Multipartite Entangled Quantum States: Transformation, Entanglement Monotones and Application

Multipartite entangled quantum states: Transformation, Entanglement monotones and Application

Wei Cui

A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Physics University of Toronto

Abstract

Multipartite entangled quantum states: Transformation, Entanglement monotones and Application

Wei Cui Doctor of Philosophy Graduate Department of Physics University of Toronto 2013

Entanglement is one of the fundamental features of quantum information science. Though bipartite entanglement has been analyzed thoroughly in theory and shown to be an important resource in quantum computation and communication protocols, the theory of entanglement shared between more than two parties, which is called multipartite entanglement, is still not complete. Specifically, the classification of multipartite entanglement and the transformation property between different multipartite states by local operators and classical communications (LOCC) are two fundamental questions in the theory of multipartite entanglement. In this thesis, we present results related to the LOCC transformation between multipartite entangled states. Firstly, we investigate the bounds on the LOCC transformation probability between multipartite states, especially the GHZ class states. By analyzing the involvement of 3-tangle and other entanglement measures under weak two-outcome measurement, we derive explicit upper and lower bound on the transformation probability between GHZ class states. After that, we also analyze the transformation between N-party W type states, which is a special class of multipartite entangled states that has an explicit unique expression and a set of analytical entanglement monotones. We present a necessary and sufficient condition for a known upper bound of transformation probability between two N-party W type states to be achieved. We also further investigate a novel entanglement transformation protocol, the random distillation, which transforms multipartite entanglement into bipartite entanglement

ii shared by a non-deterministic pair of parties. We find upper bounds for the random distillation protocol for general N-party W type states and find the condition for the upper bounds to be achieved. What is surprising is that the upper bounds correspond to entanglement monotones that can be increased by Separable Operators (SEP), which gives the first set of analytical entanglement monotones that can be increased by SEP. Finally, we investigate the idea of a new class of multipartite entangled states, the Absolutely Maximal Entangled (AME) states, which is characterized by the fact that any bipartition of the states would give a maximal entangled state between the two sets. The relationship between AME states and Quantum secret sharing (QSS) protocols is exhibited and the application of AME states in novel quantum communication protocols is also explored.

iii Acknowledgements

Firstly, I want to thank my supervisor, Prof Hoi-Kwong Lo. During these five years, his endless help and inspired discussion guided me to explore the fantastic world of quantum information theory, which was a really exciting and enjoyable journey because of him. He taught me not only in science but also in other aspects of life. He showed my how to do presentation, how to improve my English, and more importantly, how to treat and work with other people. All the above and his kind help on my nonacademic life will be valuable and remembered for a lifelong time. Secondly, I really appreciate the advices and suggestions from my committee members, Daniel James and Aephraim Steinberg. It has been my great pleasure to work with a group of pleasant and brilliant colleagues. I want to show my acknowledgement to Eric Chitambar, Wolfram Helwig, Bing Qi, Christian Weedbrook, Xiongfeng Ma, Benjamin Fortescue, Yi Zhao, Yuemeng Chi, Viacheslav Burenkov, Feihu Xu, Kero Lau, Zhiyuan Tang, Felix Liao, and He Xu. Special thanks to Eric Chitambar for his brilliant discussions and endless passion on the subject. And to Bing Qi for his support on both of my academic and nonacademic life. I have benefited a great deal from the discussion with many excellent scientists. Specif- ically, I wish to thank Lin Chen, Daniel Gottesman, Fred Fung, Debbie Leung, Jonathan Oppenheim, and David Gosset. I would like to thank Viacheslav Burenkov for his suggestions and proof reading. Responsibility for any remaining mistakes rests entirely with the author. Also, I wish to thank Krystyna Biel and Diane Silva for their great job in adminis- trative help. The help from the Center of International Experience, family care office and family housing of the University of Toronto is also acknowledged. With their help, I had a really harmonious life with my family while studying in the University of Toronto as an international student. Finally, the love and support from my family is greatly appreciated. This thesis is dedicated to my parents, my wife Bilian, and my lovely son Stephen.

iv Contents

1 Introduction 1 1.1 Our results ...... 2 1.1.1 List of papers and presentations ...... 4

2 Background Information 6 2.1 Entanglement ...... 6 2.1.1 Entanglement in quantum physics ...... 6 2.1.2 Entanglement in Hilbert space ...... 7 2.1.3 Entanglement as a resource ...... 8 2.2 Quantum operations and entanglement measures ...... 10 2.2.1 Quantum Operators ...... 10 2.2.2 Local Operators and Classical Communications ...... 11 2.2.3 Separable operators ...... 13 2.2.4 Entanglement measures for pure bipartite states ...... 14 2.2.5 Entanglement measures for mixed states ...... 17 2.3 Multipartite entangled pure states ...... 19 2.3.1 Tripartite entangled states ...... 19 2.3.2 W type entangled states ...... 21

3 LOCC transformation bounds between multipartite pure states 23 3.1 Introduction ...... 24 3.2 Upper Bound for the Conversion from GHZ state to a GHZ class state . 25 3.3 Failure Branch ...... 30 3.3.1 Conservation of interference term ...... 31 3.3.2 Conservation of normalization ...... 32 3.4 Upper Bound for a general case ...... 35 3.4.1 interference term and the maximal value of the 3-tangle of a GHZ- class state ...... 35

v 3.4.2 "stop and reconstruct" procedure ...... 36 3.4.3 Example: GHZ φ = γ( 000 + aaa ) ...... 38 | i → | i | i | i 3.4.4 general case ...... 45 3.5 Lower Bound for the Transformation ...... 48 3.6 Summary and Concluding Remarks ...... 56

4 Optimal entanglement transformations among N-qubit W-type states 57 4.1 Introduction ...... 57 4.2 Upper bounds ...... 59 4.3 Lower bounds ...... 63 4.4 General Features of symmetric transformations ...... 66 4.5 Conclusion ...... 68

5 Random distillation for W type states 69 5.1 Introduction ...... 69 5.2 Previous results and notation ...... 74 5.2.1 The generalized Fortescue-Lo protocol ...... 74 5.2.2 Additional notation and the Kintas-Turgut monotones ...... 75 5.3 The least party out protocol ...... 76

5.3.1 Phase I: Remove x0 component ...... 76 5.3.2 Phase II: Equal or vanish (e/v) subroutine ...... 77 5.3.3 Phase III: Obtaining EPR pairs ...... 77 5.4 Main results: The LPO protocol on multipartite W type states ...... 81 5.4.1 Summary of results ...... 81 5.4.2 Three qubits ...... 81 5.4.3 Four qubits ...... 83 5.4.4 n qubits and the entanglement monotones ...... 89 5.4.5 Interpretation of monotones ...... 92 5.5 SEP VS LOCC ...... 93 5.5.1 Random distillation by Separable transformations ...... 93 5.5.2 Comparison between SEP and LOCC ...... 96

5.6 Applicaiton to the transformation φ WN ...... 98 | i1,··· ,N → | i 5.7 Conclusion ...... 99 5.7.1 Open questions and concluding remarks ...... 99

6 Absolutely maximal entangled state and quantum secret sharing 102 6.1 Introduction ...... 102

vi 6.2 Deﬁnition of AME states ...... 104 6.3 Parallel Teleportation ...... 105 6.4 Quantum Secret Sharing...... 107 6.5 Conclusion ...... 110

7 Conclusion 112 7.1 Future work ...... 113 7.2 Concluding words ...... 114

8 Appendix 115 8.1 Appendix: Proof of Theorem 5 ...... 115 8.2 Appendix: proof of Theorem 14 ...... 117

8.3 Dual solution to WN distillation by SEP ...... 119 | i Bibliography 122

vii List of Figures

2.1 Structure of states that can be obtained from W3 state by SLOCC. The

ﬁrst level is the true W3 type state which is also the genuine W class state. The second level are bipartite entangled states, such as (AB)-C ( ψ φ ), (AC)-B ( ψ φ ) and (BC)-A ( ψ φ ), and the third | iAB | iC | iAC | iB | iBC | iA level is the product state φ1 φ2 φ3 ...... 22 | iA | iB | iC

3.1 mapping type 1. c 2010 American Physical Society ...... 27

3.2 mapping type 2. c 2010 American Physical Society ...... 27

y 1 U 3 3.3 The value of p as a function of a. In this figure, a = ( y−1 ) . So when a goes from 0 to 1, y goes from 0 to . Note that as y goes to infinity, a ∞ goes to 1. We express the value as a function of a because it will be easier for us to combine different graphs into one graph later. c 2010 American

Physical Society...... 34 3.4 "stop and reconstruct" for a two-outcome measurement. c 2010 American

Physical Society...... 36 3.5 The original protocol written in the many two-outcome measurements form. c 2010 American Physical Society...... 37

3.6 "stop and reconstruct" for general protocol, I stands for the interference term. c 2010 American Physical Society...... 38

3.7 The new protocol, which can reconstruct the original one. c 2010 American

Physical Society...... 39

3.8 the relation between p¯s and p¯τ . c 2010 American Physical Society. . . 43 ABC 3.9 The upper bound for the transformation ...... 44

3.10 upper bound of transformation probability from φ to ψ ...... 47 | i | i 3.11 Four-step method. c 2010 American Physical Society...... 49

viii 4.1 An LOCC transformation tree from x to y . For example, the branch | i | i traversing edges e(1,1) to e(n,1) is a success branch, while the branch from e(1,1) to e(n,2) is a failure branch. Edge e(n−1,1) is an intermediate edge. c 2010 American Physical Society...... 60

4.2 The diﬀerence in maximum transformation probabilities when only one

party measures [pmax(s)] versus an identical ﬁlter by all parties [qmax(s)]. c 2010 American Physical Society...... 66

5.1 A speciﬁed-pair versus random-pair distillation. For random distillations, it is convenient to combine all the desired outcomes into one conﬁguration graph G = (V,E) whose edge set encodes the target pairs. Here, the target pairs are AB and AC. The ” ” indicates equivalent representations. ≡ c 2011 American Physical Society...... 71

5.2 An N = 8 example of the "complete-type" distillations considered by Fortescue and Lo in [39]. Such a transformation is a success if any two parties become EPR entangled, and this can be achieved with a probability arbitrarily close to 1. Previous research has not considered more general types of conﬁguration graphs than this. c 2011 American Physical Society. 72

5.3 In Sec 5.4 we show that the optimal LOCC probability of achieving this transformation is 2/3, thus resolving an open problem in Refs. [38]. The

initial state is W4 = 1/2( 1000 + 0100 + 0010 + 0001 ). c 2011 | i | i | i | i | i American Physical Society...... 72

ix 1 5.4 Equal or vanish subroutine (Phase II) for the normalized state 1+3α (α, α, α, 1) and the conﬁguration graph with edges AB, AC, AD, BC . 1. David’s { } component is largest and Alice is a connected party to him with a lesser component value. She performs an e/v measurement. 2. For the outcome "vanish" (right branch) she is separated from the system, and since David is not connected to either Bob or Charlie, he immediately removes himself from the system leaving ψ(BC) with some probability 1. For the outcome | i "equal" (left branch) the components of all other parties receive a factor of α, and Alice’s component is now maximum equaling David’s. Bob is a connected party to Alice with a lesser component value and he performs an e/v measurement. 3. Again, either Bob vanishes (right branch) or all other components except his receive a factor of α. In both cases, Charlie is then a connected party to Bob with a lesser component value and he performs an e/v measurement. 4. The ﬁnal outcome states along these (ABD) (ACD) AD branches are W4 , W , W , and ψ . c 2011 American | i | 3 i | 3 i | i Physical Society...... 79

(S) 5.5 Phase III receives an input state W|S| and a conﬁguration graph G. Party k performs an e/v measurement. One outcome is a standard W state with party k removed, and the other is the state 1 (α, , α, 1, α, , α). |S|pα ··· ··· Phase II is applied on this state outputting either W states or a product (failure) state. Phase III will next be initiated on each of the W states, (S0) 0 and for any W state W|S0| with S < S , the transformation success | | | | 0 (S ) ¯0 probability from this point onward is given by PIII (W|S0| ,G S ); this value (S\) is already known by recursion. However, for the state W|S| , performing Phase III again will generate an indeﬁnite loop, but one whose overall success probability converges to f(α) [see Eqs. 5.13 and 5.14]. c 2011 1−α|S|−1 American Physical Society...... 80

5.6 (Left) Conﬁguration G∧. (Right) Conﬁguration G∆. An upper bound on the success probability is given by Eqs. 5.17 and 5.19, respectively, which

is eﬀectively tight when x0 = 0. For W3 , these probabilties are 2/3 and | i 1, respectively. c 2011 American Physical Society...... 82

0 5.7 Let GI , GI , and G”I be the ﬁrst, second, and third of the above conﬁgura- tions, respectively. An upper bound on the success probability is given by

Eq. 5.20 which is eﬀectively tight when x0 = 0. For W4 , this probability | i is 1/4 for each conﬁguration. c 2011 American Physical Society...... 83

x 5.8 Let GII be the above conﬁguration. An upper bound on the success prob-

ability is given by Eq. 5.21 which is eﬀectively tight when x0 = 0. For

W4 , this probability is 3/4. c 2011 American Physical Society...... 84 | i 5.9 Let GIII be any of the above conﬁgurations. In each of these, (A,C) and (B,D) are unconnected pairs. An upper bound on the success probability

is given by Eq. 5.23 which is eﬀectively tight when x0 = 0. For W4 , | i this probability is 2/3 for each of these conﬁgurations. c 2011 American

Physical Society...... 84

5.10 Let GIV be the above configuration. We say two parties are edge complementary if their nodes have a different number of connected edges. For example, A is edge complementary to both C and D. An upper bound on the success probability is given by Eq 5.30 which is effectively tight when

x0 = 0. For W4 , this probability is 5/6. c 2011 American Physical Society. 85 | i 5.11 Let GV be the above conﬁguration. An upper bound on the success proba-

bility is given by Eq 5.32 which is eﬀectively tight when x0 = 0. For W4 , | i this probability is 1. c 2011 American Physical Society...... 87

5.12 Let GVI be the above conﬁguration. For W4 , the LPO protocol gives a | i success probability of 1 (3 + √3).We conjecture this to be optimal. c 2011 6 American Physical Society...... 87 5.13 Distillation conﬁgurations for η vs κ. Top: A "combing-type" distillation:

when x0 = 0, 2η(x) is the optimal probability for a random distillation in

which party n1 shares one-half of each EPR pair. Bottom: A "complete-

type" distillation: when x0 = 0, κ(x) gives the optimal probability for a random distillation in which the target pairs are any two of the parties. c 2011 American Physical Society...... 92

5.14 LOCC vs SEP for the maximum probability of obtaining an EPR pair between any two parties as a function of s when the initial state is √s 100 + q | i 1−s ( 010 + 001 ). The LOCC probability is 2(1 s) (1 s)2/4s.A 2 | i | i − − − gap of 12.5% exists between SEP and LOCC. c 2012 American Physical

Society...... 96 5.15 LOCC vs SEP for the maximum probability of party 1 becoming EPR q 1 entangled as a function of N when the initial state is 2 10 0 + q | ··· i 1 ( 010 0 + + 0 01 ). The LOCC probability is 1 (1 2(1−N) | ··· i ··· | ··· i − − 1 )N−1. A gap of 37% exists between SEP and LOCC. c 2012 American N−1 Physical Society...... 97

xi 5.16 The relative difference between the optimal separable operation and the LPO protocol. The configuration graph consists of N disjoint pairs. Sep- q 1 arable operations perform as PSEP = N whereas the LPO protocol 2 obtains the rate of PLP O = 2N−1 . We conjecture that the LPO protocol is LOCC optimal for this configuration graph, as it is known to be when N = 4. c 2012 American Physical Society...... 101

6.1 Parallel Teleportation scenarios of Theorem 20. Scenario (i) is on the left, and (ii) on the right. Parties in A perform joint quantum operations, parties in B only local quantum operations. c 2012 American Physical

Society...... 105 6.2 (Color online) After D (blue) performs her teleportation operation, any set of m parties (red), A, A0, A00 etc., can recover the teleported state. Any set of parties with m 1 or less parties (any set consisting only of green − parties) cannot gain any information about the teleported state. c 2012

American Physical Society...... 108

xii Chapter 1

Introduction

Quantum information, as a field that employs entanglement, which is the most impres- sive feature of quantum physics and a resource to accomplish novel computation and communication protocols, has been experiencing rapid development since the 1980’-s. 1 The famous EPR pair, which was expressed as a Bell state, φ = √ ( 00 + 11 )AB, | iAB 2 | i | i has been shown to be the resource for various quantum information protocols that are impossible under classical physics. EPR state shared between parties A and B cannot be written as a direct product of two states of parties A and B respectively, which is now the most important feature of the so called entangled state. Other than EPR pairs, there are infinitely many entangled quantum states that can be shared between several parties. It would be rewarding to investigate the entanglement property of these states and also their application in quantum information protocols. Also, under real experimental conditions, one cannot guarantee the generation of a perfect EPR state, and it is necessary to analyze how one can transform a quantum entangled state into an EPR state that can be directly used in quantum information protocols. Because of that, the quantification of entanglement for an entangled state and the transformation property between two entangled states are two main research directions in entanglement theory. Based on the fact that protocols employing EPR pairs are generally for nonlocal scenarios, the transformation between entangled states is restricted under the scenario of local operators and classical communications (LOCC). Under this scenario, the entanglement theory for a bipartite pure state, which is an entangled state shared between two parties, is completely constructed. The EPR state is shown to be the most powerful resource among pure states shared by two qubits since it could be transformed into any other pure state shared by two qubits with probability 1. The optimal transformation probability between any two bipartite states is also derived.

1 Chapter 1. Introduction 2

However, the corresponding results for higher dimensions and more parties, which is called multipartite entangled state, are still lacking. The mathematical structure of multipartite entangled states turns out to be much more complicated than the case of bipartite states. For example, tripartite states, which are the quantum states shared between three parties, can be classiﬁed into two classes while states from diﬀerent classes cannot be transformed into each other with nonzero probability. Another question is the application of multipartite entangled states in novel quantum protocols. Here the interest is on the protocols that could reveal the advantage of multipartite entanglement over bipartite entanglement. That is to say, for some protocols, employing the multipartite entangled states directly would achieve a better result than converting the multipartite entangled state into EPR pairs and then using the EPR pairs for the protocol.

1.1 Our results

During my Ph. D study, I mainly worked on the LOCC transformation probability between pure multipartite entangled states. In the following I will provide a quick overview of my work, which is also an outline of this thesis. In chapter 2, I will provide the background information on entanglement theory that is related to my research. In chapter 3, upper and lower bounds on the transformation probability between multipartite entangled states will be derived. The result is mainly related to the transformation between tripartite states while some of our results can be generalized into more parties. There is still a gap between the upper bound and lower bound we found, which future work will investigate. The result of this chapter was published in [33]. As the first author, I proposed the four-step-method and discovered the upper bound and lower bound for the transformation probability. In chapter 4, another type of multipartite entangled state, the W-type state, is analyzed. Based on the explicit form of a general W-type state given in [52], we derive the lower bound on transformation probability between any two W-type states under LOCC. Also, we find the condition under which the lower bound is actually optimal. The result of this chapter was published in [31]. As the first author, I proposed the LOCC transformation protocol and discovered the necessary and sufficient condition for the bound to be achieved. In chapter 5, the question regarding the conversion from multipartite entangled state into bipartite state is explored. Specifically, we consider the following question: given a Chapter 1. Introduction 3

multipartite state, how can it be converted into EPR pairs shared between any two parties? This protocol, called random distillation, was first proposed in [39]. Here we find upper bound for this type of transformation by discovering a new type of entanglement monotones. One surprising result is that the entanglement monotones discovered can be increased by separable operators (to be defined in chapter 2), which gives the first set of analytic entanglement monotones that can be increased by separable operators. The result of this chapter was published in [32][23][19]. As the first author of [32] and the second author of [23] and [19], I proposed the least party out protocol and discovered the entanglement monotones for three and four qubit systems.

In chapter 6, a new type of multipartite entangled states, which is called the absolutely maximal entangled states (AME states) (to be deﬁned in chapter 6), is studied. This type of state is characterized by the property that any bipartition of the parties could lead to a maximal entangled state between the two sets of parties. The close relationship between AME states and quantum secret sharing protocols will be exhibited. Also, the possible application of AME states is proposed. The result of this chapter was published in [48]. As the second author, I collaborated on the analysis of the multipartite teleportation protocol and on the proof of the one-to-one correspondence relationship between AME state and quantum secret sharing protocol (to be deﬁned in chapter 6).

In chapter 7, a summary of my Ph.D research is provided, and the future work that can be developed from this thesis is also discussed.

In chapter 8, the appendix, we provide the detailed proofs for some important theorems in this thesis.

The significance of our work can be summarized as the following three points. Firstly, the transformation probability between multipartite pure states is a complex problem on which little work has been done before ours. Our work sheds some light on the investigation of this problem by finding various upper and lower bounds. Secondly, instead of trying to classify all types of multipartite entangled states, we focus on some specific types of multipartite entangled states and analyze their properties and potential applications, which is shown to be very helpful. Finally, our work on random distillation demonstrates that the mathematical structure of local operators and classical communication (LOCC, to be defined in chapter 2) is more complex than expected. Chapter 1. Introduction 4

1.1.1 List of papers and presentations

Papers

1. Wei Cui, Wolfram Helwig, Hoi-Kwong Lo, Bounds on the probability of transformation between multipartite pure states, Physics Review A, 81, 012111, 2010

2. Wei Cui, Eric Chitambar, Hoi-Kwong Lo, Optimal Entanglement Transformations Among N-qubit W-Class States, Physics Review A, 82, 062314, 2010

3. Wei Cui, Eric Chitambar, Hoi-Kwong Lo, Randomly distilling W-class states into general conﬁgurations of two-party entanglement, Phys. Rev. A 84, 052301, 2011

4. Eric Chitambar, Wei Cui, Hoi-Kwong Lo, Increasing Entanglement by Separable Operations and New Monotones for W-type Entanglement, Phys. Rev. Lett. 108, 240504, 2012. (This work was selected as a plenary talk at QIP, one of my colleagues (Eric Chitambar) did the presentation. A plenary talk is the most prestigious talk at the QIP conference, which is the most prestigious theory conference in the ﬁeld.)

5. Eric Chitambar, Wei Cui, Hoi-Kwong Lo, Entanglement monotones for W-type states, Phys. Rev. A 85, 062316, 2012

6. Wolfram Helwig, Wei Cui, Arnau Riera, Jose I. Latorre, Hoi-Kwong Lo, Absolute Maximal Entanglement and Quantum Secret Sharing, Phys. Rev. A 86, 052335, 2012

Presentations

1. August 2011, AQIS 11, Pusan National University, Pusan, Korea, "Randomly distilling W-class states into general conﬁgurations of two-party entanglement" by W. Cui, E. Chitambar, H. -K. Lo

2. March 2011, APS March Meeting, Dallas Convention Center, Dallas, Texas, United States of American, "Optimal Entanglement Transformations Among N-qubit W- Class States" by W. Cui, E. Chitambar, H. -K. Lo

3. July 2010, University of Calgary, presentation, "Bounds on the probability of transformation between Multipartite quantum states", by W. Cui, W. Helwig, H.-K. Lo

4. June 2010, CAP 2010, University of Toronto, presentation, "Bounds on the probability of transformation between Multipartite quantum states" by W. Cui, W. Helwig, H.-K. Lo Chapter 1. Introduction 5

5. January 2010, QIP 2010, ETH, Switzerland, Rump session, "Bounds on the probability of transformation between Multipartite quantum states" by W. Cui, W. Helwig, H.-K. Lo Chapter 2

Background Information

In this chapter we provide a brief summary of some important results from the ﬁeld of quantum entanglement related to our work. People who are familiar with the theory of entanglement can skip this chapter on a ﬁrst reading.

2.1 Entanglement

In this section we provide a description of entanglement, in both the physical and mathematical aspects.

2.1.1 Entanglement in quantum physics

The distinction between quantum physics and classical physics is the description of a physical system as a state with uncertain parameters. Speciﬁcally, from the uncertainty principle, it is impossible to measure the exact position and momentum of a particle simultaneously. However, in 1935, Einstein, Podolsky, and Rosen proposed a quantum state shared between two parties A and B in the following form, which is called an EPR state [35].

Z +∞ (2πi/h)(x1−x2+x0)p Φ(x1, x2) = e dp. (2.1) −∞

For this state, the two particles have a ﬁxed midpoint at x0 while their momenta should be opposite to each other. In this sense, the center of mass of the system AB would be at the original location while the total momentum for them remains zero. So, by measuring the position of A, one can determine the position of B. Or we can say the collapse of the wave function of A leads to the collapse of the wave function of B, which

6 Chapter 2. Background Information 7 appears to indicate an interaction between A and B that is faster than light. As an explanation of this feature of an EPR state, it was claimed that quantum physics is not a complete theory and that there is a set of hidden variables that actually determine the property of a physical system. The local hidden variable conjecture based on this assumption was shown to be incorrect by the experimental violation of Bell’s inequality [5]. To this day, the interpretation of quantum mechanics remainsl an open problem. However, the EPR state turns out to be a stronger resource than a classical bit from an information theory perspective. In the following subsection we will introduce the mathematical structure of general entangled states.

2.1.2 Entanglement in Hilbert space

The mathematical framework for the analysis of quantum states is the Hilbert space. Given a quantum system that could evolve as a superposition of n possible eigenstates of the Hamiltonian, one can express the state in an n-dimensional Hilbert space. A quantum state can be expressed as a density matrix in the corresponding Hilbert space. For a pure state, the density matrix can be expressed as φ φ while for a mixed state, which | i h | P is considered an ensemble of several quantum states, the density matrix is i pi φi φi . P | i h | Here we have T rρ = i pi = 1. Example 1. Let us take spin as an example. Suppose there is an electron whose spin could be up or down. We can treat the Hilbert space of spin as , . A pure state in {|↑i |↓i} this space could be written as φ = a +b (density matrix (a +b )(a∗ +b∗ ) | i |↑i |↓i |↑i |↓i h↑| h↓| where a 2 + b 2 = 1. And a mixed state could be written as ρ = a 2 + b 2 | | | | | | |↑i h↑| | | |↓i h↓| where a 2 + b 2 = 1 also. | | | | Mathematically, one can also deﬁne 0 = and 1 = . Thus the pure state and | i |↑i | i |↓i mixed state could be expressed as φ = a 0 + b 1 (density matrix (a 0 + b 1 )(a∗ 0 + | i | i | i | i | i h | b∗ 1 )) and ρ = a 2 0 0 + b 2 1 1 . h | | | | i h | | | | i h | Remark 1. One question here is how to distinguish pure states and mixed states. Math- ematically, given a hermitian matrix ρ, if the eigenvalues λi are all nonnegative and P { } i λi = 1, then this matrix can represent the density matrix of a quantum state. Further- more, if one of the eigenvalues is 1 while all the other eigenvalues are 0, the corresponding quantum state is a pure state. Otherwise, it is a mixed state. For two particles A and B, the whole system would lie in a Hilbert space given by the tensor product of the two individual Hilbert spaces.

= A B (2.2) H H ⊗ H Chapter 2. Background Information 8

The density matrix of their joint state is denoted by ρAB. If it can be written in the form of

X ρAB = piρAi ρBi , (2.3) i ⊗ P where pi 0 and pi = 1, it is called a separable state. Otherwise, it is called ≥ i entangled. The most important entangled state to consider for a bipartite system, say party A and B, is the EPR state

1 Φ = ( 00 + 11 )AB. (2.4) | iAB √2 | i | i

The above deﬁnition easily be generalized into higher dimensions and more parties.

In general, for an n-party state ρ1···n, if it can be written as

X ρ1···n = piρ1i ρni , (2.5) i ⊗ · · · P where pi 0 and pi = 1, it is called a separable state. Otherwise, it is entangled. ≥ i One thing to note is that, for a multipartite system, one can divide the parties into diﬀerent groups and talk about the entanglement property between these groups. For example, the state 1 Φ = ( 00 + 11 )AB 0 , (2.6) | iABC √2 | i | i | iC is an entangled state between A, B, and C. However, it is a separable state between the systems (AB) and C.

2.1.3 Entanglement as a resource

Now let us explain the application of a qubit from a quantum information theory perspective. We denote a classical bit as a cbit, which stands for either 0 or 1. Also, we deﬁne a two-level quantum mechanical system (a 0 +b 1 ) as 1 qubit. Another resource | i | i to consider is one in which there is an entangled state shared between the parties initially. Speciﬁcally, if Alice and Bob share an EPR pair, we denote it as 1 ebit.

Quantum superdense coding

Suppose that Alice wants to communicate two bits of information 00, 01, 10, 11 to Bob. { } One choice is to send 2 cbits via a classical channel. However, if they share an EPR Chapter 2. Background Information 9

pair (1 ebit), they have another choice that allows them to send 1 qubit instead. This protocol is called superdense coding [11]. Let us explain how the protocol works and its implication. In the beginning, Alice and Bob share an EPR state, so that we have

1 φ = ( 00 + 11 )AB. (2.7) | iAB √2 | i | i

To send the information, Alice could use an encoding scheme by implementing a local unitary operator on her subsystem, which has the following rules:

1 00 : IA φ = ( 00 + 11 )AB | iAB √2 | i | i 1 01 : XA φ = ( 10 + 01 )AB | iAB √2 | i | i (2.8) 1 10 : ZA φ = ( 00 11 )AB | iAB √2 | i − | i 1 11 : XAZA φ = ( 10 01 )AB | iAB √2 | i − | i

where X, Z are Pauli matrices deﬁned as ! ! ! 0 1 0 i 0 1 X = ,Y = − ,Z = . (2.9) 1 0 i 0 1 0

After that, Alice can send her qubit to Bob. Notice that the four resulting states are orthogonal to each other so that if Bob has the full copy of the state, he can identify which state it is via a Bell measurement. Thus Bob could recover the 2 classical bits Alice wants to send to him. In the above protocol, Alice and Bob have 1 ebit in the beginning, and they transmit 1 qubit of information. In all, they communicate 2 cbits of information. We thus have

1 qubit + 1 ebit 2 cbits. (2.10) ≥

Quantum teleportation

In quantum teleportation, Alice wants to teleport a qubit, or an unknown quantum state ( φ = a 0 + b 1 ) to Bob [7]. Supposing they share an EPR state in the beginning, | i | i | i Alice could attach the unknown state to her subsystem so we have Chapter 2. Background Information 10

1 (a 0 + b 1 )A0 ( 00 + 11 )AB | i | i √2 | i | i 1 1 = ( 00 + 11 )A0A(a 0 + b 1 )B + ( 00 11 )A0A(a 0 b 1 )B (2.11) 2 | i | i | i | i 2 | i − | i | i − | i 1 1 + ( 01 + 10 )A0A(a 1 + b 0 )B + ( 01 10 )A0A(a 1 b 0 )B 2 | i | i | i | i 2 | i − | i | i − | i Alice could make a Bell measurement on her system and communicate the measurement result (2 cbits) to Bob. With that information, Bob can recover the unknown quantum state by applying the corresponding Pauli matrices on his quantum system. During the above process, Alice and Bob possess 1 ebit in the beginning and they use 2 cbits to transmit 1 qubit of information. We thus have

1 ebit + 2 cbits 1 qubit (2.12) ≥

2.2 Quantum operations and entanglement measures

In this section we will describe the general quantum operators and especially Local Op- erators and Classical Communications (LOCC). After that we will show that LOCC provides the framework to quantify how much entanglement a system contains.

2.2.1 Quantum Operators

Given a quantum state, how can we transform it into another state? The transformation a state will undergo during a physical process is described by quantum operators. Mathematically, a quantum operator can be described by a linear and completely positive map, ψ, from the set of density operators onto itself.

Remark 2. A linear map ψ is positive if ψ(ρ) is positive for any positive ρ on the Hilbert space. And it is completely positive if ψ 1p(ρ 1p) is positive for any positive integer p. ⊗ ⊗ Mathematically, a quantum operator can always be expressed in the form [26]

X + φ(ρ) = VjρVj (2.13) j where X + Vj Vj = 1 (2.14) j Chapter 2. Background Information 11

+ + Vj ρVj Each term VjρVj can also be treated as a branch, with the resulting state + and tr(Vj ρVj ) + the probability tr(VjρVj ).

Example 2. Measurement: For example, suppose we have a quantum state φ = α 0 + p | i | i 1 α 2 1 in the 0 , 1 Hilbert space, a measurement in 0 , 1 can be performed − | | | i {| i | i} {| i | i} on the state. With probability α 2 the state will collapse onto 0 while with probability | | | i 1 α 2 the state will collapse onto 1 . This operator can be described as − | | | i

M(ρ) = 0 0 ρ 0 0 + 1 1 ρ 1 1 (2.15) | i h | | i h | | i h | | i h | The resulting state is given by

ρ = 0 ρ 0 0 0 + 1 ρ 1 1 1 (2.16) h | | i | i h | h | | i | i h | and the possible resulting states are 0 (density matrix 0 0 ) and 1 (density matrix | i | i h | | i 1 1 ), with | i h |

p( 0 ) = 0 ρ 0 = α 2 and p( 1 ) = 1 ρ 1 = 1 α 2 (2.17) | i h | | i | | | i h | | i − | | Since entanglement is the resource for quantum information, an important question would be how entanglement evolves under quantum operations, especially measurements. Here we want to emphasize that measurement could induce the collapse of a quantum wave function, which means the resulting state might have signiﬁcantly diﬀerent entanglement properties from the original state. For example, given an EPR state shared by Alice and Bob, φ = √1 ( 00 + 11 ), if Alice or Bob measures in 0 , 1 basis, the | iAB 2 | i | i {| i | i} state would collapse into 1 00 00 + 1 11 11 , which is a separable state. 2 | i h | 2 | i h |

2.2.2 Local Operators and Classical Communications

Since entanglement is used for the transmission of information between parties far apart from each other, we restrict the quantum operations to be locally implemented. Also, we only allow classical information to be transmitted between the distant parties. This scheme is called LOCC (Local Operators and Classical Communications), a standard scheme in which we could quantify the amount of quantum resource we have. In general, given three parties A, B, and C, one could describe an LOCC protocol in the following way: Party A makes a local operator and passes the information regarding this operator and the measurement result (classical information) to Bob and Charlie. Based on the information from Alice, Bob or Charlie could implement another local Chapter 2. Background Information 12 operator, and so on. Finally, their joint state would end up being some density matrix, which is the resulting state of this LOCC protocol. Suppose that we use LOCC to transform a state from φ into ψ . If there is a | i | i positive probability less than 1 with which the transformation can be successful, then the corresponding protocol is called SLOCC (Stochastic Local Operators and Classical Communications) protocol.

Two-outcome weak measurement decomposition of LOCC

If one party performs a local measurement that has several possible outcomes, the state could collapse into a new state far from the original state. This phenomenon is an obstacle for the investigation of the behavior of some quantitative parameters under LOCC since it is not continuous and hard to formulate mathematically. To overcome this problem, one could ﬁrst decompose a local measurement into several two-outcome measurements. This is shown in [3]. Another important result is that any two-outcome measurement could be decomposed into many steps of two-outcome weak measurements, while during each step the resulting quantum states are almost identical with the original state. The idea is similar to a random walk, where one needs to stop when the resulting state becomes one of the two resulting states of the original two-outcome measurement [63]. By using the above two techniques, one can discuss a general LOCC protocol under the restriction of two-outcome weak measurement, during which the state can be seen as changing continuously. This method will be explored further in chapters 3, 4, and 5.

Entanglement Monotone

Given a quantum state, how much entanglement does it contain? To answer this question, one needs a quantitative measure for the amount of entanglement a state possesses. In general, entanglement is used for nonlocal missions. That is to say, many parties share a quantum system, while they can only perform local operations on their subsystems. Entanglement, as a resource, can only be consumed, rather than created, during this process. In this sense, entanglement monotone, as a quantiﬁcation of how much entanglement one quantum system has, is deﬁned in the following way:

Deﬁnition 1. For a quantum system ρ, any magnitude µ(ρ) that does not increase on average under local transformations and classical communications is called entanglement monotone (EM) [77]. Chapter 2. Background Information 13

An important application of entanglement monotone is that it can be used to bound the optimal transformation probability between two quantum states ρ and ρ0 under LOCC. More precisely, the optimal successful probability for the conversion from ρ to ρ0 LOCC under LOCC, denoted by P (ρ ρ0), is given by −−−→

LOCC 0 µ(ρ) P (ρ ρ ) = minµ (2.18) −−−→ µ(ρ0)

where the minimization is to be performed over the set of all EMs [77]. It is straightforward to see that the optimal transformation probability should be upper bounded by µ(ρ) any µ(ρ0) . At the same time, the transformation probability itself is also an entanglement monotone, and this upper bound is hence tight. In general, it is not easy to ﬁnd the optimal probability for an LOCC transformation. However, the known entanglement monotones could be used to ﬁnd an upper bound on the transformation probability. Also, if any transformation probability coincides with a known bound obtained from some entanglement monotone, we can say for certain that it is optimal.

2.2.3 Separable operators

One drawback of LOCC is that it is hard to be analyzed mathematically, because it does not have an explicit analytical deﬁnition. Also, its mathematical structure is very complex. For a good summary of the mathematical structure of LOCC, we refer to [22]. To overcome this problem, separable operators (SEP) were introduced. Mathe- matically, SEP on an n-party state is deﬁned as

k X + Ω(ρ) = AiρAi (2.19) i=1 where k X + Ai Ai = 1 (2.20) i=1 and

Aj = A1 A2 An (2.21) j ⊗ j ⊗ · · · ⊗ j where is a local operator implemented by party k, which means + should be Akj Akj ρAkj a positive matrix deﬁned on the Hilbert space of one party. The motivation for the introduction of SEP for quantum information is the fact that separable operators have an explicit mathematical structure and can be analyzed Chapter 2. Background Information 14 numerically by programs like semi-deﬁnite programming [75]. Also, since every LOCC protocol can also be implemented by SEP, SEP can be used to identify entanglement monotones for a quantum system. More concretely, if a physical quantity could not be increased by SEP, then it is impossible for it to be increased by any LOCC protocol and it is an entanglement monotone. However, the converse is not true: there are entanglement monotones that can be increased by SEP [24] [23]. However, given an SEP, it is unclear how to check whether it can or cannot be implemented by LOCC. What we can do is that, if an SEP can accomplish a mission that is impossible by LOCC, one can be sure that this SEP cannot be implemented by any LOCC protocol. For example, separable operations that can increase some entanglement monotone [23], or distinguished states that could not be perfectly distinguished by LOCC [9], cannot be implemented by LOCC. A general method to check whether a protocol with separable operations can be implemented by LOCC within a given number of rounds was presented in [30]. In chapter 5, we will show the gaps between LOCC and SEP for some quantum information protocols.

2.2.4 Entanglement measures for pure bipartite states

In this section we review the known results for the entanglement properties of pure bipartite states.

Schmidt decomposition and Schmidt number

Bipartite pure states have been analyzed thoroughly in quantum information theory. This partly comes from the existence of Schmidt decomposition for bipartite states. Given any bipartite pure state, we can always write it as

k X φ = √al il il (2.22) | iAB | iA | iB l=1 Pk where al = 1 and i1, , in is a set of orthogonal state vectors. This form is called l=1 { ··· } Schmidt decomposition. For each bipartite pure state, one can uniquely determine the values of its Schmidt coeﬃcients (together with their degeneracies) [62]. Based on this decomposition, the LOCC transformation rules are well developed.

Remark 3. Notice that uniqueness of Schmidt decomposition means that the set of co- eﬃcients are unique, the actual states may not be unique if some of the coeﬃcients are degenerate. Chapter 2. Background Information 15

Given a bipartite pure state φ , to find its Schmidt decomposition? One first | iAB needs to compute the reduced density matrix for one party, say ρA. The eigenvalues and eigenvectors of ρA are als and il s in Eq 2.22 respectively. | iA 1 1 Example 3. For example, given a pure state φ = √ 00 + ( 10 + 11 )AB, let | iAB 2 | iAB 2 | i | i us find its Schmidt decomposition. Firstly, by tracing out party B, one can find that

1 √2 1 ρA = 0B ρAB 0B + 1B ρAB 1B = 0 0 + ( 0 1 + 1 0 ) + 1 1 . (2.23) h | | i h | | i 2 | i h | 4 | i h | | i | i 2 | i h |

By diagonalizing it, we can ﬁnd the eigenvalues and corresponding eigenvectors of ρA are

1 √2 1 1 √2 1 a1 = : i1 = ( 0 1 ); a2 = + : i2 = ( 0 + 1 ) (2.24) 2 − 4 | iA √2 | i − | i 2 4 | iA √2 | i | i

Similarly, for ρB, we have the same eigenvalues and the corresponding eigenvectors are

1 1 √ √ i1 B = p ( 0 (1 + 2) 1 ); i2 B = p ( 0 + ( 2 1) 1 ). (2.25) | i 4 + 2√2 | i − | i | i 4 2√2 | i − | i − Finally, the Schmidt decomposition is given by

φ = √a1 i1 i1 + √a2 i2 i2 (2.26) | iAB | iA | iB | iA | iB Remark 4. Note that for a quantum system with more parties, Schmidt decomposition does not always exist.

Also, given a bipartite pure state

k X φ = √al il il , (2.27) | iAB | iA | iB l=1 the Schmidt number is deﬁned as k, the number of non-zero terms in the Schmidt decomposition. A generalization of Schmidt number is called Schmidt rank, which is the minimum number of non-zero terms needed to write a multipartite state as the superposition of product states. In the following subsections, we will review some entanglement monotones for bipartite states and the transformation rules between any two bipartite states. Chapter 2. Background Information 16

Entropy of entanglement

Entropy of entanglement is one of the most important entanglement measures of bipartite pure states since it is deﬁned from the information theoretic perspective. For a pure state φ , the entropy of entanglement ( φ ) is deﬁned as the von Neumann entropy of the | iAB | i reduced density matrix for either party, or we have

( φ ) = S(ρA) = S(ρB) (2.28) | i

where S(ρ) = T rρ log ρ, ρA = T rB( φ φ ), ρB = T rA( φ φ ). − 2 | iAB h | | iAB h | Entropy of entanglement is closely related to entanglement concentration and entanglement dilution [6]. More concretely, given N copies of a bipartite pure state φ , | i asymptotically (in the limit of large N) one can use LOCC transformation to concentrate them into N( φ ) copies of EPR pairs. Also, given N copies of EPR pairs, asymptotically | i they can be distilled into N copies of φ . (|φi) | i

Transformation probability between the states

Based on the Schmidt decomposition of bipartite pure states, the optimal LOCC transformation probability between any two bipartite pure states is discovered in [77]. Given two pure bipartite states

φ = √a1 i1i1 + √a2 i2i2 + √an inin , ai ai+1 0, (2.29) | i | i | i ··· | i ≥ ≥ and

p p p ψ = b1 j1j1 + b2 j2j2 + bn jnjn , bi bi+1 0, (2.30) | i | i | i ··· | i ≥ ≥

where i0, i1, , in and j0, j1, , jn are two sets of orthogonal vectors, the optimal { ··· } { ··· } transformation probability from φ to ψ under LOCC is given by | i | i Pn LOCC i=l ai P (φ ψ) = min Pn . (2.31) −−−→ l∈[1,n] i=l bi

Pn Pn Notice that if we have ai bi for any given l, the transformation can i=l ≥ i=l be done by LOCC with probability 1. This is called the majorization relationship and was discovered in [61]. Based on this result, the EPR state serves as the maximal entangled state for bipartite quantum system since it can be transformed deterministically into any other pure bipartite state of two qubits under LOCC. Chapter 2. Background Information 17

2.2.5 Entanglement measures for mixed states

The situation becomes more complex for mixed states. In general, a mixed state could be written as

X ρ = pi φi φi (2.32) i | i h | P where i pi = 1. To quantify the entanglement of ρ, one natural choice is to consider the corresponding entanglement measures for the pure states in this ensemble, and deﬁne the corresponding P entanglement measure for mixed state as piE( φi ). i | i However, one needs to note that the decomposition of a mixed state into an ensemble of pure states is not unique, which means that we need to consider all possible decompositions and ﬁnd the one that yields the minimum value. Or we have

X E(ρ) = min piE( φi ) (2.33) {pi,|φii} i | i

Entanglement of formation and entanglement cost

If we choose entropy of entanglement as the corresponding entanglement measure for pure state, we can deﬁne entanglement of formation as [10]

X Ef (ρ) = min pi( φi ) (2.34) {pi,|φii} i | i

Note that in large N limit, one can prepare N copies of state φi with N( φ ) copies | i | i of EPR pairs. In general, one can prepare all states in this ensemble with EPR pairs and combine them together to form ρ. Thus we achieve an operational interpretation of entanglement of formation: with NEf (ρ) copies of EPR pairs, one can prepare N copies of ρ using LOCC in the limit of large N. This deﬁnition leads to another concept called entanglement cost, which is the minimum value of the average number of EPR pairs needed to prepare one copy of a state using LOCC [46]. If n EPR pairs are needed to produce m copies of a state ρ, then ⊗m n EC (ρ ) = min m . Here the minimum value is chosen from all possible LOCC protocols. Entanglement of formation is not always equal to entanglement cost, but in the large m limit, they are equal. Or we have

1 ⊗m Ec(ρ) = lim Ef (ρ ). (2.35) m→∞ m Chapter 2. Background Information 18

Distillable entanglement and Bound entanglement

Conversely, distillable entanglement is deﬁned as the number of EPR pairs one can distill from a given state [8]. In particular, suppose one can distill n copies of EPR pairs n from m copies of state ρ, one has ED(ρ) = max limm→∞ m . Here we need to consider all possible LOCC protocols to maximize the number of EPR pairs produced. Due to the complexity of LOCC protocols, there is no analytical expression for distillable entanglement. However, PPT criterion can be used to check if distillable entanglement is zero for a bipartite mixed state [65][50]. In general, entanglement cost is larger than distillable entanglement. The extreme condition is that for some states, the distillable entanglement is zero while the entanglement cost is positive, which means EPR pairs needs to be consumed to prepare the state while no EPR pairs can be distilled back from that state. This is called bound entanglement [50].

Concurrence

Concurrence is a widely used entanglement measure defined for mixed states of a two- qubit system [80]. We firstly introduce the definition of the "spin-off" density matrix between AB as ∗ ρÃB = (σy σy)ρ (σy σy), (2.36) ⊗ AB ⊗ based on which concurrence is defined as

Deﬁnition 2. Concurrence between A and B for a density matrix ρAB, is

CAB = max λ1 λ2 λ3 λ4, 0 (2.37) { − − − } where λis are the square roots of the eigenvalues of ρABρ˜AB in decreasing order (λ1 ≥ λ2 λ3 λ4). ≥ ≥

Up to now, concurrence is the only known analytical entanglement monotone for mixed states. Under a general noisy environment, a pure state will be transformed into a mixed state because of decoherence. To quantify the involvement of entanglement for this quantum system, the change in concurrence serves as an important criterion. For example, entanglement sudden death was based on the calculation of concurrence under the eﬀect of classical noise [85]. Chapter 2. Background Information 19

2.3 Multipartite entangled pure states

While bipartite pure entangled states have been analyzed thoroughly in theory, the generalization of these results into multipartite pure states remainsl an open problem.

2.3.1 Tripartite entangled states

For a general tripartite pure entangled state, the Schmidt decomposition might not always exist. Instead, a unique form called generalized Schmidt decomposition of tripartite states was discovered in [2] as

φ = λ0 000 + λ1 100 + λ2 101 + λ3 110 + λ4 111 (2.38) | i | i | i | i | i | i 2 P where λi 0, 0 φ π, µi λ , µi = 1. ≥ ≤ ≤ ≡ i i Based on this form, truly entangled three-qubit states (that is to say, other than product states or bipartite entangled states that are separable under the bipartition A-

BC, AB-C and C-AB) can be divided into two classes [34]. When λ4 = 0, it is called 6 a GHZ class state; otherwise, it is called a W class state. States that belong to diﬀerent classes cannot be transformed into each other by LOCC even with some nonzero probability. However, the optimal transformation probability between states in the same class is still a open problem. Further generalization of this result turns out to be a very complex problem. Actually, for four-qubit entangled states, no unique form has been proposed yet and it has been proved that four-qubit entangled states can be classiﬁed into nine families [76], between which the transformation rules are not yet known.

Entanglement monotones for tripartite pure entangled states

Given a tripartite state shared between A, B, and C, the entanglement can be shared by A and B, A and C, A and BC, etc. Is there any entanglement that is shared by ABC altogether? If so, how can we quantify it? The answer lies in concurrence and a quantity called 3-tangle [29].

Firstly, it should be clarified that the concurrence between A and B, CAB, is not an entanglement monotone. In fact, it can be increased significantly by LOCC. For example, 1 for a GHZ state GHZ = √ ( 000 + 111 )ABC , if Charlie performs a measurement | iABC 2 | i | i on the basis + , and communicates the measurement result to Alice and Bob, AB {| i |−i} will end up sharing an EPR pair. Notice that for the original state, the density matrix Chapter 2. Background Information 20 for AB is 1 ρAB = ( 00 00 + 11 11 ) (2.39) 2 | i h | | i h | which is a separable state with CAB = 0. But for the final EPR state, we clearly have

CAB = 1. During this transformation, CAB has been increased from 0 to 1 with Charlie’s assistance.

However, the concurrence between A and BC, CA(BC) is an entanglement monotone, similarly for CB(AC) and CC(AB). Also, based on concurrence, one can deﬁne 3-tangle - 2 the entanglement shared by ABC altogether - as the diﬀerence between CA,BC and the 2 2 summation of CAB and CAC , or we have

2 2 2 τABC = C C C . (2.40) A(BC) − AB − AC

Remark 5. Also, we can deﬁne τABC using diﬀerent orders of the parties, which will give the same result. Or we have

2 2 2 2 2 2 τABC = C C C = C C C . (2.41) B(AC) − BA − BC C(AB) − CA − CB

τABC is shown to be an entanglement monotone [29]. From the deﬁnition of τABC we can see that it quantiﬁes the entanglement not attributed to any two-party entanglement, or we can say that it is the genuine entanglement shared by three parties altogether. τABC serves as a clear distinction between the GHZ class state and W class state because τABC is zero for any W class state. Or we can say, the entanglement of a W class state can all be attributed to two-party entanglement [34]. Because of that, the transformation probability from a W class state to any GHZ class state is always zero. In chapter 3, we will show the result we obtain for upper and lower bounds of the transformation probability between GHZ class states.

Schmidt rank for tripartite pure entangled states

Another concept that can be used to distinguish GHZ class and W class states is the Schmidt rank. As we deﬁned earlier, it is the minimum number of product states needed in the superposition form of a pure state. For a GHZ class state, the Schmidt rank is 2 while any W class state has Schmidt rank 3. Notice that Schmidt rank can not even be increased by SLOCC, which means it can not increase even with a nonzero probability. Since W class states have a higher Schmidt rank than any GHZ class state, the transformation probability from a GHZ class state to a W class state is also zero [34]. Chapter 2. Background Information 21

Remark 6. For tripartite pure states with higher dimensions, it is proved that the calculation of Schmidt rank is an NP hard problem [21].

2.3.2 W type entangled states

Multipartite entangled states have a very complex mathematical structure when the number of parties is higher than three. However, for a specific class of multipartite states, the W-type states, a unique form was discovered and the transformation rules between these states turn out to be very explicit [52]. In the following we will introduce the definition of W type state as a generalization of W state. We will also describe the entanglement monotones and transformation rules associated with W type states. From the definition of the W state

1 W = ( 100 + 010 + 001 ) (2.42) | i √3 | i | i | i one can easily generalize it into more parties as

1 Wn = ( 10 0 + 010 0 + + 0 01 ) (2.43) | i √n | ··· i | ··· i ··· | ··· i which is called standard n-party W state. Then we can give the following deﬁnition of W-type state:

Deﬁnition 3. W type states are the states that can be obtained by SLOCC from a standard n-party W state, where n is any integer no less than three.

Remark 7. Notice that based on this deﬁnition, product states also belong to W type states since one can easily obtain a product state from an n-party W state by making a measurement on some parties. An illustration of W type state for three qubits is shown in 2.1.

It has been shown that W-type states have a very explicit unique form as

φ( x ) = √x0 0 0 + √x1 10 0 + √x2 010 0 + + xn 0 01 . (2.44) | −→ i | ··· i | ··· i | ··· i ··· | i | ··· i Pn where xi 0 and xi = 1. It was proved that the above form is unique when at ≥ i=0 least three xis are nonzero [52]. Also, we can use a vector notation for the above state

x = (x1, , xn). (2.45) −→ ··· Chapter 2. Background Information 22

Figure 2.1: Structure of states that can be obtained from W3 state by SLOCC. The ﬁrst level is the true W3 type state which is also the genuine W class state. The second level are bipartite entangled states, such as (AB)-C ( ψ φ ), (AC)-B ( ψ φ ) and | iAB | iC | iAC | iB (BC)-A ( ψ φ ), and the third level is the product state φ1 φ2 φ3 . | iBC | iA | iA | iB | iC

LOCC transformation rule for W-type states

In the following we will show the behavior of W-type state under LOCC transformation. Since any LOCC transformation will turn a W-type state into another W-type state, we

can simply discuss the change to xis. Given an initial W-type state x = (x1, , xn), −→ ··· consider a local operator being applied on party j. This leads to several possible ﬁnal states −→xk with corresponding probability pk, and we have [52]

(i) xk,i = skxi where i = 0, j; 6 P (ii) k pksk = 1; P (iii) pk√skxk,0 √x0. k ≥

It is not hard to see that all xis where i > 0 can never increase on average under LOCC, which means they are all entanglement monotones. Further investigation for the transformation probability between W type states will be shown in chapter 4. Also, by analyzing random distillation protocol for W type states, we ﬁnd new entanglement monotones for W type states that can be increased by SEP [20]. The detail of this result will be discussed in chapter 5 of this thesis. Chapter 3

LOCC transformation bounds between multipartite pure states

In this chapter, the upper and lower bounds on the transformation probability between multipartite pure states will be derived. For tripartite pure states, it is well known that there are two inequivalent classes of genuine tripartite entangled states, namely the Greenberger-Horne-Zeilinger (GHZ) class and the W class. Any two states within the same class can be transformed into each other via stochastic local operations and classical communication with a nonzero probability. The optimal conversion probability, however, is only known for special cases. Here, lower and upper bounds are derived for the optimal probability of transformation from a GHZ state to other states of the GHZ class. A key idea in the derivation of the upper bounds is to consider the action of the local operations and classical communications (LOCC) protocol on a diﬀerent input state, namely √1 000 111 ), and to demand that the probability of an outcome 2 | i − | i remains bounded by 1. We also ﬁnd an upper bound for more general cases by using the constraints of the so-called interference term and 3-tangle. Moreover, some of the results are generalized to the case in which each party holds a higher dimensional system. 1 In particular, the GHZ state generalized to three qutrits, that is, GHZ3 = √ ( 000 + | i 3 | i 111 + 222 ) shared among three parties can be transformed to any tripartite three-qubit | i | i pure state with probability 1 via LOCC. Some of our results can also be generalized to the case of a multipartite state shared by more than three parties. The content of this chapter is mainly based on [33].

23 Chapter 3. LOCC transformation bounds between multipartite pure states24

3.1 Introduction

Entanglement is the most peculiar feature that distinguishes quantum physics from classical physics and lies at the heart of quantum information theory. Thus it is important to get a good understanding of entanglement properties of quantum states. These properties are well understood for bipartite pure states. In the standard distant laboratory paradigm, suppose two distant parties, Alice and Bob, shared a bipartite entangled state. They may apply local operations and classical communications (LOCC) to convert it into another partite state. Bennett et al [6] has answered the question for the rate of LOCC transformation between bipartite pure states. It is quantified by the von Neumman entropy of a reduced density matrix. For the single-copy case, the optimal conversion probabilities are known for any pure state transformation [56, 61, 77]. For an LOCC transformation protocol, if it can succeed with probability 1, we call it deterministic, if it can only succeed with a nonzero probability smaller than 1, we call it stochastic, or SLOCC (Stochastic Local Operators and Classical Communications). For mixed states, the question of what the optimal rate of transformations is between them is still largely open. For multipartite states, however, the problem is much more complicated. There exist different types of entanglement and therefore the transformations are rather involved. For the case of tripartite pure three qubit states, a characterization into six different entanglement classes, of which two contain true tripartite entanglement, exists [34]. One is the GHZ class state, which is defined as

iϕ φGHZ = √K(cδ 0 0 0 + sδe ϕA ϕB ϕC ) (3.1) | i | i | i | i | i | i | i where

ϕA = cα 0 + sα 1 (3.2) | i | i | i ϕB = cβ 0 + sβ 1 (3.3) | i | i | i ϕC = cγ 0 + sγ 1 (3.4) | i | i | i

−1 1 and K=(1 + 2cδsδcαcβcγcφ) [ , ), cδ = cos δ, sδ = sin δ, the same for α, β, γ, φ. ∈ 2 ∞ The range for the parameters are δ (0, π ], α, β, γ (0, π ] and ϕ [0, 2π). ∈ 4 ∈ 2 ∈ Another is W class state, which is deﬁned as a state that is unitarily equivalent to

φ = (√c 0 + √d 1 ) 00 + 0 (√a 01 + √b 10 ) (3.5) | i | i | i | i | i | i | i Chapter 3. LOCC transformation bounds between multipartite pure states25

with c + d + a + b = 1. A transformation between any two states of the same class is always possible with nonzero probability. However, here comes the key point. The optimal conversion between the states within the same class of genuine tripartite entangled states is not known. Incidentally, a similar characterization into nine different classes exists for four qubits [76]. In 2000, the optimal rate of distillation of a GHZ state from any GHZ-class state was found [2]. Very recently, a necessary and sufficient condition for deterministically (i.e., with probability 1) transforming multipartite qubit states with Schmidt rank 2 [36] have been given [74]. In this chapter, we present new upper and lower bounds for multipartite entanglement transformations. In particular, we focus on transformations among states with the same Schmidt rank [36]. We put an emphasis on the transformation from a GHZ state to a GHZ-class state. But our upper bound can also be generalized to general transformations from one GHZ class state to another. And some of the results are derived for the more general case of higher dimensions and more parties. Especially, we find that all tripartite pure three qubit states can be transformed from 3-term GHZ state √1 ( 000 + 111 + 222 ) 3 | i | i | i with probability 1. This is a new result. Moreover, some of the general theorems for deterministic transformation are also derived. This Chapter is structured as follows. In Section 3.2, we derive upper bounds for the transformation of the GHZ-state to any other state in the GHZ-class. The upper bounds are only non-trivial for a subclass of the GHZ-class. Thus Section 3.3 and 3.4 use a different approach that results in upper bounds for a wider class of states. More specific, for any GHZ class state which does not have a known way to be transformed from GHZ state with probability 1, we can find a nontrivial upper bound for the probability of this transformation. And our upper bound can also be effective for the transformation from a GHZ class state to a large class of other GHZ class states. Lower bounds for the transformation of higher dimensional GHZ-states distributed among three or more parties to states with the same Schmidt rank are given in Section 3.5.

3.2 Upper Bound for the Conversion from GHZ state to a GHZ class state

In this section, we derive an upper bound for the conversion of the GHZ-state to any other state of the GHZ-class via LOCC. This upper bound will be nontrivial (i.e., smaller Chapter 3. LOCC transformation bounds between multipartite pure states26

than 1) for ϕ ( 1 π, 3 π). The transformation under consideration is given by ∈ 2 2

GHZ = √1 ( 000 + 111 ) | i 2 | i | i LOCC iϕ Ψ = √K(cδ 0 0 0 + sδe ϕA ϕB ϕC ), (3.6) −→ | i | i | i | i | i | i | i with the parameters deﬁned in introduction.

The LOCC operation is represented by Kraus operators Oi = Ai Bi Ci . In { ⊗ ⊗ } the following we will refer to diﬀerent Kraus operators of the LOCC protocol as diﬀerent

branches. Furthermore, a branch Oi GHZ = Φ is called a success branch if Φ Ψ , | i | i | i ∝ | i and a failure branch if there exists no LOCC-operation that can transform Φ into | i Ψ with a non-zero probability, if a branch is neither success nor failure, we call it an | i undecided branch. An optimal protocol only consists of success and failure branches. For the following analysis we ﬁrst recall two known results [34, 2]

with αi (0, 1) and aibici aibici = 1. ∈ h | i

b) This product state decomposition, i.e., the set (α1, a1b1c1 ), (α2, a2b2c2 ) is unique { | i | i } [1].

This result leads to

Lemma 2. For a successful LOCC operation within the GHZ-class,

Ψ = α1 a1b1c1 + α2 a2b2c2 | i | i | i LOCC Ψ0 = α0 a0 b0 c0 + α0 a0 b0 c0 , (3.8) −→ | i 1 | 1 1 1i 2 | 2 2 2i

described by the operator O1, we must either have the mapping

0 α1 0 0 0 O1 a1b1c1 = o1 a1b1c1 (3.9) | i α1 | i 0 α2 0 0 0 O1 a2b2c2 = o1 a2b2c2 (3.10) | i α2 | i Chapter 3. LOCC transformation bounds between multipartite pure states27

0 α2 0 0 0 O1 a1b1c1 = o1 a2b2c2 (3.11) | i α1 | i 0 α1 0 0 0 O1 a2b2c2 = o1 a1b1c1 (3.12) | i α2 | i

with some proportionality constant o1, which can be chosen to be real. See Figure 3.1, Figure 3.2.

Figure 3.1: mapping type 1. c 2010 American Physical Society

Figure 3.2: mapping type 2. c 2010 American Physical Society

Proof: Since a LOCC Kraus operator is always of the form O1 = A1 B1 C1, a ⊗ ⊗ product state is always transformed into a product state. With that observation and the Chapter 3. LOCC transformation bounds between multipartite pure states28

fact that the two-term product decomposition of a tripartite GHZ-class state is unique (Lemma 1), Lemma 2 follows.

Theorem 1. An upper bound for the conversion probability for

GHZ = √1 ( 000 + 111 ) | i 2 | i | i iϕ Ψ = √K(cδ 000 + sδe ϕAϕBϕC ), (3.13) → | i | i | i where the parameters are deﬁned in Equation (3.6), is given by

1 + 2c s c c c c p min 1, δ δ α β γ ϕ (3.14) ≤ 1 2cδsδcαcβcγcϕ −

Idea of the Proof: From Lemma 2 we know that, for a success branch, each product state (in the Schmidt term) of the input states have to be mapped to a product state of the output state. This allows us to infer how the same LOCC protocol acts on the phase ﬂipped GHZ state, i.e., √1 ( 000 111 ). From the requirement that the sum of 2 | i − | i the probabilities for the output states have to sum to 1 for this transformation, we can derive a bound for the parameters arising in the original transformation. This gives a bound on the successful transformation probability.

Proof: Consider the optimal LOCC strategy, given by the Kraus operators Oi =

Ai Bi Ci. According to Lemma 2, there are two possibilities to have a successful ⊗ ⊗ branch. They are

Oi 000 = oicδ 000 (3.15) | i | i iϕ Oi 111 = oie sδ ϕAϕBϕC (3.16) | i | i

for i = 1, . . . , n1, and

iϕ Oi 000 = oie sδ ϕAϕBϕC (3.17) | i | i Oi 111 = oicδ 000 (3.18) | i | i

for i = n1 + 1, . . . , n1 + n2. Both cases give the desired transformation

1 iϕ oi Oi GHZ = oi(cδ 000 + e sδ ϕAϕBϕC ) = Ψ (3.19) | i √2 | i | i √2K | i Chapter 3. LOCC transformation bounds between multipartite pure states29

for i = 1, . . . , n1 + n2. The successful conversion probability is then given by

n +n 1 X1 2 p = o2. (3.20) 2K i i=1

Pn1+n2 2 To get an upper bound for i=1 oi , we consider how

1 ( 000 111 ) (3.21) √2 | i − | i

behaves when put through the Kraus Operator Oi. We see that

1 Oi √ ( 000 111 ) 2 | i − | i 1 iϕ oi 0 = √ oi(cδ 000 e sδ ϕAϕBϕC ) = √ Ψ (3.22) 2 | i − | i 2K0 | i with 0 0 iϕ Ψ = √K (cδ 000 e sδ ϕAϕBϕC ), (3.23) | i | i − | i 0 1 where K = , for i = 1, . . . , n1 + n2 (up to an overall minus sign for i = n1 + 1−2cδsδcαcβ cγ cϕ 1 Pn1+n2 2 1, . . . n1 + n2). Thus the conversion probability for this process is given by 2K0 i=1 oi . Being a probability, this has to be bounded by 1, giving Pn1+n2 o2 2K0. This together i=1 i ≤ with Equation (3.20) gives the upper bound

K0 1 + 2c s c c c c p = δ δ α β γ ϕ (3.24) ≤ K 1 2cδsδcαcβcγcϕ − for the process described by Equation (3.13).

Special Case: Regarding the special case, where we have ϕA = ϕB = ϕC , cα = | i | i | i = = , = 0, and = = √1 , i.e., cβ cγ λa ϕ cδ sδ 2

1 Ψ = ( 000 aaa ), (3.25) | i √2p1 λ3 | i − | i − a we get 3 1 λa p − 3 . (3.26) ≤ 1 + λa

Theorem 1 gives a non-trivial upper bound for the transformation from the GHZ- state to a GHZ-class state for all values of ϕ with cos ϕ < 0, i.e., φ ( π , 3π ). This nicely ∈ 2 2 shows, that unlike in the bipartite case, where the maximally entangled EPR-state can Chapter 3. LOCC transformation bounds between multipartite pure states30 be tranformed into any other pure two qubit state with probability one, the GHZ-state, which exhibits maximal genuine tripartite entanglement as it maximizes the 3-tangle [29] and tracing out one qubit results in a totally mixed state, cannot be transformed to all other states in the same class with probability one.

3.3 Failure Branch

Recall in the last section that Eq. 3.14 gives a trivial bound for the case φ ( π , 3π ). ∈ 2 2 Here, we will derive a useful bound for a larger class of states: we find a upper bound π 3π nontrivial for all the cases except φ = or and 000 ϕAϕBϕC = 0. In fact, it was 2 2 h | i shown that these two kinds of transformations can succeed with probability 1 [74]. Our proof has two important ingredients, namely, the conservation of a new quantity defined as "interference term" under positive operator valued measures (POVMs) and that the three tangle is an entanglement monotone, which we will discuss in detail in the following. The idea of our discussion is that, firstly, recall our definition of "failure branch" as one can not be successful with any nonzero probability, we will prove the weight summation of the so-called interference terms and normalizations of all the branches in an LOCC protocol should be constant during the transformation, which is included in section 3.3. After that, we find that three tangle is bounded for a fixed interference term which will be defined in this section. Then, we try to see the whole process from the weak measurement aspect, which divides the whole process into many infinitesimal steps and each step changes the state very little. That is to say, the state is changing continuously. Then we stop in the middle and investigate whether there will be a new upper bound. Surprisingly we find there are some new upper bounds and these upper bounds will still be effective in the following steps, even when we reach the end. So it can be used to derive a new upper bound for the supremum success probability of the whole LOCC protocol. Detailed discussion will be showed in section 3.4.

Theorem 2. For the transformation from GHZ to GHZ-class state φ , failure branches | i should end with a state with at least one parties’ reduced matrix with rank 1.

Proof: Suppose we would like to get a GHZ-class state φ = √K(cδ 0 0 0 + iϕ | i | i | i | i sδe ϕA ϕB ϕC ), where 0A is linearly independent of ϕA , the same for B and C. If | i | i | i | i | i 0 0 there is a state whose reduced density matrices are all of full rank, φ = √K (cδ 0 0 0 + 0 iϕ0 0 0 0 0 | i | i | i | i s e ϕ ϕ ϕ ), where 0A is linearly independent of ϕ , the same for B and C. δ | Ai | Bi | C i | i | Ai Chapter 3. LOCC transformation bounds between multipartite pure states31

Then it is easy to see, the equation

OA 0 = 0 (3.27) | i | i 0 OA ϕ = ϕA (3.28) | Ai | i always has a non-trivial solution, the same for B and C. That means we can always transform this state into φ with nonzero probability. | i

3.3.1 Conservation of interference term

To go further, we want to use the following property of the LOCC Kraus operators. For P + a complete set of Kraus operators Oi = Ai Bi Ci, we have O Oi = 1. ⊗ ⊗ i Suppose that a Kraus operator O satisﬁes

O 000 = α a1b1c1 (3.29) | i | i

O 111 = β a2b2c2 (3.30) | i | i with a1b1c1 a1b1c1 = a2b2c2 a2b2c2 = 1. h | i h | i Then it can transform = √1 ( 000 + 111 ) into = √1 ( + GHZ 2 ψ 2p α a1b1c1 1 | i | i | i | i | i β a2b2c2 , where √ is the normalization factor and one can check p is exactly the | i 2p probability of getting ψ . From here we deﬁne interference term and normalization in | i the following:

Deﬁnition 4. For a normalized GHZ-class state γ where γ γ = 1, written in the 1 | i h | i form γ = √ (α a1b1c1 + β a2b2c2 ), suppose a1b1c1 a2b2c2 = k, then we call the real | i 2 | i | i h | i part of α∗βk the interference term I of γ . | i It is easy to see if an operator O transforms GHZ to a state ψ , the interference + | i | i term of ψ is in fact the real part of <000|O O|111> , where p is the probability of the | i p branch corresponding to operator O.

Remark 8. In fact, one can ﬁnd I = 1 1 ( α 2 + β 2). − 2 | | | | Remark 9. Note also that < I 1. In other words, it can be unbounded below. −∞ ≤ This fact will become important in our discussion in Section 3.4.

Remark 10. Notice that a failure branch gives a state that is outside the GHZ class. For such a state, the actual value of interference term depends not only on the state itself, Chapter 3. LOCC transformation bounds between multipartite pure states32

but also on the particular Kraus operator, Oi, and the initial state, φi, used to reach the state. So, when we talk about the interference term of failure branches of an SLOCC transformation, we need to be careful: We are not talking about the interference term of the state given by the failure branches, but the interference term determined by the whole transformation protocol.

Theorem 3. For a complete set of LOCCs which transforms GHZ state to other states, in which the operators are Oi , the weighted sum of the interference terms in all the { } branches should be zero.

X 0 = p(Oi GHZ )I(Oi GHZ ) (3.31) | i | i

where p(Oi GHZ ) is the probability of branch corresponding to the Kraus operator Oi, | i and I(Oi GHZ ) denotes the interference term I for a state Oi GHZ . | i | i

Proof: Suppose the corresponding complete set of Kraus operators consists of Oi = Ai Bi Ci. P + ⊗ ⊗ Then we have Oi Oi = Id. So, we should have

From the deﬁnition of interference term I we know the real part of the right side of Equation (3.32) is exactly the weighted sum of I of each branch. As the right side of Equation (3.32) is equal to zero, its real part should also be zero, which means for a transformation from GHZ to other states, the average value of the interference terms | i of all the states we get in each branch should be zero. We call this the conservation of interference term.

3.3.2 Conservation of normalization

Deﬁnition 5. For a two-term tripartite state , written in the form = √1 ( + γ γ 2 α a1b1c1 1 2 2 | i | i | i β a2b2c2 , then we call ( α + β ) the normalization of γ . | i 2 | | | | | i Easy to see if an operator O transforms GHZ to the state ψ , the normalization of + + | i | i ψ is in fact <000|O O|000>+<111|O O|111> , where p is the probability. And because OρO+ | i 2p is positive, normalization should be always no less than zero. Chapter 3. LOCC transformation bounds between multipartite pure states33

Suppose the corresponding complete set of Kraus operators consists of Oi = Ai P + { ⊗ Bi Ci . Then we have O Oi = I. So we should have ⊗ } i

1 = < GHZ GHZ > | = 1 (< 000 + < 111 )( 000 > + 111 >) 2 | | | | = 1 ( 000 000 + 111 111 ) 2 h | i h | i 1 P + P + = 2 ( 000 Oi Oi 000 + 111 Oi Oi 111 ) h | | i + h | + | i P <000|Oi Oi|000>+<111|Oi Oi|111> = p(Oi GHZ ) (3.33) | i 2p(Oi|GHZi)

From the deﬁnition of normalization we know it is exactly the weighed sum of the normalization of each branch. That is to say, for a transformation from GHZ to other | i states, the average value of the normalization of all the states we get in each branch should be 1. And recall that normalization can be no less than zero. So each term in the summation should be no larger than 1, which means for each branch, the product of its probability and the normalization of the state it gets should be no larger than 1. In fact, the conservation of normalization can be derived from conservation of interference term. However, conservation of the normalization also gives the following. For each branch, the product of its probability and the normalization of the state it gets should be no larger than 1. The fact is also useful in determining the upper bound of transformation probability. The basic idea is that, if we know the state we want and the state failure branch gives, equations (3.32) and (3.33) combined with the fact that the summation of probability should be one can give us some implication about the supremum success probability. For example, we can have the following theorem:

Theorem 4. For a transformation protocol from GHZ state to a GHZ-class state φ | i whose interference term is x, which is positive (negative), if there exists a y (y > 0), such that, the interference term of all the failure branches are larger than -y (smaller than y), we have an upper bound for its successful probability ps in the following: if x > 0:

U y ps p ( y) = . (3.34) ≤ − x + y if x < 0: U y ps p (y) = . (3.35) ≤ −x y − Chapter 3. LOCC transformation bounds between multipartite pure states34

See

0.8

0.6 p

0.4

0.2

pu 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 a

y 1 U 3 Figure 3.3: The value of p as a function of a. In this figure, a = ( y−1 ) . So when a goes from 0 to 1, y goes from 0 to . Note that as y goes to infinity, a goes to 1. We express the value as a function of a because∞ it will be easier for us to combine different graphs into one graph later. c 2010 American Physical Society.

Proof: Take x > 0, suppose there are n failure branches, whose probabilities are pf , pf , , pf , and the corresponding interference terms are y1, y2, yn, then we have 1 2 · n − − ·−

P psx pf yi = 0 (3.36) − i P ps + pfi = 1 (3.37)

Rewrite it in the following form,

0 psx pfty = 0 (3.38) − ps + pft = 1 (3.39)

P p y P 0 fi i where pft = pf and y = . The solution of it is i pft

y0 p = (3.40) s x + y0

As the interference term of all the failure branches are larger than -y, we have y0 < y, U y then we can get ps < p ( y) = . The discussion for the case when x < 0 is similar. − x+y

Remark 11. Recall the range of the I can be < I 1, which means I can be −∞ ≤ Chapter 3. LOCC transformation bounds between multipartite pure states35 unbounded below. Then in the x > 0 case, if the I of the failure branch goes to , or we −∞ can say y goes to , we will have pU ( y) arbitrary close to 1. Therefore, theorem 4 alone ∞ − is not enough for establishing a non-trivial upper bound. To derive a non-trivial upper bound, we need to ﬁnd some additional constraints which are related to the interference term. In fact, this is what we will do in section 3.4.

3.4 Upper Bound for a general case

In this section, we will find an upper bound in a more general case. Recall the problem of theorem 4 is that the interference can be unbounded below. So we would like to find an additional constraint. It turns out that the fact that the 3-tangle, a measure of tripartite entanglement introduced in [29], is an entanglement monotone (i.e., it cannot increase on average under LOCCs) is precisely what we need [34]. Our strategy is that, for any possible transformation protocol, we would like to construct a new protocol that has the following two properties: 1. It has an upper bound for the maximal successful probability of transformation which is obviously smaller than one; 2. We can reconstruct the original protocol from this new protocol, which means the successful probability of this new protocol can be no less than the original one. The way we construct such a protocol is given in 3.4.2 and the bound of it will be given in 3.4.3, in which we deal with a special example: the transformation from GHZ state to a special GHZ class state φ = γ( 000 + aaa ). In 3.4.4, we will generalize this bound to | i | i | i more general cases, where we find for any transformation from one GHZ-class state φ to | i another GHZ-class state ψ with different interference terms, we can find an nontrivial | i upper bound for the successful probability.

3.4.1 interference term and the maximal value of the 3-tangle of a GHZ-class state

Now consider such a question: Suppose we have an unknown GHZ class state φGHZ = iϕ | i √K(cδ 0 0 0 + sδe ϕA ϕB ϕC ) with a given interference term f, what is the | i | i | i | i | i | i maximal value of the 3-tangle τABC [34] ?

Theorem 5. For a GHZ class φGHZ , if its interference term is I, then the maximal (1−a2)3 | i f 1 3 value of its 3-tangle is (1+a3)2 , where a = ( 1−f ) .

The proof will be given in the appendix 8.1. Chapter 3. LOCC transformation bounds between multipartite pure states36

3.4.2 "stop and reconstruct" procedure

From [63], we know every measurement can be seen as constructed by many inﬁnitesimal steps of weak measurement, that is, a measurement which only slightly changes the original state. From this view, to get a better understanding of the transformation

protocol, we would like to try to reduce the case where a failure branch gives an I > I0

(some prescribed value) to the case where an undecided branch has I = I0. That is to say, we are using a reduction idea. First we need to answer the following question: Can we stop at some intermediate point and reconstruct the original measurement? It turns out that the answer is yes. In fact, from [63], the following theorem follows easily.

Figure 3.4: "stop and reconstruct" for a two-outcome measurement. c 2010 American Physical Society.

Theorem 6. A two-outcome measurement M1,M2 can be reconstructed by stopping at { } −2x 0 −2x 0 an immediate step √1 e M1, √1 + e M (x) and a reconstructing measurement 0 0 { − 0 p + } M (x, + ),M (x, ) , where M = M M1 and { ∞ −∞ } 1 1

0 q 1+tanh(x) 0 M (x, + ) = I+tanh(x)(M 02−M 02) M2, (3.41) ∞ 2 1 0 q 1−tanh(x) 0 M (x, ) = I+tanh(x)(M 02−M 02) M1 (3.42) −∞ 2 1 See Figure 3.4 for a graphic discription.

0 0 Proof: Firstly, from polar decomposition we have M1 = U1M1,M2 = U2M2, where U1 Chapter 3. LOCC transformation bounds between multipartite pure states37

0 p + 0 0 0 and U2 are unitary, M2 = M2 M2. Then M1,M2 is also a measurement. As M1 and 0 { } M2 are positive, it can be reconstructed from inﬁnitesimal steps .[63] Secondly, instead of q I+tanh(x)(M 02−M 02) measure M 0 ,M 0 , we stop at M 0(x) = 2 1 before we reach M 0 , that is { 1 2} 2 2 to say, we perform measurement √1 e−2xM 0 , √1 + e−2xM 0(x) . The eﬀect is we still { − 1 } 0 0+ −2x 0 0+ got M ρM /p1 but the probability become √1 e p1, but instead of get M ρM /p2, 1 1 − 2 2 we get M 0(x)ρM 0+(x)/p(x) where p(x) = T r(M 0(x)ρM 0+(x)). Thirdly, we do nothing to the M 0 branch, but do a POVM M 0(x, + ),M 0(x, ) . 1 { ∞ −∞ } On the M’(x) branch, it is easy to prove that,

M 0(x, )M 0(x) = e−xM 0 ,M 0(x, )M 0(x) = M 0 , (3.43) ∞ 1 −∞ 2

So in total, we perform a POVM √1 e−2xM 0 , e−xM 0 ,M 0 , that is just the same as { − 1 1 2} M 0 ,M 0 . Finally, if we get the result of measurement M 0 (M 0 ), perform a unitary { 1 2} 1 2 transformation U1(U2), we can reconstruct M1,M2 with a stop in the middle. QED. { } However, a protocol may contain many measurements and measurements with more than two outcomes, can we still use this method to stop in the middle and reconstruct everything?

Figure 3.5: The original protocol written in the many two-outcome measurements form. c 2010 American Physical Society.

The answer is yes. To show this, first we need to rewrite every measurement in the protocol into a sequence of two-outcome measurements [3], see Figure 3.5. Then the Chapter 3. LOCC transformation bounds between multipartite pure states38 protocol consists of only two-outcome measurements. So the "stop and reconstruct" can work for each of them. The only thing is that, now, each two-outcome measurement may be related to many other two-outcome measurements, so during the "stop and reconstruct" process, many measurements might be affected. How can we be sure we can reconstruct everything? For this problem, notice that these two-outcome measurements are all in order. Then when we do the "stop and reconstruct", the principle is that we should always stop at the earlier two-outcome measurement first. Moreover, we need to reconstruct the earlier ones first. See Figure 3.6.

Figure 3.6: "stop and reconstruct" for general protocol, I stands for the interference term. c 2010 American Physical Society.

3.4.3 Example: GHZ φ = γ( 000 + aaa ) | i → | i | i | i Now, we want to ﬁnd an upper bound for the success probability of the transformation.

Theorem 7. Suppose we have a SLOCC transformation protocol from GHZ to φ = | i | i γ( 000 + aaa ), where a = c 0 + √1 c2 1 and c (0, 1]. Suppose the successful | i | i | i | i − | i ∈ probability is pm. Then we can always ﬁnd a protocol consisting of only successful and failure branches which has a successful probability no less than pm.

Proof: If the protocol is in that form, we do nothing. If the protocol has some branches which are neither successful nor failure. Then we do nothing to the successful or failure branches. However, for the undecided branches, from the deﬁnition of it we know we can Chapter 3. LOCC transformation bounds between multipartite pure states39

always ﬁnd a POVM that can transform it into the desired state with nonzero probability

δp. Then the total successful probability is pm + δp, which is higher than pm. In all, we can always ﬁnd a protocol consisting of only successful and failure branches which have

a successful probability no less than pm. QED. Now modify the protocol we get in the first step in the following way: Suppose we can find at least one failure branch that have interference term smaller than -y, where y 0. then we can find a x, where 0 x y. As our initial interference ≥ ≤ ≤ term is zero, now we can use the weak measurement idea to let all the branches stop if its interference term reaches -x and do nothing to the branches which never reach -x. And we can get a new protocol in Figure 3.7

Figure 3.7: The new protocol, which can reconstruct the original one. c 2010 American Physical Society.

Remark 12. Note that to make this new protocol work, we have applied the intermediate value theorem. That is to say, we implicitly assume that the interference terms, I, of the two intermediate states speciﬁed in Theorem 5, are continuous functions of x. This assumption works because, from [63], we know M(x, δx) changes the state given by M(x) GHZ very little, or we can say it is a weak measurement. While from the | i expression of interference term Equation (8.1) in Appendix, we know interference term is a continuous function of the parameters of the state. Then, as the state changes very little under the weak measurement, the interference term also changes continuously. Then we get a new protocol. It has two properties: Chapter 3. LOCC transformation bounds between multipartite pure states40

1) There are three kinds of branches: failure branches with interference term larger than or equal to -x, successful branches and the branches neither successful nor failure with interference term -x. 2) From the "stop and reconstruct" part, we know we can reconstruct the original protocol by performing LOCCs (may be a sequence of measurements) just on these branches which have interference term -x and do nothing on other branches. That is to say, just do LOCCs on the -x branches, we can get a total successful probability no less than the original one. So, if we have an upper bound of successful probability for the new protocol, that should also be an upper bound for the original one. Then we can ﬁnd the upper bound for this new protocol. Now, the protocol consists of three kinds of branches: successful branches, failure branches with interference term larger than or equal to -x, and undecided branches with interference term -x. The total successful probability of this protocol consists of two probability: the already existing

successful branches’ total probability pse and the probability we can transform from the -x branches to the states we want.

Theorem 8. As in Theorem 7, we consider a SLOCC transformation from GHZ to | i φ = γ( 000 + aaa ). For all the possible new protocols shown in Figure 3.7, there is | i | i | i an upper bound for the success probability

p¯s( x) = pas( x) + pu( x) pm(s x) (3.44) − − − ∗ | − where pas( x) is the already successful branches in this condition, while pu( x) is the − − probability of the undecided branches with interference term -x. And pm(s x) is the | − maximal probability to transform a GHZ-class state with interference term -x into the destination state φs. And we will get a3 1−a3 pas = 3 3 , pu( x) = 1 pas, c + a 1+c3 1−a3 − − max(τABC (φ|I(φ)=−x)) pm(s x) = min( , 1) | − τABC (φs) a3 where a is the solution of the equation x = 1−a3 , τABC stands for the 3-tangle.

We ﬁrstly consider the case that there exists no failure branches with interference term larger than -x, later we will show the other case can only give an upper bound smaller than in this case.

Lemma 3. If the new protocol shown in Figure 3.7 consists of only successful branches and branches with interference term -x (no failure branches with interference term larger Chapter 3. LOCC transformation bounds between multipartite pure states41

than -x), it has an upper bound

p¯s( x) = pas( x) + pu( x) pm(s x) (3.45) − − − ∗ | − where the parameters are deﬁned in Theorem 8. Proof: For the already existing successful branches, the total probability is determined a3 by -x and the conservation of interference term. As x = 1−a3 , we have

pas( x)I( φs ) pu( x)x = 0 (3.46) − | i − − pas( x) + pu( x) = 1 (3.47) − −

a3 1−a3 Solving the above two equations, we can ﬁnd pas( x) = 3 3 . c + a − 1+c3 1−a3 For the maximum value of ps( x), using the 3-tangle idea, we know it is bounded by − pu( x) pm(s x) where − ∗ | −

max(τABC (φ I(φ) = x)) pm(s x) = min( | − , 1) (3.48) | − τABC (φs)

Then we ﬁnd an upper bound for the successful probability of this new protocol when there is no failure branch having interference term larger than -x:

p¯s( x) = pas( x) + pu( x) pm(s x) (3.49) − − − ∗ | −

Remark 13. To show it is really an upper bound for the successful probability for the new protocol, we need to show if there is any other failure branch with interference term larger than -x, we can only get a successful probability smaller than this. Proof of Theorem 8: To prove this theorem, we just need to prove the following: If the new protocol contains a failure branch which has an interference term larger than -x, it has an upper bound for the success probability smaller than what we get in Lemma 3. Consider the conservation of interference term, now we have:

0 P 0 p ( x)I( φs ) + pfiI( φf ) p ( x)x = 0 (3.50) as − | i | i i − u − 0 P 0 p ( x) + pfi + p ( x) = 1 (3.51) as − u − Chapter 3. LOCC transformation bounds between multipartite pure states42

which can be rewritten as

0 0 0 p ( x)I( φs ) p”( x )x = 0 (3.52) as − | i − − p0 ( x) + p”( x0) = 1 (3.53) as − − where

P 0 0 pfiI(|φfi i)−pu(−x)x x = P 0 (3.54) pfi+pu(−x) 0 P 0 p”( x ) = pfi + p ( x) (3.55) − u −

0 0 As I( φfi ) > x, we know x > x, so pas( x) < pas( x). Let the diﬀerence |0 i − − − − − 0 between p ( x) and pas( x) be δs, then we have δs = pas( x) p ( x), so δs > 0 and as − − − − as − 0 P 0 0 p”( x ) = pu( x) + δs. As pfi > 0, we have p ( x) < p”( x ) = pu( x) + δs. So the − − − − − total successful probability in this case is

Here, in the second last step, we have used the fact that pm(s x) can not be larger | − than one. So we know it is really an upper bound for the successful probability for the new protocol, which should also be an upper bound for the successful probability for the original protocol, and an upper bound for the transformation protocols which contains at least on failure branch which has interference term smaller than -x ( or we can say it 0 passes -x). So we have p¯ < p¯s( x), which means p¯s( x) is an upper bound for the new s − − protocol.

Corollary 1. As in Theorem 7, we consider a SLOCC transformation from GHZ to | i φ = γ( 000 + aaa ). If a protocol contains at least one failure branch whose interference | i | i | i Chapter 3. LOCC transformation bounds between multipartite pure states43

term is smaller than -y, its successful probability should be bounded by all the p¯s( x), − where 0 x y. ≤ ≤ Proof: If we see every branch from the weak measurement idea. We will ﬁnd the interference term should change continuously, so we can stop at any point between 0 and -y. For each point we choose, we can get an upper bound. And all the upper bounds should be the upper bounds of the original branch.

Corollary 2. For a LOCC transformation protocol from GHZ to φ = γ( 000 + aaa ), | i | i | i | i if the minimum interference term of all the failure branches is -z, then its successful probability should be bounded by

U pbound( z) = min(p ( z), p¯τ ( z)) (3.57) − − ABC −

where p¯τ ( z)) = min0≤x≤z(p ¯s( x)). ABC − − Proof: If the minimum interference term is -z, then from Theorem 4, we know there U z is an upper bound p ( z) = , which is in fact pas( z). As it is bounded by − I(|φsi)+z − all the p¯s( x), where 0 x z, we can ﬁnd another upper bound p¯τ ( z)) = − ≤ ≤ ABC − min0≤x≤z(p ¯s( x)). See Figure 3.8 for the relation between p¯s and p¯τ . Then the − ABC minimum of these two bounds is also an upper bound, which we call pbound( z). −

1.01

0.99

0.98

p 0.97

0.96

0.95

0.94 p± ± s pτ 0.93 abc 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 a

Figure 3.8: the relation between p¯s and p¯τ . c 2010 American Physical Society. ABC Chapter 3. LOCC transformation bounds between multipartite pure states44

1 b c

0.8

0.6 p

0.4

0.2 pu p± ± s pτ 0 abc 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 a

Figure 3.9: The upper bound for the transformation x 1 In this ﬁgure, a = ( ) 3 . So when a goes from 0 to 1, x goes from 0 to . The − x−1 ∞ dashed line is the plot of p¯s as a function of -x, the dot line is the plot of p¯τABC , the U solid line is the plot of p . Notice that point b corresponds to the minimum value of p¯s,

before b, p¯s decreases monotonically. So before b, p¯τABC is the same as p¯s, while after b, U p¯τABC remains to be the value of p¯s at b. Another thing is that before c, p is smaller U than p¯τABC , while after c, p¯τABC is smaller than p . So the ﬁnal plot we get for the upper bound is the solid line before c and the dot line after c, which we call upper bound line. The meaning of this upper bound line is that, for a given transformation protocol, if the y 1 3 smallest interference term of all the failure branches is -y, let k = ( y−1 ) , the success probability can not be larger than the corresponding point in the− upper bound line. Consider all of the possible protocols (a goes from 0 to 1), the upper bound of the transformation probability is the largest value of the points on the upper bound line, which is just the minimum value of p¯s. c 2010 American Physical Society.

Chapter 3. LOCC transformation bounds between multipartite pure states45

Theorem 9. An upper bound of LOCC transformation from GHZ state to a speciﬁc GHZ

class state φ = γ( 000 + aaa ) is the maximum value of pbound( z) where z [0, + ). | i | i | i − ∈ ∞ And it is in fact the minimum value of p¯s( z), where z [0, + ). − ∈ ∞

Proof: The basic picture of our proof can be represented in Figure 3.9. Now we consider all the possible transformation protocols. Then the value of the minimum interference term -z may vary from 0 to . (We can always ﬁnd a protocol ∞ giving a very small value of -z, while its successful probability is still bounded.) Easy to

see an upper bound is the maximum value of pbound( z) for all the possible values of z, − where z [0, ). In fact, we can ﬁnd the upper bound we get for this transformation is ∈ ∞ the minimum value of p¯s( z), where z [0, + ). − ∈ ∞

(1−c2)3 Put τABC (φ) = (1+c3)2 , in to the equation, we can get the upper bound. The analytic value is hard to get, if we put c=0.5. The minimum value of p¯s( x) = pas( x)+pu( x) − − − ∗ pm(s x) is 0.9604 at x=1.13062, which is less than 1. | −

3.4.4 general case

In the above, we have considered an upper bound for the special case of GHZ φ = | i → | i γ( 000 + aaa ) to ﬁnd the upper bound for it. Now, we will consider two more general | i | i cases. First, we will consider the transformation GHZ φGHZ = √K(cδ 0 0 0 + iϕ | i → | i | i | i | i sδe ϕA ϕB ϕC ), which is the general GHZ class state; Second, we will consider a | i | i | i general GHZ class state to another general GHZ class state. iϕ 1. GHZ φGHZ = √K(cδ 0 0 0 + sδe ϕA ϕB ϕC ). In this case, we just | i → | i | i | i | i | i | i | i need to change the expression for the interference term and 3-tangle of the destination state into

2cαcβ cγ sδcδcϕ Interference term : I(φs) = (3.58) (1+2cαcβ cγ sδcδ) 2 2 2 2 2 4sαsβ sγ sδcδ 3 tangle : τABC (φs) = 2 (3.59) − (1+2cαcβ cγ sδcδ)

Then we can use the similar process, except changing the corresponding value of Interference term and 3-tangle, see the following for details. Firstly, using the "stop and reconstruct" method to get the new protocol with only successful branches and undecided branches with interference term x. We have Chapter 3. LOCC transformation bounds between multipartite pure states46

pas(x)I( φs ) + pu(x)x = 0 (3.60) | i pas(x) + pu(x) = 1 (3.61)

We have

x pas(x) = = 1 pu(x) (3.62) x−I(|φsi) − max(τABC (φ|I(φ)=x)) pm(s x) = min( , 1) (3.63) | τABC (φs)

Then, the supremum success probability of this new protocol should be bounded by

p¯s(x) = pas(x) + pu(x) pm(s x) (3.64) ∗ |

1 Consider all possible protocols, we ﬁnd the minimum of p¯s(x) where x [0, ] if ∈ 2 I( φs ) < 0 and x ( , 0] if I( φs ) > 0 is an upper bound of the success probability | i ∈ −∞ | i of this transformation.

Remark 14. Suppose we want to transform a GHZ state to a GHZ-class state φGHZ = iϕ | i √K(cδ 0 0 0 +sδe ϕA ϕB ϕC ). If I( φGHZ ) = 0, we can always ﬁnd a nontrivial | i | i | i | i | i | i | i 6 upper bound. However, for the case where I( φGHZ ) = 0, we will get a trivial upper bound | i π 3π 1. This condition consists of 2 possibilities: 1. 000 ϕAϕBϕC = 0; 2. ϕ = or . h | i 2 2 In fact, in the paper [74], they have provided a protocol for such a transformation with success probability 1.

2. A general GHZ class state to another general GHZ class state. In this case, the interference term is still conserved, but the initial value should be the interference term of the initial state.

pas(x)I( φs ) + pu(x)x = Iinitial (3.65) | i pas(x) + pu(x) = 1 (3.66)

We have

Iinitial−x pas(x) = = 1 pu(x) (3.67) I(|φsi)−x − max(τABC (φ|I(φ)=x)) pm(s x) = min( , 1) (3.68) | τABC (φs) Chapter 3. LOCC transformation bounds between multipartite pure states47

Then, the supremum success probability of this new protocol should be bounded by

p¯s(x) =p ¯s(x) + pu(x) pm(s x) (3.69) ∗ |

1 Consider all possible protocols, we ﬁnd the minimum of p¯s(x) where x [Iinitial, ] if ∈ 2 I( φs ) < Iinitial and x ( ,Iinitial] if I( φs ) > Iinitial is an upper bound of the success | i ∈ −∞ | i probability of this transformation.

Example 4. An upper bound for the transformation from φ = γ( 000 + abc ) where | i | i | i 0 a = 0.1, 0 b = 0.2, 0 c = 0.2, to ψ = γ0( 000 + a0b0c0 ) where 0 a0 = 0.4, h | i h | i h | i | i | i | i h | i 0 b0 = 0.5, 0 c0 = 0.6 is 0.9593. See Figure 3.10. h | i h | i

0.8

0.6 p

0.4

0.2 pu p± ± s pτ 0 abc 0 0.2 0.4 0.6 0.8 1 a

Figure 3.10: upper bound of transformation probability from φ to ψ . x 1 | i | i 3 In this ﬁgure, a = ( x−1 ) where x is the maximum value of the interference term for all the failure branches.− So when a goes from 0 to 1, x goes from 0 to . The dashed ∞ line is the plot of p¯s as a function of -x, the dot line is the plot of p¯τABC , the solid line is the plot of pU . The upper bound of the transformation probability is the minimum value of p¯s. c 2010 American Physical Society.

Lemma 4. For a transformation from a tripartite state φi to another tripartite state | i φs , if the interference term of φi is not equal with φs , we can get an upper bound for | i | i | i the supremum of the successful probability which is less than 1.

Proof: If the interference term is not equal, put into the Equation 3.65 , we know the maximal successful probability pas(x) can not reach one. Otherwise the conservation of Chapter 3. LOCC transformation bounds between multipartite pure states48

interference term will be violated. In fact, as the magnitude of the interference term of

branches where we stop become larger and larger, pas(x) gets closer and closer to one. However, at the same time the maximal 3-tangle of these branches will go to zero. As the destination state is in GHZ class, its 3-tangle is not zero. So we can always ﬁnd one

x , when the magnitude square of the interference term reaches it, max(τABC (φ I(φ) = | | || | x )) < τABC (φs), then it will give an upper bound for this transformation which is smaller | | than 1.

Remark 15. One may naturally ask a question: If the interference terms of two states are the same, can we give an upper bound for the transformation probability? In this case, the above lemma can not give a nontrivial upper bound. However, we can still use other entanglement monotones, such as 3-tangle, to give an upper bound for the transformation from one to another.

Example 5. Consider the transformation from φ = γ( 000 + abc ) where 0 a = | i | i | i h | i 0.2, 0 b = 0.4, 0 c = 0.8, to ψ = γ0( 000 + a0b0c0 ) where 0 a0 = 0.4, 0 b0 = h | i h | i | i | i | i h | i h | i 0.4, 0 c0 = 0.4. One can check that I( φ ) = I( ψ ) = 0.0602. So naively we can h | i | i | i only get a trivial upper bound for the transformation between them. However, notice

that τABC ( φ ) = 0.2564 and τABC (ψ) = 0.5235, we can get an upper bound for the | i transformation from φ to ψ which is | i | i

τABC ( φ ) 0.2546 | i = = 0.4863 < 1 (3.70) τABC ( ψ ) 0.5235 | i 3.5 Lower Bound for the Transformation

After the discussion about the upper bound, we have a question: Is this upper bound tight or not? Or can we ﬁnd a transformation protocol that can reach such a transformation probability? In [21], they have provided a straight forward protocol: That is, if we want to transform

GHZ = √1 ( 000 + 111 ) | i 2 | i | i LOCC Ψ = √K( 000 + ϕA ϕB ϕC ), (3.71) −→ | i | i | i | i | i q We can let Alice perform the measurement A , I 1 A+A , where { kAk − kAk2 }

A 0 = 0 ,A 1 = ϕA (3.72) | i | i | i | i Chapter 3. LOCC transformation bounds between multipartite pure states49

3 and similarly for Bob and Charlie. Then the ﬁnal successful probability is (1+cαcβ cγ ) . (1+cα)(1+cβ )(1+cγ ) In the following, we will provide a transformation protocol that can transform GHZ- state generalized to n parties and m dimensions to other states with the same dimension and Schmidt rank with a probability higher than the straight forward protocol . However, there is still a gap between the upper bound and the lower bound we get. A surprising result derived from that is all tripartite pure 3-qubit states can be transformed from GHZ-state generalized to 3 parties and 3 dimensions by LOCC with probability 1, which was not known before. We have found a lower bound for the maximum value of the transformation probability

from GHZ state to Ψ = γ(α a1b1c1 + β a2b2c2 ) by an explicit method, which we call | i | i | i four-step method.

Figure 3.11: Four-step method. c 2010 American Physical Society.

See Figure 3.11 for the process, in the first step, we transform GHZ state into GHZ0 = α 000 + β 111 which has the same coefficient of terms (but different states) | i | i | i 0 as Ψ . Secondly, we transform GHZ into φb1 = α 0b10 + β 1b21 . Then we trans- | i | i | i | i | i form φb1 into φc1 = α 0b1c1 +β 1b2c2 . We will show the first three steps can be done | i | i | i | i with probability 1. Then if a1 a2 is zero, φc1 is unitary equivalent with Ψ so we can h | i | i | i get Ψ with probability 1. In other cases, we can still get Ψ with a higher probability | i | i than the previous result in [21] by performing an appropriate measurement. In fact, the first step is a generalization of Nielsen Majorization result [61] and Lo- Popescu’s [56] result for the maximum probability of distilling a maximally entangled Chapter 3. LOCC transformation bounds between multipartite pure states50

state. It has been noted previously in [81].

Deﬁnition 6. A GHZ-like (aka Schmidt decomposable) state is a tripartite state that can be written in the form X Ψ = λi i i i (3.73) | i i | i | i | i Theorem 1 of Lo-Popescu also holds for the GHZ-like state, because it gives a bound for the case where the Bob-Charlie alliance is allowed to perform any (non-local) operations, and when the allowed operations are restricted to the subclass of local operations, the upper bound still has to hold. Theorem 2(a) of Lo-Popescu can in the same way be applied to GHZ-like (Schmidt decomposable) states, because all unitary transformations on Bob’s side involve only a relabeling of the basis states ( i j ), and therefore extending it to GHZ-like states | i ↔ | i just changes this step to ( i i j j ), which can also be done by local unitaries only. | i | i ↔ | i | i Theorem 2(b) generalizes to GHZ-like states as well, because here Alice performs all the operations and Bob either has to perform no operation on his state at all (result "success"), or he has to discard it completely (result "failure"), which can also be done if Bob’s state is distributed among Bob and Charlie. In [81], it was shown that Nielsen’s majorization idea generalized to more parties can be applied to GHZ-like state, which means the transformation

m 1 X GHZmn = i i ... i √m 1 2 n | i i=1 | i | i | i m (3.74) X Ψ = αi i 1 i 2 ... i n → | i i=1 | i | i | i

can success with probability 1. For the second and third step, we have the following lemma.

Lemma 5. [74] The GHZ state can be transformed to Ψ = γ(α a1b1c1 + β a2b2c2 ), | i | i | i where Ψ Ψ = 1 with probability 1, if a1b1c1 and a2b2c2 are orthogonal to each other. h | i | i | i

Proof: Suppose a1 and a2 are orthogonal to each other. If we choose the basis in | i | i which a1b1c1 = 000 then we write φ = γ(α 000 + β 1 A (d1 0 + d2 1 )B(e1 0 + | i | 2 i 2 | i2 2 | i | i | i | i | i e2 1 )C ), where d1 + d2 = 1 and e1 + e2 = 1, in this case we can see γ = 1 | i | | | | | | | | Then we can do the transformation in the following way. Firstly, use the result of the ﬁrst step to transform GHZ into GHZ0 = α 000 + | i | i | i β 111 with probability 1. | i Chapter 3. LOCC transformation bounds between multipartite pure states51

Secondly, Bob performs a POVM ! ! 1 1 d1 1 1 d1 M1 = ,M2 = − , (3.75) √2 0 d2 √2 0 d2 −

1 Then with probability 2 we get φb1 = α 000 + β 1 A (d1 0 + d2 1 )B 1 C , and with 1 | i | i | i | i | i | i probability we get φb2 = α 000 β 1 (d1 0 + d2 1 )B 1 . If we get φb2 , Alice 2 | i | i − | iA | i | i | iC | i perform a unitary transformation ! 1 0 UA = , (3.76) 0 1 −

then we get φb1 , too. So, with probability 1 we get φb1 . | i | i Thirdly, Charlie performs a POVM ! ! 1 1 e1 1 1 e1 M1 = ,M2 = − , (3.77) √2 0 e2 √2 0 e2 −

1 Then with probability 2 we get φc1 = α 000 + β 1 A (d1 0 + d2 1 )B(e1 0 + e2 1 )C , | i | i | i | i 1| i | i | i which is exactly the φ we want to get, and with probability we can get φc = | i 2 | 2 i α 000 β 1 (d1 0 +d2 1 )B(e1 0 +e2 1 )C . If we get φc , again, Alice can perform | i− | iA | i | i | i | i | 2 i a unitary transformation ! 1 0 UA = , (3.78) 0 1 − to get φc1 , too. So with probability 1 we can get φc1 . Then we can get Ψ = α a1b1c1 + | i | i | i | i β a2b2c2 with certainty. | i

Then we show the ﬁrst three steps can be done with probability 1. For the last step, we have the following lemma.

2 2 Lemma 6. For Ψ = γ(α a1b1c1 + β a2b2c2 ), where α + β = 1 and γ is a normal- | i | i | i ization factor, if < a1 a2 >=λa, < b1 b2 >=λb,< c1 c2 >=λc, then there exists an LOCC | | | transformation protocol from GHZ state to ψ such that the probability of success is at | i least 1+2αβλaλbλc , where λ = min(λ , λ , λ ). 1+λm m a b c

p 2 p 2 Proof: Firstly, we have ψ = γ(α 000 +β(λa 0 + 1 λ 1 )A(λb 0 + 1 λ 1 )B(λc 0 + | i | i | i − a | i | i − b | i | i p 2 √ 1 1 λc 1 )C ), where γ = . From theorem 1, we transform GHZ state to − | i 1+2αβλaλbλc p 2 p 2 ξ = α 000 + β 1 (λb 0 + 1 λ 1 )B (λc 0 + 1 λ 1 )C , with probability 1. | i | i | iA | i − b | i | i − c | i Chapter 3. LOCC transformation bounds between multipartite pure states52

Then from ξ , Alice can do a POVM | i ! 1 1 λa M1 = √ , (3.79) 1+λa p 2 0 1 λa − ! √ 1 1 λa M2 = √ − , (3.80) 1+λa 0 0

So we have probability 1+2αβλaλbλc to get ψ , and the other branch will give a state in 1+λa | i which the rank of ρa is 1 so that the probability to get ψ from it is zero. Then the | i total probability is 1+2αβλaλbλc . However, we can do a permutation of A, B and C so that 1+λa the probability can also be 1+2αβλaλbλc and 1+2αβλaλbλc . And the maximum probability 1+λb 1+λc corresponds to min(λa, λb, λc).

Example 6. Again take the transformation from GHZ to φ = γ( 000 + aaa ), where | i | i | i | i a = c 0 + √1 c2 1 and c (0, 1] as an example, using the protocol we provide, we | i | i − | i ∈1+c3 can get a successful probability 1+c , let c=0.5, we have ps = 0.7500. In comparison, the upper bound we get in section 3 is 0.9604. There is still a gap between these two values. How to reduce it is still an open problem.

Now we will generalize the result of lemma 5 into higher dimensions and more parties. Suppose we are concerned with the transformation from the GHZ-state gener- m √1 P alized to n parties and m dimensions, GHZmn = m i=1 i 1 i 2 ... i n to ψ = Pm | i | i | i | i | i γ( i=1 αi k1i k2i ...kni ). The basic idea of our protocol can be divided into three steps: | i Pm Firstly, we would like to transform GHZmn into Ψ = αi i i i which is called | i | i i=1 | i | i | i GHZ-like (or Schmidt decomposable) state. Secondly, we transform Ψ into ψn = Pm | i | i αi i1k2 ki ...ki . We will show these two steps can be done with probability 1. i=1 | i 3 n i Finally, if for at least m-1 terms of ψ , there can be at least one party with a state | i that is orthogonal to this party’s state in every other term, our protocol can transform GHZmn into ψ with probability 1. In other cases, our protocol can be done with a | i | i probability higher than what have been known before. The second and third step for the case when for at least m-1 terms of ψ , there can | i be at least one party with a state that is orthogonal to this party’s state in every other term are incorporated in the following theorem.

Theorem 10. The GHZ-state generalized to n parties and m dimensions, GHZmn = 1 Pm Pm | i √ i i ... i , can be transformed to ψ = αi k1 k2 ...kn with probability m i=1 | i1 | i2 | in | i i=1 | i i i i 1, if p, i= p, j with kj kj = 0 for l = i. This means for at least m-1 terms, ∃ ∀ 6 ∃ h i | l i ∀ 6 Chapter 3. LOCC transformation bounds between multipartite pure states53

there has to be at least one party with a state that is orthogonal to this party’s state in every other term.

Proof: The basic idea is we at first make the coefficient of each term equal to the corresponding terms of the destination state. Then let party 2 perform a POVM which transforms the state into many states in such a form: for each term, the party 2 part of the term is the same with the destination state, but the term’s coefficient maybe of the same or the opposite sign of the destination state. Then by introducing a unitary transformation on the party 1, we can make all the coefficients the same with the destination state. Keep doing this for party 3, , n. Finally, do a similar POVM on party 1, we can · get many states in such a form: the corresponding terms are the same, but the coefficients may be of the same or the opposite sign. Then we can use unitary transformation to transform all the states into the destination state. Exact process is in the following: Pm Firstly, using the result of [81] to get GHZmn = γ( αi i i ... i ) where | i i=1 | i1 | i2 | in γ is a normalization factor. But then, the POVMs should be modified. We call the parties 1,2, , n party 1, party 2, and so on. Take Party 2 as an example, suppose ··· Pi with a unitary transformation, k2 = a2 j , in which a2 = 1, then party 2 can | i i j=0 ij | i2 00 operate a POVM

  1 a2 a2 10 ··· (m−1)0  0  1  a211 a2(m−1)1  M0 =  . . ···. .  , √2m−1  . . . .   . . . .  0 0 a2 ··· (m−1)(m−1)   1 a2 a2 10 · · · − (m−1)0  0  1  a211 a2(m−1)1  M1 =  . . · ·. · − .  , √2m−1  . . . .   . . . .  (3.81) 0 0 a2 · · · − (m−1)(m−1) . .   1 a2 a2 − 10 · · · − (m−1)0  0  1  a211 a2(m−1)1  M2m−1−1 =  . − . · ·. · − .  √2m−1  . . . .   . . . .  0 0 a2 · · · − (m−1)(m−1) Pm Then we can get a state unitary equivalent with ψ2 = i=1 αi i1ki2 i3...in with prob- 1 | i | i ability 2m−1 and with the same probability we get other states which are diﬀerent with ψ2 just because some terms have an opposite sign than the corresponding terms in ψ2 . | i | i Chapter 3. LOCC transformation bounds between multipartite pure states54

And we do a unitary transformation on party 1 to transform all branches into ψ2 . For | i party 3,4, n, we can use similar method, so at last we can get a state unitary equivalent ···Pm with ψn = αi i1k2 ki ...ki . Then we ﬁnish the second step with probability 1. | i i=1 | i 3 n i 1 After that, party 1 can perform a similar POVM and with probability 2m−1 we get ψ which we want and with the same probability we get other states which are diﬀerent | i with ψ just because some terms have an opposite sign than the corresponding terms in | i ψ . | i However, if for at least m-1 terms, there is at least one party with a state, which we call ki , that is orthogonal to this party’s state in every other term, we can introduce a | j i minus sign for this term by a unitary transformation of party j which transforms ki to | j i ki and do nothing to all the other states orthogonal to ki . Thus we can introduce − | j i | j i a minus sign for these m-1 terms just by unitary transformation. For the only one term which does not have this property (if it exists), we can introduce a minus sign for every other term and then multiply -1 for the whole wave function. Then, we can get ψ with | i probability 1.

Remark 16. The condition we require in Lemma 10 is diﬀerent from each term is orthogonal to other ones. In fact, it is a stronger requirement than orthogonality. See the example below: for a state φ = √1 [ 000 + 1 ( 0 + 1 ) 0 + ( 0 + 1 ) 0 1 ], it is easy | i 3 | i | i | i | i | i | i | i | i | i to check each term is orthogonal to another in this state. But we do not know how to introduce a minus sign for any term because the condition in Lemma 1 is not satisﬁed.

There is an open question: Can this condition in Lemma 2 be "if and only if" in higher dimensions? To make it also "only if", there are two problems: 1. Is the form we write the state still unique in higher dimensions? We know, for a 2-term tripartite state

in which each party has rank 2, if we write it in the form ψ = γ(α 000 + β(λa 0 + | i | i | i p 2 p 2 p 2 1 λ 1 )A(λb 0 + 1 λ 1 )B(λc 0 + 1 λ 1 )C ), the result should be unique. − a | i | i − b | i | i − c | i It is also true for a 3-term tripartite state (the W-class state) [?], but the similar result for the higher dimension conditions have not been proved. 2. Can a state in which all terms are orthogonal to each other be transformed from GHZ like state with probability 1? We know, our protocol can only work for a stronger requirement. However, for states with orthogonal terms, the inner product is also zero, so how can we prove the probability can not be one in this case is another problem.

Corollary 3. All tripartite pure three qubit states can be transformed from 3-term GHZ state √1 ( 000 + 111 + 222 ) with probability 1. 3 | i | i | i Chapter 3. LOCC transformation bounds between multipartite pure states55

Proof: From the paper [1], we know any tripartite pure state can be written as

iφ Φ =λ0 000 + λ1e 100 + λ2 101 > | i | i | i | + λ3 110 + λ4 111 , (3.82) | i | i 2 X λi 0, 0 φ π, µi λ , µi = 1 ≥ ≤ ≤ ≡ i And shown in [1], if Charlie introduces a unitary transformation

−iφ −iφ ! 1 λ1e λ2e U = , (3.83) √µ1 + µ2 λ2 λ1 − we can get

1 −iφ −iφ Ψ = [e λ0λ1 000 + e λ0λ2 001 | i √µ1 + µ2 | i | i 2 2 (3.84) + (λ + λ ) 100 + (λ1λ3 + λ2λ4) 110 1 2 | i | i + (λ2λ3 λ1λ4) 111 ] − | i which is unitary equivalent with the state we want. If we combine the first and second term into a term and do the same for third and fifth term we can get Ψ >= 00 (a 0 + b 1 ) + d 100 + 11 (c 0 + e 1 ), easy to see, | | i | i | i | i | i | i | i if we consider it as a 3-term state, it satisfies the condition we required for Lemma 2, (Alice in first term and Bob in third term), so we can transform it from 3-term GHZ generalized state with probability 1.

Pm Pm 2 Theorem 11. For a general ψ = γ( αi k1 k2 ...kn ), where α = 1 and γ is | i i=1 | i i i i i=1 i the normalization factor. There exists an LOCC transformation protocol from the GHZ- m √1 P state generalized to n parties and m dimensions, GHZmn = m i=1 i 1 i 2 ... i n to | 1 i | i | i | i ψ such that the probability of success is at least max( 2 ), where | i γ||Ai||   1 ai ai ai 10 ··· (m−2)0 (m−1)0  0  1  ai11 ai(m−2)1 ai(m−1)1  Ai =  . . ···. . .  (3.85) √2m−1  . . . . .   . . . . .  0 0 0 ai ··· (m−1)(m−1) Proof: To generalize theorem 2 to general m-term n-party states, we can ﬁrstly get q + Pm A1 A1 A1 ψn = αi i1ki ki ...ki with certainty. And Alice perform a POVM , 1 2 , | i i=1 | 2 3 n i { ||A1|| − ||A1|| } Chapter 3. LOCC transformation bounds between multipartite pure states56

where   1 a1 a1 a1 10 ··· (m−2)0 (m−1)0  0  1  a111 a1(m−2)1 a1(m−1)1  A1 =  . . ···. . .  (3.86) √2m−1  . . . . .   . . . . .  0 0 0 a1 ··· (m−1)(m−1) 1 After calculation we can find the successful probability is 2 . Similarly, we can γ||A1|| choose other party to finish the final step and find the best one which give the maximum transformation probability.

3.6 Summary and Concluding Remarks

We derive upper bound and lower bound for the supremum transformation probability from GHZ state to GHZ-class state. In the derivation of the upper bounds, we consider the action of the LOCC protocol on a different input state, namely 1/√2[ 000 111 ], | i − | i and demand that the probability of an outcome remains bounded by 1. By considering the constraints of the interference term and 3-tangle, we find an upper bound for more general cases. For the lower bound, we construct a new transformation protocol: the four- step method to do the transformation. Before that, there was no nontrivial upper bound known for this transformation. Based on the previous results of weak measurement, we construct a "stop and reconstruct" method which may be very useful in the analyzation of the LOCC transformation protocols. The lower bound is generalized into higher dimension. During the discussion of lower bound, we find all tripartite pure 3-qubit states 1 can be transformed from GHZ23 = √ ( 000 + 111 + 222 ) with probability 1. This | i 3 | i | i | i is a new result. There are still open questions and possible future generalization of the result we have, which is mentioned during the above discussion. To summarize, firstly, there is still a gap between the upper bound and lower bound we get. How to further decrease the gap and finally find the optimum transformation protocol are still open questions. Secondly, we want to generalize the upper bound we get to higher dimension. To do this, we need firstly make sure the GHZ-class state generalized into the higher dimension is still unique so that we can still talk about the inner product. We also need to find the corresponding entanglement monotone for higher dimension case. Finally, the result of Corollary 1 is very surprising, one question is whether it can be generalized into higher dimension case. To do this, we need to analyze the Generalized Schmidt Decomposition in higher dimension [72] or other forms to express the state in higher dimensions [76]. Chapter 4

Optimal entanglement transformations among N-qubit W-type states

In this chapter, we investigate the physically allowed probabilities for transforming one N-partite W-type state to another by means of local operations assisted with classical communication. an upper bound for the maximum probability of transforming two such states was given in [52]. Here, we provide a simple sufficient and necessary condition for when this upper bound can be satisfied and, thus, when optimality of state transformation can be achieved. Our discussion involves obtaining lower bounds for the transformation of arbitrary W-type states and showing precisely when this bound saturates the bound of [52]. Finally, we consider the question of transforming symmetric W-type states and find that, in general, the optimal one-shot procedure for converting two symmetric states requires a non symmetric filter by all the parties. The content of this chapter is mainly based on [31]

4.1 Introduction

As deﬁned in chapter 2, W type states are of the form √x0 00 0 + √x1 10 0 + | ··· i | ··· i √xn 0 01 . W-type states represent a very important family of states since they ··· | ··· i possess a high degree of robustness with respect to loss of entanglement [15] and nonlocal correlations [69] in the presence of noise. Furthermore, many speciﬁc quantum cryptography and communication protocols have been designed which utilize W-type entanglement (see [78] and references therein). Experimental setups have been proposed for the production of multiqubit W states [4] with the generation of W4 already realized | i [79]. Finally, the geometric measure of entanglement for W-type states has been studied [73], and for the special case of three qubits, partial results concerning optimality of

57 Chapter 4. Optimal entanglement transformations among N-qubit W-type states58

LOCC conversion rates have been obtained [84]. In the general multiparty setting, Kintas, and Turgut recently made significant progress in understanding LOCC transformations of W-type states [52]. They prove an upper bound on the optimal probability of converting two such states, and the first part of our article derives a necessary condition for when this rate can be achieved. We then move on to construct a general procedure for converting two W-type states that provides a lower bound on the optimal conversion probability. The necessary condition for achieving Kinta, and Turguts’ bound we obtain in the first part turns out to be sufficient when using our constructed protocol, thus proving optimality. Since much of our analysis relies on results reported in [52], we try to stay as consistent as possible with the notation established there. In the final section, we turn to the problem of converting symmetric W-type states; that is, those states that remain invariant under a permutation of parties. It has been shown that two multiqubit symmetric states are related by a reversible SLOCC transformation if and only if the transformation can be accomplished by a permutation-invariant SLOCC filtering operation [59]. In other words, if ψ and N | i φ are N-qubit SLOCC equivalent symmetric states, Ai ψ = φ , then there exists | i ⊗i=1 | i | i an operator M such that M ⊗N ψ = φ . However, one question still open is whether | i | i the same probability of transformation can be achieved in the symmetric case. If a filter N Ai succeeds in transformation with probability p, does there necessarily exist a sym- ⊗i=1 metric filter M ⊗N that transforms with the same probability? We show that in general the answer is no and often the transformation can be achieved with a greater probability when only a single party acts nontrivially. At the same time, we further observe the single party strategy to not be optimal in general. These results nicely demonstrate the complexity in analyzing issues of LOCC optimality as no simple general result appears to exist, even in the symmetric multiqubit case.

⊗N ⊗i−1 ⊗n−i For simplicity, we introduce the notation −→0 = 0 and −→i = 0 1 0 so | i | i N | i | i | i | i that the N-party W state can be expressed as = √1 P . A state is deﬁned WN N i=1 −→i ψ | i | i N | i as a W-type state if there exist invertible operators Ai such that Ai WN = ψ . ⊗i=1 | i | i Equivalently, a state is in the W type if it is SLOCC equivalent to the state WN ! | i a b [34]. Each Ai can be written in the form UiAi , with Ai = and a,c being 0 c PN real. Then every W-type state ψ is of the form √x0 −→i up to the application of | i i=0 | i N PN local unitaries; i.e. ψ = Ui √x0 −→i for unitaries Ui. Furthermore, for three | i ⊗i=1 i=0 | i or more parties the coeﬃcients xi are unique to each W-type state [52]. To easily see N PN N PN p 0 PN this, observe that Ui √xi −→i = Vi x −→i implies √xi −→i = ⊗i=1 i=0 | i ⊗i=1 i=0 i | i i=0 | i N PN p 0 x −→i for some unitaries Wi . But this means that each party’s reduced state ⊗i=1 i=0 i | i Chapter 4. Optimal entanglement transformations among N-qubit W-type states59

q is the totally mixed state, which is possible only for the bipartite state 1 ( 01 + 10 ). 2 | i | i Thus, we can unambiguously represent every multipartite W-type state by −→x ,where PN | i x = (x1, ..., xN ) is its unique coeﬃcient vector with x0 = 1 xi. |−→i − i=1 The main result presented in [52] is that whenever party k performs a measurement on

state x , for each outcome λ occurring with probability pλ, the components transform |−→i as

xk xj sλxj for j = k, 0, xk (4.1) → 6 → tλ

P P pλ such that pλsλ = 1 and 1. From these relations, it follows that under any λ λ tλ ≤ LOCC transformation with outcomes indexed by λ, the vector components are nonincreasing on average:

X xi pλxi,λ (4.2) ≥ Consequently, for the transformation x y , the maximum probability of success |−→i → |−→i pmax is bounded by

pmax min ri , (4.3) ≤ i { } xi where ri = . In the remainder of this Chapter we assume that r1 r2 , rN , as yi ≤ ≤ · · · any other ordering can be accounted for by relabeling. Thus, pmax r1 and the next two ≤ sections prove the following result.

Theorem 12. For W-type states x and y with optimal probability r1 if and only if |−→i |−→i r2 r0. ≥

4.2 Upper bounds

We begin with a more detailed description of a general W-type LOCC transformation. We can model every m-round LOCC protocol transforming x y by a tree (Fig. |−→i → |−→i 4.1) split into m segments, with e(ij) denoting the jth edge in the ith segment. The tree begins with a single node representing the initial state x and ends after the mth |−→i segment, with each final node representing a different outcome state. We say a branch is any unidirectional connected path that traverses the entire length of the tree. It is a success branch if its final state is y ; otherwise, the branch is a failure branch. An |−→i edge is called an intermediate edge if it contains at least one successful branch traveling through it; otherwise it is called a failure edge. Let I(i−1,j) denote the set of indices such that k I(i−1,j) iff e(i,k) is an intermediate edge connected to edge e(i−1,j). Likewise, let ∈ Chapter 4. Optimal entanglement transformations among N-qubit W-type states60

F (i−1,j) denote the set of indices such that k F (i−1,j) iﬀ e(i,k) is a failure edge connected ∈ to edge e(i−1,j). The set I(0,0) (resp. F (0,0)) will contain the indices corresponding to the intermediate (resp. failure) edges connected to the starting node of the tree. Finally, (i,j) (i,j) (i,j) denote the state obtained following edge e by −→x , with components xl , and | (i,j) i let pi,j be the probability of moving along edge e . From the deﬁnitions, we have P P k∈I(i−1,j) pi,k + k∈F (i−1,j) pi,k = 1 for every i,j.

Figure 4.1: An LOCC transformation tree from x to y . For example, the branch traversing edges e(1,1) to e(n,1) is a success branch,| whilei | thei branch from e(1,1) to e(n,2) is a failure branch. Edge e(n−1,1) is an intermediate edge. c 2010 American Physical Society.

With this formalism, we can systematically calculate the total success probability of obtaining y from x . A branch is successful if and only if it travels only along inter- |−→i |−→i mediate edges. So the total probability is given by summing over all possible intermediate edge paths. This value is given by

X X P ( −→x −→y ) = p1,k1 pm,km | i → | i (0,0) ··· (m−1,k ) k1∈I km∈I m−1 X X Y (4.4) = pi,ki (0,0) ··· (m−1,k ) k1∈I km∈I m−1

Starting from state x and repeatedly applying Eq. 4.2, we have |−→i Chapter 4. Optimal entanglement transformations among N-qubit W-type states61

X (1,k1) X (1,k1) xl p1,k1 xl + p1,k1 xl ≥ (0,0) (0,0) k1∈I k1∈F

X X (2,k2) X (2,k2) p1,k1 ( p2,k2 xl + p2,k2 xl ) ≥ (0,0) (1,k ) (1,k ) k1∈I k2∈I 1 k2∈F 1

X (1,k1) + p1,k1 xl (0,0) ··· k1∈F m X X Y (m,km) pi,ki xl ≥ (0,0) ··· (m1,k ) (4.5) k1∈I km∈I m1 i=1 m X X X Y (m,km) + pi,k x ··· i l (0,0) (m2,k ) (m−1),k i=1 k1∈I km−1∈I m2 km∈F m−1 X X X + (0,0) ··· (m3,k ) (m−2),k k1∈I km−1∈I m3 km−1∈F m−2 m−1 Y (m1,km1) X (1,k1) pi,ki xl + p1,k1 xl . × ··· (0,0) i=1 k1∈F

(m−1,km−1) This equation is quite informative since we know that, for km I , we have (m,km) ∈ xl = yl . Then, by dividing both sides of Eq. 4.5 by yl and using Eq. 4.4, we have

rl P ( x y ) + F ailure Edges, (4.6) ≥ |−→i → |−→i where "failure edges" refers to the non-negative quantity of all but the ﬁrst term in the ﬁnal inequality of Eq. 4.5. Physically, it is the average of the lth component of all failure states produced after some measurement on a success branch.

For P ( −→x −→y ) = r1, this requires strict equalities in Eq. 4.5, and furthermore, | i → | i (i,j) the failure edge terms must vanish. The latter condition means that x1 = 0 for every failure edge e(i,j) connected to a success branch. We combine these results in the following lemma.

Lemma 7. If −→x −→y by LOCC with probability r1, then (i,j) | i → | i (i,j) (i,j) (i) if e is an edge connected to a success branch, x1 > 0 implies e is an intermediate edge, and (i−1,j) P (i,k) (ii) x1 = k∈I(i1,j) pi,kx1 for every i,j. For any i,j, we must have P ( −→xi,j x1 xl | i → i,j i,k1 y ) = 1 . |−→i y ≤ yl Proof. Note that

X P ( −→x −→y ) = P ( −→x −→xi,j )P ( −→xi,j −→y ) (4.7) | i → | i j | i → | i | i → | i Chapter 4. Optimal entanglement transformations among N-qubit W-type states62

For i = 1, we have

X P ( x y ) = P ( x x1,k )P ( x1,k y ) |−→i → |−→i |−→i → |−−→1 i |−−→1 i → |−→i k1∈I0,0 1 (4.8) X x1,k x1 ( ) 1 = P −→x −−→x1,k1 1 ≤ | i → | i y y1 k1∈I0,0

where the ﬁnal equality follows from the previous theorem. So for P ( x y ) = x1 , −→ −→ yl x1 | i → | i 1,k1 we must have P ( −−→x1,k1 −→y ) = y1 . Now we proceed by induction by ﬁrst assuming | i →1 | i xm,j that P ( xm,j y ) = 1 . We have |−−→i → |−→i y

P ( xm,j y ) = P ( xm,j xm+1,k )P ( xm+1,k y ) |−−→i → |−→i |−−→i → |−−−−→i |−−−−→i → |−→i 1 1 X xm+1,k xm,j (4.9) P ( xm,j xm+1,k ) = , ≤ |−−→i → |−−−−→i y1 y1 k∈Im,j

x1 xl m+1,k m+1,k1 which by our inductive assumption implies P ( xm+1,k y ) = 1 l . |−−−−→i → |−→i y ≤ y Combining relations Eq. 4.1 with this lemma, the following becomes apparent.

Corollary 4. If P ( x y ) by LOCC with probability r1, there always exists at least |−→i → |−→i one success branch such that the measurements along each edge satisfy sλtλ 1. ≥

Proof. For any success branch, assume that in the first-round measurement, sλtλ < P P pλ 1 for all λ. Then 1 = pλsλ < , which is impossible. Hence, there must λ λ tλ be some outcome with sλtλ 1, and since the first component of the initial state is ≥ nonzero (it must be or else party 1 would be unentangled with all other parties), the first component of the resultant state will likewise be nonzero. Consequently, by Lemma 7, the edge corresponding to outcome λ is an intermediate edge. Consider the round 2 measurement performed along this edge and repeat the previous argument. This can be done subsequently for all m rounds, thus identifying a success branch in which all edges

correspond to measurement outcomes satisfying sλtλ 1. ≥ ∗ Now for any transformation occurring with probability r1, let p denote one of the success branches described by this corollary and let its edges be e(i,vi). Along p∗ we can divide the protocol into three parts, encoded by index sets A, B, and C, where i A if ∈ party 1 performs ameasurement along e(i,vi), i B if party 2 performs a measurement ∈ (i,vi) along ei,v , and i C if neither party 1 nor party 2 performs a measurement along e . i ∈ From Eq. 4.1, we have the following transformations during each edge in p∗: Chapter 4. Optimal entanglement transformations among N-qubit W-type states63

(i−1,vi1) (i,vi) x1 (i,vi) (i,vi−1) (i,vi) (i−1,vi−1) i A : x1 = , x2 = six2 , x0 = six0 , ∈ ti (i−1,vi1) (i,vi) (i,vi−1) (i,vi) x2 (i,vi) (i−1,vi−1) (4.10) i B : x1 = six1 , x2 = , x0 six0 , ∈ ti ≥ (i,vi) (i,vi−1) (i,vi) (i,vi−1) (i,vi) (i−1,vi−1) i C : x = six , x = six , x six . ∈ 1 1 2 2 0 ≥ 0 This implies the following relationship between the initial and the ﬁnal components:

Y 1 Y 1 Y y1 = sjskx1, y2 = si skx2, y0 sisjskx0. (4.11) ti tj ≥ i∈A,j∈B,k∈C i∈A,j∈B,k∈C i∈A,j∈B,k∈C

Substituting y1 into y0 yields

Y y1 y0 siti x0. (4.12) ≥ x1 i∈A

while dividing y1 by y2 gives

y1 Y Y x1 x1 siti = siti (4.13) y2 x2 ≥ x2 i∈A i∈B ∗ where the last inequality follows from the fact that siti 1 along every edge in p . Then ≥ substituting Eq. 4.12 into 4.13 gives the bound r2 r0. ≥

4.3 Lower bounds

In this section, we construct a speciﬁc protocol to obtain a lower bound for the maximum probability of transforming two W-type states. In the protocol, there will be a single success branch, with edges e(i) and states x (i) , whose kth component is x(i) . Only two |−→ i k types of measurements will be performed for each acting party k: Type 1 (T1), which has 1 an outcome λ such that tλ = pλ and x0,λ = sλx0; and Type 2 (T2), in which sλ = for pλ some outcome. From [52], T1 and T2 measurements can always be performed on state

pλ x for any choice of pλ and sλ as long pλsλ 1 and 1. Consequently, whenever |−→i ≤ tλ ≤ xk > yk, a T2 measurement can be performed by party k with sλ = pλ = 1 and tλ = rk. In this case, the coordinates of parties 1 through n do not change on average, and so, by normalization, neither does the 0th coordinate. By explicitly solving for the scale

factor tλ on party k, we have that in a T2 measurement by party k on state x (i−1) , the |−→ i Chapter 4. Optimal entanglement transformations among N-qubit W-type states64 coordinates change as

(i−1) (i−1) (i) xj (i) 1 xk xj = for j = k, xk = 1 − . (4.14) pi 6 − pi For the transformation of −→x to −→y it is suﬃcient to reach some round i in which (i) | (ii) | i rk 1 for all k 1. When rk > 1, as previously noted, the kth party can deter- ≥ ≥ (i+1) ministically transform the state such that rk = 1 and all nonzero components are unchanged. The basic idea of the protocol described here is to systematically raise each (1) kth component closer to yk one at a time in a "piggyback" fashion, where r1 is ﬁrst (2) increased and made equal to r2 , and both of them are then increased and made equal (3) to r , , etc. Eventually, each kth component will be raised to yk or possibly greater. 3 ··· The next simple lemma provides the tools for a precise implementation of this idea.

(i−1) (i−1) (i−1) (i) Lemma 8. (i) If rk+1 r0 rk , there exists a −→x and a T1 measurement ≥ ≥ | i (i−1) (i−1) (i) (i) (i) (i) rk by party k transforming −→x −→x such that rk+1 rk = r0 and sipi = (i−1) . | i → | i ≥ r0 (ii) If r(i−1) r(i−1) r(i−1) , there exists a x i and a T2 measurement by party k k+1 ≥ k ≥ k1 |−→ i transforming x (i−1) x (i) such that r(i) r(i) = r(i) . |−→ i → |−→ i k+1 ≥ k k−1 (i−1) (i) rk (i) (i−1) (i−1) Proof. (i) In any T1 measurement we have r = and r = sir sir = k pi 0 0 k+1 (i−1) ≤ (i) rk rk+1. Setting these equal gives sipi = (i−1) 1, from which any choice of si and pi r0 ≤ r(i−1) satisfying this provides a realizable protocol. (ii) For a T2 measurement, r(i) = k−1 k−1 pi (i−1) ≤ rk+1 (i) (i) 1 (i−1) = r and ykr = 1 [1 ykr ]. Equality is achieved with the choice pi k+1 k pi k (i−1) (i−1) − − pi = 1 yk(r r ). − k − k−1

We nowstate the result of the protocol as a theorem and then give its proof by constructing the transformation procedure.

Theorem 13. Let x and y be two W-type states with r1 r2 rN , where |−→i |−→i ≤ ≤ · · · ≤ xk rk = and yk = 0 for k = 0, ,N. If r1 r0, then x can be converted to y with yk 6 ··· ≥ |−→i |−→i probability r1. Otherwise, let h be the largest integer such that r0 > rh. Then x can be |−→i converted to y with probability |−→i

rh−1 r1 rh( ) . (4.15) r0 ··· r0 Proof:. The protocol can be divided into two parts: in the ﬁrst, only T1 measurements are performed, and in the second, only T2 measurements. If r1 > r0, proceed to the Chapter 4. Optimal entanglement transformations among N-qubit W-type states65

(1) second part and let h = 1, p1 = 1, and ri = ri for all parties i. In round 1, party h (1) (1) rh performs a T1 such that r = r with s1p1 = . In round 2, party h - 1 performs an h 0 r0 (1) (2) (2) (2) rh−1 rh−1 T1 such that r = r = r with s2p2 = (1) = . This process is continued for h h−1 0 h r0 r0 (h) (h−1) rh (h) (h) rounds. The end result is r = shr = shsh−1sh−2 s2 , and rn r h h p1 h+1 (h) (h) (h) ··· ≥ · · · ≥ ≥ r = r = r , where the last inequality is always tight when r0 r1. The next h h−1 · · · ≥ 0 ≥ part of the protocol now begins, with only T2 measurements performed. In round h + 1, (h+1) (h+1) (h+1) party h + 1 performs a T2 with probability ph+1 such that rh+1 = rh = r0 (h+1) 1 (h) · · · ≥ , with r = r .Next, party h + 2 performs a T2 with probability ph+2 such that h ph+1 h r(h+2) = r(h+2) = r(h+2) = 1 r(h). This process is continued until right before some h+2 h+1 h ph+2ph+1 h (l−1) (l−1) round l in which rl−1 1 and rl 1. Note that such a round will exist because in ≤ (j) (j) ≥ (j) every round j satisfying r0 rN , there is always some component i = 0 with ri 1. ≤ 6 (≥l−1) Returning to the protocol, in round l, party l applies a T2 measurement with pl = rl−1 . As a result, r(l) 1 for all i 1 since r(l) = 1 for 1 i < l, r(l) 1 by Eq 4.14, and i ≥ ≥ i ≤ ≥ r(l) r(l) for j l. The total probability is j ≥ l ≥

p1p2 pl−1 (h) (h) p1p2 pl−1pl = ··· rh = p1 phrh ··· ph+1ph+2 pl−1 ··· ··· rh = p1 phshsh−1sh−2 (4.16) ··· ··· p1 rh−1 r1 = rh( ) ( ). r0 ··· r0

For this protocol to be suitable for any W-type transformation, we must consider the

cases when yk = 0. If yk = 0 for k 1, then the kth party simply disentangles itself ≥ with probability 1 from the rest of the system and the preceding protocol is performed 0 0 0 on the N- 1 party state −→x , where x = x0 + xk and x = xj, with j 1. If yk = 0 | i 0 j ≥ for k = 0, then party i, speciﬁed by xi = maxj≥1 xj , performs the ﬁlter M = with { } 2xi success probability λ(1 x0), where λ = 2 . This changes the coordinates − x0+2xi+√x0+4xix0 xi as xi , and hence the constructed protocol can be implemented with an overall → 1−x0 success probability of λr1.

Our protocol is most general in that it and the derived success probability apply to all W-type transformations, even those whose target state is not N-partite entangled. As an example, we compute the probability for an arbitrary W-state distillation. Chapter 4. Optimal entanglement transformations among N-qubit W-type states66

Corollary 5. Let x be an N-party W-type state. Then |−→i

2xN x1N Pmax( x −−→WN ) (4.17) −→ p 2 | i → | i ≥ x0 + 2xN + x0 + 4xN x0

Observe that in Theorem 13, the lower bound becomes r1 whenever r2 r0. Com- ≥ bined with the results in the previous section, we see that Pmax( x y ) = r1 if and |−→i → |−→i only if r2 r0. ≥

4.4 General Features of symmetric transformations

The symmetric W-type states constitute a one-parameter family of states,which makes them easier to analyze. Any such state can be represented as s = √1 s −→0 + p s PN | i − | i −→i . Note that the state WN corresponds to s = 1. The optimal proba- N i=1 | i | i bility of converting s t , at least by a one-shot measurement, can be numerically | i → | i computed by brute force using Lagrange multipliers. However, in this section we are less concerned with analytic expressions for optimal conversion probabilities and more concerned with general properties of transforming symmetric states.

Figure 4.2: The diﬀerence in maximum transformation probabilities when only one party measures [pmax(s)] versus an identical ﬁlter by all parties [qmax(s)]. c 2010 American Physical Society.

In particular, it was recently shown that one multiqubit symmetric state can be reversibly converted into another if and only if the transformation is feasible by a protocol in which each party performs the same one-shot measurement. However, one question that was not investigated is whether the optimal one-shot success probability can always Chapter 4. Optimal entanglement transformations among N-qubit W-type states67 be obtained by a symmetric ﬁlter. Here, we answer this question by examining the transformation of an arbitrary tripartite symmetric W-type state to the target state

W3 . An optimal symmetric one-shot measurement with success probability q can be | i + expressed as (A A A) s = √q W3 , with the operator A satisfying A A I and ⊗ ⊗ | i | i ≤ det(I A+A) = 0. For comparison, we consider the same transformation when only − a single party acts nontrivially: (A I I) s = √p W3 . Note that any conversion ⊗ ⊗ | i | i among symmetric W-type states can always be achieved with a nonzero probability by the action of just a single party up to a local basis change. Studying the difference in optimal conversion rates between a multiparty symmetric filter and a single-party filter is of interest because it sheds light on two competing intuitions. On the one hand, when more parties act, the "work of conversion" can be distributed, and in light of the overall symmetry, it seems reasonable to expect that it is best for this work to be shared equally in the form of identical filters. On the other hand, if only one party performs a measurement, there are fewer possibilities for failure. Here we show that neither of these intuitions is true in general. ! a b In both cases, without loss of generality we can take A to have the form A = . 0 c Then the comparative optimization problems become

max p subject to : rs (4.18) a√1 s + b = 0, c√s = √p, a = c, − 3 and

max q subject to : rs (4.19) a3√1 s + 3ba2 = 0, a2c√s = √q, − 3 with both satisfying the common constraint that b2 = (1 a2)(1 c2). The respective − − solutions as functions of s are

1 p pmax(s) = (3 s 3(1 s)(3 + s)), (4.20) 2 − − −

[3 + 9s β(s)]2[ 3 + 3s + β(s)] qmax(s) = − − , (4.21) 48(1 + 2s)[1 s + β(s)] − p where β(s) = 3(1 s)(3 + 5s). − The difference pmax(s) qmax(s) is plotted in Fig. 4.2 as s varies between 0 and 1. − From it, we see that, in general, the optimal strategy for transforming two symmetric Chapter 4. Optimal entanglement transformations among N-qubit W-type states68 states involves neither a symmetric measurement nor the action of just a single party. For the particular class of transformations we consider here, the two strategies have 3 √ the same maximum efficiency only when s = 61 (3 + 8 3). However, the difference in optimal probabilities is never greater than 1.4%. It should also be noted that neither of these schemes may be the overall optimal protocol.While further numerical analysis could provide an answer to this question,we do not pursue it here, as we consider the reported result of greater interest.

4.5 Conclusion

In this chapter,we have investigated the LOCC convertibility of N-party W-type states. For a large family of transformations, we have proven their optimal conversion rate to

xi achieve the upper bound of mini . The question of transforming symmetric states was { yi } considered in the context of W-type states. Despite the necessity of SLOCC equivalent states to be related by a symmetric measurement, we have found that this symmetry cannot be extended to the measurement achieving optimality. A future direction of research might involve considering theW-type transformations when r0 > r2. However, preliminary numerical work on this problem has revealed the computation to be quite unyielding. It would also be interesting to know when the lower bound of Theorem 13 is optimal. Also, by analyzing random distillation of W type states into genereal bipartite entangled states, a better upper bound can be derived, which will be shown in next chapter. Chapter 5

Random distillation for W type states

In this chapter, we investigate the task of converting a single N-qubit W-type state (of

the form √x0 00 0 + √x1 10 0 + + √xN 0 01 ) into maximum entangle- | ··· i | ··· i ··· | ··· i ment shared between two random parties, which is called random distillation protocol. Previous studies in random distillation have not considered how the particular choice of target pairs affects the transformation, and here we develop a strategy for distilling into general configurations of target pairs. We completely solve the problem of determining the optimal distillation probability for all three-qubit configurations and

most four-qubit configurations when x0 = 0. Two special scenarios for general n-qubit configurations are also solved. As an additional application of our results, we present new upper bounds for converting a generic W-type state into the standard W state 1 WN = √ ( 10 0 + + 0 01 ). Finally, we find the corresponding transforma- | i N | ··· i ··· | ··· i tion probability by SEP, which is significantly higher than the case of LOCC. This result shows that the entanglement monotone we find can actually be increased by SEP significantly. This is the first analytical entanglement monotone that can be increased by SEP. The content of this chapter is mainly based on [32][23][19].

5.1 Introduction

In quantum-information processing, the two-qubit Einstein- Podolsky-Rosen (EPR) state q ψ = 1 ( 00 + 11 ) provides a key resource for performing nonclassical tasks such as | i 2 | i | i teleportation [8] and super-dense coding [11]. Thus, for a multipartite state φ , it is | i1,··· ,N important to know the optimal ways in which EPR entanglement can be obtained between two parties without having to introduce any more entanglement into the system. This latter constraint is known as the local operations and classical communication (LOCC) constraint because it requires each party to perform only local quantum operations (LO)

69 Chapter 5. Random distillation for W type states 70

while coordinating their operations through classical communication (CC). In general, the optimal conversion of φ into bipartite entanglement depends on which two final | i1,··· ,N parties are left sharing the entanglement. One scenario to consider is when two specific parties are designated as the target pair, and the transformation is considered a success if and only if these two parties end up sharing the state ψ . A transformation of this | i sort is known as a specified-pair distillation. In this setting, an important problem is to

determine the greatest success probability pij for which the conversion

φ ψ(ij) (5.1) | i1,··· ,N → | i is possible by LOCC. Here, ψ(ij) denotes an EPR pair between parties i and j , and | i it is assumed that all other parties are in some product state. While no full solution to this problem is known, some partial results exist [44, 25]. A more general question can be posed by allowing the two EPR-entangled parties to vary across the different outcomes in the transformation. Any transformation of this form is known as a random- pair distillation (or just simply random distillation) because the final two entangled parties are a priori unspecified. Additional constraints to the problem can be added by demanding that the possible target pairs be limited only to some particular subset of all possible pairs. For example, in the random distillation of Fig 5.1, the transformation is considered a success only if AB or AC obtain an EPR pair, and not if BC become EPR entangled. For an N-party system, a random distillation can be written as

(ij) φ pij, ψ (i,j)∈E (5.2) | i1,··· ,N → { | i} (ij) where pij is the probability of obtaining ψ and E [N] [N] is some designated set | i ∈ × of target bipartite pairs. The transformation is considered a success if an EPR state is obtained by any pair in E. A convenient way to represent random distillations is through a conﬁguration graph

G = (V,E). Each party k is identiﬁed with a node vk V , and an edge ejk E connects ∈ ∈ vj and vk if parties j and k form a desired target pair in the distillation (see Fig. 5.1). It should be emphasized that we are strictly dealing with a single copy of φ , | i1,··· ,N and each edge corresponds to one possible outcome. Variations to this question in the asymptotic regime have been studied elsewhere [70, 82]. Given some graph G and initial state φ , the greatest success probability is given by | i1,··· ,N

X P (φ, G) := sup pij, (5.3) (i,j)∈E Chapter 5. Random distillation for W type states 71

Figure 5.1: A specified-pair versus random-pair distillation. For random distillations, it is convenient to combine all the desired outcomes into one configuration graph G = (V,E) whose edge set encodes the target pairs. Here, the target pairs are AB and AC. The ” ” indicates equivalent representations. c 2011 American Physical Society. ≡ where the supremum is taken across all LOCC protocols. The subject of single-copy random distillation was first introduced and subsequently studied by Fortescue and Lo [39, 40]. One prominent finding of their work is that random distillations are, in general, strictly more powerful than specified-pair distillations. q 1 Perhaps themost dramatic example of this is the N-qubit state WN = ( 10 0 + | i N | ··· i 01 0 + + 00 01 ) and its random distillation into EPR pairs shared between | ··· i ··· | ··· i any two parties (see Fig. 5.2). In terms of the terminology introduced above, the initial state is WN , and the configuration graph is complete (each node connected to every | i other) such that the conversion is a success if any two parties become EPR entangled. Fortescue and Lo were able to show that this transformation can be completed with probability arbitrarily close to 1 [39]. On the other hand, for any two parties, the optimal specified-pair distillation probability is 2/N. In this chapter, we turn to one largely unexplored question in Fortescue and Lo’s work which is the random distillation to general configuration graphs, and not just complete graphs. Specifically, we consider how the target configuration graph affects the random distillation in terms of overall success probability as well as the actual LOCC protocol the parties implement during the transformation. For example, one particular problem we are able to solve is the four-qubit random distillation depicted in Fig 5.3 which was left as an open problem in Refs [38]. Our focus is on the single-copy random distillation of N-party W-type states, which is the collection of all states reversibly obtainable from WN with a nonzero probability by | i Chapter 5. Random distillation for W type states 72

Figure 5.2: An N = 8 example of the "complete-type" distillations considered by Fortes- cue and Lo in [39]. Such a transformation is a success if any two parties become EPR entangled, and this can be achieved with a probability arbitrarily close to 1. Previous research has not considered more general types of conﬁguration graphs than this. c 2011 American Physical Society.

Figure 5.3: In Sec 5.4 we show that the optimal LOCC probability of achieving this transformation is 2/3, thus resolving an open problem in Refs. [38]. The initial state is W4 = 1/2( 1000 + 0100 + 0010 + 0001 ). c 2011 American Physical Society. | i | i | i | i | i Chapter 5. Random distillation for W type states 73

LOCC. The choice to limit investigation to this class of states is motivated by multiple factors. First, from an experimental perspective, W-type entanglement seems relatively easier to generate than other forms of multipartite entanglement, with the state W4 | i already being realized in the laboratory [79]. In N >4 qubit systems, setups have also been proposed for the production of W-type states [4]. And for the particular task of random distillation, Fortescue has devised an experimental implementation of W- type random distillation using currently available technology, e.g., ion trap quantum computers [38]. Second, as we see in the next section, W-type states have a very simple structure which allows us to carefully analyze their behavior under LOCC evolution. Finally, a large amount of previous research conducted by Fortescue and Lo on random distillations involved W-type states. Thus, there is an established point of comparison for new results on the subject. The following is a summary of our results and an outline the structure of this chapter. (i) In Sec 5.2, we begin by reviewing the results of the Fortescue-Lo Protocol and a described generalization, as well as some related work by Kintas and Turgut on entanglement transformations within the W type [52]. (ii) In Sec 5.3 we construct the "Least Party Out" Protocol which distills an arbitrary N-qubit W-type state given some target configuration G. The protocol is similar in nature to the Fortescue-Lo Protocol but we show it to be strictly stronger. (iii) In Sec 5.4, we apply our protocol to multipartite systems to obtain the main results of this chapter. Every possible three- and four-qubit configuration graph is considered, and we introduce new four-qubit entanglement monotones to show that our protocol is optimal in most cases when x0 = 0. Also, regarding n-qubit system where n>4, two special scenarios, the complete distillation and the combing-type distillation, are investigated. (iv) In Sec 5.5, we compare the transformation probability between SEP and LOCC protocol. We firstly derive the upper bound for random distillation implemented by SEP, and then we show that for some special states, the random distillation probability under SEP can achieve 1 while the corresponding result under LOCC is significantly lower than 1. Given the fact that the upperbounds we have are actually entanglement monotones, the first analytic entanglement monotones that can be increased by SEP are discovered. (iv) In Sec 5.6, we further apply our results to study the transformation φ | i1,··· ,N → WN , where φ is a generic W type state. New upper bounds on the optimal conversion | i | i probability are obtained. (v) In the Conclusion, we summarize our results and discuss the possible further research directions on this topic. Chapter 5. Random distillation for W type states 74

5.2 Previous results and notation

5.2.1 The generalized Fortescue-Lo protocol

In [39], Fortescue and Lo developed a protocol which randomly distills the state WN = q | i 1 ( 10 0 + 010 0 + + 0 01 ) according to a complete configuration graph N | ··· i | ··· i ··· | ··· i with success probability arbitrarily close to 1. We briefly review the case when N = 3. For some > 0, the parties locally perform the measurement given by M1 = diag[√1 , 1] − and M2 = diag[√, 0]. If all parties obtain outcome "1", the final state is the original

state W3 . The parties then repeat the same measurement again. On the other hand | i if only two parties obtain outcome 1, then this pair is left EPR entangled. But, if one or fewer parties obtain outcome 1, all entanglement is destroyed and transformation is a failure. With the possible recursive step, this protocol can continue for an indefinite number of measurement rounds. In the end, the total probability of obtaining some EPR pair is 1- O(). For N >3, the protocol generalizes and likewise the probability of success is 1- O(). Here, the probabilities are distributed equally among all possible pairs; that is, with probability 1 O(), any two parties i and j obtain an EPR pair. C(N,2) − In Ref. [38], Fortescue briefly considered the problem of applying their protocol to more general configuration graphs, but only a limited discussion is given. Nevertheless, for a general outcome configuration graph G = (V,E), we can here describe an obvious

way to apply the Fortescue-Lo protocol. Starting with the state WN , it is converted | i with equal probability into the C(N,N 1) diﬀerent WN−1 states. Here the diﬀerence − | i between these states lies in which of the N parties are entangled. If all the entangled

parties in a particular WN−1 state are connected according to the graph G, then the | i state is broken into EPR pairs with probability arbitrarily close to 1. Otherwise, it is

converted into the C(N 1,N 2) diﬀerent WN−2 states. This process continues until − − | i W3 states are obtained. Either all these parties sharing the W3 state are connected in | i | i G or at most two are. In the former case, EPR pairs are obtained with probability 1, ≈ whereas in the former the distillation can be completed with probability 2/3.

We let PFL(WN ,G) denote the distillation success probability of this generalized

Fortescue-Lo protocol for some conﬁguration G. Obviously P (WN ,G) PFL(WN ,G). ≥ The Least Party Out protocol described in the next section is able to obtain a greater

success probability than PFL(WN ,G) in general and thus tighten the lower bound on

P (WN ,G). Chapter 5. Random distillation for W type states 75

5.2.2 Additional notation and the Kintas-Turgut monotones

In an N-partite system, if a "standard" W state WM is shared among parties S [N] := | i ∈ 1, 2, ,N with S = M, we often explicate this by writing W (S) . Equivalently we { ··· } | | | |S| i can write this state as W (T ) , where T = [N] S. Also, we deﬁne | |S| i r1 W (ij) := ( 01 + 10 ), (5.4) | 2 i 2 | i | i which is local unitarily (LU) equivalent to φ(ij) . | i We often represent a generic W-type state √x0 0 0 + √x1 010 0 + + | ··· i | ··· i ··· √xn 0 01 by an N-component vector, | ··· i

−→x = (x1, x2, ..., xN ) (5.5) PN 0 and x0 = 1 i=1 xi . More importantly, even after a basis change 0 0 and 0 − | i → | i 1 1 the component values √xi always remain unchanged for N 3 [52]. When N | i → | i ≥ = 2, uniqueness can be ensured by demanding that x0 = 0 and x1 x2. Therefore, for any ≥ number of parties, the vector −→x uniquely speciﬁes the state up to an LU transformation. For the state −→x , we denote

xn1 = max xk. (5.6) 1≤k≤N By disregarding LU transformations and decomposing a general measurement into a sequence of binary outcome measurements [3], we can assume that a local measurement by party k consists of two upper triangular matrices M (k),M (k) whose entries are { 1 2 } ! ! (k) √a1 b1 (k) √a2 b2 M1 = ,M2 = , (5.7) 0 √c1 0 √c2

with a1 +a2 = 1 and c1 +c2 1, where equality is achieved by the latter if and only if (k) (k) ≤ M1 and M2 are both diagonal. It is easy to see that this measurement by party k on

state √x0 0 0 + √x1 010 0 + + √xn 0 01 will transform the components | ··· i | ··· i ··· | ··· i as

cλ aλ xk xk, xj xλ,j, 1 j = k N, (5.8) → pλ → pλ ≤ 6 ≤

with pλ being the probability that outcome λ occurs. From this it is easy to see the following, X X x0 pλxλ,0, xi pλxλ,i, (5.9) ≤ ≥ λ λ Chapter 5. Random distillation for W type states 76

for all 1 i N. We refer to these as the K-T monotones after Kintas and Turgut who ≤ ≤ ﬁrst proved the inequalities [52].

5.3 The least party out protocol

Here we describe our W-type random distillation protocol for a given configuration graph G. It’s called the "Least Party Out" (LPO) protocol because it involves systematically removing parties from the N-party entanglement with a probability that decreases according to the number of edges connected to the party’s node in G. For some group of parties S, we let G S denote the subgraph of G obtained by removing the nodes \ corresponding to the parties in S. Our protocol can be divided into three phases. Phase I takes a generic W-type state 0 0 −→x and converts it into a state −→x such that x0 = 0. Phase II converts an x0 = 0 W-type state into standard W states W (S) for 2 S N using an "equal or vanish" (e/v) | |S| i ≤ | | ≤ measuring scheme. Phase III then converts the standard W states into EPR pairs given by the configuration graph G. Phase III is largely inspired by the Fortescue-Lo protocol in that it involves an indefinite round measurement procedure: each party performs a measurement which, with some probability, leaves the state invariant and thus subject to a repeated round of identical measurement, which again leaves the state invariant with some probability, etc.

5.3.1 Phase I: Remove x0 component

Input (−→x , G) where −→x is an N-partite W-type state and G is some conﬁguration graph with N nodes. If x0 = 0, proceed to Phase II. Otherwise, choose some party n1 with the

largest component value to perform the measurement Eq 5.7 with a1 = c1 = λ, b1 =

q x0 λ , and c2 = 0. The values for a2, b2, and λ are ﬁxed by the measurement being − x1 complete [33]. Outcome 1 occurs with probability

2xn (1 x0) 1 − , (5.10) p 2 x0 + 2xn1 + x0 + 4xn1 x0

and the resultant state has no zeroth component. For outcome 2, the state is either a product state, in which case the protocol halts as a failure, or the state is entangled with

party n1’s component being zero and the state still having a zeroth component. In both

cases, redeﬁne x as the postmeasurement state, but set G as G n1 only after outcome −→ \ 2. Repeat Phase I with input (−→x , G). Chapter 5. Random distillation for W type states 77

5.3.2 Phase II: Equal or vanish (e/v) subroutine

Input ( x ,G) where x0 = 0 and x is shared between 2 S N parties. −→ −→ ≤ | | ≤ (1) If there does not exist an isolated node vk in G (one without any outgoing edges), proceed to the next step (2). Otherwise, when |S| = 2 the protocol halts as a failure, and (k) when |S| > 2, the "isolated" party "k" performs the disentangling measurement M1 = (k) diag[1, 0] and M2 = diag[0, 1]. If outcome 1 occurs, redefine −→x as the postmeasurement state and set G as G k; repeat the e/v subroutine on input ( x ,G). If outcome 2 happens, \ −→ the protocol terminates as a failure. (S0) (2) If every component in x is maximal, then x is a standard W state W 0 and −→ −→ | |S | i proceed to Phase III. Otherwise, choose some party k such that (i) xk is nonmaximal, and (ii) party k is connected to some party whose component is maximal. If no party satisfies both these conditions, then choose some party k which just satisfies condition (i). He/she

(k) q xk then performs a two-outcome measurement with operators M1 = diag[ , 1] and xn1

(k) q xk M2 = diag[ 1 x , 0]. Party k’s component value equals the maximum upon outcome − n1 1 and vanishes upon outcome 2. In both cases, redeﬁne −→x as the postmeasurement state, but set G as G k only after outcome 2. Repeat the e/v subroutine on the newinput ( x ,G) \ −→ (see Fig. 5.4). (S0) For Phase II input (−→x ,G), the ﬁnal success states of the e/v subroutine are W|S0| 0 0| i for 2 S S where for each party * such that x∗ = xn , either S or no ≤ | | ≤ | | 1 ∗ ∈ party in S’ is connected to * in G. The latter case occurs when there is only one party with a maximum component and all parties connected to * measure a "vanish" outcome; consequently, * becomes an isolated party and removes itself from the system via step (1) above. Let

(S0) (S0) λ−→ (W 0 ) := the probability or obtaining W 0 via the e/v subroutine for input ( x , G). x ,G |S | | |S | i −→ (5.11)

Note that λ−→x ,G is a smooth function of the component values xi and can be explicitly computed from the measurement operators given above. For example, λ−→x ,G(WN ) = Q x k6=n1 k N N−2 . Also, λWN ,G(WM ) = δMN . xn1

5.3.3 Phase III: Obtaining EPR pairs

(S) Input (W|S| ,G) with G having |S| nodes and at least one edge (Fig. 5.5). If |S| = 2, the state is an EPR pair and protocol halts as a success. Otherwise, Phase III of the protocol is deﬁned recursively such that for |S| > 2, the procedure depends on a predeﬁned random Chapter 5. Random distillation for W type states 78

(S0) distillation protocol for W 0 with |S’| < |S|. Let | |S | i

(S) (S0) PIII (W ,G) := the probability of distilling W 0 into G via the P hase III procedure. |S| | |S | i (5.12)

If |S| = 2, set PIII (W2,G) = 1 by deﬁnition. For |S| > 2, identify the party k whose node in G has a least number of connected edges. He/she performs the measurement with operators given by M (k) = diag[√α, 1] and M (k) = diag[√1 α, 0] where α is determined 1 2 − according to the discussion below Eq 5.14. Outcome 2 occurs with probability (1 α) |S|−1 − |S| and the resultant state is W (k) . Phase III is then repeated on this state and the reduced | |S|−1i graph G k. \ 1+α(|S|−1) Outcome 1 happens with probability pα = |S| and the postmeasurement state is 1 y α, which (up to a permutation between party 1 and k) takes the form (1, α, α, , α). −→ |S|pα ··· Party k then has the largest component value in −→y α, and the e/v subroutine (Phase II) is (S0) performed on the input (−→y α,G). The e/v subroutine will either output the states W|S0| , 0 | i 0 (S ) (S) where S < S with respective probabilities λ−→ (W 0 ), or the original state W , | | | | y α,G |S | | |S| i with probability α|S|−1. In the former case, the Phase III procedure is performed on (S0) ¯0 the input (W 0 ,G S ). Accounting for all states with |S’| < |S|, their total distillation |S | \ success probability is

S 1 (k¯) 1 + α( S 1) f(α) = (1 α)| | − PIII (W ,G k) + | | − − S |S|−1 \ S | | | | (5.13) X (S0) (S0) ¯0 λ−→ (W 0 )PIII (W 0 ,G S ), × y α,G |S | |S | \ 2≤|S0|<|S| where the sum is taken over all subsets S’ such that 2 S0 < S and either k S0 or ≤ | | | | ∈ (S) no party in S’ is connected to k in G. If the e/v subroutine outputs the original W|S| , (S) | i repeat Phase III again on the same input (W|S| ,G). This will generate an indeﬁnite loop in which for each cycle, the probability of distillation success is f(α), and the probability of continuing on for another cycle is α|S|−1. Therefore, the total success probability across P∞ |S|−1 r all cycles is given by the geometric sum r=0[α ] f(α). To maximize this value, we set

(S) f(α) PIII (W|S| ,G) = sup |S|−1 . (5.14) 0≤α<1 1 α − This determines the original value of α in the Phase III measurement operators: if Eq. 5.14 obtains its supremum in the interval [0,1), then α is chosen to be any of these critical points; if the supremum is obtained in the limit α 1, then α = 1 for → − any desired > 0. The smaller the value of is, the closer the success probability Chapter 5. Random distillation for W type states 79

1 Figure 5.4: Equal or vanish subroutine (Phase II) for the normalized state 1+3α (α, α, α, 1) and the conﬁguration graph with edges AB, AC, AD, BC . 1. David’s component is largest and Alice is a connected party to him{ with a lesser component} value. She performs an e/v measurement. 2. For the outcome "vanish" (right branch) she is separated from the system, and since David is not connected to either Bob or Charlie, he immediately removes himself from the system leaving ψ(BC) with some probability 1. For the outcome "equal" (left branch) the components of| all otheri parties receive a factor of α, and Alice’s component is now maximum equaling David’s. Bob is a connected party to Alice with a lesser component value and he performs an e/v measurement. 3. Again, either Bob vanishes (right branch) or all other components except his receive a factor of α. In both cases, Charlie is then a connected party to Bob with a lesser component value and he performs an e/v measurement. 4. The ﬁnal outcome states along these branches are (ABD) (ACD) AD W4 , W , W , and ψ . c 2011 American Physical Society. | i | 3 i | 3 i | i Chapter 5. Random distillation for W type states 80

(S) approaches PIII (W|S| ,G). Observe that when the supremum is obtained at α = 0, the ﬁrst measurement by party k will deterministically disentangle the party from the rest of the system.

(S) Figure 5.5: Phase III receives an input state W|S| and a conﬁguration graph G. Party k performs an e/v measurement. One outcome is a standard W state with party k removed, and the other is the state 1 (α, , α, 1, α, , α). Phase II is applied on this state |S|pα ··· ··· outputting either W states or a product (failure) state. Phase III will next be initiated (S0) 0 on each of the W states, and for any W state W|S0| with S < S , the transformation | | | 0 | (S ) ¯0 success probability from this point onward is given by PIII (W|S0| ,G S ); this value is (S) \ already known by recursion. However, for the state W|S| , performing Phase III again will generate an indeﬁnite loop, but one whose overall success probability converges to f(α) [see Eqs. 5.13 and 5.14]. c 2011 American Physical Society. 1−α|S|−1

It is also important to note that the optimization of Eq. 5.14 can always be eﬃ- (k) ciently performed. By the recursive construction, the values for PIII (W|S|−1,G k) and 0 \ (S ) 0 PIII (W|S0| ,G S ) are just real numbers known a priori. Furthermore, the functions (S0) \ −→ λ y α,G(W|S0| ) are smooth functions of α, and thus, Eq. 5.14 represents a single-variable smooth function whose supremum value can be easily computed. In total then, for a

stateN-partite state −→x with x0 = 0 and conﬁguration graph G, the overall success prob- Chapter 5. Random distillation for W type states 81 ability of the LPO protocol is given by

X (S0) (S0) ¯0 PLP O( x , G) := λ−→ (W 0 )PIII (W 0 ,G S ), (5.15) −→ x ,G |S | |S | \ 2≤|S0|≤N where the sum is taken over all subsets S’ such that 2 S0 N and, for each party * 0 ≤ | | ≤ such that x∗ = xn , either S or no party in S’ is connected to * in G. 1 ∗ ∈

5.4 Main results: The LPO protocol on multipartite W type states

5.4.1 Summary of results

Before working through the LPO protocol in detail in threeand four-qubit systems, we summarize the overall results. For three qubits, the possible configuration graphs are depicted in Fig 5.6, and upper bounds on the transformation success probabilities are given by Eqs. 5.17 and 5.19, respectively. In both cases, when x0 = 0 these bounds can be approached arbitrarily close. For four qubits, there are six different families of configurations depicted in Figs 5.7,

5.8, 5.9, 5.10, 5.11. For states with x0 = 0, we have completely solved the random distillation problem for all configurations except Configuration VI. Upper bounds on Configurations I-V are given by Eqs 5.20, 5.21, 5.23, 5.30, 5.32, respectively.

5.4.2 Three qubits

As a ﬁrst example of the LPO protocol, consider the state W3 and the conﬁgura- | i tion graph given by G∧ in Fig. 5.6. In Phase III of the protocol, we can choose the "least party" to be either Bob or Charlie (say it’s Charlie). He performs a measurement as described above, and either ψ(AB) is obtained or the postmeasurement state is 1 | i (AC) −→y α = 2α+1 (α, α, 1). For the latter, the e/v subroutine obtains ψ with probability (AC) 2α(1−α) | i f(α) −→ 2 2 λ y α,G∧ (W2 ) = 2α+1 . Thus, we have fα) = 3 (1 α) + 3 (1 α)α and therefore 1−α2 2 − − takes a constant value of 3 . Hence, α can be chosen as 0 in Charlie’s measurement and

2 P (W ,G ) = . (5.16) III 3 ∧ 3

For a more general state −→x = (xA, xB, xC ) with xA xB xC and x0 = 0, we (AB) (AC) ≥ ≥ −→ xC −→ xB −→ xB xC have λ x ,G (W2 ) = 2xB(1 ), λ x ,G (W2 ) = 2xC (1 ), λ x ,G (W3) = 3 . ∧ − xA ∧ − xA ∧ xA Chapter 5. Random distillation for W type states 82

Figure 5.6: (Left) Configuration G∧. (Right) Configuration G∆. An upper bound on the success probability is given by Eqs. 5.17 and 5.19, respectively, which is effectively tight when x0 = 0. For W3 , these probabilties are 2/3 and 1, respectively. c 2011 American Physical Society. | i

Therefore, by Eq. 5.15, the distillation probability is given by

xC xB xBxC xBxC PLP O(−→x , G∧) = 2xB(1 ) + 2xC (1 ) + 2 = 2xB + 2xC 2 . (5.17) − xA − xA xA − xA

One might wonder if this probability is optimal. It turns out that the answer is yes. See Eq. 5.21 below and the discussion there.

On the other hand, consider conﬁguration G∆ given in Fig 5.6. We can still choose Charlie as the "least" party, and this time, the possible EPR success states of the e/v (AC) (BC) 2α(1−α) subroutine are Ψ(AC) and Ψ(BC) . We have λ−→ (W ) = λ−→ (W ) = | i | i yα,G∆ 2 yα,G∆ 2 2α+1 and so f(α) = 2 (1 α) + 4 (1 α)α. Thus, 3 − 3 −

2 (1 α)(1 + 2α) PLP O(W3,G∆) = sup − 2 = 1 (5.18) 0≤α<1 3 1 α − This value is obtained in the limit α 1 which means the LPO protocol calls for → inﬁnitesimal measurements with α = 1 . Hence, for three qubits, the LPO protocol − reduces to the Fortescue-Lo protocol for distilling the state W3 . | i Since PIII (W 3,G∆) = 1 for the three-edge conﬁguration, when considering the state

x = (xA, xB, xC ) with xA xB xC and x0 = 0, we obtain the distillation probability −→ ≥ ≥

xC xB xBxC xBxC PLP O(−→x , G∆) = 2xB(1 ) + 2xC (1 ) + 3 = 2xB + 2xC . (5.19) − xA − xA xA − xA

Just as with the conﬁguration graph G∧, this probability is optimal as we see from Eq. 5.21 below. We now turn to four qubits where, unlike the two cases just examined, conﬁgurations exist for which f(α) obtains a maximum in the interval (0,1). Chapter 5. Random distillation for W type states 83

5.4.3 Four qubits

Next, we apply the LPO protocol to four-qubit W-type states. We only consider a subset of possible conﬁguration graphs, but any other can be obtained by a permutation of parties.

Conﬁgurations I (Fig 5.7) For a generic W-type state −→x = (xA, xB, xC , xD), whenever xA < xj for some j B,C,D , an upper bound on distilling to any of these conﬁgura- ∈ { } tions is 2xA by the K-T monotones. However, when xA is the largest component value, we have

0 Figure 5.7: Let GI , GI , and G”I be the first, second, and third of the above configurations, respectively. An upper bound on the success probability is given by Eq. 5.20 which is effectively tight when x0 = 0. For W4 , this probability is 1/4 for each configuration. c 2011 American Physical Society. | i

P ( x , GI ) 2xB, −→ ≤ 0 (xA xB)(xA xC ) P (−→x , GI ) 2xA 2 − − , (5.20) ≤ − xA (xA xB)(xA xC )(xA xD) P (−→x , G”I ) 2xA 2 − −2 − , ≤ − xA as proven in Ref. [18]. When x0 = 0, these are precisely the rates obtained by the LPO protocol, and so our protocol is optimal for such states. Note that setting xD = 0 proves Eq. 5.17 to be optimal.

Conﬁguration II (Fig 5.8) For a generic W-type state −→x = (xA, xB, xC , xD), if we assume without loss of generality that xA xB xC , then we have ≥ ≥

(xA xB)(xA xC ) P (−→x , GII ) 1 x0 xD − − . (5.21) ≤ − − − xA

When x0 = 0, the LPO protocol can approach this upper bound arbitrarily close. Note that this also proves Eq 5.17 to be optimal. Conﬁgurations III (Fig 5.9) A feature common to all of these conﬁgurations is that, for each party, there is at Chapter 5. Random distillation for W type states 84

Figure 5.8: Let GII be the above conﬁguration. An upper bound on the success probability is given by Eq. 5.21 which is eﬀectively tight when x0 = 0. For W4 , this probability is 3/4. c 2011 American Physical Society. | i

Figure 5.9: Let GIII be any of the above configurations. In each of these, (A,C) and (B,D) are unconnected pairs. An upper bound on the success probability is given by Eq. 5.23 which is effectively tight when x0 = 0. For W4 , this probability is 2/3 for each of these configurations. c 2011 American Physical| Society.i

least one other party to whom he/she is not connected. We refer to such a pair as unconnected. For example (A,C) and (B,D) form unconnected pairs in each of the above conﬁgurations. We introduce the following entanglement monotones to put an upper bound on the probability for transformations of Conﬁgurations III. For a genericW-type 0 state −→x = (xA, xB, xC , xD), let n1 be some party whose component is maximum, n1 the party unconnected to n1 with largest component value, and p and p’ the other two

parties. For definitiveness, if party n1 has two unconnected parties (which is possible in 0 the first two of Configurations III), take p’ to be the other one besides n1. Define the function 0 0 0 xpxp 2 xpxp xn1 ( ) = 2 + 2 0 2 + (5.22) τ −→x xp xp 2 . − xn1 3 xn1

Note that τ(W4) = 2/3.

Theorem 14. The function τ is an entanglement monotone.

Proof. See the Appendix 8.2. Chapter 5. Random distillation for W type states 85

As a result of this theorem, we have that for a state −→x ,

P ( x , GIII ) τ( x ). (5.23) −→ ≤ −→

For the LPO protocol, ﬁrst consider the initial state W4 . In each of the conﬁgurations, | i either party A or D can be chosen as the "least" party. Regardless of the choice, we have

X (ij) λ−→ (W ) = (4α + 2α2)(1 α). (5.24) yα,GIII 2 − (i,j)∈E Here E denotes the edge set of whatever conﬁguration is considered. By Eq. 5.16, we (BCD) also know that PIII (W ,G A) = 2/3. Thus, 3 \ 1 1 2 2 + α + 2 α 2 PIII (W4,GIII ) = sup 2 = , (5.25) 0≤α<1 1 + α + α 3 which obtains this value as α 1. Thus, for any > 0, →

2/3 < P (W4,GIII ) 2/3. (5.26) − ≤

When x = 0 and a state −→x = (xA, xB, xC , xD) is considered, it is straightforward to (S) −→ compute the probabilities for λ x ,GIII (W|S| ). Doing so and using Eq 5.15 shows that the probability τ(−→x ) can be approached arbitrarily close using the LPO protocol. Conﬁguration IV (Fig 5.10)

Figure 5.10: Let GIV be the above configuration. We say two parties are edge complementary if their nodes have a different number of connected edges. For example, A is edge complementary to both C and D. An upper bound on the success probability is given by Eq 5.30 which is effectively tight when x0 = 0. For W4 , this probability is 5/6. c 2011 American Physical Society. | i

We first consider the transformation of the standardW state W4 . The Phase II | i Chapter 5. Random distillation for W type states 86 probabilities are (AD) 1 λ−→ (W ) = 2α(1 α)2, yα,GIV 2 1 + 3α − (BD) 1 λ−→ (W ) = 2α(1 α)2, yα,GIV 2 1 + 3α − (ABD) 1 λ−→ (W ) = 3α2(1 α), (5.27) yα,GIV 3 1 + 3α − (ACD) 1 λ−→ (W ) = 3α2(1 α), yα,GIV 3 1 + 3α − (BCD) 1 λ−→ (W ) = 3α2(1 α), yα,GIV 3 1 + 3α − This gives 3 3α2 f(α) 4 + α + 4 PIII (W4,GIV ) = sup 3 = 2 = 5/6, (5.28) 0≤α<1 1 α 1 + α + α − for which the value is obtained as α 1. → Consider a generic W-type state −→x = (xA, xB, xC , xD). We say that a party is edge complementary to a party if their corresponding nodes in GIV have a different number of connected edges. For the particular configuration GIV , let n1 denote some party with the 0 largest component, n1 the party edge complementary to n1 with the largest component, and e2 and e3 the other two parties having two and three edges, respectively. Define the function:

x 0 x x x x 0  xe n e2 4 e2 e3 n 2 + ( + 0 )(2 3 ) 2 1 + 1 if has three connected edges,  xe3 xe2 xn1 x x 3 x2 n1 Γ( x ) = − n1 − n1 n1 −→ xn0 xe3 xe2 xe3 xn0 2 0 + 2 1 + 1 if has two connected edges.  xn1 xe3 x 3x2 n1 − n1 n1 (5.29)

Note that Γ(W4) = 5/6.

Theorem 15. The function Γ is an entanglement monotone.

Proof. The proof is nearly identical in structure to the one given for τ in the Appendix 8.2.

As a result, it immediately follows that

P ( x , GIV ) Γ( x ). (5.30) −→ ≤ −→ And just as in the case of Conﬁgurations III, the LPO protocol can approach this upper bound arbitrarily close whenever x0 = 0. Highlighting the standard W state, we have Chapter 5. Random distillation for W type states 87

that, for any > 0,

5/6 < P (W4,GIV ) 5/6. (5.31) − ≤ It should be noted that 5/6 is the also the optimal transformation probability if one considers transformations within the more general class of separable operations. Conﬁguration V (Fig 5.11)

Figure 5.11: Let GV be the above conﬁguration. An upper bound on the success probability is given by Eq 5.32 which is eﬀectively tight when x0 = 0. For W4 , this probability is 1. c 2011 American Physical Society. | i

For the state W4 , the Fortescue-Lo protocol achieves this distillation conﬁguration | i with probability arbitrarily close to 1. For more general states, the result is:

(xA xB)(xA xC )(xA xD) P (−→x , GV ) 1 x0 − −2 − , (5.32) ≤ − − xA

where we have assumed without loss of generality that xA is the largest component value.

The LPO protocol can achieve this probability as close as desired whenever x0 = 0. Conﬁguration VI (Fig 5.12)

Figure 5.12: Let GVI be the above conﬁguration. For W4 , the LPO protocol gives a 1 √ | i success probability of 6 (3 + 3).We conjecture this to be optimal. c 2011 American Physical Society.

For this conﬁguration, we only work out the Phase III calculation for the standard

W state W4 . In this case, David is the least party and he measures ﬁrst. Out- | i come 2 is the state W (ABC) obtained with probability 3/4(1 α); from here, we have | 3 i − Chapter 5. Random distillation for W type states 88

(ABC) 1 PIII (W ,GVI D) = 1. Outcome 1 is the state (α, α, α, 1). The ensuing e/v 3 \ 1+3α subroutine is described in Fig. 5.10. We have

(BC) 1 λ−→ (W ) = 2α(1 α), yα,GVI 2 1 + 3α − (AD) 1 −→ 2 λyα,GVI (W2 ) = 2α(1 α) , 1 + 3α − (5.33) (ABD) 1 λ−→ (W ) = 3α2(1 α), yα,GVI 3 1 + 3α − (ACD) 1 λ−→ (W ) = 3α2(1 α), yα,GVI 3 1 + 3α − This gives

2 f(α) 3 + α + α 1 4 2 √ PIII (W4,GVI ) = sup 3 = 2 = (3 + 3), (5.34) 0≤α<1 1 α 1 + α + α 6 − which obtains this maximum when α = 1 (√3 1). The generalized Fortescue-Lo protocol 2 − gives a rate of 3/4 so we see an improvement in our protocol. For an upper bound, it is known that this transformation cannot be accomplished with any probability greater than 5/6 by the more general class of separable operations. Thus, we summarize our result by

3 + √3 5 PFL(W4,GVI ) < PLP O(W4,GVI ) P (W4,GVI ) . (5.35) ≈ 6 ≤ ≤ 6

We use the ” ” symbol for the PLP O value since it can be approached arbitrarily close. ≈ While we conjecture that this protocol is optimal for the state W4 and the conﬁgu- | i ration graph GVI , unfortunately it does not appear optimal for more general four-qubit states. Indeed, suppose that we begin with state −→x = (xA, xB, xC , xD) with x0 = 0 and xA xB xC xD. The LPO protocol says that we should ﬁrst perform the e/v ≥ ≥ ≥ subroutine with respect to party 1 and then implement Phase III on the state W4 . The | i total probability is then given by Eq 5.15. Explicitly computing it yields

xBxD xBxC xC xD 3 + 2√3 xBxC xD PLP O(−→x , GVI ) = 2(xB + xC + xD) 2 2 + 2 . (5.36) xA − xA − xA 3 xA

Now suppose we have xA = 1 3t and xB = xC = xD = t with t < 1/4. Since Alice’s − component is strictly greater than all other components, she can make a weak measurement such that her component value is still the largest in both postmeasurement states.

Speciﬁcally, when she performs the measurement given by Eq 5.7 with (a1, c1, a2, c2) in Chapter 5. Random distillation for W type states 89

some neighborhood of (1/2,1/2,1/2,1/2), the average change in PLP O is

2 2 t 12 8√3 ∆PLP O = (a c) (20 t − ), (5.37) − − 1 3t − 1 3t − − which can be positive for t close to 1/4. Therefore, PLP O cannot be the optimal probability for the initial state −→x since a weak measurement by Alice increases the overall transformation probability.

It should be emphasized that for the transformation of W4 according to the LPO | i protocol, we never encounter a state like −→x . The only time Alice’s component is larger than David’s is after David performs an e/v measurement and his component value is

zero. Consequently, we still believe the protocol to be optimal for W4 . | i

5.4.4 n qubits and the entanglement monotones

The generalization of above results into n parties would in general be diﬃcult. However, for some special cases, the result turns out to be very explicit. In the following we will ﬁrstly introduce the new entanglement monotones and then show that they are upper bounds for corresponding transformation.

The entanglement monotones

In this section, we introduce new entanglement monotones on the N-qubit W type of

states. We deﬁne our monotones as follows. For an N-party W type state (x1, x2, , xN ), ··· set n1, n2, , nN = 1, 2, ,N such that xn xn xn and consider the { ··· } { ··· } 1 ≥ 2 · · · ≥ N continuous functions:

N 1 N−2 Y η(−→x ) = xn1 ( ) (xn1 xni ), − xn1 − i=2 (5.38) N X κ(−→x ) = xni + η(−→x ) i=2

Theorem 16. (I) η is nonincreasing on average for any single local measurement in which n1 is the same value for the initial and all possible ﬁnal states, (II) κ is an entanglement monotone. It is strictly decreasing on average for any nontrivial measurement by party n1. The three-qubit form of this theorem has been proven. Here, in the general case, our proof technique will be very similar. Chapter 5. Random distillation for W type states 90

Proof: (I) We consider case-by-case measurements of each party under the conditions

of I. The function η transforms as η ηλ for λ = 1, 2, and we are interested in the → average change, ηλ = p1η1 + p2η2, under inﬁnitesimal measurements. First, suppose that

party n1 measures. According to Eq. 5.8, the average change in η is

N N Y a1xn Y a2xn η( x ) = c x (1 (1 i )) + c x (1 (1 i )). (5.39) −→λ 1 n1 c x 2 n1 c x − i=2 − 1 n1 − i=2 − 2 n1

We demonstrate that in the weak-measurement setting, this quantity is maximized by

equality of the upper bound: c1 + c2 = 1. Indeed, we have

N N N ∂ηλ Y xni X xni Y xnj a1=a2=1/2,c1=c2=1/2 = xn1 (1 (1 )) (1 ) (5.40) ∂cλ | { − − xn − xn − xn } i=2 1 i=2 1 j6=i 1

for λ = 1, 2, and it suﬃces to show that this expression is strictly positive. Now if we

diﬀerentiate Eq 5.19 with respect to any xnk , we obtain

N N N N Y xn Y xn X xn Y xnj X xn Y xnj (1 i ) (1 i )+ i (1 ) = i (1 ) 0(2 k N), −xn − −xn xn −xn xn −xn ≥ ≤ ≤ i6=k 1 i6=k 1 i6=k 1 j6=i,k 1 i6=k 1 j6=i,k 1 (5.41)

and since Eq 5.41 vanishes when xnk = 0 for all nk, it follows that for nonzero values of

xnk , Eq 5.41 is strictly positive. Thus, the maximal change in η occurs when c1 + c2 = 1. As we are interested in this upper bound, we assume the measurement is characterized

by a a1, 1 a = a2, c c1, and 1 c = c2. We then have ≡ − ≡ −

N N N Y xni Y axni Y (1 a)xni η η(−→x λ) = xn1 (1 )+cxn1 (1 )+(1 c)xn1 (1 − ). (5.42) − − xn − cxn − − (1 c)xn i=2 1 i=2 1 i=2 − 1 Expanding this to second order about the point (a,c) =(1/2,1/2) yields

N 2 X xni xnj Y xnl η η(−→x λ) 4(a c) (1 ) 0. (5.43) − ≈ − xn − xn ≥ i,j 1 l6=i,j 1

And this expression will be positive whenever a = c, which is whenever party n1 6 performs a nontrivial measurement. In the case in which party ni performs a measurement for some i > 1, η changes as Chapter 5. Random distillation for W type states 91

N Y xnj Y xnj η(−→x λ) = xn1 (a1xn1 c1xni ) (1 ) (a2xn1 c2xni ) (1 ) η(−→x ). (5.44) − − − xn − − − xn ≤ j6=i 1 j6=i 1

(II) We can always decompose a general transformation into a sequence of weak measurements for which either each measurement satisfies the conditions of I or its premeasurement state −→y satisfies yn1 = yn2 . In the first case, κ is monotonic by part I PN and the fact that i=2 xni is nonincreasing on average by the K-T monotones. In the

second case, we have κ( y ) = 1 y0. Since 1 yλ,0 is an upper bound on κ( y λ) for each −→ − − −→ of the postmeasurement states −→y λ, and 1 y0 is nonincreasing on average by the K-T P − monotones, it follows that κ( y ) pλκ( y λ). Thus, κ is an entanglement monotone −→ ≥ λ −→ in general.

Theorem 15 also applies to any ﬁxed collection of subsystems. Indeed for N-qubit systems, let S denote some subset of parties, and consider the unnormalized state −→s which has |S| components, each belonging to a diﬀerent party in S. Then Theorem 15 also holds for the functions η(−→s ) and κ(−→s ). The proof of this is exactly the same as above, with the added note that whenever a measurement is performed by a party not in S, η(−→s ) and κ(−→s ) remain invariant on average. For example, in a four-party system, let S be parties 1, 2, and 3. Now for any four-

qubit state x , take xmax, xmid, xmin = x1, x2, x3 such that xmax xmid xmin. −→ { } { } ≥ ≥ Then the function

2xmid + 2xmin xmidxmin − (5.45) xmax is an entanglement monotone. The condition in part I of Theorem 15 can be extended beyond single measurements.

Corollary 6. Suppose the transformation −→x pi, −→y i is possible by LOCC, where P → { } n1( x ) = n1( y i) for all i.Then η( x ) piη( y i). −→ −→ −→ ≥ i −→

Proof. We can partition any transformation into sections where n1(−→x ) is the largest component and where it is not. By weak measurement theory, we can assume that when passing from one section to the other, we always ﬁrst obtain a state −→s on the border such that ( ) = −→ . Therefore, since the ( ) component is always monotonic by η −→s sn1( x ) n1 −→x the K-T monotones, we have that η will not have increased on average within any region Chapter 5. Random distillation for W type states 92 for which n1(−→x ) is not the largest component. For sections when n1(−→x ) is the largest, we knowthat η is monotonic by part I of the previous theorem.

5.4.5 Interpretation of monotones

A natural question is whether the functions η and κ possess any physical interpretation. Here we show that for states −→x having x0 = 0, 2η(−→x ) gives the optimal probability for transformation when the conﬁguration graph G consists of all edges connected to node vn1 (−→x ). We refer to this as a "combing transformation" since it represents a single-copy version of the entanglement combing procedure described in Ref. [82]. On the other hand, κ(−→x ) gives the optimal probability when G is complete, i.e., each vertex is connected to every other one (see Fig 5.13). The following theorem gives a precise statement of this result.

Figure 5.13: Distillation conﬁgurations for η vs κ. Top: A "combing-type" distillation: when x0 = 0, 2η(x) is the optimal probability for a random distillation in which party n1 shares one-half of each EPR pair. Bottom: A "complete-type" distillation: when x0 = 0, κ(x) gives the optimal probability for a random distillation in which the target pairs are any two of the parties. c 2011 American Physical Society.

Chapter 5. Random distillation for W type states 93

Theorem 17. For an N-party W state x = (x1, x2, , xN ), let Ptot be the optimal total −→ ··· probability of obtaining an EPR pair by LOCC, and Pk the optimal total probability of party k becoming EPR entangled. Then

(I) Ptot < κ(−→x ), and (II)  2xk if xk < xl for some l, Pk (5.46) ≤ 2η( x ) if xk xl for all l. −→ ≥ When x0 = 0, the upper bound in I can be approached arbitrarily closely, while in II it can be achieved exactly.

Proof: First recall that κ(ψ(ij)) = 1. Then the upper bounds follow from Theorem 2 and the K-T monotones. Assume now that x0 = 0, then using LPO protocol it is easy to show that the upper bounds are eﬀectively tight.

For a combing-type distillation, when xk xl for some party l, 2xk is known to be ≥ an achievable rate [74, 33]. When xk > xl for all parties l, the procedure is for each party to perform an e/v measurement (in any order), except that when the ﬁrst party l obtains an "equal" outcome, a nonrandom EPR distillation is made between party k and party l. This occurs with total probability 2xl , and a completely analogous inductive argument to the one given above shows that this full measurement scheme succeeds with probability exactly equal to η(−→x ).

5.5 SEP VS LOCC

In this section, we compare the random distillation probability achieved by LOCC and SEP protocols. By showing that SEP could achieve significantly higher probability than LOCC in some cases, we find the first analytical entanglement monotone that can be increased by SEP.

5.5.1 Random distillation by Separable transformations

In this section we derive the conditions for which random distillaltion is possible by SEPs when the initial state is a W-type state with x0 = 0. As shown in the following lemma, the unique structure of such states allows for a major simpliﬁcation in the analysis. Chapter 5. Random distillation for W type states 94

(1) (N) Lemma 9. Suppose that Πλ := M M λ=1,··· ,t corresponds to a complete { λ ⊗ · · · ⊗ λ } measurement that achieves random distillation with probabilities p12, , pN−1,N when ··· φ 1,··· ,N = √x1 10 0 + + √xN 0 01 . Then up to LU operations, there exists | i | ···(1) i ··· (N) | ··· i a measurement Mλ Mλ λ=1,··· ,2t, that achieves the transformation with the { ⊗ · · · ⊗ (k}) same probabilities and with each Mλ being diagonal. ! (k) (k) √aλk bλk Proof. Up to an LU operation, each Mλ takes the form Mλ = so 0 √cλk that

! X X akλ √aλkbλk Π+Π = N = (5.47) λ λ k=1 ∗ 2 I. ⊗ √aλkb bλk + cλk λ λ λk | | ! (k) √aλk 0 Let Mλ = . It is straightforward to see that the operators Πλ := 0 √cλk { N (k) M λ=1···t correspond to an incomplete measurement that achieves the transfor- ⊗k=1 λ } mation with the same probabilities as the Πλ λ=1···t. From Eq 5.47, the collection of ! { } N 0 0 separable operators k=1 λ=1···t can be combined with Πλ λ=1···t to form a {⊗ 0 bλk } { } set which corresponds to a complete| | measurement.

One immediate consequence of this lemma is that for any incomplete separable trans- P + P + formation of the form with Π Πλ < I, we can always assume that I Π Πλ λ λ − λ λ has a diagonal representation and is therefore separable. As a result, when φ is a | i1···N W-type state, it is suﬃcient to consider the feasible probabilities of transformation under incomplete separable transformations. (1) (N) Now for measurement Πλ := Mλ Mλ λ=1···t, if we let Sij denote the set { ⊗ · ·(ij ·) ⊗ } of all outcomes λ such that Πλ φ ψ , we can form a Choi matrix Ωij for each | i1···N ∝ | i edge (i, j) E of graph G [51]: ∈

X N (ii0) + Ωij = Πλ I( ψ )(Π ) I. (5.48) ⊗ ⊗i=1 λ ⊗ λ∈Sij

Here, Πλ acts on systems 1, 2, ,N, while I is the identity acting on their copies 1’, ··· 2’, , N’. The Πλ can be taken as diagonal matrices so that Ωij has support only on the ··· span of i1i1 0 , i2i2 0 , , iN iN 0 i ,i ,··· ,i ∈{0,1}. Furthermore, since all parties {| i11 | i22 ··· | iNN } 1 2 N besides i and j hold pure states in the end, M (k) must be a rank 1 matrix for k = i, j and λ 6 λ Sij . Thus, up to LUs and a permutation of spaces, Ωij has the form ∈ Chapter 5. Random distillation for W type states 95

(ii0jj0) (ii0jj0) Ω(ij) = κ 0 0 , (5.49) ⊗ | i h | (ii0jj0) where κ is eﬀectively a separable 2 2 density matrix having support on mm ii0 nn jj0 m,n∈{0,1}; 0 0 ⊗ {| i | i } equivalently, κ(ii jj ) has a positive partial transpose [36]. In terms of the Choi matrix, (ij) the condition of obtaining ψ with probability pij given by | i

(10···N 0) (ij) (ij) tr10,··· ,N 0 (Ωijφ ) = pijψ 0 0 . (5.50) ⊗ | i h | 0 0 Here we use the fact that φ(1 ···N ) is taken to have only real components. Finally, the P P + constraint that Π Πλ I is captured by (i,j)∈E λ∈Sij λ ≤ X tr1···N (Ωij) I. (5.51) ≤ (i,j)∈E This construction is completely reversible such that, given matrices satisfying the above conditions, we can always construct a separable measurement facilitating transformation [27]. Thus the necessary and suﬃcient conditions for a feasible separable map are 4 4 0 0 × complex matrices κ(ii jj ) for all (i, j) E which satisfy ∈

0 0 0 0 κ(ii jj ) 0, [κ(ii jj )]Γi0j0 0. (5.52) ≥ ≥ This is a semideﬁnite feasibility problem which can be eﬃciently solved using a variety of numerical tools [75]. Furthermore, duality theory can be used to analytically prove instances of infeasibility. We perform such an analysis in the Appendix 8.3 for the initial state φ1,··· ,N = WN . The result is given by the following theorem, which also provides | i | i an LOCC upper bound.

Theorem 18. For φ1···N = WN , transformation with graph representation G is pos- | i | i sible by SEPs if and only if

2 N X 2 N X p 1, pij 1, 1 k N. (5.53) 4 ij ≤ 2 ≤ ≤ ≤ (i,j)∈E (i,j)∈Ek

where Ek is the set of all edges connected with vertex vk.

Pn 2 1 Pt 2 Remark 17. In practice, it may be helpful to use the inequality x ( xi) i=1 i ≥ n i=1 so that the ﬁrst constraint in the above equation becomes

2 N X 2 ( pij) 1 (5.54) 4 E ≤ | | (i,j)∈E Chapter 5. Random distillation for W type states 96

5.5.2 Comparison between SEP and LOCC

For three-qubit W-type state

For a three-qubit W-type state √xA 100 + √xB 010 + √xC 001 with 1/2 xA | i | i | i ≤ ≤ xB xC , the following separable measurement operators randomly distill the state with ≤ probability one:

MAB = (λCB 0 0 + 1 1 ) (λCA 0 0 + 1 1 ) 0 0 | i h | | i h | ⊗ | i h | | i h | ⊗ | i h | MAC = (λBC 0 0 + 1 1 ) 0 0 (λBA 0 0 + 1 1 ) | i h | | i h | ⊗ | i h | ⊗ | i h | | i h | MBC = 0 0 (λAC 0 0 + 1 1 ) (λAB 0 0 + 1 1 ) (5.55) | i h | ⊗ | i h | | i h | ⊗ | i h | | i h | 2 2 2 2 2 2 1/2 M000 = (1 λ λ λ λ λ λ ) 000 000 − CA CB − BA BC − AB AC | i h | M111 = 111 111 ; | i h |

q 1−2xi where λij = . As an example, consider the distillation rates on the one parameter 2xj q 1−s 1 1 family of states φs = √s 100 + ( 010 + 001 ) for s . The LOCC optimal | i | i 2 | i | i 3 ≤ ≤ 2 (1−s)2 probability is given by κ(φs) = 2(1 s) . A comparison of SEP and LOCC is − − 4s depicted in Fig 5.14. It shows that the entanglement monotone κ(φs) can be increased by SEP. When s = 0.5, the gap is around 12.5%.

Figure 5.14: LOCC vs SEP for the maximum probability of obtaining an EPR pair q between any two parties as a function of s when the initial state is √s 100 + 1−s ( 010 + | i 2 | i 001 ). The LOCC probability is 2(1 s) (1 s)2/4s. A gap of 12.5% exists between |SEPi and LOCC. c 2012 American Physical− − Society.−

Chapter 5. Random distillation for W type states 97

For n-party W-type state

q 1 q 1 The state we consider is φ1/2 = 10 0 + ( 010 0 + + 0 01 ). | i1···N 2 | ··· i 2(1−N) | ··· i ··· | ··· i By LOCC, the optimal probability for a combing-type transformation is

1 N−1 −1 2η(φ1/2) = 1 (1 ) 1 e , (5.56) − − N 1 → − − where we have taken the limit for large N. However, it is easy to see that the following SEPs (deﬁned up to a reordering of spaces) represent a complete measurement which, with total probability 1, will obtain an EPR pair shared by the ﬁrst party:

r (1) 1 (k) (k) N (j) Mk = I 0 0 + 1 1 0 0 for 1 < k N, ⊗ N 1 | i h | | i h | ⊗j6=1,k | i h | ≤ v − u N (5.57) u X + M0 = tI Mk Mk. − i=1

We plot this separation between LOCC and SEP as a function of N in Fig 5.15. We can see the gap between SEP and LOCC increases as N increases. As N becomes large, the gap could be around 37%.

Figure 5.15: LOCC vs SEP for the maximum probability of party 1 becoming EPR q q entangled as a function of N when the initial state is 1 10 0 + 1 ( 010 0 + 2 | ··· i 2(1−N) | ··· i 1 N−1 + 0 01 ). The LOCC probability is 1 (1 N−1 ) . A gap of 37% exists between ···SEP| and··· LOCC.i c 2012 American Physical− Society.−

Chapter 5. Random distillation for W type states 98

5.6 Applicaiton to the transformation φ 1, ,N WN | i ··· → | i For a generic W-type state φ , there has been promising progress on the SLOCC | i1,··· ,N transformation of φ WN since the discovery of the unique form possessed by | i1,··· ,N → | i multipartite W-type states [52]. However, the upper bound on the transformation success

probability determined by the K-T monotones is not tight when the x0 component of the initial state is not zero. A canonical example of this is the transformation of W-type

state x = (tx1, tx2, , txn) into y = (x1, x2, , xn) for 0 < t < 1, which cannot be −→ ··· −→ ··· accomplished with probability t and thus does not saturate the K-T monotones [52]. In the following, we improve on the general upper bound of the Nt set by the K-T

monotones for the transformation −→t WN , where −→t = (t, t, , t). We do this by | i → | i ··· ﬁrst considering the random distillation of −→t into EPR pairs between party 1 and any other party.

Lemma 10. The optimal LOCC success probability for randomly distilling the N-partite W-type state −→t = (t, t, ..., t), 0 t 1 into EPR pairs between party 1 and any other ≤ ≤ N party is upper bounded by p p 1 1 4(N 1)t2 (5.58) ≤ − − − Proof. The proof is straightforward. We can "merge" together all parties other than party 1 so that we have a state unitarily equivalent to ψ = √1 Nt 00 + √t 10 + |qi − | i | i p 1−√1−4(N−1)t2 (N 1)t 01 , whose smallest Schmidt component is . Therefore, an − | i 2 upper bound on the probability for distilling EPRs across the bipartition 1 : 23 N is p ··· 1 1 4(N 1)t2. − − −

The following theorem then shows the desired result.

Theorem 19. The optimal LOCC transformation probability from N-partite W-type state 1 1 1 1 −→t = (t, t, ..., t), 0 t , into the standard W state WN = ( , , , )) is upper ≤ ≤ N | i N N ··· N bounded by

N p 2 P (−→t WN ) [1 1 4(N 1)t ] < Nt. (5.59) → ≤ 2 − − −

Proof. We know that the optimal random-pair transformation of WN into an EPR | 2 i state shared between party 1 and some other party has probability N . If the trans- N p 2 formation probability from −→t into WN is higher than [1 1 4(N 1)t ], then | i 2 − − − we can ﬁrst transform −→t into WN and then distill EPR pairs between party 1 and | i Chapter 5. Random distillation for W type states 99

p the other parties with an overall successful probability larger than 1 1 4(N 1)t2, − − − contradicting with Lemma 10. Then to ﬁnish proving the theorem, we must show that N p 2 1 2 [1 1 4(N 1)t ] < Nt when 0 < t < N . It is an elementary optimization exercise − − − p 1 to see that 1 2t 1 4(N 1)t2 < 0 whenever 0 < t < . − − − − N

This "grouping" argument given for state −→t can be generalized to any φ having | i1,·,N x0 = 0 in order to obtain an upper bound on the transformation probability of φ 6 | i1,·,N → WN . While our upper bound is an improvement over the K-T monotones, it is not tight | i in general. Proving optimal transformation probability when x0 == 0 remains an open 6 problem.

5.7 Conclusion

To conclude this chapter, let us ﬁrst summarize the overall idea of the "Least Party Out"

(LPO) protocol. Given a generic W-type state, we first remove the x0 component with some probability. We then proceed to symmetrize by converting to standard W states W (S) . This is what the equal or vanish subroutine accomplishes, and it does so in such | |S| i a way that the symmetry exists only between parties connected by G or any subgraph of G. Finally, given a standard W state, the desired EPR pairings are obtained by removing parties from the entanglement in order of their connectivity in G, the least parties being removed first. For three-qubit random distillations, our protocol is optimal, and for four qubits, it is proven optimal when x0 = 0 for all but Configuration VI, although it still may be optimal for the standard W state. In proving optimality for Configurations III and IV, the strategy was to compute the general expression for the LPO probability when x0 = 0 and then show that this expression is an entanglement monotone. We have applied the same approach to study random distillations in systems with a greater number of parties. Unfortunately, the general expression for the LPO probability becomes quite complicated. This can be explicitly seen from Eq 5.15 in which the number of terms in the sum scales as O(2N ) for a general configuration graph G.

5.7.1 Open questions and concluding remarks

I. An obvious unresolved problem is to complete the four-qubit picture by solving the random distillation of Configuration VI. We know the LPO protocol is not optimal for Chapter 5. Random distillation for W type states 100 nonstandard W-type states, but it is not clear why this is the case. One possibility is the existence of k-cliques (a set of k nodes all connected to one another) and the fact that all but one party belongs to a 3-clique. While Configurations IV and V also have 3-cliques, each party belongs to at least one. This may be the reason why the protocol behaves optimally in these two cases. Understanding precisely the limitations of our protocol for Configuration VI may also prove helpful when considering the same configuration of random distillations for more general states beyond the W type.

II. Another open problem is to generalize some of our results to a larger number of parties, especially the random distillations whose configuration graphs have relatively few edges. For example, consider the first graph in Configurations III for which we know the LPO protocol reduces to the Fortescue-Lo protocol, and it is optimal. For a six-qubit system, this configuration generalizes to three disjoint pairings of the parties: (1,2), (3,4), and (5,6). If, for this configuration, we perform the LPO protocol on the state (x1, x2, x3, x4, x5, x6) with x1 x2 x6, the resultant probability function is ≥ ≥ · · · ≥ x2x4 x2x6 x4x6 x2x4x6 x4x5x6 τ6 = 2(x2 + x4 + x6 ) + 2 + 2 2 − x1 − x1 − x3 x1x3 x 3 (5.60) 2x2x3x4 2x2x5x6 x2x4x5x6 14x2x3x4x5x6 + 2 + 2 2 2 4 . 3x1 3x1 − x1x3 − 15x1

We strongly suspect that this probability is optimal, but we have no proof at this point. As in the four-qubit case, the LPO protocol reduces to the generalized Fortescue-Lo

protocol. Note that for the state W6 , the success probability is 2/5. | i

The generalization of this conﬁguration to 2N qubits consists of a graph G2N with N disjoint pairings. Intuitively, the LPO protocol will again reduce to the generalized Fortescue- Lo protocol since there exists no particular least party. That is, the procedure will be for each party to perform weak measurements to randomly obtain three-qubit W

states W3 from which an EPR state can be obtained by a specified pair with probability | i 1 2/3. One particular trio will obtain a W state with probability C(2N,3) , and there are a total of 2N - 2 trios in which this particular duo can belong. And finally, there are N possible pairs. Thus, the total probability of some specified pair (i,j) obtaining an EPR state is 1 2 2 PLP O(W2N ,G2N ) = (2N 2) N = . (5.61) C(2N, 3) − 3 2N 1 − What is particularly interesting is when this transformation is compared to the optimal distillation probability by separable operations. As shown in Ref. [14], this probability Chapter 5. Random distillation for W type states 101 is given by r 1 P (W ,G ) = . (5.62) SEP 2N 2N N Thus, if the LPO procedure is optimal for this particular transformation, whichwe strongly believe it is, then we see that the performance gap between separable operations and LOCC grows arbitrarily large. We depict this relative difference in Fig 5.16. For exam- 2 ple, in the four-qubit case where the LPO procedure is optimal, we have PLOCC = 3 < q 1 PSEP = 2 .

Figure 5.16: The relative difference between the optimal separable operation and the LPO protocol. The configuration graph consists of N disjoint pairs. Separable operations q 1 2 perform as PSEP = N whereas the LPO protocol obtains the rate of PLP O = 2N−1 . We conjecture that the LPO protocol is LOCC optimal for this configuration graph, as it is known to be when N = 4. c 2012 American Physical Society.

III. Beyond the W type of states, very little is known about single-copy random distillations. Partial extensions of the Fortescue-Lo protocol to symmetric Dicke states have been made; it has been shown that even within the three qubit GHZ class (i.e. those p states obtainable from 1/2( 000 + 111 ) by stochastic LOCC), distilling to randomly | i | i chosen pairs outperforms distilling to a specified pair [40]. Nevertheless how the topology of the outcome configuration graph G affects these transformations has yet to be studied in general.We hope the results of this article shed light on this question and provide a new insight into the structure of multipartite entanglement. Chapter 6

Absolutely maximal entangled state and quantum secret sharing

In this chapter, we study the existence of absolutely maximally entangled (AME) states in quantum mechanics and its applications to quantum information. AME states are characterized by being maximally entangled for all bipartitions of the system and exhibit genuine multipartite entanglement. With such states, we present a novel parallel teleportation protocol which teleports multiple quantum states between groups of senders and receivers. The notable features of this protocol are that (i) the partition into senders and receivers can be chosen after the state has been distributed, and (ii) one group has to perform joint quantum operations while the parties of the other group only have to act locally on their system. We also prove the equivalence between pure state quantum secret sharing schemes and AME states with an even number of parties. This equivalence implies the existence of AME states for an arbitrary number of parties based on known results about the existence of quantum secret sharing schemes. The content of this chapter is mainly based on [48].

6.1 Introduction

Entanglement is at the core of the power of quantum information processing and has been extensively studied for few qubits. The classiﬁcation of entanglement classes for three and four qubits is well understood [34, 76, 49, 55, 57, 2, 16] and canonical forms of pure states under local unitary transformations of each local Hilbert space have also been analyzed [2, 54, 53]. As the number of local quantum degrees of freedom increases, our understanding of entanglement gets poorer. The number of independent invariants that classify entanglement grows exponentially and it is unclear which purpose each

102 Chapter 6. Absolutely maximal entangled state and quantum secret sharing103 category of entanglement serves [60, 41]. In recent years, there has been an important progress in the classification of the maximally multipartite entangled states composed of qubits [64, 17, 37, 45, 16]. Nevertheless, a complete understanding of the structure, classification and usefulness of quantum states with the largest possible entanglement for arbitrary dimension is still missing. Another motivation for studying multipartite entanglement is its connection to other apparently unrelated areas of physics, like string theory and black-holes [13, 14]. Quantum teleportation is one of the most intriguing utilizations of entanglement. It allows distant parties, who share a resource of entanglement, to transport a quantum state from one party to the other by only exchanging classical information and using up said entanglement. The first proposal of such a protocol used the resource of bipartite entanglement between two parties [7]. Later teleportation protocols using genuine multipartite entanglement between more than two parties were proposed theoretically for four qubit entanglement [83], and experimentally in the form of open-destination teleportation for five qubits [86]. This chapter is devoted to initiate the study of a class of states with genuine multipartite entanglement. These states, which we call absolutely maximally entangled (AME) states, are defined as having the strict maximal entanglement in all bipartitions of the system. Up until now, AME states have been thought to be a rather limited concept, because only few AME states exist for qubits, specifically no AME states exist for four, or eight and more qubits [45, 67]. In this work, we consider the qudit problem, and show that AME states exist for any number of parties by choosing an appropriate qudit dimension. The fact that AME states contain maximal entanglement makes them the natural candidates to implement novel multipartite communication protocols. Indeed, we shall here show how they can be used to implement novel parallel teleportation scenarios that postpone the choice of senders and receivers until after the state has been distributed. These protocols require that either the senders or receivers perform joint quantum operations, while the respective other parties only have to act locally on their systems. We further establish a one-to-one correspondence between pure state quantum secret sharing (QSS) schemes [28, 42] and even-party AME states, which also proves the existence of AME states for any number of parties given an appropriate choice of the system dimensions. This follows from the existence of pure state QSS schemes for any odd number of parties [28]. It should be mentioned that, while our parallel teleportation protocol is different from the aforementioned open-destination teleportation, it is also possible to implement open-destination teleportation with AME states [47]. Chapter 6. Absolutely maximal entangled state and quantum secret sharing104

6.2 Deﬁnition of AME states

An AME(n, d) state (absolutely maximally entangled state) of n qudits of dimension d, ⊗n ψ Cd , is a pure state for which every bipartition of the system into the sets B and | i ∈ A, with m = B A = n m, is strictly maximally entangled such that | | ≤ | | −

S(ρB) = m log2 d . (6.1)

Consequently, every partition of m local degrees of freedom is represented by a reduced density matrix proportional to the identity

1 n ρB = T rA ψ ψ = Idm , 1 m . (6.2) | ih | dm ≤ ≤ 2

n In practice, to detect an AME state it is suﬃcient to check that all the bn/2c possible bipartitions of n/2 qudits are maximally entangled, since all subsequent traces of the b c identity matrix are again identity matrices. A state is an AME state iﬀ it can be written as

1 X AME = k1 km φ(k) , (6.3) m B1 Bm A | i √d m | i · · · | i | i k∈Zd

0 with φ(k) φ(k ) = δkk0 , for every partition into m = B A = n m disjoint sets B h | i | | ≤ | | − and A. Two obvious examples of AME states are the Einstein-Rosen-Podolsky (EPR) and the Greenberger-Horne-Zeilinger (GHZ) states for two and three qubits, respectively. In both cases, the entanglement entropy is maximal for all their partitions. It has been proven that there are no absolutely maximally entangled states for four qubits [45]. AME states exist for ﬁve and six qubits [12], and a possible form for them will be given later in 7. No AME states exist for eight or more qubits [45, 67]. The existence of an AME(7, 2) state is still an open question, but it has been conjectured in Ref [12] that no such state exists. By increasing the party dimension, AME states can be found for these cases in which no qubit AME states exist. We remark, however, that, although we will show that for each n, AME(n, d) states exist for some appropriate choice of d, ﬁnding the conditions for the existence of AME(n,d) states, depending on n and d, is generally a non-trivial problem. In a future publication [47], we will show that, interestingly, a special class of AME states can be constructed from certain classical error correcting codes, namely those that satisfy the singleton bound [58]. Chapter 6. Absolutely maximal entangled state and quantum secret sharing105

A B A B

Figure 6.1: Parallel Teleportation scenarios of Theorem 20. Scenario (i) is on the left, and (ii) on the right. Parties in A perform joint quantum operations, parties in B only local quantum operations. c 2012 American Physical Society.

6.3 Parallel Teleportation

The maximal entanglement property of an AME(n, d) state for any bipartition into the sets A and B can be used to teleport quantum states between those two sets. In contrast to the teleportation scenario where A and B share a maximally entangled state that is not an AME state, in the AME scenario the sets A and B do not have to be specified when the state is created, but instead can be chosen after the AME state has been distributed. There are essentially three different ways in which the teleportation can be performed, depending on which parties can perform joint quantum operations, and which are separated and only able to perform local operations on their own quantum systems. In the first case, the parties within each set, A and B, are able to perform joint quantum operations. A standard teleportation of an arbitrary dm-dimensional state, where m = min( A , B ), can be performed in either direction. | | | | In the second case, the sending parties A can perform a joint quantum operation, but the parties in B are only able to perform local quantum operations. Additionally we require m = B A = n m. Then one qudit can be teleported from A to each of the | | ≤ | | − parties in B, and thus in total m qudits are teleported from A to B. This is illustrated in the left hand side of Figure 6.1. In the third and probably the most interesting case, the sending parties can only perform local operations, but the receiving parties can perform joint quantum operations. In this case, a teleportation is possible if the number of receiving parties is larger or equal n/2. Hence, sticking to our convention m = B A , we now consider a teleportation | | ≤ | | from B to A. See the right hand side of Figure 6.1 for an illustration. The first scenario is just a straightforward teleportation between maximally entangled parties. The second and third scenarios are presented in the following theorem. Chapter 6. Absolutely maximal entangled state and quantum secret sharing106

Theorem 20. Given an AME(n, d) state, and a bipartition of the n parties into the sets A and B such that m = B A = n m, then the following two parallel teleportation | | ≤ | | − scenarios are possible

(i) A can teleport one qudit to each party in B by performing a joint quantum operation and communicating two classical “dits" to each party in B. Each party in B can then locally recover their respective qudit with a local operation.

(ii) Each party in B can locally teleport one qudit to A. After receiving the measurement outcomes, consisting of two “dits” of classical information from each party in B, the parties in A are able to recover all m qudits by performing a joint quantum operation.

Proof. In both scenarios the parties in set A perform a joint quantum operation to transform the AME state into m d-dimensional EPR pairs. Then these pairs are used to teleport m qudits from the sending to the receiving parties. This is done by performing the joint unitary operation

UA φ(k) = k1 km 0 0 . (6.4) | iA | iA1 · · · | iAm | iA

on the initial AME(n, d) state

1 X Φ = k1 km φ(k) , (6.5) m B1 Bm A | i √d m | i · · · | i | i k∈Zd

0 with φ(k) φ(k ) = δkk0 . This results in the state h | i

UA Φ = Ψ Ψ 0 0 , (6.6) | i | iB1A1 · · · | iBmAm | iA

where Ψ = P i i are d-dimensional EPR pairs. These EPR pairs can now be used to | i | i | i teleport a qudit from Ai to Bi in case (i) (Bi to Ai in case (ii)). This requires Ai (Bi) to perform a generalized Bell measurement on her qudit and the qudit she wants to teleport,

and communicate the measurement result to Bi (Ai). This amounts to sending the classical information of two “dits" for each EPR pair. Upon reception of the measurement

result, Bi (Ai) can recover the teleported qudit by performing an appropriate unitary on his qudit. Chapter 6. Absolutely maximal entangled state and quantum secret sharing107

6.4 Quantum Secret Sharing.

The last teleportation scenario suggests a close relationship between AME states and quantum secret sharing (QSS) schemes [28]. In a QSS protocol [28, 42], a dealer encodes a secret S in a quantum state that is shared among n players in such a way that only special subsets of players are able to recover the secret. The set of all subsets that are able to recover the secret form the access structure and the set of all subsets that can gain no information about the secret form the adversary structure. If the encoded state is a pure state, we call it a pure state QSS scheme. We are only interested in pure state QSS schemes here. Additionally, we restrict our attention to threshold QSS schemes [28], which means that the access structure is formed by all sets that contain k or more number of parties, while any set with less than k parties cannot obtain any information about the secret. Thus k is the threshold number of parties required to recover the secret. Such a QSS scheme is denoted as a ((k, n)) threshold QSS scheme. For pure state threshold QSS schemes, n and k are always related by n = 2k 1. − To see the relation between AME states and threshold QSS schemes, we consider an AME(2m, d) state with an even number of parties and divide the parties into two sets

A = A1,...,Am and B = D,B1,...,Bm−1 of equal size m. In set B we have singled { } { } out one party D, which will act as the dealer of the QSS scheme. Now we perform the protocol of 20 (ii), but only D B performs the final teleportation operation. Also ∈ note that the unitary operation in 6.4 and the Bell measurement by the dealer commute. Thus, D can first perform her Bell measurement, effectively encoding the teleported qudit onto the residual AME state, from which it can be recovered by the players in A. Furthermore, instead of the bipartition into the sets A and B, we could have equally well chosen any other bipartition into two sets A0 and B0 of cardinality m with D B0. ∈ Then, without changing the operations that D has to perform, the parties in A0 are able to recover the teleported qudit (see 6.2 for an illustration). Thus, any set of at least m of the residual 2m 1 parties without D can recover − the teleported state, given that the measurement outcome is broadcasted to all parties. Furthermore, the no-cloning theorem guarantees that any set of less than m players cannot gain any information about the state [42]. Hence we accomplished to construct a ((m, 2m 1)) threshold QSS scheme from an AME(2m, d) state. − Before stating the theorem that formulates this observation concisely, we shortly re- P view how a QSS protocol works. A secret of dimension d, S = ai i , is encoded P | i | i into the state ai Φi which is shared by the players such that each authorized set can | i Chapter 6. Absolutely maximal entangled state and quantum secret sharing108

D D D B/D

A0 A00 A

B0/D B00/D

Figure 6.2: (Color online) After D (blue) performs her teleportation operation, any set of m parties (red), A, A0, A00 etc., can recover the teleported state. Any set of parties with m 1 or less parties (any set consisting only of green parties) cannot gain any information− about the teleported state. c 2012 American Physical Society.

deterministically recover S from its reduced state, while the reduced state of unautho- | i rized sets is independent of the encoded secret. We call Φi the basis states of the QSS | i scheme, and we show in [47] that they are AME states for pure state threshold QSS schemes with equal share and dimension size.

Theorem 21. There is a one to one correspondence between an AME(2m, d) state and a pure state ((m, 2m 1)) threshold QSS scheme, whose share and secret dimensions are − d.

Proof. AME to QSS: For any bipartition into parties A = A1,...,Am and B = { } D,B1,...,Bm−1 , the AME(2m, d) states has the form { } 1 X Φ = i k1 km−1 φ(i, k) , m D B1 Bm−1 A | i √d m | i | i · · · | i | i (i,k)∈Zd

0 with φ(k, i) φ(k , j) = δkk0 δij. We deﬁne the QSS basis states h | i

Φi = √d D i Φ | i h | i 1 X = k1 km−1 φ(k, i) . (6.7) m−1 B A √d m−1 | ··· i | i k∈Zd

A secret encoded as X X a = ai i ai Φi , (6.8) | i | i → | i Chapter 6. Absolutely maximal entangled state and quantum secret sharing109 satisﬁes the requirement of a threshold QSS scheme, because the parties B have a completely mixed states, independent of the encoded secret. Additionally, the set A, which can be chosen to be any set of n players, can restore the secret a by performing the | i joint unitary operation

UA φ(k, i) = k1 km−1 i . (6.9) | iA | iA1 · · · | iAm−1 | iAm

QSS to AME: For any bipartition into m authorized parties A = A1,...,Am and { } m 1 unauthorized parties B = B1,...,Bm−1 , the AME basis states of the QSS scheme − { } can be written in the form

1 X Φi = k1 km−1 φ(k, i) , m−1 B1 Bm−1 A | i √d m−1 | i · · · | i | i k∈Zd

0 where φ(k, i) φ(k , i) = δkk0 , because the states are AME states, and φ(k, i) φ(k, j) = h | i h | i δij, because the authorized parties can recover the secret deterministically. Thus,

0 φ(k, i) φ(k , j) = δkk0 δij. (6.10) h | i From these basis states, deﬁne the state

1 X Φ = i Φi | i √d | i | i i∈Zd 1 X = i k1 km−1 φ(k, i) . m D B1 Bm−1 √d m | i | i · · · | i | i (i,k)∈Zd

Because of Equation (6.10), Φ is a maximally entangled state with respect to the bi- | i partition B D vs. A. Since the original bipartition into A and B was arbitrary, Φ ∪ { } | i is maximally entangled with respect to any bipartition into two cardinality m sets and thus is an AME(2m, d) state.

Since it is known that ((m, 2m 1)) threshold QSS scheme exist for any number of − m and an appropriate choice of d [28], 21 proves the existence of AME states for any number of parties.

Example 7. In this example, we show how the ﬁve qubit code can be used to construct AME(5, 2) and AME(6, 2) states. From the ﬁve qubit code a ((3, 5)) threshold QSS scheme Chapter 6. Absolutely maximal entangled state and quantum secret sharing110 can be constructed [28]. The corresponding basis states are

1 0L = ( 00000 + 10010 + 01001 + 10100 | i 4 | i | i | i | i + 01010 11011 00110 11000 | i − | i − | i − | i (6.11) 11101 00011 11110 01111 − | i − | i − | i − | i 10001 01100 10111 + 00101 ), − | i − | i − | i | i 1 1L = ( 11111 + 01101 + 10110 + 01011 | i 4 | i | i | i | i + 10101 00100 11001 00111 | i − | i − | i − | i (6.12) 00010 11100 00001 10000 − | i − | i − | i − | i 01110 10011 01000 + 11010 ). − | i − | i − | i | i These states are AME(5, 2) states as required. Following the receipe of 21, we obtain the AME(6, 2) state

1 Φ = [ 0 0L + 1 1L ] | i √2 | i | i | i | i 1 = [ 000 ( + + + + ) 4 | i | − i |− −i + 001 ( + + + + ) | i − | − −i |− i + 010 ( + + + ) | i | −i − |− − i (6.13) + 011 ( + + + ) | i − | i − |− − −i + 100 ( + + + + ) | i − | i |− − −i + 101 ( + + + ) | i − | −i − |− − i + 110 ( + + + ) | i − | − −i − |− i + 111 ( + + + + )]. | i − | − i |− −i 6.5 Conclusion

In this chapter, we have introduced AME states, a class of highly entangled states, for n qudits shared among n locally separated parties. Previous investigations of maximal entanglement showed that AME states do not exist when the number of qubits is eight or larger. Here we proved the existence of AME states for any number of parties with the appropriate qudit dimension. Moreover, we have shown how they can be utilized in diﬀerent parallel teleportation scenarios, which require some parties to perform joint quantum operations, while others’ capabilities may be restricted to local operations. In Chapter 6. Absolutely maximal entangled state and quantum secret sharing111 those scenarios the advantage of AME states over less entangled states like a collection of EPR pairs lies in the fact that the partition into senders and receivers may be chosen after the state has been distributed. Furthermore, we have investigated the relationship of AME states with QSS schemes and established a one-to-one correspondence between even party AME states and pure state threshold QSS schemes. This correspondence allows us to prove the existence of AME states for any number of parties with the appropriate dimension. We also want to point out that some of the work presented in this chapter was previously known as folklores in the quantum error correction community, see chapter 7 of Preskill’s quantum computation lecture notes [66], and the problem sets in Gottesman’s quantum error correction lecture notes [43]. In future work we further explore this very intuitive approach to develop new communication protocols from AME states as well as extending the range of QSS schemes that can be derived from AME states. For instance, instead of assigning the role of the dealer to only one of the parties in the AME state, we can imagine multiple dealers who encode independent secrets onto the residual AME states, resulting in QSS schemes with more involved access structures. The established connection to QSS schemes also conﬁrms a relation between AME states and quantum error correction codes that was already suggested in Ref. [68]. A better understanding of this relation will allow us to construct new quantum error correction codes from AME states as well as deriving AME states from already known quantum codes. This might also shed light upon the open question of existence of AME states for a given number of parties and system dimension. Chapter 7

Conclusion

We have worked on various topics related to quantum information theory, especially on the entanglement properties of multipartite systems, such as the transformation rules between different pure multipartite quantum states, investigation of new entanglement monotones, and special classes of multipartite entanglement states. Regarding the LOCC transformation probabilities between different pure multipartite quantum states, we firstly worked on the transformation between GHZ class states. By exploiting the feature of the unique expression of GHZ class state, we get an upper bound for the transformation probability. The result is also generalized into more parties and higher dimensions. Noticing the importance of unique expression for the multipartite pure states during our analysis, we investigated the transformation between W type states. In previous work [52], an upper bound for the transformation probability was given. However, the condition required for the upper bound to be achieved is not clear, and the transformation probability when the upper bound cannot be achieved was not discussed either. We discovered a necessary and sufficient condition for the upper bound in [52] to be achieved [31]. Also, we proposed an LOCC protocol for the transformation when the upper bound can not be achieved, which provides a lower bound for the transformation probability. After that, we generalize the novel random distillation protocol into general W type state. By analyzing the change of the state under two-outcome weak measurements, the entanglement monotones that exactly correspond with various random distillation scenarios are identified. Also, when x0 = 0, the upper bounds derived from these entanglement monotones can be achieved, which means they are optimal. This result solves several open problems for random distillation. Besides that, the entanglement monotones discovered are the only analytic monotones that can be increased by SEP, which provides a better understanding of the difference between LOCC and SEP.

112 Chapter 7. Conclusion 113

To extend our knowledge of multipartite pure states into more parties, we studied a new type of entanglement states, namely the Absolutely Maximal Entangled (AME) states, which is deﬁned as the multipartite pure quantum state that is maximally entangled under any bipartition. The closely relationship between AME states and Quan- tum secret sharing protocols and Quantum error correction codes is established, which connects with diﬀerent aspects of quantum information science. Also, the practical application of AME states is explored. AME states are proved to be useful in parallel teleportation, which is a novel quantum communication protocol.

7.1 Future work

All of the above projects can be extended further. For the LOCC transformation between GHZ class states, one of the open problems is the optimal transformation probability between any two GHZ class states. In [71], the necessary and suﬃcient condition for deterministic LOCC transformation between any two three-qubit pure states is given. However, the optimal transformation probability is still unknown, and we know that there is still a gap between the upper bound and lower bound we have. One potential direction would be to construct more eﬃcient transformation protocols and check whether the corresponding successful probability could be an entanglement monotone. The open problem regarding W class state is more straightforward. When the coef-

ficient x0 is zero for the initial state, the transformation rule between any two W class states turns out to be very explicit. However, whenever x0 is nonzero, the transformation becomes very complicated. One possible protocol would be to discard the x0 term with a non-deterministic measurement in the first step and treat the following steps as a transformation from a W class state with x0 = 0 into another W class state. By using this protocol, we could get a lower bound for the transformation probability with an analytic expression. However, this lower bound is shown not to be optimal in general. By analyzing the random distillation problem, a new upper bound (which nevertheless has a gap with the lower bound) was discovered. Regarding the random distillation project, there are several open problems. Firstly, since most of the random distillation results are based on the initial state as a W type state, whenever x0 is nonzero, we do not know the optimal transformation probability. Secondly, the generalization of random distillation into more general cases is also interesting, and consists of the generalization into GHZ class states and higher dimension systems. Thirdly, there are still several open cases where LPO does not give an entanglement monotone. Also, when the number of parties grows, it becomes harder to check Chapter 7. Conclusion 114 whether one quantity is an entanglement monotone. Finally, up to now, we discuss random distillation under the restriction of pure states. What will the result be for mixed states and multi-copy cases? For the multi-copy case where N is asymptotically large, one special case where the distillation configuration is between one specific party and all the other parties (which is called entanglement combing) is solved in [82]. But for a general random distillation configuration, the optimal result is still unclear. For AME states, there are two main open problems regarding theoretical clarification and practical application respectively. Firstly, in the theoretical part, since AME states are closely related to quantum secret sharing and quantum error correction code, one natural question is what the relation is between AME states and Graph states, which is also closely related to quantum error-correction code. Regarding qubit system, it is not hard to check that all AME states are graph states. However, it is still not clear whether the same condition holds for higher dimensions. Secondly, the interesting entanglement property captured by AME states leads to several interesting quantum communication protocols. But how can these be implemented in experiment? Specifically, how is one to prepare these AME states in experiment? Also, the existence of other valuable quantum information protocols based on AME states is still under investigation.

7.2 Concluding words

In all, we worked on various aspects of multipartite pure quantum states. Our work shows that multipartite entanglement has a much richer mathematical structure than bipartite entanglement and could be more powerful for novel quantum information protocols. Though a complete solution of the problems related to multipartite entanglement is still lacking, using two-outcome weak measurement to detect entanglement monotones has been proven to be an eﬀective tool. Also, focusing on some speciﬁc types of multipartite entangled states is another way to simplify this problem and discover new protocols. In all, we believe that the investigation of multipartite entanglement problem is a rewarding process that could lead to new understanding of quantum information. Chapter 8

Appendix

8.1 Appendix: Proof of Theorem 5

Proof: From the formula of the two quantity:

I = 2cαcβ cγ sδcδcϕ (8.1) 1+2cαcβ cγ sδcδcϕ 2 2 2 2 2 2 2 4cαcβ cγ sδcδcϕ I = 2 (8.2) (1+2cαcβ cγ sδcδcϕ) 2 2 2 2 2 4sαsβ sγ sδcδ τABC = 2 (8.3) (1+2cαcβ cγ sδcδcϕ) (8.4)

We have

2 2 2 2 sαsβ sγ τABC = I 2 2 2 2 cαcβ cγ cϕ 2 2 2 2 (1−cα)(1−cβ )(1−cγ ) = I 2 2 2 2 (8.5) cαcβ cγ cϕ

We consider the condition when I > 0 at ﬁrst. √ 2 Firstly, we consider the condition when sδ = cδ = 2 , cϕ = 1. In this case, we have I I I 1 3 3 cαcβcγ = 1−I . let 1−I = a where a = ( 1−I ) . Then we have

2 2 2 2 (1−cα)(1−cβ )(1−cγ ) τABC = I 2 2 2 cαcβ cγ 6 (1−c2 )(1−c2 )(1− a ) α β c2 c2 2 α β = I a6 (8.6)

Take partial derivation of cα and cβ we can ﬁnd this expression reaches its maximum

115 Chapter 8. Appendix 116

value when cα = cβ = cγ = a and the corresponding maximum value of 3-tangle is 2 2 (1−a2)3 3 3 3 τABC = ((1 I) I ) = 3 2 . 0 − − (1+a ) Now we will show in other cases when sδ = cδorcϕ < 1, we can only get a 3-tangle 6 smaller than τABC0 . 1 If sδ = cδ, we will have sδcδ < 2 , then from the expression of I we can ﬁnd cαcβcγcϕ > I 63 3 3 = a , then as cϕ 1, we also have cαcβcγ = b > a . And also take the partial 1−I ≤ derivation of (1 c2 )(1 c2 )(1 c2 ) we can ﬁnd its maximum value is (1 b2)3 < (1 a2)3. − α − β − γ − − Finally we have

2 2 2 2 (1−cα)(1−cβ )(1−cγ ) τABC = I 2 2 2 2 cαcβ cγ cϕ 2 (1−a2)3 (1−a2)3 < I a6 = (1+a3)2 = τABC0 (8.7)

That is to say, when sδ = cδ, τABC is always smaller than τABC . Now let us consider √ 0 26 the case when sδ = cδ = 2 , but cϕ < 1. I 3 Then again we have cαcβcγcϕ = 1−I = a . But as cϕ < 1, we still have cαcβcγ = d3 > a3. And also take the partial derivation of (1 c2 )(1 c2 )(1 c2 ) we can ﬁnd its − α − β − γ maximum value is (1 d2)3 < (1 a2)3. So we have − −

2 2 2 2 3 2 (1−cα)(1−cβ )(1−cγ ) 2 (1−d ) τABC = I 2 2 2 2 = I 6 cαcβ cγ cϕ a 2 (1−a2)3 (1−a2)3 < I a6 = (1+a3)2 = τABC0 (8.8)

Then we show, for the interference term I > 0, we have

(1 a2)3 max(τABC (φ I(φ) = I > 0)) = − (8.9) | (1 + a3)2

When I 0, the discussion is almost the same. Except that, we need to consider the √ ≤ 2 I 03 condition sδ = cδ = 2 , cϕ = 1 first and find cαcβcγ = 1−I = a > 0. Then easy to − (1−a02)3 − find the corresponding maximum value is (1−a03)2 . And use the same tricks one can show it is the maximum value of the 3-tangle. One thing to notice is that, the expression of a’ and a is just opposite to each other. So if we let a = I = a0 when I 0, we will get 1−I − ≤ (1 a02)3 (1 a2)3 max(τABC (φ I(φ) = I 0)) = − = − (8.10) | ≤ (1 a03)2 (1 + a3)2 − Chapter 8. Appendix 117

Then in all we have

(1 a02)3 (1 a2)3 max(τABC (φ I(φ) = I) = − = − (8.11) | (1 a03)2 (1 + a3)2 − .

8.2 Appendix: proof of Theorem 14

We consider case-by-case measurements in which each party acts according to Eq 5.7.

The function τ transforms as τ τλ for λ 1, 2 , and we are interested in the average → ∈ { } change: τλ = p1τ1 +p2τ2. By the universality of weak measurements [9] [63], it is suﬃcient

to prove ÏĎ monotonic in the weak measurement setting, i.e., with (a1, c1, a2, c2) in some neighborhood of (1/2,1/2,1/2,1/2).We consider three cases.

Case I, 0 . First consider when party performs a measurement. We can xn1 > xn1 n1 0 assume the measurement is weak enough such that n1, n1, p, and p’ are the same for both premeasurement and postmeasurement states. Consider the measurement outcome

λ 0, 1 with aλ > cλ. Then ∈ { } 2 0 0 aλ 1 aλ xn1 xpxp pλτλ = 2aλxp + 2aλxp0 2 (1 ) . (8.12) − cλ − 3 cλ xn1 xn1

∂τλ We have a =1/2,c =1/2 0 which implies that τ τλ will be minimized by the choice ∂cλ | λ λ ≥ − c1 + c2 = 1. Then writing c c1 < a a1, we have that ≡ ≡

2 2 3 3 0 0 0 a (1 a) xpxp 2 a (1 a) xpxp xn1 τ τλ = 2( 1 + + − ) + (1 2 − 2 ) 2 ) 0. (8.13) − − c (1 c) xn 3 − c − (1 c) x ≥ − 1 − n1 Expanding this expression about the point (1/2,1/2) to second order gives

0 0 0 0 1 2 1 2 1 1 xn1 xpxp 2 xn1 xpxp τ τλ 8[(a ) +(c ) 2(a )(c )](1 ) = 8(a c) (1 ) 0. − ≈ − 2 − 2 − − 2 − 2 − xn1 xn1 − − xn1 xn1 ≥ (8.14)

Now consider when the other parties measure. Since the coeﬃcient of xp is non-negative, 0 the monotonicity of τ when party p measures follows from the K-T monotones. For n1 and p’, there are two possibilities.

Subcase, 0 0 . Here we can assume the measurements are weak enough such that xn1 > xp Chapter 8. Appendix 118

their ordering does not change. Then since the coeﬃcients of 0 and 0 are non-negative, xp xn1 the K-T monotones imply the monotonicity of τ.

x x2 xpxp0 2 p p0 Subcase, 0 = 0 . We have = 2 + 2 0 2 + . It is easy to see that xn1 xp τ xp xp x 3 x2 − n1 n1 again τ τλ is minimized when c1 +c2 = 1. So parameterizing the measurement by a and − c with a > c, we have that the average change in xn0 is (a + 1 c)xp0 while the average 1 − change in xp0 is (1 a + c)xp0 . It follows that −

xpxp0 τ τλ = 2xp0 (a c) 2(a c) 0. (8.15) − − − − xn1 ≥

xp0 x 0 2 n1 Case II, xn1 = xp. Here, τ = 2xp + . When either party n1 or p measures 3 xn1 aλ cλ aλ with aλ > cλ, the new components are xλ,n = xp, xλ,p = xp, xλ,n0 = xp0 , and 1 pλ pλ 1 pλ aλ xλ,p0 = xn0 . Thus, pλ 1

0 0 2cλ xp xn1 = 2( ) 0 + 2 + (8.16) pλτλ aλ cλ xn1 cλxp . − 3 xp

∂τλ Since xp xn0 , we have a =1/2,c =1/2 0. Again, this means that τ τλ will be ≥ 1 ∂cλ | λ λ ≥ − minimized by the choice c1 + c2 = 1. Taking a > c, we have that

2 3 (1 a) 2 (1 a) xpxp0 = 2[1 (1 )] 2(1 ) 0 + (1 ) (8.17) τ τλ c a xp c − xn1 c − 2 , − − − − − − − 1 c 3 − − (1 c) xn0 − − 1 which to ﬁrst order about the point (1/2,1/2) takes the form

2 0 ( 0 ) xpxp xnp xn1 2(a c)(xp 2xn0 + ) 2(a c) − 0. (8.18) 1 0 − − xn1 ≥ − xnp ≥

If either 0 or 0 measures, then the monotonicity of follows from the K-T monotones. xp xn1 τ

4 xpxp0 0 Case III, = 0 . We have = 2 + 2 0 . When either party or xn1 xn1 τ xp xp 3 x n1 n1 − n1 measures, parties p and p’ remain the same. With aλ > cλ, we have

xpxp0 2 xpxp0 pλτ = 2aλxp + 2aλxp0 2aλ + cλ , (8.19) − xn1 3 xn1

∂τλ which has a =1/2,c =1/2 0. So again we assume c1 = 1 c2 c < a and we ﬁnd that ∂cλ | λ λ ≥ − ≡

2 3 2 (1 a) xpxp0 2 (1 a) xpxp0 τ τλ = 2( a − ) (c + − 2 ) , (8.20) − − 3 − − 1 c xn − 3 (1 c) xn − 1 − 1 Chapter 8. Appendix 119 which to third order about the point (1/2,1/2) takes the form

0 8 1 1 1 1 1 1 x x 0 8 xnp xn [(a )3 (c )3] 8[(c )(a )2 (a )(c )2] p p = (a c)3 p 0. 3 − 2 − − 2 − − 2 − 2 − − 2 − 2 xn1 3 − xn1 ≥ (8.21)

Finally, since the coeﬃcients of xp and xp0 are positive in τ, by the K-T monotones, τ is monotonic when either of these parties measures.

8.3 Dual solution to W distillation by SEP | N i

We begin by writing Eqs. 5.51 and 5.52 in standard semideﬁnite programming form. Fix some encoding function φ:E |E| and deﬁne the matrices: →

        1 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0         0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 F1 =     ,F2 =     , 0 0 0 0 ⊕ 0 0 0 0 0 0 0 0 ⊕ 0 0 0 0         0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0         0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 1         0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 F3 =     ,F4 =     , 1 0 0 0 ⊕ 1 0 0 0 0 0 0 0 ⊕ 0 0 0 0         0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0         0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0         0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 F5 =     ,F6 =     , 0 0 0 0 ⊕ 0 0 0 0 0 0 0 1 ⊕ 0 0 0 1         0 1 0 0 0 1 0 0 0 0 1 0 0 0 1 0     0 0 0 0 0 0 0 0     0 0 0 0 0 0 0 0 F7 =     , 0 0 0 0 ⊕ 0 0 0 0     0 0 0 1 0 0 0 1 (8.22) Chapter 8. Appendix 120

|E| φ(i,j)−1 |E| (ij) M M M G = [ 1] [0] [0]4×4 F1 [0]4×4, 1 − ⊕ k=1 k=1 k=φ(i,j)+1 |E| φ(i,j)−1 |E| (ij) M M M G = [0] [0] [0]4×4 F2 [0]4×4, 2 ⊕ k=1 k=1 k=φ(i,j)+1 |E| φ(i,j)−1 |E| (ij) M M M G = [0] [0] [0]4×4 F3 [0]4×4, 3 ⊕ k=1 k=1 k=φ(i,j)+1 (8.23) ··· φ(i,j)−1 |E| φ(i,j)−1 |E| (ij) M M M M G = [0] [0] [ 1] [0] [0]4×4 F7 [0]4×4, 7 ⊕ − ⊕ k=1 k=φi,j+1 k=1 k=φ(i,j)+1

   Npij  0 0 0 0 0 0 0 2 |E| |E|  Npij Npij   Npij  M M 0 2 2 0  0 2 0 0  G0 = [1] [1]     . 0 Npij Npij 0 ⊕  0 0 Npij 0  k=1 φ(i,j)=1  2 2   2  Npij 0 0 0 0 2 0 0 0

Then Eqs. 5.51 and 5.52 are captured by the existence of x(ij) C such that k ∈

7 X X (ij) (ij) G0 + x G 0, (8.24) m m ≥ (i,j)∈E m=1 with the additional constraints that

X Npij 1 for 1 k N. (8.25) 2 ≤ ≤ ≤ (i,j)∈Ek The dual problem to this asks

(i,j) (i,j) max tr(ZG0), s.t. 0 = tr(ZG ) for all G ,Z 0. (8.26) − m m ≥

A critical relationship between the dual and the primal formulations is that if (8.24) (ij) can be satisfied for some xk , then for any Z satisfying the constraints of (8.24), we must have tr(ZG0) 0. Thus infeasibility is proven by the existence of some Z 0 such that (ij) ≥ (i,j) ≥ tr(ZGm ) = 0 for all Gm and tr(ZG0) < 0. We construct a certificate for infeasibility as follows. For each (i, j) E, define the matrix ∈ Chapter 8. Appendix 121

φ(i,j)−1 2 2 |E| φ(i,j)−1 (ij) 1 M N pij M M Z = [ ] [0] [ ] [0] [0]8×8 [0]4×4 E ⊕ 4 ⊕ | | k=1 k=φ(i,j)+1 k=1  −Np  1 0 0 ij 2 (8.27)   |E|  0 0 0 0  M   [0]8×8. ⊕  0 0 0 0  k=φ(i,j)+1  2 2  −Npij N pij 2 0 0 4 The claim is that the matrix

X Z := Z(ij) (8.28) (i,j)∈E

N 2 P 2 is dual feasible with tr(ZG0) < 0 whenever 4 (i,j) Epij > 1. Indeed, it can easily (ij) ∈ be seen that Z 0 and tr[ZGm ] = 0 for 1 m 7 and (i, j) E. And ﬁnally, ≥ ≤ ≤ ∈ 2 2 N X 2 N X 2 tr[ZG0] = 1 + p p < 0. (8.29) 4 ij − 2 ij (i,j)∈E (i,j)∈E

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