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Journal articles on the topic 'Unitary rank'

Author: Grafiati

Published: 9 March 2023

Last updated: 31 July 2025

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1

Delvaux, Steven, and Marc Van Barel. "Unitary rank structured matrices." Journal of Computational and Applied Mathematics 215, no. 1 (2008): 49–78. http://dx.doi.org/10.1016/j.cam.2007.03.020.

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2

Elliott, George A., and Zhichao Liu. "Distance between unitary orbits in C∗-algebras with stable rank one and real rank zero." Journal of Operator Theory 86, no. 2 (2021): 299–316. http://dx.doi.org/10.7900/jot.2020apr21.2306.

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Let A be a C∗-algebra with stable rank one and real rank zero. In this paper, it is shown that the usual distance dU defined on the approximate unitary equivalence classes (or unitary orbits) of the positive elements in A is equal to the distance dW defined on morphisms from Cuntz semigroup of C0(0,1] to the Cuntz semigrout of A.

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3

Hanzer, Marcela. "Rank one reducibility for unitary groups." Glasnik matematicki 46, no. 1 (2011): 121–48. http://dx.doi.org/10.3336/gm.46.1.12.

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4

Flaschka, Hermann, and John Millson. "Bending Flows for Sums of Rank One Matrices." Canadian Journal of Mathematics 57, no. 1 (2005): 114–58. http://dx.doi.org/10.4153/cjm-2005-006-3.

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AbstractWe study certain symplectic quotients of n-fold products of complex projective m-space by the unitary group acting diagonally. After studying nonemptiness and smoothness of these quotients we construct the action-angle variables, defined on an open dense subset, of an integrable Hamiltonian system. The semiclassical quantization of this system reporduces formulas from the representation theory of the unitary group.

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5

Dudas, Olivier, and Gunter Malle. "Decomposition matrices for low-rank unitary groups." Proceedings of the London Mathematical Society 110, no. 6 (2015): 1517–57. http://dx.doi.org/10.1112/plms/pdv008.

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6

Bevilacqua, Roberto, Enrico Bozzo, and Gianna M. Del Corso. "Transformations to rank structures by unitary similarity." Linear Algebra and its Applications 402 (June 2005): 126–34. http://dx.doi.org/10.1016/j.laa.2004.12.029.

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7

Jiang, Yunjiang, Lon H. Mitchell, and Sivaram K. Narayan. "Unitary matrix digraphs and minimum semidefinite rank." Linear Algebra and its Applications 428, no. 7 (2008): 1685–95. http://dx.doi.org/10.1016/j.laa.2007.10.031.

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8

Delvaux, Steven, and Marc Van Barel. "Eigenvalue computation for unitary rank structured matrices." Journal of Computational and Applied Mathematics 213, no. 1 (2008): 268–87. http://dx.doi.org/10.1016/j.cam.2007.01.006.

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9

Barnet-Lamb, Thomas, Toby Gee, and David Geraghty. "Serre weights for rank two unitary groups." Mathematische Annalen 356, no. 4 (2013): 1551–98. http://dx.doi.org/10.1007/s00208-012-0893-y.

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10

Ye, Lynnelle. "Slopes in eigenvarieties for definite unitary groups." Compositio Mathematica 160, no. 1 (2023): 52–89. http://dx.doi.org/10.1112/s0010437x23007534.

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We generalize bounds of Liu–Wan–Xiao for slopes in eigencurves for definite unitary groups of rank $2$ to slopes in eigenvarieties for definite unitary groups of any rank. We show that for a definite unitary group of rank $n$ , the Newton polygon of the characteristic power series of the $U_p$ Hecke operator has exact growth rate $x^{1+2/{n(n-1)}}$ , times a constant proportional to the distance of the weight from the boundary of weight space. The proof goes through the classification of forms associated to principal series representations. We also give a consequence for the geometry of these

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11

PARIS, MATTEO G. A. "UNITARY LOCAL INVARIANCE." International Journal of Quantum Information 03, no. 04 (2005): 655–59. http://dx.doi.org/10.1142/s0219749905001523.

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We address unitary local (UL) invariance of bipartite pure states. Given a bipartite state |Ψ〉〉 = ∑ij ψij |i〉1 ⊗ |j〉2 the complete characterization of the class of local unitaries U1 ⊗ U2 for which U1 ⊗ U2|Ψ〉〉 = |Ψ〉〉 is obtained. The two relevant parameters are the rank of the matrix Ψ, [Ψ]ij = ψij, and the number of its equal singular values, i.e. the degeneracy of the eigenvalues of the partial traces of |Ψ〉〉.

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12

Ringrose, J. R. "Exponential length and exponential rank in C*-algebras." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 121, no. 1-2 (1992): 55–71. http://dx.doi.org/10.1017/s0308210500014141.

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SynopsisIn an operator algebra, the general element of the connected component of the unitary group can beexpressed as a finite product of exponential unitary elements. The recently introduced concept of exponential rank is defined in terms of the number of exponentials required for this purpose. The present paper is concerned with a concept of exponential length, determined not by the number of exponentials but by the sum of the norms of their self-adjoint logarithms. Knowledge of the exponential length of an algebra provides an upper bound for its exponential rank (but not conversely). This

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13

Ng, P. W., and P. Skoufranis. "Closed Convex Hulls of Unitary Orbits in Certain Simple Real Rank Zero C* -algebras." Canadian Journal of Mathematics 69, no. 5 (2017): 1109–42. http://dx.doi.org/10.4153/cjm-2016-045-5.

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AbstractIn this paper, we characterize the closures of convex hulls of unitary orbits of self-adjoint operators in unital, separable, simple C* -algebras with non-trivial tracial simplex, real rank zero, stable rank one, and strict comparison of projections with respect to tracial states. In addition, an upper bound for the number of unitary conjugates in a convex combination needed to approximate a self-adjoint are obtained.

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14

Putnam, Ian F., and Mikael Rordam. "The maximum unitary rank of some $C^*$-algebras." MATHEMATICA SCANDINAVICA 63 (June 1, 1988): 297. http://dx.doi.org/10.7146/math.scand.a-12242.

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15

Douglas, Ronald G., and Constanze Liaw. "A geometric approach to finite rank unitary perturbations." Indiana University Mathematics Journal 62, no. 1 (2013): 333–54. http://dx.doi.org/10.1512/iumj.2013.62.5028.

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16

Wong, W. J. "Rank 1 preservers on the unitary Lie ring." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 49, no. 3 (1990): 399–417. http://dx.doi.org/10.1017/s1446788700032419.

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AbstractThe surjective additive maps on the Lie ring of skew-Hermitian linear transformations on a finite-dimensional vector space over a division ring which preserve the set of rank 1 elements are determined. As an application, maps preserving commuting pairs of transformations are determined.

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17

Baranov, Anton, Vladimir Kapustin, and Andrei Lishanskii. "On hypercyclic rank one perturbations of unitary operators." Mathematische Nachrichten 292, no. 5 (2018): 961–68. http://dx.doi.org/10.1002/mana.201800242.

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18

Voronetsky, Egor. "Injective stability for odd unitary K1." Journal of Group Theory 23, no. 5 (2020): 781–800. http://dx.doi.org/10.1515/jgth-2020-0013.

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AbstractWe give a new purely algebraic approach to odd unitary groups using odd form rings. Using these objects, we give a self-contained proof of injective stability for the odd unitary {K_{1}}-functor under the stable rank condition.

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19

Hu, Mengyao, Lin Chen, and Yize Sun. "Mutually unbiased bases containing a complex Hadamard matrix of Schmidt rank three." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 476, no. 2235 (2020): 20190754. http://dx.doi.org/10.1098/rspa.2019.0754.

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Constructing four six-dimensional mutually unbiased bases (MUBs) is an open problem in quantum physics and measurement. We investigate the existence of four MUBs including the identity, and a complex Hadamard matrix (CHM) of Schmidt rank three. The CHM is equivalent to a controlled unitary operation on the qubit-qutrit system via local unitary transformation I 2 ⊗ V and I 2 ⊗ W . We show that V and W have no zero entry, and apply it to exclude constructed examples as members of MUBs. We further show that the maximum of entangling power of controlled unitary operation is log 2 3 ebits. We deriv

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20

Choi, Man-Duen, John A. Holbrook, David W. Kribs, and Karol Życzkowski. "Higher-rank numerical ranges of unitary and normal matrices." Operators and Matrices, no. 3 (2007): 409–26. http://dx.doi.org/10.7153/oam-01-24.

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21

Blok, Rieuwert J., and Bruce N. Cooperstein. "The generating rank of the unitary and symplectic Grassmannians." Journal of Combinatorial Theory, Series A 119, no. 1 (2012): 1–13. http://dx.doi.org/10.1016/j.jcta.2011.07.002.

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22

Shkarin, Stanislav. "A hypercyclic finite rank perturbation of a unitary operator." Mathematische Annalen 348, no. 2 (2010): 379–93. http://dx.doi.org/10.1007/s00208-010-0479-5.

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23

Bevilacqua, Roberto, Gianna M. Del Corso, and Luca Gemignani. "Fast QR iterations for unitary plus low rank matrices." Numerische Mathematik 144, no. 1 (2019): 23–53. http://dx.doi.org/10.1007/s00211-019-01080-4.

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24

Grivaux, Sophie. "A hypercyclic rank one perturbation of a unitary operator." Mathematische Nachrichten 285, no. 5-6 (2012): 533–44. http://dx.doi.org/10.1002/mana.201000112.

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25

Jiang, Siyuan, Ming Jin, Shuai Liu, and Zhiping Lin. "A Nyström-based low-rank unitary MVDR beamforming scheme." Signal Processing 220 (July 2024): 109433. http://dx.doi.org/10.1016/j.sigpro.2024.109433.

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26

Henn, Andreas, Michael Hentschel, Aloys Krieg, and Gabriele Nebe. "On the Classification of Lattices Over Which Are Even Unimodular -Lattices of Rank 32." International Journal of Mathematics and Mathematical Sciences 2013 (2013): 1–4. http://dx.doi.org/10.1155/2013/837080.

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27

Poguntke, Detlev. "Unitary Representations of Lie Groups and Operators of Finite Rank." Annals of Mathematics 140, no. 3 (1994): 503. http://dx.doi.org/10.2307/2118617.

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28

Riva, Enea, and Lidia Stoppino. "The slope of fibred surfaces: Unitary rank and Clifford index." Proceedings of the London Mathematical Society 124, no. 1 (2022): 83–105. http://dx.doi.org/10.1112/plms.12424.

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29

Rordam, Mikael. "Advances in the Theory of Unitary Rank and Regular Approximation." Annals of Mathematics 128, no. 1 (1988): 153. http://dx.doi.org/10.2307/1971465.

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30

Cao, Huiqin, Jianbei An, and Jiwen Zeng. "The essential 2-rank of general linear and unitary groups." Journal of Pure and Applied Algebra 225, no. 3 (2021): 106518. http://dx.doi.org/10.1016/j.jpaa.2020.106518.

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31

Zhu, Fu-hai, and Ke Liang. "Dirac cohomology of unitary representations of equal rank exceptional groups." Science in China Series A: Mathematics 50, no. 4 (2007): 515–20. http://dx.doi.org/10.1007/s11425-007-2081-1.

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32

Larson, Hannah K. "Pseudo-unitary non-self-dual fusion categories of rank 4." Journal of Algebra 415 (October 2014): 184–213. http://dx.doi.org/10.1016/j.jalgebra.2014.05.032.

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33

Chen, Ke, Xin Lu, and Kang Zuo. "The Oort Conjecture for Shimura Curves of Small Unitary Rank." Communications in Mathematics and Statistics 6, no. 3 (2018): 249–68. http://dx.doi.org/10.1007/s40304-018-0155-8.

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34

Chen, Lin, and Li Yu. "On the Schmidt-rank-three bipartite and multipartite unitary operator." Annals of Physics 351 (December 2014): 682–703. http://dx.doi.org/10.1016/j.aop.2014.09.026.

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35

Xu, Luogen, and James K. Freericks. "Efficient Application of the Factorized form of the Unitary Coupled-Cluster Ansatz for the Variational Quantum Eigensolver Algorithm by Using Linear Combination of Unitaries." Symmetry 15, no. 7 (2023): 1429. http://dx.doi.org/10.3390/sym15071429.

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The variational quantum eigensolver is one of the most promising algorithms for near-term quantum computers. It has the potential to solve quantum chemistry problems involving strongly correlated electrons with relatively low-depth circuits, which are otherwise difficult to solve on classical computers. The variational eigenstate is constructed from a number of factorized unitary coupled-cluster terms applied onto an initial (single-reference) state. Current algorithms for applying one of these operators to a quantum state require a number of operations that scale exponentially with the rank o

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36

Ghosh, Sagar, and Gadadhar Misra. "On the M�bius invariant principal functions of Pincus." Opuscula Mathematica 44, no. 3 (2024): 391–407. http://dx.doi.org/10.7494/opmath.2024.44.3.391.

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37

LIN, HUAXIN. "UNITARIES IN A SIMPLE C*-ALGEBRA OF TRACIAL RANK ONE." International Journal of Mathematics 21, no. 10 (2010): 1267–81. http://dx.doi.org/10.1142/s0129167x10006446.

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Let A be a unital separable simple infinite dimensional C*-algebra with tracial rank not more than one and with the tracial state space T(A) and let U(A) be the unitary group of A. Suppose that u ∈ U0(A), the connected component of U(A) containing the identity. We show that, for any ϵ > 0, there exists a self-adjoint element h ∈ As.a such that [Formula: see text] We also study the problem when u can be approximated by unitaries in A with finite spectrum. Denote by CU(A) the closure of the subgroup of unitary group of U(A) generated by its commutators. It is known that CU(A) ⊂ U0(A). Denote

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38

Cantier, Laurent. "A unitary Cuntz semigroup for C⁎-algebras of stable rank one." Journal of Functional Analysis 281, no. 9 (2021): 109175. http://dx.doi.org/10.1016/j.jfa.2021.109175.

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39

Chen, Shangdi, Qin Xu, and Junmei Zhang. "Constructions of rank metric codes under actions of the unitary groups." Linear Algebra and its Applications 645 (July 2022): 293–306. http://dx.doi.org/10.1016/j.laa.2022.03.031.

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40

Boyadzhiev, Khristo N. "A trace formula for two unitary operators with rank one commutator." Proceedings of the American Mathematical Society 113, no. 1 (1991): 157. http://dx.doi.org/10.1090/s0002-9939-1991-1057950-x.

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41

Scaramuzzi, Roberto. "A notion of rank for unitary representations of general linear groups." Transactions of the American Mathematical Society 319, no. 1 (1990): 349–79. http://dx.doi.org/10.1090/s0002-9947-1990-0958900-8.

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42

Lin, Huaxin. "Approximate unitary equivalence in simple $C^{*}$-algebras of tracial rank one." Transactions of the American Mathematical Society 364, no. 4 (2012): 2021–86. http://dx.doi.org/10.1090/s0002-9947-2011-05431-0.

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43

Grbac, Neven, and Joachim Schwermer. "Eisenstein series for rank one unitary groups and some cohomological applications." Advances in Mathematics 376 (January 2021): 107438. http://dx.doi.org/10.1016/j.aim.2020.107438.

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44

Kozioł, Karol, and Stefano Morra. "Serre weight conjectures for p-adic unitary groups of rank 2." Algebra & Number Theory 16, no. 9 (2022): 2005–97. http://dx.doi.org/10.2140/ant.2022.16.2005.

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45

Thiel, Hannes. "Diffuse traces and Haar unitaries." American Journal of Mathematics 146, no. 5 (2024): 1305–37. http://dx.doi.org/10.1353/ajm.2024.a937944.

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abstract: We show that a tracial state on a unital \ca{} admits a Haar unitary if and only if it is diffuse, if and only if it does not dominate a tracial functional that factors through a finite-dimensional quotient. It follows that a unital \ca{} has no finite-dimensional representations if and only if each of its tracial states admits a Haar unitary. More generally, we study when nontracial states admit Haar unitaries. In particular, we show that every state on a unital, simple, infinite-dimensional \ca{} admits a Haar unitary. We obtain applications to the structure of reduced free product

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46

Evans, Mogoi N., and Samuel B. Apima. "Norm-Attainable Operators in Hilbert Spaces: Probabilistic and Finite-Rank Perspectives." Asian Journal of Probability and Statistics 27, no. 6 (2025): 41–49. https://doi.org/10.9734/ajpas/2025/v27i6765.

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On this note, we investigate norm-attainable operators in Hilbert spaces, focusing on probabilistic and finite-rank perspectives. We present key results concerning the existence and properties of norm-attaining vectors, particularly for compact and finite-rank operators. Using spectral theory and concentration of measure, we show that norm-attaining vectors form compact subspaces in the unit sphere. Additionally, we explore how unitary transformations affect these vectors and discuss the implications for operator theory and functional analysis.

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47

Nien, Chufeng. "Klyachko Models for General Linear Groups of Rank 5 over a p-Adic Field." Canadian Journal of Mathematics 61, no. 1 (2009): 222–40. http://dx.doi.org/10.4153/cjm-2009-011-2.

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Abstract. This paper shows the existence and uniqueness of Klyachko models for irreducible unitary representations of GL5 (ℱ), where ℱ is a p-adic field. It is an extension of the work of Heumos and Rallis on GL4(ℱ).

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48

Zhang, Pingping, Hu Yang, and Hanyu Li. "Relative and Absolute Perturbation Bounds for Weighted Polar Decomposition." Journal of Applied Mathematics 2012 (2012): 1–15. http://dx.doi.org/10.1155/2012/219025.

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Some new perturbation bounds for both weighted unitary polar factors and generalized nonnegative polar factors of the weighted polar decompositions are presented without the restriction thatAand its perturbed matrixA˜have the same rank. These bounds improve the corresponding recent results.

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49

Hurley, Ted. "Unique builders for classes of matrices." Special Matrices 9, no. 1 (2021): 52–65. http://dx.doi.org/10.1515/spma-2020-0122.

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Abstract Basic matrices are defined which provide unique building blocks for the class of normal matrices which include the classes of unitary and Hermitian matrices. Unique builders for quantum logic gates are hence derived as a quantum logic gates is represented by, or is said to be, a unitary matrix. An efficient algorithm for expressing an idempotent as a unique sum of rank 1 idempotents with increasing initial zeros is derived. This is used to derive a unique form for mixed matrices. A number of (further) applications are given: for example (i) U is a symmetric unitary matrix if and only

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50

González-Alonso, Víctor, Lidia Stoppino, and Sara Torelli. "On the rank of the flat unitary summand of the Hodge bundle." Transactions of the American Mathematical Society 372, no. 12 (2019): 8663–77. http://dx.doi.org/10.1090/tran/7868.

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