In this seminar report, we present in detail the proof of a recent result due to J. Briët and T. Vidick, improving an estimate in a 2008 paper by D. Pérez-Garc\'ıa, M. Wolf, C. Palazuelos, I. Villanueva, and M. Junge, estimating the growth of the deviation in the tripartite Bell inequality. The proof requires a delicate estimate of the norms of certain trilinear (or $d$-linear) forms on Hilbert space with coefficients in the second Gaussian Wiener chaos. Let $E^n_{\vee}$ (resp. $E^n_{\min}$) denote $ \ell_1^n \otimes \ell_1^n\otimes \ell_1^n$ equipped with the injective (resp. minimal) tensor norm. Here $ \ell_1^n$ is equipped with its maximal operator space structure. The Briët-Vidick method yields that the identity map $I_n$ satisfies (for some $c>0$) $\|I_n:\ E^n_{\vee}\to E^n_{\min}\|\ge c n^{1/4} (\log n)^{-3/2}.$ Let $S^n_2$ denote the (Hilbert) space of $n\times n$-matrices equipped with the Hilbert-Schmidt norm. While a lower bound closer to $n^{1/2} $ is still open, their method produces an interesting, asymptotically almost sharp, related estimate for the map $J_n:\ S^n_2\stackrel{\vee}{\otimes} S^n_2\stackrel{\vee}{\otimes}S^n_2 \to \ell_2^{n^3} \stackrel{\vee}{\otimes} \ell_2^{n^3} $ taking $e_{i,j}\otimes e_{k,l}\otimes e_{m,n}$ to $e_{[i,k,m],[j,l,n]}$.
In this seminar report, we present in detail the proof of a recent result due to J. Briët and T. Vidick, improving an estimate in a 2008 paper by D. Pérez-Garc'ıa, M. Wolf, C. Palazuelos, I. Villanueva, and M. Junge, estimating the growth of the deviation in the tripartite Bell inequality. The proof requires a delicate estimate of the norms of certain trilinear (or d-linear) forms on Hilbert space with coefficients in the second Gaussian Wiener chaos. Let E∨n (resp. Eminn) denote ℓ1n⊗ℓ1n⊗ℓ1n equipped with the injective (resp. minimal) tensor norm. Here ℓ1n is equipped with its maximal operator space structure. The Briët-Vidick method yields that the identity map In satisfies (for some c>0) ∥In:E∨n→Eminn∥≥cn1/4(logn)−3/2. Let S2n denote the (Hilbert) space of n×n-matrices equipped with the Hilbert-Schmidt norm. While a lower bound closer to n1/2 is still open, their method produces an interesting, asymptotically almost sharp, related estimate for the map Jn:S2n⊗∨S2n⊗∨S2n→ℓ2n3⊗∨ℓ2n3 taking ei,j⊗ek,l⊗em,n to e[i,k,m],[j,l,n].
Let Φ be a super-operator, i.e., a linear mapping of the form Φ:L(F)→L(G) for finite dimensional Hilbert spaces F and G. This paper considers basic properties of the super-operator norms defined by ∥Φ∥q→p=sup{∥Φ(X)∥p/∥X∥q:X=0}, induced by Schatten norms for 1≤p,q≤∞. These super-operator norms arise in various contexts in the study of quantum information. In this paper it is proved that if Φ is completely positive, the value of the supremum in the definition of ∥Φ∥q→p is achieved by a positive semidefinite operator X, answering a question recently posed by King and Ruskai~\cite{KingR04}. However, for any choice of p∈[1,∞], there exists a super-operator Φ that is the {\em difference} of two completely positive, trace-preserving super-operators such that all Hermitian X fail to achieve the supremum in the definition of ∥Φ∥1→p. Also considered are the properties of the above norms for super-operators tensored with the identity super-operator. In particular, it is proved that for all p≥2, q≤2, and arbitrary Φ, the norm ∥Φ∥q→p is stable under tensoring Φ with the identity super-operator, meaning that ∥Φ∥q→p=∥Φ⊗I∥q→p. For 1≤p<2, the norm ∥Φ∥1→p may fail to be stable with respect to tensoring Φ with the identity super-operator as just described, but ∥Φ⊗I∥1→p is stable in this sense for I the identity super-operator on L(H) for $\op{dim}(\mathcal{H}) = \op{dim}(\mathcal{F})$. This generalizes and simplifies a proof due to Kitaev \cite{Kitaev97} that established this fact for the case p=1.
The main result of the paper states that if 0=v∈Cm⊗Cn, d=min{m,n}, and p=SR(v), then for every 1≤q≤d, \begin{equation*} \min\left\{OSR(\mathbf U): \begin{array}{l} \mathbf U\in\mathbb M_m\otimes\mathbb M_n\text{ is unitary}, SR(\mathbf Uv)=q \end{array}\right\} = \left\lceil\max\left\{\frac pq,\frac qp\right\}\right\rceil . \end{equation*} Here OSR(U) denotes the operator Schmidt rank of U. Furthermore, if U∈Mm⊗Mn is unitary and OSR(U)≤k, then, for every unit vector x∈Cm⊗Cn and every 1≤r≤d, \begin{equation*} F_{\min\{kr,d\}}(\mathbf Ux)\ge F_r(x) \quad\text{and}\quad F_{\min\{kr,d\}}(x)\ge F_r(\mathbf Ux), \end{equation*} where Fr(x) denotes the sum of the r largest squared Schmidt coefficients of x. This result is stronger than rank reachability as it controls the entire cumulative Schmidt profile. We also establish a majorization relation and use it to obtain sharp bounds on Schatten-norm distortion and on the change of Rényi entropies under unitaries with bounded operator Schmidt rank.
We give a simple proof of the matrix Spencer conjecture up to poly-logarithmic rank: given symmetric d×d matrices A1,…,An each with ∥Ai∥op≤1 and rank at most n/log3n, one can efficiently find ±1 signs x1,…,xn such that their signed sum has spectral norm ∥∑i=1nxiAi∥op=O(n). This result also implies a logn−Ω(loglogn) qubit lower bound for quantum random access codes encoding n classical bits with advantage ≫1/n. Our proof uses the recent refinement of the non-commutative Khintchine inequality in [Bandeira, Boedihardjo, van Handel, 2022] for random matrices with correlated Gaussian entries.
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 Greenberger-Horne-Zeilinger (GHZ) states is always bounded so that, in contrast to many other contexts, GHZ states do not lead to extremal quantum correlations in this case. In order to derive all these physical consequences, we will have to obtain new mathematical results in the theories of operator spaces and tensor norms. In particular, we will prove the existence of bounded but not completely bounded trilinear forms from commutative C*-algebras. Finally, we will relate the existence of diagonal states leading to unbounded violations with a long-standing open problem in the context of Banach algebras.