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

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, (2001) T. F. Jordan, A. Shaji, and E. C. G. Sudarshan, Phys. Rev. A 70, (2004). K. Życzkowski and I. Bengtsson, Open. Sys. Info. Dyn. 11, 3 (2004).

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

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

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

Non-CP maps: physics Unitary evolution & partial trace

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

CP map Non-CP maps: examples so what?

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

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

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

Example: transposition Sudarshan form: Negative eigenvalue:

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

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

Non-CP map: assume

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