Norms and embeddings of classes of positive semidefinite matrices

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1 Norms and embeddings of classes of positive semidefinite matrices Radu Balan Department of Mathematics, Center for Scientific Computation and Mathematical Modeling and the Norbert Wiener Center for Applied Harmonic Analysis University of Maryland, College Park, MD AMS Fall Southeastern Sectional Meeting Special Session on Applied Harmonic Analysis: Frames, Samplings and Applications, II University of Central Florida, Orlando, FL September 23, 2017, 5:30pm-5:50pm

2 This material is based upon work supported in part by ARO under Contract No. W911NF and NSF under Grant No. DMS Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation. Joint work with: Yang Wang (HKST), Dongmian Zou (UMD/IMA).

3 Table of Contents: 1 Quantum Tomography and Phase Retrieval 2 Metrics on Matrix Spaces and Spaces of Rays 3 Bi-Lipschitz Stability: Topological and Functional Analytic Approach

4 Quantum Tomography Notations Let H = C n. The quotient space of unnormalized rays Ĥ = C n /T 1, with classes induced by x y if there is real ϕ with x = e iϕ y. The projective space P(H) = {ˆx, x = 1}. The set of (lowe rank) quantum states St(H) = {T = T 0, tr(t ) = 1}.

5 Quantum Tomography Notations Let H = C n. The quotient space of unnormalized rays Ĥ = C n /T 1, with classes induced by x y if there is real ϕ with x = e iϕ y. The projective space P(H) = {ˆx, x = 1}. The set of (lowe rank) quantum states St(H) = {T = T 0, tr(t ) = 1}. St r (H) = {T = T 0, tr(t ) = 1, rank(t ) r}. P(H) St 1 (H) represent the pure states. Ĥ S 1,0 := {xx, x H}.

6 Quantum Tomography Notations Let H = C n. The quotient space of unnormalized rays Ĥ = C n /T 1, with classes induced by x y if there is real ϕ with x = e iϕ y. The projective space P(H) = {ˆx, x = 1}. The set of (lowe rank) quantum states St(H) = {T = T 0, tr(t ) = 1}. St r (H) = {T = T 0, tr(t ) = 1, rank(t ) r}. P(H) St 1 (H) represent the pure states. Ĥ S 1,0 := {xx, x H}. Given a set of p.s.d. operators F = {F 1,, F m } on H, consider two maps ( ) α : Sym(H) R m, α(t ) = tr(tf k ) 1 k m β : Sym(H) R m, β(t ) = (tr(tf k )) 1 k m

7 Quantum Tomography Notations Let H = C n. The quotient space of unnormalized rays Ĥ = C n /T 1, with classes induced by x y if there is real ϕ with x = e iϕ y. The projective space P(H) = {ˆx, x = 1}. The set of (lowe rank) quantum states St(H) = {T = T 0, tr(t ) = 1}. St r (H) = {T = T 0, tr(t ) = 1, rank(t ) r}. P(H) St 1 (H) represent the pure states. Ĥ S 1,0 := {xx, x H}. Given a set of p.s.d. operators F = {F 1,, F m } on H, consider two maps ( ) α : Sym(H) R m, α(t ) = tr(tf k ) 1 k m β : Sym(H) R m, β(t ) = (tr(tf k )) 1 k m Quantum Tomography: reconstruct T St(H) from β(t ). Phase Retrieval: estimate x Ĥ from α(xx ) when F k = f k f k.

8 Problem Statement Today we shall discuss the following problem. Assume the maps ( ) α : S r,0 R m, α(t ) = T, F k 1 k m are injective. Here β : S r,0 R m, β(t ) = ( T, F k ) 1 k m St r (H) S r,0 := {T = T 0, rank(t ) r} We want to find bi-lipschitz properties of these maps and understand if their left inverses can be extended to entire R m are Lipschitz maps.

9 Metric Structures on Ĥ and Sym(H) Norm Induced Metric Fix 1 p. The matrix-norm induced distance on Sym(H): d p : Sym(H) Sym(H) R, d p (X, Y ) = X Y p, the p-norm of the singular values. On Ĥ it induces the metric In the case p = 2 we obtain d p : Ĥ Ĥ R, d p (ˆx, ŷ) = xx yy p d 2 (X, Y ) = X Y 2 F, d 2(x, y) = x 4 + y 4 2 x, y 2

10 Metric Structures on Ĥ and Sym(H) Natural Metric The natural metric D p : Ĥ Ĥ R, D p (ˆx, ŷ) = min x e iϕ y ϕ p with the usual p-norm on C n. In the case p = 2 we obtain D 2 (ˆx, ŷ) = On Sym + (H), the natural metric lifts to x 2 + y 2 2 x, y D p : Sym + (H) Sym + (H) R, D p (X, Y ) = min VV = X WW = Y V WU p.

11 Metric Structures on Sym(H) Natural metric vs. Bures/Helinger Let X, Y Sym + (H). For the natural distance we choose p = 2: Fact: natural(x, Y ) = D natural (X, Y ) = min VV = X WW = Y V W F min X 1/2 Y 1/2 U F = tr(x) + tr(y ) 2 X 1/2 Y 1/2 1 U U(n)

12 Metric Structures on Sym(H) Natural metric vs. Bures/Helinger Let X, Y Sym + (H). For the natural distance we choose p = 2: Fact: natural(x, Y ) = D natural (X, Y ) = min VV = X WW = Y V W F min X 1/2 Y 1/2 U F = tr(x) + tr(y ) 2 X 1/2 Y 1/2 1 U U(n) Another distance: Bures/Helinger distance: D Bures (X, Y ) = X 1/2 Y 1/2 F = d 2 (X 1/2, Y 1/2 )

13 Metric Structures on Sym(H) Natural metric vs. Bures/Helinger Let X, Y Sym + (H). For the natural distance we choose p = 2: Fact: natural(x, Y ) = D natural (X, Y ) = min VV = X WW = Y V W F min X 1/2 Y 1/2 U F = tr(x) + tr(y ) 2 X 1/2 Y 1/2 1 U U(n) Another distance: Bures/Helinger distance: D Bures (X, Y ) = X 1/2 Y 1/2 F = d 2 (X 1/2, Y 1/2 ) A consequence of the Arithmetic-Geometric Mean Inequality [BK00]: 1 2 D2 Bures(X, Y ) D 2 natural(x, Y ) D 2 Bures(X, Y ).

14 Metric Structures p-dependency Lemma (BZ16) 1 (d p ) 1 p are equivalent metrics and the identity map i : (Ĥ, d p ) (Ĥ, d q ), i(x) = x has Lipschitz constant Lip d p,q,n = max(1, 2 1 q 1 p ). 2 The metric space (Ĥ, d p ) is isometrically isomorphic to S 1,0 endowed with the p-norm via κ β : Ĥ S1,0, x κ β (x) = xx. Lemma (BZ16) 1 (D p ) 1 p are equivalent metrics and the identity map i : (Ĥ, D p ) (Ĥ, D q ), i(x) = x has Lipschitz constant Lip D p,q,n = max(1, n 1 q 1 p ). 2 The metric space (Ĥ, D 2 ) is Lipschitz isomorphic to S 1,0 endowed with the 2-norm via κ α : Ĥ S1,0, x κ α (x) = 1 x xx.

15 Metric Structures Distinct Structures Two different structures: topologically equivalent, BUT the metrics are NOT equivalent: Lemma (BZ16) The identity map i : (Ĥ, D p) (Ĥ, d p), i(x) = x is continuous but it is not Lipschitz continuous. Likewise, the identity map i : (Ĥ, d p ) (Ĥ, D p ), i(x) = x is continuous but it is not Lipschitz continuous. Hence the induced topologies on (Ĥ, D p ) and (Ĥ, d p ) are the same, but the corresponding metrics are not Lipschitz equivalent. Obviously, the same result holds for (Sym + (H), D natural ) and (Sym + (H), d 2 ).

16 Lipschitz Stability in Phase Retrieval Lipschitz inversion: α Theorem (BZ16) Assume F is a phase retrievable frame for H. Then: 1 The map α : (Ĥ, D 2 ) (R m, 2 ) is bi-lipschitz. Let A 0, B 0 denote its Lipschitz constants: for every x, y H: A 0 min ϕ x e iϕ y 2 2 m k=1 x, f k y, f k 2 B 0 min ϕ x e iϕ y There is a Lipschitz map ω : (R m, 2 ) (Ĥ, D 2 ) so that: (i) ω(α(x)) = x for every x Ĥ, and (ii) its Lipschitz constant is Lip(ω) A0 = 8.24 A0.

17 Lipschitz Stability in Phase Retrieval Lipschitz inversion: β Theorem (BZ16) Assume F is a phase retrievable frame for H. Then: 1 The map β : (Ĥ, d 1 ) (R m, 2 ) is bi-lipschitz. Let a 0, b 0 denote its Lipschitz constants: for every x, y H: a 0 xx yy 2 m 1 x, f k 2 y, f k 2 2 b0 xx yy 2 1. k=1 2 There is a Lipschitz map ψ : (R m, 2 ) (Ĥ, d 1 ) so that: (i) ψ(β(x)) = x for every x Ĥ, and (ii) its Lipschitz constant is Lip(ψ) a0 = 8.24 a0.

18 Stability Results in Quantum Tomography Bi-Lipschitz properties of α and β on Quantum States Fix a closed subset S Sym + (H). For instance S = St(H), or St r (H), or S r,0. Theorem Assume F = {F 1,, F m } Sym + (H) so that α S and β S are injective. Then there are constants a 0, A 0, b 0, B 0 > 0 so that for every X, Y S, m A 0 Dnatural(X, 2 2 Y ) X, F k Y, F k B 0 Dnatural(X, 2 Y ) k=1 a 0 X Y 2 m F X, F k Y, F k 2 b 0 X Y 2 F. k=1

19 Next Results Lipschitz inversion of α and β on Quantum States Consider the measurement map β : (St r (H), d 1 ) (R m, 2 ), β(t ) = (tr(tf k )) 1 k m where St r (H) = {T = T 0, tr(t ) = 1, rank(t ) r}. If r = n := dim(h) then St n (H) = St(H) is a compact convex set, hence a Lipschitz retract. Conjecture: If r < n then St r (H) is not contractible hence not a Lipschitz retract.

20 Next Results Lipschitz inversion of α and β on Quantum States Consider the measurement map β : (St r (H), d 1 ) (R m, 2 ), β(t ) = (tr(tf k )) 1 k m where St r (H) = {T = T 0, tr(t ) = 1, rank(t ) r}. If r = n := dim(h) then St n (H) = St(H) is a compact convex set, hence a Lipschitz retract. Conjecture: If r < n then St r (H) is not contractible hence not a Lipschitz retract. If conjecture is true, it follows that even if β is injective on rank r quantum states, it cannot admit a Lipschitz (or even continuous) left inverse defined globally on R m.

21 Next Results Lipschitz inversion of α and β on Quantum States Consider the measurement map β : (St r (H), d 1 ) (R m, 2 ), β(t ) = (tr(tf k )) 1 k m where St r (H) = {T = T 0, tr(t ) = 1, rank(t ) r}. If r = n := dim(h) then St n (H) = St(H) is a compact convex set, hence a Lipschitz retract. Conjecture: If r < n then St r (H) is not contractible hence not a Lipschitz retract. If conjecture is true, it follows that even if β is injective on rank r quantum states, it cannot admit a Lipschitz (or even continuous) left inverse defined globally on R m. A similar result should hold true for α : (St r (H), D 2 ) (R m, 2 ), α(t ) = ( tr(tf k )) 1 k m.

22 References B. Alexeev, A. S. Bandeira, M. Fickus, D. G. Mixon, Phase Retrieval with Polarization, SIAM J. Imaging Sci., 7 (1) (2014), R. Balan, P. Casazza, D. Edidin, On signal reconstruction without phase, Appl.Comput.Harmon.Anal. 20 (2006), R. Balan, B. Bodmann, P. Casazza, D. Edidin, Painless reconstruction from Magnitudes of Frame Coefficients, J.Fourier Anal.Applic., 15 (4) (2009), R. Balan, Reconstruction of Signals from Magnitudes of Frame Representations, arxiv submission arxiv: (2012). R. Balan, D. Zou, On Lipschitz Inversion of Nonlinear Redundant Representations, Contemporary Mathematics 650, (2015).

23 R. Balan, The Fisher Information Matrix and the Cramer-Rao Lower Bound in a Non-Additive White Gaussian Noise Model for the Phase Retrieval Problem, proceedings of SampTA R. Balan, Y. Wang, Invertibility and Robustness of Phaseless Reconstruction, Appl.Comput.Harm.Anal. vol. 38(3), (2015). R. Balan, Reconstruction of Signals from Magnitudes of Redundant Representations: The Complex Case, Found. of Comput. Math. vol. 16(3), (2016). R. Balan, D. Zou, On Lipschitz Analysis and Lipschitz Synthesis for the Phase Retrieval Problem, Linear Algebra and Applications 496, (2016). A.S. Bandeira, J. Cahill, D.G. Mixon, A.A. Nelson, Saving phase: Injectivity and stability for phase retrieval, Appl.Comput.Harmon.Anal. vol.37, (2014).

24 R. Bhatia, F. Kittaneh, Notes on matrix arithmetic-geometric mean inequalities, Lin. Alg. Appl. 308 (2000), [KVW15] M. Kech, P. Vrana, M.M. Wolf, The role of topology in quantum tomography, J.Phys.A: Math. Thoer. 48 (2015)