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Mihály Weiner

Publications and source records attributed to Mihály Weiner.

28 records · Page 2Linked to original sources
arXiv2016

Local equivalence of representations of Diff$^+(S^1)$ corresponding to different highest weights

Let $c,h$ and $c,\tilde{h}$ be two admissible pairs of central charge and highest weight for ${\rm Diff}^+(S^1)$. It is shown here that the positive energy irreducible projective unitary representations $U_{c,h}$ and $U_{c,\tilde{h}}$ of the group ${\rm Diff}^+(S^1)$ are locally equivalent. This means that for any $I\Subset S^1$ open proper interval, there exists a unitary operator $W_I$ such that $W_I U_{c,h}(γ)W_I^* = U_{c,\tilde{h}}(γ)$ for all $γ\in {\rm Diff}^+(S^1)$ which act identically on $I^c\equiv S^1\setminus I$ (i.e. which can "displace" or "move" points only in $I$). This result extends and completes earlier ones that dealt with only certain regions of the "$c,h$-plane", and closes the gap in the full classification of superselection sectors of Virasoro nets.

math-ph↗
arXiv2014

Classical information storage in an $n$-level quantum system

A game is played by a team of two --- say Alice and Bob --- in which the value of a random variable $x$ is revealed to Alice only, who cannot freely communicate with Bob. Instead, she is given a quantum $n$-level system, respectively a classical $n$-state system, which she can put in possession of Bob in any state she wishes. We evaluate how successfully they managed to store and recover the value of $x$ in the used system by requiring Bob to specify a value $z$ and giving a reward of value $ f(x,z)$ to the team. We show that whatever the probability distribution of $x$ and the reward function $f$ are, when using a quantum $n$-level system, the maximum expected reward obtainable with the best possible team strategy is equal to that obtainable with the use of a classical $n$-state system. The proof relies on mixed discriminants of positive matrices and --- perhaps surprisingly --- an application of the Supply--Demand Theorem for bipartite graphs. As a corollary, we get an infinite set of new, dimension dependent inequalities regarding positive operator valued measures and density operators on complex $n$-space. As a further corollary, we see that the greatest value, with respect to a given distribution of $x$, of the mutual information $I(x;z)$ that is obtainable using an $n$-level quantum system equals the analogous maximum for a classical $n$-state system.

cs.IT↗
arXiv2012

Thermal States in Conformal QFT. I

We analyze the set of locally normal KMS states w.r.t. the translation group for a local conformal net A of von Neumann algebras on R. In this first part, we focus on completely rational net A. Our main result here states that, if A is completely rational, there exists exactly one locally normal KMS state ϕ. Moreover, ϕis canonically constructed by a geometric procedure. A crucial rôle is played by the analysis of the "thermal completion net" associated with a locally normal KMS state. A similar uniqueness result holds for KMS states of two-dimensional local conformal nets w.r.t. the time-translation one-parameter group.

math-ph↗
arXiv2010

A gap for the maximum number of mutually unbiased bases

A collection of pairwise mutually unbiased bases (in short: MUB) in d>1 dimensions may consist of at most d+1 bases. Such "complete" collections are known to exists in C^d when d is a power of a prime. However, in general little is known about the maximal number N(d) of bases that a collection of MUBs in C^d can have. In this work it is proved that a collection of d MUBs in C^d can be always completed. Hence N(d) cannot be d and when d>1 we have a dichotomy: either N(d)=d+1 (so that there exists a complete collection of MUBs), or N(d)\leq d-1. In the course of the proof an interesting new characterization is given for a linear subspace of M_d(C) to be a subalgebra.

math-ph↗
arXiv2010

An algebraic Haag's theorem

Under natural conditions (such as split property and geometric modular action of wedge algebras) it is shown that the unitary equivalence class of the net of local (von Neumann) algebras in the vacuum sector associated to double cones with bases on a fixed space-like hyperplane completely determines an algebraic QFT model. More precisely, if for two models there is unitary connecting all of these algebras, then --- without assuming that this unitary also connects their respective vacuum states or spacetime symmetry representations --- it follows that the two models are equivalent. This result might be viewed as an algebraic version of the celebrated theorem of Rudolf Haag about problems regarding the so-called "interaction-picture" in QFT. Original motivation of the author for finding such an algebraic version came from conformal chiral QFT. Both the chiral case as well as a related conjecture about standard half-sided modular inclusions will be also discussed.

math-ph↗
arXiv2010

On vector configurations that can be realized in the cone of positive matrices

Let $v_1$,..., $v_n$ be $n$ vectors in an inner product space. Can we find a natural number $d$ and positive (semidefinite) complex matrices $A_1$,..., $A_n$ of size $d \times d$ such that ${\rm Tr}(A_kA_l)= $ for all $k,l=1,..., n$? For such matrices to exist, one must have $ \geq 0$ for all $k,l=1,..., n$. We prove that if $n<5$ then this trivial necessary condition is also a sufficient one and find an appropriate example showing that from $n=5$ this is not so --- even if we allowed realizations by positive operators in a von Neumann algebra with a faithful normal tracial state. The fact that the first such example occurs at $n=5$ is similar to what one has in the well-investigated problem of positive factorization of positive (semidefinite) matrices. If the matrix $( )$ has a positive factorization, then matrices $A_1$,..., $A_n$ as above exist. However, as we show by a large class of examples constructed with the help of the Clifford algebra, the converse implication is false.

math.OA↗
arXiv2010

On quasi-orthogonal systems of matrix algebras

In this work it is shown that certain interesting types of quasi-orthogonal system of subalgebras (whose existence cannot be ruled out by the trivial necessary conditions) cannot exist. In particular, it is proved that there is no quasi-orthogonal decomposition of M_n(C)\otimes M_n(C)\equiv M_{n^2}(C) into a number of maximal abelian subalgebras and factors isomorphic to M_n(C) in which the number of factors would be 1 or 3. In addition, some new tools are introduced, too: for example, a quantity c(A,B), which measures "how close" the subalgebras A,B \subset M_n(C) are to being quasi-orthogonal. It is shown that in the main cases of interest, c(A',B') - where A' and B' are the commutants of A and B, respectively - can be determined by c(A,B) and the dimensions of A and B. The corresponding formula is used to find some further obstructions regarding quasi-orthogonal systems.

math-ph↗
arXiv2007

Restricting positive energy representations of ${\rm Diff}^+(S^1)$ to the stabilizer of $n$ points

Let $G_n \subset {\rm Diff}^+(S^1)$ be the stabilizer of $n$ given points of $S^1$. How much information do we lose if we restrict a positive energy representation $U^c_h$ associated to an admissible pair $(c,h)$ of the central charge and lowest energy, to the subgroup $G_n$? The question, and a part of the answer originate in chiral conformal QFT. The value of $c$ can be easily ``recovered'' from such a restriction; the hard question concerns the value of $h$. If $c\leq 1$, then there is no loss of information, and accordingly, all of these restrictions are irreducible. In this work it is shown that $U^c_{h}|_{G_n}$ is always irreducible for $n=1$, and if $h=0$, it is irreducible at least up to $n\leq 3$. Moreover, an example is given for certain values $c>1$ and $h,\tilde{h}>0$ such that $U^c_{h}|_{G_1}\simeq U^c_{\tilde{h}}|_{G_1}$. It follows that for these values $U^c_{h}|_{G_n}$ cannot be irreducible for $n\geq 2$. For further values of $c,h$ and $n$, the question is left open. Nevertheless, the example already shows, that in general, local and global intertwiners in a QFT model may not be equivalent.

math.FA↗
arXiv2005

Conformal Covariance and Positivity of Energy in Charged Sectors

It has been recently noted that the diffeomorphism covariance of a Chiral Conformal QFT in the vacuum sector automatically ensures Möbius covariance in all charged sectors. In this article it is shown that the diffeomorphism covariance and the positivity of the energy in the vacuum sector even ensure the positivity of the energy in the charged sectors. The main observation of this paper is that the positivity of the energy -- at least in case of a Chiral Conformal QFT -- is a local concept: it is related to the fact that the energy density, when smeared with some local nonnegative test functions, remains bounded from below (with the bound depending on the test function). The presented proof relies in an essential way on recently developed methods concerning the smearing of the stress-energy tensor on nonsmooth functions.

math-ph↗