Complete Log-Sobolev inequalities

Complete Log-Sobolev inequalities

Li Gao

Technical University of Munich (TUM)

ANR Workshop on NC analysis on groups and quantum groups June 17, 2021

Joint work with Cambyse Rouz´e (TUM). arXiv:2102.04146

Quantum Markov semigroup

(M, τ)finite von Neumann algebra with n.f. trace τ.

Aquantum Markov semigroup (QMS)(Tt)t≥0:M → Mis a continuous family of maps satisfying

Ttis normal unital completely positive T_t◦T_s=T_s+t, T₀=id

t7→Tt(x)isw^∗-continuous for everyx∈ M.

(symmetric): τ(T_t(x)^∗y) =τ(x^∗T_t(y)).

Generator: Ax= lim

t→0−1

t(Tt(x)−x), Tt=e^−At.

Examples: Classical Markov semigroup: M=L∞(Ω, µ)probability space. E.g. A= ∆Laplacian on a compact Riemannian manifold(M, g). Quantum physics: M= (B(H),tr)dissipative open quantum systems. E.g. LindbladianA(x) =−P

j[a_j,[a_j, x]], a_j∈B(H)

Operator algebras: group von Neumann algebras, quantum group and q-deformed Gaussian.

Quantum Markov semigroup

(M, τ)finite von Neumann algebra with n.f. trace τ.

Aquantum Markov semigroup (QMS)(Tt)t≥0:M → Mis a continuous family of maps satisfying

Ttis normal unital completely positive T_t◦T_s=T_s+t, T₀=id

t7→Tt(x)isw^∗-continuous for everyx∈ M.

(symmetric): τ(T_t(x)^∗y) =τ(x^∗T_t(y)).

Generator: Ax= lim

t→0−1

t(Tt(x)−x), Tt=e^−At. Examples:

Classical Markov semigroup: M=L∞(Ω, µ)probability space.

E.g. A= ∆ Laplacian on a compact Riemannian manifold(M, g).

Quantum physics: M= (B(H),tr)dissipative open quantum systems.

E.g. LindbladianA(x) =−P

j[a_j,[a_j, x]], a_j∈B(H)

Operator algebras: group von Neumann algebras, quantum group and q-deformed Gaussian.

Convergence and decoherence

ConsiderM= (M_d,tr), T_t=e^−At:M_d→M_d.

Fixed point space (subalgebra)N ={x∈M_d|T_t(x) =x,∀t}.

State spaceS(M_d) ={ρ∈M_d|ρ≥0,tr(ρ) = 1}. Then T_t(ρ)−→E_N(ρ)ast→ ∞

E_N :M_d→ N conditional expectation, i.e. ∀x∈ N,tr(xρ) =tr(xE_N(ρ)).

How fast does the convergenceTt(ρ)−→E_N(ρ)happen?

Mixing time

tmix = inf{t >0| kTt(ρ)−EN(ρ)k1≤1/2,∀ stateρ}

Spectral gap

λ_gap:= inf

EN(x)=0

tr(x^∗Ax) kxk2

minimal positive eigenvalueA onL₂(M_d). kTt(x)−E_N(x)k2≤e^−λgaptkx−E_N(x)k2⇒tmix .λ_gap¹ O(lnd).

Convergence and decoherence

ConsiderM= (M_d,tr), T_t=e^−At:M_d→M_d.

Fixed point space (subalgebra)N ={x∈M_d|T_t(x) =x,∀t}.

State spaceS(M_d) ={ρ∈M_d|ρ≥0,tr(ρ) = 1}. Then T_t(ρ)−→E_N(ρ)ast→ ∞

E_N :M_d→ N conditional expectation, i.e. ∀x∈ N,tr(xρ) =tr(xE_N(ρ)).

How fast does the convergenceTt(ρ)−→E_N(ρ)happen?

Mixing time

tmix = inf{t >0| kTt(ρ)−EN(ρ)k1≤1/2,∀ stateρ}

Spectral gap

λ_gap:= inf

EN(x)=0

tr(x^∗Ax) kxk2

minimal positive eigenvalueA onL₂(M_d).

kTt(x)−E_N(x)k2≤e^−λgaptkx−E_N(x)k2⇒tmix .λ_gap¹ O(lnd).

Modified Log-Sobolev Inequality

Relative entropy: for two quantum statesρ, σ, D(ρ||σ) :=tr(ρlogρ−ρlogσ) ThenT_t(ρ)→E_N(ρ) =⇒D(T_t(ρ)||EN(ρ))→0.

Definition (Bardet ’17)

We sayTt=e^−Atsatisfiesλ-modified log-Sobolev inequality(λ-MLSI) for λ >0if

2λ D(ρ||EN(ρ))≤tr(Aρlnρ), ∀ρ∈S(Md).

Fisher information: IA(ρ) :=tr(A(ρ) lnρ) =−_dt^d|t=0D(Tt(ρ)||EN(ρ)). Thenλ-MLSI means:

d

dtD(T_t(ρ)||EN(ρ))≤ −2λD(Tt(ρ)||EN(ρ))

⇐⇒D(Tt(ρ)||E_N(ρ))≤e^−2λtD(ρ||E_N(ρ)).

λ-MLSI⇐⇒ Exponential decay of relative entropy⇒t_mix≤ ¹_λO(ln(lnd))

Modified Log-Sobolev Inequality

Relative entropy: for two quantum statesρ, σ, D(ρ||σ) :=tr(ρlogρ−ρlogσ) ThenT_t(ρ)→E_N(ρ) =⇒D(T_t(ρ)||EN(ρ))→0.

Definition (Bardet ’17)

We sayTt=e^−Atsatisfiesλ-modified log-Sobolev inequality(λ-MLSI) for λ >0if

2λ D(ρ||EN(ρ))≤tr(Aρlnρ), ∀ρ∈S(Md).

Fisher information: IA(ρ) :=tr(A(ρ) lnρ) =−_dt^d|t=0D(Tt(ρ)||EN(ρ)).

Thenλ-MLSI means:

d

dtD(T_t(ρ)||EN(ρ))≤ −2λD(Tt(ρ)||EN(ρ))

⇐⇒D(Tt(ρ)||E_N(ρ))≤e^−2λtD(ρ||E_N(ρ)).

λ-MLSI⇐⇒ Exponential decay of relative entropy⇒tmix≤_λ¹O(ln(lnd))

Connection to other functional inequalities

L₂-Log-Sobolev inequality (LSI): for an ergodic semigroupT_t=e^−At (N =C1),

tr(x²lnx²)−tr(x²) lntr(x²)≤ 1

λtr(x^∗Ax)

⇐⇒Hypercontractivity: (Olkiewicz-Zegarlinski ’99) kTt:L2→Lpk≤1forp≤1 +e^2λt

(Temme-Kastoryano ’13 & Bardet ’17) For the optimal constant, λLSI≤λMLSI≤λgap

Moreover,

MLSI=⇒Transport cost inequality =⇒concentration of measure.

(Datta-Rouz´e, Carlen-Maas ’18)

Tensorization and complete MLSI

Tensorization

Tt, St:L_∞(Ω)→L_∞(Ω)hasλ-MLSI=⇒Tt⊗Stλ-MLSI (same for LSI) Unknownfor MLSI in quantum cases. True for LSI onM₂(King ’14).

Definition

Tt=e^−Atsatisfiesλ-complete log-Sobolev inequality (λ-CLSI)forλ >0 if idM_n⊗Ttsatisfiesλ-MLSI for alln≥1.

Tensorization of CLSI

T_t, S_t:M_d→M_d both hasλ-CLSI=⇒T_t⊗S_thasλ-CLSI Applications inquantum lattice spin system.

Estimating decay ofentanglement: for a bipartite stateρonM_n⊗M_d, D(idn⊗Tt(ρ)||idn⊗E_N(ρ))→0

Tensorization and complete MLSI

Tensorization

Tt, St:L_∞(Ω)→L_∞(Ω)hasλ-MLSI=⇒Tt⊗Stλ-MLSI (same for LSI) Unknownfor MLSI in quantum cases. True for LSI onM₂(King ’14).

Definition

Tt=e^−Atsatisfiesλ-complete log-Sobolev inequality (λ-CLSI)forλ >0 if idM_n⊗Ttsatisfiesλ-MLSI for alln≥1.

Tensorization of CLSI

T_t, S_t:M_d→M_d both hasλ-CLSI=⇒T_t⊗S_thasλ-CLSI Applications inquantum lattice spin system.

Estimating decay ofentanglement: for a bipartite stateρonM_n⊗M_d, D(idn⊗Tt(ρ)||idn⊗E_N(ρ))→0

Example: Depolarizing semigroup

Qubit depolarizing semigroup:

T_t:M₂→M₂, T_t(ρ) =e^−tρ+ (1−e^−t)tr(ρ)1 2 Optimal constant:

1/2≤λ_CLSI(T_t)≤λ_MLSI(T_t⊗id_M2)<λ_MLSI(T_t) =λ_LSI(T_t) = 1. In particular,λ_CLSI6=λ_MLSI.

λ_LSI(T_t⊗id_M2) = 0. In general, LSI/Hypercontractivity fails for non ergodicT_twheneverN is noncommutative, hence for id_Mn⊗T_t.

Question

Does every finite dimensionalTt:Md→Md has λ_CLSI:= inf

n λ_MLSI(id_Mn⊗T_t)>0?

(Junge-Li-LaRacuente, ’20) True for classical casesTt:l_∞^d →l^d_∞.

Example: Depolarizing semigroup

Qubit depolarizing semigroup:

T_t:M₂→M₂, T_t(ρ) =e^−tρ+ (1−e^−t)tr(ρ)1 2 Optimal constant:

1/2≤λ_CLSI(T_t)≤λ_MLSI(T_t⊗id_M2)<λ_MLSI(T_t) =λ_LSI(T_t) = 1. In particular,λ_CLSI6=λ_MLSI.

λ_LSI(T_t⊗id_M2) = 0. In general, LSI/Hypercontractivity fails for non ergodicT_twheneverN is noncommutative, hence for id_Mn⊗T_t.

Question

Does every finite dimensionalTt:Md→Md has λCLSI:= inf

n λMLSI(idMn⊗Tt)>0?

(Junge-Li-LaRacuente, ’20) True for classical casesTt:l_∞^d →l^d_∞.

Finite dimensional results

Theorem. (G.-Rouz´ e, preprint ’21)

LetTt=e^−At:Md →Md be a symmetric Quantum Markov semigroup and N be the fixed point subalgebra. Then

λ_gap

2Ccb(Md:N) ≤λCLSI ≤λgap

C(M_d:N)Pimsner-Popa index.

Remain valid forGNS-symmetricsemigroup: for a faithful stateσ, tr(T_t(x)^∗yσ) =tr(x^∗T_t(y)σ)

Pimsner-Popa index

N ⊂ Mfinite von Neumann algebra. EN :M → N cond. expectation.

Pimsner-Popa index

C(M:N) := inf{C >0|ρ≤CE_N(ρ)∀ρ∈ M+}

(Pimsner-Popa, ’86)C(M:N) = [M:N] for II₁ subfactor

& Explicit formula for finite dimensionalM,N.

CB-version: Ccb(M:N) = sup_nC(Mn(M) :Mn(N)). E.g. C(Md:C) =d, Ccb(Md:C) =d²and

Ccb(Md:N)≤Ccb(Md:C) =d².

Pimsner-Popa index

N ⊂ Mfinite von Neumann algebra. EN :M → N cond. expectation.

Pimsner-Popa index

C(M:N) := inf{C >0|ρ≤CE_N(ρ)∀ρ∈ M+}

(Pimsner-Popa, ’86)C(M:N) = [M:N] for II₁ subfactor

& Explicit formula for finite dimensionalM,N.

CB-version: Ccb(M:N) = sup_nC(Mn(M) :Mn(N)).

E.g. C(Md:C) =d, Ccb(Md:C) =d²and

Ccb(Md:N)≤Ccb(Md:C) =d².

Quantum χ

-divergence

Forρ∈S(M_d), defineρweightedL₂-norm kXk²_ρ−1=

Z ∞

0

tr X^∗ 1

ρ+sX 1 ρ+s

ds .

commutative case,kXk²_ρ−1=R |X|² ρ dµ

for two quantum states,kρ−σk²_σ−1:=χ2(ρ, σ)quantumχ2-divergence.

Key Lemma

Recall the relative entropyD(ρ||σ) =tr(ρlogρ−ρlogσ). i) D(ρ||σ)≤kρ−σk²_σ−1

ii) Ifρ≤Cσ,CkXk²_ρ−1≥kXk²_σ−1

Quantum χ

-divergence

Forρ∈S(M_d), defineρweightedL₂-norm kXk²_ρ−1=

Z ∞

0

tr X^∗ 1

ρ+sX 1 ρ+s

ds .

commutative case,kXk²_ρ−1=R |X|² ρ dµ

for two quantum states,kρ−σk²_σ−1:=χ2(ρ, σ)quantumχ2-divergence.

Key Lemma

Recall the relative entropyD(ρ||σ) =tr(ρlogρ−ρlogσ).

i) D(ρ||σ)≤kρ−σk²_σ−1

ii) Ifρ≤Cσ,CkXk²_ρ−1≥kXk²_σ−1

Proof of Theorem.

Denoteρ_N =E_N(ρ). Recallλ-MLSI is

2λ D(ρ||ρ_N)≤I_A(ρ) OnM_d,T_t=e^−At is given byA(x) =−Pk

j=1[a_j,[a_j, x]]. Then A=δ^∗δ , δ(x) =⊕^k_j=1[aj, x]

is a derivation. Then

I_A(ρ) =tr(Aρlogρ) =tr(δ(ρ)δ(logρ))

= Z ∞

0

tr(δ(ρ) 1

ρ+sδ(ρ) 1

ρ+s)ds=kδ(ρ)k²_ρ−1

D(ρ||ρ_N)≤ kρ−ρ_Nk²_ρ−1 N

≤ 1

λ_gap kδ(ρ−ρ_N)k²_ρ−1 N

= 1

λ_gap kδ(ρ)k²_ρ−1 N

= C

λ_gap kδ(ρ)k²_ρ−1 (ρ≤Cρ_N)

Discussion on optimality

SinceC(M_d:N)≤d, C_cb(M_d:N)≤d², we have shown λgap

2d ≤ λgap

2C(Md:N)≤λMLSI≤λgap

λ_gap

2d² ≤ λ_gap

2Ccb(Md:N) ≤λ_CLSI ≤λ_gap

ergodic case: N =C1 λ_gap(T_t)

logd+ 2 ≤λ_MLSI(T_t)≤λ_gap(T_t) (Temme-Kastoryano ’13)

Question

Can we haveλMLSI≥λgapO(_log¹_d)or evenλCLSI≥λgapO(_log¹_d)?

(Brannan-G.-Junge, ’20) Ture forSchur multiplierand some Random unitary.

Discussion on optimality

SinceC(M_d:N)≤d, C_cb(M_d:N)≤d², we have shown λgap

2d ≤ λgap

2C(Md:N)≤λMLSI≤λgap

λ_gap

2d² ≤ λ_gap

2Ccb(Md:N) ≤λ_CLSI ≤λ_gap

ergodic case: N =C1 λ_gap(T_t)

logd+ 2 ≤λ_MLSI(T_t)≤λ_gap(T_t) (Temme-Kastoryano ’13)

Question

Can we haveλ_MLSI≥λ_gapO(_log¹_d)or evenλ_CLSI≥λ_gapO(_log¹_d)?

(Brannan-G.-Junge, ’20) Ture forSchur multiplierand someRandom unitary.

Heat Semigroup

(Li-Junge-LaRacuente,’20) For a compact Riemannian Manifold(M, g), ifRicci(M)≥λ >0, the Heat semigroupH_t=e^−∆t satisfyλ-CLSI.

(Brannan-G.-Junge, ’20) The Heat semigroupHt=e^−∆ton every compact Riemannian Manifold hasλ_CLSI>0.

Example: Unit Circle

T={z∈C| |z|= 1}andHt(z^m) =e^−m2tz^m. 1

4 ln 3 ≤λCLSI(Ht)≤λMLSI(Ht) =λLSI(Ht) = 1

Question

Doesλ_CLSI=λ_MLSI for classical semigroup?

Hypo-elliptic case

(G.-Junge-LaRacuente, ’18) For a sub-Laplacian∆X =−P

jX_j² satisfyingH¨ormander condition, the subordinate semirgoupe^−∆θXt satisfy hasλCLSI>0 for0< θ <1.

Example:Possion semigroup

T={z∈C| |z|= 1}andPt(z^m) =e^−|m|tz^m.

λ_CLSI(P_t) =λ_MLSI(P_t) =λ_LSI(P_t) =λ_gap(P_t) = 1

Question

Does the sub-Laplacian∆_X itself satisfy CLSI?

Implies dimension free estimate for all tranference semigroup of∆X

No example known, evenSU(2).

Compact Quantum group

Central semigroup on compact quantum group has Ricci≥0.

kδ(T_t(x))kρ⁻¹≤e^−λtkδ(x)kT_t(ρ)⁻¹ (Ricci≥λ) Free orthogonal groupO_N⁺ & quantum permutation groupS_N⁺:

Heat semigroup haveλCLSI≥O(_Nlog¹ _N).

Group von Neumann algebra: Fourier multiplierTt(λ(g)) =e^−φ(g)tλ(g) hasλ_CLSI>0 under some growth condition onφ.

Theorem (Brannan-G.-Junge & Wirth-Zhang, ’20)

Fd free group andP_t(λ(g)) =e^−|g|tλ(g).

λ_Ricci(P_t) =λ_CLSI(P_t) =λ_gap(P_t) = 1

LSI constantλLSI(Pt)≥1.17(Junge-Palazuelos-Parcet-Perrin-Ricard) (1) Optimal Hypercontractivity forPt?

(2) Positive curvature forO⁺_N?

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