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SUMMARY:On the quantum capacity of the qubit depolarizing channel - Felix 
 Leditzky\, JILA\, University of Colorado Boulder
DTSTART:20170427T131500Z
DTEND:20170427T141500Z
UID:TALK71937@talks.cam.ac.uk
CONTACT:Steve Brierley
DESCRIPTION:The qubit depolarizing channel\, defined in terms of a single 
 depolarizing parameter p\, is a simple and widely studied quantum channel.
  However\, an exact determination of its quantum capacity has remained elu
 sive so far\, and we can only give lower and upper bounds on its capabilit
 y to transmit quantum data faithfully. In this seminar talk\, I will discu
 ss two upper bounds on the quantum capacity of the qubit depolarizing chan
 nel (in the sequel denoted D_p)\, which constitute the best known upper bo
 unds for low and high values of the depolarizing parameter p\, respectivel
 y.\n\nThe first bound is the approximate degradability bound derived by Su
 tter et al. (arXiv:1412.0980). This powerful bound uses the fact that D_p 
 is "almost degradable" for low values of p\, and hence its quantum capacit
 y is close to its (computable) channel coherent information. In joint work
  with Debbie Leung and Graeme Smith (soon to be posted on the arXiv)\, we 
 analyze this bound in greater detail. We prove that the complementary chan
 nel of the depolarizing channel (with a tweaked depolarizing parameter) se
 rves as an "approximate degrading map"\, and renders the depolarizing chan
 nel O(p^2)-approximate degradable. As a result\, the corresponding upper b
 ound derived by Sutter et al. on the quantum capacity is tangent to the ch
 annel coherent information\, which explains the power of the approximate d
 egradability approach. We can lift this result to all generalized Pauli ch
 annels\, and more generally to "low-noise" channels\, i.e.\, channels that
  are close to the identity channel with respect to the diamond norm.\n\nFo
 r large values of the depolarizing parameter p\, the best known upper boun
 d has recently been derived in arXiv:1701.03081\, which is joint work with
  Nilanjana Datta and Graeme Smith. One of our main results in arXiv:1701.0
 3081 is a general upper bound on the one-way distillable entanglement of a
  mixed bipartite state\, based on convex decompositions of this state into
  degradable and antidegradable states. Since the depolarizing channel belo
 ngs to the class of so-called teleportation-simulable channels\, quantum d
 ata transmission is equivalent to entanglement distillation using its Choi
  state. Hence\, an upper bound on the quantum capacity of D_p is obtained 
 from evaluating our bound for the Choi state of D_p\, which is a so-called
  isotropic state. Exploiting this symmetry\, we were able to phrase our ge
 neral upper bound as a non-convex optimization problem that can be solved 
 numerically\, yielding the advertised bound.
LOCATION:MR4\, Centre for Mathematical Sciences\, Wilberforce Road\, Cambr
 idge
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