Let $M\_n$ denote the set of all $n\times n$ complex matrices. Let $1\leq p<\infty.$ For $A\in M\_n$ define $\|A\|\_p:=(Tr(A^\*A)^{p/2})^{1/p}$ where $Tr$ denotes the usual trace of a matrix. Then $\|.\|\_p$ is indeed a norm. We write $S\_p^n$ to be the Banach space whose vector space is $M\_n$ and norm is given by $\|.\|\_p.$ Define the Banach-Mazur distance as $d(S\_p^n,S\_q^n):=\inf\{\|T\|\_{S\_p^n\to S\_q^n}\|T^{-1}\|\_{S\_q^n\to S\_p^n}:T:S\_p^n\to S\_q^n\ \text{is an invertible linear map}\}.$ Can anyone tell me how does the constant $d(S\_p^n,S\_q^n)$ behaves as $n\to\infty$?



Banach-Mazur distance between Schatten-$p$ classes
Same order as $d(\ell\_p^n, \ell\_q^n)$. It is $n^{1/p-1/q}$ when they are on the same side of 2, and $n^{(1/2-1/q)\vee (1/p-1/2)}$ for $1\le p<2<q\le\infty$.


See Section 45 of [Tomczak-Jaegermann's book Banach Mazur distances and Finite dimensional Operator ideals](https://sites.ualberta.ca/%7Entj/bm_book/index.html) which she made freely available at the link.


