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Entanglement in fermionic systems

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

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Entanglement in fermionic systems

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  1. Entanglement in fermionic systems M.C. Bañuls, J.I. Cirac, M.M. Wolf

  2. Goal Definition of entanglement in a system of fermions given the presence of superselection rules that affect the concept of locality

  3. 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

  4. Fermionic Systems • System of m=mA+mB fermionic modes • partition A={1, 2, … mA} • partition B={mA+1, … mA+mB} • Basic objects: creation and annihilation operators • canonical anticommutation relations • and their products, generate operators on A (B) • products of even number commute with parity

  5. Fermionic Systems • Fock representation, in terms of occupation number of each mode • isomorphic to m-qubit space • action of fermionic operators is not local • e.g. for mA=mB=1

  6. Parity SSR • Physical states and observables commute with parity operator 1x1 modes Fock representation physical states

  7. Parity SSR • Physical states and observables commute with parity operator 1x1 modes Fock representation physical states

  8. Parity SSR • Physical states and observables commute with parity operator 1x1 modes Fock representation physical states

  9. Parity SSR • Physical states and observables commute with parity operator 1x1 modes Fock representation physical states

  10. Parity SSR • Physical states and observables commute with parity operator mAxmB modes Fock representation physical states

  11. Parity SSR • Physical states and observables commute with parity operator mAxmB modes Fock representation local physical observables

  12. How to define entangled states? • Entangled states = are not separable • Separable states = convex combinations of product states • Define product states…

  13. Product states P1 P2 P3

  14. Product states P1 P2 P3 BUT when restricted to physical states ( commuting with parity)

  15. Product states Example: 1x1 modes P1 in P1 not in P2 P2

  16. We have… …two different sets of product states Use them to construct separable states For pure states, they are all the same!!

  17. Separable states S1 S2’ S2=S3 For physical states ( commuting with parity)

  18. Separable states Local measurements cannot distinguish states that produce the same expectation values for all physical local operators • define equivalent states • define separability as equivalence to separable state

  19. [S2] S1 S2’ S2=S3 Separable states For physical states ( commuting with parity)

  20. Separable states Example: 1x1 modes S2

  21. Separable states Example: 1x1 modes S2 S2’

  22. Separable states Example: 1x1 modes S2 S1= S2’ [S2]

  23. There are… …four different sets of separable states They correspond to four classes of states • different capabilities for preparation and measurement

  24.  Preparable by local operations and classical communication, restricted by parity  Convex combination of products in the Fock representation  Convex combination of states s.t. locally measurable observables factorize  All measurable correlations can be produced by one of the above

  25. Characterization Criteria in terms of usual separability

  26. Characterization Criteria in terms of usual separability [S2] S1 convex combination of products S2’ S2

  27. Characterization Criteria in terms of usual separability [S2] S1 convex combination of products S2’ S2

  28. Measures of entanglement • For S2’ and S2, the entanglement of formation can be defined • For 1x1 modes, EoF in terms of the elements of the density matrix

  29. Multiple Copies • Not all the definitions of separability are stable under taking several copies of the state • S2 and S2’ asymptotically equivalent • 1x1 modes: all of them equivalent in the limit of large N distillable states not in [S2]

  30. 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)

  31. Application to a particular case • Fermionic Hamiltonian • Reduced 2-mode density matrix calculated from • Regions of separability as a function of , ,  • EoF for S2, S2’

  32. Application to a particular case • Fermionic Hamiltonian • Reduced 2-mode density matrix calculated from • Regions of separability as a function of , ,  • EoF for S2, S2’

  33. Application to a particular case • Fermionic Hamiltonian • Reduced 2-mode density matrix calculated from • Regions of separability as a function of , ,  • EoF for S2, S2’ [S2] [S2] S2’ S2 [S2] [S2] S2’ S2

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