Showing papers on "Unitary state published in 2009"
TL;DR: In this paper, the concept of unitary 2-designs was introduced as a means of expressing operationally useful subsets of the stochastic properties of the uniform (Haar) measure on the unitary group $U({2}^{n})$ on qubits.
Abstract: We develop the concept of a unitary $t$-design as a means of expressing operationally useful subsets of the stochastic properties of the uniform (Haar) measure on the unitary group $U({2}^{n})$ on $n$ qubits. In particular, sets of unitaries forming 2-designs have wide applicability to quantum information protocols. We devise an $O(n)$-size in-place circuit construction for an approximate unitary 2-design. We then show that this can be used to construct an efficient protocol for experimentally characterizing the fidelity of a quantum process on $n$ qubits with quantum circuits of size $O(n)$ without requiring any ancilla qubits, thereby improving upon previous approaches.
615 citations
TL;DR: In this paper, irreducible representations of the unitary group are used to find a general lower bound on the size of a unitary t-design in U(d), for any d and t.
Abstract: A unitary design is a collection of unitary matrices that approximate the entire unitary group, much like a spherical design approximates the entire unit sphere. In this paper, we use irreducible representations of the unitary group to find a general lower bound on the size of a unitary t-design in U(d), for any d and t. We also introduce the notion of a unitary code--a subset of U(d) in which the trace inner product of any pair of matrices is restricted to only a small number of distinct absolute values--and give an upper bound for the size of a code with s inner product values in U(d), for any d and s. These bounds can be strengthened when the particular inner product values that occur in the code or design are known. Finally, we describe some constructions of designs: we give an upper bound on the size of the smallest weighted unitary t-design in U(d), and we catalogue some t-designs that arise from finite groups.
104 citations
TL;DR: It is proved that in small parameter regions, arbitrary unitary matrix integrals converge in the large N limit and match their formal expansion.
63 citations
TL;DR: In this paper, an analysis of optimal control landscapes for unitary transformations from a kinematic perspective in the finite-dimensional unitary matrices is extended to a dynamical one in the infinite-dimensional function space of the time-dependent external field.
Abstract: We consider the control problem of generating unitary transformations, which is especially relevant to current research in quantum information processing and computing, in contrast to the usual state-to-state or the more general observable expectation value control problems. A previous analysis of optimal control landscapes for unitary transformations from a kinematic perspective in the finite-dimensional unitary matrices is extended to a dynamical one in the infinite-dimensional function space of the time-dependent external field. The underlying dynamical landscape is defined as the Frobenius square norm of the difference between the control unitary matrix and the target matrix. A nonsingular adaptation matrix is introduced to provide additional freedom for exploring and manipulating key features, specifically the slope and curvature, of the control landscapes. The dynamical analysis reveals many essential geometric features of optimal control landscapes for unitary transformations, including bounds on the local landscape slope and curvature. Close examination of the curvatures at the critical points shows that the unitary transformation control landscapes are free of local traps and proper choices of the adaptation matrix may facilitate the search for optimal control fields producing desired unitary transformations, in particular, in the neighborhood of the global extrema.
61 citations
TL;DR: This work gives the best known simulation of sparse Hamiltonians with constant precision, and shows that a black-box unitary can be performed with bounded error using O(N2/3(log logN)4/3) queries to its matrix elements.
Abstract: We present general methods for simulating black-box Hamiltonians using quantum walks. These techniques have two main applications: simulating sparse Hamiltonians and implementing black-box unitary operations. In particular, we give the best known simulation of sparse Hamiltonians with constant precision. Our method has complexity linear in both the sparseness D (the maximum number of nonzero elements in a column) and the evolution time t, whereas previous methods had complexity scaling as D^4 and were superlinear in t. We also consider the task of implementing an arbitrary unitary operation given a black-box description of its matrix elements. Whereas standard methods for performing an explicitly specified N x N unitary operation use O(N^2) elementary gates, we show that a black-box unitary can be performed with bounded error using O(N^{2/3} (log log N)^{4/3}) queries to its matrix elements. In fact, except for pathological cases, it appears that most unitaries can be performed with only O(sqrt{N}) queries, which is optimal.
56 citations
53 citations
TL;DR: In this paper, it was shown that the exponential decay of fractional moments of the Green function implies dynamical localization, which in turn implies spectral localization in the case of unitary random operators.
Abstract: This paper establishes dynamical localization properties of certain families of unitary random operators on the d-dimensional lattice in various regimes. These operators are generalizations of one-dimensional physical models of quantum transport and draw their name from the analogy with the discrete Anderson model of solid state physics. They consist in a product of a deterministic unitary operator and a random unitary operator. The deterministic operator has a band structure, is absolutely continuous and plays the role of the discrete Laplacian. The random operator is diagonal with elements given by i.i.d. random phases distributed according to some absolutely continuous measure and plays the role of the random potential. In dimension one, these operators belong to the family of CMV-matrices in the theory of orthogonal polynomials on the unit circle. We implement the method of Aizenman-Molchanov to prove exponential decay of the fractional moments of the Green function for the unitary Anderson model in the following three regimes: In any dimension, throughout the spectrum at large disorder and near the band edges at arbitrary disorder and, in dimension one, throughout the spectrum at arbitrary disorder. We also prove that exponential decay of fractional moments of the Green function implies dynamical localization, which in turn implies spectral localization. These results complete the analogy with the self-adjoint case where dynamical localization is known to be true in the same three regimes.
49 citations
TL;DR: In this paper, the basic properties of unitary matrix integrals are studied with the help of the three matrix models: the ordinary unitary model, Brezin-Gross-Witten model and the Harish-Charndra-Itzykson-Zuber model.
Abstract: Concise review of the basic properties of unitary matrix integrals. They are studied with the help of the three matrix models: the ordinary unitary model, Brezin-Gross-Witten model and the Harish-Charndra-Itzykson-Zuber model. Especial attention is paid to the tricky sides of the story, from De Wit-t'Hooft anomaly in unitary integrals to the problem of correlators with Itzykson-Zuber measure. Of technical tools emphasized is the method of character expansions. The subject of unitary integrals remains highly under-investigated and a lot of new results are expected in this field when it attracts sufficient attention.
47 citations
TL;DR: In this paper, it was shown that the exponential decay of fractional moments of the Green function implies dynamical localization, which in turn implies spectral localization in the case of unitary random operators.
Abstract: This paper establishes dynamical localization properties of certain families of unitary random operators on the d-dimensional lattice in various regimes. These operators are generalizations of one-dimensional physical models of quantum transport and draw their name from the analogy with the discrete Anderson model of solid state physics. They consist in a product of a deterministic unitary operator and a random unitary operator. The deterministic operator has a band structure, is absolutely continuous and plays the role of the discrete Laplacian. The random operator is diagonal with elements given by i.i.d. random phases distributed according to some absolutely continuous measure and plays the role of the random potential. In dimension one, these operators belong to the family of CMV-matrices in the theory of orthogonal polynomials on the unit circle.
We implement the method of Aizenman-Molchanov to prove exponential decay of the fractional moments of the Green function for the unitary Anderson model in the following three regimes: In any dimension, throughout the spectrum at large disorder and near the band edges at arbitrary disorder and, in dimension one, throughout the spectrum at arbitrary disorder. We also prove that exponential decay of fractional moments of the Green function implies dynamical localization, which in turn implies spectral localization.
These results complete the analogy with the self-adjoint case where dynamical localization is known to be true in the same three regimes.
42 citations
TL;DR: The notion of nonmalleability of a quantum state encryption scheme (in dimension d) is introduced: in addition to the requirement that an adversary cannot learn information about the state, here it is demanded that no controlled modification of the encrypted state can be effected.
Abstract: We introduce the notion of nonmalleability of a quantum state encryption scheme (in dimension d): in addition to the requirement that an adversary cannot learn information about the state, here we demand that no controlled modification of the encrypted state can be effected. We show that such a scheme is equivalent to a unitary 2-design [Dankert, et al., e-print arXiv:quant-ph/0606161], as opposed to normal encryption which is a unitary 1-design. Our other main results include a new proof of the lower bound of (d2−1)2+1 on the number of unitaries in a 2-design [Gross, et al., J. Math. Phys. 48, 052104 (2007)], which lends itself to a generalization to approximate 2-design. Furthermore, while in prime power dimension there is a unitary 2-design with ≤d5 elements, we show that there are always approximate 2-designs with O(ϵ−2d4 log d) elements.
38 citations
TL;DR: In this paper, the authors used a cross-sectional regression from 1995 to 2004 to evaluate the impact of ethno-federal institutions on either civil conflict or separatism, and found that the effects of this institution on government quality relative to integrationist states.
Abstract: Recent empirical studies have demonstrated that ethno-linguistic diversity has a negative relationship with quality of government (QoG) In response to this challenge, states have two broad options with respect to vertical power-sharing One, they can attempt to integrate various factions by adopting a unitary, centralized constitution Two, they can accommodate minority groups with ethno-federalism, giving them a degree of regional autonomy Using numerous QoG indicators in a cross-sectional regression from 1995 to 2004, the data show that ethno-federalism outperforms its integrationist rival for each QoG measure employed in the analysis While numerous other empirical studies have examined the impact of ethno-federal institutions on either civil conflict or separatism, this is the first to assess the effects of this institution on government quality relative to integrationist states Copyright 2009, Oxford University Press
TL;DR: It is proved that in large dimensions, almost all bipartite unitaries have entangling and disentangling capacities close to maximal, and that there can be no unique ordering for unitary gates in terms of their ability to perform nonlocal tasks.
Abstract: We consider two capacity quantities associated with bipartite unitary gates: the entangling and the disentangling power. Here, we prove that these capacities are different in general by constructing an explicit example of a qubit-qutrit unitary whose entangling power is maximal (2 ebits), but whose disentangling power is strictly less. A corollary is that there can be no unique ordering for unitary gates in terms of their ability to perform nonlocal tasks. Finally, we show that in large dimensions, almost all bipartite unitaries have entangling and disentangling capacities close to maximal.
TL;DR: In this article, the authors present a simple derivation of the formula for the Hamiltonian operator that achieves the fastest possible unitary evolution between given initial and final states, and discuss how this formula is modified in pseudo-hermitian quantum mechanics and provide an explicit expression for the most general optimal speed quasi-Hermitian Hamiltonian.
Abstract: We present a simple derivation of the formula for the Hamiltonian operator(s) that achieve the fastest possible unitary evolution between given initial and final states. We discuss how this formula is modified in pseudo-Hermitian quantum mechanics and provide an explicit expression for the most general optimal-speed quasi-Hermitian Hamiltonian. Our approach allows for an explicit description of the metric (inner product) dependence of the lower bound on the travel time and the universality (metric independence) of the upper bound on the speed of unitary evolutions.
TL;DR: In this article, it was shown that any unitary irreducible representation of a Levi subgroup of GL(m,D), with m a positive integer, induces irreduceibly to GL(M,D).
Abstract: Let F be a non-Archimedean local field of characteristic 0, and let D be a finite dimensional central division algebra over F. We prove that any unitary irreducible representation of a Levi subgroup of GL(m,D), with m a positive integer, induces irreducibly to GL(m,D). This ends the classification of the unitary dual of GL(m,D) initiated by Tadic.
TL;DR: The structure of the unitary unit group of the group algebra 𝔽2kQ8 is described as a Hamiltonian group.
Abstract: The structure of the unitary unit group of the group algebra 𝔽2kQ8 is described as a Hamiltonian group.
TL;DR: In this article, the Taylor-Wiles method is used to prove modularity lifting theorems for l-adic Galois representations of any dimension satisfying a unitary type condition and a Fontaine-Laffaille type condition at l.
Abstract: We prove modularity lifting theorems for l-adic Galois representations of any dimension satisfying a unitary type condition and a Fontaine-Laffaille type condition at l. This extends the results of Clozel, Harris and Taylor, and the subsequent work by Taylor. The proof uses the Taylor-Wiles method, as improved by Diamond, Fujiwara, Kisin and Taylor, applied to Hecke algebras of unitary groups, and results of Labesse on stable base change and descent from unitary groups to GL_n.
Journal Article•
TL;DR: In this paper, the magnetic-field operators in Dirac-based one-electron equations and their behaviour under unitary transformations are considered and different possibilities of how to transform the vector-potential-containing Dirac Hamiltonian from the perspective of the most general parametrization of unitary matrices with a special focus on the final goal to employ them in perturbative treatments of molecular property calculations.
TL;DR: In this paper, the authors introduce the notion of polyhedral Kahler manifolds, even dimensional polyhedral manifolds with unitary holonomy, and prove that such manifolds are smooth complex surfaces and classify the singularities of the metric.
Abstract: In this article we introduce the notion of polyhedral Kahler manifolds, even dimensional polyhedral manifolds with unitary holonomy. We concentrate on the 4–dimensional case, prove that such manifolds are smooth complex surfaces and classify the singularities of the metric. The singularities form a divisor and the residues of the flat connection on the complement of the divisor give us a system of cohomological equations. A parabolic version of the Kobayshi–Hitchin correspondence of T Mochizuki permits us to characterize polyhedral Kahler metrics of nonnegative curvature on ℂP2 with singularities at complex line arrangements.
TL;DR: In this article, the authors present an account of the derailments of particle theory during more than 4 decades: the S-matrix bootstrap, the dual model and its string theoretic extension.
Abstract: The crossing property is perhaps the most subtle aspect of the particle-field relation. Although it is not difficult to state its content in terms of certain analytic properties relating different matrixelements of the S-matrix or formfactors, its relation to the localization- and positive energy spectral principles requires a level of insight into the inner workings of QFT which goes beyond anything which can be found in typical textbooks on QFT. This paper presents a recent account based on new ideas derived from "modular localization" including a mathematic appendix on this subject. Its main novel achievement is the proof of the crossing property of formfactors from a two-algebra generalization of the KMS condition. The main content of this article is the presentation of the derailments of particle theory during more than 4 decades: the S-matrix bootstrap, the dual model and its string theoretic extension. Rather than being related to crossing, string theory is the (only known) realization of a dynamic infinite component one-particle wave function space and its associated infinite component field. Here "dynamic" means that, unlike a mere collection of infinitely many irreducible unitary Poincar\'e group representation or free fields, the formalism contains also operators which communicate between the different irreducible Poincar\'e represenations (the levels of the "infinite tower") and set the mass/spin spectrum. Wheras in pre-string times there were unsuccessful attempts to achieve this in analogy to the O(4,2) hydrogen spectrum by the use of higher noncompact groups, the superstring in d=9+1, which uses instead (bosonic/fermionic) oscillators obtained from multicomponent chiral currents is the only known unitary positive energy solution of the dynamical infinite component pointlike localized field project.
TL;DR: In this paper, the authors argue that regional government formation in decentralized countries follows different rules than national government forming in unitary states and that the policy space in which coalitions are mapped is often two-dimensional.
Abstract: This article argues that regional government formation in decentralized countries follows different rules than national government formation in unitary states It revises some basic assumptions that classical coalition formation theory makes, positing that in multi-level settings parties do not behave as unitary actors, that the goals they pursue might vary across levels at any given time, that regional coalition formation is part of a two-level game and that the policy space in which coalitions are mapped is often two-dimensional Employing a combination of quantitative and qualitative techniques, several classical propositions are tested in light of these revised assumptions on data about Spanish regional governments We find that classical predictors do their fair share, but multi-level factors are crucial in explaining the making and breaking of regional governments
TL;DR: In this paper, it was shown that all pure entangled states of two d-dimensional quantum systems (i.e., two qudits) can be generated from an initial separable state via a universal Yang-Baxter matrix if one is assisted by local unitary transformations.
Abstract: We show that all pure entangled states of two d-dimensional quantum systems (i.e., two qudits) can be generated from an initial separable state via a universal Yang–Baxter matrix if one is assisted by local unitary transformations.
TL;DR: In this paper, the authors use methods of the general theory of congruence and *congruence for complex matrices (regularization and cosquares) to determine a unitary *congruence canonical form.
Abstract: We use methods of the general theory of congruence and *congruence for complex matrices – regularization and cosquares – to determine a unitary congruence canonical form (respectively, a unitary *congruence canonical form) for complex matrices A such that ĀA (respectively, A 2) is normal. As special cases of our canonical forms, we obtain – in a coherent and systematic way – known canonical forms for conjugate normal, congruence normal, coninvolutory, involutory, projection, λ-projection, and unitary matrices. But we also obtain canonical forms for matrices whose squares are Hermitian or normal, and other cases that do not seem to have been investigated previously. We show that the classification problems under (a) unitary *congruence when A 3 is normal, and (b) unitary congruence when AĀA is normal, are both unitarily wild, so these classification problems are hopeless.
TL;DR: It is proved that the entanglement is the key ingredient in designing the optimal experiment for comparison of unitary channels and the optimality cannot be achieved.
Abstract: We address the problem of an unambiguous comparison of a pair of unknown qudit unitary channels Using the framework of process positive operator valued measures we characterize all solutions and identify the optimal ones We prove that the entanglement is the key ingredient in designing the optimal experiment for comparison of unitary channels Without entanglement the optimality cannot be achieved The proposed scheme is also experimentally feasible
TL;DR: In this article, it was shown that for an appropriate topology, the set of weakly stable unitary groups (isometric semigroups) is of first category, while the residual set of almost stable unitsigroups is residual.
Abstract: Inspired by the classical category theorems of Halmos and Rohlin for discrete measure-preserving transformations, we prove analogous results in the abstract setting of unitary and isometric C 0 -semigroups on a separable Hilbert space. More precisely, we show that, for an appropriate topology, the set of all weakly stable unitary groups (isometric semigroups) is of first category, while the set of all almost weakly stable unitary groups (isometric semigroups) is residual.
TL;DR: In this paper, the asymptotic dynamics of quantum systems resulting from large numbers of iterations of random unitary operations is analyzed in a typically low dimensional attractor space, which is independent of the probability distribution of the unitary operation applied.
Abstract: We analyze the asymptotic dynamics of quantum systems resulting from large numbers of iterations of random unitary operations. Although, in general, these quantum operations cannot be diagonalized it is shown that their resulting asymptotic dynamics is described by a diagonalizable superoperator. We prove that this asymptotic dynamics takes place in a typically low dimensional attractor space which is independent of the probability distribution of the unitary operations applied. This vector space is spanned by all eigenvectors of the unitary operations involved which are associated with eigenvalues of unit modulus. Implications for possible asymptotic dynamics of iterated random unitary operations are presented and exemplified in an example involving random controlled-not operations acting on two qubits.
TL;DR: In this article, the asymptotic dynamics of quantum networks under repeated applications of random unitary operations was investigated and it was shown that the dynamics is generally governed by a typically low dimensional attractor space.
Abstract: We investigate the asymptotic dynamics of quantum networks under repeated applications of random unitary operations. It is shown that in the asymptotic limit of large numbers of iterations this dynamics is generally governed by a typically low dimensional attractor space. This space is determined completely by the unitary operations involved and it is independent of the probabilities with which these unitary operations are applied. Based on this general feature analytical results are presented for the asymptotic dynamics of arbitrarily large cyclic qubit networks whose nodes are coupled by randomly applied controlled-NOT operations.
Posted Content•
TL;DR: In this article, the asymptotic dynamics of quantum networks under repeated applications of random unitary operations was investigated and it was shown that the dynamics is generally governed by a typically low dimensional attractor space.
Abstract: We investigate the asymptotic dynamics of quantum networks under repeated applications of random unitary operations. It is shown that in the asymptotic limit of large numbers of iterations this dynamics is generally governed by a typically low dimensional attractor space. This space is determined completely by the unitary operations involved and it is independent of the probabilities with which these unitary operations are applied. Based on this general feature analytical results are presented for the asymptotic dynamics of arbitrarily large cyclic qubit networks whose nodes are coupled by randomly applied controlled-NOT operations.
TL;DR: In this article, Fock space realisations of unitary highest weight representations of compact and non-compact real forms of the quantum general linear superalgebra Uq (glm|n) are constructed.
TL;DR: The generalized Greenberger-Horne-Zeilinger (GHZ) states as discussed by the authors are the only pure states of qubits that are not uniquely determined (among arbitrary states, pure or mixed) by their reduced density matrices of $n\ensuremath{-1$ qubits.
Abstract: The generalized $n$-qubit Greenberger-Horne-Zeilinger (GHZ) states and their local unitary equivalents are the only pure states of $n$ qubits that are not uniquely determined (among arbitrary states, pure or mixed) by their reduced density matrices of $n\ensuremath{-}1$ qubits. Thus, the generalized GHZ states are the only ones containing information at the $n$-party level.