Role of Hypothesis Testing in Quantum Informationaqis-conf.org/2017/wp-content/uploads/2017/09/D3_01_InvitedTalk... · Role of Hypothesis Testing in Quantum Information Masahito Hayashi

Role of Hypothesis Testing in

Quantum Information

Masahito Hayashi

Graduate School of Mathematics, Nagoya University

Centre for Quantum Technologies, National University of

Singapore

(Many collaborators)

Contents

• Application to Quantum Coding Theory

• Application to Quantum Key Distribution– Classical Operations

– Finite-length Analysis with single photon source

– Finite-length Analysis with decoy method

• Application to Testing of Quantum Computation

Application to Quantum

Coding Theory

Part I

Quantum hypothesis testingQuantum unknown state

on or

n n nH

,n n

T I TnTQuantum measurement

Decision

• First error probability:

• Second error probability:

1Tr

n

n nI T T

2Tr

n

n nT T

Non-asymptotic formulas

2

1

s.t. Tr 0 ,

Tr 0

a

a

a e

e

T T

T

1 2 Tr 0a a

a

e e

T

T T

Direct part:

Converse part: Neymann Person lemma

where 0

0i

i i i i iix

x u u u u

Asymptotic Evaluation

2 1 sup log : limn n n n

B - T T T

Hiai-Petz 1991, Ogawa-Nagaoka 1998

Tomamichel-Hayashi 2013, Li 2014

1 ( ) ( ) ( ) ( )n

B nD nV o n

1 0

Tr log logD

2Tr [ log log ]V D

2

2

( ) :2

t

x ex dt

Large Deviation

Hayashi 2007,

Nagaoka 2007

Hayashi 2004,

Mosyoi-Ogawa

2015

1

0 1

( ) sup (1)

1

snr

ns

sr sDB e n o

s

1 0

2 1 inf log(1 ) : limc

n n n nB - T T T

1

0

( ) sup (1)

1

sc nr

ns

sr sDB e n o

s

1

1[logTr ]/s s

sD s

1 1

112 21

1

[log Tr( ) ] /

lim[ ( ) ] /n

s

sss

n n

sn

D s

D n

E

( ) ,X i i i i

i i

Y PYP X x P E

Quantum channel coding

(classical message)

Classical message

Encode

1, ,n

i M

n

Channel

1

Mn n

jj

Y Y

Decode (Measurement)

1 2( ) ( ) ( ) ( )

n n

ni i i i

1 2( ), ( ), , ( )

n

ni i i i

1, ,n

j M

( )x x

Mathematical formulation

• Channel:

• Code:

• Size:

• Average error:

• Trade off:

, ,n n

nM YnE

1

11 Tr

nM

n n n

n i

in

i YM

E

nM MnE

( )x x

,n n

M E E

Non-asymptotic formula

2Tr 0

4 Tr 0 ,

n n na n

Q Q Q

na n n na n

Q Q Q

na

e

e e

M e

n

n

E

E

Direct part (Possibility part):Hayashi-Nagaoka 2002

s.t. a Q nE

( ) ,

( ) ( ) .

n n

Q x

x

n n

Q x

x x

Q x x x W

Q x x x Q x W

X

X

where

1

Tr 0n n na n na

Q Qe e M

n nE E

Converse part (Impossibility part): s.t. na nE Q

Asymptotic formulas

,

logsup lim

sup min Tr 0

sup min Tr 0

max max ( )

=

=

.

n n n

n n nn

n n na n

Q Q QQa

n n na n

Q QQa

Q Q x QQ Q

x

MC

n

a e

a e

D Q x D W W

n

n

nE

EE

X

Holevo 1996, Schumacher-Westmoreland 1997

Hayashi-Nagaoka 2002

General case

DMS (i.i.d.) case

1 0 For

Exponents• Error probability

• Correct probability

:log

sup lim loglim

e

n

n

n

B r

M

rn

EE

En

*

:log

sup lim log 1lim

e

n

n

n

B r

M

rn

EE

En

0 1

1

maxmax ,1

log ( )Tr

( )

Q

eQ s

s s

Q x Q

x

Q x

x

r sB r

s

s Q x W W

W Q x W

X

X

Relations

Hayashi 2007

*

0 1

1/

maxmin ,1

log Tr ( )

Q

esQ

ss

Q x

x

r sB r

s

s Q x W

X

Nagaoka 2001

Ogawa-Nagaoka 1998

Application of

Quantum hypothesis testing

C-Q Channel Resolvability

C-Q wire-tap channel

Quantum

Key DistributionQuantum state

transmission

Quantum hypothesis testing

C-Q Channel Coding

C-Q Secure random number generation

Application to Quantum

Key Distribution

Part II

II-I. Classical Operations

for the QKD protocol

1st step for key distillation: error

correction:

2nd step for key distillation: privacy

amplification: (Sacrificing keys)

ksift krecError correction

krecUniversal

hash function

(random matrix)

kfin

Raw keys Corrected keys

Secret keys

Hardware

(optical system)

Application

mobile phone etc

Flow of quantum key distribution

Example：Toeplitz Matrix

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

1 2 1

1 3 2

2 3 ( ) ( ) 1

1 2 ( ) 1 ( )

:

1

1 0

0 1

1

X

n m n m n n

n m n m n n

n n m n n m

n n m n n m

A

a a a a

a a a a

a a a a

a a a a

: n0-m×n0 Matrix with n0-1 elementsXA

is universal2 hash functions{ }X

A

This method requires small amount of calculation.

01 1( , , ) F

n

n qX a a

Privacy amplification (PV)

Universal hash function

Brightness represents

visibility from eavesdropper.Imperfectly

secure keys（corrected keys）

Perfectly

secure keys(secret keys)

Ex. Applying

random matrix

(Toeplitz matrix)

• The trace norm distance to be bounded is that between

the true state and the ideal state ：

• Phase error correction can be virtually done by

performing privacy amplification.

How can we decide the sacrifice bit-length?

, ideal ph12 2 ,

A EP

ideal , | |: ( )

A U m E m

m

P m m m

EA,

Relation between

the trace norm distance and PV

phP : phase error probability

II-II. Finite-length Analysis

with single photon source

we estimate an upper bound of psft=(k-c)/n →

Relation among important parameters

n l

ck

Total

Number

of bits

Number

of errors

in phase

basis

sample bits

with phase basisraw keys

unknown

In order to decide the sacrificed bit length in PV, we need

the phase error rate psft(k,c)=(k-c)/n in the raw keys.

known

n lknown known

known

k c

unknown

solution

sftp̂

s tf bit1 ˆG n h f h p Dp

Generated key length：

f : efficiency of bit error correction

Hypothesis testing

(Interval estimation)

Number c obeys the hyper-geometric distribution Phg(c|k):

How to estimate

k

ln

c

l

ck

nkcP |hg

sft,p k c

sft,p k c

Interval estimation

sftp

calculation

1.Decoding error probability Spa(k,c) for phase error

correction with given k and c:

ckSkcPkS

kSkSkQ

ckSkcPkQPkQP

c

c

kk

k c

c

c

k

,|:

,max

,|

pa

0

hgav

avavEve

pahgEve

0

|phEveph

max

max

How to evaluate Pph

sft sft

ˆmin , [ ], 0

pa, 2

nh p k c nh p DS k c

Since this value does not depend on k,

the final goal is bounding it.

2.Decoding error probability Pph in phase error correction is

universally bounded:

Rigorous evaluation of key generation rate

without Gaussian approximation

Thick line：Straightforward bounds

Thin line: Gaussian integral bounds

Points: result by Tomamichel et al.

2011

key generation rate R

＝ final secure key length／raw key length n

n (raw key length) + l (number of sample bits with phase basis)

II-III. Finite-length Analysis

with decoy method

27

Single-photon and weak

coherentIt is difficult to equip single-photon

source for QKD.

time

Single photon

Ideal

Weak coherent pulse

time

Laser (real)

Single

photon

Two

photon vacuum

Stochastic

Optical fiber

I get 1 photon

with 1 qubit info.

Eve can get information

without any disturbance!

If two-photon is generated, ….

BobAlice

Eve

28

AliceBob

0

1

2 ,

No count

Normal count

No count

Normal count

No count

Normal count

0 0 0J q N

Quantum Channel

(channel parameters)

Eve

1 1 1J q N

2 2 2J q N

Eve

Eve

Eve

Eve

1r

Security analysis with known channel

( ratio of phase error)

0N

Num. of Pulse

1N

2N

0q

1q

2q

1r

Counting rates, Phase error ratio

(Type of pulse)

29

Eve’s information among

pulses detected by Bob.

Partition of raw keys

2JNumber of bits completely eavesdropped by Eve0J

1J1

( )h r

Eve knows bits information

concerning Alice’s bit.

1 1 2( )h r J J

Number of bits non-eavesdropped by Eve

Number of bits partially ( ) eavesdropped by Eve

Amount of leaked information

30

Eve’s information after key distillation

Eve’s Holevo information

for final key

1 1 2

1 1 2

( ( ) )

, , ,E 2

S h r J J

av r J JP

Number of sacrifice bits

1 1 2( )S h r J J

:S

Exponential

evaluation!

,:

avP

Average of virtual

phase error ratio

MH PRA, 76, 012329 (2007)

3/2

ideal ,1E 2 E

AE avP

leaked information

Sacrifice bit-length

Finite-length security analysis of decoy method

datadetection rates

error rate (intensity)

channel parameters

partition of raw keys

leaked information

sacrifice bit-length

Hypothesis testing(Interval estimation)

solving equations

percent point

phase error formula+ margin

Flow of calculation of sacrificed bit-length from measured data

Numerical result

Block-length of raw keys

•Blue: Asymptotic case.

•Yellow: 108.

•Purple: 107.

•Green: 5 × 106.

•Red: 3 × 106.

•Orange: 2×106.

•Pink: 106.

•

Graph of key generation rate as function of signal intensity μ2

decoy intensity μ1 = 0.1

No. of decoy pulses, No. of vacuum pulses, and No. of signal pulse with ×basis

are 10% of No. of signal pulses.

Dark count rate p0=4×10-7. Channel error rate s=0.03. α=10-3.

Measured detection rate: p=1-e-αμ+p0, Measured error rate: (s(1-e-αμ)+p0/2)/(1-e-αμ+p0)

R: key generation rate=No. generated keys/No. of transmitted signal pulses with matched basis

signal intensity

Improvement of Asymptotic key rate

decoy intensity μ1 = 0.1

Dark count rate p0=4×10-7.

Channel error rate s=0.03. α=10-3.

Measured detection rate: p=1-e-αμ+p0,

Measured error rate:

(s(1-e-αμ)+p0/2)/(1-e-αμ+p0)

We improve the asymptotic key generation rate by estimating

the counting rate with error in the phase basis, and

the counting rate with no error in the phase basis.

Existing studies estimate the counting rate with error in the phase basis and the counting rate.

This change of the parameter to be estimated improves the estimation of the yield.

R: key generation rate=No. generated keys/No. of transmitted signal pulses with matched basis

signal intensity

Red: Our formula

Blue: Existing formula

Verification (Testing)

of Measurement-Based

Quantum Computation

Part III

How can we guarantee

computation result?If a problem is in NP, we can verify the correctness of

a solution. But a problem to be solved with a quantum

computer is not necessarily in NP.

We need to verify quantum computer.

Usually a quantum computer is composed of a

combination of so many quantum circuits.

It is not easy to predict the outcome of the

combination of so many quantum circuits.

So, its verification is not so easy.

How to resolve the difficulty of

verification of computationSince we do not know the computation outcome,

we cannot verify the computation outcome by itself.

How can we resolve this dilemma?We employ measurement-based quantum

computation (MBQC).

MBQC is composed of graph state and local

measurements. These components are known

to us. In particular, quantum correlation is given

as graph state, which is known to us.

We can verify them!

AssumptionーWhat we should trust?ー

(1) We perfectly trust measurement. So, we need

to verify only the graph state.

(3) has weakest assumption, but it works with ideal

case. (2) has stronger assumption, but it works with

realistic case.

(2) We trust measurement, but it is noisy. The noise

can be converted to noise of graph state. So, we need

to verify only the noisy graph state. This protocol

works with noisy graph state.

(3) We do not trust measurement as well as graph

state. However, it accepts only the case when the

measurement and the graph state are noiseless.

Advantage of verification of MBQCVerification process is similar to the quality guarantee

of industrial products via statistical hypothesis testing.

It is done by random sampling.

In the case of industrial mass production,

we can use the “same” random sampling.

m samples can be used commonly for k products.

This “same” sampling makes the verification of MBQC

more economical, which is suitable for industrial

products.

Even when k increase, m does not increase.

samples prodcutsm k

Components of MBQCThe following pairs realize universal

quantum computation.

(1) 2-D (or 3-D) cluster state (two-colorable)

& measurements of X,Y,X+Y

(2) Triangular lattice state (three-colorable)

& measurements of X,Z,X+Z,X-Z

Concepts of Verification (same as QKD)

Detectability: State and measurement should be

rejected when they are not properly prepared.

Acceptability: State and measurement should be

accepted when they are properly prepared.

This condition is needed for guaranteeing the precision

of computation outcome when the test is passed.

Significance level is the maximum passing

probability with incorrect state or measurements

(e.g. 5%)

Error probability is the maximum probability that the

computation outcome is incorrect with significance level

This condition is needed to accept the proper computation

outcome.

Acceptance probability is the passing probability with

correct state and measurements

Verification of MBQC with

trusted noiseless measurementSince we perfectly trust measurement, it is sufficient

to verify only the two-colorable (Black and White)

graph state by local measurements.

In two-colorable state, the Z values on one color sites

decide the X values on the other color sites.

Z measurement on Black

Z measurement on White

predicts

X measurement on Black

X measurement on White

Our verification:

We check whether X outcomes equal the prediction.

G

MH, Morimae 2015


trusted noiseless measurement

Z on Black X on White

Computation

2 1mG

Z on White X on Black

copiesm

copiesm

1 copy

Random choice

or

incorrect state

Stabilizer test

With significance level α, the probability being

incorrect computation outcome is

less than .


trusted noiseless measurementAcceptability is satisfied with .

As detectability,

holds with significance level α.

11

(2 1)G G

m

1 / (2 1)m

1

Here, we used hyper-geometric distribution.


trusted noisy measurementIn the realistic case, the measurement and the

state have noise.

Combine verification and fault tolerant MBQC.

Verifiable fault-tolerant

topologically protected

MBQC

We check whether the error

belongs to the correctable

error set .S

Fujii’s talk (Today)


untrusted measurementWe need self testing for graph state.

It is not so easy to guarantee the complex coefficient by self

test.

We employ three-colorable graph state, which needs only

measurements of X, Z and X±Z.

We verify measurements of X, Z and X±Z and graph state

by self testing.

We reduce the required number of copies.

(RUV scheme m=n^k, k>8000)

The same scaling Hajdusek et al with different method.

MH, Hajdusek (Poster Monday)

Self testing of bell state and

measurements of X, Z and X±Z Existing method employs only CHSH test.

However, it requires so many copies.

To reduce the number of copies, we propose

Hybrid method of CHSH test and stabilizer test.

Acceptance probability is 1-δ.

As detectability,

with significance level α. m:number of copies.

McKague et al

† 1/4' ( ), , ,T UT U O m T X Z X Z

† 1/8' ( )T UT U O m


untrusted measurementFor test of measurements of X, Z and X±Z in each site,

Testing of a big three-colorable graph state

Combination of testing of Bell state.Testing measurement on black sites.

Black sites are divided into 3 groups.

Sites of each group have no common

neighborhood.

One group is fixed.(blue circle)

One neighborhood is fixed (red circle)

Other sites are measured in Z basis.

3 pair of Bell states in this example.







neighborhood.











neighborhood.






untrusted measurement

Acceptance probability is 1-δ.

With significance level α, the probability of

incorrect computation outcome is a constant.

(RUV scheme

McKague )

The same scaling Hajdusek et al with different method.

4( log )m O n n : number of copies

n : size of graph state

, 8000km n k 22m n


hypergraph state

Merit of MBQC with hypergraph:

Required measurements are X,Y, and Z.

With significance level 1/n, the probability of

incorrect computation outcome is less than 1/n.

1 nne

n: size of hypergraph state.

We prepare nk+1+m copies.

Acceptance probability is greater than .

Verification method:

We apply generalized stabilizer test.

Morimae et al. 2017

Blind Quantum Computation (BQC)Verified MBQC with trusted measurement

→BQC, Alice with X,Y,Z,X+Y (X,Z,X±Z)

Bob 2(3)-colorable generates graph state

Verified MBQC with noisy trusted measurement

→BQC, Alice with noisy X,Y,Z,X+Y

Bob generates graph state

Verified MBQC with trusted measurement and

hypergraph state

→BQC, Alice with X,Y,Z

Bob generates hypergraph state

Verified MBQC with untrusted measurement

→BQC, classical Alice+4 quantum Bobs

one Bob generates graph state,

3 Bob performs measurement on each color site.

Conclusion• We have overseen how hypothesis testing is

useful in the following topics in quantum

information.

– Quantum Channel Coding

– Quantum Key Distribution

– Testing of Measurement-based Quantum

Computation

• This is because the latter two tasks require

verification process.

• F. Hiai, D. Petz, The proper formula for relative entropy and its

asymptotics in quantum probability. Comm. Math. Phys. 143,

99–114 (1991)

• T. Ogawa, H. Nagaoka, Strong converse and Stein’s lemma in

quantum hypothesis testing. IEEE Trans. Inf. Theory 46, 2428–

2433 (2000)

• M. Mosonyi, T. Ogawa, Quantum hypothesis testing and the

operational interpretation of the quantum Renyi relative

entropies. Comm. Math. Phys. 334(3), 1617–1648 (2015)

• M. Tomamochel and MH, “A Hierarchy of Information

Quantities for Finite Block Length Analysis of Quantum Tasks,”

IEEE Transactions on Information Theory, Vol. 59, No. 11,

7693–7710 (2013).

• K. Li. Second-Order Asymptotics for Quantum Hypothesis

Testing. Annals of Statistics, 42(1):171–189, (2014).

References (Simple hypothesis testing)

• MH, "Optimal sequence of quantum measurements in the

sense of Stein's lemma in quantum hypothesis testing" Journal

of Physics A: Mathematical and General, Vol.35, No.50,

pp.10759-10773 (2002).

• H. Nagaoka, and MH, "An Information-Spectrum Approach to

Classical and Quantum Hypothesis Testing for Simple

Hypotheses," IEEE Transactions on Information Theory, Vol.53,

534-549 (2007)

• T. Ogawa and MH, "On error exponents in quantum hypothesis

testing," IEEE Transactions on Information Theory, Vol.50,

No.6, pp.1368-1372 (2004)

• MH, "Error exponent in asymmetric quantum hypothesis

testing and its application to classical-quantum channel

coding,"Physical Review A, Vol.76, 062301 (2007)

References (Simple hypothesis testing)

• MH and H. Nagaoka, "General formulas for capacity of

classical-quantum channels," IEEE Transactions on

Information Theory, Vol.49, No.7, pp.1753-1768 (2003)

• MH,“Quantum wiretap channel with non-uniform random

number and its exponent of leaked information,” Proc. of ISIT

2012, pp. 895 - 899 (2012)

• MH, “General non-asymptotic and asymptotic formulas in

channel resolvability and identification capacity and its

application to wire-tap channel,” IEEE Trans. IT 52 1562-1575

(2006).

• MH, “Large deviation analysis for quantum security via

smoothing of Renyi entropy of order 2,” IEEE Trans. IT 60

6702-6732 (2014).

• MH, “Precise evaluation of leaked information with secure

randomness extraction in the presence of quantum attacker,”

Comm. Math. Phys. (2015)

References (Channel coding etc)

• MH, R.Nakayama, “Security analysis of the decoy method

with the Bennett-Brassard 1984 protocol for finite key

lengths,”New J. Phys. 16 063009 (2014).

• MH, "Practical Evaluation of Security for Quantum Key

Distribution," Physical Review A, Vol.74, 022307 (2006).

• MH,T. Tsurumaru, “Concise and Tight Security Analysis of

the Bennett-Brassard 1984 Protocol with Finite Key

Lengths,” New J. Phys. 14 093014, (2012).

References (QKD)

References (Testing of MBQC)• MH Morimae, PRL (2015)

• Fujii, MH, arXiv:1610.05216 (PRA Rapid)

• MH, Hajdusek, arXiv:1603.02195

• Morimae, Takeuchi, MH, arXiv:1701.05688

• Hajdusek, Perez-Delgado, Fitzsimons,

arXiv:1502.02563

• Reichardt, Unger, & Vazirani, Nature (2013).

• McKague, Theory of Computing 12, (2016).

• Raussendorf & Briegel, PRL (2001).

• Aharonov, Ben-Or, Eban, & Mahadev,

arXiv:1704.04487.

Role of Hypothesis Testing in Quantum Informationaqis-conf.org/2017/wp-content/uploads/2017/09/D3_01_InvitedTalk... · Role of Hypothesis Testing in Quantum Information Masahito Hayashi

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