Entropy power inequalities for qudits M¯arisOzols University of Cambridge Nilanjana Datta Koenraad Audenaert University of Cambridge Royal Holloway & Ghent output X XYl Y l 2 [0, 1] – how much of the signal gets through How noisy is the output? ? H(X l Y) ≥ lH(X) + (1 − l)H(Y) Prototypic entropy power inequality... Additive noise channel noise Y input X X Yl Y l 2 [0, 1] – how much of the signal gets through How noisy is the output? ? H(X l Y) ≥ lH(X) + (1 − l)H(Y) Prototypic entropy power inequality... Additive noise channel noise Y input output X X X Xl Y l 2 [0, 1] – how much of the signal gets through How noisy is the output? ? H(X l Y) ≥ lH(X) + (1 − l)H(Y) Prototypic entropy power inequality... Additive noise channel noise Y input output X Y XY How noisy is the output? ? H(X l Y) ≥ lH(X) + (1 − l)H(Y) Prototypic entropy power inequality... Additive noise channel noise Y input output X l X l Y l 2 [0, 1] – how much of the signal gets through XY Additive noise channel noise Y input output X l X l Y l 2 [0, 1] – how much of the signal gets through How noisy is the output? ? H(X l Y) ≥ lH(X) + (1 − l)H(Y) Prototypic entropy power inequality... = convolution = beamsplitter = partial swap f (r l s) ≥ lf (r) + (1 − l)f (s) I r, s are distributions / states cH(·) I f (·) is an entropic function such as H(·) or e I r l s interpolates between r and s where l 2 [0, 1] Entropy power inequalities Classical Quantum Shannon Koenig & Smith Continuous [Sha48] [KS14, DMG14] This work Discrete — = convolution = beamsplitter = partial swap I r, s are distributions / states cH(·) I f (·) is an entropic function such as H(·) or e I r l s interpolates between r and s where l 2 [0, 1] Entropy power inequalities Classical Quantum Shannon Koenig & Smith Continuous [Sha48] [KS14, DMG14] This work Discrete — f (r l s) ≥ lf (r) + (1 − l)f (s) = convolution = beamsplitter = partial swap Entropy power inequalities Classical Quantum Shannon Koenig & Smith Continuous [Sha48] [KS14, DMG14] This work Discrete — f (r l s) ≥ lf (r) + (1 − l)f (s) I r, s are distributions / states cH(·) I f (·) is an entropic function such as H(·) or e I r l s interpolates between r and s where l 2 [0, 1] Entropy power inequalities Classical Quantum Shannon Koenig & Smith Continuous [Sha48] [KS14, DMG14] = convolution = beamsplitter This work Discrete — = partial swap f (r l s) ≥ lf (r) + (1 − l)f (s) I r, s are distributions / states cH(·) I f (·) is an entropic function such as H(·) or e I r l s interpolates between r and s where l 2 [0, 1] Classical EPI I aX is X scaled by a: fX f2X 0 1 2 3 4 I prob. density of X + Y is the convolution of fX and fY: fX fY fX+Y * = -2 -1 0 1 2 -2 -1 0 1 2 -2 -1 0 1 2 Continuous random variables d I X is a random variable over R with prob. density function Z d fX : R ! [0, ¥) s.t. fX(x)dx = 1 Rd fX 0 1 2 3 4 0 1 2 3 4 I prob. density of X + Y is the convolution of fX and fY: fX fY fX+Y * = -2 -1 0 1 2 -2 -1 0 1 2 -2 -1 0 1 2 Continuous random variables d I X is a random variable over R with prob. density function Z d fX : R ! [0, ¥) s.t. fX(x)dx = 1 Rd I aX is X scaled by a: fX f2X 0 1 2 3 4 -2 -1 0 1 2 -2 -1 0 1 2 -2 -1 0 1 2 Continuous random variables d I X is a random variable over R with prob. density function Z d fX : R ! [0, ¥) s.t. fX(x)dx = 1 Rd I aX is X scaled by a: fX f2X I prob. density of X + Y is the convolution of fX and fY: fX fY fX+Y * = I Shannon’s EPI [Sha48]: f (X l Y) ≥ lf (X) + (1 − l)f (Y) where f (·) is H(·) or e2H(·)/d (equivalent) I Proof via Fisher info & de Bruijn’s identity [Sta59, Bla65] I Applications: I upper bounds on channel capacity [Ber74] I strengthening of the central limit theorem [Bar86] I ... Classical EPI for continuous variables I Scaled addition: p p X l Y := l X + 1 − l Y I Proof via Fisher info & de Bruijn’s identity [Sta59, Bla65] I Applications: I upper bounds on channel capacity [Ber74] I strengthening of the central limit theorem [Bar86] I ... Classical EPI for continuous variables I Scaled addition: p p X l Y := l X + 1 − l Y I Shannon’s EPI [Sha48]: f (X l Y) ≥ lf (X) + (1 − l)f (Y) where f (·) is H(·) or e2H(·)/d (equivalent) Classical EPI for continuous variables I Scaled addition: p p X l Y := l X + 1 − l Y I Shannon’s EPI [Sha48]: f (X l Y) ≥ lf (X) + (1 − l)f (Y) where f (·) is H(·) or e2H(·)/d (equivalent) I Proof via Fisher info & de Bruijn’s identity [Sta59, Bla65] I Applications: I upper bounds on channel capacity [Ber74] I strengthening of the central limit theorem [Bar86] I ... Quantum EPI I Transmissivity l: p p Bl := l I + i 1 − l X ) Ul 2 U(H ⊗ H) I Output state: † Ul(r1 ⊗ r2)Ul Beamsplitter I Action on field operators: aˆ B dˆ r1 aˆ cˆ B = B 2 U(2) bˆ cˆ bˆ dˆ r2 I Output state: † Ul(r1 ⊗ r2)Ul Beamsplitter I Action on field operators: aˆ B dˆ r1 aˆ cˆ B = B 2 U(2) bˆ cˆ bˆ dˆ r2 I Transmissivity l: p p Bl := l I + i 1 − l X ) Ul 2 U(H ⊗ H) Beamsplitter I Action on field operators: aˆ B dˆ r1 aˆ cˆ B = B 2 U(2) bˆ cˆ bˆ dˆ r2 I Transmissivity l: p p Bl := l I + i 1 − l X ) Ul 2 U(H ⊗ H) I Output state: † Ul(r1 ⊗ r2)Ul I Quantum EPI [KS14, DMG14]: f (r1 l r2) ≥ lf (r1) + (1 − l)f (r2) where f (·) is H(·) or eH(·)/d (not known to be equivalent) I Analogue, not a generalization I Proof similar to the classical case (quantum generalizations of Fisher information & de Bruijn’s identity) Continuous-variable quantum EPI aˆ B dˆ r1 bˆ cˆ r2 I Combining two states: † r1 l r2 := Tr2 Ul(r1 ⊗ r2)Ul I Analogue, not a generalization I Proof similar to the classical case (quantum generalizations of Fisher information & de Bruijn’s identity) Continuous-variable quantum EPI aˆ B dˆ r1 bˆ cˆ r2 I Combining two states: † r1 l r2 := Tr2 Ul(r1 ⊗ r2)Ul I Quantum EPI [KS14, DMG14]: f (r1 l r2) ≥ lf (r1) + (1 − l)f (r2) where f (·) is H(·) or eH(·)/d (not known to be equivalent) I Proof similar to the classical case (quantum generalizations of Fisher information & de Bruijn’s identity) Continuous-variable quantum EPI aˆ B dˆ r1 bˆ cˆ r2 I Combining two states: † r1 l r2 := Tr2 Ul(r1 ⊗ r2)Ul I Quantum EPI [KS14, DMG14]: f (r1 l r2) ≥ lf (r1) + (1 − l)f (r2) where f (·) is H(·) or eH(·)/d (not known to be equivalent) I Analogue, not a generalization Continuous-variable quantum EPI aˆ B dˆ r1 bˆ cˆ r2 I Combining two states: † r1 l r2 := Tr2 Ul(r1 ⊗ r2)Ul I Quantum EPI [KS14, DMG14]: f (r1 l r2) ≥ lf (r1) + (1 − l)f (r2) where f (·) is H(·) or eH(·)/d (not known to be equivalent) I Analogue, not a generalization I Proof similar to the classical case (quantum generalizations of Fisher information & de Bruijn’s identity) Qudit EPI I Use S as a Hamiltonian: exp(itS) = cos t I + i sin t S I Partial swap: p p Ul := l I + i 1 − l S, l 2 [0, 1] I Combining two qudits: † r1 l r2 := Tr2 Ul(r1 ⊗ r2)Ul q = lr1 + (1 − l)r2 − l(1 − l) i[r1, r2] I This operation has applications for quantum algorithms! (Lloyd, Mohseni, Rebentrost [LMR14]) Partial swap I Swap: Sji, ji = jj, ii for all i, j 2 f1, . , dg I Partial swap: p p Ul := l I + i 1 − l S, l 2 [0, 1] I Combining two qudits: † r1 l r2 := Tr2 Ul(r1 ⊗ r2)Ul q = lr1 + (1 − l)r2 − l(1 − l) i[r1, r2] I This operation has applications for quantum algorithms! (Lloyd, Mohseni, Rebentrost [LMR14]) Partial swap I Swap: Sji, ji = jj, ii for all i, j 2 f1, . , dg I Use S as a Hamiltonian: exp(itS) = cos t I + i sin t S I Combining two qudits: † r1 l r2 := Tr2 Ul(r1 ⊗ r2)Ul q = lr1 + (1 − l)r2 − l(1 − l) i[r1, r2] I This operation has applications for quantum algorithms! (Lloyd, Mohseni, Rebentrost [LMR14]) Partial swap I Swap: Sji, ji = jj, ii for all i, j 2 f1, . , dg I Use S as a Hamiltonian: exp(itS) = cos t I + i sin t S I Partial swap: p p Ul := l I + i 1 − l S, l 2 [0, 1] I This operation has applications for quantum algorithms! (Lloyd, Mohseni, Rebentrost [LMR14]) Partial swap I Swap: Sji, ji = jj, ii for all i, j 2 f1, .