Functions of self-adjoint operators in ideals of compact operators: applications to the entanglement entropy

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1 Functions of self-adjoint operators in ideals of compact operators: applications to the entanglement entropy Alexander Sobolev University College London University of Kent A Sobolev (UCL) Functions of self-adjoint operators University of Kent / 17

2 Plan Introduction Results Applications A Sobolev (UCL) Functions of self-adjoint operators University of Kent / 17

3 Introduction We discuss two groups of estimates for functions of self-adjoint operators. 1. Let A = A in a Hilbert space H, and let P be an orthogonal projection. We study W (A, P; f ) = Pf (PAP)P Pf (A)P with suitable functions f : R C. Aim: to obtain estimates for the (quasi-)norms of W (A, P; f ) in various ideals of compact operators. In particular, in the Schatten-von Neumann ideals S p with the (quasi-)norm [ ] 1 A p = s k (A) p p, 0 < p <. k=1 Here s k (A) = λ k ( A ), k = 1, 2,..., are singular values (s-values) of the operator A. Here A = (A A) 1/2. If p 1(resp. p < 1), then the above formula defines a norm (resp. quasi-norm). A Sobolev (UCL) Functions of self-adjoint operators University of Kent / 17

4 Laptev-Safarov 1996: if f W 2, and PA S 2, then tr ( Pf (PAP)P Pf (A)P ) 1 2 f L (I P)AP 2 2. Our aim is to derive similar bounds with other (quasi-)norms and functions with singularities. Typical example: f (t) = (t a) γ, γ > 0, a fixed. A Sobolev (UCL) Functions of self-adjoint operators University of Kent / 17

5 Laptev-Safarov 1996: if f W 2, and PA S 2, then tr ( Pf (PAP)P Pf (A)P ) 1 2 f L (I P)AP 2 2. Our aim is to derive similar bounds with other (quasi-)norms and functions with singularities. Typical example: f (t) = (t a) γ, γ > 0, a fixed. 2. Let A = A on H 2, B = B on H 1, and let J : H 1 H 2 be a bounded operator. Interested in the bounds for f (A)J Jf (B) in (quasi-)normed ideals of compact operators. A Sobolev (UCL) Functions of self-adjoint operators University of Kent / 17

6 Laptev-Safarov 1996: if f W 2, and PA S 2, then tr ( Pf (PAP)P Pf (A)P ) 1 2 f L (I P)AP 2 2. Our aim is to derive similar bounds with other (quasi-)norms and functions with singularities. Typical example: f (t) = (t a) γ, γ > 0, a fixed. 2. Let A = A on H 2, B = B on H 1, and let J : H 1 H 2 be a bounded operator. Interested in the bounds for f (A)J Jf (B) in (quasi-)normed ideals of compact operators. Note that for J = P, B = PAP we have Pf (A)P Pf (PAP)P = P ( f (A)J Jf (B) ) P. A Sobolev (UCL) Functions of self-adjoint operators University of Kent / 17

7 Quasi-normed ideals Let S S be a two-sided ideal of compact operators. The functional S is a symmetric quasi-norm if 1. A S > 0, A 0; 2. za S = z A S ; 3. A 1 + A 2 S κ( A 1 S + A 2 S ), κ 1; 4. XAY S X Y A S ; 5. For every one-dimensional operator A: A S = A. A Sobolev (UCL) Functions of self-adjoint operators University of Kent / 17

8 Quasi-normed ideals Let S S be a two-sided ideal of compact operators. The functional S is a symmetric quasi-norm if 1. A S > 0, A 0; 2. za S = z A S ; 3. A 1 + A 2 S κ( A 1 S + A 2 S ), κ 1; 4. XAY S X Y A S ; 5. For every one-dimensional operator A: A S = A. Useful observations: 1. A S depends only on the s-values s k (A), k = 1, 2,...! 2. Any quasi-norm has an equivalent q-norm with a suitable q (0, 1], i.e. A 1 + A 2 q S A 1 q S + A 2 q S. In fact, q is found from the equation κ = 2 q 1 1. For S = S p, p (0, 1): Yu. Rotfeld A Sobolev (UCL) Functions of self-adjoint operators University of Kent / 17

9 Survey 1. J = I, S = S 1 : f (A) f (B) 1 C A B 1, C = C(f ). (1) V. Peller 1985,90: f B 1 1 implies (1), and (1) implies f B1 11 locally. A Sobolev (UCL) Functions of self-adjoint operators University of Kent / 17

10 Survey 1. J = I, S = S 1 : f (A) f (B) 1 C A B 1, C = C(f ). (1) V. Peller 1985,90: f B 1 1 implies (1), and (1) implies f B1 11 locally. 2. D. Potapov-F. Sukochev 2011: If f Lip(R) and p > 1, then where C(f ) = C 1 f Lip. f (A) f (B) p C(f ) A B p. (2) A Sobolev (UCL) Functions of self-adjoint operators University of Kent / 17

11 Survey 1. J = I, S = S 1 : f (A) f (B) 1 C A B 1, C = C(f ). (1) V. Peller 1985,90: f B 1 1 implies (1), and (1) implies f B1 11 locally. 2. D. Potapov-F. Sukochev 2011: If f Lip(R) and p > 1, then where C(f ) = C 1 f Lip. f (A) f (B) p C(f ) A B p. (2) 3. The case p (0, 1) was studied by V. Peller 1987 for unitary A, B: f B p 1/p implies (2), and (2) implies f Bpp 1/p. A Sobolev (UCL) Functions of self-adjoint operators University of Kent / 17

12 4. Let S be a normed ideal with the majorization property, i.e. if A S, B S, and n s k (B) k=1 n s k (A), n = 1, 2,..., k=1 then B S and B S A S. M. Birman- L. Koplienko-M. Solomyak 1975: if A 0, B 0, then for any γ (0, 1): A γ B γ S A B γ S. A Sobolev (UCL) Functions of self-adjoint operators University of Kent / 17

13 5. A. Aleksandrov-V. Peller 2010: if f Λ γ = B γ, γ (0, 1) (Zygmund class), then f (A)J Jf (B) 1 γ S C(f ) J 1 γ γ AJ JB S, A Sobolev (UCL) Functions of self-adjoint operators University of Kent / 17

14 5. A. Aleksandrov-V. Peller 2010: if f Λ γ = B γ, γ (0, 1) (Zygmund class), then f (A)J Jf (B) 1 γ S C(f ) J 1 γ γ AJ JB S, under the condition that the Boyd index β(s) is < 1! For example, β(s p ) = p 1. Define the operator [A] d = d k=1 A on d k=1 H. If A S then [A] d S. Denote by β d (S) the quasi-norm of the transformer T [T ] d in S. Then the Boyd index of S is defined to be β(s) = lim d log β d (S). log d A Sobolev (UCL) Functions of self-adjoint operators University of Kent / 17

15 Results Assumptions. Operators: A = A on H 2, B = B on H 1, J : H 1 H 2. Define the form V [f, g] = (Jf, Ag) (JBf, g), f D(B) H 1, g D(A) H 2, and suppose that V [f, g] C f g, so that V defines a bounded operator from H 1 to H 2. Functions: f C n (R \ {0}), and f (k) (t) C t β k χ 1 (t), k = 0, 1,..., n, for some β > 0. Here χ 1 is the indicator of the interval ( 1, 1). A Sobolev (UCL) Functions of self-adjoint operators University of Kent / 17

16 Theorem Let S be a q-normed ideal with (n 1) 1 < q 1. Suppose that V is such that V γ S, with some γ (0, 1], γ < β. Then f (A)J Jf (B) S C J 1 γ V γ S. A Sobolev (UCL) Functions of self-adjoint operators University of Kent / 17

17 Theorem Let S be a q-normed ideal with (n 1) 1 < q 1. Suppose that V is such that V γ S, with some γ (0, 1], γ < β. Then f (A)J Jf (B) S C J 1 γ V γ S. Assume that S = S p, β > 1, J = I. Is it possible to use one of Peller s earlier results to get the bound f (A) f (B) p C V p? No! Indeed, consider f (t) = t β ζ(t), ζ C 0 (R), ζ(t) = 1, t < 1. Then f Bp,q, ν 0 < p, 0 < q < iff ν < β + p 1. For sufficiently small p, we have f / B,p. 1/p On the other hand, f Bp,p 1/p. A Sobolev (UCL) Functions of self-adjoint operators University of Kent / 17

18 Remark. If g (k) (t) C t β k χ R (t), then g(a)j Jg(B) S CR β γ J 1 γ V γ S. Apply the Theorem to f (t) = R β g(rt) with A = R 1 A, B = R 1 B. A Sobolev (UCL) Functions of self-adjoint operators University of Kent / 17

19 Remark. If g (k) (t) C t β k χ R (t), then g(a)j Jg(B) S CR β γ J 1 γ V γ S. Apply the Theorem to f (t) = R β g(rt) with A = R 1 A, B = R 1 B. For J = P and B = PAP we have V = (I P)AP. Theorem g(a)p Pg(PAP) S CR β γ (I P)AP γ S. A Sobolev (UCL) Functions of self-adjoint operators University of Kent / 17

20 Quasi-analytic extension Let f C 2 (R \ {0}) C 0 (R), and let f C 2 (C \ {0}) C 0 (C) be a function such that f (x, 0) = f (x), and 1 z z f (x, y) dxdy <, z = x + iy, and z f (x, 0) = 0, for all x 0. Then f is called a quasi-analytic extension of f. A Sobolev (UCL) Functions of self-adjoint operators University of Kent / 17

21 Quasi-analytic extension Let f C 2 (R \ {0}) C 0 (R), and let f C 2 (C \ {0}) C 0 (C) be a function such that f (x, 0) = f (x), and 1 z z f (x, y) dxdy <, z = x + iy, and z f (x, 0) = 0, for all x 0. Then f is called a quasi-analytic extension of f. Proposition (Dyn kin, Helffer-Sjöstrand) Let A be a self-adjoint operator, and let f be a quasi-analytic extension of f. Then f (A) = 1 π z f (x, y)(a x iy) 1 dxdy. A Sobolev (UCL) Functions of self-adjoint operators University of Kent / 17

22 Application Let Λ, Ω R d, be domains, and let χ Λ, χ Ω be their characteristic functions. We are interested in the operators of the form T α (a) = χ αλ a(d)χ Ω (D)χ αλ, D = i. a = a C 0 (R d ), α >> 1. We interpret T α as a pseudo-differential operator (ΨDO) with discontinuous symbol. Straightforward: T α S 1 if Λ, Ω < : T α S1 = 1 (2π) d χ αλ (x)dx χ Ω (ξ)dξ = αd (2π) d Λ Ω. Natural asymptotic problem: Asymptotics of tr g(t α ), α, with a suitable function g : g(0) = 0. A Sobolev (UCL) Functions of self-adjoint operators University of Kent / 17

23 Two-term asymptotics Conjecture (H. Widom, 1982): tr g(t α ) = α d W 0 + α d 1 log α W 1 + o(α d 1 log α), α, ( ) 1 d W 0 = W 0 (g(a)) = g(a(ξ))dxdξ, 2π Ω Λ ( ) 1 d 1 1 W 1 = 2π 4π 2 n x n ξ U(a(ξ); g)ds x ds ξ. Ω Λ U(a; g) = 1 0 g(ta) tg(a) dt. t(1 t) A Sobolev (UCL) Functions of self-adjoint operators University of Kent / 17

24 Main result Theorem (A.S. 2009, 2013) Let d 2. Suppose that Λ, Ω are bounded and Both Λ and Ω are Lipschitz. Λ is piece-wise C 1, and Ω is piece-wise C 3. Then the Conjecture holds for any g C (R) s.t. g(0) = 0. A Sobolev (UCL) Functions of self-adjoint operators University of Kent / 17

25 Main result Theorem (A.S. 2009, 2013) Let d 2. Suppose that Λ, Ω are bounded and Both Λ and Ω are Lipschitz. Λ is piece-wise C 1, and Ω is piece-wise C 3. Then the Conjecture holds for any g C (R) s.t. g(0) = 0. Theorem (A.S. 2014) Let Λ, Ω be as above. Suppose that g (k) (t) C t β k χ 1 (t), k = 0, 1, 2,..., with some β > 0. Then the Conjecture holds. Use the previous results with P = χ αλ and A = a(d)χ Ω (D). A Sobolev (UCL) Functions of self-adjoint operators University of Kent / 17

26 Crucial ingredient: Theorem Let α 2. Then for any q (0, 1]: (I χ αλ )a(d)χ Ω (D)χ αλ q S q C q α d 1 log α. Take a function ζ C 0 (R) such that ζ(t) = 1, t < 1, and ζ(t) = 0, t 2. Denote ζ δ (t) = ζ(tδ 1 ). Split g: g = g 1 + g 2 where g 1 = gζ δ, g 2 = g(1 ζ δ ) C. Then for any 0 < q < β, q 1: g 1 (T α ) χ αλ g 1 (a(d)χ Ω (D))χ αλ 1 Cδ β q α d 1 log α. For g 2 one can use the already known result. Notice that W 1 (g 1 ) C(δ β + δ). The required result follows. A Sobolev (UCL) Functions of self-adjoint operators University of Kent / 17

27 Example: the entanglement entropy Set a = 1, so T α = χ αλ χ Ω (D)χ αλ. Free Fermions at zero temperature : Ω = {ξ : E(ξ) < E F }. Entanglement entropy (von Neumann entropy) between Fermions inside αλ and outside αλ, α : We see that E α = tr h(t α ), h(t) = t log t (1 t) log(1 t), t [0, 1]. E α = αd 1 log α 12(2π) d 1 Ω Λ n x n ξ dxdξ + o(α d 1 log α). A Sobolev (UCL) Functions of self-adjoint operators University of Kent / 17

28 Example: the entanglement entropy Set a = 1, so T α = χ αλ χ Ω (D)χ αλ. Free Fermions at zero temperature : Ω = {ξ : E(ξ) < E F }. Entanglement entropy (von Neumann entropy) between Fermions inside αλ and outside αλ, α : We see that E α = tr h(t α ), h(t) = t log t (1 t) log(1 t), t [0, 1]. E α = αd 1 log α 12(2π) d 1 The Rényi entropy: Ω Λ n x n ξ dxdξ + o(α d 1 log α). η β (t) = 1 1 β log( t β + (1 t) β), β > 0. A Sobolev (UCL) Functions of self-adjoint operators University of Kent / 17