Entanglement in fermionic systems

M.C. Bañuls, J.I. Cirac, M.M. Wolf

Goal

Definition of entanglement

in a system of fermions

given the presence of superselection rules

that affect the concept of locality

Fermionic Systems

• Indistinguishability– Physical states restricted to totally

(anti)symmetric part of Hilbert space– No tensor product structure

– Second quantization language– Entanglement between modes

• Other SSR affect locality– Physical states have even or odd number of

fermions– Physical operators do not change the parity

Fermionic Systems

• System of m=mA+mB fermionic modes

– partition A={1, 2, … mA}

– partition B={mA+1, … mA+mB}

• Basic objects: creation and annihilation operators

• canonical anticommutation relations

• and their products, generate operators on ()

• products of even number commute with parity

ijji aa

†,

††11 ,,,

AA mm aaaa

}2{};1{

:1x1 Ex.

BA

†22

†11

,by

,by

aaB

aaA

Baa

Aaa

in

in

2†2

1†1

• Fock representation, in terms of occupation number of each mode

– isomorphic to m-qubit space– action of fermionic operators is not local

– e.g. for mA=mB=1

Fermionic Systems

021

2121mn†

m

n†n†m aaannn

1110

0100

2

2

†

†

a

a

011

010

001

000

21

1

2

††

†

†

aa

a

a

Parity SSR

• Physical states and observables commute with parity operator

11100100

11

10

01

00

2110N1x1 modes

Fock representation

physical states

Parity SSR

• Physical states and observables commute with parity operator

11100100

11

10

01

00

00

00

00

00

2110N1x1 modes

Fock representation

physical states

Parity SSR

• Physical states and observables commute with parity operator

11100100

11

10

01

00

00

00

00

00

2110N1x1 modes

Fock representation

physical states

Parity SSR

• Physical states and observables commute with parity operator

11100100

11

10

01

00

00

00

00

00

2110N

eveneven

oddodd

1x1 modes

Fock representation

physical states

Parity SSR

• Physical states and observables commute with parity operator

oooeeoee

oo

oe

eo

ee

00

00

00

00 eveneven

oddodd

mAxmB modes

Fock representation

physical states

Parity SSR

• Physical states and observables commute with parity operator

BA

oooeeoee

oo

oe

eo

ee

000

000

000

000

BAO

mAxmB modes

Fock representationBABA

evenevenevenevenO

BABA

oddevenoddevenO

BABA

evenoddevenoddO

BABA

oddoddoddoddO

local physical observables

How to define entangled states?

• Entangled states = are not separable

• Separable states = convex combinations of product states

• Define product states…

P1P2,P3

Product states

BA BABABA ,

tionrepresenta spaceFock in BA

P3

BA BABABA ,

P2

P1

P1P2P3

Product states

BA BABABA ,

tionrepresenta spaceFock in BA

BA BABABA ,

P2

P1

BUT when restricted to physical states ( commuting with parity)

P3

Product statesExample:

1x1 modes

100

030

030

009

161

i

i

i

i

41

43

41

43

0

0

0

0

P1

BA

in P1 not in P2

P2

b

b

a

a

10

0

10

0

…two different sets of product states

Use them to construct separable states

P1P2P3

We have…

For pure states, they are all the same!!

S2=S3

BA Pi

Pii

S2’

Separable states

BiAii

BABA kkkkk ,S1

S1S2'S2S3 For physical states ( commuting with parity)

Local measurements cannot distinguish states that produce the same expectation values for all physical local operators– define equivalent states

– define separability as equivalence to separable state

Separable states

21 BABA

[S2] 2~,~ SBABA

Separable states

S2=S3

S2’

S1

BA Pi

Pii

BiAii

BABA kkkkk ,

For physical states ( commuting with parity)

S1S2'S2S3 S2

00

00

00

00

Separable statesExample:

1x1 modes

S2

73

71

71

72

000

000

000

000

00

00

00

00

Separable statesExample:

1x1 modes

S2

73

71

71

72

000

000

000

000

S2’

41

41

41

41

41

41

000

00

00

000

00

00

00

00

Separable statesExample:

1x1 modes

S2

73

71

71

72

000

000

000

000

S2’

41

41

41

41

41

41

000

00

00

000

[S2]

S1=

154

532

51

51

51

51

532

31

00

00

00

00

There are…

…four different sets of separable states

They correspond to four classes of states– different capabilities for preparation and

measurement

S1S2'S2S3 S2

Preparable by local operations and classical communication, restricted by parity

Convex combination of products in the Fock representation

Convex combination of states s.t. locally measurable observables factorize

All measurable correlations can be produced by one of the above

S2'

S2

S1

[S2]

Characterization

Criteria in terms of usual separability

oooeeoee

oo

oe

eo

ee

00

00

00

00

Characterization

00

00

00

00

Criteria in terms of usual separability

00

00

00

00

00

00

00

00

00

00

00

00

[S2] S1

S2’ S2

convex combination of products

00

00

00

00

Characterization

00

00

00

00

Criteria in terms of usual separability

00

00

00

00

00

00

00

00[S2] S1

S2’ S2

convex combination of products

00

00

00

00

Measures of entanglement

• For S2’ and S2, the entanglement of formation can be defined

– For 1x1 modes, EoF in terms of the elements of the density matrix

)EoF()1()EoF(EoFS2

)E(min)EoF(EoF},{

S2' kkkk

Multiple Copies

• Not all the definitions of separability are stable under taking several copies of the state

• S2 and S2’ asymptotically equivalent• 1x1 modes: all of them equivalent in the limit of large N

'2S'2S2

2S2S2

1S1S2

]2S[]2S[2

PPT]2S[2

distillable states not in [S2]

To conclude…

Different definitions of entanglement between fermionic modes are possible

They are related to different physical situations, different abilities to prepare, measure the state

Different measures of entanglement

Different behaviour for several copies

More details: Phys. Rev. A 76, 022311 (2007)

Application to a particular case

• Fermionic Hamiltonian

• Reduced 2-mode density matrix calculated from

• Regions of separability as a function of , , • EoF for S2, S2’

j

jjj

jjj

jj aaaaaaH .h.c.h.c2

1 †1

††1

†

)tr(e

eH

H

Application to a particular case

• Fermionic Hamiltonian

• Reduced 2-mode density matrix calculated from

• Regions of separability as a function of , , • EoF for S2, S2’

j

jjj

jjj

jj aaaaaaH .h.c.h.c2

1 †1

††1

†

)tr(e

eH

H

Application to a particular case

• Fermionic Hamiltonian

• Reduced 2-mode density matrix calculated from

• Regions of separability as a function of , , • EoF for S2, S2’

j

jjj

jjj

jj aaaaaaH .h.c.h.c2

1 †1

††1

†

)tr(e

eH

H

[S2] [S2]

S2’S2

[S2] [S2]

S2’ S2