Additivity in classical-quantum wiretap channels ISIT 2020 - PowerPoint PPT Presentation

Additivity in classical-quantum wiretap channels ISIT 2020 *University of Sydney Arkin Tikku*, Mario Berta † , Joseph M. Renes ‡ † Imperial College London ‡ ETH Zürich

Motivation Classical Shannon Theory Basic settings fully understood (point-to-point information theory). Quantum Shannon Theory Only partially resolved. 1 − → Explore the boundaries between classical and quantum !

Background

Classical noisy channel coding M • tractable optimization problem (Blahut-Arimoto etc.) • single-letter expression Y X n n Classical channel capacity 2 y 1 M y n x 1 x n N ˆ E D N transmission rate R:= log( | M | ) log( | M | ) p ( y | x ) C := sup sup = max p X ( x ) I ( X : Y ) E D

Sending classical information over quantum channels b 1 by using entangled inputs to the channel! BUT [ Hastings ’08] showed that for random quantum channels: x Holevo information (Achievable rate) M 3 b n a n M a 1 N B | A E A n | M D ˆ ˆ M | B n N B | A ∑ ( N B | A ) p X ( x ) | x ⟩ ⟨ x | X ⊗ N B | A ( ρ x χ := max A } I ( X : B ) ω with ω XB = A ) { p X ( x ) ,ρ x χ ( N ⊗ M ) > χ ( N ) + χ ( M )

Non-additivity of the Holevo information Non-additive Holevo information: • intractable optimization problem • multi-letter expression (regularized Holevo info) n 1 1 4 Classical capacity of a quantum channel (HSW theorem) χ ( N ⊗ M ) > χ ( N ) + χ ( M ) − → coding rate can be greater than Holevo info ! n χ ( N ⊗ n ( ) C N B | A = lim B | A ) = lim max An } I ( X : B n ) ω ≥ χ ( N ) n →∞ n →∞ { p X ,ρ x

Additivity for entanglement-breaking channels Additivity of Holevo info for entanglement-breaking channels If Important sub-class: Classical-quantum/quantum-classical channels (informal) A classical-quantum channel/quantum-classical is a channel for which the inputs/outputs are restricted to be diagonal in some 5 ( 1 A ′ ⊗ N B | A ) ( | φ ⟩ ⟨ φ | A ′ A ) is separable, then χ ( N B 1 | A 1 ⊗ N B 2 | A 2 ) = χ ( N B 1 | A 1 ) + χ ( N B 2 | A 2 ) pre-fixed orthonormal basis {| x ⟩} x ∈ X .

Our setup

The classical wiretap model Private information of a classical wiretap channel 1 ( Sender ) X 6 Setup: Z ( Eavesdropper ) Y ( Legitimate receiver ) W YZ | X P 1 ( W YZ | X ) := max p UX ( u , x ) I ( U : Y ) − I ( U : Z ) ≥ max p X ( x ) I ( X : Y ) − I ( X : Z ) =: P 0 ( W YZ | X ) with p UX ( u , x ) = p X | U ( x | u ) p U ( u ) . Private capacity of a classical wiretap channel [Csiszar & Koerner ′ 78 ] n P 1 ( W ⊗ n P ( W YZ | X ) = lim YZ | X ) = P 1 ( W YZ | X ) n →∞

The quantum wiretap model ( Q. sender ) A 1 Devetak’05] Private capacity of a quantum wiretap channel [Cai et al.’04, 7 Private information of a quantum wiretap channel Setup: C ( Q. eavesdropper ) B ( Q. legitimate receiver ) W BC | A P 1 ( W BC | A ) := max A } I ( V : B ) ω − I ( V : C ) ω { p V ,ρ v with ω VBC := ∑ v p V ( v ) | v ⟩ ⟨ v | V ⊗ W BC | A ( ρ v A ) . n P 1 ( W ⊗ n P ( W BC | A ) = lim BC | A ) ≥ P 1 ( W BC | A ) n →∞

Hybrid setting ? Our question: What happens to the private info when two of the parties are restricted to be classical ? Our results (informal): 1. Quantum sender: Additive ! 2. Quantum receiver: Non-additive ! 3. Quantum eavesdropper: Non-additive ! How did we show it ? 8 Becomes additive (like for Holeveo info χ ) or stays non-additive ?

Quantum sender Theorem I: Additivity for quantum sender • Proof of direction 2) based on classical proof of additivity • no conditioning on input (quantum) systems • prove cardinality bound of auxiliary random variable 9 Let W 1 := W Y 1 Z 1 | A 1 and W 2 := W Y 2 Z 2 | A 2 . Then: P 1 ( W 1 ⊗ W 2 ) = P 1 ( W 1 ) + P 1 ( W 2 ) Proof strategy : We need to prove two directions: 1. P 1 ( W 1 ) + P 1 ( W 2 ) ≤ P 1 ( W 1 ⊗ W 2 ) (Trivial) 2. P 1 ( W 1 ) + P 1 ( W 2 ) ≥ P 1 ( W 1 ⊗ W 2 ) • use Csiszar sum identity [Csiszar & Koerner ′ 78 ]

Quantum Receiver Theorem II: Non-additivity for quantum receiver COPY B Z X BPC( r ) BEC 10 P 1 Let W BZ | X [ r ] := BPC ( r ) / BEC (( 1 − r ) 2 ) . Then, for some r ∈ [ 0 , 1 ] : ( W BZ | X [ r ] ⊗ W BZ | X [ r ] ) ( W BZ | X [ r ] ) ( W BZ | X [ r ] ) > P 1 + P 1 where BPC ( r ) : | 0 ⟩ ⟨ 0 | → | ψ ⟩ ⟨ ψ | and | 1 ⟩ ⟨ 1 | → | φ ⟩ ⟨ φ | with r := |⟨ φ | ψ ⟩| ( ( 1 − 2 r ) 2 )

11 0.54 0.542 0.544 0.546 0 2 1 2 2 Block-coding scheme with parity pre-processing: channel parameter r achievable rate P ( W ) > 0 : Quantum Receiver BPC ( r ) / BEC (( 1 − r ) 2 ) · 10 − 3 n = 1 n = 2 1 . 5 0 . 5 → | 00 ⟩ ⟨ 00 | X 2 + | 11 ⟩ ⟨ 11 | X 2 → | 01 ⟩ ⟨ 01 | X 2 + | 10 ⟩ ⟨ 10 | X 2 | 0 ⟩ ⟨ 0 | X − and | 1 ⟩ ⟨ 1 | X − .

Proof Strategy A parameter regime 1. Construct an explicit coding scheme with positive rate: gives a 12 regime of the channel. positive lower bound on private capacity P ( W [ r ]) 2. Rigorously show that P 1 ( W [ r ]) = 0 for a given parameter Step 2) leverages ideas from classical information theory: [Ulukus & Ozel ’11] Let f W ( ρ A ) = I ( X : B ) ω − I ( X : C ) ω , then P 1 ( W ) = max f W ( E V [ ρ v A ]) − E V [ f W ( ρ v A )] p V ,ρ v with ρ A = ∑ x p X ( x ) | x ⟩ ⟨ x | A and ω XBC = ∑ x p X ( x ) | x ⟩ ⟨ x | X ⊗ W BC | A ( | x ⟩ ⟨ x | A ) • P 1 ( W [ r ]) = 0 for convex f W [ r ] ( ρ X ) via Jensen’s inequality • show that second derivative of f W [ r ] ( ρ X ) is positive in desired

Quantum Eavesdropper Theorem III: Non-additivity for quantum eavesdropper COPY Y C X BSC( p ) 13 P 1 Let W YC | X [ p ] := BSC ( p ) / BPC ( 1 − 2 p ) . Then, for some r ∈ [ 0 , 1 ] : ( ) ( ) ( ) W YC | X [ p ] ⊗ W YC | X [ p ] > P 1 W YC | X [ p ] + P 1 W YC | X [ p ] where BPC ( p ) : | 0 ⟩ ⟨ 0 | → | ψ ⟩ ⟨ ψ | and | 1 ⟩ ⟨ 1 | → | φ ⟩ ⟨ φ | with r := |⟨ φ | ψ ⟩| BPC(1 − 2 p )

0.124 0.1245 0 1 channel parameter p achievable rate Block-coding scheme: repetition code + noisy-preprocessing via a bit-flip channel. We also analytically show equivalence of achievable (positive) wiretap coding rate to BB84 key rate in [Smith et al.’06 ]). 14 P ( W ) > 0 : Quantum Eavesdropper BSC ( p ) / BPC ( 1 − 2 p ) · 10 − 4 n = 1 n = 3 0 . 5

Conclusion • Explicit example channels are provided rather than randomized construction • Entanglement is neither necessary nor suffjcient in the wiretap setting for non-additivity of private info to occur • Direct corollary: hybrid setting with two quantum parties is always non-additive • How large can such additivity violations become for the private info ? 15