# What kind on transformations can be applied on density matrices?

Completely positive trace preserving maps ( CPTP ) transform a valid density matrix to another, then why do we only talk about unitary transformations on density matrices ( $\rho \to U\rho U^{\dagger}$ ) ? For example in the paper Creation of superposition of unknown quantum states they talk about CPTP maps $\delta :C^2 \otimes C^2 \to C^2$. So is it that there would be a unitary transformation $U:C^2 \otimes C^2 \to C^2 \otimes C^2$ for $\delta$ , such that if I just look at the reduced density matrix I will get the same mapping ? In general when is unitary transformation applied and when is a CPTP applied on a density matrix ?

• "We" don't only talk about unitary transformations. The most general physical transformation is CPTP. However, you can always consider a CPTP map as a unitary transformation performed in a larger Hilbert space, which results from the Stinespring dilation theorem. Jun 5, 2015 at 12:18
• I won't make this an answer because it's a little unclear what you mean by "we"--as noted, people certainly do talk about non-unitary transformations. It's possible, however, that you're talking about unitary evolution? In which case, any time the dynamics are described by a Hamiltonian, you get unitary evolution. If you have dissipative dynamics (from being coupled to a larger reservoir that you've traced out, say) you instead have Liouvillians which are not unitary, but are CPTP. This is I suppose the physical picture that goes with the above-mentioned theorem. Jun 5, 2015 at 14:09
• @MarkMitchison sorry for my language. My basics are only limited to introductory chapters of Neilson and Chuang. By 'we' I meant, in the material I read till now I came across only unitary transformations. Jun 5, 2015 at 14:40
• @sasha No apology necessary. As noted above, non-unitary CPTP transformations are useful when you want to consider one subsystem that is part of a larger system. They are also useful when you do not want to model the environment physically, but rather would like to view it as a "quantum channel" which has some effect on your state. Examples include the effect of dephasing or measurements by some external parties. The quantum channel picture is more general than dynamics generated by a Liouvillian, since it is a transformation between states rather than a continuous evolution equation. Jun 5, 2015 at 15:01