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arXiv:1211.3287 · Physical Review A

Unitary quantum gates, perfect entanglers, and unistochastic maps

Marcin Musz, Marek Kuś, Karol Życzkowski

Published Feb 13, 2013• 58 citations
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Abstract

Nonlocal properties of ensembles of quantum gates induced by the Haar measure on the unitary group are investigated. We analyze the entropy of entanglement of a unitary matrix $U$ equal to the Shannon entropy of the vector of singular values of the reshuffled matrix. Averaging the entropy over the Haar measure on $U({N}^{2})$ we find its asymptotic behavior. For two-qubit quantum gates we derive the induced probability distribution of the interaction content and show that the relative volume of the set of perfect entanglers reads $8/3\ensuremath{\pi}\ensuremath{\approx}0.85$. We establish explicit conditions under which a given one-qubit bistochastic map is unistochastic, so it can be obtained by partial trace over a one-qubit environment initially prepared in the maximally mixed state.

DOI
10.1103/physreva.87.022111

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Nonlocal properties of ensembles of quantum gates induced by the Haar measure on the unitary group are investigated. We analyze the entropy of entanglement of a unitary matrix UUU equal to the Shannon entropy of the vector of singular values of the reshuffled matrix. Averaging the entropy over the Haar measure on U(N2)U({N}^{2})U(N2) we find its asymptotic behavior. For two-qubit quantum gates we derive the induced probability distribution of the interaction content and show that the relative volume of the set of perfect entanglers reads 8/3\ensuremathπ\ensuremath≈0.858/3\ensuremath{\pi}\ensuremath{\approx}0.858/3\ensuremathπ\ensuremath≈0.85. We establish explicit conditions under which a given one-qubit bistochastic map is unistochastic, so it can be obtained by partial trace over a one-qubit environment initially prepared in the maximally mixed state.
Nonlocal properties of an ensemble of diagonal random unitary matrices of order N2{N}^{2}N2 are investigated. The average Schmidt strength of such a bipartite diagonal quantum gate is shown to scale as lnNlnNlnN, in contrast to the lnN2ln{N}^{2}lnN2 behavior characteristic of random unitary gates. Entangling power of a diagonal gate UUU is related to the von Neumann entropy of an auxiliary quantum state $\ensuremath{\rho}=A{A}^{\ifmmode\dagger\else\textdagger\fi{}}/{N}^{2}$, where the square matrix AAA is obtained by reshaping the vector of diagonal elements of UUU of length N2{N}^{2}N2 into a square matrix of order NNN. This fact provides a motivation to study the ensemble of non-Hermitian unimodular matrices AAA, with all entries of the same modulus and random phases and the ensemble of quantum states $\ensuremath{\rho}$, such that all their diagonal entries are equal to 1/N1/N1/N. Such a state is contradiagonal with respect to the computational basis, in the sense that among all unitary equivalent states it maximizes the entropy copied to the environment due to the coarse-graining process. The first four moments of the squared singular values of the unimodular ensemble are derived, based on which we conjecture a connection to a recently studied combinatorial object called the ``Borel triangle.'' This allows us to find exactly the mean von Neumann entropy for random phase density matrices and the average entanglement for the corresponding ensemble of bipartite pure states.
Long-time behavior of a unitary quantum gate UUU, acting sequentially on two subsystems of dimension NNN each, is investigated. We derive an expression describing an arbitrary iteration of a two-qubit gate making use of a link to the dynamics of a free particle in a three-dimensional (3D) billiard. Due to ergodicity of such a dynamics an average along a trajectory Vt{V}^{t}Vt stemming from a generic two-qubit gate VVV in the canonical form tends for a large ttt to the average over an ensemble of random unitary gates distributed according to the flat measure in the Weyl chamber---the minimal 3D set containing points from all orbits of locally equivalent gates. Furthermore, we show that for a large dimension NNN the mean entanglement entropy averaged along a generic trajectory coincides with the average over the ensemble of random unitary matrices distributed according to the Haar measure on U(N2)\text{U}({N}^{2})U(N2).
Entanglement properties of bipartite unitary operators are studied via their local invariants, namely the entangling power e p and a complementary quantity, the gate typicality g t .We characterize the boundaries of the set K 2 representing all two-qubit gates projected onto the plane (e p , g t ) showing that the fractional powers of the SWAP operator form a parabolic boundary of K 2 , while the other bounds are formed by two straight lines.In this way, a family of gates with extreme properties is identified and analyzed.We also show that the parabolic curve representing powers of SWAP persists in the set K N for gates of higher dimensions (N > 2).Furthermore, we study entanglement of bipartite quantum gates applied sequentially n times, and we analyze the influence of interlacing local unitary operations, which model generic Hamiltonian dynamics.An explicit formula for the entangling power of a gate applied n times averaged over random local unitary dynamics is derived for an arbitrary dimension of each subsystem.This quantity shows an exponential saturation to the value predicted by the random matrix theory, indicating "thermalization" in the entanglement properties of sequentially applied quantum gates that can have arbitrarily small, but nonzero, entanglement to begin with.The thermalization is further characterized by the spectral properties of the reshuffled and partially transposed unitary matrices.
An ensemble of random unistochastic (orthostochastic) matrices is defined by taking squared moduli of elements of random unitary (orthogonal) matrices distributed according to the Haar measure on U ( N ) (or O ( N )). An ensemble of symmetric unistochastic matrices is obtained with use of unitary symmetric matrices pertaining to the circular orthogonal ensemble. We study the distribution of complex eigenvalues of bistochastic, unistochastic and orthostochastic matrices in the complex plane. We compute averages (entropy, traces) over the ensembles of unistochastic matrices and present inequalities concerning the entropies of products of bistochastic matrices.
Abstract We study the entangling properties of multipartite unitary gates with respect to the measure of entanglement called one-tangle. Putting special emphasis on the case of three parties, we derive an analytical expression for the entangling power of an n-partite gate as an explicit function of the gate, linking the entangling power of gates acting on the n-partite Hilbert space of dimension to the entanglement of pure states in the Hilbert space of dimension . Furthermore, we evaluate its mean value averaged over the unitary and orthogonal groups, analyze the maximal entangling power and relate it to the absolutely maximally entangled (AME) states of a system with 2n parties. Finally, we provide a detailed analysis of the entangling properties of the three-qubit unitary and orthogonal gates.
We study nonlocal two-qubit operations from a geometric perspective. By applying a Cartan decomposition to su(4), we find that the geometric structure of nonlocal gates is a 3-torus. We derive the invariants for local transformations, and connect these local invariants to the coordinates of the 3-torus. Since different points on the 3-torus may correspond to the same local equivalence class, we use the Weyl group theory to reduce the symmetry. We show that the local equivalence classes of two-qubit gates are in one-to-one correspondence with the points in a tetrahedron except on the base. We then study the properties of perfect entanglers, that is, the two-qubit operations that can generate maximally entangled states from some initially separable states. We provide criteria to determine whether a given two-qubit gate is a perfect entangler and establish a geometric description of perfect entanglers by making use of the tetrahedral representation of nonlocal gates. We find that exactly half the nonlocal gates are perfect entanglers. We also investigate the nonlocal operations generated by a given Hamiltonian. We first study the gates that can be directly generated by a Hamiltonian. Then we explicitly construct a quantum circuit that contains at most three nonlocal gates generated by a two-body interaction Hamiltonian, together with at most four local gates generated by single-qubit terms. We prove that such a quantum circuit can simulate any arbitrary two-qubit gate exactly, and hence it provides an efficient implementation of universal quantum computation and simulation.