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arXiv:1710.10983 · Physical Review A

Bipartite unitary gates and billiard dynamics in the Weyl chamber

Antonio Mandarino, Tomasz Linowski, Karol Życzkowski

Published Jul 30, 2018• 21 citations
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Abstract

Long-time behavior of a unitary quantum gate $U$, acting sequentially on two subsystems of dimension $N$ 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 ${V}^{t}$ stemming from a generic two-qubit gate $V$ in the canonical form tends for a large $t$ 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 $N$ 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 $\text{U}({N}^{2})$.

DOI
10.1103/physreva.98.012335

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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).
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.
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.
We study quantum dynamics through the lens of entanglement generation and characterize the underlying unitary evolution by the distinct signatures it imprints on the time-dependent entangling power. For a unitary operator, we characterize ergodicity by the equality between its long-time-averaged entangling power and the Haar-averaged linear entropy. We define mixing more stringently as the convergence of the time-dependent entangling power itself to the Haar value at long times. Within this framework, we establish the ergodic hierarchy of dynamical behavior, showing in particular that mixing implies ergodicity, whereas ergodicity does not necessarily imply mixing. As an application, we find that two-qubit unitary gates are neither ergodic nor mixing: their long-time-averaged entangling power can take only four discrete values, none of which coincides with the Haar average. We then investigate many-body dynamics using the kicked Ising chain and find that the long-time-averaged entangling power converges to the Haar value in both integrable and nonintegrable cases, indicating ergodicity. Remarkably, however, the nonintegrable chain exhibits mixing, whereas the integrable chain, despite being ergodic, is demonstrably nonmixing. We also introduce a Lyapunov-like exponent to characterize the rate at which the time-dependent entangling power approaches its saturation value. We find that this exponent increases systematically with the degree of integrability breaking in the many-body system. Our results establish entanglement generation as a useful framework for characterizing dynamical systems and reveal qualitatively different signatures of integrability beyond conventional diagnostics.
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 $8/3\ensuremath{\pi}\ensuremath{\approx}0.85$. 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.
The entangling power of a unitary operator acting on a bipartite Hilbert space measures the entanglement it generates from product states, averaged over the inputs. A finite-dimensional unitary has a spectral decomposition U=∑a=1neiθaPaU=\sum_{a=1}^{n}e^{iθ_a}P_aU=∑a=1n​eiθa​Pa​, where eiθae^{iθ_a}eiθa​ are the eigenvalues, PaP_aPa​ the corresponding eigen-projectors, and nnn is the number of distinct eigenvalues. After removing an overall phase, the entangling power is a function on the (n−1)(n-1)(n−1)-torus of relative eigenphases at fixed spectral projectors. We prove that this function is stationary at all 2n−12^{n-1}2n−1 points on the torus where every relative phase is 000 or πππ, which we define as \textit{corners}. Up to an overall phase, UUU at each corner is a generalized reflection R=I−2QR=\mathbb{I}-2QR=I−2Q satisfying R2=IR^2=\mathbb{I}R2=I, where QQQ is the sum of spectral projectors whose relative phase is πππ. At the corner the entangling power is expressed in terms of seven local-unitary invariants of QQQ. A unitary gate UUU can be realized as a corner of some projector family if and only if U2∝IU^2\propto\mathbb{I}U2∝I, a condition satisfied by many Clifford and non-Clifford gates. We illustrate the theorem with two-qubit gates, SU(N)SU(N)SU(N) channel decompositions, and two-site spin chains, obtaining examples of minima, maxima, and saddle points. In addition, a corner that is a saddle point on the full phase torus can appear as a local maximum or minimum along different time-evolution trajectories.
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.