Zonoids and Sparsification of Quantum Measurements

Zonoids and sparsiﬁcation of quantum measurements

Guillaume AUBRUN

(joint with C´ecilia Lancien)

Universit´eLyon 1, France

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

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 Also: for a vector measure, the convex hull of the range is a zonoid.

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 n The Euclidean ball B2 is a zonoid Z n B2 = αn [−u, −u] dσ(u). Sn−1

Guillaume Aubrun (Lyon) Zonoids Marne-la-Vall´ee,Juin 2015 3 / 16 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 ( ) X {M(A); A ∈ F} = Mi : I ⊂ {1,..., N} . i∈I

POVMs

A Positive Operator-Valued Measure (POVM) is a vector measure

d M : (Ω, F) → M+(C )

d such that M(Ω) = Id. Here M+(C ) is the set of positive self-adjoint d × d matrices.

Guillaume Aubrun (Lyon) Zonoids Marne-la-Vall´ee,Juin 2015 4 / 16 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 ( ) X {M(A); A ∈ F} = Mi : I ⊂ {1,..., N} . i∈I

POVMs

A Positive Operator-Valued Measure (POVM) is a vector measure

d M : (Ω, F) → M+(C )

d such that M(Ω) = Id. Here M+(C ) 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

d M : (Ω, F) → M+(C )

d such that M(Ω) = Id. Here M+(C ) 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 ( ) X {M(A); A ∈ F} = Mi : I ⊂ {1,..., N} . i∈I

Guillaume Aubrun (Lyon) Zonoids Marne-la-Vall´ee,Juin 2015 4 / 16 Conversely, any zonoid inside K and containing ±Id comes from a POVM.

Zonoid associated to a POVM

The convex hull of the range is a zonoid

N X conv{M(A); A ∈ F} = [0, Mi ]. i=1 It is more natural to consider the 0-symmetric version

N X KM = 2 conv{M(A); A ∈ F} − Id = [−Mi , Mi ] i=1 d This is a zonotope inside K = {A ∈ M+(C ): kAk∞ 6 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

N X conv{M(A); A ∈ F} = [0, Mi ]. i=1 It is more natural to consider the 0-symmetric version

N X KM = 2 conv{M(A); A ∈ F} − Id = [−Mi , Mi ] i=1 d This is a zonotope inside K = {A ∈ M+(C ): kAk∞ 6 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 d Note that the normed space (M+(C ), k · kM ) embeds into N N `1 = (R , k · k1) (another characterization of zonotopes/zonoids). As we shall see this norm has a interpretation as distinguishability norms (Matthews–Wehner–Winter).

Support function

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

N X k∆kM = sup Tr(∆A) = | Tr ∆Mi |. A∈KM i=1

Guillaume Aubrun (Lyon) Zonoids Marne-la-Vall´ee,Juin 2015 6 / 16 As we shall see this norm has a interpretation as distinguishability norms (Matthews–Wehner–Winter).

Support function

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

N X k∆kM = sup Tr(∆A) = | Tr ∆Mi |. A∈KM i=1

d Note that the normed space (M+(C ), k · kM ) embeds into N N `1 = (R , k · k1) (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

N X k∆kM = sup Tr(∆A) = | Tr ∆Mi |. A∈KM i=1

d Note that the normed space (M+(C ), k · kM ) embeds into N N `1 = (R , k · k1) (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 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

N 1 X p = min(Tr ρM , Tr σM ) 2 i i i=1 N 1 1 X = − | Tr ρM − Tr σM | 2 4 i i i=1 1 1 = − kρ − σk 2 4 M

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 The best strategy is of course, given the outcome, to guess the most likely state. The probability of error is

N 1 X p = min(Tr ρM , Tr σM ) 2 i i i=1 N 1 1 X = − | Tr ρM − Tr σM | 2 4 i i i=1 1 1 = − kρ − σk 2 4 M

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

N 1 X p = min(Tr ρM , Tr σM ) 2 i i i=1 N 1 1 X = − | Tr ρM − Tr σM | 2 4 i i i=1 1 1 = − kρ − σk 2 4 M

Guillaume Aubrun (Lyon) Zonoids Marne-la-Vall´ee,Juin 2015 7 / 16 We would like sparsiﬁcations of Ud , i.e. POVMs M with as few outcomes as possible and such that

(1 − ε)k · kM 6 k · kUd 6 (1 + ε)k · kM .

The uniform POVM

Let Ud be the uniform POVM, deﬁned on (SCd , Borel) by Z Ud (A) = d |ψihψ| dσ(ψ). A

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 Z Ud (A) = d |ψihψ| dσ(ψ). A

We would like sparsiﬁcations of Ud , i.e. POVMs M with as few outcomes as possible and such that

(1 − ε)k · kM 6 k · kUd 6 (1 + ε)k · kM .

Guillaume Aubrun (Lyon) Zonoids Marne-la-Vall´ee,Juin 2015 8 / 16 An ε-approximate t-design is a ﬁnitely supported measure µ on SCd such that Z ⊗t (1 − ε)π 6 |ψihψ| dµ 6 (1 + ε)π. SCd

Example : ε = 0 gives an exact integration formula (cubature formula) for homogeneous polynomial of degree t.

t-designs

Start from the identity (t ∈ N) Z ⊗t 1 π := |ψihψ| dσ = P t d . dim Symt (Cd ) Sym (C ) SCd

Guillaume Aubrun (Lyon) Zonoids Marne-la-Vall´ee,Juin 2015 9 / 16 Example : ε = 0 gives an exact integration formula (cubature formula) for homogeneous polynomial of degree t.

t-designs

Start from the identity (t ∈ N) Z ⊗t 1 π := |ψihψ| dσ = P t d . dim Symt (Cd ) Sym (C ) SCd

An ε-approximate t-design is a ﬁnitely supported measure µ on SCd such that Z ⊗t (1 − ε)π 6 |ψihψ| dµ 6 (1 + ε)π. SCd

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

Start from the identity (t ∈ N) Z ⊗t 1 π := |ψihψ| dσ = P t d . dim Symt (Cd ) Sym (C ) SCd

An ε-approximate t-design is a ﬁnitely supported measure µ on SCd such that Z ⊗t (1 − ε)π 6 |ψihψ| dµ 6 (1 + ε)π. SCd

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 Idea: the 1-norm can be controlled from 2- and 4-norms

3 kX kL2 2 6 kX kL1 6 kX kL2 kX kL4

t d 4 This approach requires card supp(µ) > dim Sym (C ) = Ω(d ). n n2 n n2 Similar to√ Rudin (1960): `2 ⊂ `4 isometrically and therefore `2 ⊂ `1 with 2 distortion 3. Equivalently,√ gives a zonotope Z with n summands such n that Z ⊂ B2 ⊂ 3Z.

Sparsiﬁcation from 4-designs

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

ck · kM 6 k · kUd 6 k · kM .

Guillaume Aubrun (Lyon) Zonoids Marne-la-Vall´ee,Juin 2015 10 / 16 n n2 n n2 Similar to√ Rudin (1960): `2 ⊂ `4 isometrically and therefore `2 ⊂ `1 with 2 distortion 3. Equivalently,√ gives a zonotope Z with n summands such n that Z ⊂ B2 ⊂ 3Z.

Sparsiﬁcation from 4-designs

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

ck · kM 6 k · kUd 6 k · kM .

Idea: the 1-norm can be controlled from 2- and 4-norms

3 kX kL2 2 6 kX kL1 6 kX kL2 kX kL4

t d 4 This approach requires card supp(µ) > dim Sym (C ) = Ω(d ).

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

ck · kM 6 k · kUd 6 k · kM .

Idea: the 1-norm can be controlled from 2- and 4-norms

3 kX kL2 2 6 kX kL1 6 kX kL2 kX kL4

Guillaume Aubrun (Lyon) Zonoids Marne-la-Vall´ee,Juin 2015 10 / 16 n Figiel–Lindenstrauss–Milman (1977): given ε > 0, `2 embeds with N −2 distortion 1 + ε in `1 with N = Cε n. Equivalently, there is a zonoid Z with Cε−2n summands such that n Z ⊂ B2 ⊂ (1 + ε)Z. Proof: choose the directions of the N segments independently and uniformly at random.

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

Concentration of measure

Rudin’s result can be improved via random constructions based on the concentration of measure phenomenon. n Figiel–Lindenstrauss–Milman (1977): given ε > 0, `2 embeds with N −2 distortion 1 + ε in `1 with N = Cε n. Equivalently, there is a zonoid Z with Cε−2n summands such that n Z ⊂ B2 ⊂ (1 + ε)Z.

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

Guillaume Aubrun (Lyon) Zonoids Marne-la-Vall´ee,Juin 2015 11 / 16 The construction is random: take (ψi ) independent, uniform on the sphere SCd . Let N X S = |ψi ihψi |. i=1 The POVM is the family

−1/2 −1/2 |S ψi ihS ψi | . 16i6N

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 −2 2 such that N 6 Cε | log ε|d and

(1 − ε)k · kM 6 k · kUd 6 (1 + ε)k · kM .

2 d The size d = dim Msa(C ) 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 −2 2 such that N 6 Cε | log ε|d and

(1 − ε)k · kM 6 k · kUd 6 (1 + ε)k · kM .

2 d The size d = dim Msa(C ) is obviously optimal.

The construction is random: take (ψi ) independent, uniform on the sphere SCd . Let N X S = |ψi ihψi |. i=1 The POVM is the family

−1/2 −1/2 |S ψi ihS ψi | . 16i6N

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

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

What about derandomization ?

Guillaume Aubrun (Lyon) Zonoids Marne-la-Vall´ee,Juin 2015 13 / 16 Simple fact: if (1 − ε)k · kM 6 k · kM0 6 (1 + ε)k · kM and (1 − ε)k · kN 6 k · kN0 6 (1 + ε)k · kN , then

2 2 (1 − ε) k · kM⊗N 6 k · kM0⊗N0 6 (1 + ε) k · kM⊗N . It follows from Theorem 1 that there are optimal local sparsiﬁcations of the “local uniform POVM” LU = Ud ⊗ Ud .

Note that k · kLU is equivalent to the following norm (Lancien–Winter)

2 2 2 2 2 k∆k2(2) = (Tr ∆) + Tr2(Tr1 ∆) + Tr1(Tr2 ∆) + Tr(∆ ).

Tensor product of POVM

There is natural notion of tensor product for POVMs: given (discrete) d POVMs (Mi )i∈I and (Nj )j∈J on C , 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 Note that k · kLU is equivalent to the following norm (Lancien–Winter)

2 2 2 2 2 k∆k2(2) = (Tr ∆) + Tr2(Tr1 ∆) + Tr1(Tr2 ∆) + Tr(∆ ).

Tensor product of POVM

Simple fact: if (1 − ε)k · kM 6 k · kM0 6 (1 + ε)k · kM and (1 − ε)k · kN 6 k · kN0 6 (1 + ε)k · kN , then

2 2 (1 − ε) k · kM⊗N 6 k · kM0⊗N0 6 (1 + ε) k · kM⊗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

Simple fact: if (1 − ε)k · kM 6 k · kM0 6 (1 + ε)k · kM and (1 − ε)k · kN 6 k · kN0 6 (1 + ε)k · kN , then

2 2 (1 − ε) k · kM⊗N 6 k · kM0⊗N0 6 (1 + ε) k · kM⊗N . It follows from Theorem 1 that there are optimal local sparsiﬁcations of the “local uniform POVM” LU = Ud ⊗ Ud .

Note that k · kLU is equivalent to the following norm (Lancien–Winter)

2 2 2 2 2 k∆k2(2) = (Tr ∆) + Tr2(Tr1 ∆) + Tr1(Tr2 ∆) + Tr(∆ ).

Guillaume Aubrun (Lyon) Zonoids Marne-la-Vall´ee,Juin 2015 14 / 16 Important open question: can we remove the log n factor ?

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 −2 N 6 Cε n 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

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 M0 with −2 2 N 6 Cε d log d outcomes, in the sense

(1 − ε)k · kM0 6 k · kM 6 (1 + ε)k · kM0 . P A sub-POVM is a ﬁnite family (Mi ) of positive operators with Mi 6 Id.

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