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Thermalization of entangling power with arbitrarily weak interactions

Bhargavi Jonnadula, Prabha Mandayam, Karol Życzkowski, Arul Lakshminarayan

Published Sep 17, 2019• 1 citation
OpenAlex

Abstract

The change of the entangling power of $n$ 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 $e_p$ is analyzed along with the gate typicality $g_t$ 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 $(e_p, g_t)$ 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.

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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.
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.
It is demonstrated here that local dynamics have the ability to strongly modify the entangling power of unitary quantum gates acting on a composite system. The scenario is common to numerous physical systems, in which the time evolution involves local operators and nonlocal interactions. To distinguish between distinct classes of gates with zero entangling power we introduce a complementary quantity called gate typicality and study its properties. Analyzing multiple, say nnn, applications of any entangling operator, UUU, interlaced with random local gates we prove that both investigated quantities approach their asymptotic values in a simple exponential form. These values coincide with the averages for random nonlocal operators on the full composite space and could be significantly larger than that of Un{U}^{n}Un. This rapid convergence to equilibrium, valid for subsystems of arbitrary size, is illustrated by studying multiple actions of diagonal unitary gates and controlled unitary gates.
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).
Operator entanglement is a well-established measure of operator complexity across a system bipartition. In this work, we introduce a measure of the ability of a unitary channel to generate operator entanglement, representing an operator-level generalization of the state-space entangling power. This operator space entangling power is demonstrated to be linked to the scrambling properties of the unitary channel via the recently introduced concept of mutual averaged noncommutativity of quantum operator algebras. An upper bound for the operator space entangling power is identified that corresponds to unitary channels with scrambling properties akin to those of typical unitaries. Additionally, for Hamiltonian dynamics, we find that the short-time growth rate of the operator space entangling power matches the Gaussian scrambling rate of the bipartite out-of-time-order correlator, establishing a direct link between information scrambling and operator entanglement generation for short timescales. Finally, we examine the average growth of local operator entanglement across a symmetric bipartition of a spin chain. For dual-unitary circuits, a combination of analytical and numerical investigations demonstrates that the average growth of local operator entanglement exhibits two distinct regimes in relation to the operator space entangling power of the building-block gate.
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.