Entropy power inequalities for qudits M¯arisOzols University of Cambridge Nilanjana Datta Koenraad Audenaert University of Cambridge Royal Holloway & Ghent University = 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 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] 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 ... I Transmissivity l: p p Bl := l I + i 1 − l X ) Ul 2 U(H ⊗ H) I Combining 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 equivalent) I Analogue, not a generalization Continuous quantum EPI I Beamsplitter: aˆ B dˆ r1 aˆ cˆ B = B 2 U(2) bˆ cˆ bˆ dˆ r2 I Combining 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 equivalent) I Analogue, not a generalization Continuous quantum EPI I Beamsplitter: 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 Quantum EPI [KS14, DMG14]: f (r1 l r2) ≥ lf (r1) + (1 − l)f (r2) where f (·) is H(·) or eH(·)/d (not equivalent) I Analogue, not a generalization Continuous quantum EPI I Beamsplitter: 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 Combining states: † r1 l r2 := Tr2 Ul(r1 ⊗ r2)Ul I Analogue, not a generalization Continuous quantum EPI I Beamsplitter: 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 Combining 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 equivalent) Continuous quantum EPI I Beamsplitter: 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 Combining 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 equivalent) I Analogue, not a generalization 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] 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] 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] 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] 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 Combining two qudits: † r1 l r2 := Tr2 Ul(r1 ⊗ r2)Ul q = lr1 + (1 − l)r2 − l(1 − l) i[r1, r2] Main result Function f : D(Cd) ! R is I concave if f lr + (1 − l)s ≥ lf (r) + (1 − l)f (s) I symmetric if f (r) = s(spec(r)) for some sym. function s Theorem If f is concave and symmetric then for any r, s 2 D(Cd), l 2 [0, 1] f (r l s) ≥ lf (r) + (1 − l)f (s) Proof Main tool: majorization. We show that spec(r l s) ≺ l spec(r) + (1 − l) spec(s) Summary of EPIs f (r l s) ≥ lf (r) + (1 − l)f (s) Continuous variable Discrete Classical Quantum Quantum (d dims) (d modes) (d dims) entropy H(·) X X X entropy power c = 2/d c = 1/d 0 ≤ c ≤ 1/(log d)2 ecH(·) entropy photon c = 1/d — 0 ≤ c ≤ 1/(d − 1) number (conjectured) g−1(cH(·)) g(x) := (x + 1) log(x + 1) − x log x I Conditional version of EPI I trivial for c.v. distributions I proved for Gaussian c.v. states [Koe15] I qudit analogue. ? I Generalization to 3 or more systems I trivial for c.v. distributions I proved for c.v. states [DMLG15] I combining three states: [Ozo15] I proving the EPI. ? I Applications I upper bounding product-state classical capacity of certain channels I more. ? Open problems I Entropy photon number inequality for c.v. states I classical capacities of various bosonic channels (thermal noise, bosonic broadcast, and wiretap channels) I proved only for Gaussian states so far [Guh08] I does not seem to follow by taking d ! ¥ I Generalization to 3 or more systems I trivial for c.v. distributions I proved for c.v. states [DMLG15] I combining three states: [Ozo15] I proving the EPI. ? I Applications I upper bounding product-state classical capacity of certain channels I more.