Quantum Achievability Proof via Collision Relative Entropy

Quantum Achievability Proof via Collision Relative Entropy Salman Beigi Institute for Research in Fundamental Sciences (IPM) Tehran, Iran Setemper 8, 2014 Based on a joint work with Amin Gohari arxiv:1312.3822

Technique of Yassaee et al (ISIT 2013) x N X!Y y

Technique of Yassaee et al (ISIT 2013) m E x m N X!Y D random encoding maximum likelihood y ˆm

Technique of Yassaee et al (ISIT 2013) m E x m N X!Y D random encoding maximum likelihood y ˆm Random encoding: choose some p X pick i.i.d samples x 1,..., x M

Technique of Yassaee et al (ISIT 2013) m E x m N X!Y D random encoding stochastic likelihood y ˆm Random encoding: choose some p X pick i.i.d samples x 1,..., x M Stochastic likelihood decoder sample from p(m y) p(m, y) = 1 M p(y x m) p(m y) = p(y x m) m p(y x m )

Technique of Yassaee et al (ISIT 2013) Random codebook: C = {x 1,..., x M } 1 E C Pr[succ] = E C M p(y x p(y x m ) m) m p(y x m ) m,y

Technique of Yassaee et al (ISIT 2013) Random codebook: C = {x 1,..., x M } 1 E C Pr[succ] = E C M p(y x p(y x m ) m) m p(y x m ) m,y p(y x 1 ) = E C p(y x 1 ) m p(y x m ) y

Technique of Yassaee et al (ISIT 2013) Random codebook: C = {x 1,..., x M } 1 E C Pr[succ] = E C M p(y x p(y x m ) m) m p(y x m ) m,y p(y x 1 ) = E C p(y x 1 ) m p(y x m ) (Jensen) y p(y x 1 ) E x1 p(y x 1 ) E C\x1 m p(y x m ) y p(y x 1 ) = E x1 p(y x 1 ) p(y x 1 ) + (M 1)p(y) y

Plan for the talk Better understanding and quantum generalization of Stochastic likelihood decoder convexity Jensen s inequality Asymptotic analysis of the result

Pretty good measurement Pretty good measure for signal states {ρ 1,..., ρ M } ( ) 1 ( ) 2 1 2 E m = ρ i ρ m ρ i i i

Pretty good measurement Pretty good measure for signal states {ρ 1,..., ρ M } ( ) 1 ( ) 2 1 2 E m = ρ i ρ m ρ i If the signal states are ρ m p(y x m ) then i E m p(y x m) i p(y x i) i

Pretty good measurement Pretty good measure for signal states {ρ 1,..., ρ M } ( ) 1 ( ) 2 1 2 E m = ρ i ρ m ρ i If the signal states are ρ m p(y x m ) then i E m p(y x m) i p(y x i) The probability of successful decoding Pr[succ] = 1 ( ) 1 ( ) 2 1 tr [ρ ] 2 m ρ i ρ m ρ i M m i i i

Collision relative entropy D 2 (ρ σ) = log tr[ρσ 1 2 ρσ 1 2 ]

Collision relative entropy D 2 (ρ σ) = log tr[ρσ 1 2 ρσ 1 2 ] D α (ρ σ) = 1 [( ) α ] log tr σ 1 α 1 α 2α ρσ 2α α 1

Collision relative entropy D 2 (ρ σ) = log tr[ρσ 1 2 ρσ 1 2 ] D α (ρ σ) = 1 [( ) α ] log tr σ 1 α 1 α 2α ρσ 2α α 1 Müller-Lennert et al, arxiv:1306.3142 Wilde, Winter, Yang, arxiv:1306.1586 Frank, Lieb, arxiv:1306.5358 SB, arxiv:1306.5920 (Positivity) D 2 (ρ σ) 0. (Data processing) D 2 (ρ σ) D 2 (N (ρ) N (σ)). (Joint convexity) exp D 2 ( ) is jointly convex

Classical-Quantum channels x N X!Y x

Classical-Quantum channels m E x m N X!Y xm D random encoding pretty good measurement ˆm input distribution p(x) pick random codebook x 1,..., x M pretty good measurement at the decoder

Classical-Quantum channels m E x m N X!Y xm D random encoding pretty good measurement ˆm input distribution p(x) pick random codebook x 1,..., x M pretty good measurement at the decoder σ UXY = 1 M M m m x m x m ρ xm, m=1

c-q channel coding m E random encoding x m N X!Y xm D pretty good measurement ˆm σ UXB = Pr[succ] = 1 M M m=1 1 M m m x m x m ρ xm, [ ] M ( ) 1/2ρxm ( ) 1/2 tr ρ xm ρ xi ρ xi m=1 i i

c-q channel coding m E random encoding x m N X!Y xm D pretty good measurement ˆm σ UXB = Pr[succ] = 1 M M m=1 1 M m m x m x m ρ xm, [ ] M ( ) 1/2ρxm ( ) 1/2 tr ρ xm ρ xi ρ xi m=1 = 1 M exp D 2(σ UXY σ UX σ Y ) i i

c-q channel coding m E random encoding x m N X!Y xm D pretty good measurement ˆm σ UXB = Pr[succ] = 1 M M m=1 1 M m m x m x m ρ xm, [ ] M ( ) 1/2ρxm ( ) 1/2 tr ρ xm ρ xi ρ xi m=1 = 1 M exp D 2(σ UXY σ UX σ Y ) 1 M exp D 2(σ XY σ X σ Y ) i i

Error analysis Use the joint convexity of exp D 2 ( ) E C Pr[succ] 1 M E C exp D 2 (σ XY σ X σ Y ) 1 M exp D 2(E C σ XY E C σ X σ Y )

Error analysis Use the joint convexity of exp D 2 ( ) E C Pr[succ] 1 M E C exp D 2 (σ XY σ X σ Y ) 1 M exp D 2(E C σ XY E C σ X σ Y ) ( = M 1 exp D 2 ρ XY 1 M ρ XY + ( 1 1 ) ) ρx ρ Y, M ρ XY = x p(x) x x ρ x

Information spectrum relative entropy Π U 0 : projection on eigenspaces with non-negative eigenvalues Π U V := Π U V 0

Information spectrum relative entropy Π U 0 : projection on eigenspaces with non-negative eigenvalues Π U V := Π U V 0 D ɛ s(ρ σ) := sup { R tr ( ρπ ρ 2 R σ) ɛ }.

Information spectrum relative entropy Π U 0 : projection on eigenspaces with non-negative eigenvalues Π U V := Π U V 0 D ɛ s(ρ σ) := sup { R tr ( ρπ ρ 2 R σ) ɛ }. D ɛ s(p q) = sup { R x: p(x) 2 R q(x) p(x) ɛ } Theorem For every 0 < ɛ, λ < 1 and density matrices ρ, σ we have exp D 2 (ρ λρ + (1 λ)σ) (1 ɛ) [ λ + (1 λ) exp ( D ɛ s(ρ σ) )] 1

A one-shot achievable bound ( E C Pr[succ] M 1 exp D 2 ρ XY 1 M ρ XY + ( 1 1 ) ) ρx ρ Y M (1 ɛ) [ 1 + (M 1) exp ( D ɛ s(ρ XY ρ X ρ Y ) )] 1 Theorem M (1, ɛ) : maximum number of messages with error ɛ ɛ δ M (1, ɛ) max p(x), 0<δ<ɛ 1 ɛ exp ( D δ s(ρ XY ρ X ρ Y ) ) + 1

Asymptotic analysis Relative entropy: D(σ ρ) := tr(σ(log σ log ρ)). Information variance or the relative entropy variance V(ρ σ) := tr ( ρ(log ρ log σ D(ρ σ)) 2). Theorem [Tomamichel & Hayashi 12] For fixed 0 < ɛ < 1 and δ proportional to 1/ n we have D ɛ±δ s (ρ n σ n ) = nd(ρ σ) + nv(ρ σ)φ 1 (ɛ) + O(log n), Φ(u) := u 1 2π e 1 2 t2 dt, and Φ 1 (ɛ) = sup{u Φ(u) ɛ}.

Asymptotic analysis Theorem For 0 < ɛ < 1/2 we have log M (n, ɛ) nc + nvφ 1 (ɛ) + O(log n), where C is the channel capacity and V is the channel dispersion V = C := max I(X; Y) p(x) min V(ρ XB ρ X ρ B ) p(x) argmax I(X;Y) [Tan & Tomamichel 13]: this achievable bound is tight

Summary m E random encoding N n D pretty good measurement ˆm Write probability of successful decoding in terms of collision relative entropy

Summary m E random encoding N n D pretty good measurement ˆm Write probability of successful decoding in terms of collision relative entropy Use joint convexity of collision relative entropy

Summary m E random encoding N n D pretty good measurement ˆm Write probability of successful decoding in terms of collision relative entropy Use joint convexity of collision relative entropy Lower bound collision relative entropy with information spectrum relative entropy

Hypothesis testing Given either ρ or σ Want to decide which one Apply POVM measurement {F ρ, F σ }

Hypothesis testing Given either ρ or σ Want to decide which one Apply POVM measurement {F ρ, F σ } Pr[type I error] = Pr[error ρ] = tr(ρf σ ) Pr[type II error] = Pr[error σ] = tr(σf ρ )

Hypothesis testing Given either ρ or σ Want to decide which one Apply POVM measurement {F ρ, F σ } Pr[type I error] = Pr[error ρ] = tr(ρf σ ) Pr[type II error] = Pr[error σ] = tr(σf ρ ) β ɛ (ρ σ) = min Pr[type II error] Pr[type I error] ɛ

A one-shot hypothesis testing 1 Pr[type I error] = exp D 2 (ρ ρ + Mσ) (1 ɛ) [ 1 + M exp ( D ɛ s(ρ σ) )] 1

A one-shot hypothesis testing 1 Pr[type I error] = exp D 2 (ρ ρ + Mσ) (1 ɛ) [ 1 + M exp ( D ɛ s(ρ σ) )] 1 1 Pr[type II error] = exp D 2 (σ M 1 ρ + σ) (1 + M 1 ) 1,

A one-shot hypothesis testing 1 Pr[type I error] = exp D 2 (ρ ρ + Mσ) (1 ɛ) [ 1 + M exp ( D ɛ s(ρ σ) )] 1 1 Pr[type II error] = exp D 2 (σ M 1 ρ + σ) (1 + M 1 ) 1, Theorem log β 2ɛ (ρ σ) D ɛ s(ρ σ) + log ɛ.

A one-shot hypothesis testing 1 Pr[type I error] = exp D 2 (ρ ρ + Mσ) (1 ɛ) [ 1 + M exp ( D ɛ s(ρ σ) )] 1 1 Pr[type II error] = exp D 2 (σ M 1 ρ + σ) (1 + M 1 ) 1, Theorem Corollary log β 2ɛ (ρ σ) D ɛ s(ρ σ) + log ɛ. log β ɛ (ρ n σ n ) nd(ρ σ) + nv(ρ σ)φ 1 (ɛ) + o(n). This upper bound is tight [Tomamichel & Hayashi 12, Li 12]

Source compression with side information x x Alice Bob

Source compression with side information x Alice m = h(x) x Bob

Source compression with side information x Alice m = h(x) x Bob {F m x 0 : x0 2 X} ρ XY = x p(x) x x ρ x

Source compression with side information x Alice m = h(x) x Bob {F m x 0 : x0 2 X} ρ XY = x p(x) x x ρ x Pr[succ] = x,m p X(x)1 {m=h(x)} tr(ρ x F m x )

Source compression with side information x Alice m = h(x) x Bob {F m x 0 : x0 2 X} ρ XY = x p(x) x x ρ x Pr[succ] = x,m p X(x)1 {m=h(x)} tr(ρ x F m x ) pick a random h : X {1,..., M}

Source compression with side information x Alice m = h(x) x Bob {F m x 0 : x0 2 X} ρ XY = x p(x) x x ρ x Pr[succ] = x,m p X(x)1 {m=h(x)} tr(ρ x F m x ) pick a random h : X {1,..., M} Let {F m x : x X } be the pretty good measurement for {p(x)ρ x x X, h(x) = m}

Source compression with side information σ XB = ( x p(x)1 {h(x)=1} x x ρ x, ) ( ) τ XB = x 1 {h(x)=1} x x x p(x )1 {h(x )=1}ρ x. E h Pr[succ] = ME h exp D 2 (σ XB τ XB )

Source compression with side information σ XB = ( x p(x)1 {h(x)=1} x x ρ x, ) ( ) τ XB = x 1 {h(x)=1} x x x p(x )1 {h(x )=1}ρ x. E h Pr[succ] = ME h exp D 2 (σ XB τ XB ) Theorem There exists some h : X {1,..., M} s.t. ( ( Pr[succ] exp D 2 ρ XB 1 ) 1 ρxb + 1 ) M M I X ρ B 1 ɛ 1 + M 1 exp ( D ɛ s(ρ XB I X ρ B ) )

Our contribution Reformulation of the technique of Yassaee et al (2013) Stochastic likelihood = pretty good measurement Collision relative entropy: D 2 ( ) Information spectrum relative entropy: D ɛ s( ) c-q channel coding quantum hypothesis testing source compression of quantum side information

What else? Quantum capacity: achievability of coherent information?

What else? Quantum capacity: achievability of coherent information? Entanglement-assisted capacity [Datta, Tomamichel, Wilde 14]

What else? Quantum capacity: achievability of coherent information? Entanglement-assisted capacity [Datta, Tomamichel, Wilde 14] Yassaee et al apply this method to Gelfand-Pinsker (coding over a state-dependent channel) Broadcast channel (Marton bound) Multiple access channel (MAC)

Thanks for your attention!