Maximizing Quantum Entanglement Characterization Using Nilpotent Polynomials

1. CHARACTERIZATION OF MULTIPARTITE ENTANGLEMENT FOR A QUANTUM SYSTEM IN PURE STATE VIA NILPOTENT POLYNOMIALSV.M.Akulin Orsay Kolymbari, Crete, Greece 2 September, 2013 Aikaterini Mandilara, Lorenza Viola, Anfrei Smilga

2. One of key messages: Entanglement is not an observation independent characteristic, like the most of physical quantities, -- entanglement in a system depends on what an obseravator considers as different parts of the system. ENTENGLEMENT DEPENDS ON THE PARTITION VERSCHRÄNKUNG ENTANGLEMENT INTRICATION ПЕРЕПУТЫВАНИЕ(PEREPUTYVANIE)

3. Product state vs entanged state

4. ENTANGLEMENT DEPENDS ON THE PARTITION

5. Energy representation. Two-level systems We want to have something like that: ?????? 1 2

6. CREDO Entanglementis Entanglement is not What do we want from a quantity which characterizes the entanglement? A physicalquantity Dependent on how do we define the partition, that is, what elements does the assembly consists of A quantity independent of the question we addressing We want it be an extensive quantity – an analog of the thermodynamic potentials, that is in the situation where two parts are unentangled the characteristic of the sum should be a sum of characteristics element assembly=ensemble-statistics

7. Two ways: • Find invariants of local transformations that take the same values on the orbit. It is possible to do for <6 qubits, but……….. • Determine amarker, that is a state representing the orbit What we have to keep in mind constructing a quantity characterizing the entanglement Local transformations preserve entanglement marker

8. Start with invariants and introduce the NOTATIONS Two, three, four, etc. qubits

9. Invariants of local transformations vs orbit markers 3-tangle For 3-level elements the invariants are unknown

10. What is the alternative?Or how to introduce a reasonable entanglement description given a reference and marker states? marker

11. Canonic state -1 Canonic state of the orbit Reference state Coset dimension Maximum population of the reference state + some phase requirements: We specify 2n out of 3n parameters, the remaining n parameters are phases of second highest states

12. How to introduce nilpotent polynomials for entanglament description Normalization to unit vacuum state amplitude for technical convenience X X C

13. nilpotent Extensive characteristic -- nilpotential Is also a polynomial! 2 3 N F=1+nilpotent, ln F=nilpotent-(nilpotent) /2+(nilpotent) /3….+(nilpotent) /N. n finite Taylor series, N~2 N+1 (sN+sN-1+sN-3+…+s5+s4+s3+s2+s1) =…+sNsN-1s²N-3….s5s4s3s2s1...+

14. Nilpotential is convenient, but not unique. However it becomes unique for the canonic state!

15. Unambiguous extensive characteristic -- tanglemeter • We would like not only to know whether or not qubits are entangled, but also • Answer the questions: • How much are they entangled? • In which way are they entangled? Depends on D parameters One real parameterD=1 111 D=5 110 101 011 010 100 001 000 011->3; 101->5 etc. D=18

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

17. Canonical states for 3-level systems (qutrits) Maximum population of maximum correlated states Tanglemeter is constructed by the analogy to qubits 2 qutrits 3 qitrits qubit and 2 qutrits

18. - Dynamic equation for nilpotential Schrödinger equation Operators in terms of nilpotent variables Similar to coherent states of harmonic oscillator su(2) operators in terms of nilpotent variables

19. Dynamic equation for nilpotential Infinitesimal transformation Universal evolution of quantum computer ????????? H=H(x,p)

20. Tanglemeter coefficient dynamics during Grover’s search algorithm

21. Entanglement characterization for mixed states. Problem is to find this state, but once it is found – the characterization can be done in the same way as for the pure states

22. Summary • 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). • Notion of the canonic states allows one to unambiguously characterize quantum entanglement with the help of the tanglemeter (nilpotential of the canonic state). • Dynamic equation for nilpotential can be derived. • This technique introduced for qubits, can be generalized on the case of multilevel systems, on the case where the number of operators in the algebra is less than the number of levels, and on the case of indirect measurements. • One can generalize the characterization on the case of mixed states, onse the canonic state is properly defiened.

23. Generalized entanglement, formalism SKEEP

24. Entanglement of quantum assemblies in pure states, general case element assembly=ensemble-statistics • 1.An algebraic description • 2.A reference state • 3.A proper representation of the generic state c + - L n,i L n,i L n,i Ln For an assembly of qubits nilpotent

25. Indirect measurements 1 D=6

26. Tanglemeter from dynamic equations for the nilpotential In order to put f to the canonic form 010 100 001 000 X Dynamic equation close to the canonic state canonic state Condition of maximum population <0 With this technique one can reproduce the results of F. Verstraete on the entanglement classes for 4 qubits

28. Generalized entanglement, idea Entanglement within a single element 1/2 Local transformation Canonic state 1/2

29. Tanglemeters for Spin-1 systems.

30. SKEEP Generalized entanglement, formalism

31. How to tailor qubits entanglement? We consider the simplest nontrivial case of the tanglemeter bkm<<1; bkmr<<< bkm exp{ bij } b-small does not mean that entanglement is small since the order of the matrix in the exponent can be very large. All order correlations differ from zero, and not small.

32. Specific problem and technical details We assume that each atom change its population just a little. It does not mean that the quantum state of the assembly changes a little and the correlations of the atomic states can be found by the perturbation theory

33. The result After having detected 0 photons in the cavity one gets an entangled state with the tanglemeter bnm fc= entangled ensemble of atoms vacuum indirect direct

34. Entanglement in a quantum assembly of qubits in a pure state • An algebraic description • A reference state • A proper representation of the generic state nilpotent element assembly=ensemble-statistics

35. Why the tanglemeter is useful? 1. It contains all information about the entanglement and no extras. 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.

36. We want the entanglement characterization to be somehow similar to statistical correlations.