Zonoids and sparsification of quantum measurements Guillaume AUBRUN - - PowerPoint PPT Presentation

Zonoids and sparsification of quantum measurements

Guillaume AUBRUN

(joint with C´ ecilia Lancien)

Universit´ e Lyon 1, France

Guillaume Aubrun (Lyon) Zonoids Marne-la-Vall´ ee, Juin 2015 1 / 16

Lyapounov convexity theorem

Let µ : (Ω, F) be a vector measure, non-atomic. Then {µ(A) : A ∈ F} ⊂ Rn is a convex set.

Guillaume Aubrun (Lyon) Zonoids Marne-la-Vall´ ee, Juin 2015 2 / 16

Lyapounov convexity theorem

Let µ : (Ω, F) be a vector measure, non-atomic. Then {µ(A) : A ∈ F} ⊂ Rn is a convex set. Such convex sets are called zonoids. Equivalently, a zonoid is a limit of zonotopes. A zonotope is a finite Minkowski sum of segments. The Minkowski sum is A + B = {a + b : a ∈ A, b ∈ B}.

Guillaume Aubrun (Lyon) Zonoids Marne-la-Vall´ ee, Juin 2015 2 / 16

Lyapounov convexity theorem

Let µ : (Ω, F) be a vector measure, non-atomic. Then {µ(A) : A ∈ F} ⊂ Rn is a convex set. Such convex sets are called zonoids. Equivalently, a zonoid is a limit of zonotopes. A zonotope is a finite Minkowski sum of segments. The Minkowski sum is A + B = {a + b : a ∈ A, b ∈ B}. Also: for a vector measure, the convex hull of the range is a zonoid.

Guillaume Aubrun (Lyon) Zonoids Marne-la-Vall´ ee, Juin 2015 2 / 16

Zonoids

1 The cube is a zonoid. 2 The octahedron is not a zonoid. 3 Any planar compact convex set with a center of symmetry is a zonoid. 4 The Euclidean ball Bn

2 is a zonoid

Bn

2 = αn

• Sn−1[−u, −u] dσ(u).

Guillaume Aubrun (Lyon) Zonoids Marne-la-Vall´ ee, Juin 2015 3 / 16

POVMs

A Positive Operator-Valued Measure (POVM) is a vector measure M : (Ω, F) → M+(Cd) such that M(Ω) = Id. Here M+(Cd) is the set of positive self-adjoint d × d matrices.

Guillaume Aubrun (Lyon) Zonoids Marne-la-Vall´ ee, Juin 2015 4 / 16

POVMs

A Positive Operator-Valued Measure (POVM) is a vector measure M : (Ω, F) → M+(Cd) such that M(Ω) = Id. Here M+(Cd) is the set of positive self-adjoint d × d matrices. POVMs corresponds to quantum measurements.

Guillaume Aubrun (Lyon) Zonoids Marne-la-Vall´ ee, Juin 2015 4 / 16

POVMs

A Positive Operator-Valued Measure (POVM) is a vector measure M : (Ω, F) → M+(Cd) such that M(Ω) = Id. Here M+(Cd) is the set of positive self-adjoint d × d matrices. POVMs corresponds to quantum measurements. We often consider the special case of discrete POVMs (=the purely atomic case). They are given by operators (M1, . . . , MN), where Mi 0 and M1 + · · · + MN = Id. The range is {M(A) ; A ∈ F} =

• i∈I

Mi : I ⊂ {1, . . . , N}

• .

Guillaume Aubrun (Lyon) Zonoids Marne-la-Vall´ ee, Juin 2015 4 / 16

Zonoid associated to a POVM

The convex hull of the range is a zonoid conv{M(A) ; A ∈ F} =

N

• i=1

[0, Mi]. It is more natural to consider the 0-symmetric version KM = 2 conv{M(A) ; A ∈ F} − Id =

N

• i=1

[−Mi, Mi] This is a zonotope inside K = {A ∈ M+(Cd) : A∞ 1.

Guillaume Aubrun (Lyon) Zonoids Marne-la-Vall´ ee, Juin 2015 5 / 16

Zonoid associated to a POVM

The convex hull of the range is a zonoid conv{M(A) ; A ∈ F} =

N

• i=1

[0, Mi]. It is more natural to consider the 0-symmetric version KM = 2 conv{M(A) ; A ∈ F} − Id =

N

• i=1

[−Mi, Mi] This is a zonotope inside K = {A ∈ M+(Cd) : A∞ 1. Conversely, any zonoid inside K and containing ±Id comes from a POVM.

Guillaume Aubrun (Lyon) Zonoids Marne-la-Vall´ ee, Juin 2015 5 / 16

Support function

Given a POVM M, the support function of the zonoid KM is a norm ∆M = sup

A∈KM

Tr(∆A) =

N

• i=1

| Tr ∆Mi|.

Guillaume Aubrun (Lyon) Zonoids Marne-la-Vall´ ee, Juin 2015 6 / 16

Support function

Given a POVM M, the support function of the zonoid KM is a norm ∆M = sup

A∈KM

Tr(∆A) =

N

• i=1

| Tr ∆Mi|. Note that the normed space (M+(Cd), · M) embeds into ℓN

1 = (RN, · 1) (another characterization of zonotopes/zonoids).

Guillaume Aubrun (Lyon) Zonoids Marne-la-Vall´ ee, Juin 2015 6 / 16

Support function

Given a POVM M, the support function of the zonoid KM is a norm ∆M = sup

A∈KM

Tr(∆A) =

N

• i=1

| Tr ∆Mi|. Note that the normed space (M+(Cd), · M) embeds into ℓN

1 = (RN, · 1) (another characterization of zonotopes/zonoids).

As we shall see this norm has a interpretation as distinguishability norms (Matthews–Wehner–Winter).

Guillaume Aubrun (Lyon) Zonoids Marne-la-Vall´ ee, Juin 2015 6 / 16

State discrimination

Let ρ, σ two quantum states on Cd. A referee chooses ρ or σ with equal

• probability. You have to guess which was chosen using the POVM M with

a single sample.

Guillaume Aubrun (Lyon) Zonoids Marne-la-Vall´ ee, Juin 2015 7 / 16

State discrimination

Let ρ, σ two quantum states on Cd. A referee chooses ρ or σ with equal

• probability. You have to guess which was chosen using the POVM M with

a single sample. Born’s rule: if ρ was chosen, outcome i is output with probability Tr ρMi; if σ was chosen, outcome i is output with probability Tr σMi.

Guillaume Aubrun (Lyon) Zonoids Marne-la-Vall´ ee, Juin 2015 7 / 16

State discrimination

Let ρ, σ two quantum states on Cd. A referee chooses ρ or σ with equal

• probability. You have to guess which was chosen using the POVM M with

a single sample. Born’s rule: if ρ was chosen, outcome i is output with probability Tr ρMi; if σ was chosen, outcome i is output with probability Tr σMi. The best strategy is of course, given the outcome, to guess the most likely

• state. The probability of error is

p = 1 2

N

• i=1

min(Tr ρMi, Tr σMi) = 1 2 − 1 4

N

• i=1

| Tr ρMi − Tr σMi| = 1 2 − 1 4ρ − σM

Guillaume Aubrun (Lyon) Zonoids Marne-la-Vall´ ee, Juin 2015 7 / 16

The uniform POVM

Let Ud be the uniform POVM, defined on (SCd, Borel) by Ud(A) = d

• A

|ψψ| dσ(ψ).

Guillaume Aubrun (Lyon) Zonoids Marne-la-Vall´ ee, Juin 2015 8 / 16

The uniform POVM

Let Ud be the uniform POVM, defined on (SCd, Borel) by Ud(A) = d

• A

|ψψ| dσ(ψ). We would like sparsifications of Ud, i.e. POVMs M with as few outcomes as possible and such that (1 − ε) · M · Ud (1 + ε) · M.

Guillaume Aubrun (Lyon) Zonoids Marne-la-Vall´ ee, Juin 2015 8 / 16

t-designs

Start from the identity (t ∈ N) π :=

• SCd

|ψψ|⊗t dσ = 1 dim Symt(Cd)PSymt(Cd).

Guillaume Aubrun (Lyon) Zonoids Marne-la-Vall´ ee, Juin 2015 9 / 16

t-designs

Start from the identity (t ∈ N) π :=

• SCd

|ψψ|⊗t dσ = 1 dim Symt(Cd)PSymt(Cd). An ε-approximate t-design is a finitely supported measure µ on SCd such that (1 − ε)π

• SCd

|ψψ|⊗t dµ (1 + ε)π.

Guillaume Aubrun (Lyon) Zonoids Marne-la-Vall´ ee, Juin 2015 9 / 16

t-designs

Start from the identity (t ∈ N) π :=

• SCd

|ψψ|⊗t dσ = 1 dim Symt(Cd)PSymt(Cd). An ε-approximate t-design is a finitely supported measure µ on SCd such that (1 − ε)π

• SCd

|ψψ|⊗t dµ (1 + ε)π. Example : ε = 0 gives an exact integration formula (cubature formula) for homogeneous polynomial of degree t.

Guillaume Aubrun (Lyon) Zonoids Marne-la-Vall´ ee, Juin 2015 9 / 16

Sparsification from 4-designs

Ambainis–Emerson (2007) showed that if µ is a (exact or approximate) 4-design, then the corresponding POVM M satisfies c · M · Ud · M.

Guillaume Aubrun (Lyon) Zonoids Marne-la-Vall´ ee, Juin 2015 10 / 16

Sparsification from 4-designs

Ambainis–Emerson (2007) showed that if µ is a (exact or approximate) 4-design, then the corresponding POVM M satisfies c · M · Ud · M. Idea: the 1-norm can be controlled from 2- and 4-norms X3

L2

X2

L4

XL1 XL2 This approach requires card supp(µ) dim Symt(Cd) = Ω(d4).

Guillaume Aubrun (Lyon) Zonoids Marne-la-Vall´ ee, Juin 2015 10 / 16

Sparsification from 4-designs

Ambainis–Emerson (2007) showed that if µ is a (exact or approximate) 4-design, then the corresponding POVM M satisfies c · M · Ud · M. Idea: the 1-norm can be controlled from 2- and 4-norms X3

L2

X2

L4

XL1 XL2 This approach requires card supp(µ) dim Symt(Cd) = Ω(d4). Similar to Rudin (1960): ℓn

2 ⊂ ℓn2 4 isometrically and therefore ℓn 2 ⊂ ℓn2 1 with

distortion √

• 3. Equivalently, gives a zonotope Z with n2 summands such

that Z ⊂ Bn

2 ⊂

√ 3Z.

Guillaume Aubrun (Lyon) Zonoids Marne-la-Vall´ ee, Juin 2015 10 / 16

Concentration of measure

Rudin’s result can be improved via random constructions based on the concentration of measure phenomenon.

Guillaume Aubrun (Lyon) Zonoids Marne-la-Vall´ ee, Juin 2015 11 / 16

Concentration of measure

Rudin’s result can be improved via random constructions based on the concentration of measure phenomenon. Figiel–Lindenstrauss–Milman (1977): given ε > 0, ℓn

2 embeds with

distortion 1 + ε in ℓN

1 with N = Cε−2n.

Equivalently, there is a zonoid Z with Cε−2n summands such that Z ⊂ Bn

2 ⊂ (1 + ε)Z.

Guillaume Aubrun (Lyon) Zonoids Marne-la-Vall´ ee, Juin 2015 11 / 16

Concentration of measure

Rudin’s result can be improved via random constructions based on the concentration of measure phenomenon. Figiel–Lindenstrauss–Milman (1977): given ε > 0, ℓn

2 embeds with

distortion 1 + ε in ℓN

1 with N = Cε−2n.

Equivalently, there is a zonoid Z with Cε−2n summands such that Z ⊂ Bn

2 ⊂ (1 + ε)Z.

Proof: choose the directions of the N segments independently and uniformly at random.

Guillaume Aubrun (Lyon) Zonoids Marne-la-Vall´ ee, Juin 2015 11 / 16

Theorem 1: optimal sparsifications of the uniform POVM

Theorem (A.-Lancien)

Given d ∈ N and ε ∈ (0, 1), there is a POVM M on Cd with N outcomes such that N Cε−2| log ε|d2 and (1 − ε) · M · Ud (1 + ε) · M. The size d2 = dim Msa(Cd) is obviously optimal.

Guillaume Aubrun (Lyon) Zonoids Marne-la-Vall´ ee, Juin 2015 12 / 16

Theorem 1: optimal sparsifications of the uniform POVM

Theorem (A.-Lancien)

Given d ∈ N and ε ∈ (0, 1), there is a POVM M on Cd with N outcomes such that N Cε−2| log ε|d2 and (1 − ε) · M · Ud (1 + ε) · M. The size d2 = dim Msa(Cd) is obviously optimal. The construction is random: take (ψi) independent, uniform on the sphere

• SCd. Let

S =

N

• i=1

|ψiψi|. The POVM is the family

• |S−1/2ψiS−1/2ψi|
• 1iN .

Guillaume Aubrun (Lyon) Zonoids Marne-la-Vall´ ee, Juin 2015 12 / 16

Theorem 1, ideas of the proof

The proof uses standard tools

1 Net arguments (discrete approximation of the unit sphere) 2 Deviation inequalities for sum of sub-exponential random variables. 3 Random matrix estimates to show that the matrix S is

well-conditioned.

Guillaume Aubrun (Lyon) Zonoids Marne-la-Vall´ ee, Juin 2015 13 / 16

Theorem 1, ideas of the proof

The proof uses standard tools

1 Net arguments (discrete approximation of the unit sphere) 2 Deviation inequalities for sum of sub-exponential random variables. 3 Random matrix estimates to show that the matrix S is

well-conditioned. What about derandomization ?

Guillaume Aubrun (Lyon) Zonoids Marne-la-Vall´ ee, Juin 2015 13 / 16

Tensor product of POVM

There is natural notion of tensor product for POVMs: given (discrete) POVMs (Mi)i∈I and (Nj)j∈J on Cd, consider (Mi ⊗ Nj)i∈I,j∈J. Accordingly there is a notion of tensor products for zonoids.

Guillaume Aubrun (Lyon) Zonoids Marne-la-Vall´ ee, Juin 2015 14 / 16

Tensor product of POVM

There is natural notion of tensor product for POVMs: given (discrete) POVMs (Mi)i∈I and (Nj)j∈J on Cd, consider (Mi ⊗ Nj)i∈I,j∈J. Accordingly there is a notion of tensor products for zonoids. Simple fact: if (1 − ε) · M · M′ (1 + ε) · M and (1 − ε) · N · N′ (1 + ε) · N, then (1 − ε)2 · M⊗N · M′⊗N′ (1 + ε)2 · M⊗N. It follows from Theorem 1 that there are optimal local sparsifications of the“local uniform POVM”LU = Ud ⊗ Ud.

Guillaume Aubrun (Lyon) Zonoids Marne-la-Vall´ ee, Juin 2015 14 / 16

Tensor product of POVM

There is natural notion of tensor product for POVMs: given (discrete) POVMs (Mi)i∈I and (Nj)j∈J on Cd, consider (Mi ⊗ Nj)i∈I,j∈J. Accordingly there is a notion of tensor products for zonoids. Simple fact: if (1 − ε) · M · M′ (1 + ε) · M and (1 − ε) · N · N′ (1 + ε) · N, then (1 − ε)2 · M⊗N · M′⊗N′ (1 + ε)2 · M⊗N. It follows from Theorem 1 that there are optimal local sparsifications of the“local uniform POVM”LU = Ud ⊗ Ud. Note that · LU is equivalent to the following norm (Lancien–Winter) ∆2

2(2) = (Tr ∆)2 + Tr2(Tr1 ∆)2 + Tr1(Tr2 ∆)2 + Tr(∆2).

Guillaume Aubrun (Lyon) Zonoids Marne-la-Vall´ ee, Juin 2015 14 / 16

Approximation of zonoids by zonotopes

A series of results from the late ’80s (Schechtman, Bourgain–Lindenstrauss–Milman, Talagrand) culminating in the following: Any zonoid K ⊂ Rn can be ε-approximated by a zonotope Z with N Cε−2n log n summands, in the sense K ⊂ Z ⊂ (1 + ε)K.

Guillaume Aubrun (Lyon) Zonoids Marne-la-Vall´ ee, Juin 2015 15 / 16

Approximation of zonoids by zonotopes

A series of results from the late ’80s (Schechtman, Bourgain–Lindenstrauss–Milman, Talagrand) culminating in the following: Any zonoid K ⊂ Rn can be ε-approximated by a zonotope Z with N Cε−2n log n summands, in the sense K ⊂ Z ⊂ (1 + ε)K. Important open question: can we remove the log n factor ?

Guillaume Aubrun (Lyon) Zonoids Marne-la-Vall´ ee, Juin 2015 15 / 16

Approximation of any POVM

The POVM version of the previous theorem is the following.

Theorem (A.-Lancien)

Any POVM M on Cd can be ε-approximated by a sub-POVM M′ with N Cε−2d2 log d outcomes, in the sense (1 − ε) · M′ · M (1 + ε) · M′. A sub-POVM is a finite family (Mi) of positive operators with Mi Id.

Guillaume Aubrun (Lyon) Zonoids Marne-la-Vall´ ee, Juin 2015 16 / 16