Entangled states and entanglement criterion

Energy representation. Two-level systems

12??????

What does one need for this?1. An algebraic description2. A reference state3. A proper representation of the generic state

nilpotent

We want to have something like that:

How to introduce nilpotent polynomials

Normalization to unit vacuum state amplitude for technical convenience

X

Extensive characteristic -- nilpotential

F=1+nilpotent, ln F=nilpotent-(nilpotent) /2+(nilpotent) /3….+(nilpotent) /N.

nilpotent

2 3 N

Is also a polynomial!

finite Taylor series, N~2n

Canonic statewhat is needed in order to make the nilpotential unique

Reference state Canonic state of the orbit

Maximum population of the reference state+ some phase requirements

Local transformations

Coset dimension

-1

C X

We would like not only to know whether or not qubits are entangled, but alsoAnswer the questions:1) How much are they entangled?2) In which way are they entangled?

Unambiguous extensive characteristic -- tanglemeter

Depends on D parameters

011->3; 101->5 etc.

111101 011110010 001100000

One real parameter D=1

D=5

D=18

An alternative: Invariants of local transformations vs orbit markers

2 qubits

3 qubits

4 qubits

Why the tanglemeter is useful?1. It contains all information about the entanglement.2. It is extensive: tanglemeter of a system is a sum of the tanglemeters of not entangled parts.3. Other characteristics can be expressed in terms of tanglemeter.

4. Tanglemeter gives one an idea about the structure of the canonical state, where all local transformation invariants take the most simple form. This helps to construct

multipartite entanglement measures: i|i|²-one of them.

Dynamic equation for nilpotential-Schrödinger

equation

su(2) operatorsin terms of nilpotent variables

Infinitesimal transformation

Similar to coherent states of harmonic oscillator

?????????H=H(x,p)

Universal evolution of quantum computer

Tanglemeter from dynamic equations for the nilpotential

<0

In order to put f to the canonic form

Dynamic equationclose to the canonicstate

010 001100000

X

Condition of maximum population

canonicstate

Beyond the qubits, qutrits

XCommuting nilpotent variables from the Cartan subalgebra L+

f =ln

Lz, L+, L-

nilpotential

entanglement criterion

su(3)

Lz L+

Canonical states for 3-level systems (qutrits)

Maximum populationof maximum correlatedstates

2 qutrits

3 qitrits

qubit and2 qutrits

Generalized entanglement

Spin-1 systems.

Summary

1. Nilpotent polynomials offer an adequate extensive description of the entanglement. Simple entanglement criterion exists in terms of nilpotentials (logarithms of nilpotent polynomials representing the quantum states).

2. Notion of the canonic states allows one to unambiguously characterize quantum entanglement with the help of the tanglemeter (nilpotential of the canonic state)

3. Dynamic equation for nilpotential can be derived.4. The technique, initially introduced for qubits, can be

generalized on both the case of multilevel systems and the case of subalgebras (where the number of operators in the subalgebra is less than the square of the number of levels).