arXiv subjects

Armin Uhlmann

Publications and source records attributed to Armin Uhlmann.

At least 19 recordsLinked to original sources

arXiv2016

Anti- (Conjugate) Linearity

This is an introduction to antilinear operators. In following E.P.Wigner the terminus "antilinear" is used as it is standard in Physics. Mathematicians prefer to say "conjugate linear". By restricting to finite-dimensional complex-linear spaces, the exposition becomes elementary in the functional analytic sense. Nevertheless it shows the amazing differences to the linear case. Basics of antilinearity is explained in sections 2, 3,4, 7 and in subsection 1.2: Spectrum, canonical Hermitian form, antilinear rank one and two operators, the Hermitian adjoint, classification of antilinear normal operators, (skew) conjugations, involutions, and acq-lines, i.e. the antilinear counterparts of 1-parameter operator groups. Applications include the representation of the Lagrangian Grassmannian by conjugations, its covering by acq-lines, as well as results on equivalence relations. After remembering elementary Tomita-Takesaki theory, antilinear maps, assiciated associated to a two-partite quantum system, are defined. By allowing to write modular objects as twisted products of pairs of them, they open some new ways to express EPR and teleportation tasks. The appendix presents a look onto the rich structure of antilinear operator spaces.

quant-ph↗

arXiv2016

Transition Probability (Fidelity) and Its Relatives

Transition Probability (fidelity) for pairs of density operators can be defined as "functor" in the hierarchy of "all" quantum systems and also within any quantum system. The introduction of "amplitudes" for density operators allows for a more intuitive treatment of these quantities, also pointing to a natural parallel transport. The latter is governed by a remarkable gauge theory with strong relations to the Riemann-Bures metric.

quant-ph↗

arXiv2014

The number of orthogonal conjugations

After a short introduction to anti-linearity, bounds for the number of orthogonal (skew) conjugations are proved. They are saturated if the dimension of the Hilbert space is a power of two. For the other dimensions this is an open problem.

quant-ph↗

arXiv2011

Roofs and Convexity

Convex roof extensions are widely used to create entanglement measures in quantum information theory. The aim of the article is to present some tools which could be helpful for their treatment. Sections 2 and 3 introduce into the subject. It follows descriptions of Wootter's method, of the "subtraction procedure", and examples on how to use symmetries.

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arXiv2009

Concurrence and Entanglement Entropy of Stochastic 1-Qubit Maps

Explicit expressions for the concurrence of all positive and trace-preserving ("stochastic") 1-qubit maps are presented. We construct the relevant convex roof patterns by a new method. We conclude that two component optimal decompositions always exist. Our results can be transferred to 2xN-quantumsystems providing the concurrence for all rank two density operators as well as lower and upper bounds for their entanglement of formation. We apply these results to a study of the entanglement entropy of 1-qubit stochastic maps which preserve axial symmetry. Using analytic and numeric results we analyze the bifurcation patterns appearing in the convex roof of optimal decompositions and give results for the one-shot (Holevo-Schumacher-Westmoreland) capacity of those maps.

quant-ph↗

arXiv2009

An Entropy Inequality

Let $S(ρ)=- Tr (ρ\logρ)$ be the von Neumann entropy of an $N$-dimensional quantum state $ρ$ and $e_2(ρ)$ the second elementary symmetric polynomial of the eigenvalues of $ρ$. We prove the inequality $S(ρ) \le c(N) \sqrt{e_2(ρ)} $ where $c(N)=\log(N) \sqrt{\frac{2N}{N-1}}$. This generalizes an inequality given by Fuchs and Graaf \cite{fuchsgraaf} for the case of one qubit, i.e., N=2. Equality is achieved if and only if $ρ$ is either a pure or the maximally mixed state. This inequality delivers new bounds for quantities of interest in quantum information theory, such as upper bounds for the minimum output entropy and the entanglement of formation as well as a lower bound for the Holevo channel capacity.

quant-ph↗

arXiv2008

Concurrence of Stochastic 1-Qubit Maps

Explicit expressions for the concurrence of all positive and trace-preserving ("stochastic") 1-qubit maps are presented. By a new method we find the relevant convex roof pattern. We conclude that two component optimal decompositions always exist. Our results can be transferred to $2 \times n$-quantum systems providing the concurrence for all rank two density operators as well as a lower bound for their entanglement of formation.

quant-ph↗

arXiv2007

Three-tangle for mixtures of generalized GHZ and generalized W states

We give a complete solution for the three-tangle of mixed three-qubit states composed of a generalized GHZ state, a|000>+b|111>, and a generalized W state, c|001>+d|010>+f|100>. Using the methods introduced by Lohmayer et al. we provide explicit expressions for the mixed-state three-tangle and the corresponding optimal decompositions for this more general case. Moreover, as a special case we obtain a general solution for a family of states consisting of a generalized GHZ state and an orthogonal product state.

quant-ph↗

arXiv2007

Tangles of superpositions and the convex-roof extension

We discuss aspects of the convex-roof extension of multipartite entanglement measures, that is, $SL(2,\CC)$ invariant tangles. We highlight two key concepts that contain valuable information about the tangle of a density matrix: the {\em zero-polytope} is a convex set of density matrices with vanishing tangle whereas the {\em convex characteristic curve} readily provides a non-trivial lower bound for the convex roof and serves as a tool for constructing the convex roof outside the zero-polytope. Both concepts are derived from the tangle for superpositions of the eigenstates of the density matrix. We illustrate their application by considering examples of density matrices for two-qubit and three-qubit states of rank 2, thereby pointing out both the power and the limitations of the concepts.

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arXiv2006

Entangled three-qubit states without concurrence and three-tangle

We provide a complete analysis of mixed three-qubit states composed of a GHZ state and a W state orthogonal to the former. We present optimal decompositions and convex roofs for the three-tangle. Further, we provide an analytical method to decide whether or not an arbitrary rank-2 state of three qubits has vanishing three-tangle. These results highlight intriguing differences compared to the properties of two-qubit mixed states, and may serve as a quantitative reference for future studies of entanglement in multipartite mixed states. By studying the Coffman-Kundu-Wootters inequality we find that, while the amounts of inequivalent entanglement types strictly add up for pure states, this ``monogamy'' can be lifted for mixed states by virtue of vanishing tangle measures.

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arXiv2006

On Concurrence and Entanglement of Rank Two Channels

Concurrence and further entanglement quantifyers can be computed explicitly for channels of rank two if representable by just two Kraus operators. Almost all details are available for the subclass of rank two 1-qubit-channels. There is a simple geometric picture beyond, explaining nicely the role of anti-linearity.

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arXiv2004

Antilinearity in Bipartite Quantum Systems and Imperfect Quantum Teleportation

Antilinearity is quite natural in bipartite quantum systems. There is a one-to-one correspondence between vectors and certain antilinear maps, here called EPR-maps. Some of their properties and uses, including the factorization of quantum teleportation maps, is explained. There is an elementary link to twisted Kronecker products and to the modular objects of Tomita and Takesaki.

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arXiv2003

Quantum information transfer from one system to another one

The topics of the paper are: a) Some anti-linear maps governing EPR tasks if no reference bases are distinguished. b) Imperfect teleportation and the composition rule. The ancilla is supposed pure but otherwise arbitrary. c) Quantum telportation with distributed measurements. d) Remarks on EPR with a mixed state, triggered by a Lueders measurement.

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arXiv2003

Concurrence and foliations induced by some 1-qubit channels

We start with a short introduction to the roof concept. An elementary discussion of phase-damping channels shows the role of anti-linear operators in representing their concurrence. A general expression for some concurrences is derived. We apply it to 1-qubit channels of length two, getting induced foliations of the state space, the optimal decompositions, and the entropy of a state with respect to these channels. For amplitude-damping channels one obtains an expression for the Holevo capacity allowing for easy numerical calculations.

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arXiv2003

On 1-qubit channels

The entropy H_T(rho) of a state rho with respect to a channel T and the Holevo capacity of the channel require the solution of difficult variational problems. For a class of 1-qubit channels, which contains all the extremal ones, the problem can be significantly simplified by associating an Hermitian antilinear operator theta to every channel of the considered class. The concurrence of the channel can be expressed by theta and turns out to be a flat roof. This allows to write down an explicit expression for H_T. Its maximum would give the Holevo (1-shot) capacity.

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arXiv2002

On Bures-Distance and -Algebraic Transition Probability between Inner Derived Positive Linear Forms over W-Algebra

On a W*-algebra M, for given two positive linear forms f,g and algebra elements a,b a variational expression for the Bures-distance d_B(f^a,g^b) between the inner derived positive linear forms f^a=f(a* . a) and g^b=g(b* . b) is obtained. Along with the proof of the formula also some earlier result of S.Gudder on non-commutative probability will be slightly extended. Also, the given expression of the Bures-distance nicely relates to some system of seminorms proposed by D.Buchholz and which occured along with the problem of estimating the so-called `weak intertwiners' in algebraic quantum field theory. In the last part some optimization problem will be considered.

math-ph↗

arXiv2000

On the LOCC Classification of Bipartite Density Matrices

We provide a unifying framework for exact, probabilistic, and approximate conversions by local operations and classical communication (LOCC) between bipartite states. This framework allows us to formulate necessary and sufficient conditions for LOCC conversions from pure states to mixed states and it provides necessary conditions for LOCC conversions between mixed states. The central idea is the introduction of convex sets for exact, probabilistic, and approximate conversions, which are closed under LOCC operations and which are largely characterized by simple properties of pure states.

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arXiv2000

Simultaneous decomposition of two states

Simultaneous decompositions of a pair of states into pure ones are examined. There are privileged decompositions which are distinguished from all the other ones.

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