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arXiv:1909.08139 · Physical Review Research

Entanglement measures of bipartite quantum gates and their thermalization under arbitrary interaction strength

Bhargavi Jonnadula, Prabha Mandayam, Karol Życzkowski, Arul Lakshminarayan

Published Oct 23, 2020• 28 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevresearch.2.043126

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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 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.
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).
We introduce the concept of entanglement features of unitary gates, as a collection of exponentiated entanglement entropies over all bipartitions of input and output channels. We obtained the general formula for time-dependent nthn\mathrm{th}nth-R\'enyi entanglement features for unitary gates generated by random Hamiltonian. In particular, we propose an Ising formulation for the second R\'enyi entanglement features of random Hamiltonian dynamics, which admits a holographic tensor network interpretation. As a general description of entanglement properties, we show that the entanglement features can be applied to several dynamical measures of thermalization, including the out-of-time-order correlation and the entanglement growth after a quantum quench. We also analyze the Yoshida-Kitaev probabilistic protocol for random Hamiltonian dynamics.
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.
Abstract We study bipartite unitary operators which stay invariant under the local actions of diagonal unitary and orthogonal groups. We investigate structural properties of these operators, arguing that the diagonal symmetry makes them suitable for analytical study. As a first application, we construct large new families of dual unitary gates in arbitrary finite dimensions, which are important toy models for entanglement spreading in quantum circuits. We then analyze the non-local nature of these invariant operators, both in discrete (operator Schmidt rank) and continuous (entangling power) settings. Our scrutiny reveals that these operators can be used to simulate any bipartite unitary gate via stochastic local operations and classical communication. Furthermore, we establish a one-to-one connection between the set of local diagonal unitary invariant dual unitary operators with maximum entangling power and the set of complex Hadamard matrices. Finally, we discuss distinguishability of unitary operators in the setting of the stated diagonal symmetry.