Entanglement Classes and Measures for 4-qubits (as they emerge from

1
Entanglement Classes and Measures for
4-qubits(as they emerge from the entanglement
description with nilpotent polynomials)

• quant-ph/0508234
• ? Aikaterini Mandilara
• Lab Aime Cotton,
  CNRS, Orsay, France.
• Vladimir Akulin
• Lab Aime Cotton,
  CNRS, Orsay, France
• Andrei Smilga
• Subatech,
  Nantes, France
• Lorenza Viola
• Dartmouth
  College, U.S.
  
• 
  
  
• 
  
  
• 
  Eilat,
  Feb 2006

2
Outline

• Writing a quantum state as a nilpotent
  polynomial. Nilpotential. Tanglemeter.
• Entanglement classes (sl-orbits)?
  sl-tanglemeter.
• Entanglement measures ? coefficients of the
  tanglemeter.
• Conclusions. Open questions.

Su-orbit
3
From quantum states to nilpotential

Nilpotential
?Extensive property
Product states become sum
?Dynamics
4
From nilpotential to tanglemeter
1 2 3 4 . n
A state/nilpotential of N qubits An orbit of
states All the states in the orbit Should have
the same Entanglement description
.
SU(2) SU(2) SU(2) SU(2) SU(2)
3 parameters each one
How many parameters for the orbit marker?
Tanglemeter
Physical condition Maximize
Method use feedback in dynamical
equations
5
More general, non-unitary, reversible, local
operations

• nonselective

LOCC operations local operations assisted by
classical communication

• selective

SLOCC stochastic LOCC (Bennet et al, PRA 63,
012307)
Indirect measurement
a
s
If ignore the normalization divide by det(M)
SLOCC described by SL(2,C)
generators
Entanglement Classes set of states which are
equivalent under local SLOCC operations

• Three qubits can be entangled in two inequivalent
  ways W. Dur et al,PRA 62, 062314, (2000)
• Four qubits can be entangled in nine different
  ways F. Verstraete et al, PRA 65 052112 (2002).

6
sl-tanglemeter?Entanglement Classes
1 2 3 4 . n
A state/nilpotential of N qubits An sl-orbit
of states Merging different su-orbits together.
.
SL(2,C) SL(2,C) SL(2,C) SL(2,C) .. SL(2,C)
6 parameters each one
How many parameters for the orbit marker?
In general.. Sl-Tanglemeter..
sl-orbit marker
Physical condition?
Method use feedback in dynamical
equations
7
(No Transcript)
8
(No Transcript)
9
su orbits tanglemeter su-orbit marker
sl orbits (entanglement classes) sl-tanglemeter
sl-orbit marker
2 qubits
3 qubits
A. Miyake 03
4 qubits
family of general orbits
10
Entanglement Measures
In order to compare different su-orbits in the
same sl-orbit or different sl-orbits in
the same general family of orbits
SU- Measures
SL-Measures

• (Give 0 for separable state) and 1 for maximally
  entangled state of the sl-orbit
• Invariant under local
• SU operations and nonincreasing
• under LOCC transformations

• Give 1 for the maximally entangled state of the
  family of the sl-orbits
• Invariant under local
• SLOCC operations

Polynomial invariants on the amplitudes of the
states
2 ways to construct invariants
Invariant coefficients of the tanglemeter
But, which su-invariants are decreasing under
LOCC?
The poly-inv. which are sl-invariants
11
sl-tanglemeter for 4 qubits
SU- Measures
SL-Measures
Polynomial invariants
Only to be used in the states Belonging to the
states above
Tanglemeters coefficients

• We start we the normalized state
• We apply sl-transformations to put in the
  sl-canonic form.
• The normalization of the state
• give us a measure on nonunitarity/distance of the
• Initial state to the maximal entangled state.

12
Conclusions

• With sl-tanglemeter we can at least identify the
  most general class of entanglement for N qubits.
  It can be generalized to ensembles of quDits.
• Investigate a little bit more in the special
  classes and their applications.
• We introduced the idea of sl-invariant measures
  that
• extends the idea of su-measures.
• Tanglemeters coefficients can serve as
  invariants for construction of measures.

13
Acknowledgements

• My advisor in WashU
• J. W. Clark
• The coworkers on this project
• V. M. Akulin
• A. V. Smilga
• Lorenza Viola
• Prof. G. Kurizki