ness of entanglement and the difficulty of obtaining asymptotic results. These

progresses motivate us to introduce a useful notion of the SLOCC entanglement

transformation rate in the following way:

R(|ψ〉, |φ〉) = supmn

: |ψ〉⊗n SLOCC−→ |φ〉⊗m.

This quantity intuitively characterizes the optimal number of copies of |φ〉 one

can obtain from a single copy of |ψ〉 under SLOCC, in an asymptotic setting.

22


For multi-partite cases (more than three parties), Schmidt decomposition of

general states also does not exist. The calculation of tensor becomes extremely

complicated. We have to consider very special states.

rk(|WN〉) = N , where |WN〉 is an N -qubit generalization of |W 〉, i.e.,

|WN〉 = |0〉⊗N−1|1〉+ |0〉⊗N−2|1〉|0〉+ · · ·+ |1〉|0〉⊗N−1.

We can prove a stronger version:

Lemma 6. For any complex number λ, rk(|WN〉+ λ|0〉⊗N) = N .

Before proceed to the proof, let us point the a very useful fact: |WN〉 and

|WN〉+ λ|0〉⊗N are SLOCC equivalent. Hence they have the same rank. As it is

clear that rk(|WN〉) ≤ N , we also have

rk(|WN〉+ λ|0〉⊗N) ≤ N.

In the following we only need to show that N is a lower bound.

Proof. For N = 1, 2, the result trivially holds. Assume that we have proven the

case of N ≤ T , say rk(|Wm〉 + |0〉⊗m) = m for any m ≤ T . Let us consider the

case of N = T + 1.

Let us prove that T + 1 is a lower bound by contradiction. If there exist

product states |φj〉 ∈ H1 ⊗ · · · ⊗ HT+1, 1 ≤ j ≤ T , s.t.

|WT+1〉+ λ|0〉T =T∑j=1

|φj〉.

More precisely, assume |φj〉 = |ψj1〉 ⊗ · · · ⊗ |ψj(T+1)〉 and |ψji〉 ∈ Hi, then

|WT+1〉+ λ|0〉⊗T+1 =T∑j=1

|ψj1〉 ⊗ · · · ⊗ |ψj(T+1)〉.

24


After checking the space HT+1, we can see there exists j0, s.t. |ψj0(T+1)〉 6= |0〉.There is no harm for us to assume that j0 = T . Then, ∃|ΦT+1〉 ∈ HT+1, s.t.

〈ΦT+1|ψT (T+1)〉 = 0 and |ΦT+1〉 6= |1〉. So, 〈ΦT+1|0〉 6= 0. Now,

〈ΦT+1|(|WT+1〉+ λ|0〉⊗T+1〉) = 〈ΦT+1|0〉|WT 〉+ (λ〈ΦT+1|0〉+ 〈ΦT+1|1〉)|0〉⊗T .

Because of the hypothesis, the rank of this state should be T. However,

〈ΦT+1|(|WT+1〉+ λ|0〉⊗T+1〉)

= 〈ΦT+1|(T∑j=1

|ψj1〉 ⊗ · · · ⊗ |ψj(T+1)〉)

=T∑j=1

〈ΦT+1|ψj(T+1)〉|ψj1〉 ⊗ · · · ⊗ |ψj(T )〉

=T−1∑j=1

〈ΦT+1|ψj(T+1)〉|ψj1〉 ⊗ · · · ⊗ |ψj(T )〉.

This tensor rank of the above state is strictly less than T , thus contradicts

our hypothesis for the case of T . That completes the proof. ut

For Lemma 6, we have rk(|W3〉) = 3. Hence,

R(|GHZ〉, |W 〉) = (log2 rk∞(|φ〉))−1 ≥ (log2 3)−1 ≈ 0.63093.

Let us abbreviate |W3〉 to |W 〉. We also know that

rk(|W 〉⊗2) = 7.

This is because |W 〉⊗2 = |007〉+ |070〉+ |700〉+1

4((|0〉+ |1〉+ |2〉)⊗3

+ (|0〉 − |1〉 − |2〉)⊗3 + (−|0〉+ |1〉 − |2〉)⊗3 + (−|0〉 − |1〉+ |2〉)⊗3)and rk(|W 〉⊗n) ≥ 2n+1 − 1 [6].

This implies

R(|GHZ〉, |W 〉) ≥ (log2

√7)−1 ≈ 0.7124.

25


Let us show rk(|W3〉⊗3) ≤ 16. We have

|W3〉⊗3

= Psym(|0〉⊗2|7〉) + Psym(|1〉|2〉|4〉)+ Psym(|0〉|1〉|6〉) + Psym(|0〉|2〉|5〉) + Psym(|0〉|4〉|3〉)= Psym(|0〉⊗2|7〉) +

1

4(|1〉+ |2〉+ |4〉)⊗3

+ Psym(|0〉 (|3〉+ |5〉) (|2〉+ |4〉 − |1〉)) +1

4(|2〉+ |4〉 − |1〉)⊗3

+ Psym(|0〉 (|3〉+ |6〉) (|1〉+ |4〉 − |2〉)) +1

4(|1〉+ |4〉 − |2〉)⊗3

+ Psym(|0〉 (|6〉+ |5〉) (|2〉+ |1〉 − |4〉)) +1

4(|2〉+ |1〉 − |4〉)⊗3.

Since we have the following nice result from [32]:

rk(Psym(|a〉|b〉|c〉) +1

4|a〉⊗3) ≤ 4,

we can readily figure out that

rk(Psym(|0〉 (|3〉+ |5〉) (|2〉+ |4〉 − |1〉)) + (|2〉+ |4〉 − |1〉)3) ≤ 4.

Hence, we obtain the following result from [7]:

rk(|W3〉⊗3) ≤ 16.

This means

R(|GHZ〉, |W 〉) ≥ (log23√

16)−1 = 0.75.

By applying the techniques of polynomial ranks, we can further obtain the

following results reported in [7]:

rk(|WN〉⊗n) ≥ (N − 1)2n −N + 2, rk(|WN〉⊗2) = 3N − 2.

26


B. R(|GHZ〉, |W 〉)=1

In this section, we prove the validity of the above conjecture by constructing an

SLOCC transformation from n+o(n) copies of GHZ-state to n copies of W -state,

that is, one can obtain 1 copy of W -state, from 1 copy of GHZ-state by SLOCC,

asymptotically. To reach our goal, we introduce a class of tripartite states which

an n-copiesW state can be written into the sum of at most n2 items of. Each state

of this class can be obtained by applying SLOCC operations on an n-copiesGHZ-

state. Then, we show that R(|GHZ〉N , |W 〉N), the entanglement transformation

rate, is also 1 forN -partite state |GHZ〉N = 1√2(|00 · · · 0〉+|11 · · · 1〉) and |W 〉N =

1√N

(|0 · · · 01〉+ · · ·+ |10 · · · 0〉).Our first result is the following,

Theorem 7. For three-qubit state system, we have

R(|GHZ〉, |W 〉) = 1.

That is, for sufficient large n, one can transform n + o(n) copies of GHZ-state

to n copies of W -state by SLOCC. An immediate consequence is that the GHZ

state is asymptotically stronger than the W state under SLOCC, although they

are incomparable at the single copy level.

Generally, for N-partite systems,

R(|GHZ〉N , |W 〉N) = 1.

Again one can obtain |W 〉N from |GHZ〉N at a rate 1 by SLOCC.

For convenience, in the following discussions we omit an unimportant normal-

ized factor and denote directly |W 〉 = |100〉 + |010〉 + |001〉. Before proving the

27


validity of Theorem 7 for three-qubit systems, we first introduce a class of tri-

partite states, |[a, b, c]〉n for any triple (a, b, c) of nonnegative integers such that

a+ b+ c = n. Let B be the following set

B := (a, b, c) : a+ b+ c = n, 0 ≤ a, b, c ≤ n.

For any (a, b, c) ∈ B, one can define an unnormalized state

|[a, b, c]〉n =∑

i⊕j⊕k=(11···1)n,i∈A(a),j∈A(b),k∈A(c)

|i〉|j〉|k〉,

where ⊕ is the bitwise addition modulo 2, (11 · · · 1)n stands for the n−bit string

with ‘1’ in all n positions, and A(·) represents the set of the n−bit strings with

the same Hamming weight, i.e.,

A(l) = i : h(i) = l, i ∈ Zn2,

where Z2 = 0, 1 and the Hamming weight h(i) of an n-bit string i simply

represents the number of ‘1’s in the string.

We can verify the following equation,

|W 〉⊗n =∑

(i,j,k)∈S

|i〉|j〉|k〉 =∑

(a,b,c)∈B

|[a, b, c]〉n, (3)

where i, j, k are n-bit strings, and S is the subset of Zn2 × Zn2 × Zn2 ,

S = (i, j, k) : i⊕ j ⊕ k = (11 · · · 1)n, itjtkt = 0 for any t,

with it the tth bit of i. Namely, S is the set of (i, j, k) such that on each 1 ≤ t ≤ n,

there is only a single ‘1’ in the t-th bit of i, j, and k. The first equality in Eq. (3)

28


follows by calculating |W 〉⊗n in computational basis, while the second equality

follows by observing that

S = (i, j, k) : i⊕ j ⊕ k = (11 · · · 1)n, h(i) + h(j) + h(k) = n.

The following property of tripartite states |[a, b, c]〉n is extremely useful in

proving Theorem 1,

Lemma 8. Any state |[a, b, c]〉n can be obtained from |GHZ〉⊗n by SLOCC.

Proof:— To see the validity of this lemma, we need the following identity,

|[a, b, c]〉n =1

2n

2n−1∑l=0

(∑i∈A(a)

(−1)l·i|i〉)⊗ (∑j∈A(b)

(−1)l·j|j〉)⊗ (∑k∈A(c)

(−1)l·k|k〉).

Here i = (11 · · · 1)n ⊕ (i1i2 · · · in), and l · i is the bitwise inner product of two

n−bit strings l and i. The above identity can be verified by a direct calculation.

We construct three 2n × 2n matrices

E =2n−1∑l=0

∑i∈A(a)

(−1)l·i|i〉〈l|,

F =2n−1∑l=0

∑j∈A(b)

(−1)l·j|j〉〈l|,

G =2n−1∑l=0

∑k∈A(c)

(−1)l·k|k〉〈l|,

Now, it is direct to verify that

(E ⊗ F ⊗G)|GHZ〉⊗n = 2n/2|[a, b, c]〉n,

29


C. Generalization to Dicke states

Both |W 〉 and |GHZ〉 are symmetric states, i.e., those invariant under any

permutation of its parties. Studying the entanglement measure of such states

has attracted a lot of attention[22, 28]. The entanglement transformation prop-

erties of such states are studied [7, 41] and some families of multi-qubit SLOCC

equivalent states are realized by using symmetric states [42, 43].

In order to generalize Theorem 1 to Dicke states, we need evaluate the entan-

glement transformation rate of Dicke states by SLOCC. A general lower bound

is given as follows.

Theorem 9. For Dicke state |D(j1, · · · , jd)〉 with j1 ≥ · · · ≥ jd and N =∑d

i=1 ji,

R(|GHZ〉N , |D(j1, · · · , jd)〉) ≥ (d∑i=2

log2 (ji + 1))−1.

The bound is tight for Dicke state with j1 ≥∑d

i=2 ji.

Proof. The proof of the lower bound part is the direct generalization of Theorem

7.We prove that:

|GHZ〉⊗mNSLOCC−→ |D(j1, j2, · · · , jd)〉⊗n

holds if

2m ≥d∏

k=2

(njk +N − 1

N − 1

)(jk + 1)n.

Before presenting the proof, we introduce some notations: T is used to denote

|1〉, |2〉, · · · , |d〉⊗n, where the tensor product S1 ⊗ S2 of two sets S1 and S2 is

31


defined as |s1〉 ⊗ |s2〉 : |si〉 ∈ Si. Now we associate any |α〉 ∈ T with a d − 1

dimensional vector of natural numbers ~v = (l2, l3, · · · , ld) iff the number of |k〉appearing in |α〉 is lk for all 2 ≤ k ≤ d, ~v is called the characteristic vector of

|α〉, written as C(|α〉) = ~v.

Now we divide T into disjoint subsets A~v according to their characteristic

vector: A~v = |α〉 : C(|α〉) = ~v.Define the B as the following set

B = (~v1, ~v2, · · · , ~vd) : ~v1 + ~v2 + · · ·+ ~vd = (nj2, nj3, · · · , njd),

where ~lk are all vectors of natural numbers.

Now we decompose |D(j1, j2, · · · , jd)〉⊗n according to the computational basis,

and rearrange the elements according to characteristic vectors as follows

|D(j1, j2, · · · , jd)〉⊗n =∑

(~v1,~v2,··· ,~vd)∈B

|[~v1, ~v2, · · · , ~vd]〉n.

Note that here |[~v1, ~v2, · · · , ~vd]〉n is not the superposition of all |α1〉⊗|α2〉 · · ·⊗|αd〉with C(|αk〉) = ~vk, we also require that |α1〉 ⊗ |α2〉 · · · ⊗ |αd〉 does appear in the

decomposition of |D(j1, j2, · · · , jd)〉⊗n.

By noticing that |B| is polynomial of n, we only need to show for any

(~v1, ~v2, · · · , ~vd) ∈ B with 2m ≥d∏

k=2

(jk + 1)n

|GHZ〉⊗mNSLOCC−→ |[~v1, ~v2, · · · , ~vd]〉n.

This is proved by verifying the following equation with wt := e2πit , µ(L) :=

d∏k=2

w∑ni=1 lk,i

jk+1 , and f(α,L) :=n∏i=1

wlsi,ijsi+1 for |α〉 = |s1〉 ⊗ |s2〉 · · · ⊗ |sn〉 with si ∈

32


1, 2 · · · d,|[~v1, ~v2, · · · , ~vd]〉n = 1

d∏k=2

(jk+1)n

∑jdld,1=0···

ld,n=0

· · ·∑j2

l2,1=0···

l2,n=0

µ(L) ·

(∑

C(|α1〉)=~v1f(α1, L)|α1〉)⊗ · · · ⊗ (

∑C(|αd〉)=~vd f(αd, L)|αd〉). ut

In order to show the tightness of the above bound for Dicke state with j1 ≥Πdi=2(ji + 1), we regard |D(j1, · · · , jd)〉 as a bipartite state |ψ〉 by arranging the

first r parties of |D(j1, · · · , jd)〉 into a single party, say Alice, and the rest N − rparties into another single party, Bob, where r = bN/2c. From the definition,

we know that the tensor rank of |D(j1, · · · , jd)〉 is not less than that of |ψ〉.Now, we apply local operator M⊗N on |ψ〉, where M maps |α1〉|α2〉 · · · |αr〉 into

|1〉⊗µ1|2〉⊗µ2 · · · |d〉⊗µd , where µi are the multiplicity of i among α1, α2, · · · , αr.The definition of N is similar.

Observe that the tensor rank of (MA ⊗NB)|ψ〉 equals to f which is the car-

dinality of the following set,

(β1, β2, · · · , βd) : 0 ≤ βi ≤ ji,d∑i=1

βi = r.

For Dicke state with j1 ≥∑d

i=2 ji, we observe that j1 ≥ N/2 ≥ r. Thus the

constraint 0 ≤ β1 ≤ j1 in the above set is automatically satisfied and the cardi-

nality is totally determined by other βi such that i ≥ 2. By a simple counting

arguments we know the cardinality is given by Πdi=2(ji + 1), which is also the

tensor rank of the bipartite state. Noticing that the tensor rank of bipartite

pure state is multiplicative, and tensor rank is a strictly non-increasing quantity

33


under SLOCC, we conclude that

R(|GHZN〉, |D(j1, · · · , jd)〉) ≤ (d∑i=2

log2 (ji + 1))−1,

for j1 ≥ Πdi=2(ji + 1). Thus, (

∑di=2 log2 (ji + 1))−1 is the tight bound for such

Dicke state.

By applying the above technique to the N−partite state |D(1, · · · , 1)〉, we

have the following result about the SLOCC transformation,

Lemma 10. The tensor rank of |D(1, · · · , 1)〉 is not less than(

NbN/2c

). Thus,

|GHZN〉⊗mSLOCC−→ |D(1, · · · , 1)〉 =⇒ 2m ≥

(N

bN/2c

),

Together with the known result that the tensor rank of |D(1, · · · , 1)〉 is not

more than ≤ 2N−1 from [33], we can conclude that its tensor rank is 2N(1+o(1)).

D. Alternative proof

In fact, there exists a easier way to prove R(|GHZ〉, |W 〉) = 1. The following

result was essentially by Bini [35].

Theorem 11. [35] If an N-partite tensor|ψ〉 can be written as |ψ〉+s∑j=1

λj|φj〉 =

b∑j=1

|f(j, λ)〉⊗N , then

rk(|ψ〉⊗n) ≤ (1 + s n)brk(|ψ〉)n.

Let us use this theorem in our case and show the decomposition of |W3〉⊗n in

detail.

34


Theorem 12. rk (|W3〉⊗n) ≤ (n+ 1)2n.

Proof. ∀λ 6= 0, |W3〉+ λ2|111〉 = 12λ

[(|0〉+ λ|1〉)⊗3 − (|0〉 − λ|1〉)⊗3].Due to non-zero Vandermonde determinant, for any nonzero pairwise distinct

λk, 1 ≤ k ≤ n+1, we can always have the solutions µj, 1 ≤ j ≤ n+1, such that1 1 · · · 1

λ21 λ22 · · · λ2n+1

......

. . ....

λ2n1 λ2n2 · · · λ2nn+1

µ1

µ2

...

µn+1

=

1

0...

0

.Then, let us show show the decomposition of |W3〉⊗n:

|W3〉⊗n

= |W3〉⊗n + 0×n−1∑j=0

Psym(|W3〉⊗j|111〉⊗n−j)

= |W3〉⊗nn+1∑k=1

µk +n−1∑j=0

Psym(|W3〉⊗j|111〉⊗n−j)n+1∑k=1

µkλ2n−2jk

=n+1∑k=1

n∑j=0

µkλ2n−2jk Psym(|W3〉⊗j|111〉⊗n−j)

=n+1∑k=1

µk(|W3〉+ λ2k|111〉)⊗n

=n+1∑k=1

µk 12λk

[(|0〉+ λk|1〉)⊗3 − (|0〉 − λk|1〉)⊗3]⊗n.

Hence, rk (|W3〉⊗n) ≤ (n+ 1)2n. ut

For Dicke state |Ψ〉 = |D(j1, · · · , jd)〉 with j1 ≥ · · · ≥ jd and N =∑d

i=1 ji,

brk(|D(j1, · · · , jd)〉) ≤∏d

i=2(ji + 1) [32]. We can also use Theorem 11 to obtain

that an upper bound of rk(|Ψ〉⊗n) is O((∏d

i=2(ji + 1))n).

35


E. Computational complexity of matrix permanent

The motivation for studying the tensor rank of |D(1, · · · , 1)〉 is the connec-

tion between its tensor rank and that of matrix permanent, a homogeneous

polynomial. It is worth noting that tensor rank of homogeneous polynomial has

already been extensively studied in algebraic complexity theory [30, 32, 36–40].

A homogeneous polynomial is a multi-variables polynomial whose nonzero terms

(monomials) all have the same degree.The tensor rank of a homogeneous poly-

nomial P (x1, · · · , xn) is defined as the smallest number r such that P (x1, ...xn)

can be written as the sum of r terms of Li(x1, · · · , xn), where each Li(· · · ) is the

product of d homogeneous linear forms with d the degree of the polynomial.

The permanent of matrix X = (xi,j)N×N is defined as

perm(X) =∑σ∈Sn

N∏i=1

xi,σ(i).

The sum here extends over all elements σ of the symmetric group SN ; i.e. over

all permutations of the numbers 1, 2, · · · , N .

It is easy to see that the tensor rank of matrix permanent is defined as the

minimum number rk such that

perm(X) =rk∑j=1

N∏i=1

Li,j(X),

where Li,j(X) is a linear function of X and Li,j(0) = 0.

The tensor rank of matrix permanent is still unknown, and it relates to the

central problem of computational complexity theory–circuit lower bounds. One

possible direction to study this problem is to restrict the form of∏N

i=1 Li,j(X), for

36


instance, assume that∏N

i=1 Li,j(X) satisfy multilinear property, see [38–40]. In

this case, it is known that the tensor rank of matrix permanent is lower bounded

by 2NΩ(1)

, where Ω(1) is some non-zero constant.

For a more restricted form, each Li,j(X) only depends on the i-th row of X

and j, we can obtain a better lower bound as follows.

Theorem 13. Let X be an N ×N matrix with permanent

perm(X) =

k(N)∑j=1

N∏i=1

N∑k=1

a(j)i,kxi,k.

Then k(N) ≥(

NbN/2c

).

Proof:— Indeed, since each Li,j(X) only depends on the i-th row of X and j,

we can easily verify that

|D(1, · · · , 1)〉 =

k(N)∑j=1

N∏i=1

(N∑k=1

a(j)i,k |k〉).

The proof is completed by applying Lemma 10.

Most part of Section III is the results form our publication [18].

37

IV COMMON RESOURCE VIA LOCAL OPERATIONS AND CLASSICALCOMMUNICATION

IV. COMMON RESOURCE VIA LOCAL OPERATIONS AND

CLASSICAL COMMUNICATION

Given a set of multipartite entangled states, can we find a common state to

prepare them by local operations and classical communication? Such a state, if

exists, will be a common resource for the given set of states. We completely solve

this problem for bipartite pure states case by explicitly constructing a unique

optimal common resource state for any given set of states. In the multipartite

setting, the general problem becomes quite complicated, and we focus on finding

nontrivial common resources for the whole multi-partite state space of given

dimensions. We show that |GHZ3〉 = 1/√

3(|000〉+ |111〉+ |222〉) is a nontrivial

common resource for 3-qubit systems. We also show some properties of the non-

trivial common resource of two N -qubit pure states. We will also study the

common resource in multipartite space system.

The problem of transforming one entangled state to another one by LOCC

is of central importance in quantum entanglement theory. The majorization

characterization can be extended to a class of multipartite pure states having

Schmidt decompositions [2] [31]. Unfortunately, the Schmidt decomposition for

a generic multipartite pure state does not exist. It is still an open problem to

determine whether a general multi-partite pure state can be transformed into

another one by LOCC.

Despite the overall complexity of multipartite entanglement transformations,

we can often find entangled states that can be transformed into any other state

in the same state space by LOCC. These kind of states are called maximally

entangled states, and they exist in spaces if and only if the dimension of one

39


subsystem is no less than the product of dimensions of all other subsystems [8].

Bell states are one such example, which can be transformed into any pure state in

two-qubit systems. As a simple corollary of the dimensionality bound, there is no

maximally entangled state in three-qubit systems. In fact, it is well-known that

three-qubit states can be entangled in two different ways, one class consisting of

so-called W-type states and other consisting of GHZ-type states [4].

In this section, we generalize the notion of maximally entangled states

with respect to LOCC transformations. The problem we study can be best

described through the following scenario. Assume that Alice and Bob are

going to implement a series of quantum information tasks, each one requiring a

different entangled state to perform. However, instead of sharing a multitude of

different states, they wish to share only one type of entangled state and then

transform this state into a different form as needed. So the question is: for a

given set of pure entangled states, is there a certain state which can be locally

transformed into all of them by LOCC? Below, we give a complete solution

to this problem for bipartite systems. In addition, we study the case when

the set of target states is the entire state space for some given dimensions.

When the dimension of one subsystem in this target space is not smaller than

the product of all the other subsystem dimensions, there always exists some

(perhaps higher-dimensional) state that can obtain all states in the target

space [8]. However, this dimensionality condition turns out not to be necessary.

Interestingly, we find a non-trivial state |GHZ3〉 which can be transformed into

any three-qubit pure state by LOCC.

40


Let us now formulate our problem more precisely. Let S = |ψ1〉, |ψ2〉, · · · be a set of (multipartite) states, possibly infinite. A common resource state |ψ〉to S can be transformed into any state in S by LOCC. We say |ψ〉 an optimal

common resource (OCR) if for any other common resource |φ〉 we have either |φ〉can be transformed into |ψ〉 by LOCC, or |φ〉 and |ψ〉 are not comparable under

LOCC.

A. Optimal common resource of bipartite pure states

In general, it is a hard problem to find OCR for a set of multipartite states.

For bipartite pure states, majorization characterizes the LOCC transformation

between two pure state [2].

Nielsen’s result together with the properties of majorization leads us to an

explicit construction of the “unique’’ OCR of a set of bipartite pure states.

Theorem 14. Let S = |φi〉, i ∈ I be a set of d⊗ d pure states, where I is an

index set (finite or infinite). Assume that the Schmidt coefficient vector of |φi〉is given by λφi = (x

(i)1 , · · · , x

(i)d ). Then if I is a finite set, the OCR state |ψ〉 for

S always exists and λψ is unique. |ψ〉 is given by λψ = (y1, · · · , yd), where

yk = mini∈I

k∑j=1

x(i)j −min

i∈I

k−1∑j=1

x(i)j .

Furthermore, if I is infinite, the min sign in above equations should be replaced

with inf.

Let us consider an example to demonstrate the application of the above

theorem. Let the d−dimensional bipartite target set be Sa = |φ〉|λφ1 ≥ a

41


, where a ≥ 1/d. Then an OCR |ψ〉 for Sa can be chosen as |ψ〉 such that

λψ = (a, 1−ad−1 ,

1−ad−1 , · · · ,

1−ad−1). The maximal entangled state 1√

d

d−1∑k=0

|kk〉 is always

a common resource but usually not optimal.

According to Nielsen’s majorization criterion for entanglement transforma-

tion, Theorem 14 is essentially due to the following lemma:

Lemma 15. Suppose that X(k) = (x(k)1 , x

(k)2 , · · · , x(k)d are a set of d–

dimensional vectors where x(k)1 ≥ x

(k)2 ≥ · · · ≥ x

(k)d . There always ex-

ists an optimal vector Y = (y1, · · · , yd) such that Y ≺ X(k) for

any k. Furthermore, Y = (y1, · · · , yd) can be chosen as yk =

min(k∑j=1

x(1)j ,

k∑j=1

x(2)j , · · · ,

k∑j=1

x(n)j ) − min(

k−1∑j=1

x(1)j ,

k−1∑j=1

x(2)j , · · · ,

k−1∑j=1

x(n)j ). Further-

more, if n is infinite, yk = inf(k∑j=1

x(h)j |h = 1, 2 · · · )− inf(

k−1∑j=1

x(h)j |h = 1, 2 · · · ).

Proof. Let us first consider the case that n is finite.

We will complete the proof via the following 3 steps:

Step 1: yk ≥ yk+1.

Suppose min(k∑j=1

x(1)j ,

k∑j=1

x(2)j , · · · ,

k∑j=1

x(n)j ) =

k∑j=1

x(t)j . We have,

yk+1

= min(k+1∑j=1

x(1)j ,

k+1∑j=1

x(2)j , · · · ,

k+1∑j=1

x(n)j )−min(

k∑j=1

x(1)j ,

k∑j=1

x(2)j , · · · ,

k∑j=1

x(n)j )

≤ x(t)j+1 ≤ x

(t)j

≤ min(k∑j=1

x(1)j ,

k∑j=1

x(2)j , · · · ,

k∑j=1

x(n)j )−min(

k−1∑j=1

x(1)j ,

k−1∑j=1

x(2)j , · · · ,

k−1∑j=1

x(n)j )

= yk.

Notice that:k∑j=1

yj = min(k∑j=1

x(1)j ,

k∑j=1

x(2)j , · · · ,

k∑j=1

x(n)j ),

42


so ∀k,∃f(k),k∑j=1

yj =k∑j=1

x(f(k))j .

Step 2: ∀m, Y ≺ X(m).

It is obvious to see thatk∑j=1

yj ≤k∑j=1

x(m)j .

Step 3: If ∀k, Z ≺ X(k), then Z ≺ Y.

Otherwise, if ∃k0,k0∑j=1

zj>k0∑j=1

yj.

k0∑j=1

zj>k0∑j=1

yj =k0∑j=1

x(f(k0))j .

This is a contradiction against with Z ≺ X(f(k0)).

If n is infinite, we modify our proof as follows:

Notice that: ∀k,m,k∑j=1

yj = inf(k∑j=1

x(h)j ) ≤

k∑j=1

x(m)j .

Step 1: yk ≥ yk+1.

Suppose inf(k∑j=1

x(h)j ) = lim

h→∞

k∑j=1

x(g(h))j , where

k∑j=1

x(g(h))j is in descending or-

der. We can also find a sub–sequence (f(h)) ⊂ sequence(g(h)), such that

inf(k∑j=1

x(h)j ) = lim

h→∞

k∑j=1

x(f(h))j , also lim

h→∞x(f(h))k and lim

h→∞x(f(h))k+1 exist. This is be-

cause any infinite bounded sequence must have a monotonic convergent sub–

sequence with a limit.

yk+1 = inf(k+1∑j=1

x(h)j )− inf(

k∑j=1

x(h)j ) = inf(

k+1∑j=1

x(h)j )− lim

h→∞

k∑j=1

xf(h)j

≤ limh→∞

k+1∑j=1

xf(h)j − lim

h→∞

k∑j=1

xf(h)j

≤ limh→∞

xf(h)k+1 ≤ lim

h→∞xf(h)k

≤ limh→∞

k∑j=1

xf(h)j − inf(

k−1∑j=1

x(h)j )

43


= inf(k∑j=1

x(h)j )− inf(

k−1∑j=1

x(h)j )

= yk.

Step 2: ∀m, Y ≺ X(m).

Step 3: If ∀k, Z ≺ X(k), then Z ≺ Y.

Otherwise, if ∃k0,k0∑j=1

zj>k0∑j=1

yj.

k0∑j=1

zj>k0∑j=1

yj = inf(k∑j=1

x(h)j ).

This contradicts the fact that ∀k, Z ≺ X(k). ut

B. Optimal common resource of a special kind of multi-partite pure

states

Xin and Duan pointed out that Nielsen’s majorization result can be extended

to a special class of multi-partite pure states with a generalized Schmidt decom-

positions in 2007 [31].

Theorem 16. [31] Suppose n people,Alice, Bob, ..., and Dana, share an n-partite

pure state |ψ〉 which has a generalized Schmidt decomposition as follows:

|ψ〉 =r∑

k=1

√xk

n⊗j=1

|k〉j, (4)

where for any 1 ≤ j ≤ n, |k〉j|1 ≤ k ≤ r is an orthonormal basis in the j-th

subsystem, and λ = (x1, . . . , xr) represents the Schmidt coefficient vector with

non-increasing order. These people want to transform |ψ〉 to the following state

44


|φ〉 by LOCC:

|φ〉 =r∑

k=1

√yk

n⊗j=1

|k′〉j. (5)

|ψ〉 can be transformed to |φ〉 via LOCC, if and only if λψ ≺ λφ.

We further generalize this theorem to the following case: for a⊗n

j=1 dj quan-

tum system, we say an orthogonal ensemble of states E = |Ψi〉N−1k=0 is locally

permutation invariant if there exists local unitary representation⊗n

j=1 U(g)j

of the permutation group SN such that

eiφgkn⊗j=1

U(g)j |Ψk〉 := |Ψσg(k)〉 ∈ E , ∀g ∈ Sn and arbitrary phases φgk.

Theorem 17. Consider two tripartite states |ψ〉 =∑N−1

k=0

√µk|k〉|Ψk〉 and

|ψ〉 =∑N−1

k=0

√λk|k〉|Ψk〉, where the |Ψk〉 is an orthogonal locally permuta-

tion invariant ensemble. Then |ψ〉 can be transformed into |φ〉 by LOCC iff

(uk) ≺ (λk).

Proof. Let ρλ (resp. ρµ) denote Alice’s reduced state in |ψ〉 (resp. |φ〉). Pro-

ceeding analogously as above, the condition (uk) ≺ (λk) implies that

N−1∑k=0

µj|k〉〈k| =∑g∈SN

pgPg

(N−1∑k=0

λk|k〉〈k|

)P †g =

∑g∈SN

pgPgρλP†g , (6)

where the Pg are permutation matrices acting on the computational basis

and pg is some distribution. Define the set of Kraus operators Mg =

45


√pgρ

1/2λ P †g ρ

−1/2µ g∈SN . Then

Mg ⊗ I⊗ I|ψ〉 =√pg

N−1∑k=0

(ρ1/2λ P †g |k〉)|Ψk〉

=√pg

N−1∑k=0

(ρ1/2λ |σ

−1g (k)〉)|Ψk〉

=√pg

N−1∑k=0

ρ1/2λ |k〉|Ψσg(k)〉

=√pg

N−1∑k=0

√λk|k〉

n⊗j=1

U(g)j |Ψk〉

=√pg I

n⊗j=1

U(g)j |ψ〉. (7)

Necessity of the condition (uk) ≺ (λk) follows from the bipartite theory. ut

For the set of states which are all orthogonal locally permutation invariant

ensemble, we can use Theorem 14 to find its optimal common resource.

C. A non-trivial common resource of 3-qubit system: |GHZ3〉

We shall now move to multipartite setting, and consider the problem “what

is the common resource of the whole system”. For N -partite quantum system

d1⊗· · ·⊗dn where d1 ≥ d2 ≥ · · · ≥ dn, the maximal entangled state exists, in the

sense that all other states in the system can be obtained from the state by LOCC,

if and only if d1 ≥n∏i=2

di [8]. For instance, the state 12(|000〉+|101〉+|210〉+|311〉)

is an OCR of tripartite H4⊗H2⊗H2 system. Interestingly, the OCR exists even

46


if any sub-system’s dimension is less than the product of other sub-system’s

dimensions.

Theorem 18. |GHZ3〉 = 1√3(|000〉 + |111〉 + |222〉) is a common resource of

3-qubit system.

If |ψ〉 can be transformed into |φ〉 via LOCC, we denote this as

|ψ〉 LOCC−→ |φ〉.

If two pure states can be transformed into each one via stochastic local op-

erations and classical communication (SLOCC), they are in the same SLOCC

equivalence class, namely, they are SLOCC-equivalent. Three-qubit pure states

can be divided into product states, W-type and GHZ-type states [4]. GHZ

(1/√

2)(|000〉+ |111〉) state can be transformed into any product states.

Hence, we divide the proof into two cases according to the target states are

W-type or GHZ-type states. The case of GHZ can be further divide into two

sub-cases: orthogonal GHZ and non-orthogonal GHZ.

The following lemma has been shown in [31]. This also is our step i) in all

the cases.

Lemma 19. ∀z0, z1, z2,2∑

k=0

|zk|2 = 1,

|GHZ3〉LOCC−→ z0|000〉+ z1|111〉+ z2|222〉.

Proof. Alice takes the following measurement and sends the result to both Bob

and Charlie,

M1 = z0|0〉〈0|+ z1|1〉〈1|+ z2|2〉〈2|,

47


M2 = z0|0〉〈1|+ z1|1〉〈2|+ z2|2〉〈0|,M3 = z0|0〉〈2|+ z1|1〉〈0|+ z2|2〉〈1|.

The, Bob and Charlie make some unitary operations based on Alice’s mea-

surement outcome, which transform the state to z0|000〉+ z1|111〉+ z2|222〉.If Alice’s outcome is 1, Bob and Charlie do nothing.

If Alice’s outcome is 2, both Bob and Charlie take a unitary operation U =

|0〉〈1|+ |1〉〈2|+ |2〉〈0| on their own subsystems.

If Alice’s outcome is 1, Bob and Charlie take a unitary operation U = |0〉〈2|+|1〉〈0|+ |2〉〈1| on their own subsystems.

ut

a. Protocol of entanglement transformation from |GHZ3〉 to W-type states

If the target state |φ〉 is SLOCC–equivalent to W state, |φ〉 can be written as

x0|000〉+x1|100〉+x2|010〉+x3|001〉, where xk are all positive real numbers and3∑

k=0

x2k = 1 [25]. We can transform |GHZ3〉 to |φ〉 by three following steps:

i) |GHZ3〉LOCC−→

√x20 + x21|000〉+ x2|111〉+ x3|222〉,

ii)√x20 + x21|000〉+x2|111〉+x3|222〉 LOCC−→

√x20 + x21|100〉+x2|010〉+x3|001〉,

iii)√x20 + x21|100〉+ x2|010〉+ x3|001〉 LOCC−→ |φ〉.

Step i) is lemma 19 which z0 =√x20 + x21, z1 = x2 and z2 = x3.

Step ii) :√x20 + x21|000〉+ x2|111〉+ x3|222〉 LOCC−→

√x20 + x21|100〉+ x2|010〉+ x3|001〉

48


Alice takes the measurement:

M1 = (|1〉〈0|+ |0〉〈1|+ |0〉〈2|)/√

2, M2 = (|1〉〈0|+ |0〉〈2| − |0〉〈1|)/√

2.

Bob takes the measurement:

M1 = (|1〉〈1|+ |0〉〈0|+ |0〉〈2|)/√

2, M2 = (|1〉〈1|+ |0〉〈0| − |0〉〈2|)/√

2.

Charlie takes the measurement

M1 = (|1〉〈2|+ |0〉〈1|+ |0〉〈0|)/√

2, M2 = (|1〉〈2|+ |0〉〈1| − |0〉〈0|)/√

2.

Alice transmits her result to Bob.

Bob transmits his result to Charlie.

Charlie transmits its result to Alice.

If they get result 1, they continue. Otherwise, if the gotten result is 2, he or

she should take a Z−operation, Z = |0〉〈0| − |1〉〈1|.Now, the state is

√x20 + x21|100〉+ x2|010〉+ x3|001〉.

Step iii) :√x20 + x21|100〉+ x2|010〉+ x3|001〉 LOCC−→ |φ〉.

(Step iii) is also researched by Kintas and Turgut in 2010 [25].)


M1 = 1√2(|0〉〈0|+ x1√

x20+x

21

|1〉〈1|+ x0√x2

0+x21

|0〉〈1|),

M2 = 1√2(|0〉〈0|+ x1√

x20+x

21

|1〉〈1| − x0√x2

0+x21

|0〉〈1|).Now, the state is (±x0|000〉+ x1|100〉+ x2|010〉+ x3|001〉).If Alice’s result is 1, we already get the target.

If it is 2, Alice transmits 2 to Bob and Charlie. Bob and Charlie make unitary

operation Z = |0〉〈0| − |1〉〈1| on their own part.

49


Then, Alice makes an unitary operation: −Z = −|0〉〈0|+ |1〉〈1|.Finally, the state is x0|000〉+ x1|100〉+ x2|010〉+ x3|001〉.

b. Protocol of entanglement transformation from |GHZ3〉 to GHZ-type

states

Without loss of generality, a GHZ-type pure states |φ〉 can be written

as x|000〉 + y|φAφBφC〉, where a0 = 〈0|φA〉, a1 = 〈1|φA〉, b0 = 〈0|φB〉, b0 =

〈0|φB〉, c0 = 〈0|φC〉, c0 = 〈0|φC〉. a0, a1, b0, b1, c0 and c1 are all real numbers.

GHZ-type pure states can be divided into two kinds: orthogonal or non-

orthogonal. |φ〉 is an orthogonal GHZ state, if a0b0c0 = 0. Otherwise, if a0b0c0 6=0, |φ〉 is an non-orthogonal GHZ state.

The LOCC protocols from |GHZ3〉 to orthogonal GHZ state can be divided

into two steps (suppose c0 = 0, |φC〉 = |1〉, thus, |x|2 + |y|2 = 1):

i) |GHZ3〉LOCC−→ |GHZ2〉,

ii) |GHZ2〉LOCC−→ |φ〉.

Step i) is for lemma 19 with z0 = z1 = 1/√

2 and z2 = 0. This is a protocol

|GHZ3〉LOCC−→ |GHZ2〉.

Step ii) can be farther divided as following:

(Step ii is also researched by Turgut, Gul and Pak in 2010 [24].)

|GHZ2〉LOCC−→ x|000〉+ y|111〉 LOCC−→ x|000〉+ y|φA11〉

50


LOCC−→ x|000〉+ y|φAφB1〉 = |φ〉.

For |GHZ2〉LOCC−→ (x|000〉+ y|111〉) :

Alice takes the measurement: M1 = x|0〉〈0|+y|1〉〈1|, M2 = x|1〉〈1|+y|0〉〈0|.If output is “2”, Alice take an X−operation, X = |1〉〈0|+ |0〉〈1|.If output is “1”, they continue.

For x|000〉+ y|111〉 LOCC−→ x|000〉+ y|φA11〉 :


M1 =√22

(|0〉〈0|+ |φA〉〈1|),M2 =√22

(|0〉〈0| − |φA〉〈1|).If output is “2”, Alice transmits the result to Charlie and Charlie takes an

Z−operation, Z = |0〉〈0| − |1〉〈1|.If output is “1”, they continue.

The protocol for x|000〉 + y|φA11〉 LOCC−→ x|000〉 + y|φAφB1〉 is similar. Bob

takes the measurement: M1 =√22

(|0〉〈0|+ |φB〉〈1|), M2 =√22

(|0〉〈0| − |φB〉〈1|).If output is “1”, finish. If output is “2”, Bob transmits the result to Charlie and

Charlie takes an Z−operation, Z = |0〉〈0| − |1〉〈1|.

The LOCC protocols from |GHZ3〉 to non-orthogonal GHZ state can be

divided to 4 steps: (this is the most complicated part of the whole protocol)

i) |GHZ3〉LOCC−→ |Ψ1〉,

ii) |Ψ1〉LOCC−→ |Ψ2〉,

iii) |Ψ2〉LOCC−→ x|00φC〉+ y|φAφB0〉,

51


iv) x|00φC〉+ y|φAφB0〉 LOCC−→ x|000〉+ y|φAφBφC〉,where |Ψ1〉 =

√|xc0 + ya0b0|2 + |ya1b0|2 |000〉+ |xc1||111〉+ |yb1||222〉

and |Ψ2〉 =√|xc0 + ya0b0|2 + |ya1b0|2 |000〉+ |xc1||101〉+ |yb1||210〉.

Step i) is for lemma 19, with z0 =√|xc0 + ya0b0|2 + |ya1b0|2,

z1 = |xc1| and z2 = |yb1|.Step ii) |Ψ1〉

LOCC−→ |Ψ2〉:Bob takes the measurement:

M1 =√22

(|1〉〈2|+ |0〉〈0|+ |0〉〈1|),M2 =√22

(|1〉〈2|+ |0〉〈0| − |0〉〈1|).Charlie takes the measurement:

M1 =√22

(|1〉〈1|+ |0〉〈2|+ |0〉〈0|),M2 =√22

(|1〉〈1| − |0〉〈2|+ |0〉〈0|).They transmit their results (1 or 2) to Alice. Suppose Bob gets β and Charlie

gets α. Then, Alice takes the unitary operation

M = |0〉〈0| − (−1)β|1〉〈1| − (−1)α|2〉〈2|.Now, the state is |Ψ2〉.Step iii): Alice does the measurement

M1 = ( (xc0+ya0b0)|0〉+ya1b0|1〉√|xc0+ya0b0|2+|ya1b0|2

〈0|+ xc1|xc1| |0〉〈1|+

yb1|yb1|(a0|0〉+ a1|1〉)〈2| )/2,


〈0|+ −xc1|xc1| |0〉〈1|+

yb1|yb1|(a0|0〉+ a1|1〉)〈2| )/2,


〈0|+ xc1|xc1| |0〉〈1|+

−yb1|yb1| (a0|0〉+ a1|1〉)〈2| )/2,


〈0|+ −xc1|xc1| |0〉〈1|+

−yb1|yb1| (a0|0〉+ a1|1〉)〈2| )/2

The resulting states are all LOCC-equivalent to ((xc0 + ya0b0)|0〉 +

ya1b0|1〉)|00〉+ xc1|001〉+ yb1(a0|0〉+ a1|1〉)|10〉 = x|00φC〉+ y|φAφB0〉.It is also LOCC-equivalent to x|000〉 + y|φAφBφC〉, which can be done in next

step.

52


Step iv): Charlie takes a local unitary operation: |φC〉〈0|+ (c1|0〉 − c0|1〉)〈1|.Finally, the state is x|000〉+ y|φAφBφC〉.

In conclusion, we introduce a notion of optimal common resource for a set of

entangled states, and explicitly construct it for any bipartite pure state set. We

also show that |GHZ3〉 state is a nontrivial common resource for 3-qubit sys-

tem, and conjecture its optimality. We hope this problem will stimulate further

research interest in entanglement transformation theory.

53

V COMMON RESOURCE VIA STOCHASTIC LOCAL OPERATIONSAND CLASSICAL COMMUNICATION

V. COMMON RESOURCE VIA STOCHASTIC LOCAL

OPERATIONS AND CLASSICAL COMMUNICATION

Let S = |ψ1〉, |ψ2〉, · · · be a set of (multipartite) states, possibly infinite. A

common resource state |ψ〉 to S is the state which can be transformed into any

state in S by SLOCC. We say |ψ〉 an optimal common resource (OCR) if for any

other common resource |φ〉 we have either |φ〉 can be transformed into |ψ〉 by

SLOCC, or |φ〉 and |ψ〉 are not comparable under SLOCC.

Without loss of generality, a⊗b denotes b subsystems with dimension a. Let

me ignore unimportant normalization coefficients because we will only talk about

SLOCC protocols in this section. We find that some 3 ⊗ 2⊗3 state is optimal

for two four-qubit symmetric states. For two general N -qubit states, we need a

4⊗ 2⊗N−1 state or a 3⊗N−2 ⊗ 2⊗ 2 state.

In fact, if two N -qubit pure states |ψ1〉 and |ψ2〉 can be SLOCC transformed

by the same state |ψ0〉 and state |ψ0〉 is a pure state with local rank 3⊗ 2⊗N−1,

the intersection between their support in H⊗N−12 space should not be empty.

Theorem 20. The common resource |ψ0〉 of two N−qubit pure states |ψ1〉 and

|ψ2〉 is with local rank 3⊗t ⊗ 2⊗N−t if and only if there exists a (N − t)-qubit |φ〉which is SLOCC-equivalent to a state in the support of |ψ1〉 and also to a state in

the support of |ψ2〉, where partial trace of these two supports are taken in those

H⊗t3 subsystems.

Proof. If the common resource |ψ0〉 of two N -qubit pure states |ψ1〉 and |ψ2〉is with local rank 3⊗t ⊗ 2⊗N−t, without loss of generality, suppose |ψ1〉 =⊗t

j=1Ej1|ψ0〉 and |ψ2〉 =⊗t

j=1Ej2|ψ0〉. Because |ψ1〉 and |ψ2〉 are two N -qubit

55


pure states. All Ej0 and Ej1 are rank 2. In H3 system, for any two rank 2 matri-

ces Ej0 and Ej1, there always exists |µj〉 and |νj〉 such that E†j1|µj〉 = E†j2|νj〉 6= 0

(notice that we ignore unimportant positive coefficients before E†j1|µj〉).

(t⊗

j=1

〈µj|)|ψ1〉 = (t⊗

j=1

〈µj|Ej1)|ψ0〉 = (t⊗

j=1

〈νj|Ej2)|ψ0〉 = (t⊗

j=1

〈νj|)|ψ2〉. (8)

Denote this (N−t)-qubit entangled state as |φ〉. |φ〉 is in the support |ψ1〉 and

also in the support of |ψ2〉, where partial trace of these two supports are taken

in those H⊗t3 subsystems.

The proof of another direction is as following.

For convenience of reading, use |j〉 to denote a t-qubits product state. The

number m at the k-th bit of j’s binary form means |m〉 on the k-th qubit sub-

system of |j〉 respectively. For example, when t = 4, |0〉 = |0000〉, |6〉 = |0110〉and |13〉 = |1101〉.

For two N−qubit pure states |ψ1〉 and |ψ2〉, if there exists a (N − t)-qubit |φ〉which is SLOCC-equivalent to a state in the support of |ψ1〉 and also to a state in

the support of |ψ2〉, where partial trace of these two supports are taken in those

H⊗t2 subsystems, without loss of generality, suppose |ψ1〉 = |0〉|φ〉+∑2t−1

j=1 |j〉|ψ1,j〉and |ψ2〉 = |0〉|φ〉 +

∑2t−1j=1 |j〉|ψ2,j〉, where |ψ1,j〉 and |ψ2,j〉 are all (N − t)-qubit

states.

Suppose E = |0〉〈0| + |2〉〈1|. Put operator E⊗t in the front t subsystems onto

|ψ2〉. We are going to prove that |ψ0〉 = |ψ1〉 + E⊗t ⊗ I⊗N−t|ψ2〉 − |0〉|φ〉 is a

common resource of two N−qubit pure states |ψ1〉 and |ψ2〉, alos |ψ0〉 with local

rank 3⊗t ⊗ 2⊗N−t.

56


Suppose A1 = |0〉〈0| + |1〉〈1| and A2 = |0〉〈0| + |1〉〈2|. A⊗t1 and A⊗t2 are acted

on those rank 3 subsystems.

We finish the whole proof after checking |ψ1〉 = A⊗t1 ⊗ I⊗N−t|ψ0〉 and

|ψ2〉 = A⊗t2 ⊗ I⊗N−t|ψ0〉. ut

For any two N -qubit pure states |ψ0〉 and |ψ1〉, one can always find some

4 ⊗ 2⊗n−1 state as their common resource, because |ψ0〉 + M ⊗ I⊗N−1|ψ1〉 is

their 4⊗ 2⊗n−1 common resource state, where M = |2〉〈0|+ |3〉〈1|. We can find a

different recourse: a 3⊗m⊗ 2n−m state. We show our research case by case based

on number of qubits.

Any two 2-qubit entangled pure states are SLOCC-equivalent.

A. Tripartite entangled pure states

There are 25 SLOCC equivalence classes in 2⊗ 3⊗ 6 system. This result is in

Ref. [44]. Because of the property of SLOCC, without loss of generality, we use

states to denote their equivalence class and ignore the unimportant coefficients

of these states.

Product state with local rank 1 ⊗ 1 ⊗ 1 are all SLOCC-equivalent, namely,

|000〉. In 1⊗2⊗2 system, there exists a SLOCC-equivalence class: |000〉+ |011〉.Similar for 2⊗ 1⊗ 2 system and 2⊗ 2⊗ 1 system.

In 2⊗ 2⊗ 2 system, there exist two more SLOCC-equivalence classes:

GHZ: |000〉+ |111〉 (ABC − 1) and W: |001〉+ |010〉+ |100〉 (ABC − 2).

In 2⊗ 2⊗ 3 system, there exist two more SLOCC-equivalence classes: |000〉+|011〉+ |102〉 (ABC − 3) and |000〉+ |011〉+ |101〉+ |112〉 (ABC − 4).

57


In 2⊗ 2⊗ 4 system, there exists one more SLOCC-equivalence class:

|000〉+|011〉+|102〉+|113〉 (ABC−5) which is so called the maximally entangled

state in 2⊗ 2⊗ 4 system.

In 2⊗ 3⊗ 2 system, there exist two more SLOCC-equivalence classes: |000〉+|011〉+ |120〉 (ABC − 6) and |000〉+ |011〉+ |101〉+ |121〉 (ABC − 7).

In 2⊗ 3⊗ 3 system, there exist six more SLOCC-equivalence classes:

|000〉+ |011〉+ |122〉 (ABC − 9),

|000〉+ |011〉+ |111〉+ |122〉 (ABC − 8),

|000〉+ |011〉+ |022〉+ |101〉 (ABC − 10),

|001〉+ |100〉+ |111〉+ |022〉+ |122〉 (ABC − 11),

|001〉+ |012〉+ |100〉+ |111〉+ |122〉 (ABC − 12) and

|000〉+ |011〉+ |102〉+ |121〉 (ABC − 13).

In 2⊗ 3⊗ 4 system, there exist five more SLOCC-equivalence classes:

|000〉+ |011〉+ |022〉+ |103〉 (ABC − 14),

|001〉+ |100〉+ |112〉+ |123〉+ |023〉 (ABC − 15),

|001〉+ |013〉+ |100〉+ |112〉+ |123 (ABC − 16)〉,|001〉+ |012〉+ |100〉+ |111〉+ |123〉 (ABC − 17) and

|001〉+ |012〉+ |023〉+ |100〉+ |111〉+ |122〉 (ABC − 18).

In 2⊗ 3⊗ 5 system, there exist two more SLOCC-equivalence classes:

|001〉+ |013〉+ |100〉+ |112〉+ |124〉 (ABC − 19) and

|001〉+ |013〉+ |024〉+ |100〉+ |112〉+ |123〉 (ABC − 20).

In 2⊗ 3⊗ 6 system, there exists one more SLOCC-equivalence class:

|000〉 + |011〉 + |022〉 + |103〉 + |114〉 + |125〉 (ABC − 21) which is so called the

maximally entangled state in 2⊗ 3⊗ 6 system.

58


The relationship of them is as following:

Figure 5.1. Partial order relation of SLOCC-equalvence classes in 2 ⊗ 3 ⊗ 6

system [44].

59


In following discussion, a m ⊗ n ⊗ p state |ψ〉, m,n, p ≥ 2, denotes the

tripartite state with local rank m,n and p. By checking the above map, we can

prove the following lemmas.

Lemma 21.

1) For any 2⊗ 2⊗ 3 entangled pure state |ψ〉 and any pure states |φ〉 in 3-qubit

system, the SLOCC protocol from |ψ〉 to |φ〉 always exist.

2) Any 2 ⊗ 2 ⊗ 3 entangled pure states can be transformed from a 2 ⊗ 3 ⊗ 3

state in the following SLOCC−equivalence class: |000〉 + |011〉 + |111〉 + |122〉,|000〉+ |011〉+ |022〉+ |101〉+ |122〉 and |000〉+ |011〉+ |022〉+ |101〉+ |112〉.3) Any 2⊗ 3⊗ 3 entangled pure states can be transformed from a 2⊗ 3⊗ 5 state

in the following SLOCC−equivalence class: |001〉+ |013〉+ |100〉+ |112〉+ |124〉and |001〉+ |013〉+ |024〉+ |100〉+ |112〉+ |123〉.

No 2⊗ 3⊗ 4 state is a common resource for 2⊗ 3⊗ 3 system.

4) Any 2 ⊗ 3 ⊗ 4 entangled pure states can be transformed from a 2 ⊗ 3 ⊗ 6

maximally entangled state.

No 2⊗ 3⊗ 5 state is a common resource for 2⊗ 3⊗ 4 system.

Proof. 1) we see both ABC − 3 and ABC − 4 can be transformed to either

ABC − 1 or ABC − 2 via SLOCC.

2) ABC−8, ABC−11 and ABC−12 can be transformed to either ABC−3

or ABC − 4 via SLOCC.

3) Neither ABC − 14 nor ABC − 16 can be transformed into ABC − 8 via

SLOCC. Neither ABC − 15 nor ABC − 18 can be transformed into ABC − 10

via SLOCC. ABC − 17 cannot be transformed into ABC − 13 via SLOCC.

We notice that if a state |ψ〉 can be transformed into ABC−15 and ABC−16

60


via SLOCC, |ψ〉 will be a common resource for all 2⊗ 3⊗ 3 pure state. Hence,

ABC − 19 and ABC − 20 can be transformed to any 2⊗ 3⊗ 3 states.

4) 6 = 2 × 3, so ABC − 21 is a maximally entangled state. Of cause, it is a

common resource for 2⊗ 3⊗ 4 system.

ABC − 19 cannot be transformed into ABC − 18 via SLOCC and ABC − 20

cannot be transformed into ABC − 14 via SLOCC. Hence, no 2⊗ 3⊗ 5 state is

a common resource for 2⊗ 3⊗ 4 system. ut

In fact, we also have the following lemmas.

Lemma 22. A tripartite entangled pure state |φ〉 can be transformed to a 2⊗2⊗3,

2⊗ 3⊗ 2 or 3⊗ 2⊗ 2 entangled pure state via SLOCC, if and only if a local rank

of |φ〉 is at least 3.

Proof. Suppose local rank of the third subsystem in |φ〉 is at least 3, denoted by

rC(|φ〉) ≥ 3. Take the Schmidt decomposition on AB − C separation:

|φ〉 =∑

j |φj〉AB|φj〉C , where |φj〉C are orthogonal to each other and |φj〉AB are

also orthogonal to each other. As rA(|φ〉) ≥ 2 and rB(|φ〉) ≥ 2, without loss of

generality, we can suppose∑3

j=1 |φj〉AB|φj〉C , denoted by |φ〉, is the state which

local ranks are all at least 2 and rC(|φ〉) ≥ 3.

We can also take the Schmidt decomposition on A−BC or B−AC separation.

Hence, we can suppose the local ranks of the tripartite entangled pure state |φ〉is at most 3.

If |φ〉 is already a 2 ⊗ 2 ⊗ 3, 2 ⊗ 3 ⊗ 2 or 3 ⊗ 2 ⊗ 2 entangled pure state, we

finish the proof. Because of relationship in SLOCC equivalence class of 2⊗ 3⊗ 6

system (section“Previous result”), if a 2⊗ 3⊗ 3, 3⊗ 2⊗ 3 or 3⊗ 3⊗ 2 entangled

pure state, we also finish the proof [44].

61


If |φ〉 is a 3⊗ 3⊗ 3 entangled pure state, take the Schmidt decomposition on

AB−C separation: |φ〉 =3∑j=1

|φj〉AB|φj〉C , rk(|φ1〉AB) ≥ rk(|φ2〉AB) ≥ rk(|φ3〉AB).

If rk(|φ1〉AB) = 3, take |1〉〈1| + |2〉〈2| on its third subsystem. The result will

be a 3⊗ 3⊗ 2 state. We can transform it into a 3⊗ 2⊗ 2 state [44].

If rk(|φ1〉AB) = 2, then suppose |φ1〉AB = |0〉A|0〉B + |1〉A|1〉B. rB(|φ〉) = 3,

so 〈2|B|φ2〉 6= 0 or 〈2|B|φ3〉 6= 0. Suppose 〈2|B|φ2〉 6= 0. Take |1〉〈1|+ |2〉〈2| on its

third subsystem. The result will be a 3 ⊗ 3 ⊗ 2 or 2 ⊗ 3 ⊗ 2 state. No matter

what the state is, it can be transformed it into a 2⊗ 3⊗ 2 state via SLOCC [44].

If rk(|φ1〉AB) = 1, then rk(|φ2〉AB) = rk(|φ3〉AB) = 1. |φ〉 is SLOCC-equivalent

to |000〉+ |111〉+ |222〉. Take |1〉〈1|+ |2〉〈2| on its third subsystem. The result will

be a 3⊗ 3⊗ 2 state. It can be transformed it into a 2⊗ 3⊗ 2 state via SLOCC

[44]. ut

A tripartite entangled pure state is a tripartite pure state which any of its

local rank is at least 2. This lemma implies the following theorem.

Theorem 23. A tripartite entangled pure state can be SLOCC transformed into

any pure state in 3-qubit system, if and only if one of its local ranks is at least 3.

Proof. Three qubit can be entangled in two different ways, which is GHZ-type

states and W-type states [4]. Namely, they are in the different SLOCC equiva-

lence classes. Hence, in 3-qubit system, there is no common resource of the whole

system. If a tripartite entangled pure state can be SLOCC transformed into any

pure state in 3-qubit system, one of its local ranks is at least 3. Otherwise, it will

be in 3-qubit system, which will cause contradiction.

If one of a tripartite entangled pure state’s local ranks is at least 3, it can be

62


transformed into a 2 ⊗ 2 ⊗ 3, 2 ⊗ 3 ⊗ 2 or 3 ⊗ 2 ⊗ 2 entangled pure state via

SLOCC by Lemma 22. Any one of these three type states can be transformed

into any pure state in 3-qubit system by Lemma 21.

ut

B. Multi-partite qubit pure states

For 4-qubit case, we should use a 4⊗ 2⊗ 2⊗ 2 or a 3⊗ 3⊗ 2⊗ 2 state.

Not all two 4-qubit states can transformed by a 3⊗ 2⊗3. Consider |GHZ4〉 =

|0000〉 + |1111〉 and |ψ〉 = |0000〉 + |1100〉 + |0110〉 + |0011〉 + |1001〉. They

cannot be transformed from a 3⊗ 2⊗ 2⊗ 2 state via SLOCC. The reason is, any

3 subsystems’ support of |GHZ4〉 only has product states or GHZ-type states.

However, in any 3-qubit support of |ψ〉 doesn’t have product states or |GHZ3〉, i.

e. span|000〉+ |110〉+ |011〉, |100〉+ |001〉 has no product states or GHZ-type

states.

Based on 4-qubit example, we have examples for more general case.

Suppose |ψ1〉 = |0〉⊗N−3|W 〉+|1〉⊗N−3|W ′〉 and |ψ2〉 = |GHZN〉, where |W ′〉 =

a|100〉+ b|010〉+ c|001〉 and (a− b)(b− c)(c− a) 6= 0. In the last 3 subsystems,

the support of |ψ1〉 only contains W -type and 1 ⊗ 2 ⊗ 2 (also, 2 ⊗ 1 ⊗ 2 and

2⊗2⊗1 ) states. However, the support of |ψ2〉 in the last 3 subsystems contains

|GHZ〉-type states and product states. Hence, the common result of these two

states is not with local rank 3⊗N−32⊗3.

This is a good example for the following theorem:

Theorem 24. No 3⊗N−32⊗3 state is a common resource for the whole N-qubit

63


system.

Hence, the common resource in kind 3⊗a2⊗N−a of N -qubit system should have

the property a ≥ N − 2. Before checking whether equality can be achieved, we

introduce a lemma firstly.

Lemma 25. [45] There always exists a product state in the span of any 2-qubit

pure states.

Therefore, we have have this following theorem:

Theorem 26. For any two given N-qubit pure states, one of their SLOCC com-

mon resources is a pure state with local rank 3⊗N−22⊗2.

Before jumping into the disscusion of symmetric states, let us review a useful

tool to check whether a 3-qubit state is a GHZ-type state. A 3-qubit pure state

|ψ〉 can be written as∑7

k=0 ak|k〉, where |k〉 = |jAjBjC〉, k = 4jA + 2jB + jC is

the value of a binary number (jAjBjC).

Theorem 27. [23] A 3-qubit pure state |ψ〉 =∑7

k=0 ak|k〉 is a GHZ-type state,

if and only if (a0a7 − a2a5 + a1a6 − a3a4)2 − 4(a2a4 − a0a6)(a3a5 − a1a7) 6= 0.

We can prove the following lemma directly by using Theorem 27.

Lemma 28. If any two symmetric W -type states are not SLOCC equivalent,

there exists at least one other type pure states in the span of these two states.

Proof. (By contradiction.) Denote two SLOCC in-equivalent symmetric W -type

states as |ψ0〉 and |ψ1〉. By the symmetric property, there exists operator E =

EA = EB = EC , such that EA⊗EB ⊗EC |ψ1〉 = |W 〉. Suppose E ⊗E ⊗E|ψ0〉 =

64


b0|000〉+ b1(|001〉+ |010〉+ |100〉) + b2(|011〉+ |101〉+ |110〉) + b3|111〉. According

to Theorem 27 [23], b20b23 + 4b0b

32 + 4b31b3 − 6b0b1b2b3 − 3b21b

22 = 0. Hence, we also

have:

∀λ, b20b23 + 4b0b32 + 4(b1 + λ)3b3 − 6b0(b1 + λ)b2b3 − 3(b1 + λ)2b22 = 0.

If this lemma is not true, b3 = b2 = 0.

E ⊗ E ⊗ E|ψ0〉 = b0|000〉+ b1(|001〉+ |010〉+ |100〉).

This means a product state is in their span or ∃b1, |ψ0〉 = b1|ψ1〉.However, the second possibility is impossible because it is a contradiction

to |ψ0〉 and |ψ1〉 are not SLOCC equivalent. The first possibility also causes a

contradiction.

Therefore, this lemma is true. ut

Now, we can prove a property of N -qubit symmetric entangled pure state

case:

Theorem 29. When N > 3, any 3-qubit support of an N-qubit symmetric en-

tangled pure state cannot just have W -type state in it.

Proof. (By contradiction.) There are at least two states in any 3-qubit support of

an entangled N -qubit pure state. Suppose they are both W -type states, denoted

as |ψ0〉 and |ψ1〉. Because of Lemma 28, in the span of |ψ0〉 and |ψ1〉, there exists

at least one other type pure states in the span of these two states. This is a

contraction. ut

In 3-qubit symmetric system, there are only 3 SLOCC-equavelence classes:

W , GHZ and product states. Hence, any two 4-qubit symmetric entangled

65


states will have a common SLOCC-equavelence state in their 3-qubit subsystems’

support by Theorem 29. This means if the optimal common resource of any two

given N -qubit symmetric states is in the system with local rank 3⊗t ⊗ 2N−t, t is

at most N − 3.

Before extending this property to general cases, let’s introduce a lemma.

Lemma 30. For any t < [N−12

], the tensor rank of any state in the following set

span (Psym(|0〉⊗[N2 ]−t, |1〉⊗[N+12

]), Psym(|0〉⊗[N2 ]−t+1, |1〉⊗[N+12

]−1), · · · ,Psym(⊗|0〉[N2 ], |1〉⊗[N+1

2]−t) is at least 3.

Proof. When t < [N−12

], t ≤ [N−32

], so [N+12

] − t ≥ 2 and [N2

] − t ≥ 1. Hence,

|0〉⊗n and states with kind of |0〉n−1|1〉 are not in this set. Hence, any state

|φ〉 in this set can be written as[(N−3)/2]∑

j=2

λjPsym(|0〉⊗[N2 ]−j, |1〉⊗j). This means

〈0|⊗n|φ〉 = 〈1|⊗n|φ〉 = 0, and for any kind of |0〉n−1|1〉 state |ψ〉, 〈ψ|φ〉 = 0.

We have: rk(|φ〉) 6= 1, otherwise |φ〉 = |µ〉⊗n. However, 〈0|⊗n|φ〉 = 0 =

〈1|⊗n|φ〉 means 〈0|µ〉 = 〈1|µ〉 = 0. This is a contradiction with |µ〉 6= 0.

We also have: rk(|φ〉) 6= 2, otherwise, because 〈1|⊗n|φ〉 = 0, without loss of

generality, we suppose |φ〉 =⊗n

k=1(ak|0〉+ |1〉)−⊗n

k=1(bk|0〉+ |1〉). However, for

any kind of |0〉n−1|1〉 state |ψ〉, 〈ψ|φ〉 = 0. This means for any k, ak = bk. It is a

contradiction with |φ〉 6= 0.

Hence, rk(|φ〉) ≥ 3.

ut

Theorem 31. If an N-qubit symmetric system’s optimal common resource is a

kind of 3⊗t ⊗ 2⊗N−t state, t ≥ [N−12

].

66


Proof. This is more general. Let us check |GHZn〉 = |0〉⊗n + |1〉⊗n and

Psym(|0〉⊗[N2 ], |1〉⊗[N+12

]).

When t < [N−12

], 2⊗N−t support of |GHZn〉 just contains product states and

|GHZN−t〉 state which is a rank 2 state.

However, 2⊗N−t support of Psym(|0〉⊗[N2 ], |1〉⊗[N+12

]) is span (Psym(|0〉⊗[N2 ]−t,

|1〉⊗[N+12

]), Psym(|0〉⊗[N2 ]−t+1, |1〉⊗[N+12

]−1), · · · , Psym(⊗|0〉[N2 ], |1〉⊗[N+12

]−t).

The rank of all the state in this set is at least 3 by Lemma 30. Hence,

|GHZn〉 = |0〉⊗n + |1〉⊗n and Psym(|0〉⊗[N2 ], |1〉⊗[N+12

]) do not have a common

3⊗t ⊗ 2⊗N−t resource.

ut

In summary, we show that two 4-qubit pure symmetric states can be obtained

from some 3⊗2⊗2⊗2 state by stochastic local operations and classical commu-

nication. For two given N -qubit pure states, we can always find some 3⊗N−22⊗2

state is their resource. The number N − 2 is optimal in the condition that we

can just expand the dimensions of the spaces one by one. We also show some

properties of optimal resource of two N -qubit pure states.

67

VI CONCLUSION

VI. CONCLUSION

We introduce a notion of entanglement transformation rate to characterize

the asymptotic comparability of two multi-partite pure entangled states under

stochastic local operations and classical communication (SLOCC). For two well

known SLOCC inequivalent three-qubit states GHZ and W, we show that the

entanglement transformation rate from GHZ to W is exactly 1. That means that

we can obtain one copy of W-state, from one copy of GHZ-state by SLOCC,

asymptotically. We then apply similar techniques to obtain a lower bound on

the entanglement transformation rates from an N-partite GHZ-state to a class

of Dicke states, and prove the tightness of this bound for some special cases

that naturally generalize the W state. A new lower bound on the tensor rank

of matrix permanent is also obtained by evaluating the the tensor rank of Dicke

states.

It is completely solved that how to find an optimal common resource for

bipartite pure states case by explicitly constructing a unique optimal common

resource state for any given set of states via LOCC. In the multi-partite set-

ting, the general problem becomes quite complicated, and we focus on finding

non-trivial common resources for the whole multi-partite state space of given di-

mensions. We show that |GHZ3〉 = (1/√

3)(|000〉+ |111〉+ |222〉) is a non-trivial

common resource for 3-qubit systems via LOCC.

We also show some properties of the non-trivial common resource of two N -

qubit pure states, N -qubit systems, and symmetric systems via SLOCC.

The most interesting open problem is what is the optimal common resource

of 3-qubit system via LOCC. Is it with local rank 2⊗ 2⊗ 3 or 2⊗ 3⊗ 3? As we

69

VI CONCLUSION

know, any common resource of 3-qubit system via LOCC has at least one local

rank not less than 3. However, we have not found any state, with local rank

2⊗ 2⊗ 3 or 2⊗ 3⊗ 3, which is a common resource of 3-qubit system via LOCC.

We are also interested in a conjecture put forward by Comon et al.: for an

arbitrary symmetric tensor, is its symmetric tensor rank always the same as

its tensor rank [30]? This has been an open problem for many years. Many

various special cases have been studied, and confirmed the above conjecture. In

bipartite cases, it is true that symmetric tensor rank is always equal to tensor

tank, by a smart application of the Schmidt decomposition. We find neither the

counterexamples nor the proof in the multi-partite cases.

Another open problem is to evaluate the exact value of the tensor rank of

|W 〉⊗n for n ≥ 3. We have found a few lower bound 2n+1 − 1 and an upper

bound (n + 1)2n. Unfortunately, no definite result is known when n > 2.This

problem has been studied extensively, and a number of partial results have been

obtained. It relates to the problem so-call polynomial rank of homogeneous poly-

nomials, because |W 〉 is a symmetric pure state. For an a degree 3 homogeneous

polynomials, we can solve some special cases [33] [34]. If someday we can solve

the polynomial rank problem completely, even just for degree three, we may

know the answer.

70

VI CONCLUSION

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