Thesis
Reference
Measurement dependence, limited detection and more: problems and applications of quantum nonlocality
PUETZ, Gilles
Abstract
The work in this thesis focuses on the subject of quantum nonlocality. The fact that the principle of locality is violated in nature is among the most counter-intuitive revelations of the past century. The original proof of the nonlocal nature of quantum mechanics relies on the so-called measurement independence assumption. This assumption has come under scrutiny in recent years, mostly because nonlocality and its implications have found their way into the realm of potential applications. In this thesis, the measurement independence assumption is relaxed and replaced by a form of limited measurement dependence. The resulting set is analyzed and it is shown that the nonlocal features of quantum mechanics resist arbitrary lack of free choice. Additionally, a new perspective on how to deal with the so-called detection loophole is provided by the definition of limited detection locality. Finally, work on applications of quantum nonlocality, specifically entanglement certification and quantification is presented.
PUETZ, Gilles. Measurement dependence, limited detection and more: problems and applications of quantum nonlocality. Thèse de doctorat : Univ. Genève, 2016, no. Sc. 4937
URN : urn:nbn:ch:unige-852444
DOI : 10.13097/archive-ouverte/unige:85244
Available at:
http://archive-ouverte.unige.ch/unige:85244
Disclaimer: layout of this document may differ from the published version.
DETECTION AND MORE: PROBLEMS AND APPLICATIONS OF QUANTUM NONLOCALITY
TH` ESE
pr´esent´ee `a la Facult´e des sciences de l’Universit´e de Gen`eve pour obtenir le grade de Docteur `es sciences, mention physique
par
Gilles P¨ utz
de
Belvaux (Luxembourg)
Th`ese No 4937
GEN`EVE
Centre d’impression de l’universit´e de Gen`eve 29/06/2016
The work in this thesis focuses on the subject of quantum nonlocality. The fact that the principle of locality is violated in nature is among the most counter-intuitive revelations of the past century and has in recent years spawned the field of device independent information processing. The original proof of the nonlocal nature of quantum mechanics by John Bell in 1964 relies on the so-called measurement in- dependence assumption. This assumption has come under scrutiny in recent years, in part since nonlocality and its implications seem implausible, but mostly because they have found their way into the realm of potential applications, specifically for privacy related tasks.
In this thesis, the measurement independence assumption is relaxed and re- placed by a form of limited measurement dependence. The set of all measurement- dependent local (MDL) distributions is investigated and shown to be a polytope, a geometric structure that can be efficiently analyzed. Furthermore, it is shown that the nonlocality of quantum mechanics persists even when only arbitrarily small amounts of measurement independence are assumed; in other words, it resists al- most arbitrary lack of free choice. It is shown that all linear inequalities for the well-studied standard local set can be transformed into valid inequalities for the MDL set. A method for handling potential memory effects and finite statistics in an implementation is also provided. Finally, the set of correlations in a measurement- dependent world restricted solely by the nonsignaling principle is introduced.
The results of the study of measurement-dependent locality are shown to be applicable to a problem that experiments attempting to demonstrate nonlocality encounter regularly: the detection loophole. The detection loophole is open unless sufficiently high detection efficiencies can be reached. Finding reasonable assump- tions that alleviate this and other challenges while maintaining the power of non- locality constitutes the field of semi-device-independent information processing. In the present work, such an alleviation is achieved by introducing the limited detec- tion assumption. The set of correlations arising from standard locality paired with this assumption, called the limited detection local (LDL) set, is shown to form a polytope. Quantum correlations are presented that are nonlocal, independent of the overall detection efficiency, as long as an arbitrarily small limitation on detection efficiency is assumed. A link between LDL and MDL is made explicit.
One of the applications of nonlocality is in device-independent entanglement cer- tification. Quantum entanglement is, especially in the multipartite case, an active area of research. Nonlocality and quantum entanglement are strongly linked; only entangled quantum states can produce nonlocal correlations. A local inequality
the depth of entanglement in the multipartite case is introduced. Furthermore, the noise-resistance of the nonlocality of a large class of multipartite entangled states, the W and Dicke states, is analyzed. Evidence is provided indicating that states with a low number of excitations are most resistant.
Finally, nonlocality is applied to the concept of quantum networks. In this case, local correlations that are restricted to the same structure as the quantum network are of interest. The corresponding sets are non-convex and thereby difficult to an- alyze. A technique that can transform previously known inequalities for a given network into new inequalities for larger networks is demonstrated.
In summary, two problems encountered in the demonstration and use of nonlo- cality are studied and tools to solve them are provided. Additionally, the nonlocality of quantum mechanics is considered in three different scenarios: its use as a witness for quantum entanglement, its resistance to noise and its application to networks.
Cette th`ese porte sur le sujet de la nonlocalit´e en m´ecanique quantique. Le fait que le principe de localit´e est viol´e dans la nature est l’une des r´ev´elations les plus contre-intuitives du si`ecle dernier et ce qui a donn´e naissance au domaine de traˆıtement d’information quantique ind´ependamment de la r´ealisation (device- independent quantum information processing). La preuve originale de la nonlo- calit´e de la m´ecanique quantique par John Bell en 1964 n´ecessite l’hypoth`ese de l’ind´ependence du choix des mesures (measurement independence). Cette hypoth`ese est devenue un sujet d’int´erˆet ces derni`eres ann´ees, en partie parce que la nonlocalit´e et ses implications paraissent invraisemblable, mais surtout parce qu’elles ont com- menc´e `a d´eboucher sur des applications potentielles, en particulier dans le domaine de la protection des donn´ees priv´es.
Dans cette th`ese, l’hypoth`ese de l’ind´ependence du choix des mesures est affaiblie en la rempla¸cant par d´ependence du choix des mesures limit´ee. L’ensemble de toutes les distributions locales avec cette d´ependence du choix des mesures (measurement dependent local, MDL) est examin´e, ce qui a permit de l’exprimer sous la forme d’un polytope, c’est-`a-dire une structure g´eom´etrique qui peut ˆetre analys´ee efficacement.
En plus, on prouve que la nonlocalit´e en m´ecanique quantique persiste mˆeme si une quantit´e arbitrairement petite d’ind´ependence du choix mesures est suppos´ee; en d’autres termes elle persiste malgr´e l’absence de libre arbitre. Au-del`a, on montre que toutes les in´egalit´es lin´eaires de la localit´e standard peuvent ˆetre transform´ee en in´egalit´es pour l’ensemble MDL. De plus, une m´ethode pour tenir en compte des effets de m´emoire et des effets de statistique finie dus `a une impl´ementation physique est aussi d´eriv´ee. Finalement, on introduit l’ensemble des correlations possible dans le cas de d´ependence du choix des mesures et restreint uniquement par le principe de nonsignaling.
Par la suite, on montre comment les r´esultats de l’´etude de l’ensemble MDL peuvent ˆetre appliqu´es `a un probl`eme r´eguli`erement rencontr´e dans les experiences tentant de d´emontrer les effets de nonlocalit´e qui est l’´echappatoire sur la d´etection (detection lopphole). L’´echappatoire sur la d´etection est ouverte si l’efficacit´e de d´etection n’est pas suffisamment ´elev´ee. Trouver des hypoth`eses raisonables qui per- mettent de relˆacher la contrainte sur cette ´echappatoire est l’un des sujets d’´etudes du domaine de traˆıtement d’information quantique quasi-ind´ependamment de la r´ealisation (semi-device-independent quantum information processing). Dans ce tra- vail, une telle hypoth`ese est propos´ee, appel´ee l’hypoth`ese de la d´etection limit´ee. On prouve que l’ensemble des correlations locales satisfaisant la d´etection limit´ee (lim- ited detection local, LDL) peut s’exprimer sour la forme d’un polytope. On pr´esente des correlations quantiques qui sont nonlocales malgr´e une d´etection limit´ee arbi-
entre LDL et MDL est explicit´e.
Une des applications de la nonlocalit´e est donn´ee par la certification d’intrication ind´ependamment de la r´ealisation (device-independent entanglement certification).
L’intrication quantique et en particulier l’intrication multipartite sont des domaines de recherches actifs. La nonlocalit´e et l’intrication quantique sont fortement li´ees car seuls les ´etats quantiques intriqu´es peuvent produire des correlations nonlocales.
Une in´egalit´e locale qui peut d´emontrer de mani`ere ind´ependante de la r´ealisation non seulement la pr´esence, mais aussi la profondeur de l’intrication est pr´esent´ee. De plus, la r´esistance au bruit de la nonlocalit´e est analys´ee pour une grande cat´egorie d’´etats multipartites comme les ´etats W et Dicke. On donne de fortes indications que les ´etats avec un petit nombre d’excitations sont le plus r´esistant.
Finalement, le concepte de nonlocalit´e est appliqu´e aux r´esaux quantiques. Dans ce cas, on s’est int´eress´e aux correlations locales restreintes `a la mˆeme structure que le r´esaux quantique. Les ensembles correspondants sont non-convexes et d`es lors difficile `a analyser. Une technique qui permet de transformer des in´egalit´es connues pour un r´esaux donn´e dans de nouvelles in´egalit´es pour des r´esaux plus larges est pr´esent´ee.
En r´esum´e, on d´etaille deux probl`emes confront´es lors de d´emonstrations et d’utilisation de la nonlocalit´e et on pr´esente des moyens pour les r´esoudre. Par la suite, on consid`ere la nonlocalit´e de la m´ecanique quantique dans trois sc´enarios diff´erents: son utilit´e pour la certification d’intrication, sa r´esistance au bruit et son application aux r´esaux quantiques.
Je remercie l’universit´e de Gen`eve, le professeur Gisin, mes coll`egues, mes amis et ma famille. Pour les remerciements plus compl`ets, veuillez voir la section ”Acknowl- edgements”.
It is not often that one gets the chance to freely and publicly thank all the people who are essential to one’s success and well-being. I shall take full advantage of this opportunity.
My first thanks clearly has to go to my family. Even neglecting the fact that it is difficult to achieve anything without being born, my family has been highly involved in getting me to the point where I could take on a PhD thesis. From making sure that I never lacked support for anything I showed the slightest interest in, be that a sport, programming or learning about dinosaurs, to exposing me to books and knowledge as a child, my parents, grandparents, brother and sister have been on top of their family-game. For that and many other things they have my gratitude and love.
Moving from the people who supported me from a young age to the other end of the time-spectrum, I would like to extend my thanks to my supervisor, Prof.
Nicolas Gisin. From talking to fellow PhD students who were not so lucky with their advisor, I realized that it is usually not a lack of expertise, vision or genius that they express dissatisfaction with, but a scarcity of support or guidance. It is thus this part in particular that I wish to highlight. Between being highly available to the point where a meeting usually occurs on the same day if not in the same hour that I thought of it, providing a work environment that ensures collaborations within the group, promoting his students by sharing talks at conferences, making sure that publications occur by pushing for them when the work is ready, and having the right balance between giving the freedom to be creative with the guidance to be productive, Nicolas is without a doubt the best supervisor I could have asked for.
I would further like to thank Prof. Renato Renner for being the person who first got me interested in quantum information theory and for introducing me to the opportunity of a PhD between the University of Geneva and ETH Zurich. The fact that the time of the Zurich group meeting was taking into account my availability despite me working mostly in Geneva is something I most definitely do not take for granted.
My thanks also go to my coworkers, collaborators and fellow group members in Geneva and Zurich. I would specifically like to mention Yeong-Cherng Liang and Roger Colbeck, who helped and supported me a lot in the first years of my PhD, as well as Denis Rosset for being a well of technical ideas, Anthony Martin for his patience with a theorist in an experimental environment, Nicolas Brunner for his general support, and Tomer Barnea for being an office-mate and friend who enjoys
When it comes to my friends, I would first and foremost like to mention Jeffrey Roiy Otmi. It is fair to say that after my family he has probably had the biggest impact on my life. Countless hours of talking until late at night about life, the universe and everything has laid the foundation for a friendship whose support I will never doubt.
Yasmine Sinno has been a constant source of support since we met a few years ago. In fact, she has made it the farthest through this thesis and has helped me avoid certain humiliation in the eyes of my jury by catching many typos and terrible English expressions. More importantly however, I want to thank her for being a close friend and constant companion these past years.
Christian Bolesch is probably the main reason I wistfully think of my years as a Bachelor student at ETH Zurich. I am still trying to figure out which one of us was made brilliant and which one insane by the study overkill in HPV G4.
L´ıdia del Rio I would like to thank for managing to fill the roles of friend and partially mentor, in quantum information theory and in general, simultaneously.
Long walks with Jonah, boardgames, Battlestar Galactica, swordfighting... I am looking forward to what’s next.
Further thanks go out to Aleksey Fomins, Lea Kr¨amer Gabriel, who besides being a friend gave me great comments on how to improve the writing of this the- sis, Evelyn Stilp, who will play a crucial role in getting me back into shape after this writing marathon, Johan ˚Aberg, who was a great supervisor during my master thesis, Johanie Uccelli, Sascha Maxeiner, Henrik Ronellenfitsch, Mischa Stocklin, Cornelia R¨osler, Normand Beaudry, Sebastian Lienert, Andreas Hermann, Alex Ae- berli, Konrad Schwenke, Rotem Arnon-Friedmann, Sandra Stupar and Ralph Silva.
Finally, I would like to thank Kaitlin Schaal, who has been incredibly supportive of me over the past year and especially over the past weeks of constant writing. Her affection, positive outlook and general fascination and curiosity about everything, from astrophysics and evolution to the pneumostome of a slug we saw on our walk in the woods, have majorly contributed to keeping me mentally sane and hopeful in this period of my life, and hopefully for all that are to come.
1 Introduction 1
1.1 Notations, definitions and mathematical background . . . 3
1.1.1 Finite convex polytopes . . . 3
1.1.2 Probabilities and random variables . . . 5
2 Nonlocality 9 2.1 The local polytope . . . 14
2.2 The nonsignaling polytope . . . 20
3 Measurement dependent locality 23 3.1 The (2,2,2)-scenario . . . 33
3.2 (`, h)-MDL inequalities from standard locality inequalities . . . 37
3.3 Measurement-dependent nonlocality for ` = 0: the (2, n,2) scenario . 40 3.4 Dealing with memory and finite statistics: the non-i.i.d. case . . . 41
3.5 Measurement dependence with independent sources . . . 43
3.6 Non-uniform inputs . . . 45
3.7 Measurement dependent nonsignaling . . . 46
4 Limited detection locality 49 4.1 Link to measurement dependent locality . . . 56
5 Other works 61 5.1 Entanglement depth from nonlocality . . . 62
5.2 W-state . . . 65
5.3 Nonlocality in quantum networks . . . 69
6 Conclusion and Outlook 73 6.1 Conclusion . . . 73
6.2 Outlook . . . 75
A Facets of MDL polytopes 85
B Papers published in the course of this thesis 95
Introduction
The science of physics is concerned with the understanding, prediction and formal- ization of the natural behavior of the universe. For many physicists, myself included, the true fun begins when our observations challenge our most basic intuitions. Some might say that this is especially true when the observations are not explained by current theories either, forcing us to go beyond our established and comfortable concepts. While I would agree that those are the most exciting times, the most enjoyable ones, to me, are when theory and experiment come together perfectly and stump us completely. The math is there, the observation confirms it and it still just does not seem to make sense. The concept underlying this thesis is such a case.
The early 20th century saw the advent of two big revolutions in physics that have since then been the cause of a lot of fun and frustration for the physics community:
the theories of relativity and quantum mechanics. The theory of relativity is only of marginal concern to this thesis, but quantum mechanics is at its core. While
‘Schr¨odingers cat’, an implication of quantum superposition, is probably the most well-known example of quantum weirdness, it is quantum nonlocality, a consequence of quantum entanglement, that we concern ourselves with in this work.
Locality is a very basic and intuitive concept in physics stating that objects only interact with their immediate surroundings. Causal effects happen locally and propagate at some finite speed through space-time. The first well-known theory that seems to be at odds with this principle actually predates the scientific revolutions of the 20th century by quite a bit: Newtonian gravity. In Newton’s theory of gravitation, masses interact with each other at a distance. If one were to move one mass, the effect would be instantaneously felt by the second mass, independent of how far away it is. This worried already Newton himself [13]. Luckily, this flaw is overcome by Einstein’s theory of general relativity. It is thus an almost ironic coincidence that Einstein himself played a key part in the development of another theory that suffering from such nonlocal effects in the form of quantum mechanics.
In quantum mechanics, two subsystems can be prepared in such a way that it is impossible to fully describe the composite system by looking at them separately;
there is more to a system than the composition of its parts. This is quantum entanglement. A side effect of entanglement is that revealing information about one of the subsystems by performing a measurement appears to have an immediate impact on the other. This idea is again at odds with our intuition of locality, as was already established by Einstein-Podolsky-Rosen in 1935 [27] . This ‘spooky action at a distance’, as Einstein referred to it, is what led many scientist to believe that
another, deeper theory would eventually be found to reproduce the predictions of quantum mechanics while being local at its core. In 1964, John Bell formalized locality in terms of observations and showed that it is possible to decide whether or not the predictions of a theory or the observations in an experiment could be explained locally [11]. Surprisingly, his work revealed some predictions of quantum mechanics to lack a local explanation; as a theory of information quantum mechanics is inherently nonlocal. The remaining hope for saving locality by proving quantum mechanics wrong was put to rest by the experiment of Aspect in 1982 [4], and more conclusively in 2015 [31, 35, 52]: according to Bell’s definition of locality, the world is nonlocal.
This fact is not only highly interesting from a fundamental point of view, but is the core idea behind a new framework of applications called device independent quantum information processing (DIQIP) [10, 18, 22]. The concept is quite fascinat- ing: it is possible to use quantum devices to perform tasks without knowing how they are built. We explain the rough idea behind DIQIP in Chapter 2.
In short, quantum nonlocality is at the intersection of fascinating fundamental physics and new frameworks of applications. Within this thesis, we study several naturally arising problems of nonlocality and provide potential solutions. The orga- nization of the thesis is as follows.
Chapter 2: Nonlocality – Since it is the underlying theme of the thesis, it is vital to have an understanding of the concept of nonlocality. Chapter 2 is devoted solely to this. We introduce the ideas of outcome independence, parameter indepen- dence and measurement independence, which together define locality. The results within the chapter have been previously derived: the set of local correlations has a specific form, called a polytope, implying that linear inequalities can witness non- local correlations. This is used to show that quantum mechanics is nonlocal. The way in which we derive that the local set is a polytope is however new and based on a theorem we introduced in [48]. Since the theorem will play a crucial role in later chapters, we apply it to re-derive this previously known result. Finally, we also introduce the set of non-signaling correlations, i.e. the set of correlations that does cannot be used for communication.
Chapter 3: Measurement dependent locality – Among the three assump- tions that locality is comprised of, the measurement independence assumption stands out since there is no way to guarantee that it holds. This is troublesome since the interesting implications of nonlocality only hold if we assume that it is the outcome independence assumption which is violated. While the measurement independence assumption cannot simply be removed, we show in chapter 3 that it can be weakened.
We define and study the set of measurement dependent correlations, proving that it too forms a polytope. Using this, we show that the nonlocality of quantum mechan- ics resists an arbitrary amount of measurement dependence, sometimes also called an arbitrary lack of free choice. This work has previously been published in [48]. An experimental implementation demonstrating this fact was published in [2]. We fur- ther report on our work published in [46], which extends the study of measurement dependent locality to more general scenarios and introduces the independent sources
assumption. Finally, we show how to deal with measurement dependence when finite statistics and memory effects are taken into account, which is an unpublished result.
Chapter 4: Limited detection locality – One of the biggest problems that experiments and applications of nonlocality are constantly faced with is that im- plementations are never perfect. Specifically, the fact that losses and non-detection events seem unavoidable is one of the main reasons that it took 51 years since Bell’s paper in 1964 to perform a loophole free nonlocality experiment. In chapter 4, we approach this problem in a different manner: instead of trying to find ways to re- duce the losses in an implementation, we introduce an assumption which, together with outcome, parameter and measurement independence, leads to the concept of limited detection locality. We analyze the set of limited detection local correlations and show that the assumption is quite powerful: even at its weakest, it allows for nonlocality to be detected independently of how lossy the setup is.
Chapter 5: Other works – We use this chapter to introduce other works that were accomplished in the course of this thesis and published in [6, 40, 50]. These works showcase three different aspects of nonlocality. In Section 5.1, we show how nonlocalty can be used to witness interesting quantum properties, in this case so- called k-partite entanglement. In Section 5.2, we analyze how resistant to noise the nonlocality inherent in a specific class of quantum states is. Finally, in Section 5.3, we apply the ideas behind nonlocality to an altered setup in the form of quantum networks.
In the following, we provide a brief introduction to two of the main mathematical concepts that are necessary to understand the rest of this thesis.
1.1 Notations, definitions and mathematical back- ground
Besides a decent understanding of the mathematics behind quantum mechanics, this thesis requires at least a basic understanding of probabilities and polytopes. While we can of course not provide a complete treatise in this section, we do present a short introduction to polytopes in section 1.1.1 and one to probabilities in section 1.1.2. Furthermore, we explain the notation that is used in the rest of the thesis.
Experts may want to skip this section and only return to it if questions arise or notations are unclear in the later parts of the thesis.
1.1.1 Finite convex polytopes
A finite convex polytope is a convex geometric structure with a finite number of extremal points, called vertices. Since it is convex it is fully described by the set of vertices; all other elements that are part of the set can be obtained by convex combi- nations. Equivalently, a finite convex polytope can be described as the intersection of a finite set of halfspaces. The delimiting hyperplanes defining the halfspaces are
Figure 1.1: A polytope in 2 dimensions. The set of vertices is given by V ={vi}5i=1
and the facets are given by the solid black lines. In 2 dimensions, every facet is given by 2 vertices. This is however not the case in general dimensions. By determining in which halfspace with respect to each facet a point lies, it is easy to determine if it is inside or outside the polytope. The pointq lies on the wrong side of the facet (v2v3) and is thus not part of the polytope. The point p on the other hand lies on the correct side of each facet and is thereby a member of the polytope. It is often computationally difficult to find all of the facets. Hyperplanes like the red dashed line can in such cases still be useful, as it would still allow one to determine that the point q is not a member of the polytope.
referred to as the facets of the polytope. Fig. 1.1 illustrates an example in two di- mensions. For a more complete introduction to polytopes we refer to Gr¨unbaum [32].
Vertices: For a given polytope P, the set of its n vertices is denoted by VP = {vPi }ni=1. Every point that is a member of the polytope can be written as a convex combination of these vertices;p∈ P ⇔p=P
iαivPi with αi ≥0,P
iαi = 1. A ver- tex cannot be expressed as a combination of the other vertices, i.e. vPj =P
iαivPi ⇔ αi =δij. Besides being able to generate every point of the polytope, the set of ver- tices can be useful when optimizing convex quantities: any convex function from elements of the polytope toR will achieve each of its global extrema at at least one of the vertices. In fact, for each vertex there exists a linear function that witnesses the vertex, i.e. ∀i ∃ci such thatci·vPi < ci·p ∀p∈ P, p6=vPi .
Facets: We denote by FP ={Fi}mi=1 the set of facets of a given polytopeP. The facets can be written as linear equalities given by a vector f and a real number B with F ≡f ·p=B. Since by definition the elements of the polytope are defined to lie in the same halfspace with respect to each facet, it follows thatp∈ P ⇔f·p≤B
∀(f, B) ∈ FP where · denotes the scalar product1. The facets are useful to deter- mine whether a point p lies inside the polytope or not; one only needs to check whether prespects all the inequalities given byFP. We will often call an inequality a facet, by which we mean that the delimiting hyperplane of the halfspace given by the inequality is a facet.
It is possible to find FP given VP and vice versa. We will not elaborate on how
1If the polytope lies in the halfspace given byf·p≥B for a given facet, one can simply choose f → −f and B→ −B in order to obtain the given form.
this is done here, suffice to say that it can be done using programs such as, for instance, the porta-library [19]. Unfortunately, performing this transformation is computationally expensive. It is therefore useful to establish inequalities that, even though they are not facets, hold for all the elements of the polytope. A violation of such an inequality implies then that a point is not a member of the polytope.
Within the context of this thesis, we are mostly interested in certifying precisely such non-memberships.
Equality constraints: Besides the inequality constraints given by the facets, a polytope can also satisfy some equality constraints. Consider for instance the poly- tope in R3 given by the set of vertices
(1,0,0),(0,1,0),(0,0,1) . This polytope fulfills the equality constraint x+y+z = 1 where x, y, z are the coordinates in R3. Even though this polytope is embedded in R3, it is only two dimensional. A simple variable elimination, e.g. z = 1−x−y, leads to a two dimensional representation with no equality constraints, given by the vertices
(1,0),(0,1),(0,0) . Due to the equality constraint, inequalities and facets of this polytope can be expressed in sev- eral equivalent ways. The inequality x+y≤ 1 for example is equivalent under the equality constraint to the inequality z ≥ 0. To circumvent this problem of having the same inequalites and facets expressed in different forms, it can be useful to es- tablish a canonical form for them. In the example at hand, if we require that the canonical form of an inequality or facet has B = 0, thenz ≥0 is the only remaining form.
Polytope cuts: Consider a polytope P. If we cut P with a hyperplane, we get another polytope in a lower dimension. This corresponds to imposing additional equality constraints. If the set of facets FP is known, then imposing such equal- ity constraints is straightforward; eliminating variables will yield the facets of the new polytope together with some superfluous hyperplanes. Consider a cube in 3 dimensions, given by
x≥0, x≤1, y≥0, y≤1, z ≥0, z ≤1.
If we want to perform a cut by imposing the additional equality constraint x−y+ 2z = 1, we can express x= 1 +y−2z in the facet defining inequalities, leading to
1 +y−2z ≥0, 1 +y−2z ≤1, y≥0, y≤1, z ≥0, z ≤1.
While the facets of the new polytope are a subset of these 6 inequalities, not all of them are facets. In fact y ≤1 and 1 +y−2z ≥0 together implyz ≤1, and y≥0 together with 1 +y−2z ≤ 1 implies z ≥ 0. The new polytope has only 4 facets given by
1 +y−2z ≥0, 1 +y−2z≤1, y≥0, y≤1.
If only the set of vertices is given for a polytope, then in order to perform a cut we first find the facet representation and use it to perform the cut.
1.1.2 Probabilities and random variables
Quantum mechanics and more specifically quantum information theory make heavy use of the concept of probability distributions. The study of probabilities itself goes
beyond what is required to understand this work. We provide here a basic intro- duction. For illustration purposes, the case of a die-throw will serve as an example.
Probability space: A probability space is given by a triplet (Ω, E, P).
• Ω is the set of all outcomes. In the case of a standard 6-sided-die-throw, this set is given by {1,2,3,4,5,6}.
• E is a Σ-algebra2 on Ω. It represents all events that one is interested in. This varies with the purpose of the die-throw. It could be that one is only interested in the parity of the result, which givesE =
∅,{1,3,5},{2,4,6},{1,2,3,4,5,6} . In another case one may want to know whether the result was a 6 or not, resulting in E =
∅,{1,2,3,4,5},{6},{1,2,3,4,5,6} . In general one can simply take E to be the powerset of Ω, i.e. the set of all subsets of Ω, which corresponds to all possible events.
• P is a probability measure, meaning a function P : E → [0,1] giving the probability of the event happening. It has to satisfy two requirements. First, P(Ω) = 1, which translates to saying that some outcome within the set of possible outcomes was produced with certainty. Second, for e1, e2 ∈ E with e1∩e2 =∅,P(e1∪e2) =P(e1)+P(e2), meaning that if two events are mutually exclusive, then the probability of either one or the other occurring is given by the sum of their individual probabilities. In the case of fair die being thrown, one hasP({1}) =. . .=P({6}) = 16 and the probability of other events can be deduced from these using the second property. For example, the probability of getting an odd outcome is given byP({1,3,5}) =P({1})+P({3})+P({5}) =
1
2 since the events ’the die shows 1’, ’the die shows 3’ and ’the die shows 5’ are mutually exclusive.
Random variable: A random variableV is a functionV : Ω→AV from the set of all outcomes Ω of a probability space (Ω, E, P) to a measurable setAV, which we call the alphabet of this random variable. The random variable inherits a probability dis- tribution PV from the probability space, given by PV(v) =P({ω∈Ω :V(ω) = v)}. Let us look at the example of rolling a fair die and playing a game where a player gets points equal to twice the number that they rolled. We can then define the random variable V as the function V(1) = 2, V(2) = 4, . . ., V(6) = 12. Even further than just asking what the probability of gaining a certain amount of points is, we can also ask what the probability of getting points within a certain range is. For example, the probability that a player gets strictly more than 3 but strictly less than 7 points on their die-roll is given by PV(3 < V < 7) = P({ω ∈ Ω : 3 <
V(ω) < 7}) = P({2,3}) = 13. Furthermore, random variables are useful in order to select the outcomes of one object in a probability space spanning several. An easy example is the case of throwing two dice, where the outcome set is given by {(1,1),(1,2), . . . ,(6,6)}. If we want to conveniently consider only one die or the other, we can define two random variablesD1 andD2, whereD1 maps any outcome of the two dice onto the outcome of the first die, i.e. D1((d1, d2)) =d1 and similarly for D2.
2A Σ-Algebra is a set of subsets that is closed under complement, union and intersection.
Joint distribution: Two random variables V1 : Ω →A1 and V2 : Ω→ A2 on the same probability space (Ω, E, P) have a joint probability distributionPV1V2 given by PV1V2(v1, v2) = P({ω ∈ Ω : V1(ω) = v1, textV2(ω) = v2}). This corresponds to the probability that V1 takes the value v1 and V2 takes the value v2 at the same time.
The definition trivially generalizes to the probability for both of the variables to be within a certain range and of course to an arbitrary number of random variables.
Within the previous example of rolling two dice, we can therefore determine the probability that the first die shows 3 pips while the second die shows 5 or more:
PD1D2(D1 = 3, D2 ≥ 5) = P({(3,5),(3,6)}) = 181. Note that it is also technically possible to ask for a relationship between the two random variables. For example, PD1D2(D1 =D2) =P({(1,1),(2,2),(3,3),(4,4),(5,5),(6,6)}) = 16 is the probability that the two dice show the same number of pips.
Marginal distribution: Given the joint distribution PV1V2 of two random vari- ables V1 and V2, we can compute the distributions PV1 and PV2 of the individual random variables directly without having to resort back to the distribution P of the underlying probability space. This is done by simple integration or summation over the random variable we wish to eliminate, i.e. PV1(v1) = R
F2dv2PV1V2(v1, v2) and vice-versa.3 Within the context of this joint distribution, the distributions PV1 and PV2 are called the marginal distributions.
Conditional distribution: Given the joint distributionPV1V2 of two random vari- ablesV1 andV2, we define the distribution ofV1 conditioned onV2 byPV1|V2(v1|v2) =
PV1V2(v1,v2)
PV2(v2) for PV2(v2) 6= 0. This corresponds to the probability that V1 takes value v1 if V2 takes value v2. Let us reconsider the example of throwing two dice, but now we define the random variable S to be the sum of the number of pips of the two dice. The joint distribution PD1S can easily be computed. The conditional distribution allows us to determine what the probability is that the first dice shows 3 pips given that we know that the sum is larger than 7, i.e.
PD1|S(D1 = 3|S ≥7) = PD1SP(D1=3,S≥7)
S(S≥7) = 1/127/12 = 17.
Independent distributions: The reason we did not use D1 andD2 as the example for conditional distributions is because they are in fact independent. Two random variables are said to be independent if knowledge about one of them tells us nothing about the other. The pips of the two dice in a fair dice-throw are a good example of this; knowing how many pips one of them shows gives us no information about the other. This manifests directly in the conditional distribution. Two random vari- ablesV1 andV2 are independent if and only if PV1|V2(v1|v2) =PV1(v1) for all v1 ∈A1, v2 ∈A2. This was clearly not the case when we looked at the distribution ofD1 and S.
Extended distribution: Throughout this work, a question that will arise several times is whether a given distribution can be written as the marginal of another one which fulfills some conditions. However, this second distribution does not need to be defined on the same event-algebra. Let (Ω, E, P) be a probability space with a
3In the case where A2 is a discrete set, the integral is replaced by a sum over all the elements ofA2.
random variableV : Ω→A. Let Ω0 andE0 be an outcome-set and an event-algebra.
We define the extended outcome-set ˜Ω = {(ω, ω0) : ω ∈ Ω, ω0 ∈ Ω0} and extended event-algebra ˜E =
{(ωi, ωj0)}i,j : {ωi}i ∈ E,{ωj0}j ∈ E0 . Let ˜P be a probability distribution on ( ˜Ω,E) such that ( ˜˜ Ω,E,˜ P˜) is a probability space. We define the random variable ˜V : ˜Ω → A with ˜V((ω, ω0)) = V(ω), which we call the extension of V, and an arbitrary random variable V0 : ˜Ω → A0. We say that PV V˜ 0 is an ex- tended distribution of PV if PV(v) = PV˜(v) ∀v ∈A. As an example, let PD be the distribution of a single die roll, with a well-established probability space. Then the distribution PD1D2 of rolling two dice, which we considered above, is an extended distribution ofPD with ˜D=D1, as is the distributionPD1S.
The notation so far has been rather cumbersome. To alleviate this, we only specify the random variables if they are not clear from context. In general, we use uppercase letters to denote random variables and the corresponding lowercase letters for a generic value within its alphabet. We thereby shorten PV(v) toP(v). We also abuse notation by denoting the alphabet of a random variable by the same letter as the random variable; for example, we writeR
V dv instead ofR
AV dv when we want to integrate over the full alphabet ofV. Whether the random variable or the alphabet is meant is clear from context. Finally, the notation for extended distributions is far less confusing if the random variable and its extension are denoted by the same letter.
Nonlocality
As is so often the case, nonlocality is best explained as a game. Two players, called Alice and Bob, participate in a cooperative game show. The game they have to play is well-known in advance. They will separately enter two sealed rooms with no means of communication to the outside or between each other. They are allowed to take anything they want with them as long as it does not allow them to communicate.
Inside the rooms, they will find two balls, one black, one white. After they enter, the game master will flick a switch on each room, turning on either a green or a red light inside the corresponding room. Alice and Bob then pick one of the balls and leave the other, see Fig. 2.1. They win the game under the following conditions: if the game master turned on at least one green light (Alice and Bob both saw a green light, or Alice saw green and Bob saw red, or vice-versa), then the players win if they chose balls of the same color. However if both lights were red then they win if their chosen balls have different colors.
Figure 2.1: Two players, named Alice and Bob, enter a room each in which a black and a white ball have been previously placed. After entering the room, either a green or a red light is turned on inside each room. The players choose one of the two balls to take out of the room and leave the other. Their choice of ball is labeled by A and B and the color of the light by X and Y for Alice and Bob respectively.
Before entering the room, the players can meet and agree on a strategy, labeled by Λ.
It is of interest to determine the players’ chances of winning this game. Consider one particular strategy which they could use: independently of which light is on, both players always choose the black ball. In this case, they win in the three cases where at least one light is green, but lose when both lights are red, i.e. they win in 3 out of 4 cases. Of course one may now argue that ignoring the lights has to be
suboptimal, so let us consider a second strategy: Alice still chooses the black ball independent of her light, Bob on the other hand now chooses the black ball when he sees a green light and the white ball when he sees a red light. With this strategy they win when both lights are red (since Alice chooses black and Bob white), but lose when Alice’s light is green and Bob’s is red. So again they win in 3 out of 4 cases. Continuing this analysis quickly leads to the conclusion that it is in fact not possible to win more than 3 out of 4 games, which still holds true if Alice and Bob were to introduce some randomness into their choices, e.g. by flipping potentially biased coins. The fact that the players have to agree on a strategy before the buttons are pressed and that there is no communication between the two rooms caps their winning ratio at 3 out of 4. This can of course be proven rigorously [21]. We give a proof in Section 2.1.
Strategies such as the ones considered above are called local. Alice and Bob agree on a predetermined strategy and choose their ball based on this strategy and the input they get from the game master. We denote Alice and Bob’s respective inputs by X ∈ {red,green} and Y ∈ {red,green}, their choice of ball, or output, by A ∈ {black,white} and B ∈ {black,white} and their pre-agreed strategy by Λ.
Their joint conditional probability distribution can be written as P(ab|xy) =
Z
Λ
dλPΛ(λ)PA|XΛ(a|xλ)PB|YΛ(b|yλ). (2.1) P(ab|xy) is the probability of Alice giving outputa and Bob giving output b condi- tioned on Alice’s input beingx and Bob’s input beingy. They can choose different strategies λ with probability P(λ). Since the game master is only interested in which outputs were given for each input and not in which strategy was used to achieve this, the strategies λ are averaged out. The rest of the expression is based on three assumptions. First, the only information Bob has about Alice’s choice of ball is given by the hidden strategy and the inputs, i.e. PB|AXYΛ(b|black, xyλ) = PB|AXYΛ(b|white, xyλ) for all hidden strategiesλ, all inputsxand y and all outputs b. Second, Bob has no information about Alice’s input, so his choice of ball can- not depend on it. This implies thatPB|XYΛ(b|green, yλ) =PB|XYΛ(b|red, yλ) for all strategies λ, inputs y and outputs b. The same two conditions hold vice-versa for Alice’s knowledge about Bob’s output and input. Finally, Alice and Bob have no information about the inputs when deciding on their strategy and the inputs are not influenced by their choice of strategy, i.e. P(λ|xy) =P(λ|x0y0) for allλ, x, x0, y, y0.
As discussed above, these local strategies can only win 3 out of 4 games on average. This condition can be expressed as
WCHSH(P)≤3 with
WCHSH(P) = P(A=B|gg) +P(A =B|gr) +P(A=B|rg) +P(A 6=B|rr), (2.2) where we shortened green to g and red to r. This game is known as the CHSH game [21]. Given how natural and intuitive local strategies are, it may come as a surprise that if Alice and Bob are familiar with quantum mechanics, they can perform better. To achieve this, they prepare, in advance, two quantum systems in an entangled quantum state. They each take one of the system as well as a
measurement apparatus with them into the room. After they enter their respective room and the lights have been turned on, they choose, based on whether they see a red or a green light, a measurement to perform on their respective system.
They decide which ball to pick dependent on the outcome of this measurement. By choosing the quantum state and the measurements smartly, as shown below, the players can now win on average 3.41 out of 4 games. This is the ‘spooky’ part of quantum mechanics; despite there being no communication between the systems, they can still exhibit correlations that cannot be explained by local distributions.
Quantum mechanics is inherently nonlocal. This was first shown by Bell in his famous theorem in 1964 [11].
To verify that quantum mechanics is nonlocal, consider the quantum state
|Φ+i= 1
√2 |00i+|11i
, (2.3)
where the first system will be with Alice and the second with Bob. This state is called the maximally entangled 2-qubit state. If Alice’s light is green, she measures in the
|0i,|1i basis, if it is red, she measures in the|0i+|1i
√2 ,|0i−|√ 1i
2 basis. Similarly, Bob measures in the
cosπ8|0i+sinπ8|1i,cos5π8 |0i+sin5π8 |1i basis in case of a green light and in the
cosπ8|0i −sinπ8|1i,cos3π8 |0i+ sin3π8 |1i basis in case of a red light.
Computing the resulting correlations, we find
PQ(A=B|gg) =PQ(A=B|gr) =PQ(A=B|rg) = PQ(A6=B|rr) = 2 +√ 2 4
(2.4) Evaluating WCHSH(P) in (2.2), we find a value of 2 +√
2 ≈ 3.41 > 3. It can be shown that this is the maximal value achievable by quantum mechanics [20].
The discovery of nonlocal correlations and their existence has major conse- quences. On the one hand, it challenges the intuition of physicists used to the classical world. In fact, before John Bell’s work demonstrating quantum nonlocality in 1964 [11], many physicists argued that there was a local mechanism explaining quantum correlations that had simply not yet been discovered [27]. Bell’s theorem proves that this is not the case, at least if we assume that there is a maximal speed with which influences propagate. The concept is therefore of major interest in the realm of fundamental physics. On the other hand, nonlocality gave rise to the con- cept of device independent quantum information processing (DIQIP) [10,18,22] with many potential applications. To understand DIQIP, it is useful to first introduce the adversarial point of view.
Let us turn the game example around. The game master, which we will from now on call Eve, is trying to convince Alice and Bob that she has nonlocal quantum resources at her disposal. To demonstrate this, she builds two devices. The devices each have two buttons, one green and one red, and a screen which can turn either black or white. Alice and Bob each take one of the devices and separate them such that the devices cannot communicate, either by putting a large distance between them or by putting them in sealed rooms. They then play the game we introduced earlier, only now they are the ones asking the questions by pressing either the green or red button and the devices have to show either a black or a white screen. If the
Figure 2.2: An adversary Eve built two devices with two buttons each and a screen which can be either black or white. She gives one device each to Alice and Bob, who can now press the buttons on the device and record the color of the screen. It is Eve’s goal to convince Alice and Bob that the devices are operating in a nonlocal manner, meaning that she wants to perform well in the CHSH game (2.2).
devices manage to win, on average, more than 3 out of 4 games, then Eve was telling the truth. This type of setup is called an adversarial scenario (see Fig. 2.2).
As a consequence of this, Alice and Bob now know that Eve could not have built these devices with a preplanned strategy; any preplanned strategy Λ can only win an average of 3 out of 4 games. Alice and Bob conclude that their outcomes could not have been predetermined, but were generated only after the devices have been separated. Even further, if Alice and Bob now communicate to each other which button it was that they pressed, they can infer with high probability which color the other person saw based on the color they saw on their own screen. If the left-hand- side of (2.2) is large, then with high probability the screens show the same color if at least one of them pressed the green button and opposite colors if they both pressed the red button. So not only are their outcomes random, but they are also strongly correlated. If one knows both inputs and the output of one of the devices, one can infer with high probability the output of the other. Eve however, who only knows both inputs cannot infer either output1. If Alice and Bob wish to create random numbers or to generate a secret key that only they know, for example in order to communicate later using a one-time-pad encryption, they can use devices that were built by a third party Eve and, if they perform well in the game, even Eve, the manufacturer of the devices, cannot guess the random numbers or decipher the encryption.
A valid objection at this point is of course that Eve could have built a third device with its own buttons and screen that could be correlated in a similar manner with Alice and Bob’s devices. This is however not possible due to a fact referred to as themonogamy of nonlocal correlations [43]. In rough terms, it states that if two devices have strong nonlocal correlations between them, then a third device cannot be correlated with either. We will not elaborate on the details here, suffice to say that this is a very powerful property of nonlocal correlations; it implies that if the correlations between Alice’s and Bob’s device are nonlocal enough, then there is no
1Technically if WCHSH(P) does not evaluate to its maximal value, knowing the inputs can allow to partially guess the outputs, but this guessing probability gets lower the closer the value gets to the maximum [43]
way for a third party to know anything about their outcomes.
Since nonlocality is a property only of the observed correlations, it is not nec- essary for Alice and Bob to know anything about the inner workings of the device;
they just need to test the correlations and can infer the desired properties from them. This is why this type of information processing is called device independent.
Developing protocols that successfully fulfill these tasks is a quite daunting enter- prise. Proving that a given protocol is truely secure against any attack an adversary could perform is in general very difficult, especially if errors have to be tolerated in order to move towards actual implementations. A large body of work has been realized in the past years towards studying interesting DIQIP protocols, be it for quantum cryptography [1, 9, 28], randomness amplification [24, 29], extraction and expansion [23, 44] or certification of quantum entanglement [5, 7]. All these current and potentially yet undiscovered future applications as well as the interest for fun- damental physics warrant a continued study of nonlocal correlations.
Let us elaborate more clearly on the assumptions that go into the definition of locality. In general, any conditional probability distribution over two random variables2 A, B conditioned on two other random variablesX and Y, assuming the existence of a fifth random variable Λ, can be written as
P(ab|xy) = Z
dλP(λ|xy)P(a|xyλ)P(b|axyλ).
We say that PAB|XY is local if there exists a Λ such that the following three inde- pendence conditions are fulfilled.
• Outcome independence: Any correlation between the outputs A and B comes from their inputs X and Y as well as the common strategy Λ:
P(b|axyλ) =P(b|xyλ) ∀a, b, x, y, λ
• Parameter independence: The input on Alice’s side cannot influence Bob’s output and vice versa:
P(b|xyλ) =P(b|yλ) ∀b, x, y, λ P(a|xyλ) = P(a|xλ) ∀a, x, y, λ
• Measurement independence: The common strategy Λ is independent of the inputs X and Y:
P(λ|xy) = P(λ) ∀x, y, λ
Under these assumptions, the distribution PAB|XY is given by P(ab|xy) =
Z
dλP(λ)P(a|xλ)P(b|yλ).
This is the general form of a bipartite local distribution. In the following, we define local distributions more generally and more carefully and study the set of all local distributions.
2See section 1.1.2.
2.1 The local polytope
So far, we had two parties, Alice and Bob, using devices or playing games with two inputs and two outputs for each party. The concepts of nonlocality and de- vice independent quantum information processing are not limited to such a special case. They can be applied to any setup where distinct parties make a choice of measurement and get an outcome. We call a specific instance of such a setup a scenario.
Definition 2.1 (Scenario). Let S = N,Ω, E, A1. . . AN, X1. . . XN
with N ∈ N, Ωan outcome-set,E an event-algebra onΩandA1. . . AN, X1. . . XN random variables on Ω. Then we call S a scenario.
N is the number of parties involved in the setup, Ai and Xi denote the output and input random variables of thei-th party. In our original example of the two players Alice and Bob getting inputs in form of green (g) and red (r) lights and giving outputs by choosing black (b) or white (w) balls (see Fig. 2.1), the outcome-set Ω is given by the 16 4-tuples{(bbgg),(wbgg),(bwgg), . . . ,(wwrr)}. The input and output random variables areX(abxy) = xand A(abxy) =a for Alice andY(abxy) = yand B(abxy) =b for Bob. Within a scenario, we are interested in studying the possible probability distributions associated with the inputs and outputs.
Note that it is possible for a scenario to have different number of outcomes for different inputs in a scenario. This is very natural since each party can perform different measurements for each input and these measurements can of course have different sets of outcomes. This is included within the given definition by having an outcome-set Ω that only includes elements corresponding to input-output combina- tions that can occur. In our previous example, if Alice could pick a third yellow ball when the light is red, but not when it is green, then elements given by (y·r·) are included in while tuples given by (y·g·) are excluded from Ω.
In the case where Alice and Bob get a finite number of possible inputs and outputs, we can without loss of generality denote these by natural numbers. Instead of specifying the full set Ω, it is sufficient to specify the number of parties N, the number of inputs ni for each party i as well as the number of outputs mji for each input j and for each party i. Furthermore, in the remainder of this work, we fix the names of the random variables for the inputs and outputs. In the case of two parties, we name the parties Alice and Bob and denote the inputs by X and Y and the outputs by A and B for respectively. If N ≥ 3 parties are involved, or if we consider the general case for any number of parties, we call them Alice1 through AliceN and label there inputs and outputs by X1. . . XN and A1. . . AN respectively. Such scenarios are then fully described by giving the number of parties, the number of inputs for each party and the number of outputs for each input for each party, N,(n1, . . . , nN),((m11, . . . , mn11), . . . ,(m1N, . . . , mnNN))
. Whenever we consider a general scenario S, we will use this nomenclature for the parameters involved.
To ease notation, we introduce, for a fixed scenario, the random variables A~ = A1. . . AN) and X~ = X1. . . XN
, that is random variables containing the outputs and inputs, respectively, for all parties. In addition, since we usually consider sce- narios where all parties have the same number of inputsn and/or all inputs have the
same number of outputs m, we introduce a shorthand notation for this case as well by only indicating the number once, i.e. (N, n, m). For example, the scenario of two parties getting ternary inputs and giving four possible outputs is denoted by (2,3,4).
We can now define local distributions in a general scenario.
Definition 2.2 (Local distribution). Let S be a scenario. Then the conditional distributionPA~|X~ is local if there exists an extended distributionPA,Λ~ |X~ satisfying
P(~a|~xλ) = Y
i
P(ai|xiλ) and P(λ|~x) =P(λ)
∀~a, ~x, λ.
The set of all local distributions for a fixed scenario S is denoted by LS.
These are all the distributions that fulfill the assumptions of outcome independence, parameter independence and measurement independence introduced previously.
Given the fundamental curiosity of nonlocality and its uses for device indepen- dent quantum information processing, we are interested in finding ways to determine whether a given probability distribution PA|~X~ in a scenarioS is a member ofLS or not. PA~|X~ can be seen as a vector in an QN
i=1niQni
j=1mji-dimensional space where each dimension represents one probability element P(~a|~x). The set of local dis- tributions LS then corresponds to a geometrical object. Here, we reproduce the previously known result thatLS is a polytope using a general theorem on polytopes we proved in [48]. This theorem will prove useful for considerations in later chapters of this thesis.
The basic statement of the theorem is the following. Given a set of polytopes, combining them in a specific way, which is precisely the way in which local correla- tions are constructed, results again in a polytope. If the vertices of the composing polytopes are known, then so are the vertices of the resulting polytope. While we only use it to replicate previously known results in this chapter, in later chapters we will use it to derive crucial results of this thesis.
Theorem 2.1. Let {Pi ⊂ Rmni}Ni=1 be N polytopes for m, n1. . . nN ∈ N. Let the entries of pi ∈ Pi be denoted by pi(k, li) with k ∈ {1. . . m}, l ∈ {1. . . ni}. Let
R=
r∈Rmn1...nN :∃Λ measureable set,
∃ρ: Λ→[0,1] with Z
Λ
dλρ(λ) = 1
∃{piλ}λ∈Λ ⊆ Pi s.t.
r(k,~l) = Z
dλρ(λ) YN
i=1
piλ(k, li) . where we defined~l= (l1. . . lN). Let VPi = {vPji
i}ji be the set of vertices of Pi
