Entanglement in Symmetric States & N- Qubit Superradiance

1. Entanglement in Symmetric States &N-QubitSuperradiance 15 Minute Version Elie Wolfe S.F. YelinUniversity of Connecticut

2. 1. Given a family of states (mixtures of theDicke states) we precompile a general N-qubitentanglement criterion.2. We use the developed criterion to study entanglement in superradiance evolution which is a special case of these states. Goals: Entanglement DetectionGühne& Tóth, Physics Reports 474 (2009). Superradiance:… Theory of Collective Spontaneous EmissionGross & Haroche, Physics Reports 93 (1982). Superradiance in spin-Jparticles: Effects of multiple levels Lin & Yelin, PRA85(2012)

3. Dicke States Dicke States are normalized completely symmetrized multipartite N-qubit pure states,invariant under any permutation of the parties. The 3-qubit W state is an example of a Dicke State.

4. The Dicke States Total Spin Dicke State Form Spin Z Example: All 4 qubitDicke states.

5. Our Mixed States These are the mixtures of Dicke states that we are considering. Matrix form of such a mixed state: 3 particles scenario

6. Partitioned Form To study the separability of mixed Dicke states we partition the Hilbert space into two complementary parts, such as collections of qubitsto the left and right of a partition. We therefore express pure states as summations over the partition.

7. Via qubit #1 A familiar 4 particle Dicke State rewritten as explicit sum over the first qubit A weighted sum over Dicke states Expressed here completely in Dicke state notation

8. General Form Partitioned form of our mixed states Rho Partial Transpose!

9. (the weights) Generic weighted sum over a partition Derived by normalization conditions The proof will not fit in the margin. Explicit expression for the weights in the sum

10. PPT Conditions Nonnegativityof the eigenvalues of the partially transposed density matrix is a necessary condition for separability, known as Positive Partial Transpose, or PPT. We solve for the ρPT eigenvalues for the general N-qubit case.

11. Dirac Eigenvalues The partially transposed density matrix The proof will not fit in the margin. A completely generic pure state Eigenvalue equation Solution for the eigenvalue (with unknowns)

12. 1-Particle Solution For a given and we have equations for with unknowns and which we carefully eliminate. Eigenvalues confirmed for finite system. For the eigenvalues are:

13. The Criteria Eigenvalues of the partial transpose, for any 1 qubitbiparitioned away. Derivation complete! This is the N-qubit entanglement criterion we set out to compile. Only the ‘minus’ eigenvalue matters!

14. Superradiance Superradiance is a nonclassical phenomenon of N-particle decay from an excited state. This time evolution of this cooperative behavior is often approximated by a conveniently simple model.

15. Superradiance J= N/2

16. Time Evolution Amplitude of Π J= 3/2 Example: Time → Population density over time: Ground state and 3 excited states.

17. Yes! Always PPT • We found that the superradiant states are PPT[n=1] at ALL TIME forN≤10. • We found that the superradiant states are PPT[n=2] at ALL TIME forN=4, the first instance where single-particle bipartitioning is insufficient. • We can prove that for GENERALNthe PPT conditions associated with M=J-1 and M=J-2 (etc…) are ALWAYS SATISFIED by superradiant states!

18. Conclusions GENERAL ρ: Method of determining eigenvalues for N/nParticle ρ-Partial-Transpose Explicit PPT conditions(for n=1) SUPERRADIANCE: PPT verified for N≤10 particlesand for the highest level eigenvalues for all N. We believe PPT in general. We conjecture that the states are in fact fully separable; verified for N=2.

19. Disclaimer #1 For N>2, PPT on a partition is a necessary but not sufficient condition for biseparability along that partition. See Unextendible Product Bases and Bound EntanglementBennett et al, PRL 82(1999)

20. Disclaimer #2 To rule out entanglement entirely, a state must be simultaneouslybiseparable over all partitions. See Quantum nonlocality does not imply entanglement distillabilityVertesi & Brunner, PRL 108 (2012) • A|B|C

21. Full Derivation 1/2

22. Full Derivation 2/2

23. Example Partition

24. Legacy Slide PPT iff: Partial Transposed Form:

25. Beta Conjectures 1) 2)

26. L’Hopital’s Rule