In this paper, we are focusing on entanglement control problem in a three-qubit system. We demonstrate that vector representation of entanglement, associated with SO(6) representation of SU(4) two-qubit group, can be used to solve various control problems analytically including ( i) the transformation between a W-type states and GHZ state, and ( ii) manipulating bipartite concurrences and three-tangle under a restricted access to only two qubits, and ( iii) designing USp(4)-type quaternionic operations and quantum states.
In this paper, we are focusing on entanglement control problem in a three-qubit system. We demonstrate that vector representation of entanglement, associated with SO(6) representation of SU(4) two-qubit group, can be used to solve various control problems analytically including ( i) the transformation between a W-type states and GHZ state, and ( ii) manipulating bipartite concurrences and three-tangle under a restricted access to only two qubits, and ( iii) designing USp(4)-type quaternionic operations and quantum states.
We suggest a dynamical vector model of entanglement in a three-qubit system based on isomorphism between su(4) and so(6) Lie algebras. This model allows one to write an evolution equation for three-qubit entanglement parameters under an arbitrary pairwise qubit coupling. Generalizing a Pl\"ucker-type description of three-qubit local invariants we introduce three pairs of real-valued three-dimensional vectors (denoted here as AR,I, BR,I, and CR,I). Magnitudes of these vectors determine two- and three-qubit entanglement parameters of the system. We show that evolution of vectors A, B, and C under local SU(2) operations is identical to SO(3) evolution of single-qubit Bloch vectors of qubits a, b, and c correspondingly. At the same time, general two-qubit su(4) Hamiltonians incorporating $a\text{\ensuremath{-}}b$, $a\ensuremath{-}c$, and $b\text{\ensuremath{-}}c$ two-qubit coupling terms generate SO(6) coupling between vectors A and B, A and C, and B and C correspondingly. It turns out that dynamics of entanglement induced by different two-qubit coupling terms is entirely determined by mutual orientation of vectors A, B, C. We illustrate the power of this vector description of entanglement by solving quantum control problems involving transformations between W, Greenberg-Horne-Zeilinger, and biseparable states.
Entanglement is a physical phenomenon that links a pair, or a set of particles that correlates with each other, regardless of the distance between them. Recent researches conducted on entanglement are mostly focused on measurement and classification in multiqubit systems. Classification of two qubits will only distinguish the quantum state as either separable or entangled, and it can be done by measurement. Meanwhile, in a three-qubit system, it becomes more complex because of the structure of the three qubits itself. It is not sufficient to do measurement because the states are divided into three types, including fully separable state, biseparable state, and genuine entangled state. Therefore, the classification is needed to distinguish the type of states in the three-qubit system. This study aims to classify the entanglement of three-qubit pure states using a combination model of special unitary groups, SU(2) and SU(4), by changing the angle of selected parameters in SU(4) and acting on the separable pure state. The matrix represents SU(2) is 2×2 matrix while matrix for SU(4) is 4×4 matrix. Hence, the combination of SU(2) and SU(4) represent 8×8 matrix. This classification uses the von Neumann entropy and three tangle measurements to classify the class, respectively. The results of this study have indicated that the three-qubit pure state has been successfully classified into different classes, namely, A-B-C, A-BC, C-AB, GHZ, and W, with A-B-C being a fully separable state, A-BC and C-AB are biseparable states, and GHZ and W are genuine entangled states. The results show that this model can change separable pure states to other entanglement class after transformation is done.
This study presents an analytical investigation of entanglement quantification in three-qubit pure states through the Acín canonical representation, which serves as a generalization of the two-qubit Schmidt decomposition. Driven by the intricacies of multipartite entanglement and the shortcomings of current measures, we employ a global concurrence measure derived from the generalized concurrence of various bipartitions within the system. The characteristics of this measure are explored both analytically and numerically across a range of SLOCC entanglement classes, such as product, biseparable, GHZ-type, and W-type states. When compared to the three-tangle and tripartite negativity, our findings indicate that this measure yields complementary insights into multipartite correlations. The outcomes underscore the significance of this measure in elucidating the structure and distribution of entanglement within three-qubit quantum systems.
We construct an important missing piece in the entanglement theory of pure three-qubit states, which is a polynomial measure of W-entanglement. This works in parallel to the three-tangle, which is a polynomial measure of GHZ-entanglement, and to the bipartite concurrence, which is a polynomial measure of bipartite entanglement. We also show that these entanglement measures are ordered, the bipartite measure is larger than the W measure, which is larger than the GHZ measure. It is meaningful then to consider these three types of three-qubit entanglement, which are also ordered, bipartite is weaker than W, which is weaker than GHZ, in parallel to the order of the three kinds of equivalence classes of entangled three-qubit states.
We present a new tripartite entanglement measure for three-qubit mixed states. The new measure tr(ρ), which we refer to as the r-tangle, is given as a kind of the tangle, but has a feature which the tangle does not have; if we can derive an analytical form of $ t_{\mathrm{r}}(ρ)$ for a three-qubit mixed state ρ, we can also derive tr(ρ′) analytically for any states ρ′ which are SLOCC-equivalent to the state ρ. The concurrence of two-qubit states also satisfies the feature, but the tangle does not. These facts imply that the r-tangle tr is the appropriate three-partite counterpart of the concurrence. We also derive an analytical form of the r-tangle tr for mixtures of a generalized GHZ state and a generalized W state, and hence for all states which are SLOCC-equivalent to them.
We generalize the control power of a perfect controlled teleportation of an entangled three-qubit pure state, suggested by Li and Ghose [Phys. Rev. A 90, 052305 (2014)], to the control power of a general controlled teleportation of a multiqubit pure state. Thus, we define the minimal control power, and calculate the values of the minimal control power for a class of general three-qubit Greenberger-Horne-Zeilinger (GHZ) states and the three-qubit W class whose states have zero three-tangles. Moreover, we show that the standard three-qubit GHZ state and the standard three-qubit W state have the maximal values of the minimal control power for the two classes, respectively. This means that the minimal control power can be interpreted as not only an operational quantity of a three-qubit quantum communication but also a degree of three-qubit entanglement. In addition, we calculate the values of the minimal control power for general n-qubit GHZ states and the n-qubit W-type states.