Entropy power inequalities for qudits Entropy power inequalities for - PowerPoint PPT Presentation

Entropy power inequalities for qudits Entropy power inequalities for qudits M¯ M¯ aris Ozols aris Ozols University of Cambridge University of Cambridge Nilanjana Datta Nilanjana Datta Koenraad Audenaert Koenraad Audenaert University of Cambridge University of Cambridge Royal Holloway & Royal Holloway & Ghent University Ghent University

Entropy power inequalities Quantum Classical Shannon Koenig & Smith [Sha48] [KS14, DMG14] Continuous This work — Discrete

Entropy power inequalities Quantum Classical Shannon Koenig & Smith [Sha48] [KS14, DMG14] Continuous This work — Discrete f ( ρ ⊞ λ σ ) ≥ λ f ( ρ ) + ( 1 − λ ) f ( σ ) ◮ ρ , σ are distributions / states ◮ f ( · ) is an entropic function such as H ( · ) or e cH ( · ) ◮ ρ ⊞ λ σ interpolates between ρ and σ where λ ∈ [ 0, 1 ]

Entropy power inequalities Quantum Classical Shannon Koenig & Smith [Sha48] [KS14, DMG14] Continuous ⊞ = convolution ⊞ = beamsplitter This work — Discrete ⊞ = partial swap f ( ρ ⊞ λ σ ) ≥ λ f ( ρ ) + ( 1 − λ ) f ( σ ) ◮ ρ , σ are distributions / states ◮ f ( · ) is an entropic function such as H ( · ) or e cH ( · ) ◮ ρ ⊞ λ σ interpolates between ρ and σ where λ ∈ [ 0, 1 ]

Continuous random variables ◮ X is a random variable over R d with prob. density function � f X : R d → [ 0, ∞ ) s.t. R d f X ( x ) dx = 1 f X 0 1 2 3 4

Continuous random variables ◮ X is a random variable over R d with prob. density function � f X : R d → [ 0, ∞ ) s.t. R d f X ( x ) dx = 1 ◮ α X is X scaled by α : f X f X f 2 X 0 1 2 3 4

Continuous random variables ◮ X is a random variable over R d with prob. density function � f X : R d → [ 0, ∞ ) s.t. R d f X ( x ) dx = 1 ◮ α X is X scaled by α : f X f X f 2 X 0 1 2 3 4 ◮ prob. density of X + Y is the convolution of f X and f Y : f X f Y f X + Y = * - 2 - 1 0 1 2 - 2 - 1 0 1 2 - 2 - 1 0 1 2

Classical EPI for continuous variables ◮ Scaled addition: √ √ X ⊞ λ Y : = λ X + 1 − λ Y

Classical EPI for continuous variables ◮ Scaled addition: √ √ X ⊞ λ Y : = λ X + 1 − λ Y ◮ Shannon’s EPI [Sha48]: f ( X ⊞ λ Y ) ≥ λ f ( X ) + ( 1 − λ ) f ( Y ) where f ( · ) is H ( · ) or e 2 H ( · ) / d (equivalent)

Classical EPI for continuous variables ◮ Scaled addition: √ √ X ⊞ λ Y : = λ X + 1 − λ Y ◮ Shannon’s EPI [Sha48]: f ( X ⊞ λ Y ) ≥ λ f ( X ) + ( 1 − λ ) f ( Y ) where f ( · ) is H ( · ) or e 2 H ( · ) / d (equivalent) ◮ Proof via Fisher info & de Bruijn’s identity [Sta59, Bla65] ◮ Applications: ◮ upper bounds on channel capacity [Ber74] ◮ strengthening of the central limit theorem [Bar86] ◮ . . .

Continuous quantum EPI ◮ Beamsplitter: ˆ ˆ a B d ρ 1 � ˆ � � ˆ � a c B = B ∈ U ( 2 ) ˆ ˆ ˆ b d c ˆ b ρ 2

Continuous quantum EPI ◮ Beamsplitter: ˆ ˆ a B d ρ 1 � ˆ � � ˆ � a c B = B ∈ U ( 2 ) ˆ ˆ ˆ b d c ˆ b ρ 2 ◮ Transmissivity λ : √ √ B λ : = λ I + i 1 − λ X ⇒ U λ ∈ U ( H ⊗ H )

Continuous quantum EPI ◮ Beamsplitter: ˆ ˆ a B d ρ 1 � ˆ � � ˆ � a c B = B ∈ U ( 2 ) ˆ ˆ ˆ b d ˆ c b ρ 2 ◮ Transmissivity λ : √ √ B λ : = λ I + i 1 − λ X ⇒ U λ ∈ U ( H ⊗ H ) ◮ Combining states: U λ ( ρ 1 ⊗ ρ 2 ) U † � � ρ 1 ⊞ λ ρ 2 : = Tr 2 λ

Continuous quantum EPI ◮ Beamsplitter: ˆ ˆ a B d ρ 1 � ˆ � � ˆ � a c B = B ∈ U ( 2 ) ˆ ˆ ˆ b d c ˆ b ρ 2 ◮ Transmissivity λ : √ √ B λ : = λ I + i 1 − λ X ⇒ U λ ∈ U ( H ⊗ H ) ◮ Combining states: U λ ( ρ 1 ⊗ ρ 2 ) U † � � ρ 1 ⊞ λ ρ 2 : = Tr 2 λ ◮ Quantum EPI [KS14, DMG14]: f ( ρ 1 ⊞ λ ρ 2 ) ≥ λ f ( ρ 1 ) + ( 1 − λ ) f ( ρ 2 ) where f ( · ) is H ( · ) or e H ( · ) / d ( not equivalent)

Continuous quantum EPI ◮ Beamsplitter: ˆ ˆ a B d ρ 1 � ˆ � � ˆ � a c B = B ∈ U ( 2 ) ˆ ˆ ˆ b d c ˆ b ρ 2 ◮ Transmissivity λ : √ √ B λ : = λ I + i 1 − λ X ⇒ U λ ∈ U ( H ⊗ H ) ◮ Combining states: U λ ( ρ 1 ⊗ ρ 2 ) U † � � ρ 1 ⊞ λ ρ 2 : = Tr 2 λ ◮ Quantum EPI [KS14, DMG14]: f ( ρ 1 ⊞ λ ρ 2 ) ≥ λ f ( ρ 1 ) + ( 1 − λ ) f ( ρ 2 ) where f ( · ) is H ( · ) or e H ( · ) / d ( not equivalent) ◮ Analogue, not a generalization

Partial swap ◮ Swap: S | i , j � = | j , i � for all i , j ∈ { 1, . . . , d }

Partial swap ◮ Swap: S | i , j � = | j , i � for all i , j ∈ { 1, . . . , d } ◮ Use S as a Hamiltonian: exp ( itS ) = cos t I + i sin t S

Partial swap ◮ Swap: S | i , j � = | j , i � for all i , j ∈ { 1, . . . , d } ◮ Use S as a Hamiltonian: exp ( itS ) = cos t I + i sin t S ◮ Partial swap: √ √ U λ : = λ I + i 1 − λ S , λ ∈ [ 0, 1 ]

Partial swap ◮ Swap: S | i , j � = | j , i � for all i , j ∈ { 1, . . . , d } ◮ Use S as a Hamiltonian: exp ( itS ) = cos t I + i sin t S ◮ Partial swap: √ √ U λ : = λ I + i 1 − λ S , λ ∈ [ 0, 1 ] ◮ Combining two qudits: U λ ( ρ 1 ⊗ ρ 2 ) U † � � ρ 1 ⊞ λ ρ 2 : = Tr 2 λ � = λρ 1 + ( 1 − λ ) ρ 2 − λ ( 1 − λ ) i [ ρ 1 , ρ 2 ]

Main result Function f : D ( C d ) → R is � ≥ λ f ( ρ ) + ( 1 − λ ) f ( σ ) ◮ concave if f � λρ + ( 1 − λ ) σ ◮ symmetric if f ( ρ ) = s ( spec ( ρ )) for some sym. function s Theorem If f is concave and symmetric then for any ρ , σ ∈ D ( C d ) , λ ∈ [ 0, 1 ] f ( ρ ⊞ λ σ ) ≥ λ f ( ρ ) + ( 1 − λ ) f ( σ ) Proof Main tool: majorization. We show that spec ( ρ ⊞ λ σ ) ≺ λ spec ( ρ ) + ( 1 − λ ) spec ( σ )

Summary of EPIs f ( ρ ⊞ λ σ ) ≥ λ f ( ρ ) + ( 1 − λ ) f ( σ ) Continuous variable Discrete Classical Quantum Quantum ( d dims) ( d modes) ( d dims) entropy � � � H ( · ) entropy 0 ≤ c ≤ 1/ ( log d ) 2 c = 2/ d c = 1/ d power e cH ( · ) entropy c = 1/ d photon — 0 ≤ c ≤ 1/ ( d − 1 ) number (conjectured) g − 1 ( cH ( · )) g ( x ) : = ( x + 1 ) log ( x + 1 ) − x log x

Open problems ◮ Entropy photon number inequality for c.v. states ◮ classical capacities of various bosonic channels (thermal noise, bosonic broadcast, and wiretap channels) ◮ proved only for Gaussian states so far [Guh08] ◮ does not seem to follow by taking d → ∞

Open problems ◮ Entropy photon number inequality for c.v. states ◮ classical capacities of various bosonic channels (thermal noise, bosonic broadcast, and wiretap channels) ◮ proved only for Gaussian states so far [Guh08] ◮ does not seem to follow by taking d → ∞ ◮ Conditional version of EPI ◮ trivial for c.v. distributions ◮ proved for Gaussian c.v. states [Koe15] ◮ qudit analogue.. . ?

Open problems ◮ Entropy photon number inequality for c.v. states ◮ classical capacities of various bosonic channels (thermal noise, bosonic broadcast, and wiretap channels) ◮ proved only for Gaussian states so far [Guh08] ◮ does not seem to follow by taking d → ∞ ◮ Conditional version of EPI ◮ trivial for c.v. distributions ◮ proved for Gaussian c.v. states [Koe15] ◮ qudit analogue.. . ? ◮ Generalization to 3 or more systems ◮ trivial for c.v. distributions ◮ proved for c.v. states [DMLG15] ◮ combining three states: [Ozo15] ◮ proving the EPI.. . ?

Open problems ◮ Entropy photon number inequality for c.v. states ◮ classical capacities of various bosonic channels (thermal noise, bosonic broadcast, and wiretap channels) ◮ proved only for Gaussian states so far [Guh08] ◮ does not seem to follow by taking d → ∞ ◮ Conditional version of EPI ◮ trivial for c.v. distributions ◮ proved for Gaussian c.v. states [Koe15] ◮ qudit analogue.. . ? ◮ Generalization to 3 or more systems ◮ trivial for c.v. distributions ◮ proved for c.v. states [DMLG15] ◮ combining three states: [Ozo15] ◮ proving the EPI.. . ? ◮ Applications ◮ upper bounding product-state classical capacity of certain channels ◮ more.. . ?

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Combining 3 states U ( ρ 1 ⊗ ρ 2 ⊗ ρ 3 ) U † � � Let ρ = Tr 2,3 where U = ∑ π ∈ S 3 z π Q π is a linear combination of 3-qudit permutations. Then [Ozo15] ρ = p 1 ρ 1 + p 2 ρ 2 + p 3 ρ 3 + √ p 1 p 2 sin δ 12 i [ ρ 1 , ρ 2 ] + √ p 1 p 2 cos δ 12 ( ρ 2 ρ 3 ρ 1 + ρ 1 ρ 3 ρ 2 ) + √ p 2 p 3 sin δ 23 i [ ρ 2 , ρ 3 ] + √ p 2 p 3 cos δ 23 ( ρ 3 ρ 1 ρ 2 + ρ 2 ρ 1 ρ 3 ) + √ p 3 p 1 sin δ 31 i [ ρ 3 , ρ 1 ] + √ p 3 p 1 cos δ 31 ( ρ 1 ρ 2 ρ 3 + ρ 3 ρ 2 ρ 1 ) for some probability distribution ( p 1 , p 2 , p 3 ) and angles δ ij s.t. δ 12 + δ 23 + δ 31 = 0 √ p 1 p 2 cos δ 12 + √ p 2 p 3 cos δ 23 + √ p 3 p 1 cos δ 31 = 0 Conjecture If f is concave and symmetric then f ( ρ ) ≥ p 1 f ( ρ 1 ) + p 2 f ( ρ 2 ) + p 3 f ( ρ 3 )