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Entanglement Revealed by Macroscopic Observations

Entanglement Revealed by Macroscopic Observations. Johannes Kofler 1,2 and Č aslav Brukner 1,2 1 Institut für Experimentalphysik, Universität Wien 2 Institut für Quantenoptik und Quanteninformation, Österreichische Akadamie der Wissenschaften. Central European Workshop on Quantum Optics

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Entanglement Revealed by Macroscopic Observations

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  1. Entanglement Revealedby Macroscopic Observations Johannes Kofler1,2 and Časlav Brukner1,2 1 Institut für Experimentalphysik, Universität Wien 2 Institut für Quantenoptik und Quanteninformation, Österreichische Akadamie der Wissenschaften Central European Workshop on Quantum Optics Vienna, May 26th 2006

  2. Macroscopic Entanglement 1. Within a macro-object 2. Between two macroscopic parts Measuring internal energy, heat capacity, magnetic susceptibility … Measuring collective properties of both parts

  3. Motivation • Aim: Detect entanglement between macroscopic parts of a system by looking at collectiveoperators (macroscopic observables) for different regions • Collective operators: Sum over (average of) individual operators • Why: (i) Experimentally approachable • Fundamentally interesting: “Can collective operators be entangled?” • Part I: The Linear Harmonic Chain (ground state) • Part II: Spin Systems

  4. Part I: The Linear Harmonic Chain Hamiltonian Coupling:

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

  6. Collective Blocks of Oscillators A Collective Operators B

  7. Notation • Usual: “Mathematical” Block • Every measurement in the block is allowed in principle [Audenaert et al.], [Serafini et al.], [Botero and Reznik], … • Here: “Physical” Block • Characterized by collective operators • Measurement couples only to the block as a whole • “Information loss”

  8. Matrix of Second Moments Degree of Entanglement (Peres–Horodecki–Simon–Kim negativity)

  9. Results: Negativity  versus block size n d = 0: Entanglement for all coupling parameters  and block sizes n (macroscopic parts) Strength of entanglement decreases with decreasing coupling and increasing block size d = 1: No entanglement between two separated single oscillators (n = 1) Genuine multi-particle entanglement for n = 2, 3 and 4

  10. Periodic Blocks (Exploitation of the Area Law) d = 0 d = 1 s = 2 s = 1 s = 5 s = 2 s = 1 s = 5 • d = 0: Entanglement “scales” with the total boundary • d 1: Trade-off between many boundaries and having large subblocks

  11. Scalar Quantum Field Theory Collective Field Operators etc.

  12. Conclusions Part I (Harmonic Chain) • Entanglement between collective operators (macroscopic parts) in the linear harmonic chain is demonstrated and quantified • Neighbouring collective blocks are entangled for all block sizes and coupling constants • Genuinemulti-particle entanglement between (small) separated blocks where no individual pair is entangled • Entanglement between periodic blocks “scales” with total boundary • Phys. Rev. A 73, 022104 (2006)

  13. Part II: Spin Systems • n spins in each subsystem: • Hilbert space is exponentially large: (2n)A (2n)B (intractable) • discrepancy between quantum und classical correlations grows exponentially • subsystems totally symmetric (Dicke states)  total spins = n/2 • Hilbert space: (n+1)A (n+1)B • maximally entangled (pure) states •  entanglement scales with n,s • no analytical solutions for mixed states for s > 1 (i.e., n > 2) • recent (experimental) progress [Julsgaard et al.], [Sørensen et al.]

  14. Spin Systems can we say anything if we measure only classical observables (magnetization) and do not make assumptions (symmetry, mixedness) about the state? Collective Spin Operators (i = x,y,z)

  15. Expectation Values and Correlations Individual Spins Collective Spins

  16. Normalization ab Two Virtual Qubits a and b(44 density matrix) These „virtual“ qubits carry the collective information of the subsystems A and B they are associated with sa, sb, tab … only 3+3+9 = 16 numbers

  17. Entanglement For any convex entanglement Measure: (Negativity, Concurrence, …) pairwise collective entanglement, only collective observables average entanglement For a certain class of states (which need not be totally symmetric):

  18. Example: W-state

  19. Example: Generalized Singlet State (spin s) A B 1/s Average entanglement per pair:  1/s Number of pairs: s2

  20. Conclusions Part II (Spin Systems) • What can be said about entanglement between two spin subsystems using only collective observables and without assumptions about the state (symmetry or mixedness)? • (Normalized) Collective properties are transferred to two virtual qubits • For any convex measure the pairwise collective entanglement is a lower bound for the average entanglement • Method is not restricted to totally symmetric states • Method can be generalized to define multi-partite entanglement of M collective spins belonging to M separated samples • quant-ph/0603208

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