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M. Junge

Publications and source records attributed to M. Junge.

11 recordsLinked to original sources
arXiv2017

Embeddings of operator ideals into $\mathcal{L}_p-$spaces on finite von Neumann algebras

Let $\mathcal{L}(H)$ be the $*$-algebra of all bounded operators on an infinite dimensional Hilbert space $H$ and let $(\mathcal{I}, \|\cdot\|_{\mathcal{I}})$ be an ideal in $\mathcal{L}(H)$ equipped with a Banach norm which is distinct from the Schatten-von Neumann ideal $\mathcal{L}_p(\mathcal{H})$, $1\leq p<2$. We prove that $\mathcal{I}$ isomorphically embeds into an $L_p$-space $\mathcal{L}_p(\mathcal{R}),$ $1\leq p<2,$ (here, $\mathcal{R}$ is the hyperfinite II$_1$-factor) if its commutative core (that is, Calkin space for $\mathcal{I}$) isomorphically embeds into $L_p(0,1).$ Furthermore, we prove that an Orlicz ideal $\mathcal{L}_M(H)\neq\mathcal{L}_p(H)$ isomorphically embeds into $\mathcal{L}_p(\mathcal{R}),$ $1\leq p<2,$ if and only if it is an interpolation space for the Banach couple $(\mathcal{L}_p(H),\mathcal{L}_2(H)).$ Finally, we consider isomorphic embeddings of $(\mathcal{I}, \|\cdot\|_{\mathcal{I}})$ into $L_p$-spaces associated with arbitrary finite von Neumann algebras.

math.OA↗
arXiv2013

Rank-one Quantum Games

In this work we study rank-one quantum games. In particular, we focus on the study of the computability of the entangled value $ω^*$. We show that the value $ω^*$ can be efficiently approximated up to a multiplicative factor of 4. We also study the behavior of $ω^*$ under the parallel repetition of rank-one quantum games, showing that it does not verify a perfect parallel repetition theorem. To obtain these results, we first connect rank-one games with the mathematical theory of operator spaces. We also reprove with these new tools essentially known results about the entangled value of rank-one games with one-way communication $ω_{qow}$. In particular, we show that $ω_{qow}$ can be computed efficiently and it satisfies a perfect parallel repetition theorem.

quant-ph↗
arXiv2010

Connes' embedding problem and Tsirelson's problem

We show that Tsirelson's problem concerning the set of quantum correlations and Connes' embedding problem on finite approximations in von Neumann algebras (known to be equivalent to Kirchberg's QWEP conjecture) are essentially equivalent. Specifically, Tsirelson's problem asks whether the set of bipartite quantum correlations generated between tensor product separated systems is the same as the set of correlations between commuting C*-algebras. Connes' embedding problem asks whether any separable II$_1$ factor is a subfactor of the ultrapower of the hyperfinite II$_1$ factor. We show that an affirmative answer to Connes' question implies a positive answer to Tsirelson's. Conversely, a positve answer to a matrix valued version of Tsirelson's problem implies a positive one to Connes' problem.

math-ph↗
arXiv2009

Operator Space theory: a natural framework for Bell inequalities

In this letter we show that the field of Operator Space Theory provides a general and powerful mathematical framework for arbitrary Bell inequalities, in particular regarding the scaling of their violation within quantum mechanics. We illustrate the power of this connection by showing that bipartite quantum states with local Hilbert space dimension n can violate a Bell inequality by a factor of order $\frac{\sqrt{n}}{\log^2n}$ when observables with n possible outcomes are used. Applications to resistance to noise, Hilbert space dimension estimates and communication complexity are given.

quant-ph↗
arXiv2009

Unbounded violations of bipartite Bell Inequalities via Operator Space theory

In this work we show that bipartite quantum states with local Hilbert space dimension n can violate a Bell inequality by a factor of order $\sqrt{n}$ (up to a logarithmic factor) when observables with n possible outcomes are used. A central tool in the analysis is a close relation between this problem and operator space theory and, in particular, the very recent noncommutative $L_p$ embedding theory. As a consequence of this result, we obtain better Hilbert space dimension witnesses and quantum violations of Bell inequalities with better resistance to noise.

quant-ph↗
arXiv2007

Unbounded violation of tripartite Bell inequalities

We prove that there are tripartite quantum states (constructed from random unitaries) that can lead to arbitrarily large violations of Bell inequalities for dichotomic observables. As a consequence these states can withstand an arbitrary amount of white noise before they admit a description within a local hidden variable model. This is in sharp contrast with the bipartite case, where all violations are bounded by Grothendieck's constant. We will discuss the possibility of determining the Hilbert space dimension from the obtained violation and comment on implications for communication complexity theory. Moreover, we show that the violation obtained from generalized GHZ states is always bounded so that, in contrast to many other contexts, GHZ states do in this case not lead to extremal quantum correlations. The results are based on tools from the theories of operator spaces and tensor norms which we exploit to prove the existence of bounded but not completely bounded trilinear forms from commutative C*-algebras.

quant-ph↗
arXiv2005

On Asymptotically Symmetric Banach Spaces

We define and study asymptotically symmetric Banach spaces (a.s.) and its variations: weakly a.s. (w.a.s.) and weakly normalized a.s. (w.n.a.s.). If X is a.s. then all spreading models of X are uniformly symmetric. We show that the converse fails. We also show that w.a.s. and w.n.a.s. are not equivalent properties and that Schlumprecht's space S fails to be w.n.a.s. We show that if X is separable and has the property that every normalized weakly null sequence in X has a subsequence equivalent to the unit vector basis of c_0 then X is w.a.s.. We obtain an analogous result if c_0 is replaced by ell_1 and also show it is false if c_0 is replaced by ell_p, 1 < p < infinity. We prove that if 1 less than or equal p < infinity and the norm of the sum of (x_i)_1^n is of the order n^{1/p} for all (x_i)_1^n in the n^{th} asymptotic structure of $X$, then X contains an asymptotic ell_p, hence w.a.s. subspace.

math.FA↗
arXiv2002

Approximation Properties for Non-commutative L_p-Spaces Associated with Discrete Groups

Let $1 < p < \infty$. It is shown that if $G$ is a discrete group with the approximation property introduced by Haagerup and Kraus, then the non-commutative $L_p(VN(G))$ space has the operator space approximation property. If, in addition, the group von Neumann algebra $VN(G)$ has the QWEP, i.e. is a quotient of a $C^*$-algebra with Lance's weak expectation property, then $L_p(VN(G))$ actually has the completely contractive approximation property and the approximation maps can be chosen to be finite-rank completely contractive multipliers on $L_p(VN(G))$. Finally, we show that if $G$ is a countable discrete group having the approximation property and $VN(G)$ has the QWEP, then $L_p(VN(G))$ has a very nice local structure, i.e. it is a $\mathcal C\OL_p$ space and has a completely bounded Schauder basis.

math.OA↗
arXiv2002

On ${\OL}_{\infty}$ structure of nuclear $C^*$-algebras

We study the local operator space structure of nuclear $C^*$-algebras. It is shown that a $C^*$-algebra is nuclear if and only if it is an $\OL_{\infty, \la}$ space for some (and actually for every) $\la > 6$. The $\OL_\infty$ constant $λ$ provides an interesting invariant \[ \OL_\infty (\A) = \inf\{\la: ~ \A ~{\rm is ~ an} ~ {\OL}_{\infty, \la} ~ {\rm space}\} \] for nuclear $C^*$-algebras. Indeed, if $\A$ is a nuclear $C^*$-algebra, then we have $1\le \OL_\infty (\A) \le 6$, and if $\A$ is a unital nuclear $C^*$-algebra with $\OL_{\infty} (\A) \le (\frac {1+{\sqrt 5}}2)^{\frac 12}$, we show that $\A$ must be stably finite. We also investigate the connection between the rigid $\OL_{\infty, 1^+}$ structure and the rigid complete order $\OL_{\infty, 1^+}$ structure on $C^*$-algebras, where the latter structure has been studied by Blackadar and Kirchberg in their characterization of strong NF $C^*$-algebras. Another main result of this paper is to show that these two local structrues are actually equivalent on unital nuclear $C^*$-algebras. We obtain this by showing that if a unital (nuclear) $C^*$-algebra is a rigid ${\OL}_{\infty, 1^+}$ space, then it is inner quasi-diagonal, and thus is a strong NF algebra. It is also shown that if a unital (nuclear) $C^*$-algebra is an ${\OL}_{\infty, 1^+}$ space, then it is quasi-diagonal, and thus is an NF algebra.

math.OA↗
arXiv2002

Doob's inequality for non-commutative martingales

Let $1\le p<\8$ and $(x_n)_{\nen}$ be a sequence of positive elements in a non-commutative $L_p$ space and $(E_n)_{\nen}$ be an increasing sequence of conditional expectations, then the $L_p$ norm of \sum_n E_n(x_n) can be estimated by c_p times the $L_p$ norm of \sum_n x_n. This inequality is due to Burkholder, Davis and Gundy in the commutative case. By duality, we obtain a version of Doob's maximal inequality for $1<p\le \8$.

math.OA↗