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arXiv:2107.12653 · Physical Review A

Entangling power of symmetric two-qubit quantum gates and three-level operations

Diego Morachis Galindo, Jesús A. Maytorena

Published Jan 3, 2022• 11 citations
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Abstract

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 $\text{su}(3)$ 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.

DOI
10.1103/physreva.105.012601

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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.
Abstract We show that the geometrical features of the unimodular squared eigenvalues of the nonlocal part of a two-qubit gate in the complex plane can be used to study the nonlocal characteristics of the two-qubit gate. It has been shown that the entangling power of a two-qubit gate can be expressed in terms of the mean squared length of the chords connecting the squared eigenvalues of the nonlocal part with each other. In this paper, we provide similar expressions for the gate typicality and the linear entropy of two-qubit gates. The expression of gate typicality involves the squared lengths of three chords connecting a squared eigenvalue of the nonlocal part with the complex conjugate of the other three squared eigenvalues. The expression of linear entropy is obtained from the expressions of entangling power and gate typicality. We also express them in terms of the linear entropies of certain two-qubit states. In addition, we show that the normalized entangling power of a two-qubit gate can be provided exact description in the Argand diagram. Finally, we analyze the simplices enclosing the origin in the convex hull of the squared eigenvalues of the nonlocal part of the perfect entangling two-qubit gates to construct a pair of orthonormal product states that can be transformed into maximally entangled states, and divide the region of perfect entanglers in the Weyl chamber into different regions based on the uniqueness of the simplices enclosing the origin.
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 $a\text{\ensuremath{-}}b$, $a\ensuremath{-}c$, and $b\text{\ensuremath{-}}c$ 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.
Abstract For general symmetric multi-qubit systems, the behavior of one- and two-qubit entanglement for Dicke, spin coherent and parity-adapted (even and odd) spin coherent states is determined. These quantum correlations are quantified by linear and von Neumann entropies of the corresponding one- and two-qubit reduced density matrices of the multi-qubit system. These states play a fundamental role in the study of Hamiltonian systems written in terms of collective generators of the angular momentum algebra like, for example, the Lipkin–Meshkov–Glick (LMG) model. Here we shall use these entanglement measures as a signature to characterize the different quantum phases that appear in these models.
The change of the entangling power of nnn fixed bipartite unitary gates, describing interactions, when interlaced with local unitary operators describing monopartite evolutions, is studied as a model of the entangling power of generic Hamiltonian dynamics. A generalization of the local unitary averaged entangling power for arbitrary subsystem dimensions is derived. This quantity shows an exponential saturation to the random matrix theory (RMT) average of the bipartite space, indicating thermalization of quantum gates that could otherwise be very non-generic and have arbitrarily small, but nonzero, entanglement. The rate of approach is determined by the entangling power of the fixed bipartite unitary, which is invariant with respect to local unitaries. The thermalization is also studied numerically via the spectrum of the reshuffled and partially transposed unitary matrices, which is shown to tend to the Girko circle law expected for random Ginibre matrices. As a prelude, the entangling power epe_pep​ is analyzed along with the gate typicality gtg_tgt​ for bipartite unitary gates acting on two qubits and some higher dimensional systems. We study the structure of the set representing all unitaries projected into the plane (ep,gt)(e_p, g_t)(ep​,gt​) and characterize its boundaries which contains distinguished gates including Fourier gate, CNOT and its generalizations, swap and its fractional powers. In this way, a family of gates with extreme properties is identified and analyzed. We remark on the use of these operators as building blocks for many-body quantum systems.
We study two distinct diagnostics of bipartite quantum channels: the average fidelity between a desired state and noisy output across different classes of input states, and generation of entanglement from pure product input states, quantified by well-defined entangling power. These diagnostics are generally independent, but together they provide a more complete description of bipartite channels. We show that the fidelity averaged over any local-unitary orbit with fixed Schmidt coefficients for equal local dimensions is completely determined by the average input-output fidelity and its restriction to product inputs. We also introduce concurrence- and negativity-based entangling power for two-qubit channels, prove their convexity and monotonicity under local postprocessing, and show that, unlike the previously proposed linear-entropy quantity, they vanish for all separable channels. Representative separable and nonseparable channels are investigated. Finally, we generalize our definitions of entangling power to nonproduct inputs, and provide an analytical lower bound for the concurrence-based entanglement variation, showing its effectiveness with specific examples.
Two applications of gates from the B gate equivalence class can generate all two-qubit gates. This local equivalence class is invariant under the mirror (multiplication with the SWAP gate) operation, inverse (Hermitian conjugate) operation, and the combined inverse and mirror operations. The last two symmetries are associated with the ability of a two-qubit gate to generate the two-qubit local gates and the SWAP gate in two applications. No single local equivalence class of two-qubit gates, except the B gate equivalence class, has these two symmetries. Only the planar regions of the Weyl chamber, describing the mirror operation, contain the local equivalence classes with either one of the two symmetries. We show that there exist one-parameter families of local equivalence classes on these planes, with and without the B gate equivalence class, such that each of them can be used to construct a parameterized universal two-qubit quantum circuit that involves only two nonlocal two-qubit gates. We also discuss the implementation of the gates from a few families of local equivalence classes on superconducting quantum computers for optimal generation of all two-qubit gates. We provide upper bounds on the number of two-qubit gates required to generate an arbitrary nnn-qubit gate for two families, each of which is conjectured to generate all two-qubit gates in two applications. We show that there exists a positive correlation between the area of the convex hull of the squared eigenvalues of the nonlocal part of a parameterized two-qubit gate and the fractional volume of the Weyl chamber covered in two applications of the parameterized two-qubit gate for two families of local equivalence classes.