Quantum Information Complexity and Direct Sum Dave Touchette Universit´ e de Montr´ eal QIP 2015, Sydney, Australia touchette.dave@gmail.com Quantum Information Complexity and Direct Sum QIP 2015, Sydney, Australia 1 / 19
Interactive Quantum Communication Communication complexity setting: Alice μ Bob |Ψ Input: y Input: x T A T B m 1 m 2 m 3 ... m M Output: f(x, y) Output: f(x, y) Information-theoretic view: quantum information complexity ◮ How much quantum information to compute f on µ touchette.dave@gmail.com Quantum Information Complexity and Direct Sum QIP 2015, Sydney, Australia 2 / 19
Results Definition of quantum information complexity of task T = ( f , µ, ǫ ) Interpretation as amortized communication ◮ QIC ( T ) = AQCC ( T ) := lim n →∞ 1 n QCC ( T ⊗ n ) Properties ◮ Lower bounds communication: QIC ( T ) ≤ QCC ( T ) ⋆ No dependance on # of messages M ◮ Additivity: QIC ( T 1 ⊗ T 2 ) = QIC ( T 1 ) + QIC ( T 2 ) Application to direct sum for quantum communication ◮ Protocol compression builds on one-shot state redistribution of [BCT14] ◮ M -rounds: QCC M (( f , ǫ ) ⊗ n ) ∈ Ω( n ( δ M ) 2 QCC M ( f , ǫ + δ ) − M ) Potential application to communication lower bound ◮ Direct sum on composite functions ◮ E.g.: reduction from QIC of DISJ n to QIC of AND ◮ Conjecture for DISJ n : QCC M ( DISJ n ) ∈ Θ( n M + M ) ◮ Known bounds: O ( n M + M ) , Ω( n M 2 + M ) [AA03, JRS03] touchette.dave@gmail.com Quantum Information Complexity and Direct Sum QIP 2015, Sydney, Australia 3 / 19
Direct Sum T T T n ≈ ... (n times) T touchette.dave@gmail.com Quantum Information Complexity and Direct Sum QIP 2015, Sydney, Australia 3 / 19
Results Definition of quantum information complexity of task T = ( f , µ, ǫ ) Interpretation as amortized communication: QIC ( T ) = AQCC ( T ) Properties ◮ Lower bounds communication: QIC ( T ) ≤ QCC ( T ) ⋆ No dependance on # of messages M ◮ Additivity: QIC ( T 1 ⊗ T 2 ) = QIC ( T 1 ) + QIC ( T 2 ) Application to direct sum for quantum communication ◮ Protocol compression builds on one-shot state redistribution of [BCT14] ◮ M -rounds: QCC M (( f , ǫ ) ⊗ n ) ∈ Ω( n ( δ M ) 2 QCC M ( f , ǫ + δ ) − M ) Potential application to communication lower bound ◮ Direct sum on composite functions ◮ E.g.: reduction from QIC of DISJ n to QIC of AND ◮ Conjecture for DISJ n : QCC M ( DISJ n ) ∈ Θ( n M + M ) ◮ Known bounds: O ( n M + M ) , Ω( n M 2 + M ) [AA03, JRS03] touchette.dave@gmail.com Quantum Information Complexity and Direct Sum QIP 2015, Sydney, Australia 3 / 19
Disjointness Decomposition x 1 AND y 1 DISJ n = x 2 AND y 2 ≈ OR( x i AND y i ) ... x n AND y n touchette.dave@gmail.com Quantum Information Complexity and Direct Sum QIP 2015, Sydney, Australia 3 / 19
Results Definition of quantum information complexity of task T = ( f , µ, ǫ ) Interpretation as amortized communication: QIC ( T ) = AQCC ( T ) Properties ◮ Lower bounds communication: QIC ( T ) ≤ QCC ( T ) ⋆ No dependance on # of messages M ◮ Additivity: QIC ( T 1 ⊗ T 2 ) = QIC ( T 1 ) + QIC ( T 2 ) Application to direct sum for quantum communication ◮ Protocol compression builds on one-shot state redistribution of [BCT14] ◮ M -rounds: QCC M (( f , ǫ ) ⊗ n ) ∈ Ω( n ( δ M ) 2 QCC M ( f , ǫ + δ ) − M ) Potential application to communication lower bound ◮ Direct sum on composite functions ◮ E.g.: reduction from QIC of DISJ n to QIC of AND ◮ Conjecture for DISJ n : QCC M ( DISJ n ) ∈ Ω( n M + M ) ◮ Known bounds: O ( n M + M ) , Ω( n M 2 + M ) [AA03, JRS03] touchette.dave@gmail.com Quantum Information Complexity and Direct Sum QIP 2015, Sydney, Australia 3 / 19
Results Definition of quantum information complexity of task T = ( f , µ, ǫ ) Interpretation as amortized communication: QIC ( T ) = AQCC ( T ) Properties ◮ Lower bounds communication: QIC ( T ) ≤ QCC ( T ) ⋆ No dependance on # of messages M ◮ Additivity: QIC ( T 1 ⊗ T 2 ) = QIC ( T 1 ) + QIC ( T 2 ) Application to direct sum for quantum communication ◮ Protocol compression builds on one-shot state redistribution of [BCT14] ◮ M -rounds: QCC M (( f , ǫ ) ⊗ n ) ∈ Ω( n ( δ M ) 2 QCC M ( f , ǫ + δ ) − M ) Potential application to communication lower bound ◮ Direct sum on composite functions ◮ E.g.: reduction from QIC of DISJ n to QIC of AND ◮ Conjecture for DISJ n : QCC M ( DISJ n ) ∈ Θ( n M + M ) ◮ Known bounds: O ( n M + M ) , Ω( n M 2 + M ) [AA03, JRS03] touchette.dave@gmail.com Quantum Information Complexity and Direct Sum QIP 2015, Sydney, Australia 3 / 19
Unidirectional Classical Communication Separate into 2 prominent communication problems ◮ Compress messages with ”low information content” ◮ Transmit messages ”noiselessly” over noisy channels touchette.dave@gmail.com Quantum Information Complexity and Direct Sum QIP 2015, Sydney, Australia 4 / 19
Unidirectional Classical Communication Separate into 2 prominent communication problems ◮ Compress messages with ”low information content” ◮ Transmit messages ”noiselessly” over noisy channels touchette.dave@gmail.com Quantum Information Complexity and Direct Sum QIP 2015, Sydney, Australia 4 / 19
Information Theory How to quantify information? Shannon’s entropy! Source X of distribution p X has entropy H ( X ) = − � x p X ( x ) log( p X ( x )) bits Operational significance: optimal asymptotic rate of compression for i.i.d. copies of source X X x t ...x 2 x 1 Derived quantities: conditional entropy H ( X | Y ), mutual information I ( X : Y )... Mutual information characterizes a noisy channel’s capacity ◮ Also the optimal channel simulation rate touchette.dave@gmail.com Quantum Information Complexity and Direct Sum QIP 2015, Sydney, Australia 5 / 19
Interactive Classical Communication Communication complexity of tasks, e.g. bipartite functions or relations μ Bob Alice R Input: y Input: x R A R B S A S B m 1 m 2 m 3 ... m M Output: f(x, y) Output: f(x, y) Protocol transcript Π( x , y , r , s ) = m 1 m 2 · · · m M Can memorize whole history touchette.dave@gmail.com Quantum Information Complexity and Direct Sum QIP 2015, Sydney, Australia 6 / 19
Coding for Interactive Protocols Protocol compression ◮ Can we compress protocols that ”do not convey much information” ⋆ For many copies run in parallel? ⋆ For a single copy? ◮ What is the amount of information conveyed by a protocol? ⋆ Optimal asymptotic compression rate? touchette.dave@gmail.com Quantum Information Complexity and Direct Sum QIP 2015, Sydney, Australia 7 / 19
Protocol Compression: Information Complexity Information complexity: IC ( f , µ, ǫ ) = inf Π IC (Π , µ ) Information cost: IC (Π , µ ) = I ( X : Π | YR ) + I ( Y : Π | XR ) ◮ Amount of information each party learns about the other’s input from the transcript Important properties: ◮ Operational interpretation: n CC n ( T ⊗ n ) [BR11] 1 IC ( T ) = ACC ( T ) = lim sup n →∞ ◮ Lower bounds communication: IC ( T ) ≤ CC ( T ) ◮ Additivity: IC ( T 1 ⊗ T 2 ) = IC ( T 1 ) + IC ( T 2 ) ◮ Direct sum on composite functions, e.g. DISJ n from AND touchette.dave@gmail.com Quantum Information Complexity and Direct Sum QIP 2015, Sydney, Australia 8 / 19
Applications of Classical Information Complexity Direct sum: CC (( f , ǫ ) ⊗ n ) ≈ nCC (( f , ǫ )) Direct product: suc ( f n , µ n , o ( Cn )) < suc ( f , µ, C ) Ω( n ) Exact communication complexity bound!! ◮ E.g. CC ( DISJ n ) = 0 . 4827 · n ± o ( n ) Etc. touchette.dave@gmail.com Quantum Information Complexity and Direct Sum QIP 2015, Sydney, Australia 9 / 19
Quantum Information Theory von Neumann’s quantum entropy: H ( A ) ρ = − Tr ( ρ A log ρ A ) = H ( λ i ) for ρ A = � i λ i | i � � i | Characterizes optimal rate for quantum source compression Derived quantities defined in formal analogy to classical quantities Conditional entropy can be negative! Mutual information characterizes a channel’s entanglement-assisted capacity and optimal simulation rate touchette.dave@gmail.com Quantum Information Complexity and Direct Sum QIP 2015, Sydney, Australia 10 / 19
Interactive Quantum Communication and QIC Bob Alice μ |Ψ Input: y Input: x T A T B m 1 m 2 m 3 ... m M Output: f(x, y) Output: f(x, y) Yao: no pre-shared entanglement ψ , quantum messages m i Cleve-Buhrman: arbitrary pre-shared entanglement ψ , classical messages m i Hybrid: arbitrary pre-shared entanglement ψ , quantum messages m i Potential definition for quantum information cost: QIC (Π , µ ) = I ( X : m 1 m 2 · · · m M | Y ) + I ( Y : m 1 m 2 · · · m M | X )? No!! touchette.dave@gmail.com Quantum Information Complexity and Direct Sum QIP 2015, Sydney, Australia 11 / 19
Problems Bad QIC (Π , µ ) = I ( X : m 1 m 2 · · · m M | Y ) + I ( Y : m 1 · · · | X ) Many problems Yao model: ◮ No-cloning theorem : cannot copy m i , no transcript ◮ Can only evaluate information quantities on registers defined at same moment in time ◮ Not even well-defined! Cleve-Buhrman model: ◮ m i ’s could be completely uncorrelated to inputs ◮ e.g. teleportation at each time step ◮ Corresponding quantum information complexity is trivial touchette.dave@gmail.com Quantum Information Complexity and Direct Sum QIP 2015, Sydney, Australia 12 / 19
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