PHYSICAL REVIEW A 90, 032303 (2014) Diagonal unitary entangling gates and contradiagonal quantum states Arul Lakshminarayan* Department of Physics, Indian Institute of Technology Madras, Chennai 600036, India Zbigniew Puchała† Institute of Theoretical and Applied Informatics, Polish Academy of Sciences, Bałtycka 5, 44-100 Gliwice, Poland and Institute of Physics, Jagiellonian University, ul. Reymonta 4, 30-059 Krakow,´ Poland Karol Zyczkowski˙ ‡ Institute of Physics, Jagiellonian University, ul. Reymonta 4, 30-059 Krakow,´ Poland and Center for Theoretical Physics, Polish Academy of Sciences, Warsaw, Poland (Received 10 July 2014; published 2 September 2014) 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 ln N, 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 ρ = AA†/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 ρ, 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 PACS number(s): 03.67.Bg, 03.67.−a, 03.65.Ud, 05.30.Ch I. INTRODUCTION ρA and ρB will be typically negative under partial transpose and therefore A and B are entangled [8]. Entanglement has been at the focus of recent researches Ways to generate entangled states from initially unentan- in quantum information—for a review see [1]—as it enables gled ones via unitary operators are of natural interest, and in a range of uniquely quantum tasks such as teleportation, the context of quantum computation imply the construction quite apart from being a singular nonclassical phenomenon of appropriate gates. Investigations of entangling power of and therefore of fundamental interest. It is well appreciated a unitary quantum gate were initiated by Zanardi and co- now that many-particle pure states are typically highly en- workers [9,10], while some measures of nonlocality were tangled [2–5], and share entanglement in a manner that is analyzed in [11–16]. A typical quantum gate acting on a almost wholly of a multipartite nature. Here typicality refers composed system consisting of two N-level systems can to ensembles of pure states selected according to the uniform be represented by a random unitary matrix of order N 2. (Haar) measure. If two distinct subsystems of a pure state Nonlocal properties of such random gates were investigated are such that together they make up the entire system in a in [17]. typical pure state, then the two subsystems will be largely In this work we shall analyze a simpler ensemble of diago- entangled. In early works, Page and others [3,6] had found the nal unitary random matrices and will characterize nonlocality average entanglement, which in this case is simply the mean of the corresponding quantum gates. It is also naturally related von Neumann entropy of the reduced state, and showed that it to an ensemble of pure bipartite states in N 2-dimensional is nearly the maximum possible. space whose components in some fixed basis are of the form More recent studies have explored the distribution of iφj 2 e /N , and φi are uniformly distributed random numbers. entanglement in such complete bipartite partitions of random Such an ensemble has been recently studied as phase-random states [7]. If the two subsystems do not comprise the entire states [18], and in fact connections to diagonal quantum systems, for example two particles in a three-particle state, the circuits has been pointed out [19]. Part of the motivation for average entanglement depends on the dimensionality of the the study of diagonal quantum gates is that many Hamiltonians subsystems. Roughly speaking if the complementary space of have the structure that the basis in which the interaction is the subsystems (say A and B) is smaller, the density matrices diagonal can be chosen to be unentangled. Time evolution is then governed by unitary operators whose entangling parts are diagonal. Recently studies of measurement-based *[email protected] quantum computation have also used diagonal unitary gates †[email protected] and shown that it can still remain superior to classical ‡[email protected] computation [20,21]. 1050-2947/2014/90(3)/032303(13) 032303-1 ©2014 American Physical Society LAKSHMINARAYAN, PUCHAŁA, AND ZYCZKOWSKI˙ PHYSICAL REVIEW A 90, 032303 (2014) Also explicitly, unitary operators such as (UA ⊗ UB )UAB equivalently of the von Neumann entropy of the random occur in the study of coupled systems including kicked phase states. The procedure of contradiagonalization of a quantum tops; see the book of Haake [22]. The coupling could Hermitian matrix, which makes all diagonal elements equal, is z z A,B be of the form JAJB , where Jz are spin operators. Thus the introduced in Sec. V. In this section we study in particular the nonlocal part of the evolution is diagonal again. It is found cognate ensemble of random contradiagonal states and show numerically that the eigenstates of such operators as well as their particular properties concerning the transfer of quantum the time evolution engendered by repeated applications of such information. The paper is concluded in Sec. VI, and appendices operators can create large entanglement well approximated by review the operator Schmidt decomposition and entangling that of random states [23]. However such operators have a entropy. much smaller number of possible independent elements than HN ⊗ HN Haar distributed unitaries on . Thus it is of interest II. DIAGONAL BIPARTITE QUANTUM GATES to study the origin of such large entanglement. 2 Furthermore, we are going to study the related ensemble of Consider a diagonal unitary matrix U of order N . Each = “unimodular random matrices,” comprising complex matrices entry is assumed to be random, so that Uμν δμν exp(iφν ), whose entries all have the same modulus and randomly chosen where φν are independent random phases distributed uniformly phases. Such matrices arise from reshaping a pure phase in [0,2π). Such a matrix represents a diagonal unitary gate N N random state as defined in [18]. Note that the usage of the acting on a bipartite quantum system, a state in H ⊗ H . H ⊗ term “unimodular” concerns all the entries of a matrix, so Any matrix U acting on the composed Hilbert space N H such a matrix is not unimodular in the sense of being integer N can be represented in its operator Schmidt form, ± matrices with determinants 1. K Although the unimodular ensemble differs from the Gini- = ⊗ U k Bk Bk , (1) bre ensemble of complex, non-Hermitian matrices, with k=1 independent, normally distributed elements, it displays the 2 where the Schmidt rank K N . Note that the matrices Bk same asymptotic behavior of the level density, which covers uniformly the unit disk. On the other hand the squared singular and Bk of order N in general are nonunitary. values of unimodular matrices coincide with eigenvalues of It can be shown [24] that the Schmidt coefficients k, = certain specific quantum states of size N, the diagonal entries k 1,...,K, can be obtained as squared singular values R which are equal to 1/N. As the notion of an “antidiagonal of the reshuffled matrix U . This fact is briefly recalled matrix” has entirely different meaning, the density matrices in Appendix A, where the notation is explained. A generic with all diagonal elements equal will be called contradiagonal. diagonal matrix of size four after reshuffling forms a non- Observe that reduced density matrices of random phase Hermitian matrix of rank two, ⎡ ⎤ pure states are thus contradiagonal. We investigate properties U11 000 of such an ensemble of quantum states and discuss the ⎢ 0 U 00⎥ U = ⎣ 22 ⎦ , contradiagonalization procedure, which brings any Hermitian 00U33 0 matrix to such a basis, that all their diagonal elements do 000U44 coincide. ⎡ ⎤ One may expect that random contradiagonal states cor- U11 00U22 ⎢ 0000⎥ respond to the large entanglement of the pure bipartite U R = ⎣ ⎦ . (2) states and we show that indeed these states have larger 0000 von Neumann entropy than those sampled according to the U33 00U44 Ginibre ensemble. The unimodular ensemble, while having no For a diagonal matrix U of order N 2 the reshuffled matrix obvious invariance properties, seems to also have remarkable U R contains N(N − 1) columns and rows with all entries underlying mathematical structure. For example, the average equal to zero. Hence the nonzero singular values of U R are of the moments of the matrices in the ensemble are connected equal to the singular values of a square matrix A of size N, to polynomials with combinatorial interpretations. We evaluate obtained by reshaping the diagonal of the unitary gate, Ajk = the first four moments and use this to conjecture an exact exp(iφν ), where ν = (j − 1)N + k.