A new approach to quantum metrics. Nik Weaver. (joint work with Greg Kuperberg, in progress)

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1 A new approach to quantum metrics Nik Weaver (joint work with Greg Kuperberg, in progress)

2 Definition. A dual operator system is a linear subspace V of B(H) such that I V A V implies A V V is weak*-closed. Definition. A W*-filtration of B(H) is a family of subspaces V r, 0 r <, such that each V r is a dual operator system V r V s V r+s for all r, s 0 V r = s>r V s for all r 0. Remarks. 1. r s implies V r V s. 2. V 0 is a von Neumann algebra. 3. Every V r is a V 0 -V 0 -bimodule.

3 Definition. Let M B(H) be a von Neumann algebra. A (concrete) quantum metric on M is a W*-filtration of B(H) such that V 0 = M. Hint: d(x, x) = 0 I V r d(x, y) = d(y, x) V r = V r d(x, z) d(x, y) + d(y, z) V r V s V r+s

4 We call d(a) = inf{r : A V r } (possibly infinite) the displacement of A B(H), and we write B(H) r = {A B(H) : d(a) r} = V r.

5 Example. H = l 2 (X) and M = l (X) B(H). Let ρ be a metric on X and define B(H) r = {A B(H) : d(x, y) > r Ae x, e y = 0}. Equivalently: for all x X, Ae x is supported on ball(x; r) ( displacement at most r ). Theorem. The above is a quantum metric on M. Every quantum metric on M is of this form.

6 Definition. Let (X, µ) be a σ-finite measure space and let Ω be the positive measure subsets of X modulo null sets. A measurable pseudometric on (X, µ) is a function ρ : Ω 2 [0, ) such that ρ(s, S) = 0 ρ(s, T) = ρ(t, S) ρ( S n, T) = inf ρ(s n, T) ρ(s, U) sup T T ρ(s, T ) + ρ(t, U) for all S, T, U, S n Ω. It is a measurable metric if for all ɛ > 0 and all disjoint S and T there exist S S and T T such that µ(s S ), µ(t T ) ɛ and ρ(s, T ) > 0. Let H be separable (not essential) and let M B(H) be an abelian von Neumann algebra. Say M = L (X, µ) and let ρ be a measurable metric on (X, µ). Define B(H) r = {A B(H) : ρ(s, T) > r M χs AM χt = 0}. Theorem. The above is a quantum metric on M. Every quantum metric on M is of this form.

7 Example. M = M 2 (C). Pauli spin matrices σ x = [ ] σ y = [ ] σ z = [ 0 ] i i 0 Choose 0 < a b c such that c a + b and define B(H) =0 = span{i} B(H) a = span{i, σ x } B(H) b = span{i, σ x, σ y } B(H) c = span{i, σ x, σ y, σ z } = M 2 (C) Proposition. The above is a quantum metric on M 2 (C), and up to a change of basis every quantum metric on M 2 (C) is of this form.

8 Example. Quantum Hamming distance. H = C 2n, M = n 1 M 2(C) = B(H) B(H) k = span{a 1 A n : at most k factors are not I 2 } standard basis vector in H a string of n 0 s and 1 s Intuition: operators in B(H) k do not destroy more than k bits (really qubits ) of information. Classically, if we restrict to a set of sequences any two of which differ on more than 2k bits, the original message can be recovered (error correction). In the quantum case we restrict to a subspace of H such that any unit vector can be recovered after corruption by operators in B(H) k. (I.e., a projection P B(H) such that PAP is proportional to P for all A B(H) k.)

9 Example. Quantum tori. Fix h [0, 2π). On H = l 2 (Z 2 ) define Ue m,n = exp( i hn/2)e m+1,n V e m,n = exp(i hm/2)e m,n+1 (so that UV = e i h V U). (standard form) Let ρ be a translation-invariant metric on T 2 (T = R/2πZ) and define B(H) r = span wk {U k V l M exp(i(sm+tn)) : k, l Z and ρ((0, 0), (s, t)) r}. So B(H) =0 = W (U, V ) and B(H) =0 = M = W (Ũ, Ṽ ) where Ũe m,n = exp(i hn/2)e m+1,n Ṽ e m,n = exp( i hm/2)e m,n+1. Translation-invariant: B(H) r is stable under conjugation by M exp(i(sm+tn)) for all s, t. Theorem. The above is a translation-invariant quantum metric on W (Ũ, Ṽ ). Every translation-invariant quantum metric on W (Ũ, Ṽ ) is of this form.

10 Intrinsic characterization #1: distances between projections For projections p, q M define ρ(p, q) = sup{r : d(a) < r paq = 0}. In general cannot recover the quantum metric from this. We need matrix projections. So, for projections p, q M B(l 2 ) define ρ(p, q) = sup{r : d(a) < r p(a I)q = 0}. We can recover the W*-filtration on B(H) from the distance function on projections in M B(l 2 ) via d(a) = inf{r : ρ(p, q) r p(a I)q = 0}.

11 Definition. Let M be a von Neumann algebra and let P be the set of projections in M B(l 2 ). An (intrinsic) quantum metric on M is a function ρ : P P [0, ] such that ρ(p, 0) = ; pq 0 implies ρ(p, q) = 0; ρ(p, q) = ρ(q, p); ρ(p q, r) = min(ρ(p, r), ρ(q, r)); ρ(p, r) ρ(p, q) + ρ(q, r); for any b B(l 2 ) we have ρ(p, q ) = ρ(p, q) where p is the range projection of (1 M b )p and q is the range projection of (1 M b)q; if p λ p and q λ q weak* then ρ(p, q) lim sup ρ(p λ, q λ ) there are nets (p κ ) and (q κ ) such that p κ p, q κ 1 p, and ρ(p κ, q κ ) > 0 for all κ for all p, q, r, p λ, q λ P, where ρ(p, q) = sup{ρ(p, q ) : qq 0}. Theorem. There is a natural bijection between the intrinsic quantum metrics on M and the concrete quantum metrics on M.

12 Intrinsic characterization #2: Lipschitz numbers Distances between projections yield a Lipschitz number for any a (M B(l 2 )) sa, and conversely: { } β α L(a) = sup ρ(p (,α] (a), P [β, ) (a)) : α < β R ρ(p, q) = sup{α : a (M B(l 2 )) sa, L(a) = 1, p P (,0] (a), q P [α, ) (a)}.

13 Definition. A Lipschitz gauge for a von Neumann algebra M is a map L : (M B(l 2 )) sa [0, ] such that L(a + I) = L(a) L(αa) = α L(a) L ( a λ ) sup L(a λ ) L(uau ) = L(a) x takes the α spectral subspace of {b : L(b) 1, ran(x 1 p) ker(b)} onto the α spectral subspace of {b : L(b) 1, ran(p) ker(b)} a λ a implies L(a) lim inf L(a λ ) for all self-adjoint a, a λ M B(l 2 ) with sup a λ <, all α R, all isometries u I B(l 2 ), all projections p M B(l 2 ), and all invertibles x I B(l 2 ). Here a λ is the spectral join. Theorem. There is a natural bijection between the Lipschitz gauges for M and the quantum metrics on M. Note: L is generally not a seminorm.

14 There is an alternative notion of Lipschitz gauge. For A M define { } [A, X] L c (A) = sup : X B(H), X = 1 d(x) (commutator Lipschitz number of A) this is a seminorm it satisfies L c (AB) A L c (B) + L c (A) B L c (A) = L(A) in the abelian case (not obvious) but no obvious axiomatization