Abstractvet S 1 @the htten!von xeumnn tre lssA denote the fnh spe of ll ompt liner opertors T : 2 → 2 whose nuler norm T S 1 = ∞ j=1 σ j (T ) is (niteD where {σ j (T )} ∞ j=1 re the singulr vlues of T F e prove tht for ritrrily lrge n ∈ N there exists suset C ⊆ S 1 with |C| = n tht nnot e emedded with iEvipshitz distortion O(1) into ny n o(1) Edimensionl liner suspe of S 1 F C is not even O(1)Evipshitz quotient of ny suset of ny n o(1) Edimensionl liner suspe of S 1 F husD S 1 does not dmit dimension redution result á l tohnson nd vindenstruss @IWVRAD whih omplements the work of rrrowD wontnro nd hort @PHIIA on the limittions of quntum dimension redution under the ssumption tht the emedding into low dimensions is quntum hnnelF uh sttement ws previously known with S 1 repled y the fnh spe 1 of solutely summle sequenes vi the work of frinkmn nd ghrikr @PHHQAF sn ftD the ove set C n e tken to e the sme set s the one tht frinkmn nd ghrikr onsideredD viewed s olletion of digonl mtries in S 1 F he hllenge is to demonstrte tht C nnot e fithfully relized in n ritrry lowEdimensionl suspe of S 1 D while frinkmn nd ghrikr otined suh n ssertion only for suspes of S 1 tht onsist of digonl opertors @iFeFD suspes of 1 AF e estlish this y proving tht the wrkov PEonvexity onstnt of ny (nite dimensionl liner suspe X of S 1 is t most universl onstnt multiple of log dim(X)F
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