By Greg Kuperberg

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

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This shows that ρ = ρ˜. ˜ = Vρ . Next let V be any quantum pseudometric on M, let ρ = ρV , and let V ˜ is straightfoward. Conversely, let The inequality V ≤ V Wt = {A ∈ B(L2 (X, μ)) : Mp Vt Mq = 0 ⇒ Mp AMq = 0}, so that Vt ⊆ Wt and Vt is reﬂexive if and only if Vt = Wt . We have V˜t = {A ∈ B(L2 (X, μ)) : Mp Vt+ Mq = 0 for some > 0 ⇒ Mp AMq = 0}, and for any Wt ⊆ V˜t ⊆ Wt+ > 0. Thus if each Vt is reﬂexive then Vt = Wt ⊆ V˜t ⊆ Ws = s>t Vs = Vt s>t for all t, so that Vt = V˜t , and if some Vt is not reﬂexive then Vt Wt ⊆ V˜t ˜ if and only if each Vt is reﬂexive.

The essential computation is ¯ ⊗w A¯ ⊗ v, B = = = A, B v, w P B ∗ AP v, w Av, Bw for A, B ∈ V<δ(P )/2 and v, w ∈ ran(P ). Taking linear combinations shows that the map A¯ ⊗ v → Av is well-deﬁned and isometric on the uncompleted version of the construction, and taking completions then yields an isometry from the completed tensor product onto s<δ(P )/2 Vs (ran(P )) = ran((P )δ(P )/2 ). 5. Quantum tori We formulate a notion of translation invariant quantum pseudometrics on quantum tori. 7 of [35], it is then straightforward to deduce strong structural information about translation invariant quantum pseudometrics.

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. For example, a single photon has two basis polarization states, so a binary string of length n could be encoded as the polarization of an array of n photons.