General Von Research Articles

The drawing of borders in occupied territories of the former Polish-Lithuanian Commonwealth as an instrument of implementation of German special policies

The work aims to present German political concepts implemented in the territories of the former Grand Duchy of Lithuania occupied by the Reich during World War I. Their main contractors were Chief of the General Staff Erich Ludendorff and Field Marshal Paul von Hindenburg. The key objective of this special policy was the maximum weakening of Poles’ self-organisational capabilities.

Quantum Talagrand, KKL and Friedgut’s Theorems and the Learnability of Quantum Boolean Functions

We extend three related results from the analysis of influences of Boolean functions to the quantum setting, namely the KKL theorem, Friedgut’s Junta theorem and Talagrand’s variance inequality for geometric influences. Our results are derived by a joint use of recently studied hypercontractivity and gradient estimates. These generic tools also allow us to derive generalizations of these results in a general von Neumann algebraic setting beyond the case of the quantum hypercube, including examples in infinite dimensions relevant to quantum information theory such as continuous variables quantum systems. Finally, we comment on the implications of our results as regards to noncommutative extensions of isoperimetric type inequalities, quantum circuit complexity lower bounds and the learnability of quantum observables.

On α-z-Rényi divergence in the von Neumann algebra setting

We will investigate the α-z-Rényi divergence in the general von Neumann algebra setting based on Haagerup non-commutative Lp-spaces. In particular, we establish almost all its expected properties when 0 < α < 1 and some of them when α > 1. In an Appendix we also give an equality condition for generalized Hölder’s inequality in Haagerup non-commutative Lp-spaces.

Order type Tingley's problem for type I finite von Neumann algebras

Let M and N be von Neumann algebras, S(M)+ be the collection of all positive norm one elements in M, and PM be the projection lattice of M. Let Φ:PM→PN be a metric preserving order isomorphism and Λ:S(M)+→S(N)+ be a bijective isometry. When both M and N are type I finite, we establish that the map Φ extends to a Jordan ⁎-isomorphism from M onto N. On the other hand, if M and N are of the form ⨁n=1k0Mn, where Mn is either zero or a von Neumann algebra of type In, then the map Λ extends to a Jordan ⁎-isomorphism from M onto N. On our way, we also verify that when M and N are general von Neumann algebras, the map Λ extends to a Jordan ⁎-isomorphism if and only if Λ|PM∖{0} is a bi-orthogonality preserving bijection from PM∖{0} onto PN∖{0}.

Trace- and Improved Data-Processing Inequalities for von Neumann Algebras

We prove a version of the data-processing inequality for the relative entropy for general von Neumann algebras with an explicit lower bound involving the measured relative entropy. The inequality, which generalizes previous work by Sutter et al. on finite-dimensional density matrices, yields a bound for how well a quantum state can be recovered after it has been passed through a channel. Some natural applications of our results are in quantum field theory where the von Neumann algebras are known to be of type III. Along the way we generalize various multi-trace inequalities to general von Neumann algebras.

Channel Divergences and Complexity in Algebraic QFT

We consider a notion of divergence between quantum channels in relativistic continuum quantum field theory (QFT) that is derived from the Belavkin–Staszewski relative entropy and the concept of bimodules for general von Neumann algebras. Key concepts of the divergence that we shall prove based on a new variational formulation of that relative entropy are the subadditivity under composition and additivity under the tensor product between channels. Based on these properties, we propose to use the channel divergence relative to the trivial (identity-) channel as a novel measure of complexity. Using the properties of our channel divergence, we prove in the prerequisite generality necessary for the algebras in QFT that the corresponding complexity has several reasonable properties: (i) the complexity of a composite channel is not larger than the sum of its parts, (ii) it is additive for channels localized in spacelike separated regions, (iii) it is convex, (iv) for an N-ary measurement channel it is log N, (v) for a conditional expectation associated with an inclusion of QFTs with finite Jones index it is given by log (text {Jones Index}).

Hermitian operators and isometries on symmetric operator spaces

Let \mathcal{M} be an atomless semifinite von Neumann algebra (or an atomic von Neumann algebra with all atoms having the same trace) acting on a (not necessarily separable) Hilbert space \mathcal{H} equipped with a semifinite faithful normal trace \tau . Let E(\mathcal{M},\tau) be a symmetric operator space affiliated with \mathcal{M} , whose norm is order continuous and is not proportional to the Hilbertian norm {\parallel{\cdot}\parallel}_2 on L_2(\mathcal{M},\tau) . We obtain a general description of all bounded hermitian operators on E(\mathcal{M},\tau) . This is the first time that the description of hermitian operators on a symmetric operator space (even for a noncommutative L_p -space) is obtained in the setting of general (non-hyperfinite) von Neumann algebras. As an application, we resolve a long-standing open problem concerning the description of isometries raised in the 1980s, which generalizes and unifies numerous earlier results.

Monotonic multi-state quantum f-divergences

We use the Tomita–Takesaki modular theory and the Kubo–Ando operator mean to write down a large class of multi-state quantum f-divergences and prove that they satisfy the data processing inequality. For two states, this class includes the (α, z)-Rényi divergences, the f-divergences of Petz, and the Rényi Belavkin-Staszewski relative entropy as special cases. The method used is the interpolation theory of non-commutative Lωp spaces, and the result applies to general von Neumann algebras, including the local algebra of quantum field theory. We conjecture that these multi-state Rényi divergences have operational interpretations in terms of the optimal error probabilities in asymmetric multi-state quantum state discrimination.

Ke Li’s Lemma for Quantum Hypothesis Testing in General Von Neumann Algebras

A lemma stated by Li (Ann Statist 42:171-189, 2014) has been used in, for example, Nilanjana et al. (J Math Phys 57:062207, 2016), Kaur et al. (Phys Rev A 96:062318, 2017), Rouzé, Datta (IEEE Trans Inform Theory 64:593–612, 2018), Tomamichel, Tan (Comm Math Phys 338:103-137, 2015) and Wilde et al. (IEEE Trans Inform Theory 63:1792-1817, 2017) for various tasks in quantum hypothesis testing, data compression with quantum side information or quantum key distribution. This lemma was originally proven in finite dimension, with a direct extension to type I von Neumann algebras. Here, we show that the use of modular theory allows to give more transparent meaning to the objects constructed by the lemma, and to prove it for general von Neumann algebras. This yields a new proof of quantum Stein’s lemma with slightly weaker assumption, as well as immediate generalizations of its second-order asymptotics, for example the main results in Nilanjana et al. (J Math Phys 57:062207, 2016) and Li (Ann Statist 42:171-189, 2014).

The Strong Converse Exponent of Discriminating Infinite-Dimensional Quantum States

The sandwiched Rényi divergences of two finite-dimensional density operators quantify their asymptotic distinguishability in the strong converse domain. This establishes the sandwiched Rényi divergences as the operationally relevant ones among the infinitely many quantum extensions of the classical Rényi divergences for Rényi parameter alpha >1. The known proof of this goes by showing that the sandwiched Rényi divergence coincides with the regularized measured Rényi divergence, which in turn is proved by asymptotic pinching, a fundamentally finite-dimensional technique. Thus, while the notion of the sandwiched Rényi divergences was extended recently to density operators on an infinite-dimensional Hilbert space (in fact, even for states of an arbitrary von Neumann algebra), these quantities were so far lacking an operational interpretation similar to the finite-dimensional case, and it has also been open whether they coincide with the regularized measured Rényi divergences. In this paper we fill this gap by answering both questions in the positive for density operators on an infinite-dimensional Hilbert space, using a simple finite-dimensional approximation technique. We also initiate the study of the sandwiched Rényi divergences, and the related problem of the strong converse exponent, for pairs of positive semi-definite operators that are not necessarily trace-class (this corresponds to considering weights in a general von Neumann algebra setting). This is motivated by the need to define conditional Rényi entropies in the infinite-dimensional setting, while it might also be interesting from the purely mathematical point of view of extending the concept of Rényi (and other) divergences to settings beyond the standard one of positive trace-class operators (positive normal functionals in the von Neumann algebra setting). In this spirit, we also discuss the definition and some properties of the more general family of Rényi (alpha ,z)-divergences of positive semi-definite operators on an infinite-dimensional separable Hilbert space.

Maps preserving products of commuting elements in von Neumann algebras

A bijective map between specific structures in operator algebras is called a piecewise isomorphism if it preserves products of commuting elements in both directions. We shall show that any bicontinuous piecewise isomorphism between positive cones of invertible elements in von Neumann algebras or between unitary groups of von Neumann algebras can be described in terms of the following parameters: (i) Jordan *-isomorphism between given algebras (ii) one fixed central element in domain (or range) algebra (iii) a hermitian linear map from one algebra into the center of the other one. Especially, in case of factors any piecewise isomorphism between unitary groups is a Jordan *-isomorphism or Jordan *-isomorphism composed with inversion. This extends hitherto known results from von Neumann factors to general von Neumann algebras and brings new Jordan invariants of operator structures.

German Colonial Guerilla Warfare in German East Africa and the “China Matter” at the First World War

Robert Gaudi's book about the German guerilla warfare in East Africa led by General Paul von Lettow-Vorbeck is a monography of military history in cultural historical approach.

Multivariate trace inequalities, p-fidelity, and universal recovery beyond tracial settings

Trace inequalities are general techniques with many applications in quantum information theory, often replacing the classical functional calculus in noncommutative settings. The physics of quantum field theory and holography, however, motivates entropy inequalities in type III von Neumann algebras that lack a semifinite trace. The Haagerup and Kosaki Lp spaces enable re-expressing trace inequalities in non-tracial von Neumann algebras. In particular, we show this for the generalized Araki–Lieb–Thirring and Golden–Thompson inequalities from the work of Sutter et al. [Commun. Math. Phys. 352(1), 37 (2017)]. Then, using the Haagerup approximation method, we prove a general von Neumann algebra version of universal recovery map corrections to the data processing inequality for relative entropy. We also show subharmonicity of a logarithmic p-fidelity of recovery. Furthermore, we prove that the non-decrease of relative entropy is equivalent to the existence of an L1-isometry implementing the channel on both input states.

Nonlinear Vibration Analysis of Functionally Graded Porous Plates Reinforced by Graphene Platelets on Nonlinear Elastic Foundations

This paper presents a nonlinear vibration analysis of functionally graded graphene platelet (GPL) reinforced plates on nonlinear elastic foundations. Uniformly or non-uniformly distributed internal pores were present in the plates. Based on the modified Halpin-Tsai micromechanics model and the extended rule of mixture, the material properties were evaluated. The governing equations, coupled with the effect of nonlinear foundations, were derived by using the higher-order plate theory and general von Kármán-type equations. A two-step perturbation technique was employed to obtain the nonlinear frequency and transient response. After the present method was verified, the effects of pores, GPLs, and elastic foundations were investigated in detail. A new finding is that the influence of the porosity coefficient on the natural frequency and dynamic response is relevant to foundation parameters. Moreover, the influence of the nonlinear foundation parameter can be negligible.

Does the sovereign exist? Robert Musil’s political theology

ABSTRACT The paper discusses a possible political theological interpretation of arguments developed in Robert Musil’s The Man Without Qualities. What emerges is that Musil (or his characters) pose a fundamental challenge to the possibility of any real analogy between God and the political sovereign, as suggested by Carl Schmitt. At stake is Austria as a yet-to-be-born modern sovereign. However, the novel shows why attempts to conceive it in an image of God all fail. After surveying four such attempts, the main focus will be the discussion of Anselm of Canterbury’s existential argument in this secular context. At Diotima’s inspiration it is General von Bordwehr, a largely neglected figure, who makes the most serious attempt to argue for the political sovereign as the greatest conceivable thing. The argument is that greatness entails containing every idea and its opposite, and this yields the concept of order. But order means the frozen end of everything. Hence, instead of a living God, we end up with a political sovereign marked by death.

Approximate recoverability and relative entropy II: 2-positive channels of general von Neumann algebras

We generalize our results in paper I in this series to quantum channels between general von Neumann algebras, proving the approximate recoverability of states which undergo a small change in relative entropy through the channel. To this end, we derive a strengthened form of the quantum data processing inequality for the change in relative entropy of two states under a channel between two von Neumann algebras. Compared to the usual inequality, there is an explicit lower bound involving the fidelity between the original state and a recovery channel.

Generalized semiconfined harmonic oscillator model with a position-dependent effective mass

By using a point canonical transformation starting from the constant-mass Schr\"odinger equation for the isotonic potential, it is shown that a semiconfined harmonic oscillator model with a position-dependent mass in the BenDaniel-Duke setting and the same spectrum as the standard harmonic oscillator can be easily constructed and extended to a semiconfined shifted harmonic oscillator, which could result from the presence of a uniform gravitational field. A further generalization is proposed by considering a $m$-dependent position-dependent mass for $0<m<2$ and deriving the associated semiconfined potential. This results in a family of position-dependent mass and potential pairs, to which the original pair belongs as it corresponds to $m=1$. Finally, the potential that would result from a general von Roos kinetic energy operator is presented and the examples of the Zhu-Kroemer and Mustafa-Mazharimousavi settings are briefly discussed.

Werner von Bülow német repülő tüzér hadnagy halála

German flying lieutenant Werner von Bülow was killed by a Romanian soldier at 7 a.m. on May 8, 1920. He received a gunshot wound during the crossing from the left bank of the Maros to the Hungarian side, which came under Romanian control, which caused him such severe internal bleeding that he died of the injury on the Maros bank, on the Hungarian side. He was only 23 years old when he died. Bülow was released from French captivity and wanted to return to Germany with his companion, German flight lieutenant Franz Clemens Schiffer. Schiffer luckily survived the incident. The case gained wider publicity after the mistaken belief that Werner von Bülow was the son of General Karl von Bülow and the grandson of Chancellor Bülow arose. With the deep condolences of the population, Werner von Bülow was buried in the Lutheran cemetery in Makó at the expense of the town. A memorial column was erected on his grave from a public donation. The monument is still there today in a heavily cracked state. Werner von Bülow received the II. and the German military award of the First Class Iron Cross. Werner von Bülow is also included in the Hungarian National Archives database of World War I casualties. The article describes the circumstances of the murder, burial and memorial.

Quantum Fokker–Planck Dynamics

The Fokker-Planck equation is a partial differential equation which is a key ingredient in many models in physics. This paper aims to obtain a quantum counterpart of Fokker-Planck dynamics, as a means to describing quantum Fokker-Planck dynamics. Given that relevant models relate to the description of large systems, the quantization of the Fokker-Planck equation should be done in a manner that respects this fact, and is therefore carried out within the setting of non-commutative analysis based on general von Neumann algebras. Within this framework we present a quantization of the generalized Laplace operator, and then go on to incorporate a potential term conditioned to noncommutative analysis. In closing we then construct and examine the asymptotic behaviour of the corresponding Markov semigroups. We also present a noncommutative Csiszar-Kullback inequality formulated in terms of a notion of relative entropy, and show that for more general systems, good behaviour with respect to this notion of entropy ensures similar asymptotic behaviour of the relevant dynamics.

Recoverability for optimized quantum f-divergences

The optimized quantum f-divergences form a family of distinguishability measures that includes the quantum relative entropy and the sandwiched Rényi relative quasi-entropy as special cases. In this paper, we establish physically meaningful refinements of the data-processing inequality for the optimized f-divergence. In particular, the refinements state that the absolute difference between the optimized f-divergence and its channel-processed version is an upper bound on how well one can recover a quantum state acted upon by a quantum channel, whenever the recovery channel is taken to be a rotated Petz recovery channel. Not only do these results lead to physically meaningful refinements of the data-processing inequality for the sandwiched Rényi relative entropy, but they also have implications for perfect reversibility (i.e. quantum sufficiency) of the optimized f-divergences. Along the way, we improve upon previous physically meaningful refinements of the data-processing inequality for the standard f-divergence, as established in recent work of Carlen and Vershynina [arXiv:1710.02409, arXiv:1710.08080]. Finally, we extend the definition of the optimized f-divergence, its data-processing inequality, and all of our recoverability results to the general von Neumann algebraic setting, so that all of our results can be employed in physical settings beyond those confined to the most common finite-dimensional setting of interest in quantum information theory.