Diagonal Unitary Entangling Gates and Contradiagonal Quantum States

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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].

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. As all entries of A are expression for all of them. We numerically show that this unimodular and have a random phase, this construction defines is more than likely to be correct. Analytical continuation of an ensemble of random unimodular matrices. the moments to noninteger powers allows us to evaluate the average von Neumann entropy for this ensemble which appears to be exact for all dimensions. III. RANDOM UNIMODULAR MATRICES This paper is organized as follows. In Sec. II the ensemble Consider a complex square matrix A of size N from of diagonal unitary gates is introduced, while in Sec. III the re- the unimodular ensemble, so that (a) all entries have the lated ensemble of unimodular matrices is analyzed. Low-order same modulus, |Ajk|=1, and (b) the phases are drawn moments are calculated analytically, and a natural conjecture independently from a uniform distribution, is made for exact expressions for all moments. Numerical 1 evidence is presented for this. Nonlocality and entangling Ajk = exp(iφ),P(φ) = , for φ ∈ [0,2π). (3) powers of random unitary gates are studied in Sec. IV. An exact 2π expression based on a continuation of moments is presented Such a random matrix A could be called a pre-Hadamard as for the average Schmidt strength of diagonal unitaries, or all entries have the same modulus, so choosing an appropriate

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ensemble, deviations are visible for small matrix size. To visu- alize these effects we studied the moments of the distribution of m squared singular values Mm = x P (x)dx. Figure 2 shows a comparison of the moments M2, M3, and M4 for random matrices of the unimodular ensemble and the Hilbert-Schmidt ensemble of order N. In the latter case analytical predictions for the moments are known as the traces of the random states ρN of size N distributed according to the Hilbert-Schmidt FIG. 1. (Color online) Properties of random unimodular matrices measure and read [2,27], of order N = 100 are close√ to those of the complex Ginibre ensemble: (a) the spectrum of A/ N in the complex plane satisfies the circular 2 law of Girko; (b) the radial density P (r) of the complex eigenvalues 2N 5N + 1 Trρ2 = , Trρ3 = , grows linearly in the center of the unit disk; and (c) the distribution of N HS N 2 + 1 N HS (N 2 + 1)(N 2 + 2) squared singular values P (x) is described by the Marchenko-Pastur (5) distribution. set of phases it may become unitary, and thus belong to the and due to rescaling of the variable x one has Mm = − class of complex Hadamard matrices [25]. In our model all N m 1Trρm. Numerical data, such as those presented in Fig. 2, phases are random and noncorrelated, so a typical matrix show that for a given N the moments for the unimodular from this ensemble exhibits effects of strong nonunitarity. ensemble are smaller, so the corresponding distributions are The ensemble of unimodular matrices A with all independent, narrower, even though for large N both distributions tend to well-behaved, identically distributed entries is of the Wigner the limiting Marchenko-Pastur distribution. type. Thus this non-Hermitian ensemble or random matrices Note that the averages moments for the distribution of satisfies asymptotically the circular law of Girko [26]. squared singular values for random Ginibre matrices of a As shown in Fig. 1 already for N = 100 the spectral given size N, derived recently in [28], coincide with the density for the unimodular ensemble is close to uniform predictions (5) for the HS ensemble only in the asymptotic in the unit disk. Furthermore, the distribution of rescaled case N →∞. In the former ensemble the constraint concerns † squared singular values, x = eig(AA†)/N, is asymptoti- the average trace TrGG , while in the latter each random = cally described√ by the Marchenko-Pastur (MP) distribution matrix has a fixed trace, Trρ 1, so that this difference = 1 − ∈ asymptotically vanishes. PMP (x) π 1/x 1/4forx [0,4], characteristic to the Ginibre ensemble. The moments of this distribution are given by the Catalan numbers, while its entropy reads − 4 =− = 0 x ln xPMP (x)dx 1/2. As x Nλ, where λ denotes A. Lower order moments and a conjecture for all orders the eigenvalue of a normalized density matrix The lower order moments of the unimodular ensemble can ρ = AA†/N 2 (4) be exactly evaluated and compared to the above case. In fact we will calculate exactly moments till the fourth and conjecture = satisfying Trρ 1, the average entropy of spectrum of ρ be- an exact formula for any moment. The density matrix elements − haves asymptotically as ln N 1/2. Note that this behavior is are characteristic of random quantum states distributed uniformly with respect to the Hilbert-Schmidt measure [27] in the entire set of quantum states of a given dimension. 1 N ρ = exp i φ − φ . (6) Although for large N statistical properties of the uni- α1α2 2 α1l1 α2l1 N = modular ensemble coincide with those of a complex Ginibre l1 1

2 5 14

1.9 4.5 12

1.8 4 10 2 3 4 M M M 1.7 3.5 8

1.6 Unimodular ensemble simulation 3 Unimodular ensemble analytical HS ensemble simulation 6 1.5 HS ensemble analytical 2 2.5 5 14 4 5 10 15 20 25 30 5 10 15 20 25 30 5 10 15 20 25 30 N N N

FIG. 2. (Color online) Moments of the distribution of squared singular values P (x) for unimodular ensemble and the Hilbert-Schmidt measure as a function of the matrix size N: from left to right are shown the second moments M2, third moments M3, and fourth moments M4. Horizontal lines at M2 = C2 = 2, M3 = C3 = 5, and M4 = C4 = 14 denote the Catalan numbers, which give the corresponding moments of the MP distribution and determine the asymptotic behavior of both moments. Solid lines represent predictions (5) for the Hilbert-Schmidt measure, while the dashed lines the predictions for the unimodular ensemble.

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ρ2 1. The second moment Tr N UE values between 1 and N. Another bijection is between the The second moment while being the simplest also serves as indexes and points that are joined by noncrossing semicircles. a measure of purity of a given quantum mixed state. We have Thus the contributions from ()()() and (()()) are respectively 1 1 = N 4, 1 = N 4 − N 2. Trρ2 = exp i φ − φ + φ − φ . (7) 4 α1l1 α1l2 α2l2 α2l1 N l1 α1,α2,α3 α1 l1,l2,l3 α1α2 l1l2 The last sum is restricted in the sense that it does not include On averaging over the uniform phases the only terms that the case l = l = l that is already included in the first count. would survive are those cases for which the phase vanishes. 1 2 3 = The other three cases contribute equally: This happens if α1 α2 for arbitrary l1 and l2. Thus this case contributes 1 = N 2(1 − N)2, 3 = = 1 N 1 α1 α3 l1 l2 1 = = . (8) N 4 N 4 N where the distinct indexes are always unequal; if they are equal α1 l1l2 it will reduce to a term considered in the first two cases. Thus Indeed this is the “diagonal” contribution. The phase also van- putting them all together we get ishes if α1 = α2,butl1 = l2. This “off-diagonal” contribution is 3 1 2 Trρ UE = (5N − 6N + 2), (11) 1 1 N − 1 N 4 1 = N(N − 1)N = . (9) 4 4 2 = 2 3 = − + 2 N = N N and M3 N Tr ρN (5 6/N 2/N ). α1 α2 l1 k 2(k−1) The kth moment Trρ is of the form Pk(N)/N , As these exhaust the exclusive possibilities, the average where Pk(N) is a degree k − 1 polynomial in N whose leading second moment for the level density related to the unimodular = 1 2k term’s coefficient is Ck k+1 ( k ), the kth Catalan number. ensemble reads k−1 k Thus it follows that the moments Mk = N Trρ tend to Ck 2N − 1 Trρ2 = . (10) and hence the asymptotic density is described by the universal N UE N 2 Marchenko-Pastur distribution. Writing explicit expressions

Equivalently M2 = (2N − 1)/N, which indeed tends to 2 for the moments Mk for the unimodular ensemble we are 2 2 for large N. Also note that Trρ HS −Trρ UE = (N − in position to quantify the deviations from the asymptotic 1)2/[N 2(N 2 + 1)] > 0, indicating that the density matrices universal MP distribution. constructed from the unimodular ensemble are on average ρ4 more mixed than those from the Hilbert-Schmidt ensemble. 3. The fourth moment Tr N UE We will now evaluate explicitly the fourth moment that ρ3 2. The third moment Tr N UE presents some challenges and then conjecture an exact expres- The third moment is obtained by considering all those cases sion for Pk(N). There are 14 different parenthesizations for when the k = 4 case with 8 pairs of indexes involved. There are 3 “contractions” in each case. For instance the contractions φ − φ + φ − φ + φ − φ α1l1 α1l2 α2l2 α2l3 α3l3 α3l1 corresponding to ()(())() are l1 = l2, α2 = α3, l4 = l1. Thus there are 8 − 3 = 5 sums that are unconstrained from each vanishes for arbitrary sets of phases, where αi and li take values in 1,...,N. The indexes are ordered in such a way that of the 14 parenthesizations and hence the leading term in 8 4 5 a bijection to a standard counting problem becomes possible. N TrρN is 14N . Generalizing, to the kth moment, the − Starting from the first pair, the sign is reversed while the second number of contractions is k 1. This is seen by writing the al- index is new in the second pair. The third pair is obtained by ternating indexes as actual products and sums while respecting k − = again reversing the sign of the second but now the first index their distinctness and requiring i=1 αi (li li+1) 0 with = = becomes new, and so on. Finally when there are 6 pairs, the lk+1 l1. Setting say αnln αmlm+1 for some n and m reduces last index is the same as that of the first pair. It is clear that this the number of terms by 1. Due to periodic boundary conditions, generalizes to the moment of order k, where there are 2k such the number of contractions that will set the whole sum to 0 pairs. is k − 1. Thus it follows that in general the leading term in 2k k 2k−(k−1) = k+1 At this level a bijection to several standard problems in N Trρ is CkN CkN . 8 4 counting that involve the Catalan numbers Ck [29] is possible. Overcounting however lowers the value of N TrρN from 5 k+1 = For example that of matching 3 pairs of parentheses— 14N , and in general from CkN . Sticking to the k 4 case, ()()(),(())(),()(()),((())),(()())—is relevant to the third moment; the combined contributions from ()()()() and (()()()) which 5 − 2 2 each parenthesis represents the pair of indexes at the cor- either contract all li or all αi respectively is 2N N , N responding place. The matched pair of parentheses imply being the double count of all αi being the same and all li being that the pair of indexes are equal. The first possibility and the same. The other 12 contributions involve sums such as = its translation in terms of indexes is ()()(): α1l1 α1l2, = 2 − − 2 = = = = 1 (N 1)(N 1)N , α2l2 α2l3, α3l3 α3l1,orl1 l2 l3. The second (())(): = α1 α2 α3=α4 =α1 l1=l3 l2,l4 α1l1 = α2l3, α1l2 = α2l2, α3l3 = α3l1,orα1 = α2, l1 = l3. Similarly ()(()): α2 = α3, l1 = l2; ((())): α1 = α3, l2 = l3; (()()): where the restricted sum eliminates the cases when l2 = l4 = α1 = α2 = α3. The unconstrained indexes can take arbitrary l1 = l3, which has already been considered. While this case

032303-4 DIAGONAL UNITARY ENTANGLING GATES AND . . . PHYSICAL REVIEW A 90, 032303 (2014) corresponds to the parenthesization (())(()), the other 11 have where Cn,s is Catalan’s triangle: sums over 5 indexes with similar restrictions. 1 Thus these contribute 12N 2(N − 1)(N 2 − 1); however 11 there still remains some overcounting to be accounted for. 122 For example the above case includes instances when l = 2 1 355 l = l but = l which is also possible when the contracted 1 3 4 1 491414 indexes are α = α , and l = l = l and corresponds to 2 4 1 2 3 ... the parenthesization ()()(()). Exhaustive enumeration of these terms that originate from contracting 4 indexes reveals that which satisfies the recursion Cn,k = Cn−1,k + Cn,k−1; that is, there are 16 such terms each of which gives the entries are sums of the one to the left and the one above. = The first column of all 1 is the “boundary condition” Cn,0 1. 1 = N 2(N − 1)2. Explicit formulas for Cn,k and fn,k are available [31]: ======α1 α2 α3 α4 α1 l1 l2 l3 l4 l1 (n + k)!(n − k + 1) C = , n,k k!(n + 1)! There is no further overcounting as the cases with > 4 contractions have already been properly included. Finally 1 2n + 2 n + k 5 2 2 f = . in total the contribution is 2N − N + 12(N − 1)(N − n,k n + 1 n − k k 1)N 2 − 16N 2(N − 1)2 = 14N 5 − 28N 4 + 20N 3 − 5N 2 and we get It is then natural to conjecture that for n 1 the average moments of the level density of the random matrices of size 1 N are Tr ρ4 = (14N 3 − 28N 2 + 20N − 5). (12) N UE N 6 n−1 n 1 k n−k−1 Trρ = (−1) f − N . (14) N UE 2(n−1) n 1,k N = ρn k 0 4. A conjecture for all moments Tr N UE = The complexity of the counting problem is naturally The smallest unproven case is n 5, which can be simply read off from Borel’s triangle: increasing. However having found the polynomials P2(N) = − = 2 − + = 3 − 2N 1, P3(N) 5N 6N 2, and P4(N) 14N 5 1 4 3 2 2 Trρ = (42N − 120N + 135N − 70N + 14). 28N + 20N − 5 exactly, the following are evident: they UE N 8 = have alternating signs, satisfy Pk(1) 1, and the constants (15) (coefficients of N 0) have the absolute value 1, 2, and 5 which are themselves Catalan numbers. That Pk(1) = 1 follows Figure 3 displays the numerical calculations of moment iφ 4 5 simply as for N = 1, there is a only a single pure phase e M5 = N Trρ ,aswellasM6 and M7 from a million and the “density matrix” is simply eiφe−iφ = 1. random realizations of the ensemble and shows how well this Based on the triangle of coefficients {1,{2,1},{5,6,2}, conjecture fares. There seems to be no room for doubting the {14,28,20,5}} a search in the On-line Encyclopedia of Integer correctness of the conjecture. Sequences [30] (OEIS) returns two sequences A062991 and A calculation of the first few cumulants from the moments A234950. In the first of which is the signed version (that we Mn of the scaled variables Nλ results in (apart from κ1 = 1) encounter), it arises as generalizations of Pascal’s triangles 1 via Riordan arrays [31]. In the second, the unsigned version, κ2 = (N − 1), which the authors call Borel’s triangle [32], connections are N made to new counting problems in commutative algebra and 1 κ3 = (N − 1)(N − 2), discrete geometry. Going beyond what we have above, the first N 2 seven rows of the unsigned triangle read 1 κ =− (N − 1)(4N − 5), 4 N 3 1 1 2 21 κ5 =− (N − 1)(N − 2)(4N + 2N − 7). (16) N 4 56 2 14 28 20 5 As N →∞these tend to {1,1,1,0,−4,...}, the initial cumu- 42 120 135 70 14 lants corresponding to the moments being the Catalan numbers 132 495 770 616 252 42 {1,2,5,14,42,...}. 429 2002 4004 4368 2730 924 132 The moments of the unimodular ensemble themselves also ... seem to have combinatorial significance. For example the (n−1) moments for N = 3 are such that 3 Mn(N = 3) is the { } The left and rightmost entries are Catalan numbers. The entry integer sequence 1,5,29,181,1181,... . If we include an additional 1 corresponding to M , this sequence is found as a fn,k (n 0,k 0) of the Borel triangle is 0 column in the entry A183134 of the OEIS. Indeed the other n columns of the square array of this entry are similarly the s = fn,k = Cn,s , (13) moments for different values of N 1,2,....Thisprompts k 2(n−1) n s=0 the additional conjecture that the N Tr(ρN )isthesame

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45 140 450 400 40 120 35 350 100 300 30 80 250 5 25 6 7 M M M 60 200 20 150 40 15 Unimodular ensemble simulation 100 20 Unimodular ensemble conjecture 10 HS ensemble simulation 50 42 132 429 5 0 0 5 10 15 20 25 30 5 10 15 20 25 30 5 10 15 20 25 30 N N N

FIG. 3. (Color online) The fifth, sixth, and seventh moments, M5, M6,andM7, of the distribution of squared singular values P (x)for the unimodular ensemble as a function of the matrix size N. The points are from numerical simulation based on a million realizations while the curves are from the formula in Eq. (15). The horizontal lines are at C5 = 42, C6 = 132, and C7 = 429, further Catalan numbers and the asymptotic value of the scaled moments. = = n + − as the number of N-alphabet words of length 2n beginning Pochhammer symbols (x)0 1 and (x)n j=1(x j 1). with the first character of the alphabet by repeatedly inserting Using this we can continue the moments to noninteger powers, x doublets into the initially empty word, as this is the counting so that we have i λi for x real. Observing that the average − x − problem that is stated in the OEIS entries (see also [33]). For entropy S(ρ) UE is the limit of (1 i λi )/(x 1) as example in the case of n = 2 and N = 2, the alphabet set is x → 1+, we have that binary {a,b}. Doublets are repeated alphabets. Thus inserting = two doublets (for n 2) one gets aaaa, aabb, abba as the S(ρ) UE =−df (x)/dx|x=1, (20) three possibilities that start with a. This coincides with 2N − 1 = x = that we derived above. where f (x) TrρN Having explored the moments of the unimodular ensemble we now turn to one of our central motivations, finding the 1 (2x + 1) F (x,1 − x;2+ x;1/N). (21) entangling power of diagonal unitaries. N x−1 (x + 1) (x + 2) 2 1

IV. ENTROPY OF THE UNIMODULAR ENSEMBLE AND While it is not evident that such a continuation be exact, that NONLOCALITY OF RANDOM DIAGONAL GATES it is indeed very likely to be so is illustrated in Fig. 4, where the moments are plotted for 1 x 2 for small values of N. By construction, the average entropy of squared singular This gives us conﬁdence that the entropy found from such a values of random unimodular matrices A is equal to the average procedure is also exact. entropy of entanglement for the corresponding unitary gates To evaluate −f (1) one needs to evaluate the derivatives of R U = A . As squared singular values of A are asymptotically gamma functions and the hypergeometric function with respect described by the Marchenko-Pastur distribution, making use to their parameters. Using that (z) = (z)ψ0(z), where ψ0(z) of the Page formula [3] we infer that the mean entropy of entanglement (A4) of a random diagonal gate behaves as = = † 2 ≈ − 1 S(U) diag S(ρ AA /N ) UE ln N 1/2. (17) N=2 (n) N=2 (a) N=3 (n) This result forms approximately a half of the entropy of generic 0.9 N=3 (a) Haar random unitary matrices [17] N=4 (n) N=4 (a) N=5 (n) 0.8 S(U) Haar ≈ 2lnN − 1/2. (18) N=5 (a) )

Based on the discussion of moments in the previous x

N 0.7 section and taking them to be exact allows us to find what ρ appears to be an exact expression for the average entropy of Tr( 0.6 entanglement, which can be interpreted as entanglement in a random ensemble of states S(ρ) UE where ρ is defined in 0.5 Eq. (4) or the average entanglement of diagonal unitary gates via Eq. (A4). Using the view of state entanglement, from the last section, the expression in Eq. (14) can be used to write the 0.4 nth moment as 1 1.2 1.4 1.6 1.8 2 x n 1 2n Trρ = 2F1(n,1 − n;2+ n;1/N), N UE nN n−1 n − 1 FIG. 4. The moments for noninteger powers between 1 and 2 = (19) are plotted for the unimodular ensemble, and N 2,3,4. The points are obtained by numerical simulation using 106 realizations from the where 2F1 denotes the hypergeometric function defined ensemble, while the lines are those from the analytical expression in n ∞ (a)n(b)n z as 2F1(a,b; c; z) = = , using the notion of Eq. (21). n 0 (c)n n!

032303-6 DIAGONAL UNITARY ENTANGLING GATES AND . . . PHYSICAL REVIEW A 90, 032303 (2014) is the digamma function, we get that are exact for any dimension. The differences between the d 1 (2x + 1) 1 formula√ and numerical simulations are shown to be smaller − | = = ln N − , (22) than 1/ N , where N denotes the size of the numerical x−1 + + x 1 S S dx N (x 1) (x 2) 2 sample. For large N it is easy to see that this approaches where the origin of 1/2 is due to the fact that ψ0(3) − ψ0(2) = ln N − 1/2 with the neglected terms being of order 1/N, 1/2. The derivative of the hypergeometric function can be in agreement with what is expected from the Marchenko- evaluated using its definition as an infinite series. We have to Pastur distribution and from the Page formula [3]fortheHS compute ensemble. For instance, if N =√2, the exact density of the ∞ eigenvalues in [0,1] reads 1/(π y(1 − y)), as discussed in d (x) (1 − x) 1 m m , (23) the next section. Hence the average entropy is dx (x + 2) m!N m m=0 m 1 −2y ln y for which we need the derivatives of the Pochhammer symbols. √ = ln 4 − 1, (26) π y(1 − y) Due to the fact that we need to evaluate the derivative at x = 1 0 [and (0)m = 0form>1, while (0)0 = 1], we only need that which agrees with Eq. (25). We may compare this with the d(1 − x) /dx| = =−(m − 1)! which is easy to see from the = m x 1 exact entropy from the HS ensemble, which is S(ρ) HS definition of the symbol. Putting these together we get that N 2 − − = k=N+1 1/k (N 1)/(2N)[3]. For example for N 2this gives 1/3 which is smaller than that for the UE ensemble − d − + | 2F1(x,1 x,x 2; 1/N) x=1 − ≈ dx that is ln 4 1 0.39. Although the difference decreases with ∞ the matrix size, the relation S(ρ) HS < S(ρ) UE holds true, = 2 1 which indicates again the enhanced average entanglement in m(m + 1)(m + 2) N m m=1 the unimodular ensemble. Another measure of nonlocality of gates is the so-called 3 1 = − N − (N − 1)2 ln 1 − . (24) entangling power based on the ability of operators to create 2 N subsystem mixed states from originally unentangled pure | ⊗| The last equality is obtained as the infinite sum can be evaluated bipartite states. If ψ1 ψ2 is an unentangled state in H ⊗ H by elementary means, integrating thrice the identity 1/(1 − N N , and U is an unitary operator on this space, x) = 1 + x + x2 +···, and thus finally the average entropy is its entangling power as defined by Zanardi, Zalka, and the sum of the results in Eqs. (22) and (24): Faoro [9]is N − 1 = | ⊗| ψ1,ψ2 S(ρ) = ln N − (N − 1) − (N − 1)2 ln . ep(U) E(U ψ1 ψ2 ) , (27) UE N the average being over all product states. Any entanglement (25) measure can be used for E, the one that was used in [9] being Numerical results presented in Fig. 5 provide further the simplest useful one, the linear entropy: E(|ψ ) = 1 − 2 arguments that the above expressions for the average entropy Tr1μ , where μ ≡ Tr2|ψ ψ| is the reduced density matrix of the subsystem labeled by 1. The case of interest in the present work is one where the 3 Numerical matrix U is diagonal and hence the following is obtained: Analytical

iφαl 2.5 α|l|U|ψ1 |ψ2 =e α|ψ1 l|ψ2 , (28)

where we have used αl|U|βm =eiφαl δ δ . The reduced 2 αβ lm density matrix is

)> −

ρ 1.5 i(φα l φα l ) μα α = e 1 2 α1|ψ1 l|ψ2 ψ1|α2 ψ2|l , (29) -0.0001 1 2

2 6 10 14 18 α1,α2,l1,l2 0 2 2 2 2 2 4 6 8 10 12 14 16 18 20 ×|α1|ψ1 | |l1|ψ2 | |α2|ψ1 | |l2|ψ2 | . (30) N Averaging over the states |ψ1,2 can be done assuming that FIG. 5. The average entropy or entanglement in the unimodular they are random vectors distributed uniformly according to ensemble is plotted against various dimensionalities. The points are the invariant Haar measure. Further assuming the general case obtained by a numerical simulation using 106 realizations, while of complex random states [34], the following are the average the smooth line is from the analytical formula in Eq. (25). The of the products of two intensity components: essentially exact nature of the expression is illustrated in the inset where the difference between the numerical and analytical is shown to δ + 1 |α |ψ |2|α |ψ |2 = α1α2 . (31) be consistent with statistical ﬂuctuations from a million realizations. 1 1 2 1 N(N + 1)

032303-7 LAKSHMINARAYAN, PUCHAŁA, AND ZYCZKOWSKI˙ PHYSICAL REVIEW A 90, 032303 (2014)

Using this, it follows that function f becomes minimal, N 2 + 2N 3 + N 4Tr(ρ2) = † = | † |2 2 = D UminHUmin : f (Umin) min (VHV )ij . Trμ . (32) V ∈U(N) N 2(N + 1)2 i =j (34) The connection to Trρ2 of the last section follows from Eq. (7). Thus the entangling power of diagonal unitaries e (U) = 1 − † p In a similar way let A represent a Hermitian matrix VHV for Trμ2 is directly related to the second moment of the squared which the function f becomes maximal, singular values of the reshaped matrix. The average entangling power, now averaged over all phases in the diagonal unitary = † = | † |2 A UmaxHUmax : f (Umax) max (VHV )ij . gates, is V ∈U(N) i =j N 2 + 2N 3 + N 2(2N − 1) N − 1 2 (35) e (U) = 1 − = , p diag N 2(N + 1)2 N + 1 It is clear that the minimum f (Umin) = 0 is achieved for a (33) matrix U = Umin consisting of eigenvectors of H ,soD is a diagonal matrix of eigenvalues of H or any other matrix similar 2 = − 2 where the result Trρ UE (2N 1)/N has been used with respect to permutations D = PDPT . As the standard from the previous section. This can be compared with the procedure to find D = UHU† is called diagonalization, the = average entangling power of unitary gates [9]: ep(U) transformation of H by U , leading to the maximum in (35), − 2 2 + max (N 1) /(N 1), which is only marginally larger. The will be called contradiagonalization. For any given Hermitian entropies of full unitaries were almost twice as large as the H we shall show below how to find its contradiagonalizing diagonal ones. At the level of the purity however one still sees matrix Umax. the difference in that Trμ2 =4N/(N − 1)2 ≈ 4/N is double It is well known [50] that any matrix can be unitarily that of the reduced density matrix of typical random states in transformed to a form in which all the diagonal entries are HN ⊗ HN , which reads 2/N [2]. equal. As the trace of a Hermitian matrix H is unitarily invariant this constant reads Hjj = TrH/N for j = 1,...,N. Let F denote a complex Hadamard matrix [25]oforder V. CONTRADIAGONAL HERMITIAN MATRICES N, so that F is unitary√ and the moduli of all its entries Any ensemble of random matrices allows one to generate an are equal, |Fjk|=1/ N. As a typical example let us = ensemble of quantum states [35]. Taking a random matrix A, mention the Fourier√ matrix of size N with entries (FN )jk one writes σ = AA†/TrAA† to get a random density matrix: a exp(ij kπ/N )/ N for j,k = 0,...,N − 1. Two complex Hermitian, positive operator normalized by the trace condition Hadamard matrices are called equivalent, written H2 ∼ H1, Trσ = 1. In the case of unimodular random matrices (3) one if they are equivalent up to enphasing and permutations, † 2 † 2 has TrAA = N , hence σ = AA /N . H2 = P1E2H1E2P2.HereE1 and E2 denote diagonal unitary Observe that by construction of the unimodular matrix matrices while P1 and P2 represent permutation matrices of A the corresponding positive Wishart matrix AA† has all order N.ForN = 2,3 and N = 5 all complex Hadamard diagonal elements equal. Thus the diagonal elements of the matrices are equivalent to the Fourier matrices F2, F3, and corresponding random density matrix ρ read ρii = 1/N, F5, respectively; see [25]. where i = 1,...,N. In other words, the state ρ is represented In order to compare spectra of Hermitian matrices it is in such a particular basis {|1 ,|2 ,...,|N } that the expectation convenient to use the notion of majorization. Consider vectors values among each of the basis states are equal. Thus the of size N ordered decreasingly, x1 x2 ...xN . A vector entropy of an orthogonal measurement in this basis is maximal y is said to majorize [36] vector x, written x ≺y, if partial m m and equal to ln N. We show below that such a basis is dual sums satisfy following inequalities: i=1 xi i=1 yi for to the basis in which a given state is diagonal, in the sense = − N = N m 1,...,N 1, and additionally i=1 xi i=1 yi .A that the norm of all the off-diagonal elements is maximal. function f : RN → R is said to be Schur convex when x ≺ y † Thus any density matrix σ = AA /N 2 constructed out of a implies f (x) f (y). random unimodular matrix A has all diagonal elements equal Now we can formulate the following result. and therefore will be called contradiagonal. Proposition 1.LetH be a Hermitian matrix of order N and let V be unitary. Then (a) the maximum in (35) is obtained for Umax = FUmin A. Procedure of contradiagonalization of a matrix where Umin denotes the matrix of eigenvectors of H and F is Let H be a Hermitian matrix of order N and let G = a complex Hadamard matrix, † † = † = † VHV be a unitarily similar matrix, as VV V V IN .All (b) the matrix A = UmaxHUmax obtained in this way is matrices from this orbit share the same spectrum and possess contradiagonal, Ajj = TrH/N for j = 1,...,N, = = 2 2 the same trace, TrG TrH : t. For any fixed H we are (c) the maximum reads fmax(V ) = TrH − (TrH ) /N. going to analyze the sum of the squared moduli of off-diagonal Proof of Proposition 1. First to show item (b) we con- = | |2 elements of G and define a function f (V ) i =j Gij .Let sider first diagonal matrix H = D and take an arbitrary us denote by D any Hermitian matrix VHV† for which the complex Hadamard matrix F and find that A = FDF† is

032303-8 DIAGONAL UNITARY ENTANGLING GATES AND . . . PHYSICAL REVIEW A 90, 032303 (2014) contradiagonal, as all its diagonal elements are equal, where d is a diagonal of matrix D and q(V ) is a diagonal of matrix VDV†. It is easy to see that one obtains the maximum † = † = (FDF )ii FikDkl(F )li FikDkkFik value if the vectors are ordered in the same way; i.e., kl k | (V ) = ↓| (V ) ↓ 1 max d Pq d (q ) . (43) = |F |2D = TrD. (36) P ∈Perm ik kk N k To perform minimization over the set of unitary matrices V ∈ Consider now any Hermitian matrix H and denote by U the U(N) we note that the minimum value for the above inner product is achieved, if vector q is minimal in the majorization matrix of its eigenvectors. Thus taking a unitary matrix Umax = FU we see that the transformed matrix partial order, min † † † 2 A = U HU = FU HU F (37) ↓ (V ) ↓ di (TrD) max max min min min d |(q ) di = . (44) V ∈U(N) N N is contradiagonal, as all its diagonal elements are equal, as stated in item (b). The above minimum can be achieved if the unitary matrix V To prove item (a) we note that the sum in (35) is maximized is complex Hadamard for instance the Fourier matrix FN . if and only if the sum To summarize the proof we write † † − T 2 | |2 max min D PVDV P HS (VHV )ii (38) V ∈U(N) P ∈Perm i 1 = 2 TrD2 − (TrD)2 , (45) is minimized, since the vector of diagonal elements majorizes N the constant vector of the same sum, and the sum of squares is a Schur-convex function. We thus obtain the result. and every complex Hadamard matrix V gives the maximum The last item (c) is obtained from the definition of the and in this case one can take any permutation matrix P . function f by computing the trace of H 2 and subtracting the sum of squared elements at the diagonal. B. Contradiagonal density matrices A generalization of this procedure allowing us to find a basis in which the Hermitian matrix with spectrum y has diagonal The statements on contradiagonalization introduced above x ≺ y for arbitrary Hermitian (or normal) matrices can be now used is presented in Appendix B. † = Proposition 2 for positive-definite density matrices ρ ρ 0 normalized . Consider a family of unitarily similar = Hermitian matrices as Trρ 1. Thus a quantum state σ of size N will be called contradiagonal if σii = 1/N for i = 1,...,N. G = VDV†, (39) Spectral density for the ensemble of contradiagonal states obtained form a random unimodular matrix A by Eq. (4) where D is a given diagonal matrix; then the maximal Hilbert- is shown Fig. 6 for N = 2,3,4. For N = 2 a random con- Schmidt distance between the orbit of the unitarily similar 1 z tradiagonal density matrix takes the form σ = 1 [ ], where matrices to diagonal matrix D optimized with respect to all 2 z¯ 1 = iψ1 + iψ2 permutation matrices is given by z e e and the phases ψ1 and ψ2 are random. Thus the rescaled eigenvalues of√σ are distributed according to the (TrD)2 = − ∈ − † T 2 = 2 − arcsin law, PAs(x) 1/π x(2 x)forx (0,2), as shown max min D PVDV P HS 2 TrD , V ∈U(N) P ∈Perm N in Fig. 6. Note oscillations of the level density P (x) present for N (40) 3. Observe that the conjectures above yield all the moments and for the optimal matrix Vopt one can take any complex for any value of the matrix size N. It is then a classic moment Hadamard matrix F . problem to find the corresponding density. Curiously, there Consider an arbitrary Hermitian matrix H, such that H = exists densities that have the exact same moments on different UDU†, so its diagonalization corresponds to transforming the intervals and are found in [35] as a sequence of densities that basis by the matrix U consisting of eigenvectors. The above converges to the MP distribution. We were however unable lemma explains why representing it in the basis W = FU† can to solve this moment problem, to find the actual oscillatory be called contradiagonalization, as the matrix A = WHW† one that is found for the unimodular ensemble. While the has all diagonal elements equal and is as far from the diagonal appearance of multiple densities with the same moments is matrix as possible. known in the literature [38], this seems to be a curious case as Proof of Proposition 2. To prove the lemma we write the densities have support in (0,1]. The fact that the densities diverge at the origin makes the current moment problem not − † T 2 max min D PVDV P HS belong to the class of Haussdorf moment problems that treat V ∈U(N) P ∈Perm compact intervals and absolutely continuous densities [38]. † = max min 2TrD2 − 2TrDPV DV P T . (41) The Schur-Horn theorem states [37] that for any density V ∈U(N) P ∈Perm matrix ρ its diagonal is majorized by the spectrum, diag(ρ) ≺ Next we note that eig(ρ). The uniform vector x∗ ={1/N,...,1/N} is majorized by any other probability vector. This observation implies the max TrDPV DV †P T = max d|Pq(V ) , (42) P ∈Perm P ∈Perm following fact:

032303-9 LAKSHMINARAYAN, PUCHAŁA, AND ZYCZKOWSKI˙ PHYSICAL REVIEW A 90, 032303 (2014)

2.5 3 4 Unimodular ensemble HS ensemble (n) 3.5 2.5 2 N=2 HS ensemble (a) Marcenko-Pastur density 3 2 N=3 1.5 2.5 1.5 2 N=4 P(x) P(x) P(x) 1 1.5 1 1 0.5 0.5 0.5

0 0 0 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.5 1 1.5 2 2.5 3 3.5 4 x x x

FIG. 6. (Color online) Distribution of squared singular values P (x) for random unimodular matrices of order (a) N = 2, (b) N = 3, and (c) N = 4 (crosses). The case N = 2 is described by the arcsin distribution supported in [0,2], while the asymptotic behavior corresponds to the MP distribution represented by solid lines. The corresponding density of random states distributed according to the Hilbert-Schmidt measure is marked with dots for comparison, while the exact expressions are also plotted.

Proposition 3.Letσ denote a contradiagonal state of order unitary matrices one obtains an enphased, complex Hadamard = N, so that σii 1/N, and let U be a unitary matrix of order FN [25], which leads to a more general form of a complex N. Then the following majorization relation holds: Hermitian contradiagonal state σ with all elements of the same modulus, |σ |=1/N, and all diagonal elements equal, ≺ † ≺ ij diag(σ ) diag(UσU ) eig(σ). (46) = σjj 1/N. The above result provides an additional argument in favor As stated in Proposition 1 the sum of squared moduli of the of usage of the notion of a contradiagonal form of a matrix, of-diagonal elements is maximal if and only if the matrix is in as σ is distinguished by the majorization order (46) and is its contradiagonal form. The above is equivalent to the fact that opposite to the diagonal form of a density matrix. the sum of squared diagonal elements is minimal. On the other Let ρ denote an arbitrary density matrix, Umin the matrix hand, since the geometric mean is a Schur concave function, of its eigenvectors, and Umax = FUmin the matrix defining the product of diagonal elements of a semipositive matrix H the bases in which the state is contradiagonal. Then the is maximal if H is contradiagonal. This fact was used in the entropy of the projective measurement of ρ with respect analysis of an entanglement measure called collectibility [40]. to the basis Umax is maximal and equal to ln N. Note that this basis is hence dual to the eigenbasis of ρ for which VI. CONCLUDING REMARKS the entropy of the projective measurement is minimal and equals to the von Neumann entropy S(ρ). As each projective In this work we showed that a generic random diagonal gate measurement induces the decoherence to the system and acting on a symmetric bipartite system is strongly nonlocal and copies the information on the eigenstates of the density is amazingly efficient in generating quantum entanglement. Its matrix to the environment, the information copied in the average Schmidt strength [14] is on average smaller than this case of the measurement in the contradiagonal basis is the characteristic of a Haar random unitary gate by the factor of largest and reads ln N − S(ρ). In other words, performing two, while the mean entangling powers [9] for both ensembles a coarse-graining map, ρ → ρ = diag(ρ)onanypurestate are only marginally different. ρ =|ψ ψ|, the exchange entropy [39] is the largest if the Investigation of the ensemble of diagonal unitary gates 2 state is represented in the contradiagonal basis. of size N leads to the unimodular ensemble of matrices of Consider, for instance, a single-qubit pure state written order N with all entries of the same modulus and independent with its eigenbasis as H = diag(1,0). Making use of the real random phases. We computed first moments of the distribution of the squared singular values of these matrices and showed Hadamard matrix F2 and putting it into Eq. (37) one gets the contradiagonal state σ where that it asymptotically converges to the universal Marchenko- Pastur form. The moments have remarkable connections to = √1 11 = 1 11 combinatorial structures that have been recently studied. This F2 − and σ . (47) 2 1 1 2 11 allowed us to find the mean entanglement or von Neumann entropy exactly for the ensemble of unimodular matrices. Choosing a complex Hadamard matrix F2 enphased with However, for a finite N we reported the differences with respect an arbitrary complex phase eıφ we obtain a more general to the Hilbert-Schmidt ensemble, which produces, on average, contradiagonal state σ with less mixed states. iφ iφ iφ Squared singular values of a matrix from the unimodular = √1 e e = 1 1 e F ,σ −iφ . (48) ensemble after a suitable normalization coincide with the 2 1 −1 2 e 1 2 spectrum of a density matrix σ of order N, such that all their In higher dimensions, a density matrix |ψ ψ| corresponding diagonal elements are equal. This form of a matrix is called to a pure state of size N with spectrum diag(1,0,...,0) contradiagonal, as it is shown to be opposite to the diagonal transformed as in (37) by the Fourier matrix FN leads to a form of a matrix concerning the majorization order, the norm flat contradiagonal state σ with all elements equal, σij = 1/N. of the off-diagonal elements, and the entropy of the projective Multiplying FN from left and right by two arbitrary diagonal measurement performed on a mixed state in such a basis.

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In general, for any Hermitian matrix H of order N we have has a unique meaning if a concrete decomposition of the total shown how to find a unitary matrix Umax which brings it to dimension, L = MN, is specified. Similar reorderings of ma- = † = trices were considered by Hill et al. [44,45] while investigating the contradiagonal form H UmaxHUmax such that Hjj TrH/N. This method based on diagonalization and complex CP maps and also in [46–48] to analyze separability of mixed Hadamard matrices [25] can be easily applied for any mixed quantum states and in [49] to generate local unitary invariants. state ρ to transform it to its contradiagonal form, distinguished This operation in these latter contexts is also referred to as from the perspective of quantum-information processing. realignment. From a mathematical perspective the problem of construct- To get a better feeling of the transformation of reshuffling ing a Hermitian matrix with a given spectrum and prescribed observe that reshaping each row of an arbitrary matrix X of diagonal entries was studied in [41,42] and more recently length N 2 into a submatrix of size N and placing it according in [43]. Although several general algorithms for this task to the lexicographical order block after block produces the were analyzed in these papers, the limiting problem of a reshuffled matrix XR as defined in (A2). Let us illustrate this constant diagonal was not shown to be reducible to the procedure for the simplest case N = 2, in which any row of standard diagonalization procedure followed by a unitary the matrix X is reshaped into a 2 × 2matrix transformation with an arbitrary complex Hadamard matrix. ⎡ ⎤ X11 X12 X21X22 ⎢ ⎥ = R = ⎣ X13X14 X23 X24 ⎦ ACKNOWLEDGMENTS Ckj Xkj : . (A3) X31 X32 X41X42 It is a pleasure to thank Sarika Jalan for the invitation X33X34 X43 X44 to the CNSD, at IIT Indore, where this work was initi- It is easy to see that (XR)R = X. In general, N 3 elements of X ated. We are thankful to Paweł Horodecki for stimulating do not change their position during the operation of reshuffling; discussions and fruitful remarks. We acknowledge support these are typeset in bold in Eq. (A3). The other N 4 − N 3 from Grants No. DEC-2011/02/A/ST1/00119 (K.Z.)˙ and No. elements do. It is worth emphasizing that if a matrix X is DEC-2012/04/S/ST6/00400 (Z.P.)financed by Polish National Hermitian the reshuffled matrix XR need not to be Hermitian. Science Centre. The Schmidt coefficients of U are thus equal to squared R singular values i of the reshuffled matrix, U , equal to the APPENDIX A: RESHUFFLING AND OPERATOR eigenvalues of a positive matrix H = (U R)†U R.ThegateU is ENTANGLEMENT ENTROPY local if and only if the rank K of H is equal to 1, so that the = ⊗ In this Appendix a link between the operator Schmidt matrix can be factorized into a product form, U UA UB . decomposition and an the reshuffling of a matrix [24]is The squared Hilbert-Schmidt norm of any unitary matrix 2 || ||2 = 2 N 2 = reviewed. of order N is U N , which implies that k=1 k Consider a given unitary matrix U of size N 2 × N 2.It N 2. To characterize nonlocality of a gate U one can then belongs to the composite Hilbert-Schmidt space HHS ⊗ HHS. use the normalized vector λ of the squared singular values, 2 Let us write down its representation in a product basis in the λk := k/N , which may be interpreted as a probability vector space of matrices, of length N 2. In general, the vector of the Schmidt coefficients of a 2 2 N N unitary matrix U acting on a composite N × N system conveys = ⊗ U CmnBm Bn, (A1) information concerning the nonlocal properties of U.To m=1 n=1 characterize quantitatively the distribution of λ one uses the † where Cmn = Tr[(Bm ⊗ Bn) U]. Shannon entropy, The complex matrix C of order N 2 × N 2 need not be N 2 Hermitian or normal. The usual Schmidt decomposition hold S(U):= S(λ) =− λ ln(λ ), (A4) for this vector space and therefore the Schmidt decomposition k k k=1 of U given√ in Eq. (1) consists of K terms entering with the weights k equal to the singular values of C. called in this context entropy of entanglement of U [10], (or It will be convenient to work with the product bases in the Schmidt strength [14]), and the generalized, Renyi´ entropies HS space of matrices, generated by the identity matrix, of size ⎡ ⎤ N 2 N 2 × N 2. Each of the N 2 basis matrices B of size N × N 1 n S (U):= S (λ) = ln ⎣ (λ )q ⎦ , (A5) has only a single nonvanishing element equal to unity. Let us q q 1 − q k mμ k=1 denote Bk = B =|m μ|, where k = N(m − 1) + μ. For this choice of the basis the matrix of the coefficients C which tend to S in the limit q → 1. The entropy S0, sometimes in Eq. (A1) takes a particularly simple form, called Hartley entropy, is equal to ln K, where K denotes the mμ nν number of positive coefficients λ , and is called Schmidt rank Cmμ = Tr(B ⊗ B )U = U mn . (A2) i nν μν (or Schmidt number). The second-order Renyi´ entropy S2 is Note that both matrices U and C consist of the same entries, closely related to the linear entropy E(U) = 1 − exp(−S2) ordered in a different way. This particular reordering of a used by Zanardi in [10]. R matrix, called reshuffling [24], will be denoted as U := C.In The generalized entropies Sq are equal to zero if and only general the notion of reshuffling is well defined if a matrix X if the gate U has a product structure, so it can be obtained H ⊗ H R max = acts on a composite Hilbert space, M N . The symbol U by performing local gates. The upper bound, Sq 2lnN,is

032303-11 LAKSHMINARAYAN, PUCHAŁA, AND ZYCZKOWSKI˙ PHYSICAL REVIEW A 90, 032303 (2014) achieved, e.g., for the Fourier unitary matrix of size N 2 defined to formulate a simple lemma, which is a generalization of by Proposition 1(b). Lemma 2.LetH be a Hermitian matrix with spectrum y (N 2) 1 ≺ F := exp(i2πkl/N2). (A6) and let x y; then there exists a unitary matrix V , such that kl N the diagonal of VHV† is given by x. To show this fact it is sufficient to notice that the reshuffled To prove it we assume, without loss of generality, that H is O matrix F R remains unitary, so all its singular values are equal to in its diagonal form with vector y on diagonal, and let be = O unity, hence the Schmidt vector contains N 2 equal components an orthostochastic matrix such that x y.ByW we denote O = 2 and is maximally mixed. an orthogonal matrix such that ij Wij . Now we write T (WHW )ii = WikHkkWik = Oikyk = xi . (B1) APPENDIX B: HERMITIAN MATRICES WITH k k PRESCRIBED SPECTRUM In other words the unitary matrix V which describes the unitary We begin with a fact known as a Horn lemma [37]: transformation can be represented as a product of a unitary Lemma 1. Assume that x ≺ y; then there exists an orthos- matrix W appearing in the Horn lemma and the matrix U † tochastic matrix O such that x = Oy. The matrix O is said containing eigenvectors of H . In particular if x is flat, i.e., all xi to be orthostochastic if there exists an orthogonal matrix W, are equal, the matrix W present in the Horn lemma can be taken O = 2 such that ij Wij . as a complex Hadamard matrix and item (b) in Proposition 1 The above lemma in the case of a bistochastic matrix instead is recovered. This allows us to obtain an alternative solution of orthostochastic is well known [50], and sometimes used to the problem of constructing Hermitian matrices with the in a definition of majorization. The Horn lemma allows us prescribed spectrum, studied in [41–43].

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