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

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Channel Capacities (qu)bits … n …

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

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Regularization requiredAdditivity violationSuperactivation Classical channels Quantum channels Classical Quantum ? Status of Zero-Error Capacity Theory (trivial) [Alon 97] [Shannon 56] [Duan, 09] ?

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Superactivation of the Zero-Error Classical Capacity of Quantum Channels Theorem For 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.

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Status of Zero-Error Capacity Theory Regularization requiredAdditivity violationSuperactivation Classical channels Quantum channels Classical Quantum (trivial) [Alon 97] [Shannon 56]

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Superduperactivation of Zero-Error Capacities of Quantum Channels Theorem For any satisfying, there exist channels such that: Each channel maps and has Kraus operators. Each channel individually has no classical 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).

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1.Reduce super(duper)activation to question about existence of certain subspaces. 2.Show that such subspaces exist.

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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.

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

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

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

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

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(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 dont want.

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(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: E d is an algebraic set defined by simultaneous polynomial equations Zariski closed in the Grassmanian)

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(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, its defined by some algebraic constraints, so either the constraints are trivial, or they restrict the set to some lower-dimensional manifold.) Gr d EdEd EdEd

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(Almost) any old subspace will do! To show E d is measure zero, just have to exhibit one subspace thats not contained in E d. Gr d EdEd 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. UPB EdEd

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(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 ofgood ones is full measure ! pick a one at random! Gr d EdEd

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Reducuctio ad Subspace Want a bipartite subspace such that: pick one at random! Subspaces obeying symmetry constraints also form an algebraic set

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(Almost) any subspace will do! Want a bipartite subspace such that: full measure non-zero measure pick one at random!

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Superduperactivation of Zero-Error Capacities of Quantum Channels Translates into one additional constraint on the subspace : full measure non-zero measure pick one at random!

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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 quantum information [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.

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Status of Shannon Capacity Theory Regularization requiredAdditivity violationSuperactivation Classical channels Quantum channels Classical Quantum * probably [Shannon 48] [Shannon 48] (trivial) * [Hasting 08] ? (trivial) * [DiVincenzo, Shor, Smolin 98] [Smith, Yard 08]

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