In contrast to abstract statistical analyses in the literature, we present a concrete physical diagrammatic model of entanglement characterization and measure with its underlying discrete phase-space physics. This paper serves as a pedagogical treatment of this complex subject of entanglement measures. We review the important inherent concurrence property of entangled qubits, as well as underscore its emergent qubit behavior. From the discrete phase space point of view, concurrence translates to translation symmetry of entangled binary systems in some quantitative measure of entanglement. Although the focus is on bipartite system, the technique is readily extendable to multi-partite system of qubits, as can easily be deduced from the physical inverter-chain link model. A diagrammatic analysis of the entanglement of formation for any multi-partite qubit system is given. We show that quantum mechanical joint distribution are entanglements.
In contrast to abstract statistical analyses in the literature, we present a concrete physical diagrammatic model of entanglement characterization and measure with its underlying discrete phase-space physics. This paper serves as a pedagogical treatment of this complex subject of entanglement measures. We review the important inherent concurrence property of entangled qubits, as well as underscore its emergent qubit behavior. From the discrete phase space point of view, concurrence translates to translation symmetry of entangled binary systems in some quantitative measure of entanglement. Although the focus is on bipartite system, the technique is readily extendable to multi-partite system of qubits, as can easily be deduced from the physical inverter-chain link model. A diagrammatic analysis of the entanglement of formation for any multi-partite qubit system is given. We show that quantum mechanical joint distribution are entanglements.
In contrast to abstract statistical analyses in the literature, we present a concrete physical diagrammatic model of entanglement characterization and measure with its underlying discrete phase-space physics. This paper serves as a pedagogical treatment of this complex subject of entanglement measures. We review the important inherent concurrence property of entangled qubits, as well as underscore its emergent qubit behavior. From the discrete phase space point of view, concurrence translates to translation symmetry of entangled binary systems in some quantitative measure of entanglement. Although the focus is on bipartite system, the notion is readily extendable to multi-partite system of qubits, as can easily be deduced from the physical inverter-chain link model. A diagrammatic analysis of the entanglement of formation for any multi-partite qubit system is given
The focal point of this thesis is strongly based on quantum entanglement within Quantum Information Science. We provide an indepth analysis of entanglement quantification in both bipartite and multipartite quantum systems, with an emphasis on many-body entanglement. We examine the definitions of entanglement, introduce several entanglement measures, and examine the challenges in expanding these measures to multipartite scenarios. For bipartite systems, we detail entropy-based measures such as von Neumann and Rényi entropies, the entanglement spectrum and important algebraic metrics such as negativity and concurrence. We then elaborate on multipartite entanglement, introducing modern measures and detection techniques including geometric and global entanglement measurements, generalised multipartite concurrence, multipartite negativity, fidelity-based entanglement benchmarks, and entanglement witnesses. Some examples include Greenberger-Horne-Zeilinger (GHZ), and the W states to illustrate the distinction between biseparable and truly multi-partite-entangled states, as well as the SLOCC classification of families of entangled states. We also briefly introduce practical ways of measuring entanglement, such as the SWAP test, quantum state tomography for experimental estimation of entanglement and consequently randomised measurements. We apply these measures to specific quantum states (GHZ, W , graph states, and random states), analysing how entanglement evolves under operations and how different metrics capture distinct features of multi-qubit entanglement. Finally, we conclude with a summary of findings, a discussion of current limitations such as scalability of measures. The thesis is organised into five chapters: Introduction, Entanglement, Multipartite Entanglement Measures, Negativity vs. Three-Tangle, and Conclusion.
Quantum information and computation is one of the new fields of physics and information science. Entanglement is one of the most significant resources in quantum information processing. The most basic quantitative measure of entanglement, particularly for mixed states of a bipartite system is the entanglement of formation. This entanglement of formation is used in the concurrence, another measurement of entanglement. Here I have briefly reviewed the entanglement of formation and the concurrence of bipartite systems. I have discussed that for a three qubit pure state, if A, B and C are the three qubits, then sum of the squared concurrence between A and B and the squared concurrence between A and C never exceeds the squared concurrence between A and BC pair. The residual entanglement, obtained from the aforementioned inequality, is also discussed here. Keywords- Quantum information, entanglement, qubit.
Characterizing entanglement shareability and distribution in N-partite systems
Exploring the shareability and distribution of entanglement possesses fundamental significance in quantum information tasks. In this paper, we demonstrate that the square of bipartite entanglement measures Gq-concurrence, which is the generalization of concurrence, follows a set of hierarchical monogamy relations for any N-qubit quantum state. On the basis of these monogamy inequalities, we render two kinds of hierarchical indicators that exhibit evident advantages in the capacity of witnessing entanglement. Moreover, we show an analytical relation between Gq-concurrence and concurrence in 2⊗d systems. Furthermore, we rigorously prove that the monogamy property of squared Gq-concurrence is superior to that of squared concurrence in 2⊗d2⊗d3⊗⋯⊗dN systems. In addition, several concrete examples are provided to illustrate that for multilevel systems, the squared Gq-concurrence satisfies the monogamy relation, even if the squared concurrence does not. These results better reveal the intriguing characteristic of multilevel entanglement and provide critical insights into the entanglement distribution within multipartite quantum systems.
It is widely recognized that concurrence can be regarded as a measure of two-qubit entanglement. This paper studies the geometry of concurrence for a two-qubit system to show that the concurrence serves as a coordinate of the factor space , where is the space of normalized two-qubit states, and where G = U (1) × SU (2) × SU (2). Any monotonically increasing function of the concurrence can serve as a measure for entanglement. From the viewpoint of Riemannian geometry, a state with concurrence r is shown to be distant from the separable states by , where r ranges over 0 ⩽ r ⩽ 1. In addition, measures of entanglement for n -qubits are discussed on the basis of a bipartite decomposition with ℓ + m = n . They are invariant under the local unitary transformation group U (2 ℓ ) × U (2 m ).
This paper, a continuation of a previous one ( Iwai 2007 J. Phys. A: Math. Theor. 40 1361 ), studies the geometry of multi-qubit entanglement with respect to bipartite partitions. An n -qubit system ( C 2 ) ⊗ n is isomorphic with , the linear space of 2 ℓ × 2 m complex matrices, where ℓ + m = n . According to the isomorphism, the local transformation group U (2 ℓ ) × U (2 m ) acts on the space of normalized states in . Let M and G denote the space of normalized states and the local transformation group acting on M , respectively. According to the orbit types, M is stratified into strata, among which a principal stratum and the sets of separable states and maximally entangled states will be identified. For , a function F ( C ) = det( I − CC *) proves to serve as a measure of entanglement, where I denotes the 2 ℓ × 2 ℓ identity matrix. The F ( C ) attains the minimal and the maximal values, respectively, on the sets of separable states and maximally entangled states, and further takes no extremal values on the principal stratum. The F ( C ) projects to a function on the factor space G \ M . A naturally defined metric on M also projects to that on G \ M , which serves to measure the distance between the separable states, F −1 (0), and the states, F −1 ( k ), of prescribed value k of measure. To be precise, one has to restrict M to the principal stratum, when projecting the metric. Three- and four-qubit systems will be studied intensively, and then multi-qubit systems discussed.