By Greg Kuperberg

ISBN-10: 0821853414

ISBN-13: 9780821853412

Quantity 215, quantity 1010 (first of five numbers).

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**Additional info for A von Neumann algebra approach to quantum metrics. Quantum relations**

**Example text**

0 We call Ak,l the (k, l) Fourier term of A. (c) For k, l ∈ N deﬁne Sk,l (A) = Ak ,l |k |≤k,|l |≤l and for N ∈ N deﬁne σN (A) = 1 N2 Sk,l (A). 0≤k,l≤N −1 In the L2 (T2 ) picture the operator Mei(mx+ny) on l2 (Z2 ) becomes translation by (−x, −y), so that θx,y is conjugation by a translation. The integral used to deﬁne Ak,l can be understood in a weak sense: for any vec2π 2π tors v, w ∈ l2 (Z2 ) we take Ak,l w, v to be 4π1 2 0 0 e−i(kx+ly) θx,y (A)w, v dxdy. In particular, if w = em,n and v = em ,n then we have Ak,l em,n , em ,n = Aem,n , em ,n 0 if m = m + k and n = n + l otherwise.

The reverse implication follows from the fact that M ⊗N = M ⊗F N since M and N are von Neumann algebras [11]. 6. INTRINSIC CHARACTERIZATION 33 The deﬁnition of the metric product can be varied. For instance, an lp product ˜ = {W ˜ t } of B(H ⊗K) (1 ≤ p < ∞) could be deﬁned as the smallest W*-ﬁltration W satisfying ˜ (sp +tp )1/p Vs ⊗F Wt ⊆ W for all s, t ≥ 0. 42 (b) and (c); the ﬁrst holds by essentially the same proof given for metric products, and the second follows from the fact that the lp product W*ﬁltration is contained in the metric product W*-ﬁltration.

Furthermore, we must have c ≤ a + b because Va Vb = (CI + Cσx )(CI + Cσx + Cσy ) ⊆ Va+b and iσx σy = σz , so that Vc ⊆ Va+b . 4. Quantum Hamming distance n Fix a natural number n and let H = C2 ∼ = C2 ⊗ · · · ⊗ C2 . If {e0 , e1 } is the 2 standard orthonormal basis of C then {ei1 ⊗ · · · ⊗ ein : each ik = 0 or 1} is an orthonormal basis for H. These basis vectors correspond to binary strings of length n. Thus the information represented by such a string can be encoded in an appropriate physical system as the state modelled by the corresponding basis vector.