Norms and embeddings of classes of positive semidefinite...

Norms and embeddings of classes of positive semidefinitematrices

Radu Balan

Department of Mathematics, Center for Scientific Computation andMathematical Modeling and the Norbert Wiener Center for Applied

Harmonic AnalysisUniversity of Maryland, College Park, MD

AMS Fall Southeastern Sectional MeetingSpecial Session on Applied Harmonic Analysis: Frames, Samplings and

Applications, IIUniversity of Central Florida, Orlando, FL

September 23, 2017, 5:30pm-5:50pm

This material is based upon work supported in part by ARO under ContractNo. W911NF-16-1-0008 and NSF under Grant No. DMS-1413249.”Any opinions, findings, and conclusions or recommendations expressed inthis material are those of the author(s) and do not necessarily reflect theviews of the National Science Foundation.”

Joint work with: Yang Wang (HKST), Dongmian Zou (UMD/IMA).

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

QTvPR Metric Structures Bi-Lipschitz Stability

Quantum TomographyNotations

Let H = Cn. The quotient space of unnormalized rays H = Cn/T 1, withclasses induced by x ∼ y if there is real ϕ with x = eiϕy . The projectivespace P(H) = {x , ‖x‖ = 1}. The set of (lowe rank) quantum states

St(H) = {T = T ∗ ≥ 0 , tr(T ) = 1}.

Str (H) = {T = T ∗ ≥ 0 , tr(T ) = 1 , rank(T ) ≤ r}.P(H)⇔ St1(H) represent the pure states. H ⇔ S1,0 := {xx∗, x ∈ H}.Given a set of p.s.d. operators F = {F1, · · · ,Fm} on H, consider two maps

α : Sym(H)→ Rm , α(T ) =(√

tr(TFk))

1≤k≤m

β : Sym(H)→ Rm , β(T ) = (tr(TFk))1≤k≤m

Quantum Tomography: reconstruct T ∈ St(H) from β(T ).Phase Retrieval: estimate x ∈ H from α(xx∗) when Fk = fk f ∗k .

Radu Balan (UMD) Quantum Norms 23 September 2017




St(H) = {T = T ∗ ≥ 0 , tr(T ) = 1}.

Str (H) = {T = T ∗ ≥ 0 , tr(T ) = 1 , rank(T ) ≤ r}.P(H)⇔ St1(H) represent the pure states. H ⇔ S1,0 := {xx∗, x ∈ H}.

Given a set of p.s.d. operators F = {F1, · · · ,Fm} on H, consider two maps

α : Sym(H)→ Rm , α(T ) =(√

tr(TFk))

1≤k≤m







St(H) = {T = T ∗ ≥ 0 , tr(T ) = 1}.

Str (H) = {T = T ∗ ≥ 0 , tr(T ) = 1 , rank(T ) ≤ r}.P(H)⇔ St1(H) represent the pure states. H ⇔ S1,0 := {xx∗, x ∈ H}.Given a set of p.s.d. operators F = {F1, · · · ,Fm} on H, consider two maps

α : Sym(H)→ Rm , α(T ) =(√

tr(TFk))

1≤k≤m





Problem Statement

Today we shall discuss the following problem. Assume the maps

α : Sr ,0 → Rm , α(T ) =(√〈T ,Fk〉

)1≤k≤m

β : Sr ,0 → Rm , β(T ) = (〈T ,Fk〉)1≤k≤m

are injective. Here

Str (H) ⊂ Sr ,0 := {T = T ∗ ≥ 0 , rank(T ) ≤ r}

We want to find bi-Lipschitz properties of these maps and understand iftheir left inverses can be extended to entire Rm are Lipschitz maps.



Metric Structures on H and Sym(H)Norm Induced Metric

Fix 1 ≤ p ≤ ∞. The matrix-norm induced distance on Sym(H):

dp : Sym(H)× Sym(H)→ R , dp(X ,Y ) = ‖X − Y ‖p,

the p-norm of the singular values.On H it induces the metric

dp : H × H → R , dp(x , y) = ‖xx∗ − yy∗‖p

In the case p = 2 we obtain

d2(X ,Y ) = ‖X − Y ‖2F , d2(x , y) =√‖x‖4 + ‖y‖4 − 2|〈x , y〉|2



Metric Structures on H and Sym(H)Natural Metric

The natural metric

Dp : H × H → R , Dp(x , y) = minϕ‖x − eiϕy‖p

with the usual p-norm on Cn. In the case p = 2 we obtain

D2(x , y) =√‖x‖2 + ‖y‖2 − 2|〈x , y〉|

On Sym+(H), the ”natural” metric lifts to

Dp : Sym+(H)× Sym+(H)→ R , Dp(X ,Y ) = minVV ∗ = X

WW ∗ = Y

‖V −WU‖p.



Metric Structures on Sym(H)Natural metric vs. Bures/Helinger

Let X ,Y ∈ Sym+(H). For the natural distance we choose p = 2:

Dnatural (X ,Y ) = minVV ∗ = X

WW ∗ = Y

‖V −W ‖F

Fact:

Dnatural (X ,Y ) = minU∈U(n)

‖X 1/2 − Y 1/2U‖F =√

tr(X ) + tr(Y )− 2‖X 1/2Y 1/2‖1

Another distance: Bures/Helinger distance:

DBures(X ,Y ) = ‖X 1/2 − Y 1/2‖F = d2(X 1/2,Y 1/2)

A consequence of the Arithmetic-Geometric Mean Inequality [BK00]:12D2

Bures(X ,Y ) ≤ D2natural (X ,Y ) ≤ D2

Bures(X ,Y ).



Metric Structuresp-dependency

Lemma (BZ16)1 (dp)1≤p≤∞ are equivalent metrics and the identity map

i : (H, dp)→ (H, dq), i(x) = x has Lipschitz constantLipd

p,q,n = max(1, 21q−

1p ).

2 The metric space (H, dp) is isometrically isomorphic to S1,0 endowedwith the p-norm via κβ : H → S1,0 , x 7→ κβ(x) = xx∗.

Lemma (BZ16)1 (Dp)1≤p≤∞ are equivalent metrics and the identity map

i : (H,Dp)→ (H,Dq), i(x) = x has Lipschitz constantLipD

p,q,n = max(1, n1q−

1p ).

2 The metric space (H,D2) is Lipschitz isomorphic to S1,0 endowedwith the 2-norm via κα : H → S1,0 , x 7→ κα(x) = 1

‖x‖xx∗.



Metric StructuresDistinct Structures

Two different structures: topologically equivalent, BUT the metrics areNOT equivalent:

Lemma (BZ16)The identity map i : (H,Dp)→ (H, dp), i(x) = x is continuous but it isnot Lipschitz continuous. Likewise, the identity mapi : (H, dp)→ (H,Dp), i(x) = x is continuous but it is not Lipschitzcontinuous. Hence the induced topologies on (H,Dp) and (H, dp) are thesame, but the corresponding metrics are not Lipschitz equivalent.

Obviously, the same result holds for (Sym+(H),Dnatural ) and(Sym+(H), d2).



Lipschitz Stability in Phase RetrievalLipschitz inversion: α

Theorem (BZ16)

Assume F is a phase retrievable frame for H. Then:1 The map α : (H,D2)→ (Rm, ‖ · ‖2) is bi-Lipschitz. Let

√A0,√

B0denote its Lipschitz constants: for every x , y ∈ H:

A0 minϕ‖x − eiϕy‖22 ≤

m∑k=1||〈x , fk〉| − |〈y , fk〉||2 ≤ B0 min

ϕ‖x − eiϕy‖22.

2 There is a Lipschitz map ω : (Rm, ‖ · ‖2)→ (H,D2) so that: (i)ω(α(x)) = x for every x ∈ H, and (ii) its Lipschitz constant isLip(ω) ≤ 4+3

√2√

A0= 8.24√

A0.



Lipschitz Stability in Phase RetrievalLipschitz inversion: β

Theorem (BZ16)

Assume F is a phase retrievable frame for H. Then:1 The map β : (H, d1)→ (Rm, ‖ · ‖2) is bi-Lipschitz. Let √a0,

√b0

denote its Lipschitz constants: for every x , y ∈ H:

a0‖xx∗ − yy∗‖21 ≤m∑

k=1

∣∣∣|〈x , fk〉|2 − |〈y , fk〉|2∣∣∣2 ≤ b0‖xx∗ − yy∗‖21.

2 There is a Lipschitz map ψ : (Rm, ‖ · ‖2)→ (H, d1) so that: (i)ψ(β(x)) = x for every x ∈ H, and (ii) its Lipschitz constant isLip(ψ) ≤ 4+3

√2√a0

= 8.24√a0.



Stability Results in Quantum TomographyBi-Lipschitz properties of α and β on Quantum States

Fix a closed subset S ⊂ Sym+(H). For instance S = St(H), or Str (H), orSr ,0.

TheoremAssume F = {F1, · · · ,Fm} ⊂ Sym+(H) so that α|S and β|S are injective.Then there are constants a0,A0, b0,B0 > 0 so that for every X ,Y ∈ S,

A0D2natural (X ,Y ) ≤

m∑k=1

∣∣∣∣√〈X ,Fk〉 −√〈Y ,Fk〉

∣∣∣∣2 ≤ B0D2natural (X ,Y )

a0‖X − Y ‖2F ≤m∑

k=1|〈X ,Fk〉 − 〈Y ,Fk〉|2 ≤ b0‖X − Y ‖2F .



Next ResultsLipschitz inversion of α and β on Quantum States

Consider the measurement map

β : (Str (H), d1)→ (Rm, ‖ · ‖2) , β(T ) = (tr(TFk))1≤k≤m

where Str (H) = {T = T ∗ ≥ 0 , tr(T ) = 1 , rank(T ) ≤ r}.If r = n := dim(H) then Stn(H) = St(H) is a compact convex set, hencea Lipschitz retract.Conjecture: If r < n then Str (H) is not contractible hence not a Lipschitzretract.

If conjecture is true, it follows that even if β is injective on rank rquantum states, it cannot admit a Lipschitz (or even continuous) leftinverse defined globally on Rm.A similar result should hold true for

α : (Str (H),D2)→ (Rm, ‖ · ‖2) , α(T ) = (√

tr(TFk))1≤k≤m.






where Str (H) = {T = T ∗ ≥ 0 , tr(T ) = 1 , rank(T ) ≤ r}.If r = n := dim(H) then Stn(H) = St(H) is a compact convex set, hencea Lipschitz retract.Conjecture: If r < n then Str (H) is not contractible hence not a Lipschitzretract.If conjecture is true, it follows that even if β is injective on rank rquantum states, it cannot admit a Lipschitz (or even continuous) leftinverse defined globally on Rm.

A similar result should hold true for

α : (Str (H),D2)→ (Rm, ‖ · ‖2) , α(T ) = (√

tr(TFk))1≤k≤m.






where Str (H) = {T = T ∗ ≥ 0 , tr(T ) = 1 , rank(T ) ≤ r}.If r = n := dim(H) then Stn(H) = St(H) is a compact convex set, hencea Lipschitz retract.Conjecture: If r < n then Str (H) is not contractible hence not a Lipschitzretract.If conjecture is true, it follows that even if β is injective on rank rquantum states, it cannot admit a Lipschitz (or even continuous) leftinverse defined globally on Rm.A similar result should hold true for

α : (Str (H),D2)→ (Rm, ‖ · ‖2) , α(T ) = (√

tr(TFk))1≤k≤m.



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