Quantum Channels Optical fibers Quantum memories Any quantum
- Slides: 40
Quantum Channels: Optical fibers Quantum memories Any quantum operation (without post selection)
Quantum Channels: Defining Properties: Optical fibers Linear Quantum memories Trace preserving Any quantum operation (without post selection) Completely positive
What does a snapshot tell us about the evolution of a system?
What does a snapshot tell us about the evolution of a system? Markovian:
What does a snapshot tell us about the evolution of a system? Markovian: t-dep. Markovian: Infinitesimal divisible:
What does a snapshot tell us about the evolution of a system? Markovian: t-dep. Markovian: Infinitesimal divisible: Divisible:
What does a snapshot tell us about the evolution of a system? Markovian: t-dep. Markovian: Infinitesimal divisible: Divisible: Indivisible: , unitary
Ideal channel
Markovian channels (2%) Ideal channel
t-dep. Markovian = inf. divisible (17%) Markovian channels (2%) Ideal channel
t-dep. Markovian = inf. divisible (17%) Markovian channels (2%) Ideal channel Indivisible channels (zero measure)
t-dep. Markovian = inf. divisible (17%) Markovian channels (2%) Ideal channel Indivisible channels (zero measure)
Matrix rep. of T “ log T ” Check Lindblad form
Matrix rep. of T with Jordan decomp. “ log T ” Check Lindblad form
Matrix rep. of T with Jordan decomp. “ log T ” Check Lindblad form
Matrix rep. of T with Jordan decomp. “ log T ” Check Lindblad form
Matrix rep. of T with Jordan decomp. “ log T ” Check Lindblad form
Matrix rep. of T with Jordan decomp. “ log T ” Check Lindblad form semidef. integer program has efficient solution [Khachiyan, Porkolab] becomes NP-hard if d is not fixed
What about ? How to compute? ?
‘Liouvillian’ of master equation non-physical evolution
‘Liouvillian’ of master equation non-physical evolution measure based on rate minimal noise
‘Liouvillian’ of master equation non-physical evolution measure based on rate minimal noise integer s. d. p.
‘Liouvillian’ of master equation non-physical evolution measure based on rate minimal noise M(T)=1 iff T Markovian basis independent Monotone w. r. t. adding noise computable geometric interpretation (minimal shrinking factor)
‘Liouvillian’ of master equation non-physical evolution measure based on rate minimal noise non-convex structure! 1 dephasing 0. 5 p/4 x-rotation 0 0. 5 1
Def. : T is infinitesimal divisible if for all e there exist channels s. t.
Def. : T is infinitesimal divisible if for all e there exist channels s. t. Thm. : A channel T is infinitesimal divisible iff
Def. : T is infinitesimal divisible if for all e there exist channels s. t. Thm. : A channel T is infinitesimal divisible iff Thm. : Infinitesimal divisibility is preserved under filtering operations with Kraus rank one
Def. : T is infinitesimal divisible if for all e there exist channels s. t. Thm. : A channel T is infinitesimal divisible iff Thm. : Infinitesimal divisibility is preserved under filtering operations with Kraus rank one Thm. : A qubit channel is infinitesimal divisible iff and with
Def. : T is infinitesimal divisible if for all e there exist channels s. t. Thm. : A channel T is infinitesimal divisible iff Thm. : Infinitesimal divisibility is preserved under filtering operations with Kraus rank one Thm. : A qubit channel is infinitesimal divisible iff and with Thm. : All extremal qubit channels are infinitesimal divisible. non-convex!
Useful:
Useful: Thm. :
Useful: Thm. :
Useful: Thm. :
Def. :
Def. : Thm. :
Def. : Thm. : There are channels not allowing any nontrivial decomposition. Proof: Take assume since Example: approx. time-reversal
Def. : Thm. :
• Wolf, Cirac, Comm. Math. Phys. (2008) • Wolf, Eisert, Cubitt, Cirac, ar. Xiv: 0711. 3172 • Cubitt, Eisert, Wolf, Cirac, t. b. p. soon Markovian channels t-dep. Markovian channels Determinants Indivisble channels 1 0. 5 0 0. 5 1
