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Multipartite entangled quantum states: Transformation, Entanglement monotones and Application by Wei Cui A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Physics University of Toronto c Copyright 2013 by Wei Cui

Multipartite entangled quantum states: Transformation ......Multipartite entangled quantum states: Transformation, Entanglement monotones and Application by Wei Cui A thesis submitted

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  • Multipartite entangled quantum states: Transformation,Entanglement monotones and Application

    by

    Wei Cui

    A thesis submitted in conformity with the requirementsfor the degree of Doctor of Philosophy

    Graduate Department of PhysicsUniversity of Toronto

    c© Copyright 2013 by Wei Cui

  • Abstract

    Multipartite entangled quantum states: Transformation, Entanglement monotones andApplication

    Wei CuiDoctor of Philosophy

    Graduate Department of PhysicsUniversity 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 multi-

    partite 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 multi-

    partite 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 probabil-

    ity 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 ran-

    dom distillation, which transforms multipartite entanglement into bipartite entanglement

    ii

  • shared by a non-deterministic pair of parties. We find upper bounds for the random dis-

    tillation 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 en-

    tanglement 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, hisendless help and inspired discussion guided me to explore the fantastic world of quantuminformation 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 howto do presentation, how to improve my English, and more importantly, how to treat andwork with other people. All the above and his kind help on my nonacademic life will bevaluable 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 col-leagues. I want to show my acknowledgement to Eric Chitambar, Wolfram Helwig, BingQi, Christian Weedbrook, Xiongfeng Ma, Benjamin Fortescue, Yi Zhao, Yuemeng Chi,Viacheslav Burenkov, Feihu Xu, Kero Lau, Zhiyuan Tang, Felix Liao, and He Xu. Specialthanks 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, JonathanOppenheim, 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 familyhousing of the University of Toronto is also acknowledged. With their help, I had areally harmonious life with my family while studying in the University of Toronto as aninternational student.

    Finally, the love and support from my family is greatly appreciated. This thesis isdedicated to my parents, my wife Bilian, and my lovely son Stephen.

    iv

  • Contents

    1 Introduction 11.1 Our results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

    1.1.1 List of papers and presentations . . . . . . . . . . . . . . . . . . . 4

    2 Background Information 62.1 Entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

    2.1.1 Entanglement in quantum physics . . . . . . . . . . . . . . . . . . 62.1.2 Entanglement in Hilbert space . . . . . . . . . . . . . . . . . . . . 72.1.3 Entanglement as a resource . . . . . . . . . . . . . . . . . . . . . 8

    2.2 Quantum operations and entanglement measures . . . . . . . . . . . . . . 102.2.1 Quantum Operators . . . . . . . . . . . . . . . . . . . . . . . . . 102.2.2 Local Operators and Classical Communications . . . . . . . . . . 112.2.3 Separable operators . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2.4 Entanglement measures for pure bipartite states . . . . . . . . . . 142.2.5 Entanglement measures for mixed states . . . . . . . . . . . . . . 17

    2.3 Multipartite entangled pure states . . . . . . . . . . . . . . . . . . . . . . 192.3.1 Tripartite entangled states . . . . . . . . . . . . . . . . . . . . . . 192.3.2 W type entangled states . . . . . . . . . . . . . . . . . . . . . . . 21

    3 LOCC transformation bounds between multipartite pure states 233.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.2 Upper Bound for the Conversion from GHZ state to a GHZ class state . 253.3 Failure Branch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

    3.3.1 Conservation of interference term . . . . . . . . . . . . . . . . . . 313.3.2 Conservation of normalization . . . . . . . . . . . . . . . . . . . . 32

    3.4 Upper Bound for a general case . . . . . . . . . . . . . . . . . . . . . . . 353.4.1 interference term and the maximal value of the 3-tangle of a GHZ-

    class state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

    v

  • 3.4.2 "stop and reconstruct" procedure . . . . . . . . . . . . . . . . . . 363.4.3 Example: |GHZ〉 → |φ〉 = γ(|000〉+ |aaa〉) . . . . . . . . . . . . . 383.4.4 general case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

    3.5 Lower Bound for the Transformation . . . . . . . . . . . . . . . . . . . . 483.6 Summary and Concluding Remarks . . . . . . . . . . . . . . . . . . . . . 56

    4 Optimal entanglement transformations among N-qubit W-type states 574.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.2 Upper bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.3 Lower bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.4 General Features of symmetric transformations . . . . . . . . . . . . . . . 664.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

    5 Random distillation for W type states 695.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695.2 Previous results and notation . . . . . . . . . . . . . . . . . . . . . . . . 74

    5.2.1 The generalized Fortescue-Lo protocol . . . . . . . . . . . . . . . 745.2.2 Additional notation and the Kintas-Turgut monotones . . . . . . 75

    5.3 The least party out protocol . . . . . . . . . . . . . . . . . . . . . . . . . 765.3.1 Phase I: Remove x0 component . . . . . . . . . . . . . . . . . . . 765.3.2 Phase II: Equal or vanish (e/v) subroutine . . . . . . . . . . . . . 775.3.3 Phase III: Obtaining EPR pairs . . . . . . . . . . . . . . . . . . . 77

    5.4 Main results: The LPO protocol on multipartite W type states . . . . . . 815.4.1 Summary of results . . . . . . . . . . . . . . . . . . . . . . . . . . 815.4.2 Three qubits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 815.4.3 Four qubits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 835.4.4 n qubits and the entanglement monotones . . . . . . . . . . . . . 895.4.5 Interpretation of monotones . . . . . . . . . . . . . . . . . . . . . 92

    5.5 SEP VS LOCC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 935.5.1 Random distillation by Separable transformations . . . . . . . . . 935.5.2 Comparison between SEP and LOCC . . . . . . . . . . . . . . . . 96

    5.6 Applicaiton to the transformation |φ〉1,··· ,N → |WN〉 . . . . . . . . . . . . 985.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

    5.7.1 Open questions and concluding remarks . . . . . . . . . . . . . . 99

    6 Absolutely maximal entangled state and quantum secret sharing 1026.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

    vi

  • 6.2 Definition of AME states . . . . . . . . . . . . . . . . . . . . . . . . . . . 1046.3 Parallel Teleportation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1056.4 Quantum Secret Sharing. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1076.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

    7 Conclusion 1127.1 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1137.2 Concluding words . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

    8 Appendix 1158.1 Appendix: Proof of Theorem 5 . . . . . . . . . . . . . . . . . . . . . . . 1158.2 Appendix: proof of Theorem 14 . . . . . . . . . . . . . . . . . . . . . . . 1178.3 Dual solution to |WN〉 distillation by SEP . . . . . . . . . . . . . . . . . 119

    Bibliography 122

    vii

  • List of Figures

    2.1 Structure of states that can be obtained from W3 state by SLOCC. Thefirst level is the true W3 type state which is also the genuine W classstate. The second level are bipartite entangled states, such as (AB)-C(|ψ〉AB |φ〉C), (AC)-B (|ψ〉AC |φ〉B) and (BC)-A (|ψ〉BC |φ〉A), and the thirdlevel is the product state |φ1〉A |φ2〉B |φ3〉C . . . . . . . . . . . . . . . . . . 22

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

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

    3.3 The value of pU as a function of a. In this figure, a = ( yy−1)

    13 . So when a

    goes from 0 to 1, y goes from 0 to ∞. Note that as y goes to infinity, agoes to 1. We express the value as a function of a because it will be easierfor us to combine different graphs into one graph later. c©2010 AmericanPhysical Society. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

    3.4 "stop and reconstruct" for a two-outcome measurement. c©2010 AmericanPhysical Society. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

    3.5 The original protocol written in the many two-outcome measurementsform. c©2010 American Physical Society. . . . . . . . . . . . . . . . . . . 37

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

    3.7 The new protocol, which can reconstruct the original one. c©2010 AmericanPhysical Society. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

    3.8 the relation between p̄s and p̄τABC . c©2010 American Physical Society. . . 43

    3.9 The upper bound for the transformation . . . . . . . . . . . . . . . . . . 44

    3.10 upper bound of transformation probability from |φ〉 to |ψ〉. . . . . . . . . 47

    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 branchtraversing edges e(1,1) to e(n,1) is a success branch, while the branch frome(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 oneparty 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 configurationgraph G = (V,E) whose edge set encodes the target pairs. Here, the targetpairs 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 byFortescue and Lo in [39]. Such a transformation is a success if any two par-ties become EPR entangled, and this can be achieved with a probabilityarbitrarily close to 1. Previous research has not considered more generaltypes 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 thistransformation is 2/3, thus resolving an open problem in Refs. [38]. Theinitial state is |W4〉 = 1/2(|1000〉 + |0100〉 + |0010〉 + |0001〉). c©2011American Physical Society. . . . . . . . . . . . . . . . . . . . . . . . . . . 72

    ix

  • 5.4 Equal or vanish subroutine (Phase II) for the normalized state 11+3α

    (α, α, α, 1)

    and the configuration graph with edges {AB,AC,AD,BC}. 1. David’scomponent is largest and Alice is a connected party to him with a lessercomponent value. She performs an e/v measurement. 2. For the outcome"vanish" (right branch) she is separated from the system, and since Davidis not connected to either Bob or Charlie, he immediately removes himselffrom the system leaving |ψ(BC)〉 with some probability 1. For the outcome"equal" (left branch) the components of all other parties receive a factorof α, and Alice’s component is now maximum equaling David’s. Bob is aconnected party to Alice with a lesser component value and he performsan e/v measurement. 3. Again, either Bob vanishes (right branch) or allother components except his receive a factor of α. In both cases, Charlieis then a connected party to Bob with a lesser component value and heperforms an e/v measurement. 4. The final outcome states along thesebranches are |W4〉, |W (ABD)3 〉, |W (ACD)3 〉, and |ψAD〉. c©2011 AmericanPhysical Society. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

    5.5 Phase III receives an input state W (S)|S| and a configuration graph G. Partyk performs an e/v measurement. One outcome is a standard W state withparty k removed, and the other is the state 1|S|pα (α, · · · , α, 1, α, · · · , α).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,and for any W state W (S

    ′)|S′| with |S ′| < |S|, the transformation success

    probability from this point onward is given by PIII(W(S′)|S′| , G\S̄ ′); this value

    is already known by recursion. However, for the state W (S)|S| , performingPhase III again will generate an indefinite loop, but one whose overallsuccess probability converges to f(α)

    1−α|S|−1 [see Eqs. 5.13 and 5.14]. c©2011American Physical Society. . . . . . . . . . . . . . . . . . . . . . . . . . . 80

    5.6 (Left) Configuration G∧. (Right) Configuration G∆. An upper bound onthe success probability is given by Eqs. 5.17 and 5.19, respectively, whichis effectively tight when x0 = 0. For |W3〉, these probabilties are 2/3 and1, respectively. c©2011 American Physical Society. . . . . . . . . . . . . . 82

    5.7 Let GI , G′I , and G”I be the first, second, and third of the above configura-tions, respectively. An upper bound on the success probability is given byEq. 5.20 which is effectively tight when x0 = 0. For |W4〉, this probabilityis 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

    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 probabilityis 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 AmericanPhysical Society. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

    5.10 Let GIV be the above configuration. We say two parties are edge comple-mentary if their nodes have a different number of connected edges. Forexample, A is edge complementary to both C and D. An upper bound onthe success probability is given by Eq 5.30 which is effectively tight whenx0 = 0. For |W4〉, this probability is 5/6. c©2011 American Physical Society. 85

    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〉,this probability is 1. c©2011 American Physical Society. . . . . . . . . . . 87

    5.12 Let GV I be the above configuration. For |W4〉, the LPO protocol gives asuccess probability of 1

    6(3 +√

    3).We conjecture this to be optimal. c©2011American 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 inwhich party n1 shares one-half of each EPR pair. Bottom: A "complete-type" distillation: when x0 = 0, κ(x) gives the optimal probability for arandom 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 be-tween any two parties as a function of s when the initial state is

    √s |100〉+√

    1−s2

    (|010〉+ |001〉). The LOCC probability is 2(1− s)− (1− s)2/4s. Agap of 12.5% exists between SEP and LOCC. c©2012 American PhysicalSociety. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

    5.15 LOCC vs SEP for the maximum probability of party 1 becoming EPRentangled as a function of N when the initial state is

    √12|10 · · · 0〉 +√

    12(1−N)(|010 · · · 0〉 + · · · + |0 · · · 01〉). The LOCC probability is 1 − (1 −

    1N−1)

    N−1. A gap of 37% exists between SEP and LOCC. c©2012 AmericanPhysical Society. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

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  • 5.16 The relative difference between the optimal separable operation and theLPO protocol. The configuration graph consists of N disjoint pairs. Sep-arable operations perform as PSEP =

    √1N

    whereas the LPO protocolobtains the rate of PLPO = 22N−1 . We conjecture that the LPO protocolis LOCC optimal for this configuration graph, as it is known to be whenN = 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 PhysicalSociety. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

    6.2 (Color online) After D (blue) performs her teleportation operation, anyset of m parties (red), A, A′, A′′ etc., can recover the teleported state. Anyset of parties with m − 1 or less parties (any set consisting only of greenparties) cannot gain any information about the teleported state. c©2012American 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 andcommunication protocols, has been experiencing rapid development since the 1980’-s.The famous EPR pair, which was expressed as a Bell state, |φ〉AB = 1√2(|00〉 + |11〉)AB,has been shown to be the resource for various quantum information protocols that areimpossible under classical physics. EPR state shared between parties A and B cannot bewritten as a direct product of two states of parties A and B respectively, which is nowthe most important feature of the so called entangled state.

    Other than EPR pairs, there are infinitely many entangled quantum states that canbe shared between several parties. It would be rewarding to investigate the entangle-ment property of these states and also their application in quantum information proto-cols. Also, under real experimental conditions, one cannot guarantee the generation ofa perfect EPR state, and it is necessary to analyze how one can transform a quantumentangled state into an EPR state that can be directly used in quantum information pro-tocols. Because of that, the quantification of entanglement for an entangled state and thetransformation property between two entangled states are two main research directionsin entanglement theory.

    Based on the fact that protocols employing EPR pairs are generally for nonlocalscenarios, the transformation between entangled states is restricted under the scenario oflocal operators and classical communications (LOCC). Under this scenario, theentanglement theory for a bipartite pure state, which is an entangled state shared betweentwo parties, is completely constructed. The EPR state is shown to be the most powerfulresource among pure states shared by two qubits since it could be transformed into anyother pure state shared by two qubits with probability 1. The optimal transformationprobability between any two bipartite states is also derived.

    1

  • Chapter 1. Introduction 2

    However, the corresponding results for higher dimensions and more parties, whichis called multipartite entangled state, are still lacking. The mathematical structureof multipartite entangled states turns out to be much more complicated than the caseof bipartite states. For example, tripartite states, which are the quantum states sharedbetween three parties, can be classified into two classes while states from different classescannot be transformed into each other with nonzero probability.

    Another question is the application of multipartite entangled states in novel quantumprotocols. Here the interest is on the protocols that could reveal the advantage of mul-tipartite entanglement over bipartite entanglement. That is to say, for some protocols,employing the multipartite entangled states directly would achieve a better result thanconverting the multipartite entangled state into EPR pairs and then using the EPR pairsfor the protocol.

    1.1 Our results

    During my Ph. D study, I mainly worked on the LOCC transformation probability be-tween pure multipartite entangled states. In the following I will provide a quick overviewof my work, which is also an outline of this thesis.

    In chapter 2, I will provide the background information on entanglement theory thatis related to my research.

    In chapter 3, upper and lower bounds on the transformation probability betweenmultipartite entangled states will be derived. The result is mainly related to the trans-formation between tripartite states while some of our results can be generalized into moreparties. There is still a gap between the upper bound and lower bound we found, whichfuture work will investigate. The result of this chapter was published in [33]. As thefirst author, I proposed the four-step-method and discovered the upper bound and lowerbound for the transformation probability.

    In chapter 4, another type of multipartite entangled state, the W-type state, is ana-lyzed. Based on the explicit form of a general W-type state given in [52], we derive thelower 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 resultof this chapter was published in [31]. As the first author, I proposed the LOCC trans-formation protocol and discovered the necessary and sufficient condition for the boundto be achieved.

    In chapter 5, the question regarding the conversion from multipartite entangled stateinto 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 par-ties? This protocol, called random distillation, was first proposed in [39]. Here we findupper bound for this type of transformation by discovering a new type of entanglementmonotones. One surprising result is that the entanglement monotones discovered can beincreased by separable operators (to be defined in chapter 2), which gives the first setof analytic entanglement monotones that can be increased by separable operators. Theresult of this chapter was published in [32][23][19]. As the first author of [32] and thesecond author of [23] and [19], I proposed the least party out protocol and discovered theentanglement monotones for three and four qubit systems.

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

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

    In chapter 8, the appendix, we provide the detailed proofs for some important theo-rems 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 onwhich little work has been done before ours. Our work sheds some light on the investi-gation of this problem by finding various upper and lower bounds. Secondly, instead oftrying to classify all types of multipartite entangled states, we focus on some specific typesof multipartite entangled states and analyze their properties and potential applications,which is shown to be very helpful. Finally, our work on random distillation demonstratesthat 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 transfor-mation between multipartite pure states, Physics Review A, 81, 012111, 2010

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

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

    4. Eric Chitambar, Wei Cui, Hoi-Kwong Lo, Increasing Entanglement by SeparableOperations 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 talkat 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-typestates, Phys. Rev. A 85, 062316, 2012

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

    Presentations

    1. August 2011, AQIS 11, Pusan National University, Pusan, Korea, "Randomly dis-tilling 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, UnitedStates 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 trans-formation between Multipartite quantum states", by W. Cui, W. Helwig, H.-K. Lo

    4. June 2010, CAP 2010, University of Toronto, presentation, "Bounds on the prob-ability 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 prob-ability 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 ofquantum entanglement related to our work. People who are familiar with the theory ofentanglement 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 math-ematical aspects.

    2.1.1 Entanglement in quantum physics

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

    Φ(x1, x2) =

    ∫ +∞−∞

    e(2πi/h)(x1−x2+x0)pdp. (2.1)

    For this state, the two particles have a fixed midpoint at x0 while their momentashould be opposite to each other. In this sense, the center of mass of the system ABwould 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 thecollapse 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 actuallydetermine the property of a physical system. The local hidden variable conjecture basedon this assumption was shown to be incorrect by the experimental violation of Bell’sinequality [5]. To this day, the interpretation of quantum mechanics remainsl an openproblem. However, the EPR state turns out to be a stronger resource than a classicalbit from an information theory perspective. In the following subsection we will introducethe 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 ofthe Hamiltonian, one can express the state in an n-dimensional Hilbert space. A quantumstate can be expressed as a density matrix in the corresponding Hilbert space. For apure state, the density matrix can be expressed as |φ〉 〈φ| while for a mixed state, whichis considered an ensemble of several quantum states, the density matrix is

    ∑i pi |φi〉 〈φi|.

    Here we have Trρ =∑

    i pi = 1.

    Example 1. Let us take spin as an example. Suppose there is an electron whose spincould be up or down. We can treat the Hilbert space of spin as {|↑〉 , |↓〉}. A pure state inthis space could be written as |φ〉 = a |↑〉+b |↓〉 (density matrix (a |↑〉+b |↓〉)(a∗ 〈↑|+b∗ 〈↓|)where |a|2 + |b|2 = 1. And a mixed state could be written as ρ = |a|2 |↑〉 〈↑| + |b|2 |↓〉 〈↓|where |a|2 + |b|2 = 1 also.

    Mathematically, one can also define |0〉 = |↑〉 and |1〉 = |↓〉. Thus the pure state andmixed state could be expressed as |φ〉 = a |0〉+ b |1〉 (density matrix (a |0〉+ b |1〉)(a∗ 〈0|+b∗ 〈1|)) and ρ = |a|2 |0〉 〈0|+ |b|2 |1〉 〈1|.

    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∑

    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 correspondingquantum 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 thetensor product of the two individual Hilbert spaces.

    H = HA ⊗HB (2.2)

  • Chapter 2. Background Information 8

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

    ρAB =∑i

    piρAi ⊗ ρBi , (2.3)

    where pi ≥ 0 and∑

    i pi = 1, it is called a separable state. Otherwise, it is calledentangled. The most important entangled state to consider for a bipartite system, sayparty A and B, is the EPR state

    |Φ〉AB =1√2

    (|00〉+ |11〉)AB. (2.4)

    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

    ρ1···n =∑i

    piρ1i ⊗ · · · ρni , (2.5)

    where pi ≥ 0 and∑

    i pi = 1, it is called a separable state. Otherwise, it is entangled.

    One thing to note is that, for a multipartite system, one can divide the parties intodifferent groups and talk about the entanglement property between these groups. Forexample, the state

    |Φ〉ABC =1√2

    (|00〉+ |11〉)AB |0〉C , (2.6)

    is an entangled state between A, B, and C. However, it is a separable state between thesystems (AB) and C.

    2.1.3 Entanglement as a resource

    Now let us explain the application of a qubit from a quantum information theory per-spective. We denote a classical bit as a cbit, which stands for either 0 or 1. Also, wedefine a two-level quantum mechanical system (a |0〉+b |1〉) as 1 qubit. Another resourceto 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. Thisprotocol is called superdense coding [11].

    Let us explain how the protocol works and its implication. In the beginning, Aliceand Bob share an EPR state, so that we have

    |φ〉AB =1√2

    (|00〉+ |11〉)AB. (2.7)

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

    00 : IA |φ〉AB =1√2

    (|00〉+ |11〉)AB

    01 : XA |φ〉AB =1√2

    (|10〉+ |01〉)AB

    10 : ZA |φ〉AB =1√2

    (|00〉 − |11〉)AB

    11 : XAZA |φ〉AB =1√2

    (|10〉 − |01〉)AB

    (2.8)

    where X, Z are Pauli matrices defined as

    X =

    (0 1

    1 0

    ), Y =

    (0 −ii 0

    ), Z =

    (0 1

    1 0

    ). (2.9)

    After that, Alice can send her qubit to Bob. Notice that the four resulting states areorthogonal to each other so that if Bob has the full copy of the state, he can identifywhich state it is via a Bell measurement. Thus Bob could recover the 2 classical bitsAlice wants to send to him.

    In the above protocol, Alice and Bob have 1 ebit in the beginning, and they transmit1 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,Alice could attach the unknown state to her subsystem so we have

  • Chapter 2. Background Information 10

    (a |0〉+ b |1〉)A′1√2

    (|00〉+ |11〉)AB

    =1

    2(|00〉+ |11〉)A′A(a |0〉+ b |1〉)B +

    1

    2(|00〉 − |11〉)A′A(a |0〉 − b |1〉)B

    +1

    2(|01〉+ |10〉)A′A(a |1〉+ b |0〉)B +

    1

    2(|01〉 − |10〉)A′A(a |1〉 − b |0〉)B

    (2.11)

    Alice could make a Bell measurement on her system and communicate the measure-ment result (2 cbits) to Bob. With that information, Bob can recover the unknownquantum 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 use2 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 LOCCprovides 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 transformationa state will undergo during a physical process is described by quantum operators.Mathematically, a quantum operator can be described by a linear and completely positivemap, 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 Hilbertspace. 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]

    φ(ρ) =∑j

    VjρV+j (2.13)

    where ∑j

    V +j Vj = 1 (2.14)

  • Chapter 2. Background Information 11

    Each term VjρV +j can also be treated as a branch, with the resulting stateVjρV

    +j

    tr(VjρV+j )

    and

    the probability tr(VjρV +j ).

    Example 2. Measurement: For example, suppose we have a quantum state |φ〉 = α |0〉+√1− |α|2 |1〉 in the {|0〉 , |1〉} Hilbert space, a measurement in {|0〉 , |1〉} can be performed

    on the state. With probability |α|2 the state will collapse onto |0〉 while with probability1− |α|2 the state will collapse onto |1〉. This operator can be described as

    M(ρ) = |0〉 〈0| ρ |0〉 〈0|+ |1〉 〈1| ρ |1〉 〈1| (2.15)

    The resulting state is given by

    ρ = 〈0|ρ|0〉 |0〉 〈0|+ 〈1|ρ|1〉 |1〉 〈1| (2.16)

    and the possible resulting states are |0〉 (density matrix |0〉 〈0|) and |1〉 (density matrix|1〉 〈1|), with

    p(|0〉) = 〈0|ρ|0〉 = |α|2 and p(|1〉) = 〈1|ρ|1〉 = 1− |α|2 (2.17)

    Since entanglement is the resource for quantum information, an important questionwould be how entanglement evolves under quantum operations, especially measurements.Here we want to emphasize that measurement could induce the collapse of a quantumwave function, which means the resulting state might have significantly different entan-glement properties from the original state. For example, given an EPR state shared byAlice and Bob, |φ〉AB = 1√2(|00〉+ |11〉), if Alice or Bob measures in {|0〉 , |1〉} basis, thestate would collapse into 1

    2|00〉 〈00|+ 1

    2|11〉 〈11|, which is a separable state.

    2.2.2 Local Operators and Classical Communications

    Since entanglement is used for the transmission of information between parties far apartfrom each other, we restrict the quantum operations to be locally implemented. Also,we only allow classical information to be transmitted between the distant parties. Thisscheme is called LOCC (Local Operators and Classical Communications), a stan-dard 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 inthe following way: Party A makes a local operator and passes the information regardingthis 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 apositive probability less than 1 with which the transformation can be successful, then thecorresponding protocol is called SLOCC (Stochastic Local Operators and ClassicalCommunications) protocol.

    Two-outcome weak measurement decomposition of LOCC

    If one party performs a local measurement that has several possible outcomes, the statecould collapse into a new state far from the original state. This phenomenon is an obstaclefor the investigation of the behavior of some quantitative parameters under LOCC sinceit 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 decomposedinto many steps of two-outcome weak measurements, while during each step the resultingquantum states are almost identical with the original state. The idea is similar to arandom walk, where one needs to stop when the resulting state becomes one of the tworesulting states of the original two-outcome measurement [63].

    By using the above two techniques, one can discuss a general LOCC protocol underthe restriction of two-outcome weak measurement, during which the state can be seen aschanging 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 partiesshare a quantum system, while they can only perform local operations on their subsys-tems. Entanglement, as a resource, can only be consumed, rather than created, duringthis process. In this sense, entanglement monotone, as a quantification of how muchentanglement 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 onaverage under local transformations and classical communications is called entanglementmonotone (EM) [77].

  • Chapter 2. Background Information 13

    An important application of entanglement monotone is that it can be used to boundthe optimal transformation probability between two quantum states ρ and ρ′ underLOCC. More precisely, the optimal successful probability for the conversion from ρ to ρ′

    under LOCC, denoted by P (ρ LOCC−−−→ ρ′), is given by

    P (ρLOCC−−−→ ρ′) = minµ

    µ(ρ)

    µ(ρ′)(2.18)

    where the minimization is to be performed over the set of all EMs [77]. It is straight-forward to see that the optimal transformation probability should be upper bounded byany µ(ρ)

    µ(ρ′). 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 onthe transformation probability. Also, if any transformation probability coincides with aknown bound obtained from some entanglement monotone, we can say for certain thatit is optimal.

    2.2.3 Separable operators

    One drawback of LOCC is that it is hard to be analyzed mathematically, because itdoes not have an explicit analytical definition. Also, its mathematical structure is verycomplex. 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

    Ω(ρ) =k∑i=1

    AiρA+i (2.19)

    wherek∑i=1

    A+i Ai = 1 (2.20)

    andAj = A1j ⊗ A2j ⊗ · · · ⊗ Anj (2.21)

    where Akj is a local operator implemented by party k, which means AkjρA+kj

    should bea positive matrix defined on the Hilbert space of one party.

    The motivation for the introduction of SEP for quantum information is the factthat 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 LOCCprotocol can also be implemented by SEP, SEP can be used to identify entanglementmonotones for a quantum system. More concretely, if a physical quantity could not beincreased by SEP, then it is impossible for it to be increased by any LOCC protocol and itis an entanglement monotone. However, the converse is not true: there are entanglementmonotones that can be increased by SEP [24] [23].

    However, given an SEP, it is unclear how to check whether it can or cannot beimplemented by LOCC. What we can do is that, if an SEP can accomplish a missionthat is impossible by LOCC, one can be sure that this SEP cannot be implemented by anyLOCC protocol. For example, separable operations that can increase some entanglementmonotone [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 protocolwith separable operations can be implemented by LOCC within a given number of roundswas presented in [30]. In chapter 5, we will show the gaps between LOCC and SEP forsome 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 purebipartite states.

    Schmidt decomposition and Schmidt number

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

    |φ〉AB =k∑l=1

    √al |il〉A |il〉B (2.22)

    where∑k

    l=1 al = 1 and {i1, · · · , in} is a set of orthogonal state vectors. This form is calledSchmidt decomposition. For each bipartite pure state, one can uniquely determinethe values of its Schmidt coefficients (together with their degeneracies) [62]. Based onthis 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 aredegenerate.

  • Chapter 2. Background Information 15

    Given a bipartite pure state |φ〉AB, to find its Schmidt decomposition? One firstneeds to compute the reduced density matrix for one party, say ρA. The eigenvalues andeigenvectors of ρA are als and |il〉As in Eq 2.22 respectively.

    Example 3. For example, given a pure state |φ〉AB = 1√2 |00〉AB +12(|10〉 + |11〉)AB, let

    us find its Schmidt decomposition.Firstly, by tracing out party B, one can find that

    ρA = 〈0B|ρAB|0B〉+ 〈1B|ρAB|1B〉 =1

    2|0〉 〈0|+

    √2

    4(|0〉 〈1|+ |1〉 |0〉) + 1

    2|1〉 〈1| . (2.23)

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

    a1 =1

    2−√

    2

    4: |i1〉A =

    1√2

    (|0〉 − |1〉); a2 =1

    2+

    √2

    4: |i2〉A =

    1√2

    (|0〉+ |1〉) (2.24)

    Similarly, for ρB, we have the same eigenvalues and the corresponding eigenvectorsare

    |i1〉B =1√

    4 + 2√

    2(|0〉 − (1 +

    √2) |1〉); |i2〉B =

    1√4− 2

    √2

    (|0〉+ (√

    2− 1) |1〉). (2.25)

    Finally, the Schmidt decomposition is given by

    |φ〉AB =√a1 |i1〉A |i1〉B +

    √a2 |i2〉A |i2〉B (2.26)

    Remark 4. Note that for a quantum system with more parties, Schmidt decompositiondoes not always exist.

    Also, given a bipartite pure state

    |φ〉AB =k∑l=1

    √al |il〉A |il〉B , (2.27)

    the Schmidt number is defined as k, the number of non-zero terms in the Schmidtdecomposition. A generalization of Schmidt number is called Schmidt rank, whichis the minimum number of non-zero terms needed to write a multipartite state as thesuperposition of product states.

    In the following subsections, we will review some entanglement monotones for bipar-tite 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 bipartitepure states since it is defined from the information theoretic perspective. For a pure state|φ〉AB, the entropy of entanglement �(|φ〉) is defined as the von Neumann entropy of thereduced density matrix for either party, or we have

    �(|φ〉) = S(ρA) = S(ρB) (2.28)

    where S(ρ) = −Trρ log2 ρ, ρA = TrB(|φ〉AB 〈φ|), ρB = TrA(|φ〉AB 〈φ|).Entropy of entanglement is closely related to entanglement concentration and

    entanglement dilution [6]. More concretely, given N copies of a bipartite pure state |φ〉,asymptotically (in the limit of large N) one can use LOCC transformation to concentratethem into N�(|φ〉) copies of EPR pairs. Also, given N copies of EPR pairs, asymptoticallythey can be distilled into N

    �(|φ〉) copies of |φ〉.

    Transformation probability between the states

    Based on the Schmidt decomposition of bipartite pure states, the optimal LOCC trans-formation probability between any two bipartite pure states is discovered in [77]. Giventwo pure bipartite states

    |φ〉 = √a1 |i1i1〉+√a2 |i2i2〉+ · · ·

    √an |inin〉 , ai ≥ ai+1 ≥ 0, (2.29)

    and

    |ψ〉 =√b1 |j1j1〉+

    √b2 |j2j2〉+ · · ·

    √bn |jnjn〉 , bi ≥ bi+1 ≥ 0, (2.30)

    where {i0, i1, · · · , in} and {j0, j1, · · · , jn} are two sets of orthogonal vectors, the optimaltransformation probability from |φ〉 to |ψ〉 under LOCC is given by

    P (φLOCC−−−→ ψ) = min

    l∈[1,n]

    ∑ni=l ai∑ni=l bi

    . (2.31)

    Notice that if we have∑n

    i=l ai ≥∑n

    i=l bi for any given l, the transformation canbe done by LOCC with probability 1. This is called the majorization relationshipand was discovered in [61]. Based on this result, the EPR state serves as the maximalentangled state for bipartite quantum system since it can be transformed deterministicallyinto 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 couldbe written as

    ρ =∑i

    pi |φi〉 〈φi| (2.32)

    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 correspondingentanglement measure for mixed state as

    ∑i piE(|φi〉).

    However, one needs to note that the decomposition of a mixed state into an en-semble of pure states is not unique, which means that we need to consider all possibledecompositions and find the one that yields the minimum value. Or we have

    E(ρ) = min{pi,|φi〉}

    ∑i

    piE(|φi〉) (2.33)

    Entanglement of formation and entanglement cost

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

    Ef (ρ) = min{pi,|φi〉}

    ∑i

    pi�(|φi〉) (2.34)

    Note that in large N limit, one can prepare N copies of state |φi〉 with N�(|φ〉) copiesof EPR pairs. In general, one can prepare all states in this ensemble with EPR pairsand combine them together to form ρ. Thus we achieve an operational interpretation ofentanglement of formation: with NEf (ρ) copies of EPR pairs, one can prepare N copiesof ρ using LOCC in the limit of large N.

    This definition leads to another concept called entanglement cost, which is the min-imum value of the average number of EPR pairs needed to prepare one copy of a stateusing LOCC [46]. If n EPR pairs are needed to produce m copies of a state ρ, thenEC(ρ

    ⊗m) = min nm. 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 mlimit, they are equal. Or we have

    Ec(ρ) = limm→∞

    1

    mEf (ρ

    ⊗m). (2.35)

  • Chapter 2. Background Information 18

    Distillable entanglement and Bound entanglement

    Conversely, distillable entanglement is defined as the number of EPR pairs one can distillfrom a given state [8]. In particular, suppose one can distill n copies of EPR pairsfrom m copies of state ρ, one has ED(ρ) = max limm→∞ nm . Here we need to considerall possible LOCC protocols to maximize the number of EPR pairs produced. Dueto the complexity of LOCC protocols, there is no analytical expression for distillableentanglement. However, PPT criterion can be used to check if distillable entanglementis zero for a bipartite mixed state [65][50].

    In general, entanglement cost is larger than distillable entanglement. The extremecondition is that for some states, the distillable entanglement is zero while the entan-glement cost is positive, which means EPR pairs needs to be consumed to prepare thestate while no EPR pairs can be distilled back from that state. This is called boundentanglement [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 matrixbetween AB as

    ρ̃AB = (σy ⊗ σy)ρ∗AB(σy ⊗ σy), (2.36)

    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 ρABρ̃AB in decreasing order (λ1 ≥λ2 ≥ λ3 ≥ λ4).

    Up to now, concurrence is the only known analytical entanglement monotone formixed states. Under a general noisy environment, a pure state will be transformed intoa mixed state because of decoherence. To quantify the involvement of entanglement forthis quantum system, the change in concurrence serves as an important criterion. Forexample, entanglement sudden death was based on the calculation of concurrence underthe 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 gen-eralization 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 alwaysexist. Instead, a unique form called generalized Schmidt decomposition of tripartite stateswas discovered in [2] as

    |φ〉 = λ0 |000〉+ λ1 |100〉+ λ2 |101〉+ λ3 |110〉+ λ4 |111〉 (2.38)

    where λi ≥ 0, 0 ≤ φ ≤ π, µi ≡ λ2i ,∑

    i µi = 1.

    Based on this form, truly entangled three-qubit states (that is to say, other thanproduct 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 6= 0, it is calleda GHZ class state; otherwise, it is called a W class state. States that belong todifferent classes cannot be transformed into each other by LOCC even with some non-zero probability. However, the optimal transformation probability between states in thesame 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 beenproved that four-qubit entangled states can be classified into nine families [76], betweenwhich 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 byA and B, A and C, A and BC, etc. Is there any entanglement that is shared by ABCaltogether? If so, how can we quantify it? The answer lies in concurrence and a quantitycalled 3-tangle [29].

    Firstly, it should be clarified that the concurrence between A and B, CAB, is not anentanglement monotone. In fact, it can be increased significantly by LOCC. For example,for a GHZ state |GHZ〉ABC = 1√2(|000〉+ |111〉)ABC , if Charlie performs a measurementon the basis {|+〉 , |−〉} and communicates the measurement result to Alice and Bob, ABwill end up sharing an EPR pair. Notice that for the original state, the density matrix

  • Chapter 2. Background Information 20

    for AB isρAB =

    1

    2(|00〉 〈00|+ |11〉 〈11|) (2.39)

    which is a separable state with CAB = 0. But for the final EPR state, we clearly haveCAB = 1. During this transformation, CAB has been increased from 0 to 1 with Charlie’sassistance.

    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 -the entanglement shared by ABC altogether - as the difference between C2A,BC and thesummation of C2AB and C2AC , or we have

    τABC = C2A(BC) − C2AB − C2AC . (2.40)

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

    τABC = C2B(AC) − C2BA − C2BC = C2C(AB) − C2CA − C2CB. (2.41)

    τABC is shown to be an entanglement monotone [29]. From the definition of τABC wecan 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. τABCserves as a clear distinction between the GHZ class state and W class state because τABCis zero for any W class state. Or we can say, the entanglement of a W class state canall be attributed to two-party entanglement [34]. Because of that, the transformationprobability 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 transformationprobability 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 theSchmidt rank. As we defined earlier, it is the minimum number of product statesneeded in the superposition form of a pure state. For a GHZ class state, the Schmidtrank is 2 while any W class state has Schmidt rank 3. Notice that Schmidt rank cannot even be increased by SLOCC, which means it can not increase even with a nonzeroprobability. 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 cal-culation 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 thenumber of parties is higher than three. However, for a specific class of multipartitestates, the W-type states, a unique form was discovered and the transformation rulesbetween these states turn out to be very explicit [52]. In the following we will introducethe definition of W type state as a generalization of W state. We will also describe theentanglement monotones and transformation rules associated with W type states.

    From the definition of the W state

    |W 〉 = 1√3

    (|100〉+ |010〉+ |001〉) (2.42)

    one can easily generalize it into more parties as

    |Wn〉 =1√n

    (|10 · · · 0〉+ |010 · · · 0〉+ · · ·+ |0 · · · 01〉) (2.43)

    which is called standard n-party W state. Then we can give the following definitionof W-type state:

    Definition 3. W type states are the states that can be obtained by SLOCC from a stan-dard 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 typestates since one can easily obtain a product state from an n-party W state by making ameasurement on some parties. An illustration of W type state for three qubits is shownin 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)

    where xi ≥ 0 and∑n

    i=0 xi = 1. It was proved that the above form is unique when atleast 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 firstlevel is the true W3 type state which is also the genuine W class state. The second levelare bipartite entangled states, such as (AB)-C (|ψ〉AB |φ〉C), (AC)-B (|ψ〉AC |φ〉B) and(BC)-A (|ψ〉BC |φ〉A), and the third level is the product state |φ1〉A |φ2〉B |φ3〉C .

    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, wecan 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 finalstates −→xk with corresponding probability pk, and we have [52]

    (i) xk,i = skxi where i 6= 0, j;

    (ii)∑

    k pksk = 1;

    (iii)∑

    k pk√skxk,0 ≥

    √x0.

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

  • Chapter 3

    LOCC transformation bounds betweenmultipartite pure states

    In this chapter, the upper and lower bounds on the transformation probability betweenmultipartite pure states will be derived. For tripartite pure states, it is well knownthat there are two inequivalent classes of genuine tripartite entangled states, namelythe Greenberger-Horne-Zeilinger (GHZ) class and the W class. Any two states withinthe same class can be transformed into each other via stochastic local operations andclassical communication with a nonzero probability. The optimal conversion probability,however, is only known for special cases. Here, lower and upper bounds are derivedfor the optimal probability of transformation from a GHZ state to other states of theGHZ class. A key idea in the derivation of the upper bounds is to consider the actionof the local operations and classical communications (LOCC) protocol on a differentinput state, namely 1√

    2|000〉− |111〉), and to demand that the probability of an outcome

    remains bounded by 1. We also find an upper bound for more general cases by usingthe constraints of the so-called interference term and 3-tangle. Moreover, some of theresults are generalized to the case in which each party holds a higher dimensional system.In particular, the GHZ state generalized to three qutrits, that is, |GHZ3〉 = 1√3(|000〉 +|111〉+|222〉) shared among three parties can be transformed to any tripartite three-qubitpure state with probability 1 via LOCC. Some of our results can also be generalized tothe case of a multipartite state shared by more than three parties. The content of thischapter 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 clas-sical physics and lies at the heart of quantum information theory. Thus it is importantto get a good understanding of entanglement properties of quantum states. These prop-erties are well understood for bipartite pure states. In the standard distant laboratoryparadigm, suppose two distant parties, Alice and Bob, shared a bipartite entangled state.They may apply local operations and classical communications (LOCC) to convert it intoanother partite state. Bennett et al [6] has answered the question for the rate of LOCCtransformation between bipartite pure states. It is quantified by the von Neummanentropy of a reduced density matrix. For the single-copy case, the optimal conversionprobabilities are known for any pure state transformation [56, 61, 77]. For an LOCCtransformation protocol, if it can succeed with probability 1, we call it deterministic, ifit can only succeed with a nonzero probability smaller than 1, we call it stochastic, orSLOCC (Stochastic Local Operators and Classical Communications). For mixed states,the question of what the optimal rate of transformations is between them is still largelyopen.

    For multipartite states, however, the problem is much more complicated. There existdifferent types of entanglement and therefore the transformations are rather involved.For the case of tripartite pure three qubit states, a characterization into six differententanglement classes, of which two contain true tripartite entanglement, exists [34]. Oneis the GHZ class state, which is defined as

    |φGHZ〉 =√K(cδ |0〉 |0〉 |0〉+ sδeiϕ |ϕA〉 |ϕB〉 |ϕC〉) (3.1)

    where

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

    and K=(1 + 2cδsδcαcβcγcφ)−1 ∈ [12 ,∞), cδ = cos δ, sδ = sin δ, the same for α, β, γ, φ.The range for the parameters are δ ∈ (0, π

    4], α, β, γ ∈ (0, π

    2] and ϕ ∈ [0, 2π).

    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)

  • 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 non-zero probability. However, here comes the key point. The optimal conversion betweenthe 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 statewas 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 entanglementtransformations. In particular, we focus on transformations among states with the sameSchmidt rank [36]. We put an emphasis on the transformation from a GHZ state to aGHZ-class state.

    But our upper bound can also be generalized to general transformations from oneGHZ class state to another. And some of the results are derived for the more generalcase of higher dimensions and more parties. Especially, we find that all tripartite purethree qubit states can be transformed from 3-term GHZ state 1√

    3(|000〉 + |111〉 + |222〉)

    with probability 1. This is a new result. Moreover, some of the general theorems fordeterministic transformation are also derived.

    This Chapter is structured as follows. In Section 3.2, we derive upper bounds for thetransformation of the GHZ-state to any other state in the GHZ-class. The upper boundsare only non-trivial for a subclass of the GHZ-class. Thus Section 3.3 and 3.4 use adifferent 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 GHZstate with probability 1, we can find a nontrivial upper bound for the probability ofthis transformation. And our upper bound can also be effective for the transformationfrom a GHZ class state to a large class of other GHZ class states. Lower bounds forthe transformation of higher dimensional GHZ-states distributed among three or moreparties 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 anyother 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 ∈ (12π, 3

    2π). The transformation under consideration is given by

    |GHZ〉 = 1√2(|000〉+ |111〉)

    LOCC−→ |Ψ〉 =√K(cδ |0〉 |0〉 |0〉+ sδeiϕ |ϕA〉 |ϕB〉 |ϕC〉), (3.6)

    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 differentbranches. Furthermore, a branch Oi |GHZ〉 = |Φ〉 is called a success branch if |Φ〉 ∝ |Ψ〉,and a failure branch if there exists no LOCC-operation that can transform |Φ〉 into|Ψ〉 with a non-zero probability, if a branch is neither success nor failure, we call it anundecided branch. An optimal protocol only consists of success and failure branches.

    For the following analysis we first recall two known results [34, 2]

    Lemma 1. For a GHZ-class state |Ψ〉 we have:

    a) The Schmidt rank of |Ψ〉 is 2 [34]. This means that the minimum number of productstates necessary to write |Ψ〉 as a superposition of them is 2:

    |Ψ〉 =2∑i=1

    αi |aibici〉 , (3.7)

    with αi ∈ (0, 1) and 〈aibici|aibici〉 = 1.

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

    This result leads to

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

    |Ψ〉 = α1 |a1b1c1〉+ α2 |a2b2c2〉LOCC−→ |Ψ′〉 = α′1 |a′1b′1c′1〉+ α′2 |a′2b′2c′2〉 , (3.8)

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

    O1 |a1b1c1〉 = o1α′1α1|a′1b′1c′1〉 (3.9)

    O1 |a2b2c2〉 = o1α′2α2|a′2b′2c′2〉 (3.10)

  • Chapter 3. LOCC transformation bounds between multipartite pure states27

    or

    O1 |a1b1c1〉 = o1α′2α1|a′2b′2c′2〉 (3.11)

    O1 |a2b2c2〉 = o1α′1α2|a′1b′1c′1〉 (3.12)

    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, aproduct 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√2(|000〉+ |111〉)

    → |Ψ〉 =√K(cδ |000〉+ sδeiϕ |ϕAϕBϕC〉), (3.13)

    where the parameters are defined in Equation (3.6), is given by

    p ≤ min{

    1,1 + 2cδsδcαcβcγcϕ1− 2cδsδcαcβcγcϕ

    }(3.14)

    Idea of the Proof: From Lemma 2 we know that, for a success branch, each productstate (in the Schmidt term) of the input states have to be mapped to a product stateof the output state. This allows us to infer how the same LOCC protocol acts on thephase flipped GHZ state, i.e., 1√

    2(|000〉 − |111〉). From the requirement that the sum of

    the probabilities for the output states have to sum to 1 for this transformation, we canderive a bound for the parameters arising in the original transformation. This gives abound 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 successfulbranch. They are

    Oi |000〉 = oicδ |000〉 (3.15)Oi |111〉 = oieiϕsδ |ϕAϕBϕC〉 (3.16)

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

    Oi |000〉 = oieiϕsδ |ϕAϕBϕC〉 (3.17)Oi |111〉 = oicδ |000〉 (3.18)

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

    Oi |GHZ〉 =1√2oi(cδ |000〉+ eiϕsδ |ϕAϕBϕC〉) =

    oi√2K|Ψ〉 (3.19)

  • Chapter 3. LOCC transformation bounds between multipartite pure states29

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

    p =1

    2K

    n1+n2∑i=1

    o2i . (3.20)

    To get an upper bound for∑n1+n2

    i=1 o2i , we consider how

    1√2

    (|000〉 − |111〉) (3.21)

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

    Oi1√2(|000〉 − |111〉)

    = 1√2oi(cδ |000〉 − eiϕsδ |ϕAϕBϕC〉) = oi√2K′ |Ψ

    ′〉 (3.22)

    with|Ψ′〉 =

    √K ′(cδ |000〉 − eiϕsδ |ϕAϕBϕC〉), (3.23)

    where K ′ = 11−2cδsδcαcβcγcϕ

    , for i = 1, . . . , n1 +n2 (up to an overall minus sign for i = n1 +1, . . . n1 +n2). Thus the conversion probability for this process is given by 12K′

    ∑n1+n2i=1 o

    2i .

    Being a probability, this has to be bounded by 1, giving∑n1+n2

    i=1 o2i ≤ 2K ′. This together

    with Equation (3.20) gives the upper bound

    p ≤ K′

    K=

    1 + 2cδsδcαcβcγcϕ1− 2cδsδcαcβcγcϕ

    (3.24)

    for the process described by Equation (3.13). �

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

    |Ψ〉 = 1√2√

    1− λ3a(|000〉 − |aaa〉), (3.25)

    we get

    p ≤ 1− λ3a

    1 + λ3a. (3.26)

    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 cosϕ < 0, i.e., φ ∈ (π

    2, 3π

    2). This nicely

    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 allother 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 ∈ (π2, 3π

    2).

    Here, we will derive a useful bound for a larger class of states: we find a upper boundnontrivial for all the cases except φ = π

    2or 3π

    2and 〈000|ϕAϕBϕC〉 = 0. In fact, it was

    shown that these two kinds of transformations can succeed with probability 1 [74]. Ourproof has two important ingredients, namely, the conservation of a new quantity definedas "interference term" under positive operator valued measures (POVMs) and that thethree 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 weightsummation of the so-called interference terms and normalizations of all the branches inan LOCC protocol should be constant during the transformation, which is included insection 3.3. After that, we find that three tangle is bounded for a fixed interferenceterm which will be defined in this section. Then, we try to see the whole process fromthe weak measurement aspect, which divides the whole process into many infinitesimalsteps and each step changes the state very little. That is to say, the state is changingcontinuously. Then we stop in the middle and investigate whether there will be a newupper bound. Surprisingly we find there are some new upper bounds and these upperbounds will still be effective in the following steps, even when we reach the end. So itcan be used to derive a new upper bound for the supremum success probability of thewhole LOCC protocol. Detailed discussion will be showed in section 3.4.

    Theorem 2. For the transformation from GHZ to GHZ-class state |φ〉, failure branchesshould 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〉 +

    sδeiϕ |ϕA〉 |ϕB〉 |ϕC〉), where |0A〉 is linearly independent of |ϕA〉, the same for B and C. If

    there is a state whose reduced density matrices are all of full rank, |φ〉 =√K ′(c′δ |0〉 |0〉 |0〉+

    s′δeiϕ′ |ϕ′A〉 |ϕ′B〉 |ϕ′C〉), where |0A〉 is linearly independent of |ϕ′A〉, the same for B and C.

  • Chapter 3. LOCC transformation bounds between multipartite pure states31

    Then it is easy to see, the equation

    OA |0〉 = |0〉 (3.27)OA |ϕ′A〉 = |ϕA〉 (3.28)

    always has a non-trivial solution, the same for B and C. That means we can alwaystransform this state into |φ〉 with nonzero probability. �

    3.3.1 Conservation of interference term

    To go further, we want to use the following property of the LOCC Kraus operators. Fora complete set of Kraus operators Oi = Ai ⊗Bi ⊗ Ci, we have

    ∑O+i Oi = 1.

    Suppose that a Kraus operator O satisfies

    O |000〉 = α |a1b1c1〉 (3.29)

    O |111〉 = β |a2b2c2〉 (3.30)

    with 〈a1b1c1|a1b1c1〉 = 〈a2b2c2|a2b2c2〉 = 1.Then it can transform |GHZ〉 = 1√

    2(|000〉 + |111〉) into |ψ〉 = 1√

    2p(α |a1b1c1〉 +

    β |a2b2c2〉, where 1√2p is the normalization factor and one can check p is exactly theprobability of getting |ψ〉. From here we define interference term and normalization inthe following:

    Definition 4. For a normalized GHZ-class state |γ〉 where 〈γ|γ〉 = 1, written in theform |γ〉 = 1√

    2(α |a1b1c1〉 + β |a2b2c2〉), suppose 〈a1b1c1|a2b2c2〉 = k, then we call the real

    part of α∗βk the interference term I of |γ〉.

    It is easy to see if an operator O transforms |GHZ〉 to a state |ψ〉, the interferenceterm of |ψ〉 is in fact the real part of

    p, where p is the probability of the

    branch corresponding to operator O.

    Remark 8. In fact, one can find I = 1− 12(|α|2 + |β|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 thestate. So, when we talk about the interference term of failure branches of an SLOCCtransformation, we need to be careful: We are not talking about the interference term ofthe state given by the failure branches, but the interference term determined by the wholetransformation 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 thebranches should be zero.

    0 =∑

    p(Oi |GHZ〉)I(Oi |GHZ〉) (3.31)

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

    Proof: Suppose the corresponding complete set of Kraus operators consists ofOi = Ai ⊗Bi ⊗ Ci.Then we have

    ∑O+i Oi = Id. So, we should have

    0 = < 000|111 >=< 000|I|111 >= < 000|∑O+i Oi|111 >=

    ∑< 000|O+i Oi|111 >

    =∑p(Oi |GHZ〉)

    p(Oi|GHZ〉) (3.32)

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

    3.3.2 Conservation of normalization

    Definition 5. For a two-term tripartite state |γ〉, written in the form |γ〉 = 1√2(α |a1b1c1〉+

    β |a2b2c2〉, then we call 12(|α|2 + |β|2) the normalization of |γ〉.

    Easy to see if an operator O transforms |GHZ〉 to the state |ψ〉, the normalization of|ψ〉 is in fact +

    2p, where p is the probability. And because OρO+

    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 ⊗Bi ⊗ Ci}. Then we have

    ∑O+i Oi = I. So we should have

    1 = < GHZ|GHZ >= 1

    2(< 000|+ < 111|)(|000 > +|111 >)

    = 12(〈000|000〉+ 〈111|111〉)

    = 12(∑ 〈000|O+i Oi|000〉+∑ 〈111|O+i Oi|111〉)

    =∑p(Oi |GHZ〉)+

    2p(Oi|GHZ〉) (3.33)

    From the definition of normalization we know it is exactly the weighed sum of thenormalization of each branch. That is to say, for a transformation from |GHZ〉 to otherstates, the average value of the normalization of all the states we get in each branchshould be 1. And recall that normalization can be no less than zero. So each term in thesummation should be no larger than 1, which means for each branch, the product of itsprobability 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 inter-ference term. However, conservation of the normalization also gives the following. Foreach branch, the product of its probability and the normalization of the state it getsshould be no larger than 1. The fact is also useful in determining the upper bound oftransformation 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 probabilityshould be one can give us some implication about the supremum success probability. Forexample, we can have the following theorem:

    Theorem 4. For a transformation protocol from GHZ state to a GHZ-class state |φ〉whose interference term is x, which is positive (negative), if there exists a y (y > 0), suchthat, 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:

    ps ≤ pU(−y) =y

    x+ y. (3.34)

    if x < 0:ps ≤ pU(y) = −

    y

    x− y . (3.35)

  • Chapter 3. LOCC transformation bounds between multipartite pure states34

    See

    0

    0.2

    0.4

    0.6

    0.8

    1

    0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

    p

    a

    pu

    Figure 3.3: The value of pU as a function of a. In this figure, a = ( yy−1)

    13 . 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 expressthe value as a function of a because it will be easier for us to combine different graphsinto one graph later. c©2010 American Physical Society.

    Proof: Take x > 0, suppose there are n failure branches, whose probabilities arepf1 , pf2 , ·, pfn , and the corresponding interference terms are −y1,−y2, ·−yn, then we have

    psx−∑pfiyi = 0 (3.36)

    ps +∑pfi = 1 (3.37)

    Rewrite it in the following form,

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

    where pft =∑pfi and y′ =

    ∑pfiyipft

    . The solution of it is

    ps =y′

    x+ y′(3.40)

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

    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 wecan say y goes to∞, we will have pU(−y) arbitrary close to 1. Therefore, theorem 4 aloneis not enough for establishing a non-trivial upper bound. To derive a non-trivial upperbound, we need to find some additional constraints which are related to the i