Logarithmic Sobolev Inequalities for Entropy Production - TUM · 2016. 2. 23. · Logarithmic...

Logarithmic Sobolev Inequalities for Entropy

Production

Daniel Stilck França

Technische Universität München

CEQUIP 2015

Joint work with Alexander Müller-Hermes and Michael M. Wolf

arXiv:1505.04678

Daniel Stilck França Logarithmic Sobolev Inequalities for Entropy Production

Outline

De�nition and applications of the Logarithmic Sobolev 1

constant.

The Logarithmic Sobolev 1 constant for depolarizing

semigroups and applications to the concavity of the von

Neumann entropy.

The Logarithmic Sobolev 2 constant, hypercontractivity and

LS inequalities that tensorize with applications to the entropy

production.


Outline


constant.



Neumann entropy.



production.


Outline


constant.



Neumann entropy.



production.


Logarithmic Sobolev 1 Constant

Given a Liouvillian L :Md →Md with stationary state σ ∈ D+d

we want to estimate the convergence in the relative entropy:

D(etLρ||σ

)≤ e−2α1tD (ρ||σ)

with D (ρ||σ) = tr[ρ(log(ρ)− log(σ))].



Given a Liouvillian L :Md →Md with stationary state σ ∈ D+d

we want to estimate the convergence in the relative entropy:

D(etLρ||σ

)≤ e−2α1tD (ρ||σ)

with D (ρ||σ) = tr[ρ(log(ρ)− log(σ))].



Given a primitive Liouvillian L :Md →Md with stationary state

σ ∈ D+dwe want to estimate the convergence in the relative

entropy:

D(etLρ||σ

)≤ e−2α1tD (ρ||σ) (1)

with D (ρ||σ) = tr[ρ(log(ρ)− log(σ))].The largest α1 s.t. (1) holds for all t > 0 is the Logarithmic

Sobolev 1 constant.


Entropy Production

For S(ρ) = −tr[ρ log(ρ)] the von Neumann entropy and doubly

stochastic Liouvillians (L(1) = L∗(1) = 0), a LS-1 inequality is

equivalent to:

S(etLρ)− S(ρ) ≥ (1− e−2αt)(log(d)− S(ρ))

Provides a way of quantifying the production of entropy by the

semigroup.


Entropy Production



equivalent to:



semigroup.


Entropy Production



equivalent to:



semigroup.


Why you should care

LS inequalities have already found many applications, such as:

1 If we have a family of Liouvillians de�ned on a lattice that

have a LS constant which does not scale with size of the

system, this implies:

Strong notion of stability of observables w.r.t. perturbations ofthe Liouvillian1.

1T. S. Cubitt et al. �Stability of local quantum dissipative systems�. In:ArXiv e-prints (Mar. 2013). arXiv:1303.4744 [quant-ph]


http://arxiv.org/abs/1303.4744

Why you should care





Strong notion of stability of observables w.r.t. perturbations ofthe LiouvillianArea law and exponential decay of correlations for thestationary state. 23.

2M. J. Kastoryano and J. Eisert. �Rapid mixing implies exponential decay ofcorrelations�. In: Journal of Mathematical Physics 54.10 (Oct. 2013),p. 102201. DOI: 10.1063/1.4822481. arXiv:1303.6304 [quant-ph]

3F. G. S. L. Brandao et al. �Area law for �xed points of rapidly mixingdissipative quantum systems�. In: ArXiv e-prints (May 2015).arXiv:1505.02776 [quant-ph]


http://dx.doi.org/10.1063/1.4822481



Why you should care





Strong notion of stability of observables w.r.t. perturbations ofthe LiouvillianArea law and exponential decay of correlations for thestationary state.

2 These are all consequence of rapid mixing:

||etL(ρ)− σ||1 ≤ e−α1t√2 log

(σ−1min

)

3 Re�nements of entropic inequalities.

4 Analysis of the lifetime of quantum memories.


Why you should care







||etL(ρ)− σ||1 ≤ e−α1t√2 log

(σ−1min

)3 Re�nements of entropic inequalities.



Why you should care







||etL(ρ)− σ||1 ≤ e−α1t√2 log

(σ−1min

)3 Re�nements of entropic inequalities.



Di�erential Formulation

We can express the LS-1 constant as:

α1 (L) = infρ∈D+

d

tr[L(ρ)(log(σ)− log(ρ))]

D (ρ||σ)

Hard to compute analytically! Only known for doubly stochastic,

reversible qubit Liouvillians!


Di�erential Formulation

We can express the LS-1 constant as:

α1 (L) = infρ∈D+

d

tr[L(ρ)(log(σ)− log(ρ))]

D (ρ||σ)

Hard to compute analytically! Only known for doubly stochastic,

reversible qubit Liouvillians!


LS-1 Constant for the Depolarizing Channel

Using techniques from fractional programming, we have computed

this constant for the depolarizing channels Lσ(ρ) = tr(ρ)σ − ρ,σ ∈ D+

darbitrary.

α1 (Lσ) = minx∈[0,1]

1

2

(1+

D2 (σmin||x)D2 (x ||σmin)

)where D2 is the binary relative entropy. We also have:

1 ≥ α1(Lσ) ≥1

2

(1+

√σmin(1− σmin)

)





darbitrary.

α1 (Lσ) = minx∈[0,1]

1

2

(1+


)where D2 is the binary relative entropy.

We also have:

1 ≥ α1(Lσ) ≥1

2

(1+


)





darbitrary.

α1 (Lσ) = minx∈[0,1]

1

2

(1+


)where D2 is the binary relative entropy. We also have:

1 ≥ α1(Lσ) ≥1

2

(1+


)


Application: Concavity of the von Neumann Entropy

It follows from the last result that for ρ, σ ∈ Dd and q ∈ [0, 1] wehave

S((1− q)σ + qρ)− (1− q)S(σ)− qS(ρ) ≥

max

{q(1− qc(σ))D(ρ‖σ)(1− q)(1− (1− q)c(ρ))D(σ‖ρ)

,

with

c(σ) = minx∈[0,1]

D2(σmin‖x)D2(x‖σmin)

and c(ρ) de�ned in the same way.


Similar Result by Kim and Ruskai

We have4:

S((1− q)σ + qρ)− (1− q)S(σ)− qS(ρ) ≥ (1− q)q

2||ρ− σ||21

4Isaac Kim and Mary Beth Ruskai. �Bounds on the concavity of quantumentropy�. In: Journal of Mathematical Physics 55.9, 092201 (2014), pp. �


Comparison with Similar Results

q0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.02

0.04

0.06

0.08

0.1

0.12

(a)

q0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.02

0.04

0.06

0.08

0.1

0.12

(b)

Figure : Comparison of bound the bound by Kim(red), ours (blue) andthe exact value S((1− q)σ + qρ)− (1− q)S(σ)− qS(ρ) (black).


Entropy Production and Hypercontractivity

It is desirable to have lower bounds on α1(L) that are easier toevaluate.

We will focus on doubly stochastic, reversible Liouvillians

(L = L∗,L(1) = 0).

For quantum memories it is desirable to have bounds that

tensorize, that is α1(L(n)) ≥ c and L(n) the generator of

(etL)⊗n.





(L = L∗,L(1) = 0).



(etL)⊗n.





(L = L∗,L(1) = 0).



(etL)⊗n.


Hypercontractivity

The LS-2 constant of L is de�ned as the optimal α2 > 0 s.t. for all

X ∈M+dand t > 0

d12− 1

p(t)||etLX ||p(t)||X ||2

≤ 1

holds for p(t) = 1+ e2α2t .

Interpretation: larger p emphasizes the �peaks� in the spectrum of

X . If we have a small p-norm with p large, this means the

spectrum is ��at�.

α1 (L) ≥ α2 (L)Easier to handle!


Hypercontractivity


X ∈M+dand t > 0

d12− 1

p(t)||etLX ||p(t)||X ||2

≤ 1







Hypercontractivity


X ∈M+dand t > 0

d12− 1

p(t)||etLX ||p(t)||X ||2

≤ 1







Depolarizing channels again

Using a comparison technique, we show:

||L||α2(L 1

d

)≥ α2 (L) ≥ λα2

(L 1

d

)where λ is the spectral gap of L (second smallest eigenvalue of

−L) .This inequality tensorizes.

A bound for the depolarizing channel gives a universal lower

bound in terms of the spectral gap!




||L||α2(L 1

d

)≥ α2 (L) ≥ λα2

(L 1

d


−L) .

This inequality tensorizes.






||L||α2(L 1

d

)≥ α2 (L) ≥ λα2

(L 1

d








||L||α2(L 1

d

)≥ α2 (L) ≥ λα2

(L 1

d






Lower Bound for Depolarizing Channels

Use group theoretic techniques to relate the LS-2 constant of

the depolarizing channel to the LS-2 of a classical Markov

chain with known LS-2 constant.

These stay invariant under taking tensor powers, so we obtain:

α2

(L(n)1

d

)≥

(1− 2d−2

)log(3) log(d2 − 1) + 2 (1− 2d−2)

Improves upon previous bounds5 and has the right order of

magnitude.

5Kristan Temme, Fernando Pastawski, and Michael J Kastoryano.�Hypercontractivity of quasi-free quantum semigroups�. In: Journal of PhysicsA: Mathematical and Theoretical 47.40 (2014)







α2

(L(n)1

d

)≥

(1− 2d−2

)log(3) log(d2 − 1) + 2 (1− 2d−2)


magnitude.








α2

(L(n)1

d

)≥

(1− 2d−2

)log(3) log(d2 − 1) + 2 (1− 2d−2)


magnitude.



General Doubly Stochastic Liouvillians

For any doubly stochastic Liouvillian it follows that:

α2

(L(n)

)≥ λ

(1− 2d−2

)log(3) log(d2 − 1) + 2 (1− 2d−2)

λ is its spectral gap.



In terms of the entropy production, we have that:

S((etL)⊗n

ρ)− S(ρ) ≥ (1− e−2αt)(n log(d)− S(ρ))

with α = λ(1−2d−2)

log(3) log(d2−1)+2(1−2d−2)

This inequality can be used to analyze quantum memories

subjected to doubly stochastic noise.



In terms of the entropy production, we have that:

S((etL)⊗n

ρ)− S(ρ) ≥ (1− e−2αt)(n log(d)− S(ρ))

with α = λ(1−2d−2)

log(3) log(d2−1)+2(1−2d−2)

This inequality can be used to analyze quantum memories

subjected to doubly stochastic noise.


Summary

LS inequalities are a powerful framework to show entropic

inequalities and rapid mixing.

It is di�cult to obtain analytical results. Hypercontractivity is

a valuable tool to obtain lower bounds, especially for product

channels.

The potential quality of this bound decreases as the local

dimension increases, as made explicit by the depolarizing

semigroups.

The entropy always increases exponentially fast under local,

primitive and doubly stochastic noise.


Summary





channels.



semigroups.




Summary





channels.



semigroups.




Summary





channels.



semigroups.




Bibliography

[KT13] M. J. Kastoryano and K. Temme.Quantum logarithmic Sobolev inequalities and rapid mixing.Journal of Mathematical Physics, 54(5):052202, May 2013.

[OZ99] R. Olkiewicz and B. Zegarlinski.Hypercontractivity in noncommutative lp-spaces.Journal of Functional Analysis, 161(1):246 � 285, 1999.

[KR14] I. Kim and M. B. Ruskai,Bounds on the concavity of quantum entropyJournal of Mathematical Physics, vol. 55, no. 9, 2014.

[EK13] J. Eisert and M. J. KastoryanoRapid mixing implies exponential decay of correlationsJournal of Mathematical Physics, vol. 54, 2013

[CLM+13] Cubitt, T. S. and Lucia, A. and Michalakis, S. and Perez-GarciaStability of local quantum dissipative systemsarXiv 1303.4744

[BCL+15] Brandao, F. G. S. L. and Cubitt, T. S. and Lucia, A. and Michalakis, S. and Perez-Garcia,D.Area law for �xed points of rapidly mixing dissipative quantum systemsarXiv 1505.02776

[DSC96] Diaconis, P. and Salo�-Coste, L.Logarithmic Sobolev inequalities for �nite Markov chainsThe Annals of Applied Probability, vol. 6, no. 3, 1996


Thanks!


Date post:	31-Aug-2020
Category:	Documents
Upload:	others
View:	3 times
Download:	0 times

Logarithmic Sobolev Inequalities for Entropy Production - TUM · 2016. 2. 23. · Logarithmic...

Documents