  # 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

1. 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

2. Additive noise channel noise Y input X

3. Additive noise channel noise Y input output X X

4. Additive noise channel noise Y input output X Y

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

6. 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...

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

8. 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 )

9. 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 ]

10. 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 ]

11. Classical EPI

12. 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

13. 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

14. 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

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

16. 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)

17. 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] ◮ . . .

18. Quantum EPI

19. 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

20. 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 )

21. 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

22. 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

23. 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)

24. 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

25. 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)

26. Qudit EPI

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

28. 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

29. 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 ]

30. 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 ]

31. 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])

32. 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

33. 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 )

34. 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 )

35. 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 ˜

36. 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

37. 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)

38. 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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