1. 1 Entanglement between Collective Operators in a Linear Harmonic Chain Johannes Kofler 1, Vlatko Vedral 2, Myungshik S. Kim 3, Časlav Brukner 1,4 1 University of Vienna, Austria 2 University of Leeds, United Kingdom 3 Queen’s University Belfast, United Kingdom 4 Austrian Academy of Sciences, Austria Quantum Information Theory & Technology Summer School Belfast, United Kingdom September 2, 2005

2. 2 Motivation Aim: Detect entanglement between macroscopic parts of a system by looking at collective operators (macroscopic observables) for different regions Collective operators: Sum over (average of) individual operators Why: (i) Experimentally approachable (ii) Fundamentally interesting: “Can collective operators be entangled?” Up to now: entanglement between single particles or between “mathematical” blocks Here: Entanglement between two “physical” blocks

3. 3 The Linear Harmonic Chain Coupling: Hamiltonian

4. 4  Upgrading position and momentum to operators and expansion into modes Pseudo-momentum Dispersion relation Ground state

5. 5 Two-point Vacuum Correlation Functions A. Botero, B. Reznik, Phys. Rev. A 70, 052329 (2004)

6. 6 Collective Blocks of Oscillators Collective Operators same block different blocks

7. 7 Notation Usual: “Mathematical” Block -Every measurement in the block is allowed in principle Here: “Physical” Block -Characterized by collective operators -Measurement couples only to the block as a whole -Information loss

8. 8 Microscopic Mesoscopic Macroscopic

9. 9 Entanglement between two blocks of equal size Matrix of second moments

10. 10 Degree of Entanglement between Blocks Peres–Horodecki–Simon–Kim Negativity

11. 11 Trade-off Microscopic case: p = 0 -Measurement apparatus has to be accurate to resolve individual oscillators -Absolute error in the measured numbers can be large (  1) Macroscopic case: p = 1 -Measurement apparatus can be coarse -Absolute accuracy has to scale with n –2

12. 12 Neighbouring Blocks (of equal size n) Entanglement for all coupling parameters  and block sizes n (macroscopic parts) Strength of entanglement decreases with decreasing coupling and increasing block size

13. 13 Separated Blocks (of equal size n) No entanglement between separated individual oscillators (n = 1) Genuine multi-particle entanglement for n = 2, 3 and 4

14. 14 Entanglement Detection by Uncertainty Relations then the two blocks are entangled (Duan et al)if For n A or n B > 1 this witness is weaker than the entanglement degree  Entanglement not for all neighbouring blocks No multi-particle entanglement detected

15. 15 Scalar Quantum Field Theory Collective operators etc.

16. 16 Conclusions Entanglement between collective operators (macroscopic parts) of a system is demonstrated and quantified Neighbouring blocks in the harmonic chain are entangled for all block sizes and coupling constants Genuine multi-particle entanglement between (small) separated blocks where no individual pair is entangled arXiv:quant-ph/0506236