TITLE PAGE TITLE HERE TITLE HERE DIVISIBILITY OF QUBIT CHANNELS - PowerPoint PPT Presentation

TITLE PAGE TITLE HERE TITLE HERE DIVISIBILITY OF QUBIT CHANNELS AND DYNAMICAL MAPS David Carlos Mario Ziman Davalos Pineda

PLAN 1. Channel divisibility 2. dynamical maps 3. Subsets of 1. 4. How 2. passes through 3.

QUANTUM CHANNELS time L E N N A H C Φ completely positive trace-preserving linear map

DIVISIBILITY time L E N N A H C L E N N A H C

TYPES OF DIVISIBILITY TYPES OF DIVISIBILITY in fjnite ly simal DIVISIBLE M.Wolf, J. Eisert, T. Cubitt, I. Cirac, Phys. Rev. Lett., 101, 150402 (2008) M.Wolf, I. Cirac, Comm. Math. Phys. 279, 147–168 (2008) I n d I v I s I b l e infnitesimal divisible infnitely divisible

SET OF CHANNELS - convex set Φ identity

SET OF CHANNELS infnitely-divisible Φ infnitesimal-divisible divisible indivisible identity C ∞ ⊂ C inf ⊂ C div C indivisible , e.g. qNOT

DYNAMICAL MAP t Φ → Φ t identity SET OF CHANNELS

DYNAMICAL MAP t Φ → Φ Δ=Φ·Ψ -1 t Δ Ψ identity SET OF CHANNELS

DYNAMICAL MAP t Φ → Φ Δ=Φ·Ψ -1 t t → Ψ t Δ Δ Ψ identity SET OF CHANNELS

DYNAMICAL MAP t Φ → semigroup t t → Ψ t → ε t t identity SET OF CHANNELS

DYNAMICAL MAP t Φ → semigroup t t → Ψ t P → ε t CP t identity DIVISIBILITY OF DYNAMICAL MAPS SET OF CHANNELS

DYNAMICAL MAP NP L-divisible CP-divisible P-divisible IN-divisible CP identity P for all time intervals

CHANNEL TYPES C L L-divisible C CP CP-divisible C P P-divisible C NP NP-divisible identity Achievability by dynamical maps + closure

C NP =C CHANNEL TYPES C P C L L-divisible C div C CP CP-divisible C P P-divisible C NP NP-divisible C ∞ C CP = C L C inf C indivisible C div divisible C inf infnitesimal-divisible C ∞ infnitely-divisible

C NP =C CHANNEL TYPES C P C L L-divisible C div C CP CP-divisible C P P-divisible C ∞ ⊂ C inf ⊂ C div C NP NP-divisible C ∞ C CP ⊂ = = C L ⊂ C CP ⊂ C P C L C inf C indivisible C div divisible C inf infnitesimal-divisible C ∞ infnitely-divisible

QUBIT UNITAL C P ⇔ det ≥ 0 i n d C indivisible = faces i v i s C P i b l e not C P identity C P but not divisible

QUBIT UNITAL C L C CP \ C L no tetrahedron symmetries plus tetrahedron symmetries

QUBIT UNITAL identity

QUBIT NONUNITAL

DYNAMICAL MAP L-divisible CP-divisible P-divisible NP-divisible identity dynamical phases - intervals

CHANNELS L-divisible CP-divisible divisible indivisible identity dynamical phases - pointwise

QUESTION Which transitions are allowed? identity

QUBIT UNITAL identity All types of borders exist.

QUBIT DYNAMICAL MAP identity C P C L C div qNOT Time evolution to quantum NOT T. Rybár, S. N. Filippov, M. Ziman, V. Bužek. J. Phys. B, 45, 154006 (2012 )

QUBIT DYNAMICAL MAP Jaynes-Cumming model E. T. Jaynes and F. W. Cummings. Proc. IEEE 51, 89 (1963)

GAME OVER THANK YOU THANK YOU FOR YOUR FOR YOUR ATTENTION ATTENTION D. Davalos, M. Ziman, C. Pineda, Quantum 3, 144 (2019)