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

Additivity in classical-quantum wiretap channels

ISIT 2020

Arkin Tikku*, Mario Berta†, Joseph M. Renes‡

*University of Sydney

†Imperial College London ‡ETH Zürich

Motivation

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

1

Background

Classical noisy channel coding

E M

x1 N xn N yn y1

D ˆ M

transmission rate R:= log(|M|)

n

Classical channel capacity C := sup

E

sup

D

log(|M|) n = max

pX(x) I(X : Y)

X Y p(y|x)

• single-letter expression
• tractable optimization problem (Blahut-Arimoto etc.)

2

Sending classical information over quantum channels EAn|M

M

a1 NB|A an NB|A bn b1

Dˆ

M|Bn

ˆ M

Holevo information (Achievable rate) χ ( NB|A ) := max

{pX(x),ρx

A} I(X : B)ω with ωXB =

∑

x

pX(x) |x⟩ ⟨x|X ⊗ NB|A (ρx

A)

BUT [Hastings ’08] showed that for random quantum channels: χ(N ⊗ M) > χ(N) + χ(M) by using entangled inputs to the channel!

3

Non-additivity of the Holevo information

Non-additive Holevo information: χ(N ⊗ M) > χ(N) + χ(M) − → coding rate can be greater than Holevo info ! Classical capacity of a quantum channel (HSW theorem) C ( NB|A ) = lim

n→∞

1 nχ(N ⊗n

B|A ) = lim n→∞

1 n max

{pX,ρx

An} I(X : Bn)ω ≥ χ (N)

• multi-letter expression (regularized Holevo info)
• intractable optimization problem

4

Additivity for entanglement-breaking channels

Additivity of Holevo info for entanglement-breaking channels If ( 1A′ ⊗ NB|A ) (|φ⟩ ⟨φ|A′A) is separable, then χ(NB1|A1 ⊗ NB2|A2) = χ(NB1|A1) + χ(NB2|A2) 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 pre-fixed orthonormal basis {|x⟩}x∈X.

5

Our setup

The classical wiretap model

(Sender) X

Y (Legitimate receiver) Z (Eavesdropper) WYZ|X

Setup: Private information of a classical wiretap channel P1(WYZ|X) := max

pUX(u,x) I(U : Y)−I(U : Z) ≥ max pX(x) I(X : Y)−I(X : Z) =: P0(WYZ|X)

with pUX(u, x) = pX|U(x|u)pU(u). Private capacity of a classical wiretap channel [Csiszar&Koerner′78] P(WYZ|X) = lim

n→∞

1 nP1(W⊗n

YZ|X) = P1(WYZ|X) 6

The quantum wiretap model

(Q. sender) A

B (Q. legitimate receiver) C (Q. eavesdropper) WBC|A

Setup: Private information of a quantum wiretap channel P1(WBC|A) := max

{pV,ρv

A} I(V : B)ω − I(V : C)ω

with ωVBC := ∑

v pV(v) |v⟩

⟨v|V ⊗ WBC|A (ρv

A).

Private capacity of a quantum wiretap channel [Cai et al.’04, Devetak’05] P(WBC|A) = lim

n→∞

1 nP1(W⊗n

BC|A) ≥ P1(WBC|A) 7

Hybrid setting ?

Our question: What happens to the private info when two of the parties are restricted to be classical ? Becomes additive (like for Holeveo info χ ) or stays non-additive ? Our results (informal):

• 1. Quantum sender: Additive !
• 2. Quantum receiver: Non-additive !
• 3. Quantum eavesdropper: Non-additive !

How did we show it ?

8

Quantum sender

Theorem I: Additivity for quantum sender Let W1 := WY1Z1|A1 and W2 := WY2Z2|A2. Then: P1(W1 ⊗ W2) = P1(W1) + P1(W2) Proof strategy: We need to prove two directions:

• 1. P1(W1) + P1(W2) ≤ P1(W1 ⊗ W2) (Trivial)
• 2. P1(W1) + P1(W2) ≥ P1(W1 ⊗ W2)
• Proof of direction 2) based on classical proof of additivity
• no conditioning on input (quantum) systems
• use Csiszar sum identity [Csiszar&Koerner′78]
• prove cardinality bound of auxiliary random variable

9

Quantum Receiver

Theorem II: Non-additivity for quantum receiver Let WBZ|X[r] := BPC(r)/BEC((1 − r)2). Then, for some r ∈ [0, 1]: P1 ( WBZ|X[r] ⊗ WBZ|X[r] ) > P1 ( WBZ|X[r] ) + P1 ( WBZ|X[r] ) where BPC(r) : |0⟩ ⟨0| → |ψ⟩ ⟨ψ| and |1⟩ ⟨1| → |φ⟩ ⟨φ| with r := |⟨φ|ψ⟩| BEC ( (1 − 2r)2) BPC(r)

X Z B

COPY

10

P(W) > 0: Quantum Receiver

0.54 0.542 0.544 0.546 0.5 1 1.5 2 ·10−3

channel parameter r achievable rate

BPC(r)/BEC((1 − r)2)

n = 1 n = 2

Block-coding scheme with parity pre-processing: |0⟩ ⟨0|X − → |00⟩ ⟨00|X2 + |11⟩ ⟨11|X2 2 and |1⟩ ⟨1|X − → |01⟩ ⟨01|X2 + |10⟩ ⟨10|X2 2 .

11

Proof Strategy

• 1. Construct an explicit coding scheme with positive rate: gives a

positive lower bound on private capacity P(W[r])

• 2. Rigorously show that P1(W[r]) = 0 for a given parameter

regime of the channel. Step 2) leverages ideas from classical information theory: [Ulukus&Ozel ’11] Let fW(ρA) = I(X : B)ω − I(X : C)ω, then P1(W) = max

pV,ρv

A

fW (EV[ρv

A]) − EV [fW(ρv A)]

with ρA = ∑

x pX(x) |x⟩

⟨x|A and ωXBC = ∑

x pX(x) |x⟩

⟨x|X ⊗ WBC|A (|x⟩ ⟨x|A)

• P1(W[r]) = 0 for convex fW[r](ρX) via Jensen’s inequality
• show that second derivative of fW[r](ρX) is positive in desired

parameter regime

12

Quantum Eavesdropper

Theorem III: Non-additivity for quantum eavesdropper Let WYC|X[p] := BSC(p)/BPC(1 − 2p). Then, for some r ∈ [0, 1]: P1 ( WYC|X[p] ⊗ WYC|X[p] ) > P1 ( WYC|X[p] ) + P1 ( WYC|X[p] ) where BPC(p) : |0⟩ ⟨0| → |ψ⟩ ⟨ψ| and |1⟩ ⟨1| → |φ⟩ ⟨φ| with r := |⟨φ|ψ⟩| BPC(1 − 2p) BSC(p)

X C Y

COPY

13

P(W) > 0: Quantum Eavesdropper

0.124 0.1245 0.5 1 ·10−4

channel parameter p achievable rate

BSC(p)/BPC(1 − 2p)

n = 1 n = 3

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

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