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}
Ex. 1x1 :
A  {1}; B  {2}
• Basic objects: creation and annihilation
operators

†
• canonical anticommutation relations  ai , a j    ij


•
a , a ,a
1
†
1
mA
†
mA
,a
 and their products,


A by  a1 , a1† 




B by  a2 , a2† 


generate operators on A (B)
• products of even number commute with parity
a1† a1 in A
a2† a2 in B
Fermionic Systems
• Fock representation, in terms of
occupation number of each mode
  a 
† n1
1
n1n2  nm  a
† n2
2
 
† nm
m
a
0
00  0
01  a2† 0
10  a1† 0
11  a1† a2† 0
– isomorphic to m-qubit space
– action of fermionic operators is not local
– e.g. for mA=mB=1
a 00 
†
2
01
a2† 10   11
Parity SSR
• Physical states and observables commute
with parity operator
1x1 modes
Fock representation
N 0 1 1
00 01 10

00 
physical
states

01 

 

10


11 

2
11










Parity SSR
• Physical states and observables commute
with parity operator
1x1 modes
Fock representation
N 0 1 1
00 01 10

00 
physical
states


01  0

 
0
10


11  

0
0
0
0
2
11

 

0 

0 


 

Parity SSR
• Physical states and observables commute
with parity operator
1x1 modes
Fock representation
N 0 1 1
00 01 10

00 
physical
states


01  0

 
0
10


11  

0
0




0
0
2
11

 

0 

0 


 

Parity SSR
• Physical states and observables commute
with parity operator
1x1 modes
Fock representation
N 0 1 1
00 01 10

00 
physical
states


01  0

 
0
10


11  

0
0




0
0
2
11

 

0 

0 


 

   even   even
  odd   odd
Parity SSR
• Physical states and observables commute
with parity operator
mAxmB modes
Fock representation
ee

ee 
physical
states


eo  0

 
0
oe


oo  

eo
oe
oo
0
0
 




0
0


0 

0 


 

   even   even
  odd   odd
Parity SSR
• Physical states and observables commute
with parity operator
mAxmB modes
Fock representation
ee

ee 
A  B
local physical
observables


eo  0


0
oe


oo  0

eo
0
oe
0
O  A  B 
oo
0

0
0
0

0
0
0











 Aeven  Beven O  Aeven  Beven
  A B O  A B
even
odd
even
odd
  A B O  A B
odd
even
odd
even
  A B O  A B
odd
odd
odd
odd
How to define entangled states?
• Entangled states = are not separable
• Separable states = convex combinations
of product states
• Define product states…
Product states
P1
  A  B     A  B  A  A, B  B
P2
   A   B in Fock space representa tion
P3
  A  B    A B A  A, B  B
P3, P2  P1
Product states
P1
  A  B     A  B  A  A, B  B
P2
P3
   A   B in Fock space representa tion
  A  B    A B A  A, B  B
BUT when restricted to
physical states (
commuting with parity)
P3  P2  P1
Product states
Example:
1x1 modes
P1
P2
9

0
  161 
0

i

i

3 i 0 

i 3 0

0 0 1 
0
0
 a 0  b 0 


 
 0 1 a   0 1 b

 

 34

0

0   34

0
1 
4

in P1 not in P2
   A  B
0

1 
4
We have…
…two different sets of product states
P3  P2  P1
For pure states, they are
all the same!!
Use them to construct separable states
Separable states
S1
   k k , k  A  B   k  A k B 
S2’
   i i A  i B
S2=S3
For physical states (
commuting with parity)
   i   
PA
i
PB
i
S3  S2  S2'  S1
Separable states
Local measurements cannot distinguish
states that produce the same expectation
values for all physical local operators
– define equivalent states
A B
1
 A B
2
– define separability as equivalence to
separable state
Separable states
A  B
[S2]
S1

 A  B
~  S2
,

~
   k k , k  A  B   k  A k B 
   i i A  i B
S2’
S2=S3
For physical states (
commuting with parity)
   i   
PA
i
PB
i
S3  S2  S2'  S1  S2
Separable states
Example:
 27

0
 
0

0

S2
1x1 modes


0
 
0



0
0




0
0


0

0

 
0
0
1
0
7
0
1
0
0
7
0

0
0

3 
7
Separable states
Example:
 27

0
 
0

0

S2
1x1 modes


0
 
0



0
0



0

0


0

0

 
 14

0
 
0

0

S2’
0
0
1
0
7
0
1
0
0
0
0
1
1
1
4
4
0
1
7
4
4
0
0

0
0

3 
7
0

0
0

1 
4
Separable states
Example:
 27

0
 
0

0

S2
1x1 modes


0
 
0



0
0



0

0


0

0

 
 14

0
 
0

0

S1= S2’
[S2]  1 3

 0
 
0

2
 3 5
0
0
1
0
7
0
1
0
0
0
0
1
1
1
4
4
7
1
4
4
0
0
0
0
1
1
1
5
5
0
1
5
5
0
0

0
0

3 
7
0

0
0

1 
4


0 
0 


4
15 
2
3 5
There are…
…four different sets of separable states
S3  S2  S2'  S1  S2
They correspond to four classes of states
– different capabilities for preparation and
measurement
S2  Preparable by local operations and
classical communication, restricted by
parity
S2'  Convex combination of products in
the Fock representation
S1  Convex combination of states s.t.
locally measurable observables
factorize
[S2]  All measurable correlations can be
produced by one of the above
Characterization
Criteria in terms of usual separability
ee

ee 


eo  0

 
0
oe


oo  

eo
oe
oo
0
0
 




0
0


0 

0 


 

Characterization
Criteria in terms of usual separability
[S2]
 
S2’
 



0

0







0

0




0
0




0
0
0
0




0
0

 

0 

0 


 

S1
 
convex
combination of
products

 

0 

0 


 

S2
 



0

0







0

0




0
0




0
0
0
0




0
0

 

0 

0 


 


 

0 

0 


 

Characterization
Criteria in terms of usual separability
[S2]
 
S2’
 



0

0







0

0




0
0




0
0
0
0




0
0

S1
 

0 

0 


 

 
convex
combination of
products

 

0 

0 


 

S2
 



0

0




0
0




0
0



0

0




0
0




0
0
 
  
 
0  0
 
0
0 
 

  
 

 

0 

0 


 

0
0




0
0



0

0



Measures of entanglement
• For S2’ and S2, the entanglement of
formation can be defined
EoFS2'  EoF(  )  min
{ k ,  k }

k
E(  k )
EoFS2   EoF(  )  (1   ) EoF(   )
– For 1x1 modes, EoF in terms of the elements
of the density matrix
Multiple Copies
• Not all the definitions of separability are stable
under taking several copies of the state
 2  [S2] 
   [S2]
 2  S1
   S1

2
 S2'    S2'

2
 S2    S2
 2  [S2]   PPT
distillable states not in [S2]
• S2 and S2’ asymptotically equivalent
• 1x1 modes: all of them equivalent in the limit of large N
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
1
H    a †j a j 1  h.c.    a †j a j    a †j a †j 1  h.c.
2 j
j
j
• Reduced 2-mode density matrix calculated from
 H
e

 H
tr(e )
• Regions of separability as a function of , , 
• EoF for S2, S2’
Application to a particular case
• Fermionic Hamiltonian
1
H    a †j a j 1  h.c.    a †j a j    a †j a †j 1  h.c.
2 j
j
j
• Reduced 2-mode density matrix calculated from
 H
e

 H
tr(e )
• Regions of separability as a function of , , 
• EoF for S2, S2’
Application to a particular case
• Fermionic Hamiltonian
[S2]
1
†
H    a j a j 1  h.c.    a †j a j    a †j a †j 1  h.c.
2 j S2’
j
j
S2
• Reduced 2-mode density matrix calculated from
[S2]
 H
e

 H
tr(e )
[S2]
• Regions of separability as a function of[S2]
, , 
S2’
• EoF for S2, S2’
S2