The Joint Condensed Matter Seminar (JCMS) series is organised by KTH Royal Insitute of Technology, Nordita, and Stockholm University.
On Monday, March 6th, 2023 from 11 am to 12 am we will host a seminar by Erik Aurell from KTH.
Time reversals of open quantum systems as linear involutions (or not)
Standard quantum-mechanical time inversion is an anti-unitary operation representing time-reversal symmetry. A unitary quantum map is hence changed from \(\Phi: \rho\to U\rho U^{\dagger}\) to \(\Phi^{SQTI}: \rho\to U^{\dagger}\rho U\). One can ask what would be the corresponding operation acting on any CPTP map, which would still give the standard result for unitary maps. The time reversed quantum map \(\Phi^R\) must then also be a CPTP map, and the time reversal operation \(R\) must be an involution, i.e. \(\left(\Phi^R\right)^R=\Phi\). Further, the standard result means \(\left(U \cdot U^{\dagger}\right)^R=\left(U^{\dagger} \cdot U\right)\). One class of \(R\)s fulfilling these natural requirements is standard quantum mechanical time inversion of a system and an environment. This is however contingent on the environmental representation, and is not defined in terms of only the CPTP map itself.
In this talk I consider if \(R\) can act linearly. I will give two arguments why this is probably impossible. The first is based on a theorem presented in a recent paper with Chiribella and Zyczkowski, where we considered the analogous question for quantum operations (completely positive trace-non-increasing maps). In this setting one can extend Wigner’s theorem from quantum states to quantum evolutions, such that every symmetry of the space of quantum evolutions can be decomposed into two state space symmetries that are either both unitary or both anti-unitary, and this rules out standard quantum-mechanical time inversion. The second argument is based on the fact that a linear involution on the set of all CPTP maps can be lifted to a reflection symmetry in the linear space of trace-preserving maps. If there is an \(R\) acting as a linear involution of all CPTP maps, this non-convex set hence must have a nontrivial reflection symmetry, which seems unlikely.
The talk is partly based on joint work with Giulio Chiribella and Karol Życzkowski, published as Phys. Rev. Research 3, 033028 (2021).