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 & Ghent Royal Holloway & Ghent

Additive noise channel noise Y input X

Additive noise channel noise Y input output X X

Additive noise channel noise Y input output X Y

Additive noise channel noise Y input output X ⊞ λ Y X λ λ ∈ [ 0, 1 ] – how much of the signal gets through

Additive noise channel noise Y input output X ⊞ λ Y X λ λ ∈ [ 0, 1 ] – how much of the signal gets through How noisy is the output? ? H ( X ⊞ λ Y ) ≥ λ H ( X ) + ( 1 − λ ) H ( Y ) Prototypic entropy power inequality...

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 )

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 ]

Classical EPI

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] ◮ . . .

Quantum EPI

Beamsplitter ◮ Action on field operators: 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

Beamsplitter ◮ Action on field operators: 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 )

Beamsplitter ◮ Action on field operators: 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 ) ◮ Output state: U λ ( ρ 1 ⊗ ρ 2 ) U † λ

Continuous-variable quantum EPI ˆ a ˆ B d ρ 1 ˆ c ˆ b ρ 2 ◮ Combining two states: � � U λ ( ρ 1 ⊗ ρ 2 ) U † ρ 1 ⊞ λ ρ 2 : = Tr 2 λ

Continuous-variable quantum EPI ˆ a ˆ B d ρ 1 ˆ c ˆ b ρ 2 ◮ Combining two 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 known to be equivalent)

Continuous-variable quantum EPI ˆ a ˆ B d ρ 1 ˆ c ˆ b ρ 2 ◮ Combining two 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 known to be equivalent) ◮ Analogue, not a generalization

Continuous-variable quantum EPI ˆ a ˆ B d ρ 1 ˆ c ˆ b ρ 2 ◮ Combining two 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 known to be equivalent) ◮ Analogue, not a generalization ◮ Proof similar to the classical case (quantum generalizations of Fisher information & de Bruijn’s identity)

Qudit EPI

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 ]

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 ] ◮ This operation has applications for quantum algorithms! (Lloyd, Mohseni, Rebentrost [LMR14])

Main result Theorem For any concave and symmetric function f : D ( C d ) → R , f ( ρ ⊞ λ σ ) ≥ λ f ( ρ ) + ( 1 − λ ) f ( σ ) for all λ ∈ [ 0, 1 ] and qudit states ρ and σ

Main result Theorem For any concave and symmetric function f : D ( C d ) → R , f ( ρ ⊞ λ σ ) ≥ λ f ( ρ ) + ( 1 − λ ) f ( σ ) for all λ ∈ [ 0, 1 ] and qudit states ρ and σ Relevant functions � ≥ 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 Typical example: von Neumann entorpy H ( ρ )

Proof idea Theorem f ( ρ ⊞ λ σ ) ≥ λ f ( ρ ) + ( 1 − λ ) f ( σ ) Proof Main tool: majorization. Assume we can show spec ( ρ ⊞ λ σ ) ≺ λ spec ( ρ ) + ( 1 − λ ) spec ( σ )

Proof idea Theorem f ( ρ ⊞ λ σ ) ≥ λ f ( ρ ) + ( 1 − λ ) f ( σ ) Proof Main tool: majorization. Assume we can show spec ( ρ ⊞ λ σ ) ≺ λ spec ( ρ ) + ( 1 − λ ) spec ( σ ) � � = spec λ ˜ ρ + ( 1 − λ ) ˜ σ ρ : = diag ( spec ( ρ )) . where ˜

Proof idea Theorem f ( ρ ⊞ λ σ ) ≥ λ f ( ρ ) + ( 1 − λ ) f ( σ ) Proof Main tool: majorization. Assume we can show spec ( ρ ⊞ λ σ ) ≺ λ spec ( ρ ) + ( 1 − λ ) spec ( σ ) � � = spec λ ˜ ρ + ( 1 − λ ) ˜ σ ρ : = diag ( spec ( ρ )) . Then where ˜ � � f ( ρ ⊞ λ σ ) ≥ f λ ˜ ρ + ( 1 − λ ) ˜ (Schur-concavity) σ

Proof idea Theorem f ( ρ ⊞ λ σ ) ≥ λ f ( ρ ) + ( 1 − λ ) f ( σ ) Proof Main tool: majorization. Assume we can show spec ( ρ ⊞ λ σ ) ≺ λ spec ( ρ ) + ( 1 − λ ) spec ( σ ) � � = spec λ ˜ ρ + ( 1 − λ ) ˜ σ ρ : = diag ( spec ( ρ )) . Then where ˜ � � f ( ρ ⊞ λ σ ) ≥ f λ ˜ ρ + ( 1 − λ ) ˜ (Schur-concavity) σ ≥ λ f ( ˜ ρ ) + ( 1 − λ ) f ( ˜ σ ) (concavity)

Proof idea Theorem f ( ρ ⊞ λ σ ) ≥ λ f ( ρ ) + ( 1 − λ ) f ( σ ) Proof Main tool: majorization. Assume we can show spec ( ρ ⊞ λ σ ) ≺ λ spec ( ρ ) + ( 1 − λ ) spec ( σ ) � � = spec λ ˜ ρ + ( 1 − λ ) ˜ σ ρ : = diag ( spec ( ρ )) . Then where ˜ � � f ( ρ ⊞ λ σ ) ≥ f λ ˜ ρ + ( 1 − λ ) ˜ (Schur-concavity) σ ≥ λ f ( ˜ ρ ) + ( 1 − λ ) f ( ˜ σ ) (concavity) = λ f ( ρ ) + ( 1 − λ ) f ( σ ) (symmetry)

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