Entropy Power Inequalities for Qudits M¯Arisozols University of Cambridge

Entropy power inequalities for qudits M¯arisOzols University of Cambridge

Nilanjana Datta Koenraad Audenaert

University of Cambridge Royal Holloway & Ghent output X XYλ Y

λ ∈ [0, 1] – how much of the signal gets through

How noisy is the output?

? H(X λ Y) ≥ λH(X) + (1 − λ)H(Y)

Prototypic entropy power inequality...

Additive noise channel noise Y

input X X Yλ Y

λ ∈ [0, 1] – how much of the signal gets through

How noisy is the output?

? H(X λ Y) ≥ λH(X) + (1 − λ)H(Y)

Prototypic entropy power inequality...

Additive noise channel noise Y

input output X X X Xλ Y

λ ∈ [0, 1] – how much of the signal gets through

How noisy is the output?

? H(X λ Y) ≥ λH(X) + (1 − λ)H(Y)

Prototypic entropy power inequality...

Additive noise channel noise Y

input output X Y XY

How noisy is the output?

? H(X λ Y) ≥ λH(X) + (1 − λ)H(Y)

Prototypic entropy power inequality...

Additive noise channel noise Y

input output X λ X λ Y

λ ∈ [0, 1] – how much of the signal gets through XY

Additive noise channel noise Y

input output X λ X λ Y

λ ∈ [0, 1] – how much of the signal gets through

How noisy is the output?

? H(X λ Y) ≥ λH(X) + (1 − λ)H(Y)

Prototypic entropy power inequality... = convolution = beamsplitter

= partial swap

f (ρ λ σ) ≥ λf (ρ) + (1 − λ)f (σ)

I ρ, σ are distributions / states cH(·) I f (·) is an entropic function such as H(·) or e

I ρ λ σ interpolates between ρ and σ where λ ∈ [0, 1]

Entropy power inequalities

Classical Quantum Shannon Koenig & Smith Continuous [Sha48] [KS14, DMG14]

This work Discrete — = convolution = beamsplitter

= partial swap

I ρ, σ are distributions / states cH(·) I f (·) is an entropic function such as H(·) or e

I ρ λ σ interpolates between ρ and σ where λ ∈ [0, 1]

Entropy power inequalities

Classical Quantum Shannon Koenig & Smith Continuous [Sha48] [KS14, DMG14]

This work Discrete —

f (ρ λ σ) ≥ λf (ρ) + (1 − λ)f (σ) = convolution = beamsplitter

= partial swap

Entropy power inequalities

Classical Quantum Shannon Koenig & Smith Continuous [Sha48] [KS14, DMG14]

This work Discrete —

f (ρ λ σ) ≥ λf (ρ) + (1 − λ)f (σ)

I ρ, σ are distributions / states cH(·) I f (·) is an entropic function such as H(·) or e

I ρ λ σ interpolates between ρ and σ where λ ∈ [0, 1] Entropy power inequalities

Classical Quantum Shannon Koenig & Smith Continuous [Sha48] [KS14, DMG14] = convolution = beamsplitter

This work Discrete — = partial swap

f (ρ λ σ) ≥ λf (ρ) + (1 − λ)f (σ)

I ρ, σ are distributions / states cH(·) I f (·) is an entropic function such as H(·) or e

I ρ λ σ interpolates between ρ and σ where λ ∈ [0, 1] Classical EPI I αX is X scaled by α:

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

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 αX is X scaled by α:

fX f2X

0 1 2 3 4 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 αX is X scaled by α:

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 I Shannon’s EPI [Sha48]:

f (X λ Y) ≥ λf (X) + (1 − λ)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: √ √ X λ Y := λ X + 1 − λ 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: √ √ X λ Y := λ X + 1 − λ Y

I Shannon’s EPI [Sha48]:

f (X λ Y) ≥ λf (X) + (1 − λ)f (Y)

where f (·) is H(·) or e2H(·)/d (equivalent) Classical EPI for continuous variables

I Scaled addition: √ √ X λ Y := λ X + 1 − λ Y

I Shannon’s EPI [Sha48]:

f (X λ Y) ≥ λf (X) + (1 − λ)f (Y)

I Output state: † Uλ(ρ1 ⊗ ρ2)Uλ

Beamsplitter

I Action on ﬁeld operators: aˆ B dˆ ρ1 aˆ cˆ B = B ∈ U(2) bˆ cˆ bˆ dˆ ρ2 I Output state: † Uλ(ρ1 ⊗ ρ2)Uλ

Beamsplitter

I Action on ﬁeld operators: aˆ B dˆ ρ1 aˆ cˆ B = B ∈ U(2) bˆ cˆ bˆ dˆ ρ2

I Transmissivity λ: √ √ Bλ := λ I + i 1 − λ X ⇒ Uλ ∈ U(H ⊗ H) Beamsplitter

I Action on ﬁeld operators: aˆ B dˆ ρ1 aˆ cˆ B = B ∈ U(2) bˆ cˆ bˆ dˆ ρ2

I Transmissivity λ: √ √ Bλ := λ I + i 1 − λ X ⇒ Uλ ∈ U(H ⊗ H)

I Output state: † Uλ(ρ1 ⊗ ρ2)Uλ I Quantum EPI [KS14, DMG14]:

f (ρ1 λ ρ2) ≥ λf (ρ1) + (1 − λ)f (ρ2)

Continuous-variable quantum EPI

aˆ B dˆ ρ1

bˆ cˆ ρ2

I Combining two states:

† ρ1 λ ρ2 := Tr2 Uλ(ρ1 ⊗ ρ2)Uλ 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 ρ1

bˆ cˆ ρ2

I Combining two states:

† ρ1 λ ρ2 := Tr2 Uλ(ρ1 ⊗ ρ2)Uλ

I Quantum EPI [KS14, DMG14]:

f (ρ1 λ ρ2) ≥ λf (ρ1) + (1 − λ)f (ρ2)

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 ρ1

bˆ cˆ ρ2

I Combining two states:

† ρ1 λ ρ2 := Tr2 Uλ(ρ1 ⊗ ρ2)Uλ

I Quantum EPI [KS14, DMG14]:

f (ρ1 λ ρ2) ≥ λf (ρ1) + (1 − λ)f (ρ2)

where f (·) is H(·) or eH(·)/d (not known to be equivalent) I Analogue, not a generalization Continuous-variable quantum EPI

aˆ B dˆ ρ1

bˆ cˆ ρ2

I Combining two states:

† ρ1 λ ρ2 := Tr2 Uλ(ρ1 ⊗ ρ2)Uλ

I Quantum EPI [KS14, DMG14]:

f (ρ1 λ ρ2) ≥ λf (ρ1) + (1 − λ)f (ρ2)

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: √ √ Uλ := λ I + i 1 − λ S, λ ∈ [0, 1]

I Combining two qudits:

† ρ1 λ ρ2 := Tr2 Uλ(ρ1 ⊗ ρ2)Uλ q = λρ1 + (1 − λ)ρ2 − λ(1 − λ) i[ρ1, ρ2]

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

Partial swap

I Swap: S|i, ji = |j, ii for all i, j ∈ {1, . . . , d} I Partial swap: √ √ Uλ := λ I + i 1 − λ S, λ ∈ [0, 1]

I Combining two qudits:

† ρ1 λ ρ2 := Tr2 Uλ(ρ1 ⊗ ρ2)Uλ q = λρ1 + (1 − λ)ρ2 − λ(1 − λ) i[ρ1, ρ2]

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

Partial swap

I Swap: S|i, ji = |j, ii for all i, j ∈ {1, . . . , d} I Use S as a Hamiltonian: exp(itS) = cos t I + i sin t S I Combining two qudits:

† ρ1 λ ρ2 := Tr2 Uλ(ρ1 ⊗ ρ2)Uλ q = λρ1 + (1 − λ)ρ2 − λ(1 − λ) i[ρ1, ρ2]

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

Partial swap

I Swap: S|i, ji = |j, ii for all i, j ∈ {1, . . . , d} I Use S as a Hamiltonian: exp(itS) = cos t I + i sin t S I Partial swap: √ √ Uλ := λ I + i 1 − λ S, λ ∈ [0, 1] I This operation has applications for quantum algorithms! (Lloyd, Mohseni, Rebentrost [LMR14])

Partial swap

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

I Combining two qudits:

† ρ1 λ ρ2 := Tr2 Uλ(ρ1 ⊗ ρ2)Uλ q = λρ1 + (1 − λ)ρ2 − λ(1 − λ) i[ρ1, ρ2] Partial swap

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

I Combining two qudits:

† ρ1 λ ρ2 := Tr2 Uλ(ρ1 ⊗ ρ2)Uλ q = λρ1 + (1 − λ)ρ2 − λ(1 − λ) i[ρ1, ρ2]

I This operation has applications for quantum algorithms! (Lloyd, Mohseni, Rebentrost [LMR14]) Relevant functions I concave if f λρ + (1 − λ)σ ≥ λf (ρ) + (1 − λ)f (σ) I symmetric if f (ρ) = s(spec(ρ)) for some sym. function s Typical example: von Neumann entorpy H(ρ)

Main result

Theorem For any concave and symmetric function f : D(Cd) → R,

f (ρ λ σ) ≥ λf (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(Cd) → R,

f (ρ λ σ) ≥ λf (ρ) + (1 − λ)f (σ)

for all λ ∈ [0, 1] and qudit states ρ and σ

Relevant functions I concave if f λρ + (1 − λ)σ ≥ λf (ρ) + (1 − λ)f (σ) I symmetric if f (ρ) = s(spec(ρ)) for some sym. function s Typical example: von Neumann entorpy H(ρ) ≥ λf (ρ˜) + (1 − λ)f (σ˜ ) (concavity) = λf (ρ) + (1 − λ)f (σ) (symmetry)

= specλρ˜ + (1 − λ)σ˜

where ρ˜ := diag(spec(ρ)). Then f (ρ λ σ) ≥ f λρ˜ + (1 − λ)σ˜ (Schur-concavity)

Proof idea

Theorem f (ρ λ σ) ≥ λf (ρ) + (1 − λ)f (σ)

Proof Main tool: majorization. Assume we can show

spec(ρ λ σ) ≺ λ spec(ρ) + (1 − λ) spec(σ) ≥ λf (ρ˜) + (1 − λ)f (σ˜ ) (concavity) = λf (ρ) + (1 − λ)f (σ) (symmetry)

Then f (ρ λ σ) ≥ f λρ˜ + (1 − λ)σ˜ (Schur-concavity)

Proof idea

Theorem f (ρ λ σ) ≥ λf (ρ) + (1 − λ)f (σ)

Proof Main tool: majorization. Assume we can show

spec(ρ λ σ) ≺ λ spec(ρ) + (1 − λ) spec(σ) = specλρ˜ + (1 − λ)σ˜

where ρ˜ := diag(spec(ρ)). ≥ λf (ρ˜) + (1 − λ)f (σ˜ ) (concavity) = λf (ρ) + (1 − λ)f (σ) (symmetry)

Proof idea

Theorem f (ρ λ σ) ≥ λf (ρ) + (1 − λ)f (σ)

Proof Main tool: majorization. Assume we can show

spec(ρ λ σ) ≺ λ spec(ρ) + (1 − λ) spec(σ) = specλρ˜ + (1 − λ)σ˜

where ρ˜ := diag(spec(ρ)). Then f (ρ λ σ) ≥ f λρ˜ + (1 − λ)σ˜ (Schur-concavity) = λf (ρ) + (1 − λ)f (σ) (symmetry)

Proof idea

Theorem f (ρ λ σ) ≥ λf (ρ) + (1 − λ)f (σ)

Proof Main tool: majorization. Assume we can show

spec(ρ λ σ) ≺ λ spec(ρ) + (1 − λ) spec(σ) = specλρ˜ + (1 − λ)σ˜

where ρ˜ := diag(spec(ρ)). Then f (ρ λ σ) ≥ f λρ˜ + (1 − λ)σ˜ (Schur-concavity) ≥ λf (ρ˜) + (1 − λ)f (σ˜ ) (concavity) Proof idea

Theorem f (ρ λ σ) ≥ λf (ρ) + (1 − λ)f (σ)

Proof Main tool: majorization. Assume we can show

spec(ρ λ σ) ≺ λ spec(ρ) + (1 − λ) spec(σ) = specλρ˜ + (1 − λ)σ˜

where ρ˜ := diag(spec(ρ)). Then f (ρ λ σ) ≥ f λρ˜ + (1 − λ)σ˜ (Schur-concavity) ≥ λf (ρ˜) + (1 − λ)f (σ˜ ) (concavity) = λf (ρ) + (1 − λ)f (σ) (symmetry) q = λkρkI − ρ + (1 − λ)kσkI − σ + λ(1 − λ) i[ρ, σ] = λ(X − X2) + (1 − λ)(Y − Y2) + Z†Z √ √ where X := kρkI − ρ, Y := kσkI − σ, Z := λX + i 1 − λY.

Anti-symmetric extension of operator A:

[k] A (v1 ∧ · · · ∧ vk) := (Av1 ∧ · · · ∧ vk) + ··· + (v1 ∧ · · · ∧ Avk)

[k] k Since kA k = ∑j=1 λj(A), it only remains to show that ∀k

[k] [k] [k] [k] kρ λ σ k ≤ λkρ k + (1 − λ)kσ k

We can forget about [k]’s and simply show that 0 ≤ −ρ λ σ + λkρk + (1 − λ)kσk I

Proof by Carlen, Lieb, Loss [CLL16] (from yesterday!)

Goal: spec(ρ λ σ) ≺ λ spec(ρ) + (1 − λ) spec(σ) q = λkρkI − ρ + (1 − λ)kσkI − σ + λ(1 − λ) i[ρ, σ] = λ(X − X2) + (1 − λ)(Y − Y2) + Z†Z √ √ where X := kρkI − ρ, Y := kσkI − σ, Z := λX + i 1 − λY.

[k] k Since kA k = ∑j=1 λj(A), it only remains to show that ∀k

[k] [k] [k] [k] kρ λ σ k ≤ λkρ k + (1 − λ)kσ k

We can forget about [k]’s and simply show that 0 ≤ −ρ λ σ + λkρk + (1 − λ)kσk I

Proof by Carlen, Lieb, Loss [CLL16] (from yesterday!)

Goal: spec(ρ λ σ) ≺ λ spec(ρ) + (1 − λ) spec(σ) Anti-symmetric extension of operator A:

[k] A (v1 ∧ · · · ∧ vk) := (Av1 ∧ · · · ∧ vk) + ··· + (v1 ∧ · · · ∧ Avk) q = λkρkI − ρ + (1 − λ)kσkI − σ + λ(1 − λ) i[ρ, σ] = λ(X − X2) + (1 − λ)(Y − Y2) + Z†Z √ √ where X := kρkI − ρ, Y := kσkI − σ, Z := λX + i 1 − λY.

We can forget about [k]’s and simply show that 0 ≤ −ρ λ σ + λkρk + (1 − λ)kσk I

Proof by Carlen, Lieb, Loss [CLL16] (from yesterday!)

Goal: spec(ρ λ σ) ≺ λ spec(ρ) + (1 − λ) spec(σ) Anti-symmetric extension of operator A:

[k] A (v1 ∧ · · · ∧ vk) := (Av1 ∧ · · · ∧ vk) + ··· + (v1 ∧ · · · ∧ Avk)

[k] k Since kA k = ∑j=1 λj(A), it only remains to show that ∀k

[k] [k] [k] [k] kρ λ σ k ≤ λkρ k + (1 − λ)kσ k q = λkρkI − ρ + (1 − λ)kσkI − σ + λ(1 − λ) i[ρ, σ] = λ(X − X2) + (1 − λ)(Y − Y2) + Z†Z √ √ where X := kρkI − ρ, Y := kσkI − σ, Z := λX + i 1 − λY.

Proof by Carlen, Lieb, Loss [CLL16] (from yesterday!)

Goal: spec(ρ λ σ) ≺ λ spec(ρ) + (1 − λ) spec(σ) Anti-symmetric extension of operator A:

[k] A (v1 ∧ · · · ∧ vk) := (Av1 ∧ · · · ∧ vk) + ··· + (v1 ∧ · · · ∧ Avk)

[k] k Since kA k = ∑j=1 λj(A), it only remains to show that ∀k

[k] [k] [k] [k] kρ λ σ k ≤ λkρ k + (1 − λ)kσ k

We can forget about [k]’s and simply show that 0 ≤ −ρ λ σ + λkρk + (1 − λ)kσk I = λ(X − X2) + (1 − λ)(Y − Y2) + Z†Z √ √ where X := kρkI − ρ, Y := kσkI − σ, Z := λX + i 1 − λY.

Proof by Carlen, Lieb, Loss [CLL16] (from yesterday!)

Goal: spec(ρ λ σ) ≺ λ spec(ρ) + (1 − λ) spec(σ) Anti-symmetric extension of operator A:

[k] A (v1 ∧ · · · ∧ vk) := (Av1 ∧ · · · ∧ vk) + ··· + (v1 ∧ · · · ∧ Avk)

[k] k Since kA k = ∑j=1 λj(A), it only remains to show that ∀k

[k] [k] [k] [k] kρ λ σ k ≤ λkρ k + (1 − λ)kσ k

We can forget about [k]’s and simply show that 0 ≤ −ρ λ σ + λkρk + (1 − λ)kσk I q = λkρkI − ρ + (1 − λ)kσkI − σ + λ(1 − λ) i[ρ, σ] Proof by Carlen, Lieb, Loss [CLL16] (from yesterday!)

Goal: spec(ρ λ σ) ≺ λ spec(ρ) + (1 − λ) spec(σ) Anti-symmetric extension of operator A:

[k] A (v1 ∧ · · · ∧ vk) := (Av1 ∧ · · · ∧ vk) + ··· + (v1 ∧ · · · ∧ Avk)

[k] k Since kA k = ∑j=1 λj(A), it only remains to show that ∀k

[k] [k] [k] [k] kρ λ σ k ≤ λkρ k + (1 − λ)kσ k

We can forget about [k]’s and simply show that 0 ≤ −ρ λ σ + λkρk + (1 − λ)kσk I q = λkρkI − ρ + (1 − λ)kσkI − σ + λ(1 − λ) i[ρ, σ] = λ(X − X2) + (1 − λ)(Y − Y2) + Z†Z √ √ where X := kρkI − ρ, Y := kσkI − σ, Z := λX + i 1 − λY. Photon number

I Thermal state with N photons:

∞ Ni ρ = |iihi| th ∑ ( + )i+1 i=0 N 1

I It’s entropy is g(N) := (N + 1) log(N + 1) − N log N

I Nc(ρ) := the average photon number of the thermal state that has the same entropy as ρ

Particular functions of interest

Functions of entropy

I entropy H(·) cH(·) I entropy power e −1 I entropy photon number Nc(·) := g (cH(·)) Particular functions of interest

Functions of entropy

I entropy H(·) cH(·) I entropy power e −1 I entropy photon number Nc(·) := g (cH(·))

Photon number

I Thermal state with N photons:

∞ Ni ρ = |iihi| th ∑ ( + )i+1 i=0 N 1

I It’s entropy is g(N) := (N + 1) log(N + 1) − N log N

I Nc(ρ) := the average photon number of the thermal state that has the same entropy as ρ Summary of EPIs

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(·))

f (ρ λ σ) ≥ λf (ρ) + (1 − λ)f (σ) Applications ≤ log d − (1 − λ)H(σ)

≥ (1 − λ)H(σ)

I Holevo quantity: n o χ(E) := max H ∑ piE(ρi) − ∑ piH E(ρi) {pi,ρi} i i ≤ log d − min H(E(ρ)) ρ

I Minimum output entropy:

H(Eλ,σ(ρ)) = H(ρ λ σ) ≥ λH(ρ) + (1 − λ)H(σ)

Product-state classical capacity

Eλ,σ

ρ ρ λ σ Uλ σ ≥ (1 − λ)H(σ)

≤ log d − (1 − λ)H(σ)

I Minimum output entropy:

H(Eλ,σ(ρ)) = H(ρ λ σ) ≥ λH(ρ) + (1 − λ)H(σ)

Product-state classical capacity

Eλ,σ

ρ ρ λ σ Uλ σ

I Holevo quantity: n o χ(E) := max H ∑ piE(ρi) − ∑ piH E(ρi) {pi,ρi} i i ≤ log d − min H(E(ρ)) ρ ≤ log d − (1 − λ)H(σ)

≥ (1 − λ)H(σ)

Product-state classical capacity

Eλ,σ

ρ ρ λ σ Uλ σ

I Holevo quantity: n o χ(E) := max H ∑ piE(ρi) − ∑ piH E(ρi) {pi,ρi} i i ≤ log d − min H(E(ρ)) ρ

I Minimum output entropy:

H(Eλ,σ(ρ)) = H(ρ λ σ) ≥ λH(ρ) + (1 − λ)H(σ) ≤ log d − (1 − λ)H(σ)

Product-state classical capacity

Eλ,σ

ρ ρ λ σ Uλ σ

I Holevo quantity: n o χ(E) := max H ∑ piE(ρi) − ∑ piH E(ρi) {pi,ρi} i i ≤ log d − min H(E(ρ)) ρ

I Minimum output entropy:

H(Eλ,σ(ρ)) = H(ρ λ σ) ≥ λH(ρ) + (1 − λ)H(σ) ≥ (1 − λ)H(σ) Product-state classical capacity

Eλ,σ

ρ ρ λ σ Uλ σ

I Holevo quantity: n o χ(E) := max H ∑ piE(ρi) − ∑ piH E(ρi) {pi,ρi} i i ≤ log d − min H(E(ρ)) ρ ≤ log d − (1 − λ)H(σ)

I Minimum output entropy:

H(Eλ,σ(ρ)) = H(ρ λ σ) ≥ λH(ρ) + (1 − λ)H(σ) ≥ (1 − λ)H(σ) X −X 1 I e σe = σ + [X, σ] + 2! [X, [X, σ]] + ··· −iρt iρt I e σe = σ − it[ρ, σ] + O(t2) I Partial swap:

h −iSt iSti 2 2 Tr2 e (σ ⊗ ρ)e = cos t σ + sin t ρ − i sin t cos t [ρ, σ] = σ − it[ρ, σ] + O(t2)

−iρt iρt I This approximates e σe well when t 1 I Using n copies of ρ can boost t to nt:

(σ λ ρ) λ ρ λ ··· λ ρ

Partial swap in quantum algorithms

Quantum principal component analysis [LMR14]

σ ⊗ ρ⊗n 7→ e−iρntσeiρnt −iρt iρt I e σe = σ − it[ρ, σ] + O(t2) I Partial swap:

h −iSt iSti 2 2 Tr2 e (σ ⊗ ρ)e = cos t σ + sin t ρ − i sin t cos t [ρ, σ] = σ − it[ρ, σ] + O(t2)

−iρt iρt I This approximates e σe well when t 1 I Using n copies of ρ can boost t to nt:

(σ λ ρ) λ ρ λ ··· λ ρ

Partial swap in quantum algorithms

Quantum principal component analysis [LMR14]

σ ⊗ ρ⊗n 7→ e−iρntσeiρnt

X −X 1 I e σe = σ + [X, σ] + 2! [X, [X, σ]] + ··· I Partial swap:

h −iSt iSti 2 2 Tr2 e (σ ⊗ ρ)e = cos t σ + sin t ρ − i sin t cos t [ρ, σ] = σ − it[ρ, σ] + O(t2)

−iρt iρt I This approximates e σe well when t 1 I Using n copies of ρ can boost t to nt:

(σ λ ρ) λ ρ λ ··· λ ρ

Partial swap in quantum algorithms

Quantum principal component analysis [LMR14]

σ ⊗ ρ⊗n 7→ e−iρntσeiρnt

X −X 1 I e σe = σ + [X, σ] + 2! [X, [X, σ]] + ··· −iρt iρt I e σe = σ − it[ρ, σ] + O(t2) −iρt iρt I This approximates e σe well when t 1 I Using n copies of ρ can boost t to nt:

(σ λ ρ) λ ρ λ ··· λ ρ

Partial swap in quantum algorithms

Quantum principal component analysis [LMR14]

σ ⊗ ρ⊗n 7→ e−iρntσeiρnt

X −X 1 I e σe = σ + [X, σ] + 2! [X, [X, σ]] + ··· −iρt iρt I e σe = σ − it[ρ, σ] + O(t2) I Partial swap:

h −iSt iSti 2 2 Tr2 e (σ ⊗ ρ)e = cos t σ + sin t ρ − i sin t cos t [ρ, σ] = σ − it[ρ, σ] + O(t2) I Using n copies of ρ can boost t to nt:

(σ λ ρ) λ ρ λ ··· λ ρ

Partial swap in quantum algorithms

Quantum principal component analysis [LMR14]

σ ⊗ ρ⊗n 7→ e−iρntσeiρnt

X −X 1 I e σe = σ + [X, σ] + 2! [X, [X, σ]] + ··· −iρt iρt I e σe = σ − it[ρ, σ] + O(t2) I Partial swap:

h −iSt iSti 2 2 Tr2 e (σ ⊗ ρ)e = cos t σ + sin t ρ − i sin t cos t [ρ, σ] = σ − it[ρ, σ] + O(t2)

−iρt iρt I This approximates e σe well when t 1 Partial swap in quantum algorithms

Quantum principal component analysis [LMR14]

σ ⊗ ρ⊗n 7→ e−iρntσeiρnt

X −X 1 I e σe = σ + [X, σ] + 2! [X, [X, σ]] + ··· −iρt iρt I e σe = σ − it[ρ, σ] + O(t2) I Partial swap:

h −iSt iSti 2 2 Tr2 e (σ ⊗ ρ)e = cos t σ + sin t ρ − i sin t cos t [ρ, σ] = σ − it[ρ, σ] + O(t2)

−iρt iρt I This approximates e σe well when t 1 I Using n copies of ρ can boost t to nt:

(σ λ ρ) λ ρ λ ··· λ ρ Extensions of λ I Permutations of n qudits: Cd Cd Cd Cd ... n π ∈ Sn Qπ ∈ U(d ) ...

I Question:

When is U := ∑ zπQπ unitary (zπ ∈ C)? π∈Sn

How to combine n states?

† (ρ1,..., ρn) 7→ Tr2,...,n U(ρ1 ⊗ · · · ⊗ ρn)U

I For n = 2, U is a linear combination of I and S: √ √ U = λ I + i 1 − λ S I Question:

When is U := ∑ zπQπ unitary (zπ ∈ C)? π∈Sn

How to combine n states?

† (ρ1,..., ρn) 7→ Tr2,...,n U(ρ1 ⊗ · · · ⊗ ρn)U

I For n = 2, U is a linear combination of I and S: √ √ U = λ I + i 1 − λ S

I Permutations of n qudits: Cd Cd Cd Cd ... n π ∈ Sn Qπ ∈ U(d ) ... How to combine n states?

† (ρ1,..., ρn) 7→ Tr2,...,n U(ρ1 ⊗ · · · ⊗ ρn)U

I For n = 2, U is a linear combination of I and S: √ √ U = λ I + i 1 − λ S

I Permutations of n qudits: Cd Cd Cd Cd ... n π ∈ Sn Qπ ∈ U(d ) ...

I Question:

When is U := ∑ zπQπ unitary (zπ ∈ C)? π∈Sn Proof: Represent g by Lg deﬁned as Lg|hi := |ghi. This is known as left regular representation.

? G ,→ S|G| ,→ U(|G|)

Q: When is L := ∑ zgLg unitary (zg ∈ C)? g∈G

Unitary Cayley’s Theorem

Cayley’s Theorem

Every ﬁnite group G is isomorphic to a subgroup of S|G|. ? G ,→ S|G| ,→ U(|G|)

Q: When is L := ∑ zgLg unitary (zg ∈ C)? g∈G

Unitary Cayley’s Theorem

Cayley’s Theorem

Every ﬁnite group G is isomorphic to a subgroup of S|G|.

Proof: Represent g by Lg deﬁned as Lg|hi := |ghi. This is known as left regular representation. Q: When is L := ∑ zgLg unitary (zg ∈ C)? g∈G

Unitary Cayley’s Theorem

Cayley’s Theorem

Every ﬁnite group G is isomorphic to a subgroup of S|G|.

Proof: Represent g by Lg deﬁned as Lg|hi := |ghi. This is known as left regular representation.

? G ,→ S|G| ,→ U(|G|) Unitary Cayley’s Theorem

Cayley’s Theorem

Every ﬁnite group G is isomorphic to a subgroup of S|G|.

Proof: Represent g by Lg deﬁned as Lg|hi := |ghi. This is known as left regular representation.

? G ,→ S|G| ,→ U(|G|)

Q: When is L := ∑ zgLg unitary (zg ∈ C)? g∈G h i ˆ † M Fact: Lg := FLgF = τ(g) ⊗ Idτ τ∈Gˆ " # ? ˆ = ˆ = M ( ) ⊗ ∈ (| |) L ∑ zgLg ∑ zgτ g Idτ U G g∈G τ∈Gˆ g∈G | {z } Uτ

Theorem: L = ∑g∈G zgLg is unitary iff for some Uτ ∈ U(dτ),

d z = τ Trτ(g)†U g ∑ |G| τ τ∈Gˆ

= = Note: If L ∑π∈Sn zπLπ is unitary then so is U ∑π∈Sn zπQπ

Fourier transform to the rescue!

Q: When is L := ∑ zgLg unitary (zg ∈ C)? g∈G " # ? ˆ = ˆ = M ( ) ⊗ ∈ (| |) L ∑ zgLg ∑ zgτ g Idτ U G g∈G τ∈Gˆ g∈G | {z } Uτ

Theorem: L = ∑g∈G zgLg is unitary iff for some Uτ ∈ U(dτ),

d z = τ Trτ(g)†U g ∑ |G| τ τ∈Gˆ

= = Note: If L ∑π∈Sn zπLπ is unitary then so is U ∑π∈Sn zπQπ

Fourier transform to the rescue!

Q: When is L := ∑ zgLg unitary (zg ∈ C)? g∈G h i ˆ † M Fact: Lg := FLgF = τ(g) ⊗ Idτ τ∈Gˆ Theorem: L = ∑g∈G zgLg is unitary iff for some Uτ ∈ U(dτ),

d z = τ Trτ(g)†U g ∑ |G| τ τ∈Gˆ

= = Note: If L ∑π∈Sn zπLπ is unitary then so is U ∑π∈Sn zπQπ

Fourier transform to the rescue!

Q: When is L := ∑ zgLg unitary (zg ∈ C)? g∈G h i ˆ † M Fact: Lg := FLgF = τ(g) ⊗ Idτ τ∈Gˆ " # ? ˆ = ˆ = M ( ) ⊗ ∈ (| |) L ∑ zgLg ∑ zgτ g Idτ U G g∈G τ∈Gˆ g∈G | {z } Uτ = = Note: If L ∑π∈Sn zπLπ is unitary then so is U ∑π∈Sn zπQπ

Fourier transform to the rescue!

Theorem: L = ∑g∈G zgLg is unitary iff for some Uτ ∈ U(dτ),

d z = τ Trτ(g)†U g ∑ |G| τ τ∈Gˆ Fourier transform to the rescue!

Theorem: L = ∑g∈G zgLg is unitary iff for some Uτ ∈ U(dτ),

d z = τ Trτ(g)†U g ∑ |G| τ τ∈Gˆ

= = Note: If L ∑π∈Sn zπLπ is unitary then so is U ∑π∈Sn zπQπ EPI conjecture for 3 states

If f is concave and symmetric then

f (ρ) ≥ p1 f (ρ1) + p2 f (ρ2) + p3 f (ρ3)

† where ρ = Tr2,3 U(ρ1 ⊗ ρ2 ⊗ ρ3)U is explicitly given by

ρ = p1 ρ1 + p2 ρ2 + p3 ρ3 √ + p p2 sin δ i[ρ , ρ2] √ 1 12 1 + p2p3 sin δ23 i[ρ2, ρ3] √ + p3p1 sin δ31 i[ρ3, ρ1] √ + p p2 cos δ i[ρ , i[ρ2, ρ3]] √ 1 12 1 + p2p3 cos δ23 i[i[ρ1, ρ2], ρ3]

+ + = where√ δij are subject√ to δ12 δ23 √δ31 0 and p1p2 cos δ12 + p2p3 cos δ23 + p3p1 cos δ31 = 0 I Conditional version of qudit 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 extension of λ for combining ≥ 3 states [Ozo15] I proving the EPI. . . ? I Applications I lower bounds for min output entropy & upper bounds for product-state classical capacity I more. . . ? for quantum algorithms. . . ?

Open problems

I Entropy photon number inequality for c.v. states I useful for bounding classical capacities of various bosonic channels [GGL+04, GSE07, GSE08] I proved only for Gaussian states [Guh08] I does not seem to follow from qudit EPI by taking d → ∞ I Generalization to 3 or more systems I trivial for c.v. distributions I proved for c.v. states [DMLG15] I extension of λ for combining ≥ 3 states [Ozo15] I proving the EPI. . . ? I Applications I lower bounds for min output entropy & upper bounds for product-state classical capacity I more. . . ? for quantum algorithms. . . ?

Open problems

I Entropy photon number inequality for c.v. states I useful for bounding classical capacities of various bosonic channels [GGL+04, GSE07, GSE08] I proved only for Gaussian states [Guh08] I does not seem to follow from qudit EPI by taking d → ∞ I Conditional version of qudit EPI I trivial for c.v. distributions I proved for Gaussian c.v. states [Koe15] I qudit analogue. . . ? I Applications I lower bounds for min output entropy & upper bounds for product-state classical capacity I more. . . ? for quantum algorithms. . . ?

Open problems

I Entropy photon number inequality for c.v. states I useful for bounding classical capacities of various bosonic channels [GGL+04, GSE07, GSE08] I proved only for Gaussian states [Guh08] I does not seem to follow from qudit EPI by taking d → ∞ I Conditional version of qudit 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 extension of λ for combining ≥ 3 states [Ozo15] I proving the EPI. . . ? Open problems

I Entropy photon number inequality for c.v. states I useful for bounding classical capacities of various bosonic channels [GGL+04, GSE07, GSE08] I proved only for Gaussian states [Guh08] I does not seem to follow from qudit EPI by taking d → ∞ I Conditional version of qudit 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 extension of λ for combining ≥ 3 states [Ozo15] I proving the EPI. . . ? I Applications I lower bounds for min output entropy & upper bounds for product-state classical capacity I more. . . ? for quantum algorithms. . . ? Thank you!

arXiv:1503.04213 – Qudit EPIs arXiv:1508.00860 – Unitary Cayley’s theorem BibliographyI

[Bar86] Andrew R. Barron. Entropy and the central limit theorem. The Annals of Probability, 14(1):336–342, 1986. URL: http://projecteuclid.org/euclid.aop/1176992632. [Ber74] Patrick P. Bergmans. A simple converse for broadcast channels with additive white Gaussian noise. Information Theory, IEEE Transactions on, 20(2):279–280, Mar 1974. doi:10.1109/TIT.1974.1055184. [Bla65] Nelson M. Blachman. The convolution inequality for entropy powers. Information Theory, IEEE Transactions on, 11(2):267–271, Apr 1965. doi:10.1109/TIT.1965.1053768. [CLL16] Eric A. Carlen, Elliott H. Lieb, and Michael Loss. On a quantum entropy power inequality of Audenaert, Datta and Ozols. 2016. arXiv:1603.07043. [DMG14] Giacomo De Palma, Andrea Mari, and Vittorio Giovannetti. A generalization of the entropy power inequality to bosonic quantum systems. Nature Photonics, 8(12):958–964, 2014. arXiv:1402.0404, doi:10.1038/nphoton.2014.252. BibliographyII

[DMLG15] Giacomo De Palma, Andrea Mari, Seth Lloyd, and Vittorio Giovannetti. Multimode quantum entropy power inequality. Phys. Rev. A, 91(3):032320, Mar 2015. arXiv:1408.6410, doi:10.1103/PhysRevA.91.032320. [GGL+04] Vittorio Giovannetti, Saikat Guha, Seth Lloyd, Lorenzo Maccone, Jeffrey H. Shapiro, and Horace P. Yuen. Classical capacity of the lossy bosonic channel: the exact solution. Phys. Rev. Lett., 92(2):027902, Jan 2004. arXiv:quant-ph/0308012, doi:10.1103/PhysRevLett.92.027902. [GSE07] Saikat Guha, Jeffrey H. Shapiro, and Baris I. Erkmen. Classical capacity of bosonic broadcast communication and a minimum output entropy conjecture. Phys. Rev. A, 76(3):032303, Sep 2007. arXiv:0706.3416, doi:10.1103/PhysRevA.76.032303. [GSE08] Saikat Guha, Jeffrey H. Shapiro, and Baris I. Erkmen. Capacity of the bosonic wiretap channel and the entropy photon-number inequality. In IEEE International Symposium on Information Theory, 2008 (ISIT 2008), pages 91–95, Jul 2008. arXiv:0801.0841, doi:10.1109/ISIT.2008.4594954. Bibliography III

[Guh08] Saikat Guha. Multiple-user quantum information theory for optical communication channels. PhD thesis, Dept. Electr. Eng. Comput. Sci., MIT, Cambridge, MA, USA, 2008. URL: http://hdl.handle.net/1721.1/44413. [Koe15] Robert Koenig. The conditional entropy power inequality for Gaussian quantum states. Journal of Mathematical Physics, 56(2):022201, 2015. arXiv:1304.7031, doi:10.1063/1.4906925. [KS14] Robert Konig¨ and Graeme Smith. The entropy power inequality for quantum systems. Information Theory, IEEE Transactions on, 60(3):1536–1548, Mar 2014. arXiv:1205.3409, doi:10.1109/TIT.2014.2298436. [LMR14] Seth Lloyd, Masoud Mohseni, and Patrick Rebentrost. Quantum principal component analysis. Nature Physics, 10(9):631–633, 2014. arXiv:1307.0401, doi:10.1038/nphys3029. [Ozo15] Maris Ozols. How to combine three quantum states. 2015. arXiv:1508.00860. BibliographyIV

[Sha48] Claude E. Shannon. A mathematical theory of communication. The Bell System Technical Journal, 27:623–656, Oct 1948. URL: http://cm.bell-labs.com/cm/ms/what/shannonday/ shannon1948.pdf. [Sta59] A. J. Stam. Some inequalities satisﬁed by the quantities of information of Fisher and Shannon. Information and Control, 2(2):101–112, Jun 1959. doi:10.1016/S0019-9958(59)90348-1.