Entangled states and entanglement criterion

Energy representation. Two-level systems 1 2 ?????? What does one need for this? 1.An algebraic description 2.A reference state 3.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 23N Is also a polynomial! finite Taylor series, N~2 n

Canonic state what 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 C X

We would like not only to know whether or not qubits are entangled, but also Answer 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 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) operators in 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 equation close to the canonic state X Condition of maximum population canonic state

Beyond the qubits, qutrits X Commuting nilpotent variables from the Cartan subalgebra L + f =ln L z, L +, L - nilpotential entanglement criterion su(3) LzLz L+L+

Canonical states for 3-level systems (qutrits) Maximum population of maximum correlated states 2 qutrits 3 qitrits qubit and 2 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).