### Abstract

This paper defines the squashed entanglement of a quantum channel as the maximum squashed entanglement that can be registered by a sender and receiver at the input and output of a quantum channel, respectively. A new subadditivity inequality for the original squashed entanglement measure of Christandl and Winter leads to the conclusion that the squashed entanglement of a quantum channel is an additive function of a tensor product of any two quantum channels. More importantly, this new subadditivity inequality, along with prior results of Christandl and Winter, establishes the squashed entanglement of a quantum channel as an upper bound on the quantum communication capacity of any channel assisted by unlimited forward and backward classical communication. A similar proof establishes this quantity as an upper bound on the private capacity of a quantum channel assisted by unlimited forward and backward public classical communication. This latter result is relevant as a limitation on rates achievable in quantum key distribution. As an important application, we determine that these capacities can never exceed log((1+η)/(1-η)) for a pure-loss bosonic channel for which a fraction η of the input photons make it to the output on average. The best known lower bound on these capacities is equal to log(1/(1-η)). Thus, in the high-loss regime for which η ≪ 1, this new upper bound demonstrates that the protocols corresponding to the above lower bound are nearly optimal.

Original language | English (US) |
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Article number | 6832533 |

Pages (from-to) | 4987-4998 |

Number of pages | 12 |

Journal | IEEE Transactions on Information Theory |

Volume | 60 |

Issue number | 8 |

DOIs | |

State | Published - Aug 2014 |

Externally published | Yes |

### Keywords

- Squashed entanglement
- private states
- pure-loss bosonic channel
- quantum capacity
- quantum key distribution
- secret key agreement capacity

### ASJC Scopus subject areas

- Information Systems
- Computer Science Applications
- Library and Information Sciences

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## Cite this

*IEEE Transactions on Information Theory*,

*60*(8), 4987-4998. [6832533]. https://doi.org/10.1109/TIT.2014.2330313