A semidefinite programming upper bound of quantum capacity

Abstract

Recently the power of positive partial transpose preserving (PPTp) and no-signalling (NS) codes in quantum communication has been studied. We continue with this line of research and show that the NS/PPTp/NS$\cap$PPTp codes assisted zero-error quantum capacity depends only on the non-commutative bipartite graph of the channel and the one-shot case can be computed efficiently by semidefinite programming (SDP). As an example, the activated PPTp codes assisted zero-error quantum capacity is carefully studied. We then present a general SDP upper bound $Q_{\Gamma }$ of quantum capacity and show it is always smaller than or equal to the "Partial transposition bound" introduced by Holevo and Werner, and the inequality could be strict. This upper bound is found to be additive, and thus is an upper bound of the potential PPTp assisted quantum capacity as well. We further demonstrate that $Q_{\Gamma }$ is strictly better than several previously known upper bounds for an explicit class of quantum channels. Finally, we show that $Q_{\Gamma }$ can be used to bound the super-activation of quantum capacity.

5 pages, 1 figure, v2 has improved presentation, many typos corrected

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• Semidefinite programming converse bounds for quantum communication
• Amortization does not enhance the max-Rains information of a quantum channel
• Semidefinite programming hierarchies for constrained bilinear optimization
• Pursuing the fundamental limits for quantum communication
• Capacity Estimates via comparison with TRO channels
• Geometric Rényi Divergence and its Applications in Quantum Channel Capacities
• Separation between quantum Lovász number and entanglement-assisted zero-error classical capacity
• Estimate distillable entanglement and quantum capacity by squeezing useless entanglement
• Capacity Bounds via Operator Space Methods
• Capacities of a two-parameter family of noisy Werner-Holevo channels
• An upper bound on quantum capacity of unital channels