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31
Higherrank numerical ranges of unitary and normal matrices, preprint
, 608
"... Abstract. We verify a conjecture on the structure of higherrank numerical ranges for a wide class of unitary and normal matrices. Using analytic and geometric techniques, we show precisely how the higherrank numerical ranges for a generic unitary matrix are given by complex polygons determined by ..."
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Abstract. We verify a conjecture on the structure of higherrank numerical ranges for a wide class of unitary and normal matrices. Using analytic and geometric techniques, we show precisely how the higherrank numerical ranges for a generic unitary matrix are given by complex polygons determined by the spectral structure of the matrix. We discuss applications of the results to quantum error correction, specifically to the problem of identification and construction of codes for binary unitary noise models. 1.
Geometry and the complexity of matrix multiplication
, 2007
"... Abstract. We survey results in algebraic complexity theory, focusing on matrix multiplication. Our goals are (i) to show how open questions in algebraic complexity theory are naturally posed as questions in geometry and representation theory, (ii) to motivate researchers to work on these questions, ..."
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Abstract. We survey results in algebraic complexity theory, focusing on matrix multiplication. Our goals are (i) to show how open questions in algebraic complexity theory are naturally posed as questions in geometry and representation theory, (ii) to motivate researchers to work on these questions, and (iii) to point out relations with more general problems in geometry. The key geometric objects for our study are the secant varieties of Segre varieties. We explain how these varieties are also useful for algebraic statistics, the study of phylogenetic invariants, and quantum computing.
The structure of bipartite quantum states. Insights from group theory and cryptography
, 2006
"... dissertation is the result of my own work and includes nothing which is the outcome Currently, a rethinking of the fundamental properties of quantum mechanical systems in the light of quantum computation and quantum cryptography is taking place. In this PhD thesis, I wish to contribute to this effor ..."
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Cited by 10 (1 self)
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dissertation is the result of my own work and includes nothing which is the outcome Currently, a rethinking of the fundamental properties of quantum mechanical systems in the light of quantum computation and quantum cryptography is taking place. In this PhD thesis, I wish to contribute to this effort with a study of the bipartite quantum state. Bipartite quantummechanical systems are made up of just two subsystems, A and B, yet, the quantum states that describe these systems have a rich structure. The focus is twofold: Part I studies the relations between the spectra of the joint and the reduced states, and in part II, I will analyse the amount of entanglement, or quantum correlations, present in a given state. In part I, the mathematical tools from group theory play an important role, mainly drawing on the representation theory of finite and Lie groups and the SchurWeyl duality. This duality will be used to derive a onetoone relation between the spectra of a joint quantum system AB and its parts A and B, and the Kronecker coefficients of the symmetric group. In this way the two problems are connected for the first time, which
Decoherenceinsensitive quantum communications by optimal C∗encoding
, 2006
"... The central issue in this article is to transmit a quantum state in such a way that after some decoherence occurs, most of the information can be restored by a suitable decoding operation. For this purpose, we incorporate redundancy by mapping a given initial quantum state to a messenger state on a ..."
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Cited by 8 (2 self)
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The central issue in this article is to transmit a quantum state in such a way that after some decoherence occurs, most of the information can be restored by a suitable decoding operation. For this purpose, we incorporate redundancy by mapping a given initial quantum state to a messenger state on a largerdimensional Hilbert space via a C ∗algebra embedding. Our noise model for the transmission is a phase damping channel which admits a noiseless subsystem or decoherencefree subspace. More precisely, the transmission channel is obtained from convex combinations of a set of lowest rank yes/no measurements that leave a component of the messenger state unchanged. The objective of our encoding is to distribute quantum information optimally across the noisesusceptible component of the transmission when the noiseless component is not large enough to contain all the quantum information to be transmitted. We derive simple geometric conditions for optimal encoding and construct examples of such encodings.
Cones of positive maps and their duality relations
 J. MATH. PHYS
, 2009
"... The structure of cones of positive and kpositive maps acting on a finitedimensional Hilbert space is investigated. Special emphasis is given to their duality relations to the sets of superpositive and ksuperpositive maps. We characterize kpositive and ksuperpositive maps with regard to their ..."
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Cited by 5 (0 self)
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The structure of cones of positive and kpositive maps acting on a finitedimensional Hilbert space is investigated. Special emphasis is given to their duality relations to the sets of superpositive and ksuperpositive maps. We characterize kpositive and ksuperpositive maps with regard to their properties under taking compositions. A number of results obtained for maps are also rephrased for the corresponding cones of block positive, kblock positive, separable and kseparable operators, due to the Jamio̷lkowskiChoi isomorphism. Generalizations to a situation where no such simple isomorphism is available are also made, employing the idea of mapping cones. As a side result to our discussion, we show that extreme entanglement witnesses, which are optimal, should be of special interest in entanglement studies.
The Significance of the CNumerical Range and the Local CNumerical Range
 in Quantum Control and Quantum
"... This paper shows how Cnumericalrange related new strucures may arise from practical problems in quantum control—and vice versa, how an understanding of these structures helps to tackle hot topics in quantum information. We start out with an overview on the role of Cnumerical ranges in current res ..."
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Cited by 4 (2 self)
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This paper shows how Cnumericalrange related new strucures may arise from practical problems in quantum control—and vice versa, how an understanding of these structures helps to tackle hot topics in quantum information. We start out with an overview on the role of Cnumerical ranges in current research problems in quantum theory: the quantum mechanical task of maximising the projection of a point on the unitary orbit of an initial state onto a target state C relates to the Cnumerical radius of A via maximising the trace function tr{C † UAU †}. In quantum control of N qubits one may be interested (i) in having U ∈ SU(2 N) for the entire dynamics, or (ii) in restricting the dynamics to local operations on each qubit, i.e. to the Nfold tensor product SU(2) ⊗ SU(2) ⊗ · · · ⊗ SU(2). Interestingly, the latter then leads to a novel entity, the local Cnumerical range Wloc(C, A), whose intricate geometry is neither starshaped nor simply connected in contrast to the conventional Cnumerical range. This is shown in the accompanying paper on Relative CNumerical Ranges and Local CNumerical Ranges for Application in Quantum Computing [1]. We present novel applications of the Cnumerical range in quantum control assisted by gradient flows on the local unitary group: they serve as powerful tools (1) for deciding whether a quantum interaction can be inverted in time (in a sense generalising Hahn’s
Linear preservers of Tensor product of Unitary Orbits, and Product Numerical Range
"... It is shown that the linear group of automorphism of Hermitian matrices which preserves the tensor product of unitary orbits is generated by natural automorphisms: change of an orthonormal basis in each tensor factor, partial transpose in each tensor factor, and interchanging two tensor factors of t ..."
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Cited by 4 (1 self)
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It is shown that the linear group of automorphism of Hermitian matrices which preserves the tensor product of unitary orbits is generated by natural automorphisms: change of an orthonormal basis in each tensor factor, partial transpose in each tensor factor, and interchanging two tensor factors of the same dimension. The result is then applied to show that automorphisms of the product numerical ranges have the same structure.
Duality of cones of positive maps
, 810
"... We study the socalled Kpositive linear maps from B(L) into B(H) for finite dimesional Hilbert spaces L and H and give characterizations of the dual cone of the cone of Kpositive maps. Applications are given to decomposable maps and their relation to PPTstates. ..."
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Cited by 2 (0 self)
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We study the socalled Kpositive linear maps from B(L) into B(H) for finite dimesional Hilbert spaces L and H and give characterizations of the dual cone of the cone of Kpositive maps. Applications are given to decomposable maps and their relation to PPTstates.
Geometry of State Spaces
"... In the Hilbert space description of quantum theory one has two major inputs: Firstly its linearity, expressing the superposition principle, and, secondly, the scalar product, allowing to compute transition probabilities. The scalar product defines an Euclidean geometry. One may ask for the physical ..."
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Cited by 1 (0 self)
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In the Hilbert space description of quantum theory one has two major inputs: Firstly its linearity, expressing the superposition principle, and, secondly, the scalar product, allowing to compute transition probabilities. The scalar product defines an Euclidean geometry. One may ask for the physical meaning in quantum physics of geometric constructs in this setting. As an important example we consider the length of a curve in Hilbert space and the “velocity”, i. e. the length of the tangents, with which the vector runs through the Hilbert space. The Hilbert spaces are generically complex in quantum physics: There is a multiplication with complex numbers: Two linear dependent vectors represent the same state. By restricting to unit vectors one can diminish this arbitrariness to phase factors. As a consequence, two curves of unit vectors represent the same curve of states if they differ only in phase. They are physically equivalent. Thus, considering a given curve — for instance a piece of a solution of a Schrödinger equation – one can ask for an equivalent curve of minimal length. This minimal length is the “FubiniStudy length”. The geometry, induced by the minimal length requirement in the set of vector states, is the “FubiniStudy metric”. There is a simple condition from which one can read off whether all pieces of a curve in Hilbert space fulfill the minimal length condition, so that their Euclidean and their StudyFubini length coincide piecewise: It is the parallel transport condition, defining the geometric (or Berry) phase of closed curves by the following mechanism: We replace the closed curve by changing its phases to a minimal length curve. Generically, the latter will not close. Its initial and its final point will differ by a phase factor, called the geometric phase (factor). We only touch these aspects in our essay and advice the reading of [6] instead. We discuss, as quite another application, the TamMandelstamm inequalities. The set of vector states associated to a Hilbert space can also be described as the set of its 1dimensional subspaces or, equivalently, as the set of all rank one projection operators. Geometrically it is the “projective space ” given by
QuantumBayesian Coherence ∗
, 2009
"... In a quantumBayesian take on quantum mechanics, the Born Rule cannot be interpreted as a rule for setting measurementoutcome probabilities from an objective quantum state. But if not, what is the role of the rule? In this paper, we argue that it should be seen as an empirical addition to Bayesian ..."
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In a quantumBayesian take on quantum mechanics, the Born Rule cannot be interpreted as a rule for setting measurementoutcome probabilities from an objective quantum state. But if not, what is the role of the rule? In this paper, we argue that it should be seen as an empirical addition to Bayesian reasoning itself. Particularly, we show how to view the Born Rule as a normative rule in addition to usual Dutchbook coherence. It is a rule that takes into account how one should assign probabilities to the consequences of various intended measurements on a physical system, but explicitly in terms of prior probabilities for and conditional probabilities consequent upon the imagined outcomes of a special counterfactual reference measurement. This interpretation is seen particularly clearly by representing quantum states in terms of probabilities for the outcomes of a fixed, fiducial symmetric informationally complete (SIC) measurement. We further explore the extent to which the general form of the new normative rule implies the