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View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Electronic Thesis and Dissertation Archive - Università di Pisa UNIVERSITA` DI PISA Facolt`adi Scienze MFN Corso di Laurea Specialistica in Scienze Fisiche Tesi di Laurea Specialistica Area laws for Entanglement, QMA-completeness of the local Hamiltonian problem and the simulation of Quantum Mechanics. Relatori: Prof. Ettore Vicari Prof. Pasquale Calabrese Prof. Jos´eIgnacio Latorre Candidato: Giuseppe Vitagliano Anno Accademico 2009/2010 Alla mia famiglia e a Marcella 1 Contents Table of Contents I 1 Introduction: Physics of Information1 1.1 Beyond the classical physics of information...............2 1.2 Quantitative study of entanglement...................3 2 Overview of quantum mechanics6 2.1 Axioms of Quantum Mechanics.....................6 2.2 Density matrix..............................9 2.3 Open systems: POVM, decoherence and master equation....... 12 2.4 Quantum operation formalism...................... 14 3 Quantum information and complexity: the QMA class. 18 3.1 An introduction on information theory.................. 18 3.1.1 Classical information theory................... 19 3.1.2 Classical computational models................. 25 3.1.3 Complexity theory for classical machines............ 28 3.2 Quantum information theory....................... 33 3.3 Quantum information processing.................... 42 3.3.1 Quantum complexity theory................... 47 4 Entanglement theory 51 4.1 Introduction................................ 51 4.2 Bi-partite systems............................ 56 4.2.1 Separability criteria....................... 60 4.2.2 Entanglement manipulation and distillation........... 63 4.3 Entanglement Measures......................... 64 5 Hamiltonian Models 70 5.1 Introduction on gaussian states..................... 70 5.2 Spin systems............................... 75 5.3 Free fermionic systems.......................... 77 5.3.1 Ground State........................... 81 5.4 Free bosonic systems........................... 83 5.4.1 Ground State........................... 85 5.5 Entanglement............................... 86 5.5.1 von Neumann Entropy...................... 88 2 6 Known results on the scaling of entanglement 90 6.1 Fermionic homogeneous models..................... 91 6.2 Bosonic homogeneous models...................... 96 6.3 Non-homogeneous models........................ 99 6.3.1 Random Models......................... 100 6.3.2 Non homogeneous models with particular symmetries.... 101 6.4 Other results from homogeneous models: CFT approach....... 103 6.5 Simulations of quantum states...................... 105 6.5.1 MPS................................ 106 7 XX local model which highly violates area law. 110 7.1 Area laws vs QMA-Completeness of the local Hamiltonian problem. 110 7.2 Expansion of the entanglement entropy................ 111 7.3 RSRG approach and perturbation expansion of the GS........ 116 7.4 XX local model with high scaling of ground state's entanglement.. 119 8 Numerical Datas 125 Conclusions 128 A Impossibility of a null T submatrix with local interactions 132 Bibliography 134 3 Chapter 1 Introduction: Physics of Information Since the pioneering works of Alan Turing (Tur36) and then of Claude Shannon (Sha48) they have become progressively clearer the connections between the funda- ments of physics and a mathematical theory of information1. In particular, Turing's work was originally conceived to solve a purely mathematical task (riformulable also from a physical perspective) and opened the door to the formulation of the mathe- matical concepts of information and computation, but, trying to idealize a specific practical situation (that is the algorithmic resolution of a problem), implicitly pre- supposed a particular underlying physical framework (that is classical mechanics). Automatically, then, this \translation" led to a remarkable physical result: that not all the computational problems can be solved. Similar dynamics recurred on Shannon's works, which allowed to deduce important physical results transposing the problem of communication in terms of information transmitted. So, already in the dawn of information science, appeared very profitable the interchange of views between this kind of formalization and the physics itself, starting from its founda- tions. More explicitly, the first observation that could be done, also in the light of the recent developments, is that any set of physical principles could be used as basis for the development of a mathematical model of information, i.e. a theory that tries to capture the intuitive meaning of the concept of information. Actually, the previous statement is usually rephrased, quoting Rolf Landauer, saying briefly that information is physical. Some examples that would confirm this claim, in particular, will be explicitly viewed during the present thesis. Putting ourselves in a complementar point of view, then, historically it was proved to be useful to think about fundamental questions of physis in terms of information contained in a state; in some cases the same definition of a state or of a physical theory in information-theoretical terms could be extremely useful and clarificatory. An historically important example can be given considering the famous \Maxwell's daemon", a paradoxal situation originally thought to provide evidence of the possibility of diminishing the total entropy of a closed system. Loosely speaking, that system consists in a box, separable into two compartment and containing a gas of particles: the daemon acts on the splitting surface and let the fast particles come 1We will come back later, in a specific chapter, on the meaning of this expression. 1 1 { Introduction: Physics of Information into a compartment and the slow ones into the other; in this way it makes the total entropy to decrease. This apparently paradoxal situation can be solved in a very elegant way taking account of the entropy acquired by the daemon itself, considering finite the memory at its disposal to progressively register position and velocity of the particles: the daemon will be forced to erase bits of information when its memory will be full. In the total estimate (system + daemon) it can be seen that the entropy could not decrease. In a more modern perspective, instead, the same foundations of the pillar theories, like e.g. quantum field theories and general relativity, could be better explored thinking at the information content of a state. There are a lot of possible examples of this last claim too, looking at the theory of black holes or at the features of quantum mechanics like nonlocality or the decoherence phenomenon. 1.1 Beyond the classical physics of information. Following the paradigm \information is physical" and considering the fact that at a fundamental level the physical world is quantum-mechanical, it would result quite natural to formulate a sort of extension of the theory of information, which takes account of the principles of quantum mechanics: a \quantum information theory". In fact this idea has been recently followed and has been studied some quantum aspects of information, leading to a theory that today is subject of intense research (see (Pre04; NC00)). Historically, however, the motivations, more than theoretical, has been mainly driven by situations of \practical" interest. First of all, Feynmann himself realized the enormous practical difficulties to overcome when trying to classically simulate a \truly quantum" system and, on the other side, the great advantages achievable from the development of computational techniques of pure quantum nature, that would allow to simulate experiments otherwise very difficult or expensive. In addition, many scientists began to wonder about the possibility of improving the efficiency of calculators (in the theoretical terms that will be introduced later), defining some quantum computational models, and thus, in other words, about the possibility of seriously challenging the strong Church- Turing thesis (3.1.3) with those models. This approach led also to the conception of many quantum communication protocols, that would offer a great advantage over the traditional ones and acquired renewed enthusiasm when Peter Shor found a quantum algorithm capable to solve efficiently (i.e. in polynomial time) the factoring problem, that is still believed to be efficiently untractable on a classical Turing machine. In particular, the above-mentioned problem is at the basis of the actual standard cryptographic protocols (like e.g. RSA), which's security, then, would be seriously challenged by an eventual practical implementation of Shor's algorithm. On the other side, the same quantum information provides cryptographic protocols that are perfectly secure against any kind of possible attack; an example is the so called quantum key distribution protocol. All that led to the introduction of new computational, as well as communication quantum models and of a new complexity theory, that, among other things, is part of the present thesis' work, mainly focused on the so called QMA(Quantum Merlin Arthur)2 class. Still from 2We will give a precise definition of that class in a following chapter3 2 1.2 { Quantitative study of entanglement. a practical point of view, besides the benefits just viewed, it is even possible to take account of the technological issues that we will have in a near future, when the devices will be so small that it will be inevitable to take care of quantum effects. An useful empirical

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