Transforming Bell's Inequalities into State Classifiers with

Machine Learning

Author: Yue-Chi Ma, Man-Hong Yung

Speaker: Yue-Chi Ma

Contribution of our workBuild a bridge between Bell

Inequality (physics) and Artificial Neuron Network (computer science)

Train ANN to detect entanglement with all or partial informationand without prior knowledge

It’s impossible!Quantum Mechanics is

incomplete.

What is EntanglementAlice

Source

Bob

V=∞

Application of entanglement:

Quantum computation, Teleportation,

Quantum key distribution…1935. EPR paradox

However,Entanglement was verified experimentally (Bell Inequality),

and becomes kernel resource of quantum information!

Definition of Entanglement(1) (2) ( )

sep

n

i i i i

i

p

separable

or

entangled

ENT

SEP

: qubit number

: probability

nip

2 22 2n n

Positive Partial Transpose(PPT criterion)

min min(eig( ))AT

min 0 entangled

min 0 2-qubit

n-qubit (n=3,4,…)

separable

???

Any system

, },{ , yx z z zII I

𝒙 𝒙

𝒛 𝒛

Problem reconstruct is resource-consuming

{ , , , } { , , , }x y z x y zI I

𝒚 𝒚

4 1 15n features

3 angles

Tomography

Alice Bob

1 2

y y 1 2

x x 1 2

z z 1 2

x y 1 2 ...x z

CHSH (Bell) inequality

Alice’s

measurement:Bob’s

measurement:

{ , } { , }a a b b

{ , , , }ab ab a b a b

2 angles

4 features

Source

CHSH (Bell) inequality

2 0ab ab a b a b

entangled

violatenot violate

???

ENT

SEP

Artificial Neuron Network（ANN）

2 0ab ab a b a b Input

Layer

1

Output

Layer

ab

ab

a b

a b 0.5

0.5

Class 1

Class 0

Bell to ANN!

entangled

separable

Probability

Activate

CHSH(Bell) Inequality :

1 1 1 12

0w

3w

4w

2w1w

Supervised learning

Input

Layer

1

Output

LayerOptimize

1w 4w0w 2w 3w

0.5

0.5

Class 1

Class 0entangled

separable

Probability

1

1

1

1

2

“Test Run” state

, , ,| | (1 ) / 4p p I

,| cos( / 2) | 00 sin( / 2) |11ie

min ,( ) 1 / 4 cos / 2 sin / 2BTp p

Independent of

separable

Defect of original CHSH0 0 0 0, , ( ) / 2, ( ) / 2z x x z x za a b b

p

Mis

mat

ch R

ate

wit

h P

PT

(err

or)

1 1 ~

3 2

1~ 1

2No knowledge of

Total failure

/2,

1| (| 00 |11 )

2

ie

, , ,| | (1 ) / 4p p I

Defect of Witness

0 / 4

/ 3 / 2

3 angles

Entangled (success)

Separable (success)

Entangled (fail)

, ,x y z

Experimental detection of entanglement via witness operators and

local measurements, Journal of Modern Optics, 2003, Gühne et al.

Tr( W) 0

Tr( W) < 0W is Witness iff

for allfor some

5 features

sep

ent

1 2

y y 1 2

x x 1 2

z z 1

z 2

z

Linear optimization by ANN

0 0 0 014 28 10a b a b

Analytical result

If (linear) solution exists, ANN can find it out.

0 0 0 0

0 0 0 0

/ 2,

sin cos / 2

a b a b p

a b a b p

Our ANN result

0 If trained with only

0 0 0 010 2 20 2 10a b a b

(1 2sin ) 1 0p

2 1.4

Reduced to

Trained with RandomizedNo solution

for lack of information

Our ANN result

1/ 2p p

Mis

mat

ch R

ate

(err

or)

mlCHSH

Entangled

Separable

How to improve CHSHml ? 0 0 0 0, , ( ) / 2, ( ) / 2z x x z x za a b b

yNo information about

, , ,a a b b Randomize angles

So we…

ViolateBell Inequality

max(0, )x

How to improve CHSHml ? More inequalities,

more entanglement

detected

Multi Bell to 1 ANN!hidden layer

ENT

SEP

OutputLayer

......

1

1

HiddenLayer

InputLayer

mlBell (2,4, )x

2 qubits，4 features

x hidden neurons

xBell-like Predictor

2 0ab ab a b a b

2 0

ab ab

a b a b

2 0ab ab a b a b

2 0ab ab a b a b

Not ViolateBell Inequality

Detect !

Perfect activation:

ReLu

bias

Results of Bell-like Predictor

p

Mis

mat

ch R

ate

(err

or)

Entangled

Separable

/ 2

Error occurs on the boundary

Entanglement detection

Contrast

ENT

SEP

A BENT

SEP A BENT

SEP AB

Training & test set

PPT \ Bell Inq. \ Witness Our Bell-like Predictor (ANN)

Entanglement detection Classifier

ENT

General 2-qubit system

SEP A

B

†

rand †tr( )

ij ij ija ib

Detect all entangled states

Hermitian

Normalize

Harr Measure (mix)

General 2-qubit system

{ , , , } { , , , }x y z x y zI I

Tomographic Predictor

(15 features)

2-4% error

Where the error occurs?

{ , , } { , , }I a a I b b

Bell-like Predictor

(8 features)

25% error (lack of information)

remove −𝑔 < 𝜆min < 0

Add Gap¾ Entangled ¼ Separable

𝑔 = 0 𝑔 = 0.05 𝑔 = 0.1

𝜆min

Nu

mb

er

of

en

sem

ble

sfo

r te

st

ANN results

For Tomographic Predictor:

𝑔 = 0.01𝑔

Accuracy larger than 99%

If solution exists,

ANN fit it,

even if it is non-linear.

Depends on data, hidden layer

and training time

Acc

ura

cy(o

r ra

tio

)

𝑔max = 0.5

ANN results

For Bell-like Predictor:

𝑔

Acc

ura

cy(o

r ra

tio

)

𝑔 Accuracy

More difference,

Better performance

𝑔max = 0.5

Multi-qubit systemGeneral 3-qubit detection?

We cannot label

Fortunately, we can cover (fully) separable states as much as posiible†

†tr( )i

(1) (2) ( )

sep

n

i i i i

i

p 1i

i

p Classical

noise

All data are entangled

The only constraint is Data!!!

Uniform distribution

Bipartite entanglement

Biseparable states

Fully separable states

FullySep.

𝜌𝐵|𝐴𝐶

𝜌𝐴|𝐵𝐶

𝜌𝐶|𝐴𝐵

A| sep(1 )BCp p A| (1 ) / 8BC Ip p

A| rand

A BC BC| |BC

𝜌𝐵|𝐴𝐶

𝜌𝐶|𝐴𝐵

𝜌𝐴|𝐵𝐶

Source

Fully

separable

states

+

Acc

ura

cy(o

r ra

tio

)

BC

1 2 3 4 rand| | , , ,v v v v Harr Measure (pure)

FullySep.

𝜌𝐵|𝐴𝐶

𝜌𝐴|𝐵𝐶

𝜌𝐶|𝐴𝐵

3-qubit system

ENT

SEP AB

......

1......

𝜌𝐵|𝐴𝐶

𝜌𝐴|𝐵𝐶

𝜌𝐶|𝐴𝐵

Fully Sep.

1

softmax

Error of biseparable states

Neurons in hidden layer

Mis

matc

h R

ate

wit

h P

PT

(err

or

)

Random guess

mlBell (3,4, )x mlBell (3,8, )x

mlBell (3,12, )x mlBell (3,26, )x

Tomographic Predictor

(63 features)

mlBell (3,26, )x

{ , , } { , , } { , , }I a a I b b I c c

mlBell (3,4, )x 2 abc a b c ab c a bc Mermin inequality

mlBell (3,8, )x |

| 4

abc ab c a bc a b c

a b c a bc ab c abc

Svetlichny inequality

Detect tripartite, but not bipartite

entangled states

The features of Bell-like predictors are composed of

2 random angles

3 CHSH Inequalities

mlBell (3,12, )x

Do PPT can not do (shifts UPB)

Biseparablestates

Tripartite entangled states (from shifts UPB)

Source

Tripartite entangled states

PPT can not detected!

bisep A|BC B|A C|AB1 2 3C

A|BC

A BC

i ii ip iHarr measure

(mix)

tile SLOCC

bisep

Unextendible Product Bases and Bound Entanglement

1998 Physical Review Letters, Bennette et al.

Error of UPB

Unpublished data

4-qubit system

Source Fully separable

statessep Separable

Fully separable

states

Entangled states+

sep| | (1 )p p

| Harr measure

(pure)

Entangled or not sure

ENT

SEP

4-qubit system

Tomographic

(255 features)

Bell-like

(80 features)

Entangled 99% 98%

Separable near 100% 99%

AccuracySeparable

Entangled not sureData number

In the future

Cross entropy

Supervised learning

Unsupervised learning

???

predylabel (PPT)y

predy

Application & Summary

New tool to detect entanglement with

partial information

Verify other quantum entanglement

detection methods

T H A N K S