Quantum Entanglement And The Relative Entropy Of Entanglement

Quantum entanglement has played a significant role in the field of quantuminformation and quantum computation. This attracts an increasing interest in thestudy of detection and quantification of entanglement for any quantum state. Al-though there is a clear definition of what an entangled state is, it is still di?cultto discriminate entanglement from separability for high dimension and multipar-tite mixed states. In particular, symmetric states have elegant forms to testify andquantify their entanglement. In this thesis, we mainly utilize a family of symmetricstates to construct PPT bound entangled states and derive the relative entropy ofentanglement for two kinds of di?erent symmetric states. In addition, we researchthe relationship between states of a quantum system and its subsystems.Even though bound entangled is indeed a very poor type of entanglement, itcan produce a nonclassical e?ect, enhancing quantum communication via a subtleactivation-like process. In section 3 we obtain an entangled condition for isotropic-like states by using an atomic map. A class of bound entangled states is constructedfrom the entangled condition and we show that the partial transposition of the statefrom the constructed bound entangled class is an edge state by using range criterion.The relative entropy of entanglement is one of the fundamental entanglementmeasures, as relative entropy is one of the most important functions in quantuminformation theory. In section 4 we calculate the relative entropy of entanglementfor rotationally invariant states of spin-12 and arbitrary spin-j particles or of spin-1particle and spin-j particle with integer j. Furthermore, as a by-product a lowerbound of relative entropy of entanglement and an upper bound of distillable entan-glement are presented for rotationally invariant states of spin-1 particle and spin-jparticle with half-integer j. One of the most fundamental properties of entanglement is monogamy. In sec-tion 5 we consider special two-qubit states called Ux-invariant states and give theirupper bounds of relative entropy of entanglement. The upper bound is shown tobe exact for some special Ux-invariant state according to examples. In addition, asan application we discuss the monogamy relation of relative entropy of entangle-ment by use of this upper bound of relative entropy of entanglement. We find thatmost of three-qubit W states satisfy the monogamy inequality of relative entropy ofentanglement.Entanglement can be viewed in terms of the relationship between states ofa quantum system and its subsystems. In section 6 at first we show that everyn parties pure state can be determined (among pure sates) up to a local unitarytransformation by its reduced matrices. Then we present that the local unitaryequivalence of n parties pure states is consistent with the local unitary equivalenceof their n ? 1 parties reduced density matrices.