Numerical range and random matrices

Numerical range and random matrices Karol Życzkowski in collaboration with P. Gawron, J. Miszczak, Z. Pucha la (Gliwice), C. Dunkl (Virginia), J. Holbrook (Guelph), B. Collins (Ottawa) and A. Litvak (Edmonton) Universität Bielefeld, November 7, 013 KŻ (Uni - Bielefeld) Numerical Range & Numerical Shadow 7.11.013 1 /

Numerical Range (Field of Values) Definition For any operator A acting on H N one defines its NUMERICAL RANGE (Wertevorrat) as a subset of the complex plane defined by: Λ(A) = { ψ A ψ : ψ H N, ψ ψ = 1}. (1) In physics: Rayleigh quotient, R(A) := x A x / x x Hermitian case For any hermitian operator A = A with spectrum λ 1 λ λ N its numerical range forms an interval: the set of all possible expectation values of the observable A among arbitrary pure states, Λ(A) = [λ 1, λ N ]. KŻ (Uni - Bielefeld) Numerical Range & Numerical Shadow 7.11.013 /

Numerical range and its properties Compactness Λ(A) is a compact subset of C. Convexity: Hausdorff-Toeplitz theorem - Λ(A) is a convex subset of C. Example Numerical range for random matrices of order N = 6 a) normal, b) generic (non-normal) 0.4 0. 1 0.5 0.5 1.0 e 0. 1 1 e 0.4 1 0.6 KŻ (Uni - Bielefeld) Numerical Range & Numerical Shadow 7.11.013 3 /

Numerical range for matrices of order N =. with spectrum {λ 1, λ } a) normal matrix A Λ(A) = closed interval [λ 1, λ ] b) not normal matrix A Λ(A) = elliptical disk with λ 1, λ as focal points and minor axis, d = TrAA λ 1 λ (Murnaghan, 193; Li, 1996). [ ] 0 1 Example : Jordan matrix, J =. 0 0 Its numerical range forms a circular disk, Λ(J) = D(0, r = 1/). The set of N = pure quantum states The set of N = pure quantum states normalized as ψ ψ = 1 forms the Bloch sphere, S = CP 1. A projection of the Bloch sphere onto a plane forms an ellipse, (which could be degenerated to an interval). KŻ (Uni - Bielefeld) Numerical Range & Numerical Shadow 7.11.013 4 /

Numerical range & quantum states: Set of normalized pure states of a finite size N: Let Ω N = CP N 1 denotes the set of all pure states in H N. Example: for N = one obtains Bloch sphere, Ω = CP 1. Probability distributions on complex plane supported in Λ(A) - Fix an arbitrary operator A of size N, - Generate random pure states ψ of size N (with respect to unitary invariant, Fubini Study measure) and analyze their contribution ψ A ψ to the numerical range on the complex plane... Probability distribution: (a numerical shadow ) P A (z) := is supported on Λ(A). How it looks like? Ω N dψ δ ( z ψ A ψ ) KŻ (Uni - Bielefeld) Numerical Range & Numerical Shadow 7.11.013 5 /

Normal case: a shadow of the classical simplex... Normal matrices of size N = with spectrum {λ 1, λ } Distribution P A (z) covers uniformly the interval [λ 1, λ ] on the complex plane, Examples for diagonal matrices A of size two and three 1.0 Density plot of numerical range of operator [1, 0; 0, i] 1.0 Density plot of numerical range of operator diag(1, i, i) 0.5 0.5 0.0 0.0 0.5 0.5 1.0 1.0 0.5 0.0 0.5 1.0 1.0 1.0 0.5 0.0 0.5 1.0 Normal matrices of order N = 3 with spectrum {λ 1, λ, λ 3 } Distribution P A (z) covers uniformly the triangle (λ 1, λ, λ 3 ) on the KŻ (Uni - Bielefeld) Numerical Range & Numerical Shadow 7.11.013 6 /

Examples for diagonal matrices A of size four. Observe shadow of the edges of the tetrahedron = the set of all classical mixed states for N = 4. KŻ (Uni - Bielefeld) Numerical Range & Numerical Shadow 7.11.013 7 / Normal case: a shadow of the classical simplex... Normal matrices of size N = 4 with spectrum {λ 1, λ, λ 3, λ 4 } Distribution P A (z) covers the quadrangle (λ 1, λ, λ 3, λ 4 ) with the density determined by projection of the regular tetrahedron (3 simplex) on the complex plane 1.0 Density plot of numerical range of operator diag(1, exp(iπ/3), exp(iπ/3), 1) 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0

Numerical shadow for N = Non normal matrices of size N = Distribution P A (z) covers entire (eliptical) disk on the complex plane with non-uniform (!) density determined by projection of (empty!) Bloch sphere, S = CP 1 on the complex plane. 1.0 Density plot of numerical range of operator [0, 1; 0, 1/].0 Density plot of numerical range of operator [0, 1; 0, 1] 1.5 0.5 1.0 0.5 0.0 0.0 0.5 0.5 1.0 1.5 1.0 1.0 0.5 0.0 0.5 1.0.0.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5.0 Examples for non-normal matrices A of size two. Note the shadow of the projective space CP 1 = S = set of quantum pure states for N =. KŻ (Uni - Bielefeld) Numerical Range & Numerical Shadow 7.11.013 8 /

Shadows of three dimensional objects... KŻ (Uni - Bielefeld) Numerical Range & Numerical Shadow 7.11.013 9 /

Quantum States and Numerical Range/Shadow Classical States & normal matrices Proposition 1.Let C N denote the set of classical states (probability distributions) of size N, which forms the regular simplex N 1 in R N 1. Then the set of similar images of orthogonal projections of C N on a plane is equivalent to the set of all possible numerical ranges Λ(A) of all normal matrices A of order N (such that AA = A A). Quantum States & non normal matrices Proposition.Let M N denotes the set of quantum states (hermitian positive matrices, ρ = ρ 0, of size N normalized as Tr ρ = 1), embedded in R N 1 with respect to Euclidean geometry induced by Hilbert-Schmidt distance. Then the set of similar images of orthogonal projections M N on a -plane is equivalent to the set of all possible numerical ranges Λ(A) of all matrices A of order N. KŻ (Uni - Bielefeld) Numerical Range & Numerical Shadow 7.11.013 10 /

Normal case: a shadow of the 4-D simplex... Normal matrices of size N = 5 with spectrum (λ 1, λ, λ 3, λ 4, λ 5 ) Distribution P A (z) covers the pentagon {exp(πj/5)}, j = 0,..., 4 with the density determined by projection of the 4-D simplex on the complex plane KŻ (Uni - Bielefeld) Numerical Range & Numerical Shadow 7.11.013 11 /

How complex / real projective space looks like? Not normal matrices of size N = 3 Distribution PA (z) covers the numerical range Λ(A) on the complex plane with a non-uniform density Density plot of numerical range of operator O5 subject to complex states Density plot of numerical range of operator O5 subject to real states 0.5 0.5 im 1.0 im 1.0 0.0 0.0 0.5 0.5 1.0 0.5 re 0.0 0.5 1.0 C P 0.5 re 0.0 R 0.5 P shadow of the projective space Ω3 = (left) and (right) for a N = 3 non-normal matrix A = [0, 1, 1; 0, i, 1; 0, 0, 1] KZ (Uni - Bielefeld) Numerical Range & Numerical Shadow 7.11.013 1 /

Numerical range for matrices of order N = 3. Classification by Keeler, Rodman, Spitkovsky 1997 Numerical range Λ of a 3 3 matrix A forms: a) Λ(A) is a compact set of an ovular shape (which contains three eigenvalues!) the generic case, b) a compact set with one flat part (e.g. convex hull of a cardioid), c) a compact set with two flat parts (e.g. convex hull of an ellipse and a point outside it), d) triangle of eigenvalues, Λ(A) = (λ 1, λ, λ 3 ) for any normal matrix A. These four cases describe the shape of possible projections of the complex projective space CP (embedded into R 8 ) onto a plane. KŻ (Uni - Bielefeld) Numerical Range & Numerical Shadow 7.11.013 13 /

Shadows of typical matrices of size N = 3 correspond to the four classes of N = 3 numerical ranges by Keeler et al. KŻ (Uni - Bielefeld) Numerical Range & Numerical Shadow 7.11.013 14 /

Unitary dynamics in the shadow (as a background) Examplary unitary dynamics of N = 3 pure state in CP ψ(t) = U ψ(0) = e iht ψ(0) (green line) z(t) := ψ(t) A ψ(t) = ψ(0) U AU ψ(t) superimposed on the numerical shadow of three exemplary matrices A 1, A and A 3 of size N = 3. KŻ (Uni - Bielefeld) Numerical Range & Numerical Shadow 7.11.013 15 /

Shadows of typical matrices of size N = 4 [Classification of numerical ranges for N = 4 is not yet complete...] KŻ (Uni - Bielefeld) Numerical Range & Numerical Shadow 7.11.013 16 /

Large N limit and random matrices I Non hermitian Ginibre matrices of size N normalized as TrGG = N 1 1 1 I 0 1 I 0 1 I 0 1 ρ r 1 0 1 R 1 0 1 R 1 0 1 R N = 10 N = 100 N = 1000 Numerical range and spectrum of random Ginibre matrices of size N. Note circular disk of eigenvalues of Girko and the non normality belt. Result I In the limit N the numerical range of a random Ginibre matrix G converges (in the Hausdorff distance) to the disk of radius. KŻ (Uni - Bielefeld) Numerical Range & Numerical Shadow 7.11.013 17 /

Large N limit and random matrices II Diagonalized Ginibre matrices of size N normalized as TrGG = N 1 1 1 I 0 I 0 I 0 ρ r 1 1 1 1 0 1 R 1 0 1 R 1 0 1 R N = 10 N = 100 N = 1000 As diagonal matrix G = ZGZ = diag ( Eig(G) ) is normal, the numerical range W (G ) forms the convex hull of its spectrum and asymptotically converges to the unit disk of Girko. Note absence of the non normality belt! KŻ (Uni - Bielefeld) Numerical Range & Numerical Shadow 7.11.013 18 /

Large N limit and random matrices III Strictly tri diagonal random matrices T (with T ij = ξ for i > j) of size N normalized as TrTT = N 1 1 1 I 0 1 I 0 1 I 0 1 r 1 0 1 R 1 0 1 R 1 0 1 R N = 10 N = 100 N = 1000 The spectrum of T by definition includes the origin, {0}. Its numerical range W (T ) forms a disk with a thick non normality belt. Result II In the limit N the numerical range of a triangular matrix T converges (in the Hausdorff distance) to the disk of radius. KŻ (Uni - Bielefeld) Numerical Range & Numerical Shadow 7.11.013 19 /

Large N limit and random matrices Theorem (CGLZ 013, arxiv:1309.603) Let R > 0 and let {X n } n be a sequence of complex random matrices or order n such that for every θ R with probability one Then with probability one lim Re( e iθ ) X n = R n lim d ( H W (Xn ), D(0, R) ) = 0. n Examples: For random Ginibre matrix G and random triagonal matrix T one can show that the norm does not depend on the phase θ and (upon the normalization used) converges to R =. KŻ (Uni - Bielefeld) Numerical Range & Numerical Shadow 7.11.013 0 /

Concluding Remarks Standard Numerical Range (NR) is a useful algebraic tool... We introduce a concept of numerical shadow of an operator. If A is normal one obtains images of the set of classical mixed states, and the set of quantum pure states plays the role for the generic, non normal A. The set of all possible projections of the N 1 dimensional set of quantum states of order N onto a plane is equivalent (up to shift and rescaling) to the set of numerical ranges of matrices of order N. projections of the set of classical states (i.e. probability simplex) are equivalent to the set of numerical ranges of all normal matrices of a given order. Numerical range of a large random Ginibre matrix forms a disk of radius R =. KŻ (Uni - Bielefeld) Numerical Range & Numerical Shadow 7.11.013 1 /

Some References C. F. Dunkl, P. Gawron, J. A. Holbrook, Z. Pucha la and K. Życzkowski, Numerical shadows: Measures and densities on the numerical range, Lin. Algebra Appl 434, 04-080 (011). C. F. Dunkl, P. Gawron, J. A. Holbrook, J. Miszczak, Z. Pucha la and K. Życzkowski, Numerical shadow and geometry of quantum states, J. Phys. A 44 335301 (19pp) (011). Z. Pucha la, J. A. Miszczak, P. Gawron, C. F. Dunkl, J. A. Holbrook, and K. Życzkowski, Restricted numerical shadow and geometry of quantum entanglement, J. Phys. A 45, 415309 (8pp) (01). A. Litvak. B. Collins, P. Gawron and K. Życzkowski, Numerical range of random matrices, preprint arxiv:1309.603 T. Gallay and D. Serre, Numerical measure of a complex matrix, Commun. Pure Apl. Math. 65, 87-336 (01). KŻ (Uni - Bielefeld) Numerical Range & Numerical Shadow 7.11.013 /