By Greg Kuperberg

ISBN-10: 0821853414

ISBN-13: 9780821853412

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

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

**Sample text**

Then wk∗ , the weak* closure of the algebraic product taken [t] times, set Vt = V · . . · V where [t] is the greatest integer ≤ t and with the convention that the empty product is M . We call V = {Vt } the quantum graph metric associated to V. We are most interested in the case where M = B(H), V0 = CI, and V is any dual operator system in B(H). 4. Let M ⊆ B(H) be a von Neumann algebra and let V be a dual operator system that is a bimodule over M . Then the quantum graph metric is the smallest quantum metric V on M such that V1 = V.

We can now characterize all quantum metrics on M2 (C). 6. Let 0 ≤ a ≤ b ≤ c ≤ ∞ satisfy where ⎧ CI if ⎪ ⎪ ⎪ ⎨ CI + Cσx if Vt = ⎪ + Cσ if CI + Cσ x y ⎪ ⎪ ⎩ CI + Cσx + Cσy + Cσz if c ≤ a + b and let V = {Vt } 0≤t 0 and it is reﬂexive if and only if b = c. Up to a change of orthonormal basis every quantum pseudometric on M2 (C) is of this form. Proof. It is elementary to check that V is a quantum pseudometric, that it is a quantum metric if and only if a > 0, and that CI and CI + Cσx are reﬂexive but CI + Cσx + Cσy is not, so that V is reﬂexive if and only if b = c.

7. 7. 9. Comparing this with the deﬁnition of R ˜ t ⊆ Rt+ for all > 0. , R ρ = ρ˜. CHAPTER 3 Examples Our new deﬁnition of quantum metrics supports a wide variety of examples. This is also true of the previously proposed deﬁnitions mentioned in the introduction, and indeed the main classes of examples in the diﬀerent cases substantially overlap. If anything, a complaint could be made that the previous deﬁnitions are too broad. 16). The metric aspect of error correcting quantum codes provided the original motivation behind our new approach.

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