A Quantum Multiparty Packing Lemma and the Relay Channel Dawei Ding - PowerPoint PPT Presentation

A Quantum Multiparty Packing Lemma and the Relay Channel Dawei Ding Stanford University Joint with Hrant Gharibyan, Patrick Hayden, and Michael Walter

Outline 1 Introduction Relay channel Relay channel definition Multihop bound Quantum multiparty packing lemma Packing lemma statement Conclusion Dawei Ding (Stanford University) |

Introduction Message packing 2 Input Output Dawei Ding (Stanford University) |

Introduction Message packing 3 Raw Encoded Packed Decoded Dawei Ding (Stanford University) |

Introduction Communication as message packing 4 Taken from Mark Wilde’s From Classical to Quantum Shannon Theory Dawei Ding (Stanford University) |

Introduction Encoding messages into quantum systems 5 ◮ Classical-quantum black box: Input is classical, output is quantum Dawei Ding (Stanford University) |

Introduction Encoding messages into quantum systems 5 ◮ Classical-quantum black box: Input is classical, output is quantum Theorem (Holevo-Schumacher-Westmoreland theorem) The classical capacity of a quantum channel N A → B with separable encodings is C ( N ) = max I ( X ; B ) ρ { p X ,ρ ( x ) A } where � p X ( x ) | x � � x | X ⊗ ρ ( x ) ρ XB ≡ B . x Dawei Ding (Stanford University) |

Introduction Encoding messages into quantum systems 6 ◮ Network information theory: multiple senders and/or multiple receivers Dawei Ding (Stanford University) |

Introduction Encoding messages into quantum systems 6 ◮ Network information theory: multiple senders and/or multiple receivers ◮ Classical-quantum channels: Multiple classical inputs, quantum output Dawei Ding (Stanford University) |

Introduction Encoding messages into quantum systems 6 ◮ Network information theory: multiple senders and/or multiple receivers ◮ Classical-quantum channels: Multiple classical inputs, quantum output ◮ Mostly open for quantum channels, especially one-shot Dawei Ding (Stanford University) |

Introduction Encoding messages into quantum systems 6 ◮ Network information theory: multiple senders and/or multiple receivers ◮ Classical-quantum channels: Multiple classical inputs, quantum output ◮ Mostly open for quantum channels, especially one-shot ◮ Multiparty packing: encoding multiple messages M j into a quantum system B via multiple classical systems X v Dawei Ding (Stanford University) |

Introduction Encoding messages into quantum systems 6 ◮ Network information theory: multiple senders and/or multiple receivers ◮ Classical-quantum channels: Multiple classical inputs, quantum output ◮ Mostly open for quantum channels, especially one-shot ◮ Multiparty packing: encoding multiple messages M j into a quantum system B via multiple classical systems X v ◮ Multiple senders: multiple access channel, relay channel Dawei Ding (Stanford University) |

Introduction Encoding messages into quantum systems 6 ◮ Network information theory: multiple senders and/or multiple receivers ◮ Classical-quantum channels: Multiple classical inputs, quantum output ◮ Mostly open for quantum channels, especially one-shot ◮ Multiparty packing: encoding multiple messages M j into a quantum system B via multiple classical systems X v ◮ Multiple senders: multiple access channel, relay channel ◮ Assuming classical-quantum channel, codebook is of the form { x v ( m ) } v ∈ V , m ∈ M , where M = × j ∈ J M j Dawei Ding (Stanford University) |

Relay channel Relay channel definition 7 Relay Receiver Sender ◮ N X 1 X 2 → B 2 B 3 : X 1 × X 2 → H B 2 ⊗ H B 3 , ( x 1 , x 2 ) �→ ρ ( x 1 x 2 ) B 2 B 3 (SWV, 2012) Dawei Ding (Stanford University) |

Relay channel Relay channel definition 7 Relay Receiver Sender ◮ N X 1 X 2 → B 2 B 3 : X 1 × X 2 → H B 2 ⊗ H B 3 , ( x 1 , x 2 ) �→ ρ ( x 1 x 2 ) B 2 B 3 (SWV, 2012) ◮ Relay’s transmission affects relay’s system, sender’s transmission affects receiver’s: More general than concatenated channels! Dawei Ding (Stanford University) |

Relay channel definition Multihop bound 8 Quantum generalization of classical protocols Dawei Ding (Stanford University) |

Relay channel definition Multihop bound 8 Quantum generalization of classical protocols ◮ Multihop Dawei Ding (Stanford University) |

Relay channel definition Multihop bound 8 Quantum generalization of classical protocols ◮ Multihop ◮ Treat relay channel as concatenated channel: sender transmits message to relay, relay transmits decoded message to receiver Dawei Ding (Stanford University) |

Relay channel definition Multihop bound 9 Random codebook b � C = { ( x 1 ) j ( m j ) , ( x 2 ) j ( m j − 1 ) } m j ∈ M j , m j − 1 ∈ M j − 1 j = 1 Dawei Ding (Stanford University) |

Relay channel definition Multihop bound 9 Random codebook b � C = { ( x 1 ) j ( m j ) , ( x 2 ) j ( m j − 1 ) } m j ∈ M j , m j − 1 ∈ M j − 1 j = 1 Code: Dawei Ding (Stanford University) |

Relay channel Remarks 10 ◮ Coherent multihop, decode forward, partial decode forward ◮ All straightforward generalizations of classical protocols Dawei Ding (Stanford University) |

Quantum multiparty packing lemma Multiplex Bayesian networks 11 Random codebooks for network communication Dawei Ding (Stanford University) |

Quantum multiparty packing lemma Multiplex Bayesian networks 11 Random codebooks for network communication ◮ Has structure depending on network setting and protocol chosen Dawei Ding (Stanford University) |

Quantum multiparty packing lemma Multiplex Bayesian networks 11 Random codebooks for network communication ◮ Has structure depending on network setting and protocol chosen ◮ How to represent mathematically? Dawei Ding (Stanford University) |

Quantum multiparty packing lemma Multiplex Bayesian networks 11 Random codebooks for network communication ◮ Has structure depending on network setting and protocol chosen ◮ How to represent mathematically? ◮ Muliplex Bayesian network Dawei Ding (Stanford University) |

Quantum multiparty packing lemma Multiplex Bayesian networks 12 Detailed definition: Dawei Ding (Stanford University) |

Quantum multiparty packing lemma Multiplex Bayesian networks 12 Detailed definition: ◮ Let X be a Bayesian network with respect to (DAG) G = ( V , E ) Dawei Ding (Stanford University) |

Quantum multiparty packing lemma Multiplex Bayesian networks 12 Detailed definition: ◮ Let X be a Bayesian network with respect to (DAG) G = ( V , E ) ◮ Bayesian network: statistical model that represents a set of random variables and their conditional dependencies via a DAG Dawei Ding (Stanford University) |

Quantum multiparty packing lemma Multiplex Bayesian networks 12 Detailed definition: ◮ Let X be a Bayesian network with respect to (DAG) G = ( V , E ) ◮ Bayesian network: statistical model that represents a set of random variables and their conditional dependencies via a DAG ◮ X composed of X v for v ∈ V Dawei Ding (Stanford University) |

Quantum multiparty packing lemma Multiplex Bayesian networks 12 Detailed definition: ◮ Let X be a Bayesian network with respect to (DAG) G = ( V , E ) ◮ Bayesian network: statistical model that represents a set of random variables and their conditional dependencies via a DAG ◮ X composed of X v for v ∈ V ◮ For v ∈ V , let pa ( v ) ≡ { v ′ ∈ V | ( v ′ , v ) ∈ E } Dawei Ding (Stanford University) |

Quantum multiparty packing lemma Multiplex Bayesian networks 12 Detailed definition: ◮ Let X be a Bayesian network with respect to (DAG) G = ( V , E ) ◮ Bayesian network: statistical model that represents a set of random variables and their conditional dependencies via a DAG ◮ X composed of X v for v ∈ V ◮ For v ∈ V , let pa ( v ) ≡ { v ′ ∈ V | ( v ′ , v ) ∈ E } ◮ Message sets: let J be an index set labeling the multiple message sets Dawei Ding (Stanford University) |

Quantum multiparty packing lemma Multiplex Bayesian networks 12 Detailed definition: ◮ Let X be a Bayesian network with respect to (DAG) G = ( V , E ) ◮ Bayesian network: statistical model that represents a set of random variables and their conditional dependencies via a DAG ◮ X composed of X v for v ∈ V ◮ For v ∈ V , let pa ( v ) ≡ { v ′ ∈ V | ( v ′ , v ) ∈ E } ◮ Message sets: let J be an index set labeling the multiple message sets ◮ Let ind : V → P ( J ) denote the (indices of) the message sets the random variable X v will be generated over Dawei Ding (Stanford University) |

Quantum multiparty packing lemma Multiplex Bayesian networks 12 Detailed definition: ◮ Let X be a Bayesian network with respect to (DAG) G = ( V , E ) ◮ Bayesian network: statistical model that represents a set of random variables and their conditional dependencies via a DAG ◮ X composed of X v for v ∈ V ◮ For v ∈ V , let pa ( v ) ≡ { v ′ ∈ V | ( v ′ , v ) ∈ E } ◮ Message sets: let J be an index set labeling the multiple message sets ◮ Let ind : V → P ( J ) denote the (indices of) the message sets the random variable X v will be generated over ◮ Index inheritance: For v ∈ V , ind ( v ′ ) ⊆ ind ( v ) for all v ′ ∈ pa ( v ) Dawei Ding (Stanford University) |