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arXiv:2003.14390 · Physical Review A

Vector properties of entanglement in a three-qubit system

Dmitry B. Uskov, Paul M. Alsing

Published Sep 1, 2020• 4 citations
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Abstract

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 ${\mathbf{A}}_{R,I}$, ${\mathbf{B}}_{R,I}$, and ${\mathbf{C}}_{R,I}$). Magnitudes of these vectors determine two- and three-qubit entanglement parameters of the system. We show that evolution of vectors $\mathbf{A}$, $\mathbf{B}$, and $\mathbf{C}$ 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 $\mathbf{A}$ and $\mathbf{B}$, $\mathbf{A}$ and $\mathbf{C}$, and $\mathbf{B}$ and $\mathbf{C}$ correspondingly. It turns out that dynamics of entanglement induced by different two-qubit coupling terms is entirely determined by mutual orientation of vectors $\mathbf{A}$, $\mathbf{B}$, $\mathbf{C}$. 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.

DOI
10.1103/physreva.102.032401

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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{\mathbf{A}}_{R,I}AR,I​, BR,I{\mathbf{B}}_{R,I}BR,I​, and CR,I{\mathbf{C}}_{R,I}CR,I​). Magnitudes of these vectors determine two- and three-qubit entanglement parameters of the system. We show that evolution of vectors A\mathbf{A}A, B\mathbf{B}B, and C\mathbf{C}C under local SU(2) operations is identical to SO(3) evolution of single-qubit Bloch vectors of qubits aaa, bbb, and ccc correspondingly. At the same time, general two-qubit su(4) Hamiltonians incorporating a\ensuremath-ba\text{\ensuremath{-}}ba\ensuremath-b, a\ensuremath−ca\ensuremath{-}ca\ensuremath−c, and b\ensuremath-cb\text{\ensuremath{-}}cb\ensuremath-c two-qubit coupling terms generate SO(6) coupling between vectors A\mathbf{A}A and B\mathbf{B}B, A\mathbf{A}A and C\mathbf{C}C, and B\mathbf{B}B and C\mathbf{C}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\mathbf{A}A, B\mathbf{B}B, C\mathbf{C}C. We illustrate the power of this vector description of entanglement by solving quantum control problems involving transformations between WWW, Greenberg-Horne-Zeilinger, and biseparable states.
The capacity of a quantum gate to produce entangled states on a bipartite system is quantified in terms of the entangling power. This quantity is defined as the average of the linear entropy of entanglement of the states produced after applying a quantum gate over the whole set of separable states. Here we focus on symmetric two-qubit quantum gates, acting on the symmetric two-qubit space, and calculate the entangling power in terms of the appropriate local invariant. A geometric description of the local equivalence classes of gates is given in terms of the su(3)\text{su}(3)su(3) Lie algebra root vectors. These vectors define a primitive cell with hexagonal symmetry on a plane, and through the Weyl group the minimum area on the plane containing the whole set of locally equivalent quantum gates is identified. We give conditions to determine when a given quantum gate produces maximally entangled states from separable ones (perfect entanglers). We find that these gates correspond to one-fourth of the whole set of locally distinct quantum gates. The formalism developed here is applicable to general three-level systems. Via the Majorana representation, qutrit transformations can be regarded as having entangling power and hence classified as perfect and nonperfect entanglers and be grouped into local-equivalence classes of the associated symmetric two-qubit space. The results are illustrated by an anisotropic Heisenberg model, the Lipkin-Meshkov-Glick model, and two coupled quantized oscillators with cross-Kerr interaction, which we use to obtain three-level gates relevant in qutrit quantum computation.
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.
Background. To operate a quantum computer, a set of universal gates must be implemented, such as a two-cubit gate of the controlled negation type plus one-cubit spins. As a universal alternative, three-cubic-bit gates may be used. In this regard, it seems very relevant to investigate the dynamics of three-qubit systems in microwave resonators, in particular to study the most efficient schemes for generating, controlling, and monitoring entangled qubit states. Aim. To investigate the features of the dynamics of entangled pairs of qubits for a system in which two qubits are locked in a single-mode resonator and interact with the thermal field mode, and the third qubit is in a free state. Methods. To analyze the dynamics of the considered system, the solution of the quantum Liouville equation for the full density matrix is investigated. The exact solution of the above equation in the case of initial biseparable states of the qubits is found. The exact solution of the evolution equation is used to calculate the criterion of entanglement of qubit pairs – negativity. Numerical simulations of negativity for biseparable qubit states as well as different values of the thermal field intensity of the resonator have been carried out. Results. It is shown that for intense thermal fields of the resonator the effect of instantaneous death of entanglement is observed, while the time intervals between death and revival of entanglement of qubits depend essentially on the choice of their initial biseparable state. It is found that for one of the biseparable states, entanglement of qubits trapped in the resonator does not occur at any field intensities of the resonator. Conclusion. It is found that the peculiarities of the dynamics of entanglement of qubits, in particular the time intervals between the death and birth of entangled qubits, are determined by the choice of the initial biseparable state of qubits, as well as by the values of the field intensity of the resonator. The results obtained can be used to effectively control and manage the degree of qubit entanglement in three-qubit systems in microwave resonators.
The first characterization of mixed-state entanglement was achieved for two-qubit states in Werner's seminal work [Phys. Rev. A 40, 4277 (1989)]. A physically important extension concerns mixtures of a pure entangled state [such as the Greenberger-Horne-Zeilinger (GHZ) state] and the unpolarized state. These mixed states serve as benchmark for the robustness of multipartite entanglement. They share the symmetries of the GHZ state. We call such states GHZ symmetric. Here we give a complete description of the entanglement in the family of three-qubit GHZ-symmetric states and, in particular, of the three-qubit generalized Werner states. Our method relies on the appropriate parametrization of the states and on the invariance of entanglement properties under general local operations. An application is the definition of a symmetrization witness for the entanglement class of arbitrary three-qubit states.
The entanglement properties of a two-qubit system are analyzed from the point of view of Lie algebras and geometry. We deal with $4\ifmmode\times\else\texttimes\fi{}4$ matrices belonging to a particular class having seven free parameters, constructed from the imposed symmetries on two-qubit states expanded in terms of a basis whose elements belong to a subalgebra of su(4). The entanglement or separability character of the state is determined and measured by analyzing the eigenvalues of an auxiliary matrix obtained from the original state after performing on it a nonunitary operation. In terms of its eigenvalues, we define two squared distances having a Minkowski metric, ${s}^{2}={t}^{2}\ensuremath{-}{\stackrel{P\vec}{V}}^{2}$, which become a measure of the entanglement. The Minkowski metric is a signature of that class of two-qubit states (matrices) and also of the auxiliary matrices to be constructed. The squared distance is invariant by any unitary transformation on the two-qubit state. We illustrate the theory with examples.