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arXiv:1407.1169 · Physical Review A

Diagonal unitary entangling gates and contradiagonal quantum states

Arul Lakshminarayan, Zbigniew Puchała, Karol Życzkowski

Published Sep 2, 2014• 29 citations
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Abstract

Nonlocal properties of an ensemble of diagonal random unitary matrices of order ${N}^{2}$ are investigated. The average Schmidt strength of such a bipartite diagonal quantum gate is shown to scale as $lnN$, in contrast to the $ln{N}^{2}$ behavior characteristic of random unitary gates. Entangling power of a diagonal gate $U$ 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 $A$ is obtained by reshaping the vector of diagonal elements of $U$ of length ${N}^{2}$ into a square matrix of order $N$. This fact provides a motivation to study the ensemble of non-Hermitian unimodular matrices $A$, 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 $1/N$. 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.

DOI
10.1103/physreva.90.032303

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We analyze the properties of entangled random pure states of a quantum system partitioned into two smaller subsystems of dimensions N and M. Framing the problem in terms of random matrices with a fixed-trace constraint, we establish, for arbitrary N≤M, a general relation between the n-point densities and the cross moments of the eigenvalues of the reduced density matrix, i.e., the so-called Schmidt eigenvalues, and the analogous functionals of the eigenvalues of the Wishart-Laguerre ensemble of the random matrix theory. This allows us to derive explicit expressions for two-level densities, and also an exact expression for the variance of von Neumann entropy at finite N,M. Then, we focus on the moments E{K^{a}} of the Schmidt number K, the reciprocal of the purity. This is a random variable supported on [1,N], which quantifies the number of degrees of freedom effectively contributing to the entanglement. We derive a wealth of analytical results for E{K^{a}} for N=2 and 3 and arbitrary M, and also for square N=M systems by spotting for the latter a connection with the probability P(x_{min}^{GUE}≥sqrt[2N]ξ) that the smallest eigenvalue x_{min}^{GUE} of an N×N matrix belonging to the Gaussian unitary ensemble is larger than sqrt[2N]ξ. As a by-product, we present an exact asymptotic expansion for P(x_{min}^{GUE}≥sqrt[2N]ξ) for finite N as ξ→∞. Our results are corroborated by numerical simulations whenever possible, with excellent agreement.
Quantum entanglement plays a crucial role in quantum information, quantum teleportation and quantum computation. The information about the entanglement content between subsystems of the composite system is encoded in the Schmidt eigenvalues. We derive here closed expressions for the spectral density of Schmidt eigenvalues for all three invariant classes of random matrix ensembles. We also obtain exact results for average von Neumann entropy. We find that maximum average entanglement is achieved if the system belongs to the symplectic invariant class.
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We study the scaling of the convergence of several statistical properties of a recently introduced random unitary circuit ensemble towards their limits given by the circular unitary ensemble. Our study includes the full distribution of the absolute square of a matrix element, moments of that distribution up to order eight, as well as correlators containing up to 16 matrix elements in a given column of the unitary matrices. Our numerical scaling analysis shows that all of these quantities can be reproduced efficiently, with a number of random gates which scales at most as ${n}_{q}{[\mathrm{ln}({n}_{q}∕ϵ)]}^{\ensuremath{\nu}}$ with the number of qubits nq{n}_{q}nq​ for a given fixed precision ϵϵϵ and $\ensuremath{\nu}>0$.
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).