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} ⊂ R n 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} ⊂ R n 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} ⊂ R n 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 B n 2 is a zonoid � B n 2 = α n S n − 1 [ − u , − u ] d σ ( 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 + ( C d ) such that M (Ω) = Id . Here M + ( C d ) 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 + ( C d ) such that M (Ω) = Id . Here M + ( C d ) 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 + ( C d ) such that M (Ω) = Id . Here M + ( C d ) 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 ( M 1 , . . . , M N ), where M i � 0 and M 1 + · · · + M N = Id . The range is �� � { M ( A ) ; A ∈ F} = M i : I ⊂ { 1 , . . . , N } . i ∈ I 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 N � conv { M ( A ) ; A ∈ F} = [0 , M i ] . i =1 It is more natural to consider the 0-symmetric version N � K M = 2 conv { M ( A ) ; A ∈ F} − Id = [ − M i , M i ] i =1 This is a zonotope inside K = { A ∈ M + ( C d ) : � 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 N � conv { M ( A ) ; A ∈ F} = [0 , M i ] . i =1 It is more natural to consider the 0-symmetric version N � K M = 2 conv { M ( A ) ; A ∈ F} − Id = [ − M i , M i ] i =1 This is a zonotope inside K = { A ∈ M + ( C d ) : � 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 K M is a norm N � � ∆ � M = sup Tr(∆ A ) = | Tr ∆ M i | . A ∈ K M i =1 Guillaume Aubrun (Lyon) Zonoids Marne-la-Vall´ ee, Juin 2015 6 / 16
Support function Given a POVM M , the support function of the zonoid K M is a norm N � � ∆ � M = sup Tr(∆ A ) = | Tr ∆ M i | . A ∈ K M i =1 Note that the normed space ( M + ( C d ) , � · � M ) embeds into ℓ N 1 = ( R N , � · � 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 K M is a norm N � � ∆ � M = sup Tr(∆ A ) = | Tr ∆ M i | . A ∈ K M i =1 Note that the normed space ( M + ( C d ) , � · � M ) embeds into ℓ N 1 = ( R N , � · � 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 C d . 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 C d . 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 ρ M i ; if σ was chosen, outcome i is output with probability Tr σ M i . Guillaume Aubrun (Lyon) Zonoids Marne-la-Vall´ ee, Juin 2015 7 / 16
State discrimination Let ρ, σ two quantum states on C d . 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 ρ M i ; if σ was chosen, outcome i is output with probability Tr σ M i . The best strategy is of course, given the outcome, to guess the most likely state. The probability of error is N 1 � p = min(Tr ρ M i , Tr σ M i ) 2 i =1 N 1 2 − 1 � = | Tr ρ M i − Tr σ M i | 4 i =1 1 2 − 1 = 4 � ρ − σ � M Guillaume Aubrun (Lyon) Zonoids Marne-la-Vall´ ee, Juin 2015 7 / 16
The uniform POVM Let U d be the uniform POVM, defined on ( S C d , Borel ) by � U d ( A ) = d | ψ �� ψ | d σ ( ψ ) . A Guillaume Aubrun (Lyon) Zonoids Marne-la-Vall´ ee, Juin 2015 8 / 16
The uniform POVM Let U d be the uniform POVM, defined on ( S C d , Borel ) by � U d ( A ) = d | ψ �� ψ | d σ ( ψ ) . A We would like sparsifications of U d , i.e. POVMs M with as few outcomes as possible and such that (1 − ε ) � · � M � � · � U d � (1 + ε ) � · � M . Guillaume Aubrun (Lyon) Zonoids Marne-la-Vall´ ee, Juin 2015 8 / 16
t -designs Start from the identity ( t ∈ N ) � 1 | ψ �� ψ | ⊗ t d σ = π := dim Sym t ( C d ) P Sym t ( C d ) . S C d Guillaume Aubrun (Lyon) Zonoids Marne-la-Vall´ ee, Juin 2015 9 / 16
t -designs Start from the identity ( t ∈ N ) � 1 | ψ �� ψ | ⊗ t d σ = π := dim Sym t ( C d ) P Sym t ( C d ) . S C d An ε -approximate t -design is a finitely supported measure µ on S C d such that � | ψ �� ψ | ⊗ t d µ � (1 + ε ) π. (1 − ε ) π � S C d Guillaume Aubrun (Lyon) Zonoids Marne-la-Vall´ ee, Juin 2015 9 / 16
t -designs Start from the identity ( t ∈ N ) � 1 | ψ �� ψ | ⊗ t d σ = π := dim Sym t ( C d ) P Sym t ( C d ) . S C d An ε -approximate t -design is a finitely supported measure µ on S C d such that � | ψ �� ψ | ⊗ t d µ � (1 + ε ) π. (1 − ε ) π � S C d 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 � � · � U d � � · � 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 � � · � U d � � · � M . Idea: the 1-norm can be controlled from 2- and 4-norms � X � 3 L 2 � � X � L 1 � � X � L 2 � X � 2 L 4 This approach requires card supp( µ ) � dim Sym t ( C d ) = Ω( d 4 ). 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 � � · � U d � � · � M . Idea: the 1-norm can be controlled from 2- and 4-norms � X � 3 L 2 � � X � L 1 � � X � L 2 � X � 2 L 4 This approach requires card supp( µ ) � dim Sym t ( C d ) = Ω( d 4 ). 2 ⊂ ℓ n 2 2 ⊂ ℓ n 2 Similar to Rudin (1960): ℓ n 4 isometrically and therefore ℓ n 1 with √ 3. Equivalently, gives a zonotope Z with n 2 summands such distortion √ that Z ⊂ B n 2 ⊂ 3 Z . 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
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