a new definition for the quantum conditional rényi...

A new definition for the quantumconditional Renyi entropy

Frederic DupuisAarhus University

May 21, 2014

Frederic Dupuis Quantum Renyi entropies May 21, 2014 1 / 29

The papers

This talk will contain results from four papers:Martin Muller-Lennert, Frederic Dupuis, Oleg Szehr, SergeFehr, Marco Tomamichel. On quantum Renyi entropies: anew definition and some properties. 1306.3142Mark M. Wilde, Andreas Winter, Dong Yang. Strongconverse for the classical capacity ofentanglement-breaking channels. 1306.1586Rupert L. Frank and Elliott H. Lieb. Monotonicity of arelative Renyi entropy. 1306.5358Salman Beigi. Quantum Renyi divergence satisfies dataprocessing inequality. 1306.5920


Entropy

Shannon entropy:

Hppq :“ ´ÿ

i

pi log pi

Immensely useful in information theory, appears everywhere.Why this function in particular?


Renyi’s axioms

Shannon entropy can be derived axiomatically [Renyi, 1961].Let H be a function from unnormalized probability distributionsto the reals, and suppose it satisfies:

1 Continuity: Hppq is continuous.2 Symmetry: Hppq “ Hpπppqq for any permutation π3 Normalization: Hpt1

2uq “ 1

4 Additivity: Hpp� qq “ Hppq `Hpqq

5 Mean value: Hpp� qq “ TrrpsTrrp`qs

Hppq ` TrrqsTrrp`qs

Hpqq

Then Hppq is the Shannon entropy of the distribution p.


Renyi’s axioms

Why? (1), (3) and (4) impose that

Hptxuq “ ´ log2 x

[Erdos 1946]. The mean-value property does the rest:

Hptp1, . . . , pnuq “ Hptp1u� tp2, . . . , pnuq

“p1

TrrpsHptp1uq `

Trrtp2, . . . , pnus

TrrpsHptp2, . . . , pnuq

“ÿ

i

piTrrps

Hptpiuq

“ ´ÿ

i

piTrrps

log2 pi.


Renyi’s axioms

The mean-value axiom seems oddly specific. What if we relaxit?

1 Continuity: Hppq is continuous.2 Symmetry: Hppq “ Hpπppqq for any permutation π3 Normalization: Hpt1

2uq “ 1

4 Additivity: Hpp� qq “ Hppq `Hpqq5 General Mean:gpHpp� qqq “ Trrps

Trrp`qsgpHppqq ` Trrqs

Trrp`qsgpHpqqq, where g is

some strictly monotonic function.


Renyi’s axioms

Different choices of g yield interesting entropies:gpxq “ x: Shannon entropygαpxq “ 2p1´αqx: Renyi entropy of order α:

Hαppq :“1

1´ αlog

1

Trrps

ÿ

i

pαi .


Renyi entropy

The Renyi entropy unifies a lot of other entropies:αÑ 1: Shannon entropyα “ 2: Collision entropyαÑ 8: Min-entropy: Hminppq “ ´ log 1

Trrpsmaxi pi

α “ 12: Max-entropy: Hmaxppq “ 2 log 1

Trrps

ř

i

?pi.


Divergence and conditional entropy

What about conditional entropy, mutual information, etc? Thesecan be defined via a notion of divergence (“relative entropy”).For Shannon entropy:

Dpp}qq :“ÿ

i

piplog pi ´ log qiq.

Then,Hppq :“ Dpp}1q

where 1 “ t1, 1, . . . , 1u.


Conditional entropy and mutual information

Conditional entropy:

HpX|Y qp :“ ´DppXY }1X � pY q “ ´minqY

DppXY }1X � qY q.

Mutual information:

IpX;Y qp :“ DppXY }pX � pY q “ minqY

DppXY }pX � qY q.


Renyi’s axioms for divergence

Can we generalize these? Yes, by axiomatizing the divergence:1 Continuity: Dpp}qq is continuous in p, q.2 Symmetry: Dpp}qq “ Dpπppq}πpqqq for any permutation π.3 Normalization: Dpt1u}t1

2uq “ 1.

4 Order: If p ě q, then Dpp}qq ě 0; if p ď q, then Dpp}qq ď 0.5 Additivity: Dpp� r}q � sq “ Dpp}qq `Dpr}sq for allp, q, r, s ě 0 such that q Ţ p and s Ţ r.

6 General Mean:

gpDpp� r}q� sqq “Trrps

Trrp` rsgpDpq}sqq`

Trrrs

Trrp` rsgpDpr}sqq.


Renyi’s axioms for divergence

Different choices of g yield corresponding divergences:gpxq “ x: Usual relative entropygpxq “ 2p1´αqx: Renyi divergence:

Dαpp}qq “1

α ´ 1log

1

Trrps

ÿ

i

pαi q1´αi .

We can define conditional entropy and mutual information thisway:

HαpX|Y qp :“ ´minqY

DαppXY }1X � qY q

IαpX;Y qp :“ minqY

DαppXY }pX � qY q.


Quantum divergence

Can we make this quantum? We can “quantize” the axioms. Letρ, σ be positive semidefinite.

1 Continuity: Dpρ}σq is continuous in ρ, σ.2 Symmetry: Dpρ}σq “ DpUρU :}UσU :q for any unitary U .3 Normalization: Dpr1s}r1

2sq “ 1.

4 Order: If ρ ě σ, then Dpρ}σq ě 0; if ρ ď σ, then Dpρ}σq ď 0.5 Additivity: Dpρ� τ}σ � ωq “ Dpρ}σq `Dpτ}ωq for allρ, σ, τ, ω ě 0 such that σ Ţ ρ and ω Ţ τ .

6 General Mean:

gpDpρ�τ}σ�ωqq “Trrρs

Trrρ` τ sgpDpρ}σqq`

Trrτ s

Trrρ` τ sgpDpτ}ωqq.


Quantum divergence

Does this isolate a unique function?For commuting ρ and σ, this must reduce to thecorresponding g-divergence.What if ρ and σ don’t commute? No, multiple definitionssatisfy the axioms. For example:

D1αpρ}σq :“1

α ´ 1log

1

TrrρsTrrρασ1´α

s

D2αpρ}σq :“1

α ´ 1log

1

TrrρsTrrpσp1´αq{4ρα{2σp1´αq{4q2s.

for α between 1 and 2.


Quantum divergence

How do we pick the right version?Add an axiom: If you have ideas, let me know!Next best thing: pick a version that happens to satisfy othernice properties.


Quantum divergence

What nice properties?Corresponds to usual quantum entropies:

For αÑ 1, get the usual relative entropy.For αÑ8, get Dmax.For αÑ 2, get the collision entropy.For α “ 1

2 , get Dmin.

Data processing: Dαpρ}σq ě DαpFpρq}Fpσqq for any CPTPmap F .Monotonicity in α: Dαpρ}σq ď Dβpρ}σq if α ď β.Duality: HαpA|Bq “ ´HβpA|Cq for 1

α` 1

β“ 2 (will be

explained later)Chain rules


The definition

We define:

Dαpρ}σq :“1

α ´ 1log

ˆ

1

TrrρsTr

”´

σ12α´ 1

2ρσ12α´ 1

2

¯αı˙

.

Contrast with

D1αpρ}σq :“1

α ´ 1log

ˆ

1

TrrρsTrrρασ1´α

s

˙

D2αpρ}σq :“1

α ´ 1log

ˆ

1

TrrρsTrrpσp1´αq{4ρα{2σp1´αq{4q2s

˙

.


Properties

Properties of the new Renyientropy


Corresponds to known entropies

For αÑ 1, get the usual relative entropy.For αÑ 8, get Dmax.For αÑ 2, get the collision entropy.For α “ 1

2, get Dmin.


α “ 8: Max-relative entropy/min-entropy

Recall that

Dmaxpρ}σq “ logmintλ : ρ ď λσu

“ log›

›

›σ´

12ρσ´

12

›

›

›

8.

Contrast with α-divergence:

Dαpρ}σq “α

α ´ 1log

ˆ

1

Trrρs1{α

›

›

›σ

12α´ 1

2ρσ12α´ 1

2

›

›

›

α

˙

.


α “ 12: Min-relative entropy/Max-entropy

Recall that

Dminpρ}σq “ 2 log

ˆ

1

TrrρsTr

”

pσ12ρσ

12 q

12

ı

˙

Contrast with α-divergence:

Dαpρ}σq :“1

α ´ 1log

ˆ

1

TrrρsTr

”´

σ12α´ 1

2ρσ12α´ 1

2

¯αı˙


α “ 2: Collision entropy

Collision entropy:

D2pρ}σq “ log1

TrrρsTr

“

pσ´1{4ρσ´1{4q2‰

.

The associated conditional entropy is used in proofs of privacyamplification, decoupling, etc.Contrast with α-divergence:

Dαpρ}σq :“1

α ´ 1log

ˆ

1

TrrρsTr

”´

σ12α´ 1

2ρσ12α´ 1

2

¯αı˙


Data processing inequality

We have thatDαpρ}σq ě DαpFpρq}Fpσqq

for any CPTP map F and any 12ď α ď 8.

For entropies, this means that

HαpA|BCq ď HαpA|Bq.


Duality

For any pure ρABC , we have that

H1pA|Bqρ “ ´H1pA|Cqρ.


Duality

Recall

HminpA|Bqρ “ ´minσDmaxpρAB}1A � σBq

HmaxpA|Bqρ “ ´minσDminpρAB}1A � σBq.

For any pure state ρABC ,

HminpA|Bqρ “ ´HmaxpA|Cqρ.


Duality

We can generalize this:

HαpA|Bqρ “ ´HβpA|Cqρ,

where 1α` 1

β“ 2.

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5


Conclusion

New definition of quantum Renyi divergenceLots of cools propertiesMilan Mosonyi’s talk: application to hypothesis testingKoenraad Audenaert’s talk: generalization of α entropies


Thank you

Thank you!


a new definition for the quantum conditional rényi...

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