Generalized measurements, quantum designs and ‘quantum simplices’ Anna Szymusiak Institute of Mathematics, Jagiellonian University, Kraków 05.11.2018 Anna Szymusiak Generalized measurements, quantum designs and ‘quantum simplices’ A generalized (discrete) quantum measurement: k POVM (positive operator valued measure) Π = fΠj gj=1, where k d X Πj 2 L(C ); Πj ≥ 0 and Πj = I: j=1 For pre-measurement state ρ, the probability of j-th outcome is tr(ρΠj ). How to quantify the indeterminacy of the measurement? The Shannon entropy of POVM Π is defined by: k X HΠ(ρ) := η(tr(ρΠj )); j=1 for an initial state ρ, where η(x) := −x ln x (x > 0), η(0) = 0. Anna Szymusiak Generalized measurements, quantum designs and ‘quantum simplices’ What are we looking for: min HΠ(τ) and max HΠ(τ): d d τ2P(C ) τ2P(C ) But that is the same as: min H(P) and max H(P); 1 1 P2∆Π P2∆Π 1 d where ∆Π := fP 2 ∆k : P = (tr(ρΠ1);:::; tr(ρΠk )) for some ρ 2 P(C )g. Let d 1 ∆Π := fP 2 ∆k : P = (tr(ρΠ1);:::; tr(ρΠk )) for some ρ 2 S(C )g = conv(∆Π): We call it the set of ‘allowed’ probabilities. 1 How do ∆Π and ∆Π look like? Anna Szymusiak Generalized measurements, quantum designs and ‘quantum simplices’ 1 How do ∆Π and ∆Π look like? 2 ∆Π ⊂ ∆k , dim ∆Π ≤ d − 1 with equality iff Π is informationally complete Simplest case: Π is a rank-one PVM (projective valued measure). Then 1 ∆Π = ∆Π = ∆k = ∆d : In general: joint algebraic numerical range of (Π1;:::; Πk ) 1 In dimension 2: ∆Π is one of the following: a sphere, an ellipsoid, a disc, an ellipse (with the interior), a segment or a point. d When ∆Π is ‘the same’ as S(C )? Anna Szymusiak Generalized measurements, quantum designs and ‘quantum simplices’ d For A; B 2 Ls(C ) we denote their Hilbert-Schmidt inner product by hhAjBii. d We say that ∆Π is isomorphic to quantum state space S(C ) if there exists α > 0 such that d hhρ − I=djσ − I=dii = αhp(ρ) − c; p(σ) − ci 8ρ, σ 2 S(C ); (1) where p(ρ) = (tr(ρΠ1);:::; tr(ρΠk )) and c = p(I=d) = (trΠ1=d;:::; trΠk =d). Theorem d k ∆Π is isomorphic to quantum state space S(C ) iff fΠi − (trΠi =d)Igi=1 is a 0 d 2 Pk 2 2 tight frame in Ls (C ). Then α = (d − 1)=( i=1(trΠi − (trΠi ) =d)). Anna Szymusiak Generalized measurements, quantum designs and ‘quantum simplices’ Frames, tight frames V – N-dimensional Hilbert space with inner product h·|−i F := ff1;:::; fmg ⊂ V F is a frame if there exist α; β > 0 such that m 2 X 2 2 αkvk ≤ jhvjfi ij ≤ βkvk for all v 2 V : i=1 F is called a tight frame if α = β. Pm An operator S := i=1 jfi ihfi j is called a frame operator. F is a tight frame iff S = αI. Anna Szymusiak Generalized measurements, quantum designs and ‘quantum simplices’ d 2 d Ls(C ) – d -dimensional real Hilbert space of selfadjoint operators on C with the Hilbert-Schmidt inner product hhAjBii = tr(AB) 0 d d 2 Ls (C ) := fA 2 Ls(C )jtrA = 0g –(d − 1)-dimensional subspace of traceless operators d d π0 : Ls(C ) 3 A 7! A − (trA=d)I 2 Ls(C ) 0 d π0 is the orthogonal projection onto Ls (C ) Theorem d k ∆Π is isomorphic to quantum state space S(C ) iff fΠi − (trΠi =d)Igi=1 is a 0 d 2 Pk 2 2 tight frame in Ls (C ). Then α = (d − 1)=( i=1(trΠi − (trΠi ) =d)). Theorem (reformulated) The following conditions are equivalent: d ∆Π is isomorphic to quantum state space S(C ) 0 d π0(Π) is a tight frame in Ls (C ) with frame constant 1/α Pk 1 i=1 jπo(Πi )iihhπ0(Πi )j = α I0. 2 Pk 2 2 Moreover, α = (d − 1)=( i=1(trΠi − (trΠi ) =d)). Anna Szymusiak Generalized measurements, quantum designs and ‘quantum simplices’ 0 d Some special cases of POVMs Π such that π0(Π) is a tight frame in Ls (C ): trΠi = d=k for i = 1;:::; k. Then k X 1 αd − k jΠ iihhΠ j = I + jIiihhIj i i α αdk i=1 In particular, Π is a tight IC-POVM [Scott ’06] trΠi = d=k and rankΠi = 1 for i = 1;:::; k. Then Π is a 2-design k POVM, i.e. pure states ρi := d Πi (i = 1;:::; k) are such that k 1 X ZZ f (tr(ρ ρ )) = f (tr(ρσ))dµ(ρ)dµ(σ) k 2 j l j;l=1 d d P(C )×P(C ) for every f : R ! R polynomial of degree t or less, where µ denotes the d unique unitarily invariant measure on P(C ). Examples: SIC-POVMs, complete MUBs Anna Szymusiak Generalized measurements, quantum designs and ‘quantum simplices’ k fΠj gj=1 is a 2-design POVM iff k X k τ = (d + 1) tr(τΠ )Π − I d j j j=1 for every τ such that τ ∗ = τ and trτ = 1. k For 2-design POVM Π, ρj = d Πj and pj (τ) = tr(τΠj ) we thus have k X pj (τ) = 1 (2) j=1 k X d(tr(τ 2) + 1) (p (τ))2 = (3) j k(d + 1) j=1 k X tr(τ + I)3 p (τ)p (τ)p (τ)tr(ρ ρ ρ ) = (4) j l m j l m (d + 1)3 j;l;m=1 Anna Szymusiak Generalized measurements, quantum designs and ‘quantum simplices’ Remark: A self-adjoint operator τ is a pure quantum state iff trτ = 1, trτ 2 = 1 and trτ 3 = 1. Theorem k Let Π be a 2-design POVM and let ρj = d Πj (j = 1;:::; k). Let k (p1;:::; pk ) 2 R be such that k Pk d pl = (d + 1) j=1 pj tr(ρj ρl ) − 1 for l = 1;:::; k, Pk 2 2d j=1 pj = k(d+1) , Pk (ρ ρ ρ ) = d+7 j;l;m=1 pj pl pmtr j l m (d+1)3 . d Pk Then (p1;:::; pk ) 2 ∆k and pl = k tr(ρl τ), where τ := (d + 1) j=1 pj ρj − I, d τ 2 P(C ). Anna Szymusiak Generalized measurements, quantum designs and ‘quantum simplices’ k (2) X 2d ∆ := fP 2 ∆ j p2 = g: k k j k(d + 1) j=1 This value is referred to as the index of coincidence and denoted IC(P). Then [Harremoes&Topsoe ’01]: 1 − p 1 − p arg max H(P) = s p; ;:::; js 2 Sk ; ( ) 2∆ 2 k − 1 k − 1 P k 1 q (d−1)(k−1) where p = k 1 + d+1 . and: arg min H(P) = fs(p; q;:::; q; 0;:::; 0)js 2 Sk g; ( ) 2∆ 2 P k | {z } | {z } m k−m−1 p p IC(P)− 1 1− rm m+ rm −1 m+1 where p = m+1 , q = m(m+1) , m = (IC(P)) and rm = 1 1 . m − m+1 Anna Szymusiak Generalized measurements, quantum designs and ‘quantum simplices’ (2) For a 2-design POVM Π we have ∆Π ⊂ ∆k and thus we get the following bounds on HΠ: min H(P) ≤ min H(P) = min HΠ(τ) ( ) d 2∆ 2 P2∆Π τ2P( ) P k C max H(P) ≥ max H(P) = max HΠ(τ) ( ) d 2∆ 2 P2∆Π τ2P( ) P k C For which 2-design POVMs these bounds are met? In other words: d for which 2 design POVM Π there exist τmin; τmax 2 P(C ) such that P(τmin) 2 arg min H(P) ( ) 2∆ 2 P k and P(τmax) 2 arg max H(P): ( ) 2∆ 2 P k Anna Szymusiak Generalized measurements, quantum designs and ‘quantum simplices’ Upper bound Theorem d Let Π be a 2-design POVM in C . Then max H(P) = max H(P) = max HΠ(τ) ( ) d 2∆ 2 P2∆Π τ2P( ) P k C if and only if Π is a SIC-POVM. Sketch of the proof. 1 d2 For a SIC-POVM Π = f d ρj gj=1 take τ = ρj . To see the converse it is enough to show that the equality holds iff k = d 2. Pk k Use the fact that j=2 ρj = d I − ρ1 and the condition k Pk d pl = (d + 1) j=1 pj tr(ρj ρl ) − 1 for l = 1;:::; k. Anna Szymusiak Generalized measurements, quantum designs and ‘quantum simplices’ Lower bound Known examples for which the lower bound is achieved: the unique (tetrahedral) SIC-POVM in dimension2 minimal configuration: ‘twin’ SIC the supersymmetric SIC-POVM in dimension3 (the Hesse configuration) minimal configuration: complete set of MUB a generic SIC-POVM in dimension3 minimal configuration: an orthonormal basis the complete set of MUB in dimension3 minimal configuration: the supersymmetric SIC the supersymmetric SIC-POVM in dimension8 (the Hoggar lines) minimal configuration: ‘twin’ SIC Known examples for which the lower bound is not achieved: SIC-POVM in dimension 4 complete sets of MUBs in even dimensions 5 configuration of 45 vectors in C (Example 18 in [Hoggar ’82]) Note that for all known examples with attained lower bound we have: k(d + 1) IC(P)−1 = 2 2d N Anna Szymusiak Generalized measurements, quantum designs and ‘quantum simplices’ −1 k(d+1) If m := (IC(P)) = 2d 2 N, then 1 1 arg min H(P) = fs( ;:::; ; 0;:::; 0)js 2 Sk g: ( ) 2∆ 2 m m P k | {z } | {z } k−m m d k d k(d+1) Let Π = f k ρj gj=1 be a 2-design POVM in C and let 2d 2 N.