Elie Wolfe * & S.F. Yelin

1. POSTDOC ? Separability inSymmetric States Elie Wolfe*& S.F. Yelin ask for preprint *elupus@gmail.com

2. Mixtureof Entangled≠an Entangled Mixture Here are some highly entangled Bell states… they can be expressed as product states… and when mixed are seen to be separable!

3. Introducing the Symmetric Basis Eigenstates The symmetric 4 qubit Hilbert space is spanned by the five such possible eigenstates 3 qubits lower state, 1 qubit upper state. Sum over all permutations & normalize. the symmetric (qubit) product states

4. General Diagonal Symmetric mixed states “Diagonal” ρ: Any ρ such that the eigenstates of ρ are themselves symmetric Highly entangled product states General Diagonal Symmetric mixed state. Which such states are separable? For sure? HereafterGDS

5. Why should you care? Dicke Model Superradiance GDSpopulations evolve from maximally excited, governed by differential equations Does superradiance time evolution lead to entangled states ?

6. Outline of Solution Goal:Construct “generic enough” SeparableDiagonalSymmetricstates (herafterSDS) which have the same general form as the GDSstates GDS mixed states(usually entangled) SDSmixed states(100% separable)

7. SeparableDiagonal Symmetric mixed states 1) Begin with a one qubit state 2) Take tensor product with itself: separable. 3) Form a symmetric separable product state 4) [SECRET SAUCE]Uniformly mix over all  5) Take convex mixture over various y Final form:Separable Diagonal Symmetricc (Ask for preprint to see proof of this form.)

8. When GDS is achievable though SDS – separable! General Diagonal Symmetric state Separable Diagonal Symmetric state UNIVERSAL DECOMPOSITION: System of polynomial equations PERFECT SEPARABALITY TEST

9. Example N=4 5th parameter The 5 polynomial equations for 4 qubits These condition ensure that the system of N+1 equations are not underdetermined

10. Superradiance Answer Proof of separability for N=4 superradiance

11. Other Ramifications We can compute maximum values for General Diagonal Symmetric populations consistent with separability Here is an example GDS state which is, counter-intuitively, completely separable. We can solve analytically for the volume of seperable states.

12. Conclusions elupus@gmail.com • Explicit separable construction! Specially tailored to: • Match the general form (uniform mixing over phase!!!!!) • Same # of degrees of freedom • It provides a necessary and sufficient separability test for General Diagonal Symmetric mixed states of N qudits POSTDOC ?

13. Proof of SDS form

14. SDS Form Derivation - 1 (1) Begin with completely generic 1qudit state (2) Repeated tensor product with itself

15. SDS Form Derivation - 2 (3a) Uniform mixture over all phi by integration (3b) Integration forces diagonal symmetry. Use “K” (3c) We can “clean up” the leading coefficients

16. SDS Form Derivation - 3 Where we left off Derivation essentially comlete, just mix over w