Quantum expanders and geometry of operator spaces (Q2510744)

Quantum expanders were introduced by \textit{M. Hastings} [``Random unitaries give quantum expanders'', Phys. Rev. B (3) 76, No. 3, 035114 (2007; \url{doi:10.1103/PhysRevA.76.032315})] and by \textit{A. Ben-Aroya} and \textit{A. Ta-Shma} [``Quantum expanders and the quantum entropy difference problem'', Preprint, \url{arXiv:quant-ph/0702129}] as sequences \(\{U^{(N)}\mid N\geq 1 \}\) of \(n\)-tuples \(U^{(N)}=(U_1^{(N)},\dots,U_n^{(N)})\) of \(N\times N\) unitary matrices such that there is an \(\varepsilon >0\) satisfying the following ``spectral gap'' condition: \[ \forall N\;\forall x\in M_N\quad \left\| \sum_{j=1}^n U_j^{(N)} (x-N^{-1}\text{trace}(x)) { U_j^{(N)}}^* \right\|_2 \leq n(1-\varepsilon)\| x-N^{-1}\text{trace}(x)\|_2, \] where \(\|\cdot\|_2\) denotes the Hilbert-Schmidt norm on \(M_N\). Usual expander graphs can be regarded as expanders in this sense: we write their adjacency matrices as sums of permutation matrices, then the spectral definition of expanders implies that these permutation matrices satisfy the condition above, if we restrict \(x\) to the set of diagonal matrices. From the text: ``We will say that two \(n\)-tuples \(u=(u_j)\) and \(v=(v_j)\) of \(N\times N\) unitary matrices are \(\delta\)-separated if \[ \forall x\in M_{N}\quad \left\|\sum_{j=1}^n u_j x {v_j}^* \right\|_2\leq n(1-\delta)\|x\|_2. \] Equivalently, this means that \[ \left\|\sum_{j=1}^n u_j \otimes \bar {v_j}\right\|\leq n(1-\delta), \] where \(\bar {v_j}\) denotes the complex conjugate of the matrix \(v_j\), and the norm is the operator norm on \(\ell_2^N \otimes \overline{ \ell_2^N} \). Let \(U(N)\subset M_N\) denote the group of unitary matrices. The main result of \S 1 asserts that for any \(0<\delta<1\) there is a constant \(\beta=\beta_\delta>0\) such that for each \(0<\varepsilon<1\), for all sufficiently large integer \(n \) (i.e., \(n\geq n_0(\varepsilon,\delta)\)), for any integer \(N\), there is a \(\delta\)-separated family \(\{u(t)\mid t\in T\}\subset U(N)^n\) of \(\varepsilon\)-quantum expanders such that \(|T|\geq \exp(\beta nN^2)\).'' This is the maximal cardinality allowed by a known upper bound. From the abstract: ``This has applications to the `growth' of certain operator spaces: It implies asymptotically sharp estimates for the growth of the multiplicity of \(M_N\)-spaces needed to represent (up to a constant \(C>1\)) the \(M_N\)-version of the \(n\)-dimensional operator Hilbert space \(OH_n\) as a direct sum of copies of \(M_N\). We show that, when \(C\) is close to 1, this multiplicity grows as \(\exp(\beta n N^2)\) for some constant \(\beta>0\). The main idea is to relate quantum expanders with `smooth' points on the matricial analogue of the Euclidean unit sphere. This generalizes to operator spaces a classical geometric result on \(n\)-dimensional Hilbert spaces (corresponding to \(N=1\)). In an appendix, we give a quick proof of an inequality (related to Hastings's previous work) on random unitary matrices that is crucial for this paper.''