1. Non completely positive maps in physics Daniel R Terno Karol Życzkowski Hilary Carteret

2. History, references, etc P. Pechukas, Phys. Rev. Lett. 73, 1060 (1994) R. Alicki, Phys. Rev. Lett. 75, 3020 (1995) P. Štelmachovič and V. Bužek, Phys. Rev. A 64, 062106 (2001) T. F. Jordan, A. Shaji, and E. C. G. Sudarshan, Phys. Rev. A 70, 052110 (2004). K. Życzkowski and I. Bengtsson, Open. Sys. Info. Dyn. 11, 3 (2004).

3. Maps & dynamical matrices Dynamical (Choi) matrix Properties If trace-preserving If unital Choi theorem Hermitian

4. Church of the larger Hilbert space CP maps & dynamical matrices Eigendecomposition Kraus form of a CP map Sudarshan form

5. Non-CP maps & extensions Explicit form of a non-unitary evolution: extensions CP map Non- CP map Note Pure states => CP evolution Unique assignment map E for all mixed states => product extension => CP evolution

6. Non-CP maps: physics Unitary evolution & partial trace

7. Defs: Non-CP maps: definitions linear assignment map Non-CP map with domain of positivity unitary transformation is physically accessible Example

8. CP map Non-CP maps: examples so what?

9. Negative eigenvalues of D : non-CP map Entanglement PPT test: No entanglement both before & after

10. NO! Non-CP maps: state-dependence? nonlinear

11. Necessary condition 1 is physically accessible, then Finite-dim extension: trivial Infinite-dim extension: no decomposition à la Bloch

12. Example: transposition Sudarshan form: Negative eigenvalue:

13. Necessary condition 2 is physically accessible with state-independent affine form & unital, then it is CP Qualification: if are state-independent, is not necessary

14. Dynamical decoupling as a non-CP map Story… Real life (almost)

15. Non-CP map: assume

16. Summary & questions Sufficient conditions Channels Accessibility=extension & unitary Necessary conditions Fake non-linearity No entanglement is required May not occur