Group theoretical study of LOCC-detection of maximally entangled - - PowerPoint PPT Presentation

Group theoretical study of LOCC-detection of maximally entangled state using hypothesis testing

Graduate School of Information Sciences Tohoku University Masahito Hayashi

MH arXiv:0810.3380 Accepted for publication for NJP.

Contents

• Motivation
• Formulation
• Testing of binomial distributions
• Global tests
• A-B locality
• Two different information sources
• A-B locality and sample locality

Contents

• Motivation
• Formulation
• Testing of binomial distributions
• Global tests
• A-B locality
• Two different information sources
• A-B locality and sample locality

Bell pair generator JPY 1,000,000

Problem

• You find a Bell pair

source in a shop.

• Shall you buy this?

Real Bell pair source ? Precision? How to check ? Tomography?, witness? Bell’s inequality? Coincidence count?

?

NO .. !!! NO .. !!! PLEASE USE OUR METHOD

PLEASE USE OUR METHOD

By K. Matsumoto

Problems with existing methods

Careless treatment of the error. Often, the conclusion is meaningless Here, we introduce notion of hypothesis test which is used in statistics. These methods are not optimal. There are more precise methods, which is easy to implement We search for optimal test. Conventional visibility and witness have bias. (The quality depends not only on the fidelity but also on the angle.) We propose symmetric testing.

Who needs precise and efficient test?

However, this is not always true.

• 1. Bell pair sharing in the long distance

(repeater, entanglement distillation)

• 2. At the development stage, some new

Bell pair sources may be very unstable.

Huge number of data is available. Even poor test gives a precise decision! Huge number

• f data

is not available. Poor test gives a bad decision! Efficient test is needed!

Contents

• Motivation
• Formulation
• Testing of binomial distributions
• Global tests
• A-B locality
• Two different information sources
• A-B locality and sample locality

Quantum Hypothesis Testing I

• Two Hypotheses
• Two-valued measurement
• Two error probabilities

1

: Accept : Accept (Reject ) T H I T H H −

1 1

: null hypothesis : alternative hypothesis H H ρ ρ ∈ ∈ S S

{ }

, M T I T = −

( ) ( )

1

First error probability , 1 Tr T ( ) Second error probability , Tr T ( ) T T α ρ ρ ρ β ρ ρ ρ − ∈ ∈ S S

• Test is called level-

if

• Test is called a UMP level-

test if

• Test is called a UMP level-

test if

( )

{ }

( )

( )

, 0

,

, , , min

T

T T I T T

α

α α

ρ α ρ α β ρ β ρ

∈

≤ ≤ ∀ ∈ ≤

S

S T

T S T S S , S ,

• T

α

( )

, T α ρ α ρ ≤ ∀ ∈ S α

( )

( )

1

, T

α

β ρ β ρ ρ = ∀ ∈ , S , S S T

1 2

, C C T α

Quantum Hypothesis Testing II

( )

( ) ( )

( )

2 1 2 1 , 0

, 1 , 1 2

, satisfies min and

C C C C T

T T T C C

α

α α

β ρ β ρ ρ β ρ β ρ

∈

= ∀ ∈ ⎧ ⎫ ⎪ ⎪ ⎨ ⎬ ⎪ ⎪ ⎩ ⎭

S

T

, S , S S S , S ,

• 1

2

, C C : conditions

Our setting

• Is the state close to the maximally entangled

state on ?

• We assume that i.i.d. condition i.e.,
• Our hypotheses are
• The set of level-

tests

{ }

,

, ,1 Tr

n n

T T I T

α ε ε

σ σ α

⊗ ≤ ≤

≤ ≤ ∀ ∈ − ≤ T S T S

• n

n

ρ σ ⊗ =

{ }

, , 1

: 1 :

A B A B c

H H

ε ε

σ σ φ σ φ ε σ

≤ ≤

∈ − ≤ ∈ S S

• 1

, d A B A B i

i i φ

− =

∑

• ,

A B

H α

Group invariance on

• U(1)-action
• SU(d)-action
• SU(d)xU(1)-action
• U(d2-1)-action

( )

, , , , i A B A B A B A B

U e I

θ θ

φ φ φ φ + −

• ( )

, ( ) U g g g g SU d ⊗ ∀ ∈

• ( , )

( ) , ( , ) ( ) (1) U g U g U g SU d U

θ

θ θ ∀ ∈ ×

• (

)

, , , , 2

( ) , ( 1)

A B A B A B A B

V g g I g U d φ φ φ φ + − ∀ ∈ −

• ,

A B

H

1 1 1 1 1 1 1 1 1 1 1

: No condition ( , ) :

• separable

( ) : 2-way LOCC ( ) :1-way LOCC ( , , , , , ) :separable on , , , , , ( , , , , , ) : 2-way LOCC on , , , , , ( , , , , ) : 1-way LOCC ,

n n n n n n n n n n

S A B A B L A B L A B A B S A A B B A A B B L A A B B A A B B L A A B B A ∅ → → … … … … … … … … … → …

• 1

, , ,

n n

A B B … → …

Locality restriction ( )

( )

,

, ,

:

• inv.

min satisfies

C n G T

T G T T C

α ε

α

β ε σ β σ

≤

⊗ ∈

⎧ ⎫ ⎨ ⎬ ⎩ ⎭

S

n T

,

• where

2

(1), ( ), ( ) (1), ( 1) G U SU d SU d U U d = × −

Contents

• Motivation
• Formulation
• Testing of binomial distributions
• Global tests
• A-B locality
• Two different information sources
• A-B locality and sample locality

Binomial distribution

• The data obeys the distribution
• The test is described by a map from

to

• The test (will be fined in the next slide)

is UMP test, i.e.,

( ) (1 )

n n k k p

n P k p p k

−

⎛ ⎞ − ⎜ ⎟ ⎝ ⎠

• k

{ }

( ) min ( ) [0, ],1 ( ) ( ) ( ) ( )

n n n q p T n n n p p k

q P T p P T P T P k T k

α

β ε ε α

=

∀ ∈ − ≤

∑

• {0,1,

, } n … [0,1] T

• ,

( , ) ( ), .

n n n q

T P q p

ε α α

β β ε ε = ∀ >

• ,

n

Tε α

• ε

H

1

H p

Definition of

, n

Tε α

• ,

, , , ,

1 if ( ) if 0 if ,

n n n n n

k l T k k l k l

ε α ε α ε α ε α ε α

γ ⎧ < ⎪ = ⎨ ⎪ > ⎩

• where and are defined as

, n ε α

γ

, n

lε α

, , ,

1 1 , ,

( ) 1 ( ), ( ) 1 ( ).

n n n

l l n n k k l n n n n k

P k P k P l P k

ε α ε α ε α

ε ε ε α ε ε α ε

α γ α

− = = − =

< − ≤ = − −

∑ ∑ ∑

H

1

H k

, n

lε α H is supported with pro

, n ε α

γ

Asymptotic theory (small deviation)

/

lim ( ) ( ) !

k n t t n t n

t P k P k e k

− →∞

=

• When the true parameter is close to 0, the

distribution goes to Poisson distribution.

'

lim ( / '/ ) ( ') [0, ], ( ') min ( ) 1 ( )

n n t T t

n t n t t t P T P T

α α α

β δ β δ δ β δ α

→∞

= ⎧ ⎫ ∀ ∈ ⎪ ⎪ ⎨ ⎬ − ≤ ⎪ ⎪ ⎩ ⎭

Contents

• Motivation
• Formulation
• Testing of binomial distributions
• Global tests
• A-B locality
• Two different information sources
• A-B locality and sample locality

Global Test ( samples)

Theorem

( )

( )

, , , ,

( ) ( ) , ,

n n G n n A B A B

p T

α α α

β ε σ β ε β φ φ ε σ ⊗ = =

where

2

(1), ( ) (1), ( 1) G U SU d U U d = × −

( )

, ,

1 ,

, ( ) ( ) ( ) ( ) ( ) + + ( ) ( ).

n n

l n n n n k l k n k k n k k n k

T T P T P T P T I T I T T T T T I T I T

ε α ε α

α ε α

ε γ

− = − −

+ − ⊗ ⊗ − ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ − ⊗ ⊗ −

∑

• n

Contents

• Motivation
• Formulation
• Testing of binomial distributions
• Global tests
• A-B locality
• Two different information sources
• A-B locality and sample locality

A-B locality (One sample)

• For a rank-one POVM

the test is constructed as follows:

• 1. Alice performs POVM .
• 2. Bob performs two-valued POVM
• 3. If is observed, accept.

Otherwise, reject

{ }

i i i

M p u u = ( )

i i i i i i

T M p u u u u ⊗ ⊗

∑

• {

}

i i i

M p u u =

{ }

,

i i i i

u u I u u −

i i

u u

A-B locality (One sample)

• Covariant POVM

where is invariant measure.

• The POVM can be realized by randomly

choosing measuring basis

• The test

is level- .

1 cov

M

1, 1 1 , cov

( ( ), ( 1))

A B

T T T M d d

ε α α

ε

−

+

• ν

α

1 cov(

) ( ) M d d d ϕ ϕ ϕ ν ϕ

• 1

cov . .

Tr ( ) 1 ( 1), 1

A B A B

T M dp d p σ φ σ φ = − + −

• 1

cov

M

( )

1 cov , , , ,

( ) 1 ( 1)

A B A B A B A B

T M d I φ φ φ φ = + + −

A-B locality (One sample)

1, ,1, ,

( ) Tr (1- ) 1- 1- if 1 1 1 = 1 if 1

C A B G

T d dp d d d d p d d

α ε α

β ε σ σ ε ε α α α ε α ε

−

= ⎧ ⎛ ⎞ ⎛ ⎞ ≤ ⎜ ⎟ ⎜ ⎟ ⎪ + + ⎪ + ⎝ ⎠ ⎝ ⎠ ⎨ ⎪ − > ⎪ + ⎩

where

2

( ), ( ) (1), ( 1) ( , ), ( ), ( ) G SU d SU d U U d C S A B L A B L A B = × − = →

A-B locality (Two samples)

• Covariant POVM

where is invariant measure, and is maximally entangled on .

2 cov

M ν

2 cov 1 2 2 † 1 2 1 2 1 2

( ) ( ) ( ) ( ) ( ) M dg dg d g g u u g g dg dg ν ν ⊗ ⊗

• (

) ( )

2 cov , , , , 2 , , , , 2 2 2 2 2 cov

( ) 1 ( 1) Tr ( ) 1 2 ( 1)

A B A B A B A B A B A B A B A B

T M d I I T M p d p d φ φ φ φ φ φ φ φ σ ⊗ = ⊗ + − − ⊗ − = − + −

1 2

A A

⊗ H H H H u

A-B locality (Two samples)

• Bell POVM on

2 Bell

M

2 2 2 2

1- 2 Tr ( ) 1 2 2

Bell

p p T M p p σ ⊗ + ≤ ≤ − +

1 2

A A

⊗ H H H H

Alice Bob Maximally entangled? Maximally entangled? 4-vauled Bell Measurement 2-valued Bell Measurement

A-B locality ( samples)

• The test

is level- .

2n

2 2 2 , 2 , cov 2

( ),2 1

n A B n

d T T T M d

ε α α

ε ε

−

⎛ ⎞ − ⎜ ⎟ − ⎝ ⎠

• α

( )

2 , 2 , , ,

lim ( ) lim , ( )

C n A B n n G n n n n

n T t

α ε α α

β δ σ β σ β δ

− ⊗ →∞ →∞

= =

where

2

(1), ( ) (1), ( 1) , ( , ), ( ), ( ) G U SU d U U d C S A B L A B L A B = × − = ∅ →

• Hence, is asymptotically UMP C G-inv. Test.

2 , , n A B

Tε α

− , ,

1

A B n A B

t n φ σ φ = −

A-B locality ( samples)

• The test

is asymptotically level- .

2n

2 2 2 , 2 , , 2

( ),2 1

n A B n Bell Bell

d T T T M d

ε α α

ε ε

−

⎛ ⎞ − ⎜ ⎟ − ⎝ ⎠

• α

( )

2 , 2 , , , ,

lim ( ) lim , ( )

C n A B n n G n Bell n n n

n T t

α ε α α

β δ σ β σ β δ

− ⊗ →∞ →∞

= =

where

2

(1), ( ) (1), ( 1) , ( , ), ( ), ( ) G U SU d U U d C S A B L A B L A B = × − = ∅ →

• Hence, is asymptotically UMP C G-inv. Test.

2 , , , n A B Bell

Tε α

− , ,

1

A B n A B

t n φ σ φ = −

Experiment

Alice Bob

Maximally entangled? Maximally entangled? 4-valued Bell Measurement 2-valued Bell Measurement Maximally entangled? Maximally entangled? 4-valued Bell Measurement 2-valued Bell Measurement Maximally entangled? Maximally entangled? 4-valued Bell Measurement 2-valued Bell Measurement 1 2 n

Contents

• Motivation
• Formulation
• Testing of binomial distributions
• Global tests
• A-B locality
• Two different information sources
• A-B locality and sample locality

Even if the two independent states are different, this test has the same performance.

Alice Bob

Maximally entangled? Maximally entangled? 4-valued Bell Measurement 2-valued Bell Measurement Maximally entangled? Maximally entangled? 4-valued Bell Measurement 2-valued Bell Measurement Maximally entangled? Maximally entangled? 4-valued Bell Measurement 2-valued Bell Measurement 1 2 n

Contents

• Motivation
• Formulation
• Testing of binomial distributions
• Global tests
• A-B locality
• Two different information sources
• A-B locality and sample locality

A-B locality and sample locality ( samples)

n

, 1 , cov

( ), 1

n A B n

d T T T M d

ε α α

ε

−

⎛ ⎞ ′ ⎜ ⎟ + ⎝ ⎠

• (

)

, , , 0, ( 1)

lim (0 ) lim , (0 )

C n A B n n G n n n n dt d t

T e e t

α α α

β σ β σ α α β

− ⊗ →∞ →∞ − + −

′ = = < =

where

2 1 1 1 1 1 1

( ), ( ) (1), ( 1) ( , , , , , ), ( , , , , , ), ( , , , , )

n n n n n n

G SU d SU d U U d C S A A B B L A A B B L A A B B = × − = … … … … … → …

Hence, is asymptotically UMP C G-inv. Test.

, 0, n A B

T α

−

′

, ,

1

A B n A B

t n φ σ φ = −