On Bell Inequalities And Quantum Entanglement

In1935, A. Einstein, B. Podolsky and N. Rosen questioned the completeness of quantummechanical description by using local realism, which was believed to be the underlyingexplanation of physical reality. Local realism theory states that there are hidden variables whichdetermine the properties of physical system prior to and independent of measurement, also anotherassumption states that these properties of one system can not be influenced by another when thesetwo system are spacelike separated. In1964, Bell proposed his theorem that quantum mechanics(QM) violates certain inequalities that any local hidden variable (LHV) theory must obey. Theseinequalities are now called Bell inequalities which provide a tool to test whether the nature inclineto LHV theory or QM description. The violations of Bell inequalities by QM, specifically byquantum entangled state, was observed by multiple experimentalists, which indicate that the QMdescription fits the nature. Quantum entanglement is a fundamental concept in QM, which means astate of a composite quantum system can not be decomposed into the direct product of states onindividual subsystems. Though the understanding of quantum entanglement is still not quite clearyet, it has already played a central role and will become more and more important with thedevelopment of quantum information theory. It has become a useful resource in many quantuminformation processes, such as quantum key distribution, quantum teleportation, quantum densecoding, quantum calculation and so on. This thesis mainly concerns Bell inequalities and itsviolation by quantum entangled state, and it is organized as follows:In Chapter1we review some background and basic concepts in quantum information theory,such as the three of five postulates in quantum mechanics, pure state, mixed state, separable state,entangled state and qubit, which is necessary for the completeness of present thesis.In Chapter2we elaborate what LHV theory is really about and how to construct Bellinequalities based on LHV assumptions. We introduce some famous Bell inequalities, such asCHSH inequality a simplest one applied to two-qubit system and its generalization CGLMPinequality. We also introduce CFRD inequality which can apply to both qubit system andcontinuous variable system. The CFRD inequality was violated by composite system with morethan10subsystems, which is a extreme hash condition for experimental observation of violation.The CFRD inequality was optimized by Ji et al. We point out that these inequalities constructed byJi et al were not legitimate Bell inequalities. Local commutators are involved in their inequalities.They believed that local commutators are zeros in LHV theory, but in QM the commutatorsgenerally are not zeros. Thus, the violation of these inequalities might just originate from this trivial fact. We demonstrate that a more general inequality based on the assumption of vanishing localcommutators in LHV is violated by a pure separable state. Thus, such a violation can not beattributed to some kinds of non-locality. Actually, the CFRD inequality was well constructed thatthere are no local commutators involved on both sides of this equality. So, whether the assumptionof vanishing local commutators in LHV theory is right or wrong, The CFRD inequality is alwaystrue since the derivation of CFRD inequality is independent of this assumption.Bell inequalities exihibit the contradiction between QM and LHV theory in a probability way.There are also some other theorems without inequalities which exihibit the contradiction in adeterministic way. In Chapter3we introduce two theorems without inequality, specifically, GHZtheorem and Hardy theorem.Besides LHV theory, there is noncontextual hidden variable (NCHV) theory which states thatcompatible observable can be assigned a value. Kochen-Specker theorem point out that no NCHVtheory can reproduce the result QM predicts. In Chapter4we shall introduce KS theorem andNCHV theory.In Chapter5we derive a new equality based on NCHV theory. It is proved straightforwardlythat any two-qubit entangled state violates this equality. For continuous variable system, we applythis equality to two-mode squeeze state and find out that all two-mode entangled state violate thisequality. We then generalize this equality to multipartite system. The generalization form isviolated by quantum entangled state too. We analytically prove that all three-qubit entangled stateviolate this generalization form. Furthermore, we find that the violations of these generalizedequalities are responsible for the violation of CFRD inequality. We notice that these equalities areviolated by mixed separable state, which contradict with Werner’s results that all mixed separablestate admit a local hidden variable model. We then realize that in the derivation of these equalitiesdeterministic model can not apply to these cases, since for a deterministic model these equalitiesalways hold true, but it may not hold true for a dispersive situation, which means these equalitiesare not valid even for hidden variable theory. It is straightforward that all pure separable statessatisfy these equalities, then if for some measurements on subsystems these equalities are violatedthe studied state must be entangled. Thus, the usefulness of these equalities is to detect pureentanglement. We can still drown the conclusion that the violation of CFRD inequality by quantumpure state originate from the violation of these generalization form.Some conclusions and prospects are given in the last chapter.