Sandwiched R´enyi divergences and the strong converse exponent of state discrimination in operator algebras

exponent of state discrimination in operator algebras

Fumio Hiai

Tohoku University

2021, Sep. (RIMS Workshop)

Recent Developments in Operator Algebras

Fumio Hiai (Tohoku University) Sandwiched R ´enyi divergences 2021, Sep. (RIMS Workshop) 1 / 23

Plan

Sandwiched R ´enyi divergences The finite dimensional case

Definitions: the von Neumann algebra case Properties ofD_α

Properties ofD^∗_α

Quantum hypothesis testing Stein’s lemma

Chernoff bound (symmetric type) Hoeffding bound (asymmetric type)

Hoeffding anti-bound (strong converse type) The

C

^∗-algebra case

The finite dimensional case H

is a finite dimensional Hilbert space.

ρ, σ ∈ B( H )

₊,

ρ , 0

.

α ∈ [0 , ∞ )

,

α , 1

. Standard R ´enyi divergence

Q

_α

( ρ∥σ ) : = Tr( ρ

^α

σ

^1−α

) , D

_α

( ρ∥σ ) : = 1

α − 1 log Q

_α

( ρ∥σ ) Tr ρ .

Sandwiched R ´enyi divergence(M ¨uller-Lennert at al., 2013;

Wilde–Winter–Yang, 2014)

Q

^∗_α

( ρ∥σ ) : = 

 Tr (

σ

^12−αα

ρσ

^12−αα

)

_α

if

0 < α < 1

or

s( ρ ) ≤ s( σ ) ,

+∞

^otherwise

,

D

^∗_α

( ρ∥σ ) : = 1 α − 1 log

Q

^∗_α

( ρ∥σ ) Tr ρ .

Note

lim

α→1

D

_α

( ρ∥σ ) = lim

α→1

D

^∗_α

( ρ∥σ ) = D( ρ∥σ ) / Tr ρ

^{, where}

D( ρ∥σ ) : = Tr( ρ log ρ − ρ log σ ) ,

relative entropy

.

Fumio Hiai (Tohoku University) Sandwiched R ´enyi divergences 2021, Sep. (RIMS Workshop) 3 / 23

Preliminaries

M

is a von Neumann algebra with the core

( N = M ⋊

_σ^φ

R, θ = σ c

^φ

, τ ) , τ ◦ θ

s

= e

^−s

τ.

For

p ∈ (0 , ∞ ]

,

L

^p

( M)

isHaagerup’s

L

^p-space, i.e.,

L

^p

( M) : = { a ∈ N e : θ

s

(a) = e

^−s/p

a , s ∈ R}.

In particular,

L

^∞

( M) = M

,

L

¹

(M) M

_∗with

ψ ∈ M

_∗

↔ h

_ψ

∈ L

¹

( M)

,

tr : L

¹

(M) → C

^by

tr(h

_ψ

) : = ψ (1)

. For

p ∈ [1 , ∞ ]

and a faithful

σ ∈ M

_∗⁺,

L

^p

(M , σ )

isKosaki’s interpolation

L

^p-space, i.e.,

L

^p

( M , σ ) : = C

_1/p

( M , L

¹

(M))

with the interpolation norm

∥ · ∥

p,σ

,

where

M , → L

¹

( M)

by

x 7→ h

¹_σ^/2

xh

¹_σ^/2with

∥ h

¹_σ^/2

xh

¹_σ^/2

∥ = ∥ x ∥

^. In particular,

L

¹

( M, σ ) = L

¹

( M)

,

L

^∞

( M, σ ) = h

^1/2_σ

Mh

^1/2_σ (

M

).

Theorem (Kosaki, 1984)

For every

p ∈ [1 , ∞ ]

and

1 / p + 1 / q = 1

,

L

^p

( M , σ ) = h

1 2q

σ

L

^p

( M)h

1 2q

σ

( ⊆ L

¹

(M)) ,

∥ h

1 2q

σ

ah

1 2q

σ

∥

p,σ

= ∥ a ∥

p

, a ∈ L

^p

( M) ,

that is,

L

^p

( M) L

^p

( M , σ )

by

a 7→ h

1 2q

σ

ah

1 2q

σ .

Note When

σ

is not faithful with

e = s( σ )

,

L

^p

(M , σ )

is still defined for

(eMe , σ|

eMe

)

, and the above theorem holds with

eL

^p

( M)e L

^p

(eMe)

in place of

L

^p

( M)

.

Fumio Hiai (Tohoku University) Sandwiched R ´enyi divergences 2021, Sep. (RIMS Workshop) 5 / 23

Definitions: the von Neumann algebra case

(M,L²(M),J = ^∗,L²(M)₊)is astandard formofM.

Forσ∈ M⁺_∗,σ(x)=tr(h_σx)=⟨h¹_σ^/2,xh¹_σ^/2⟩. Forρ, σ∈ M⁺_∗,∆ρ,σis therelative modular operator.

Definition (Kosaki, 1982; Petz, 1985)

Letρ, σ∈ M⁺_∗ andα∈[0,∞)\ {1}. For0≤α <1, Q_α(ρ∥σ) :=∥∆^α/_ρ,σ²h¹_σ^/2∥² ∈[0,+∞). Forα >1,

Q_α(ρ∥σ) :=

∥∆^α/_ρ,σ²h¹_σ^/2∥² ifs(ρ)≤ s(σ)andh¹_σ^/2∈ D(∆^α/_ρ,σ²),

+∞ otherwise.

Whenρ,0, thestandardα-R ´enyi divergenceis

D_α(ρ∥σ):= 1

α−1logQ_α(ρ∥σ) ρ(1) .

Definition (Berta–Scholz–Tomamichel, 2018; Jenˇcov ´a, 2018) The following is Jenˇcov ´a’s definition:

Let

ρ, σ ∈ M

⁺_∗ and

α ∈ [1 / 2 , ∞ ) \ { 1 }

^{. For}

α ∈ (1 , ∞ )

,

Q

^∗_α

( ρ∥σ ) : = 

 ∥ h

_ρ

∥

^α_α,σ ^if

h

_ρ

∈ L

^α

( M , σ )

(hence

s( ρ ) ≤ s( σ )

)

, +∞

^ohterwise

.

For

α ∈ [1 / 2 , 1)

,

Q

^∗_α

( ρ∥σ ) : = tr ( h

1−α 2α

σ

h

_ρ

h

1−α 2α

σ

)

_α

.

When

ρ , 0

, thesandwiched

α

-R ´enyi divergenceis

D

^∗_α

( ρ∥σ ) : = 1 α − 1 log

Q

^∗_α

( ρ∥σ ) ρ (1) .

Note In particular, for

α = 1 / 2

,

Q

^∗

1/2

( ρ∥σ ) = F( ρ, σ ) : = tr (

h

¹_σ^/2

h

_ρ

h

¹_σ^/2

)

1/2

, D

^∗

1/2

( ρ∥σ ) = − 2 log F( ρ, σ ) ρ (1) .

fidelity

Fumio Hiai (Tohoku University) Sandwiched R ´enyi divergences 2021, Sep. (RIMS Workshop) 7 / 23

Properties of D

_α

The limit

α↗lim1D_α(ρ∥σ)= D(ρ∥σ)

ρ(1) =: D1(ρ∥σ)

exists. If D_α(ρ∥σ)<+∞for someα >1, thenlim_α↘1D_α(ρ∥σ)= D1(ρ∥σ).

The functionα∈[0,∞)7→ D_α(ρ∥σ)is monotone increasing.

The map(ρ, σ)∈(M⁺_∗ \ {0})×M⁺_∗ 7→ D_α(ρ∥σ)∈(−∞,+∞]is jointly lower semicontinuous in theσ(M_∗,M)-topology when0≤α≤2, and jointly continuous in the norm topology when0 ≤α < 1.

The map(ρ, σ)∈ M⁺_∗ ×M_∗⁺7→ Q_α(ρ∥σ)

is jointly concave for0≤α <1, and jointly convex for1< α≤2.

When0≤α≤1,D_α(ρ∥σ)is jointly convex on

{(ρ, σ)∈M⁺_∗ ×M⁺_∗ :ρ(1)= c}for any fixedc>0. When0 ≤α≤2, the mapσ∈M⁺_∗ 7→ D_α(ρ∥σ)is convex for any fixedρ∈M_∗⁺,ρ,0.

When0≤α≤2, σ1 ≤σ2 =⇒ D_α(ρ∥σ1)≥ D_α(ρ∥σ2).

Monotonicity (Data-processing inequality):For everyα ∈[0,2]and any unital normal Schwarz mapγ:N→ M,

D_α(ρ◦γ∥σ◦γ)≤ D_α(ρ∥σ).

Martingale convergence:Let{Mi}i∈I be a net of increasing unital von Neumann subalgebras of Mwith M=(∪

i∈I Mi)′′. Then for every α∈[0,2],

D_α(ρ|Mi∥σ|Mi) ↗ D_α(ρ∥σ).

Let{ei}be an increasing net of projections in Mwithei ↗1. Then for every α∈[0,2],

D_α(eiρei∥eiσei) ↗ D_α(ρ∥σ).

Fumio Hiai (Tohoku University) Sandwiched R ´enyi divergences 2021, Sep. (RIMS Workshop) 9 / 23

Properties of D

^∗_α

The limit

limα↗1D^∗_α(ρ∥σ)= D(ρ∥σ)

ρ(1) =: D^∗₁(ρ∥σ) (= D1(ρ∥σ)) exists. If D^∗_α(ρ∥σ)<+∞for someα >1, thenlim_α↘1D^∗_α(ρ∥σ)= D^∗

1(ρ∥σ).

Moreover,lim_α→∞ D^∗_α(ρ∥σ)= Dmax(ρ∥σ), where

Dmax(ρ∥σ) :=log inf{t >0 :ρ≤ tσ}. max-relative entropy. The functionα∈[1/2,∞)7→ D^∗_α(ρ∥σ)is monotone increasing.

Tthe map(ρ, σ)∈(M⁺_∗ \ {0})×M⁺_∗ 7→ D^∗_α(ρ∥σ)is jointly lower semi-continuous in the norm topology when1≤α <∞, and jointly continuous in the norm topology when1/2≤α <1.

The map(ρ, σ)∈ M⁺_∗ ×M_∗⁺7→ Q^∗_α(ρ∥σ)is jointly convex for1< α <∞and jointly concave for1/2 ≤α <1.

When1/2≤α≤1, D^∗_α(ρ∥σ)is jointly convex on {(ρ, σ)∈M⁺_∗ ×M⁺_∗ :ρ(1)= c}for any fixedc>0.

When1/2≤α <∞, σ1≤σ2 =⇒ D^∗_α(ρ∥σ1)≥ D^∗_α(ρ∥σ2).

Monotonicity (DPI):For everyα∈[1/2,∞)and any unital normal positive mapγ: N→ M,

D^∗_α(ρ◦γ∥σ◦γ)≤ D^∗_α(ρ∥σ).

Martingale convergence:Assume thatMisσ-finite, and let{Mi}i∈I be a net of increasing unital von Neumann subalgebras ofMwith M=(∪

i∈I Mi)_′′

. Then for everyα∈[1/2,∞),

D^∗_α(ρ|Mi∥σ|Mi) ↗ D^∗_α(ρ∥σ).

Assume thatMisσ-finite, and let{ei}be an increasing net of projections in Mwithei↗ 1. Then for everyα∈[1/2,∞),

D^∗_α(eiρei∥eiσei) ↗ D^∗_α(ρ∥σ).

For everyα∈[1/2,∞),

D^∗_α(ρ∥σ)≤ D_α(ρ∥σ).

Fumio Hiai (Tohoku University) Sandwiched R ´enyi divergences 2021, Sep. (RIMS Workshop) 11 / 23

Proposition (Frank–Lieb, 2013¹; H, 2021²; Jenˇcov ´a, 2021³) For anyρ, σ∈M⁺_∗.

(i) For everyα∈(1,∞), Q^∗_α(ρ∥σ)= sup

x∈M₊

[αρ(x)−(α−1)tr(

h_σ^α−12α xh_σ^α−12α)_α−1^α ] .

(ii) For everyα∈[1/2,1), Q^∗_α(ρ∥σ)= inf

x∈M₊₊

[αρ(x)+(1−α)tr( h

1−α2α

σ x⁻¹h

1−α2α

σ

)_1−α^α ] .

1R. L. Frank and E. H. Lieb, Monotonicity of a relative R ´enyi entropy,J. Math. Phys.54 (2013), 122201, 5 pp.

2F. Hiai,Quantum f-Divergences in von Neumann Algebras. Reversibility of Quantum Operations, Mathematical Physics Studies, Springer, Singapore, 2021.

3A. Jenˇcov ´a, R ´enyi relative entropies and noncommutativeLp-spaces II,Ann. Henri Poincar ´e, Online first, 2021.

Quantum hypothesis testing

ρ, σ ∈ M

⁺_∗ are states.

For each

n ∈ N

^,

ρ

n

: = ρ

^⊗n,

σ

n

: = σ

^⊗nin

( M

^⊗n

)

₊

∗.

Consider the simple (i.e.,i.i.d.) hypothesis testing problem for

H

₀

: ρ

n (the null hypothesis) vs.

H

₁

: σ

n (the alternative hypothesis)

. T

_n

∈ M

^⊗nwith

0 ≤ T

_n

≤ 1

is atest.

For a test

T

_n,

α

n

(T

_n

) : = ρ

n

(1 − T

_n

)

the type I error probability

, β

n

(T

n

) : = σ

n

(T

n

)

the type II error probability

, 1 − α

n

(T

n

) : = ρ

n

(T

n

)

the type I success probability

.

Fumio Hiai (Tohoku University) Sandwiched R ´enyi divergences 2021, Sep. (RIMS Workshop) 13 / 23

finite dim. case v.N. algebra case Stein’s lemma

Chernoff type

Hoeffding type

Strong converse type

H–Petz, 1991

Ogawa–Nagaoka, 2000

H–Petz, 1991 Jakˇsi´c et al., 2012⁴ Audenaert et al., 2007

Nussbaum–Szkoła, 2009 H–Mosonyi–Ogawa, 2008

Jakˇsi´c et al., 2012

Hayashi, 2007 Nagaoka, 2006

H–Mosonyi–Ogawa, 2008

Jakˇsi´c et al., 2012

Mosonyi–Ogawa, 2015⁵ H–Mosonyi, 2021?

4V. Jakˇsi´c, Y. Ogata, C.-A. Pillet and R. Seiringer, Quantum hypothesis testing and non-equilibrium statistical mechanics,Rev. Math. Phys.24(2012), 1230002, 67 pp.

5M. Mosonyi and T. Ogawa, Quantum hypothesis testing and the operational interpretation of the quantum R ´enyi relative entropies,Comm. Math. Phys.334(2015), 1617–1648.

Stein’s lemma

Define

B(ρ∥σ) :=sup

{Tn}

{ lim inf

n→∞ −1

nlogβn(Tn) lim

n→∞αn(Tn)=0 },

B(ρ∥σ) :=sup

{T_n}

{ lim sup

n→∞ −1

nlogβn(Tn) lim

n→∞αn(Tn)=0 },

B(ρ∥σ) :=sup

{T_n}

{

nlim→∞−1

nlogβn(Tn) lim

n→∞αn(Tn)=0 }.

Moreover, for anyε∈(0,1)define β^∗_ε(ρn∥σn) := min

0≤T_n≤1{βn(Tn) :αn(Tn)≤ε}.

Theorem If D_α(ρ∥σ)<+∞for someα >1, then B(ρ∥σ)= B(ρ∥σ)= B(ρ∥σ)= lim

n→∞−1

nlogβ^∗_ε(ρn∥σn)= D(ρ∥σ).

Fumio Hiai (Tohoku University) Sandwiched R ´enyi divergences 2021, Sep. (RIMS Workshop) 15 / 23

Chernoff bound (symmetric type)

TheChernoff boundis C(ρ∥σ) :=− inf

0≤α<1logQ_α(ρ∥σ)= sup

0≤α<1

(1−α)D_α(ρ∥σ).

For anyπ∈(0,1)define

e^∗_π(ρn∥σn) := min

0≤Tn≤1{παn(Tn)+(1−π)βn(Tn)}.

Theorem

nlim→∞−1

nloge^∗_π(ρn∥σn)=C(ρ∥σ).

Hoeffding bound (asymmetric type)

Forr∈RtheHoeffding boundis Hr(ρ∥σ) := sup

0<α<1

α−1

α {r−D_α(ρ∥σ)}.

Define theasymmetric error exponentsas B_r(ρ∥σ) :=sup

{T_n}

{ lim inf

n→∞ −1

nlogαn(Tn)lim inf

n→∞ −1

nlogβn(Tn)≥ r },

Br(ρ∥σ) :=sup

{Tn}

{ lim sup

n→∞ −1

nlogαn(Tn)lim inf

n→∞ −1

nlogβn(Tn)≥r },

Br(ρ∥σ) :=sup

{T_n}

{

nlim→∞−1

nlogαn(Tn) lim inf

n→∞ −1

nlogβn(Tn)≥r }.

Moreover, define

α^∗_e−nr(ρn∥σn) := min

0≤Tn≤1{αn(Tn) :βn(Tn)≤e^−nr}.

Theorem For any r, D0(ρ∥σ),

Br(ρ∥σ)= Br(ρ∥σ)=Br(ρ∥σ)= lim

n→∞−1

nlogα^∗_e−nr(ρn∥σn)=Hr(ρ∥σ).

Fumio Hiai (Tohoku University) Sandwiched R ´enyi divergences 2021, Sep. (RIMS Workshop) 17 / 23

Hoeffding anti-bound (strong converse type)

Forr∈RtheHoeffding anti-boundis H^∗_r(ρ∥σ) :=sup

α>1

α−1

α {r− D^∗_α(ρ∥σ)}.

Define thestrong converse exponentsas scr(ρ∥σ) :=inf

{T_n}

{ lim inf

n→∞ −1

nlog(1−αn(Tn))lim inf

n→∞ −1

nlogβn(Tn)≥ r },

scr(ρ∥σ) :=inf

{T_n}

{ lim sup

n→∞ −1

nlog(1−αn(Tn)) lim inf

n→∞ −1

nlogβn(Tn)≥r },

scr(ρ∥σ) :=inf

{T_n}

{

nlim→∞−1

nlog(1−αn(Tn)) lim inf

n→∞ −1

nlogβn(Tn)≥r }.

Theorem Assume thatMisinjective. IfD^∗_α(ρ∥σ)<+∞for someα >1, then for anyr∈R,

scr(ρ∥σ)= scr(ρ∥σ)= scr(ρ∥σ)= lim

n→∞−1

nlog{1−α^∗_e−nr(ρn∥σn)}=H^∗_r(ρ∥σ).

Analogy

Large deviation theory(i.i.d. case) Quantum hypothesis testing(sc case) (Xn)^∞

n=1i.i.d. random variables ρ, σ∈ M_∗⁺states Logarithmic moment generating

function

Sandwiched R ´enyi divergence Λ(u) :=logE(e^uX1), u ∈R ψ(s) :=logQ^∗

s+1(ρ∥σ)= s D^∗

s+1(ρ∥σ), s∈(0,∞) ψ˜(u) :=(1−u)ψ( 1

1−u

), u∈(0,1)

Rate function Hoeffding anti-bound

Λ^∗(r) :=sup_u∈R{ur−Λ(u)} (Fenchel–Legendre transform)

H^∗_r(ρ∥σ) :=sup_α>1^α−_α¹{r−D^∗_α(ρ∥σ)}

=sup_u∈(0,1){ur−ψ˜(u)}, r∈R Empirical distribution Optimal (success) probability

µn:=µ¹

n

∑n

i=1Xi 1−α^∗_e−nr(ρn∥σn)

:=max_0≤Tn≤1{1−αn(Tn) :βn(Tn)≤e^−nr}

Cram ´er’s LDP Asymptotic bound

1

nlogµn(F)≈ −infr∈FΛ^∗(r) ¹_nlog(1−α^∗_e−nr(ρn∥σn))≈ −H^∗_r(ρ∥σ)

Fumio Hiai (Tohoku University) Sandwiched R ´enyi divergences 2021, Sep. (RIMS Workshop) 19 / 23

Outlook

We may conclude that the “right” quantum

α

-R ´enyi divergence is

 





D

_α

( ρ∥σ )

when

0 < α < 1 ,

D

^∗_α

( ρ∥σ )

when

α > 1 ,

D( ρ∥σ )

when

α = 1 .

The C

^∗

-algebra case

A

is a unital

C

^∗-algebra.

{π

u

, H

u

}

is the universal representation of

A

^.

ρ, σ ∈ A

^∗₊^.

ρ, σ

are the normal extensions to

π

u

( A )

^′′

A

^{∗∗such that}

ρ = ρ ◦ π

u

and

σ = σ ◦ π

u.

A representation

π : A → B( H )

is

( ρ, σ )

-normal if there exist

ρ

_π

, σ

_π

∈ ( π ( A )

^′′

)

⁺_∗ such that

ρ = ρ

_π

◦ π

and

σ = σ

_π

◦ π

.

Lemma

Let

π

^{be any}

( ρ, σ )

-normal representation

π

^of

A

^{. Then}

D

_α

( ρ∥σ ) = D

_α

( ρ

π

∥σ

π

) , α ∈ [0 , ∞ ) \ { 1 }, D

^∗_α

( ρ∥σ ) = D

^∗_α

( ρ

_π

∥σ

_π

) , α ∈ [1 / 2 , ∞ ) \ { 1 }.

Fumio Hiai (Tohoku University) Sandwiched R ´enyi divergences 2021, Sep. (RIMS Workshop) 21 / 23

TheC^∗-algebra case

Definition

Define

D

_α

( ρ∥σ ) : = D

_α

( ρ∥σ )

and

D

^∗_α

( ρ∥σ ) : = D

^∗_α

( ρ∥σ )

, extending

D

_α

( ρ∥σ )

,

D

^∗_α

( ρ∥σ )

when

ρ, σ ∈ M

⁺_∗.

Define the Hoeffding bound

H

_r

( ρ∥σ )

and the Hoeffding anti-bound

H

^∗_r

( ρ∥σ )

in the same way as in the von Neumann algebra case.

Define the exponents

sc

r

( ρ∥σ )

,

sc

_r

( ρ∥σ )

,

sc

_r

( ρ∥σ )

, etc. in the same way as in the von Neumann algebra case, where

{T

n

}

in the present case are taken as

T

n

∈ A

^⊗n(the

n

-fold minimal

C

^∗-tensor product),

0 ≤ T

_n

≤ 1

,

n ∈ N

^.

Lemma

Let

π

^{be any}

( ρ, σ )

-normal representation

π

^of

A

. Then for every

r ∈ R

^,

sc

r

( ρ∥σ ) = sc

r

( ρ

π

∥σ

π

)

etc.

Theorem

Assume that

A

^isnuclear(equivalently,

A

^∗∗is injective). If

D

^∗_α

( ρ∥σ ) < +∞

for some

α > 1

, then for any

r ∈ R

the result on the strong converse exponents in the von Neumann algebra case hold in the

C

^∗-algebra setting too.

Fumio Hiai (Tohoku University) Sandwiched R ´enyi divergences 2021, Sep. (RIMS Workshop) 23 / 23