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 = fa + b : a 2 A; b 2 Bg: 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 fµ(A): A 2 Fg ⊂ 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 fµ(A): A 2 Fg ⊂ 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 = fa + b : a 2 A; b 2 Bg: 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 fµ(A): A 2 Fg ⊂ 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 = fa + b : a 2 A; b 2 Bg: 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 fM(A); A 2 Fg = Mi : I ⊂ f1;:::; Ng : i2I 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 fM(A); A 2 Fg = Mi : I ⊂ f1;:::; Ng : i2I 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 fM(A); A 2 Fg = Mi : I ⊂ f1;:::; Ng : i2I 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 convfM(A); A 2 Fg = [0; Mi ]: i=1 It is more natural to consider the 0-symmetric version N X KM = 2 convfM(A); A 2 Fg − Id = [−Mi ; Mi ] i=1 d This is a zonotope inside K = fA 2 M+(C ): kAk1 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 convfM(A); A 2 Fg = [0; Mi ]: i=1 It is more natural to consider the 0-symmetric version N X KM = 2 convfM(A); A 2 Fg − Id = [−Mi ; Mi ] i=1 d This is a zonotope inside K = fA 2 M+(C ): kAk1 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) = j Tr ∆Mi j: A2KM 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) = j Tr ∆Mi j: A2KM 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) = j Tr ∆Mi j: A2KM 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 = − j Tr ρM − Tr σM j 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 = − j Tr ρM − Tr σM j 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 = − j Tr ρM − Tr σM j 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 − ")k · kM 6 k · kUd 6 (1 + ")k · kM : The uniform POVM Let Ud be the uniform POVM, defined on (SCd ; Borel) by Z Ud (A) = d j ih j dσ( ): 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 j ih j dσ( ): A We would like sparsifications of Ud , i.e. POVMs M with as few outcomes as possible and such that (1 − ")k · kM 6 k · kUd 6 (1 + ")k · kM : Guillaume Aubrun (Lyon) Zonoids Marne-la-Vall´ee,Juin 2015 8 / 16 An "-approximate t-design is a finitely supported measure µ on SCd such that Z ⊗t (1 − ")π 6 j ih j dµ 6 (1 + ")π: SCd Example : " = 0 gives an exact integration formula (cubature formula) for homogeneous polynomial of degree t.