Super-Duper-Activation of Quantum Zero-Error Capacities

1. Super-Duper-Activation of Quantum Zero-Error Capacities Toby Cubitt, Jenxin Chen, Aram Harrow arXiv:0906.2547 and Toby Cubitt, Graeme Smith arXiv:0912.2737

2. (qu)bits (qu)bits … … n Channel Capacities

3. (qu)bits (qu)bits … … n Zero-Error Channel Capacities

4. Status of Zero-Error Capacity Theory  [Shannon ’56]  [Alon ’97]  (trivial)  ?     [Duan, ’09] ?

5. Superactivation of the Zero-Error Classical Capacity of Quantum Channels • TheoremFor any satisfying, there exist channels such that: • Each channel maps and has Kraus operators. • Each channel individually has no zero-error classical capacity at all. • The joint channel has positive zero-error classical capacity.

6. Status of Zero-Error Capacity Theory  [Shannon ’56]  [Alon ’97]  (trivial)       Superduperactivation

7. Superduperactivation of Zero-Error Capacities of Quantum Channels • TheoremFor any satisfying, there exist channels such that: • Each channel maps and has Kraus operators. • Each channel individually has noclassical zero-error capacity • The joint channel even has positive quantum zero-error capacity (hence, also no quantum zero-error). (hence, every other capacity is also positive).

8. Proofs • Reduce super(duper)activation to question about existence of certain subspaces. • Show that such subspaces exist.

10. Reducuctio ad Subspace • Want two channels such that: A channel has positive zero-error capacity iff there exist two different inputs that are mapped to outputs states that are perfectly distinguishable.

11. The zero-error capacity is non-zero iff • Conversely, the zero-error capacity is zero iff Reducuctio ad Subspace • Want two channels such that:

12. The zero-error capacity is non-zero iff • Conversely, the zero-error capacity is zero iff Reducuctio ad Subspace • Want two channels such that:

13. Translate these into statements about the supports of the Choi-Jamiołkowski matrices of the composite maps Reducuctio ad Subspace • Want two channels such that:

14. Want a bipartite subspace such that: Reducuctio ad Subspace (This is similar to the argument for p = 0 minimum output entropies in[Cubitt, Harrow, Leung, Montanaro, Winter, CMP 284, 281 (2008)])

15. (Almost) any old subspace will do! • Def. The set of subspaces that have a tensor power whose orthogonal complement does contains a product state. = The subspaces we don’t want.

16. (Almost) any old subspace will do! • Proof intuition:Think of (unnormalised) bipartite subspaces as (projective) matrix spaces:Product states$ rank-1 matrices$ all order-2 minors vanish$ set of simultaneous polynomials$ Segre variety • Lemma:Ed is an algebraic set≡ defined by simultaneous polynomial equations≡ Zariski closed in the Grassmanian)

17. Ed (Almost) any old subspace will do! • An algebraic set set is either measure zero (in the usual Haar measure) or it is the entire space.(Intuitively, it’s defined by some algebraic constraints, so either the constraints are trivial, or they restrict the set to some lower-dimensional manifold.) Grd Grd Ed

18. Grd UPB Ed Ed (Almost) any old subspace will do! • To show Ed is measure zero, just have to exhibit one subspace that’s not contained in Ed. Grd • Use a subspace whose orthogonal complement is spanned by an unextendible product basis (UPB), which exist for a large range of dimensions (! mild dimension constraints) • Lemma: Tensor products of UPBs are again UPBs.

19. Grd Ed (Almost) any old subspace will do! • The set of “bad” subspaces is zero-measure (in the usual Haar measure on the Grassmanian), so the set of “good” ones is full measure! pick a one at random!

20. Reducuctio ad Subspace • Want a bipartite subspace such that: pick one atrandom! Subspaces obeyingsymmetry constraintsalso form an algebraic set

21. (Almost) any subspace will do! • Want a bipartite subspace such that: full measure pick one atrandom! non-zeromeasure

22. Superduperactivation

23. Superduperactivation of Zero-Error Capacities of Quantum Channels • Translates into one additional constraint on the subspace : full measure pick one atrandom! non-zeromeasure

24. Conclusions • Unwind everything(!) to get channels that give super(duper)activation of the asymptotic zero-error capacity. • Quantum channels not only behave in this extremely weird way for quantuminformation [Smith, Yard ’07],but also for classical information too. • A corollary of this result is that the regularized Rényi-0 entropy is non-additive. If we could prove the same thing for the regularized Rényi-1 (von Neumann) entropy, would prove non-additivity of the classical Shannon capacity of quantum channels.

25. Status of Shannon Capacity Theory  [Shannon ’48]  [Shannon ’48]  (trivial)  (trivial) ? * [Hasting ’08] * [DiVincenzo, Shor, Smolin ’98]   [Smith, Yard ’08] * probably