2. Entanglement Detection Gühne & Tóth, Physics Reports 474 (2009). Superradiance: … Theory of Collective Spontaneous Emission Gross & Haroche, Physics Reports 93 (1982). Superradiance in spin- J particles: Effects of multiple levels Lin & Yelin, PRA 85 (2012)

3. 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. Dicke State Form Example: All 4 qubit Dicke states. Total Spin Spin Z

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

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

7. 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. Partitioned form of our mixed states Rho Partial Transpose!

9. Generic weighted sum over a partition Explicit expression for the weights in the sum Derived by normalization conditions

10. Nonnegativity of 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 ρ PT eigenvalues for the general N-qubit case.

11. The partially transposed density matrix A completely generic pure state Eigenvalue equation Solution for the eigenvalue (with unknowns)

13. Eigenvalues of the partial transpose, for any 1 qubit biparitioned away. Derivation complete! This is the N-qubit entanglement criterion we set out to compile.

14. 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.

16. Amplitude of Π Time → Population density over time: Ground state and 3 excited states.

19. See Unextendible Product Bases and Bound Entanglement Bennett et al, PRL 82 (1999) All PPT Biseperable

20. See Quantum nonlocality does not imply entanglement distillability Vertesi & Brunner, PRL 108 (2012) A|BCAB|C A|B|C