Diagonal Unitary Matrices Research Articles

Optimized Synthesis of Circuits for Diagonal Unitary Matrices with Reflection Symmetry

Optimized Synthesis of Circuits for Diagonal Unitary Matrices with Reflection Symmetry

A graphical calculus for integration over random diagonal unitary matrices

We provide a graphical calculus for computing averages of tensor network diagrams with respect to the distribution of random vectors containing independent uniform complex phases. Our method exploits the order structure of the partially ordered set of uniform block permutations. A similar calculus is developed for random vectors consisting of independent uniform signs, based on the combinatorics of the partially ordered set of even partitions. We employ our method to extend some of the results by Johnston and MacLean on the family of local diagonal unitary invariant matrices. Furthermore, our graphical approach applies just as well to the real (orthogonal) case, where we introduce the notion of triplewise complete positivity to study the condition for separability of the relevant bipartite matrices. Finally, we analyze the twirling of linear maps between matrix algebras by independent diagonal unitary matrices, showcasing another application of our method.

Bi-Hamiltonian structure of a dynamical system introduced by Braden and Hone

We investigate the finite dimensional dynamical system derived by Braden and Hone in 1996 from the solitons of An−1 affine Toda field theory. This system of evolution equations for an Hermitian matrix L and a real diagonal matrix q with distinct eigenvalues was interpreted as a special case of the spin Ruijsenaars–Schneider models due to Krichever and Zabrodin. A decade later, Li re-derived the model from a general framework built on coboundary dynamical Poisson groupoids. This led to a Hamiltonian description of the gauge invariant content of the model, where the gauge transformations act as conjugations of L by diagonal unitary matrices. Here, we point out that the same dynamics can be interpreted also as a special case of the spin Sutherland systems obtained by reducing the free geodesic motion on symmetric spaces, studied by Pusztai and the author in 2006; the relevant symmetric space being . This construction provides an alternative Hamiltonian interpretation of the Braden–Hone dynamics. We prove that the two Poisson brackets are compatible and yield a bi-Hamiltonian description of the standard commuting flows of the model.

Multiplication of conjugacy classes, colligations, and characteristic functions of matrix argument

We extend the classical construction of operator colligations and characteristic functions. Consider the group $G$ of finite block unitary matrices of size $\alpha+\infty+...+\infty$ ($k$ times). Consider the subgroup $K=U(\infty)$, which consists of block diagonal unitary matrices (with a block 1 of size $\alpha$ and a matrix $u\in U(\infty)$ repeated $k$ times). It appears that there is a natural multiplication on the conjugacy classes $G//K$. We construct 'spectral data' of conjugacy classes, which visualize the multiplication and are sufficient for a reconstruction of a conjugacy class.

The Stationary Measure for Diagonal Quantum Walk with One Defect

This study is motivated by previous work [14]. We treat 3 types of one-dimensional quantum walks (QWs), whose time evolutions are described by diagonal unitary matrix, and diagonal unitary matrices with one defect. In this paper, we call QW defined by diagonal unitary matrices, the diagonal and we consider stationary distributions of generally 2-state diagonal QW with one defect, 3-state space-homogeneous diagonal QW, and 3-state diagonal QW with one defect. One of purposes of our study is to characterize QWs by stationary measure, which may lead to answer basic and natural question, What stationary measure is for one-dimensional QWs ?. In order to analyze stationary distribution, we focus on corresponding eigenvalue problems and definition of stationary measure.

Determination of locally perfect discrimination for two-qubit unitary operations

In the study of local discrimination for multipartite unitary operations, Duan et al. (Phys Rev Lett 100(2):020503, 2008) exhibited an ingenious expression: Any two different unitary operations $$U_1$$U1 and $$U_2$$U2 are perfectly distinguishable by local operations and classical communication in the single-run scenario if and only if 0 is in the local numerical range of $$U_1^\dag U_2$$U1?U2. However, how to determine when 0 is in the local numerical range remains unclear. So it is generally hard to decide the local discrimination of nonlocal unitary operations with a single run. In this paper, for two-qubit diagonal unitary matrices V and their local unitary equivalent matrices, we present a necessary and sufficient condition for determining whether the local numerical range is a convex set or not. The result can be used to easily judge the locally perfect distinguishability of any two unitary operations $$U_1$$U1 and $$U_2$$U2 satisfying $$U_1^\dag U_2=V$$U1?U2=V. Moreover, we design the corresponding protocol of local discrimination. Meanwhile, an interesting phenomenon is discovered: Under certain conditions with a single run, $$U_1$$U1 and $$U_2$$U2 such that $$U_1^\dag U_2=V$$U1?U2=V are locally distinguishable with certainty if and only if they are perfectly distinguishable by global operations.

Generating a state t-design by diagonal quantum circuits

We investigate protocols for generating a state t-design by using a fixed separable initial state and a diagonal-unitary t-design in the computational basis, which is a t-design of an ensemble of diagonal unitary matrices with random phases as their eigenvalues. We first show that a diagonal-unitary t-design generates a -approximate state t-design, where N is the number of qubits. We then discuss a way of improving the degree of approximation by exploiting non-diagonal gates after applying a diagonal-unitary t-design. We also show that it is necessary and sufficient to use -qubit gates with random phases to generate a diagonal-unitary t-design by diagonal quantum circuits, and that each multi-qubit diagonal gate can be replaced by a sequence of multi-qubit controlled-phase-type gates with discrete-valued random phases. Finally, we analyze the number of gates for implementing a diagonal-unitary t-design by non-diagonal two- and one-qubit gates. Our results provide a concrete application of diagonal quantum circuits in quantum informational tasks.

A Non-differential Distributed Space-Time Coding for Partially-Coherent Cooperative Communication

In a distributed space-time coding scheme, based on the relay channel model, the relay nodes co-operate to linearly process the transmitted signal from the source and forward them to the destination such that the signal at the destination appears as a space time block code. Recently, a code design criteria for achieving full diversity in a partially-coherent environment have been proposed along with codes based on differential encoding and decoding techniques. For such a set up, in this paper, a nondifferential encoding technique and construction of distributed space time block codes from unitary matrix groups at the source and a set of diagonal unitary matrices for the relays are proposed. It is shown that, the performance of our scheme is independent of the choice of unitary matrices at the relays. When the group is cyclic, a necessary and sufficient condition on the generator of the cyclic group to achieve full diversity and to minimize the pairwise error probability is proved. Various choices on the generator of cyclic group to reduce the ML decoding complexity at the destination is presented. It is also shown that, at the source, if non-cyclic abelian unitary matrix groups are used, then full-diversity can not be obtained. The presented scheme is also robust to failure of any subset of relay nodes.

On Overdiagonal Subgroups of the Hyperbolic Unitary Group for a Skew Field

In the hyperbolic unitary group over a skew field, we study the lattice of subgroups that contain all diagonal unitary matrices. For the group of degree 4, it is proved that if a form parameter is minimal, this lattice admits a standard description in terms of form net subgroups. Bibliography: 27 titles.

Construction of Matrices with Prescribed Singular Values and Eigenvalues

Two issues concerning the construction of square matrices with prescribe singular values an eigenvalues are addressed. First, a necessary and sufficient condition for the existence of an n × n complex matrix with n given nonnegative numbers as singular values an m (≤ n) given complex numbers to be m of the eigenvalues is determined. This extends the classical result of Weyl and Horn treating the case when m = n. Second, an algorithm is given to generate a triangular matrix with prescribe singular values an eigenvalues. Unlike earlier algorithms, the eigenvalues can be arranged in any prescribe order on the diagonal. A slight modification of this algorithm allows one to construct a real matrix with specified real an complex conjugate eigenvalues an specified singular values. The construction is done by multiplication by diagonal unitary matrices, permutation matrices and rotation matrices. It is numerically stable and may be useful in developing test software for numerical linear algebra packages.

Overdiagonal subgroups of the hyperbolic unitary group for a good form ring over a field

In a hyperbolic unitary group over a field, the lattice of subgroups that contain all diagonal unitary matrices allows for the standard description. The corresponding form ring must satisfy certain conditions; for this reason, for fields of characteristic 2 we do not consider the symplectic group and the defective orthogonal groups. Bibliography: 12titles.

Interpolating ensembles of random unitary matrices.

We investigate transitions between ensembles of unitary random matrices modeling changes of statistical properties of quantum chaotic systems which are periodically time dependent. The transitions simulating change of integrability are modeled with the help of numerically generated one-parameter families of ensembles interpolating between an ensemble of diagonal unitary matrices and a circular unitary or circular orthogonal ensemble. In an analogous manner we described transitions between circular orthogonal and unitary ensembles corresponding to the time-reversal symmetry-breaking perturbation of a chaotic system. In all cases we present results concerning statistics of the quasi-energy levels and eigenvectors. \textcopyright{} 1996 The American Physical Society.

Normal structure of the unitary group over a local ring

A new approach is used to describe the normal subgroups of unitary groups over the ring of integers of an unramified quadratic extension of a local field of characteristic ≠2. This method is based on the author's results concerning subgroups of unitary groups normalized by diagonal unitary matrices. By means of this method it has been possible to remove restrictions imposed by other authors in a number of papers pertaining to the same question.

Subgroups of the unitary group over a dyadic local field

Let K/k be an unramified quadratic extension of a dyadic local field of characteristic ≠2. In this paper there is studied the lattice of subgroups of the classical unitary group U(n, K), containing the subgroup D of diagonal unitary matrices. The pronormality of the subgroup D in the group U(n, K) is proved and a constructive description is obtained of its unique fan. Analogous results are formulated for the “nonclassical” unitary group U− (n, K) corresponding to a Hermitian form with matrix diag (1,...,1, π) (π is a prime element of the field k).

Subgroups of the unitary group over a local field

For the unitary group of matrices defined over an unramified quadratic extension of a local field, we give a description of the lattice of subgroups containing the group of diagonal unitary matrices (the characteristic of the residue fields is different from 2).

Optimal Conditioning of Matrices

The optimal condition number c of a matrix S is the minimum, over all diagonal matrices D, of $\| {DS} \|\| {(DS)^{ - 1} } \|$. It arises in many applications of scaling in linear systems, and has been computed in the maximum norm by Bauer. In the $\ell _2 $-norm, we establish the inequalities $b \leqq c \leqq 2b$ and disprove—by an example of order seven !—the minimax conjecture that always $b = c$. Here b is the supremum over diagonal unitary matrices of $\| {S^{ - 1} US} \|$.

Group theory of harmonic oscillators: (V). The harmonic oscillator shell model

In the present series of papers, we have dealt with the explicit construction of states for a system of n particles in a harmonic oscillator common potential. These states were classified by the irreducible representations (IR) of the symmetry group of the harmonic oscillator, i.e. the unitary group of three dimensionsU 3 and its rotational subgroupR 3. Furthermore, they are also characterized by the IR of the chain of groups U n ⊃ S n where the latter groups are respectively the unitary and symmetric group associated with the n particles. In this paper, we show that the shell structure of the n-particle states in a harmonic oscillator potential can be obtained when we characterize further the states by the IR of a group K n . This group is the semi-direct product of the group of n-dimensional diagonal unitary matrices A n and of the symmetric group, i.e. K n = A n ∧ S n . Using these chains of groups, we construct explicitly shell-model states in the harmonic oscillator potential and from them obtain the fractional parentage coefficients (fpc) in terms of the Wigner coefficients of the groups in the chain. An important result is that the fpc become much simpler when the IR ofU 3 are of one and two rows than when they have the most general form of three rows. Thus, for example, for particles in the 2d-1d shell of the harmonic oscillator, the fpc corresponding to a two-row IR ofU 3 become a product of the familiar fpc in the p-shell tabulated by Jahn and a reduced Wigner coefficient ofU 3 in theU 3⊃R 3 chain, for which many programs and tables are available. The present analysis may be useful in the many cases of physical interest when one needs the fpc in a multishell state in the harmonic oscillator potential with the IR ofU 3 not exceeding two rows.