Hidden Nonlocality Flavien Hirsch September 2012 - June 2013 University of Geneva Supervisor : Prof. Nicolas Brunner Abstract In this essay we study the hidden nonlocality which can be revealed by local filtering. We present the idea and the physical motivations, and we recapitulate all the known cases. We then construct some new examples in lower dimension and discuss the local models of the Werner states. For the bipartite qubits case, we propose some improvements on these models that allow us to simulate a larger class of states, and we construct a protocol that, from a projective-local state, gives a POVM-local one. Finally, we prove the existence of genuine hidden nonlocality with a few concrete examples, including maximal and genuine hidden nonlocality. 2 Contents Contents 3 1 Introduction 5 2 Concepts and motivations 6 2.1 Nonlocality and Bell’s game ..................... 6 2.1.1 The scenario .......................... 6 2.1.2 Classification of the probability distribution ....... 7 2.1.3 Bell’s game .......................... 9 2.2 Entanglement ............................. 11 2.2.1 Quantum formalism ..................... 11 2.2.2 Characterization of quantum states ............ 12 2.3 SLOCC and hidden nonlocality ................... 14 2.4 Notations ................................ 16 2.5 Classes of measurement ....................... 17 2.5.1 Quantum observable ..................... 18 3 Local states 19 3.1 Local models .............................. 19 3.1.1 Simulate POVMs ....................... 20 3.2 Werner state .............................. 20 3.2.1 Werner’s model ........................ 21 3.2.2 Barrett’s model ........................ 22 3.2.3 Interpretation of Werner’s and Barrett’s model ..... 22 3.3 New classes of local states ...................... 23 s 3.3.1 ra ................................ 23 3.3.2 The model ........................... 24 s 3.3.3 ra,b ............................... 26 s 3.3.4 ra,b,g .............................. 28 4 Examples of filterable states 30 4.1 General filtering and qubits filtering ................ 30 4.2 Werner state .............................. 32 4.3 Gisin’s state .............................. 33 4.4 Collins-Gisin’s state .......................... 34 4.5 Hidden nonlocality for a two-qubits state ............. 35 3 CONTENTS 4 5 Maximal and genuine hidden nonlocality 36 5.1 From projective measurements to POVMs ............. 36 5.2 Genuine hidden nonlocality ..................... 38 5.3 Maximal hidden nonlocality ..................... 39 5.4 Maximal and genuine hidden nonlocality ............. 42 5.5 Generalization of rGHNL and of the POVM protocol ....... 44 5.5.1 The protocol for a qudit ................... 44 a,b,g,d 5.5.2 rGHNL ............................. 44 6 Open questions 46 6.1 Sequential filtering .......................... 46 6.2 POVM versus projectors ....................... 48 6.3 Genuine locality ............................ 48 7 Conclusion 49 8 Acknowledgments 50 Bibliography 51 Chapter 1 Introduction For more than a century Quantum Mechanic has been studied and tested every day by the physicists. Thanks to the progress of the science and in particular those of the experimental apparatus, the precision with which one is able to verify the theories has become better everyday. So, for more than a century Quantum Mechanic has been put to the proof more and more rigor- ously, and to this day has never been faulted. The precision of the quantum theory is spectacular and it therefore seems to describe perfectly the observed phenomena; in all cases better than any other theory. Yet, however good are its predictions, Quantum Mechanic continues to defy the human intuition. Indeed, three properties of the theory are very difficult to accept: intrinsic randomness, collapse of the wavefunction and nonlocality. While the two first ones are fundamental properties of the theory and have given rise to fruitful discussions about independence of Nature and about determinism, the third one was discovered later and was considered first as an anomaly of the theory. Nowadays, we have tested and accepted nonlocality as a characteristic of Nature. It means that two systems can be well-separated in space (and in particular space-like separated) but act as if they were one single system. We strongly think that no realistic theories are able to predict the future. Knowing this, it is not astonishing that nonlocality has been vastly studied in the past thirty years. What seems less normal is the actual theoretical disorder about nonlocality. Multipartite scenarios, Bell’s inequalities, link between entanglement and local models: despite a lot of nice results and the work of brilliant people, these tasks are very partially understood, and we still lack an overview, a general formalism, an analytical method to solve the problems we have. Genuine hidden nonlocality is one of these big questions: can a fully local state becomes nonlocal after local operations? Even if some partial results are known (for projective measurements), no definitive answer has been yet given. In this work we will prove that the answer to the above question is yes. 5 Chapter 2 Concepts and motivations The aim of this first chapter is to introduce all the concepts that will be used later in this work and to motivate them physically. 2.1 Nonlocality and Bell’s game Since quantum mechanics gives probability to events, i.e. is not deterministic, we will focus ourselves on probability theory, define nonlocality and charac- terize the quantum probability distributions. Finally we will explain what we call a Bell’s game. 2.1.1 The scenario The standard situation we will focus on is the one with two parties, sharing some resources. Let’s call Alice the "left" one and Bob the "right" one for convenience. Figure 1: the general situation we consider Now Alice and Bob can perform a measurement on their respective re- sources, and obtain a result. We do not care about the exact mechanism of what they do, but our assumptions are the following ones: they can freely choose between a finite set of possible measurements and the set of the results they can obtain after the measurement is finite as well. If Alice has NA possible measurements we symbolically define MA = A , A , ..., A as the set of measurements and we call the integers num- { 1 2 NA } bers that label the measurements the inputs of Alice. We define RA = RA , RA , ..., RA as the set of the n results she can obtain and we call the { 1 2 na } A integers numbers that label the results the outputs. Similarly, the Bk represent 6 CHAPTER 2. CONCEPTS AND MOTIVATIONS 7 Bob’s measurements and the RBk his results (with set of size NB and nB, respectively). Note that it is a very general way to study two physical systems; the assumption about the finite number of inputs and outputs is used just for convenience (one can extend to countable and uncountable infinite number of possibility, in particular the quantum formalism allow such case) and we did not mention anything about the physical system nor what the measurement is. Since we do not focus on what really happens when a measurement is applied on a part, we can represent our scenario in the following way: both parties have a box, with a finite number of buttons on the top which can be pushed individually and a finite number of lights on the bottom which can light individually. What Alice and Bob can do is push one of the buttons -which therefore represent the inputs- and note which light is lit -i.e. which output they obtain. This approach is often called the device-independent approach. Figure 2: Each party has a box with possible inputs and outputs All the information about this bipartite system is contained in the prob- ability of obtaining a pair of outputs given a pair of measurements. In other terms we want to know p(a, b A , B ) (2.1) | x y where a 1, 2, ..., N , b 1, 2, ..., N That is (2.1) is the so-called joint 2{ A} 2{ B} probability that Alice obtains the result RAa given the measurement Ax and Bob obtains the result RBb given the measurement By. 2.1.2 Classification of the probability distribution From here we will rewrite (2.1) in the following form: p(a, b A, B), for con- | venience. Please keep in mind that A, respectively B, can be any measurement chosen in the set MA, respectively MB. Definition 1 (Determinism). A joint probability distribution p(a, b A, B) is de- | terministic if p(a, b A, B)=d d (2.2) | a,a0 b,b0 CHAPTER 2. CONCEPTS AND MOTIVATIONS 8 where a0 (b0) is the result that Alice (Bob) obtains after measurement A (B), and di,j = 1 if i = j, 0 otherwise. That is, a measurement determines precisely its result. Obviously this will never be the case with a quantum system. Definition 2 (Independence). A joint probability distribution p(a, b A, B) is inde- | pendent if it can be written as p(a, b A, B)=p(a A)p(b B) (2.3) | | | i.e. if the probability distribution factorizes. This corresponds to the trivial case where Alice and Bob are completely uncorrelated; there is no particular interest to study them together rather than separately in this situation. Definition 3 (Locality). A joint probability distribution p(a, b A, B) is local if it | can be written as p(a, b A, B)= p(a A, l)p(b B, l)r(l)dl (2.4) | ZL | | where L is a set, l L and r : L [0, 1] is a probability density. 2 ! The interpretation of this decomposition of the probability distribution is that Alice and Bob share some local resources, described by the variable l. The variable is said to be local because we assume that physically it is created somewhere (according to the probability distribution r(l)) and sent to Alice and Bob.