Geometry and Symmetric Coherent States of Three Qubits Systems

Oktay K. Pashaev, Zeynep Nilhan Gürkan

Motivated by the Möbius transformation for symmetric points under the generalized circle in the complex plane, the system of symmetric spin coherent states corresponding to antipodal qubit states is introduced. In terms of these states, we construct the maximally entangled complete set of two-qubit coherent states, which in the limiting cases reduces to the Bell basis. A specific property of our symmetric coherent states is that they never become unentangled for any value of ψ from the complex plane. Entanglement quantifications of our states are given by the reduced density matrix and the concurrence determinant, and it is shown that our basis is maximally entangled. Universal one- and two-qubit gates in these new coherent state basis are calculated. As an application, we find the Q symbol of the XY Z model Hamiltonian operator H as an average energy function in maximally entangled two- and three-qubit phase space. It shows regular finite-energy localized structure with specific local extremum points. The concurrence and fidelity of quantum evolution with dimerization of double periodic patterns are given.

Yaokun Wang, Lian-He Shao, Li-Zhu Ge +2 more

Tuğçe Parlakgörür, Oktay K. Pashaev

A representation of one qubit state by points in complex plane is proposed, such that the computational basis corresponds to two fixed points at a finite distance in the plane. These points represent common symmetric states for the set of quantum states on Apollonius circles. It is shown that, the Shannon entropy of one qubit state depends on ratio of probabilities and is a constant along Apollonius circles. For two qubit state and for three qubit state in Apollonius representation, the concurrence for entanglement and the Cayley hyperdeterminant for tritanglement correspondingly, are constant on the circles as well. Similar results are obtained also for n-tangle hyperdeterminant with even number of qubit states. It turns out that, for arbitrary multiple qubit state in Apollonius representation, fidelity between symmetric qubit states is also constant on Apollonius circles. According to these, the Apollonius circles are interpreted as integral curves for entanglement characteristics. The bipolar and the Cassini representations for qubit state are introduced, and their relations with qubit coherent states are established. We proposed the differential geometry for qubit states in Apollonius representation, defined by the metric on a surface in conformal coordinates, as square of the concurrence. The surfaces of the concurrence, as surfaces of revolution in Euclidean and Minkowski spaces are constructed. It is shown that, curves on these surfaces with constant Gaussian curvature becomes Cassini curves.

Oktay K. Pashaev, Aygül Koçak

Т.Э. Кренкель

Yaroslav Balytskyi, Yevgen V. Kotukh, Gennady Khalimov +1 more

Abstract We present a novel $\mathcal{PT}$-symmetric approach for discriminating three pure qubit states and its implementation by the dilation method. Our approach based on the dilation method manipulating the Hilbert space of the system by two-staged $\mathcal{PT}$-symmetric evolution expedites the three-state discrimination process at the cost of introducing an inconclusive outcome. This modification of the Hilbert space of the qubit makes it possible to eliminate one of three states in the limit $\alpha\rightarrow\pm\frac{\pi}{2}$ in the vicinity of the exceptional point corresponding to coalescing eigenvectors and eigenvalues, effectively reducing the problem to two-state discrimination, or reducing the geometry to the mirror-symmetric states for which an efficient discrimination strategy is developed. To demonstrate the effectiveness of our approach, we implement it on a superconducting quantum processor provided by IBM Quantum Experience employing an ancilla qubit. We conclude our paper by identifying potential applications, including those in quantum communication and cryptography, to facilitate further research and development.