Noncommutative Uncertainty Principle

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1 Noncommutative Uncertainty Principle Zhengwei Liu (joint with Chunlan Jiang and Jinsong Wu) Vanderbilt University The 12th East Coast Operator Algebras Symposium, Oct 12, 2014 Z. Liu (Vanderbilt) Noncommutative Uncertainty Principle Oct, / 23

2 Classical Uncertainty Principles Heisenberg uncertainty principle x p 2. Hirschman uncertainty principle: H s ( f 2 ) + H s ( ˆf 2 ) 0. Donoho-Stark uncertainty principle: supp(f ) supp(ˆf ) G. Z. Liu (Vanderbilt) Noncommutative Uncertainty Principle Oct, / 23

3 Heisenberg Uncertainty Principle Heisenberg [1927] x position; p momentum. A mathematic formulation: where ˆf (ξ) = ( x 2 f (x) 2 dx)( f (x)e 2πixξ dx. x p 2 ξ 2 ˆf (ξ) 2 dξ) f π 2, Z. Liu (Vanderbilt) Noncommutative Uncertainty Principle Oct, / 23

4 Hirschman-Beckner Uncertainty Principle Hirschman [1957], for real number group R and f 2 = 1 Shannon Entropy: H s ( f 2 ) = Hirschman s Conjecture: H s ( f 2 ) + H s ( ˆf 2 ) 0. (1) f 2 log f 2 dx. H s ( f 2 ) + H s ( ˆf 2 ) log e 2. Beckner [1975], the conjecture is true. Özaydin and Przebinda [2000], locally compact abelian group with an open compact subgroup for inequality (1). Z. Liu (Vanderbilt) Noncommutative Uncertainty Principle Oct, / 23

5 Donoho-Stark Uncertainty Principle Donoho-Stark [1989], for cyclic group G, supp(f ) supp(ˆf ) G. f, a function on G ˆf, the Fourier transform of f supp(f ) = #{x G f (x) 0}. K. Smith [1990], finite abelian group Özaydin and Przebinda [2000], locally compact abelian group with an open compact subgroup Z. Liu (Vanderbilt) Noncommutative Uncertainty Principle Oct, / 23

6 Noncommutative Uncertainty Principle Some recent results: [D. Goldstein, R. Guralnick, and I. Isaacs, 2005] finite groups [J. Crann and M. Kalantar, 2014] Kac algebras (C Hopf algebras or quantum groups in von Neumann algebraic setting) Subfactor theory naturally provides a Fourier transform over a pair of von-neumann algebras and a measurement. W. Szymanski [1994], irreducible depth-2 subfactors Kac algebras. We are going to talk about the uncertainty principle for finite index subfactors. Haussdorff-Young s inequality Young s inequality (new for Kac algebras) Uncertainty principles Minimizers (new for finite non-abelian groups) Z. Liu (Vanderbilt) Noncommutative Uncertainty Principle Oct, / 23

7 Subfactors Theorem (Jones83) {[M : N ] := dim N (L 2 (M))} = {4 cos 2 π, n = 3, 4, } [4, ]. n N M, a subfactor (of type II 1 ) with finite index Jones projection e 1 B(L 2 (M)) : L 2 (M) L 2 (N ) Basic construction M 1 =< M, e 1 > Jones tower e 1 e 2 e 3 N M M 1 M 2 Standard invariant N N N M N M 1 N M 2 M M M M 1 M M 2 Z. Liu (Vanderbilt) Noncommutative Uncertainty Principle Oct, / 23

8 Axioms of standard invariants Ocneanu s paragroups [1988] Popa s standard λ-lattices [1995] Jones subfactor planar algebras [1999] Z. Liu (Vanderbilt) Noncommutative Uncertainty Principle Oct, / 23

9 Group Subfactors Example When M = N G, for an outer action a finite abelian group G, we have [M : N ] = G, the Jones tower N N G N G Ĝ N G Ĝ G and the standard invariant C C LĜ L(Ĝ G) C C LG The 2-box spaces of the standard invariant (M M 2 and N M 1 ) recover the group G and its dual Ĝ! Moreover N M 2 provides a natural algebra to consider G and Ĝ simultaneously! Z. Liu (Vanderbilt) Noncommutative Uncertainty Principle Oct, / 23

10 A Pair of C Algebras (N M 1, M M 2 ) Measure: the (unnormalized) trace of M n p-norm: x p = tr( x p ) 1 p, p 1 Fourier transform (Ocneanu): 1-click rotation (for paragroups) Definition Fourier transform F : N M 1 M M 2, F(x) = [M : N ] 3 2 E N M (xe 2e 1 ) for x N M 1, where EM N is the trace preserving condition expectation from N to M. In subfactor planar algebras, the fourier transform is a 1-click rotation, x. Z. Liu (Vanderbilt) Noncommutative Uncertainty Principle Oct, / 23

11 Main Theorem (Haussdorf-Young s inequality) For an irreducible subfactor N M with finite index, take δ = [M : N ]. For any x, y N M 1, we have Theorem (Jiang-L-Wu) F(x) p where 2 p, 1 p + 1 q = 1. ( ) p x q (2) δ Extremal: x is called extremal, if the equality of (2) holds. All positive operators are extremal. Our proof of Haussdorf-Young s inequality also works for Popa s λ-lattice, modular tensor category etc. Z. Liu (Vanderbilt) Noncommutative Uncertainty Principle Oct, / 23

12 Main Theorem (Young s inequality) Convolution: x y = F(F 1 (x)f 1 (y)) Theorem (Jiang-L-Wu) x y r x p y q. δ where 1 p, q, r, 1 p + 1 q = 1 r + 1. Z. Liu (Vanderbilt) Noncommutative Uncertainty Principle Oct, / 23

13 Main Theorem (Hirschman-Beckner uncertainty principle) Theorem (Jiang-L-Wu) H( x 2 ) + H( F(x) 2 ) x 2 (2 log δ 4 log x 2 ), where H( x 2 ) = tr 2 ( x 2 log x 2 ) is the von Neumann entropy of x 2. A quick proof. Take the derivative of Hausdorff-Young s inequality at p = 2. Z. Liu (Vanderbilt) Noncommutative Uncertainty Principle Oct, / 23

14 Main Theorem (Donoho-Stark uncertainty principle) Theorem (Jiang-L-Wu) S(x)S(F(x)) δ 2, where S(x) is the trace of range projection of x, x 0. A quick proof. Take log on both side and apply the former Theorem at x 2 = δ 1 2. The inequality reduces to the fact that log S(x) log H( x 2 ) which follows from the concavity of t log t. Z. Liu (Vanderbilt) Noncommutative Uncertainty Principle Oct, / 23

15 Minimizer of classical Uncertainty Principles Suppose G is a finite abelian group, and H < G. Translation: f (x) f (x + y) Modulation: f (x) χ(x)f (x), where χ is a character of G Indicator function of H: h H h. Theorem (Özaydin and Przebinda) The follows are equivalent (1) H( x 2 ) + H( F(x) 2 ) = x 2 (2 log δ 4 log x 2 ) (2) S(x)S(F(x)) = δ 2 (3) x = c h H χ(h)hg, c 0, H < G, g G, χ Ĝ Remark The generalization of these concepts is not obvious in the non-commutative world. That makes extra difficulties to characterize minimizers of uncertainty principles for subfactors. Z. Liu (Vanderbilt) Noncommutative Uncertainty Principle Oct, / 23

16 Main Theorem (minimizers) A nonzero element x is called an extremal bi-partial isometry if x and F(x) are multiplies of extremal partial isometries. A projection p is called a biprojection if F(p) is a multiple of a projection. Biprojections are introduced by Bisch [1994] and studies by Bisch and Jones [1997] from planar algebra perspective. Biprojections generalize indicator functions of subgroups. We introduced a new notion, a bi-shift of a biprojection, which generalizes a translation and a modulation of the indicator function of a subgroup. Theorem (Jiang-L-Wu) The following statements are equivalent, (1) S(x)S(F(x)) = δ 2 ; (2) H( x 2 ) + H( F(x) 2 ) = x 2 (2 log δ 4 log x 2 ); (3) x is an extremal bi-partial isometry; (3 ) x is a partial isometry and F 1 (x) is extremal; (4) x is a bi-shift of a biprojection. Z. Liu (Vanderbilt) Noncommutative Uncertainty Principle Oct, / 23

17 Remarks on Minimizers To prove the theorem, we find the following key relation of a norm-1 extremal bi-partial isometry w (in planar algebras) based on Young s inequality : (w w)(w w ) = w 2 2 (w w) (w w ) δ i.e. $ $ w w $ $ w w = w 2 2 δ $ $ w w $ $ w w, where w = F 2 (w). Moreover ((w ) w)(w w ) is a biprojection. The relation is obtained by planar algebra methods. Up to now, we cannot find any other method to prove the above relation. Z. Liu (Vanderbilt) Noncommutative Uncertainty Principle Oct, / 23

18 Main results (uniqueness) Donoho and Stark 1989 noticed that the minimizer of uncertainty principles is uniquely determined by the supports of itself and its Fourier transform. This kind of result is very useful for signal recovery. It is further developed by Candes, Romberg and Tao We are considering non-commutative algebras. Both an element and its Fourier transform have two supports. Theorem (Jiang-L-Wu) The minimizer of uncertainty principles is uniquely determined by the range projections of itself and its Fourier transform. Z. Liu (Vanderbilt) Noncommutative Uncertainty Principle Oct, / 23

19 Minimizers of Groups Proposition (Jiang-L-Wu) Suppose G is a finite (non-abelian) group. Take a subgroup H, a one dimension representation χ of H, an element g G, a non-zero constant c C. Then x = c h H χ(h)hg is a bi-shift of a biprojection. Conversely any bi-shift of a biprojection is of this form. Remark Note that χ is the pull back of a character of H/[H, H], where [H, H] is the commutator subgroup. Z. Liu (Vanderbilt) Noncommutative Uncertainty Principle Oct, / 23

20 Main results (for the pair (N M k 1, M M k )) Definition (Ocneanu) For any x N M k 1, the Fourier transform F : N M k 1 M M k is F(x) = [M : N] n+1 2 E N M (xe ne n 1 e 1 ). Theorem (Jiang-L-Wu) ( 1 F(x) p δ 0 ) 1 2 p x q, 2 p <, 1 p + 1 q = 1; n 1 k=0 S(F k (x)) δ n ; n 1 H( F k (x) 2 ) x 2 (n log δ 2n log x 2 ). k=0 Z. Liu (Vanderbilt) Noncommutative Uncertainty Principle Oct, / 23

21 Tao s uncertainty principle Theorem (Tao, 2005) For the abelian group Z p, p prime, supp(f ) + supp(ˆf ) p + 1. Question Suppose P is a finite depth subfactor planar algebra with index p for some prime number p. Is the following inequality true for a non-zero 2-box x? S(x) + S(F(x)) p + 1 (3) Z. Liu (Vanderbilt) Noncommutative Uncertainty Principle Oct, / 23

22 Remarks Remark Our method is based on subfactor planar algebras. We combined its categorial and analytic properties. The method also works for (C spherical) planar algebras with multiple kinds of regions and strings. Further Researches: Applications Infinite index n-box spaces Z. Liu (Vanderbilt) Noncommutative Uncertainty Principle Oct, / 23

23 Thank you! Z. Liu (Vanderbilt) Noncommutative Uncertainty Principle Oct, / 23

24 W. Beckner, Inequalities in fourier analysis, Annals of Math. 102(1975), D. Bisch, A note on intermediate subfactors, Pacific J. Math. 163(1994), D. Bisch and V. Jones, Algebras associated to intermediate subfactors, Invent. Math. 128(1997), A. Connes, On the spatial theory of von Neumann algebras. J. Funct. Anal. 35 (1980), no. 2, 153C164. J. Crann and M. Kalantar, An uncertainty principle for unimodular quantum groups ArXiv: v1 [math-ph], E. Candes, J. Romberg and T. Tao, Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information IEEE Transactions on Information Theory. 52(2006), A. van Daele The Fourier transform in quantum group theory ArXiv:math/ v3. Z. Liu (Vanderbilt) Noncommutative Uncertainty Principle Oct, / 23

25 A. Dembo, T. M. Cover and J. A. Thomas, Information theoritc inequalities, IEEE Transactions on Information Theory, 37(1991), D.L. Donoho and P.B. Stark, Uncertainty principles and signal recovery, SIAM J. Appl. Math. 49(1989), D. Goldstein, R. Guralnick, and I. Isaacs, Inequalities for finite group permutation modules, Trans. Amer. Math. Soc. 357(2005), G.H. Hardy, A theorem concerning Fourier transforms, Journal of the London Mathematical Society, 8(1933), no.3, I.I. Hirschman, A note on entropy, Amer. Jour. Math. 79(1957), L. Hörmander, Notions of convexity, Birkhäuser, V. Jones, Planar algebras, I, New Zealand Journal of Math. QA/ V. Jones, Index for subfactors, Invent. Math. 72(1983), Z. Liu (Vanderbilt) Noncommutative Uncertainty Principle Oct, / 23

26 V. Jones, Quadratic tangles in planar algebras, Duke Math. J. 161(2012), B. Kahng, Fourier transform on locally compact quantum group ArXiv: v1. H. Kosaki, Applications of the complex interpolation method to a von Neumann algebra: noncommutative L p -spaces J. Funct. Anal. 56(1984), Z. Liu, Exchange relation planar algebras of small rank, arxiv: v2. A. Ocneanu, Quantized groups, string algebras and Galois theory for algebras, Operator algebras and applications, Vol. 2, London Math. Soc. Lecture Note Ser., vol. 136, Cambridge Univ. Press, Cambridge, 1988, pp M. Özaydm and T. Przebinda, An entropy-based uncertainty principle for a locally compact abelian group, J. Funct. Anal. 215(2004), Z. Liu (Vanderbilt) Noncommutative Uncertainty Principle Oct, / 23

27 S. Popa, An axiomatization of the lattice of higher relative commutants, Invent. Math. 120 (1995), K. T. Smith, The uncertainty principle on groups, SIAM J. Appl. Math. 50(1990), W. Szymanski, Finite index subfactors and Hopf algebra crossed products Proc. Amer. Math. Soc. 120(1994), no. 2, T. Tao, An uncertainty principle for cyclic groups of prime order, Math. Res. Lett. 12 (2005), no H. Wenzl, On sequences of projections, C.R. Math. Rep. Acad. Sci. Canada. 9(1987), no. 1, 5-9. Q. Xu, Operator spaces and noncommutative L p : The part on noncommutative L p -spaces, Lectures in the Summer school on Banach spaces and operator spaces. Z. Liu (Vanderbilt) Noncommutative Uncertainty Principle Oct, / 23