Generalized Measurements, Quantum Designs and `Quantum Simplices'

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 = {Πj }j=1, where

k d X Πj ∈ 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 τ∈P(C ) τ∈P(C )

But that is the same as:

min H(P) and max H(P), 1 1 P∈∆Π P∈∆Π 1 d where ∆Π := {P ∈ ∆k : P = (tr(ρΠ1),..., tr(ρΠk )) for some ρ ∈ P(C )}.

Let

d 1 ∆Π := {P ∈ ∆k : P = (tr(ρΠ1),..., tr(ρΠk )) for some ρ ∈ S(C )} = 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 ∈ Ls(C ) we denote their Hilbert-Schmidt inner product by hhA|Bii.

d We say that ∆Π is isomorphic to quantum state space S(C ) if there exists α > 0 such that

d hhρ − I/d|σ − I/dii = αhp(ρ) − c, p(σ) − ci ∀ρ, σ ∈ 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 {Πi − (trΠi /d)I}i=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 := {f1,..., fm} ⊂ V

F is a frame if there exist α, β > 0 such that

m 2 X 2 2 αkvk ≤ |hv|fi i| ≤ βkvk for all v ∈ V . i=1 F is called a tight frame if α = β.

Pm An operator S := i=1 |fi ihfi | 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 hhA|Bii = tr(AB)

0 d d 2 Ls (C ) := {A ∈ Ls(C )|trA = 0} –(d − 1)-dimensional subspace of traceless operators

d d π0 : Ls(C ) 3 A 7→ A − (trA/d)I ∈ 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 {Πi − (trΠi /d)I}i=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 |πo(Πi )iihhπ0(Πi )| = 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 |Π iihhΠ | = I + |IiihhI| 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 {Πj }j=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 ) ∈ 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 ) ∈ ∆k and pl = k tr(ρl τ), where τ := (d + 1) j=1 pj ρj − I, d τ ∈ P(C ).

Anna Szymusiak Generalized measurements, quantum designs and ‘quantum simplices’ k (2) X 2d ∆ := {P ∈ ∆ | p2 = }. 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, ,..., |s ∈ Sk , ( ) ∈∆ 2 k − 1 k − 1 P k 1 q (d−1)(k−1) where p = k 1 + d+1 . and:

arg min H(P) = {s(p, q,..., q, 0,..., 0)|s ∈ Sk }, ( ) ∈∆ 2 P k | {z } | {z } m k−m−1

√ √ 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 P∈∆Π τ∈P( ) P k C

max H(P) ≥ max H(P) = max HΠ(τ) ( ) d ∈∆ 2 P∈∆Π τ∈P( ) 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 ∈ P(C ) such that

P(τmin) ∈ arg min H(P) ( ) ∈∆ 2 P k and P(τmax) ∈ arg max H(P). ( ) ∈∆ 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 P∈∆Π τ∈P( ) P k C if and only if Π is a SIC-POVM.

Sketch of the proof.

1 d2 For a SIC-POVM Π = { d ρj }j=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 = ∈ 2d N

Anna Szymusiak Generalized measurements, quantum designs and ‘quantum simplices’ −1 k(d+1) If m := (IC(P)) = 2d ∈ N, then 1 1 arg min H(P) = {s( ,..., , 0,..., 0)|s ∈ Sk }. ( ) ∈∆ 2 m m P k | {z } | {z } k−m m

d k d k(d+1) Let Π = { k ρj }j=1 be a 2-design POVM in C and let 2d ∈ N. Then the following conditions are equivalent:

min H(P) = min H(P) = min HΠ(τ) ( ) d ∈∆ 2 P∈∆Π τ∈P( ) P k C k(d−1) 2d P There exists J ⊂ {1, 2,..., k} such that |J| = 2d and k ρj is an j∈J d orthogonal projection onto (d − 1)-dimensional supspace of C . d Let ρj = |ψj ihψj | for some ψj ∈ C . Then there exists J ⊂ {1, 2,..., k} k(d−1) such that |J| = 2d , dim(span{ψj }j∈J ) = d − 1 and {ψj }j∈J is a normalized tight frame in span{ψj }j∈J . k(d−1) P 1 There exists J ⊂ {1, 2,..., k} such that |J| = 2d and pj (τ) ≤ 2 for j∈J d all τ ∈ P(C ).

Anna Szymusiak Generalized measurements, quantum designs and ‘quantum simplices’ ‘Qplex’ – a quantum simplex

Quantum bayesianizm [Appleby, Fuchs, Stacey & Zhu, 2016] d2 ”Bureau of Standards" measurement: SIC-POVM E = {Ej }j=1 m an arbitrary POVM F = {Fj }j=1

let r(j|i) = tr(Ei Fj )/d d for τ ∈ S(C ) let pτ (j) = tr(τEj ) and qτ (j) = tr(τFj ). URGLEICHUNG:

d2 X qτ (j) = (αpτ (i) − β)r(j|i) i=1 with α = d + 1 and β = 1/d. 2 d QPLEX= {(pτ (1),..., pτ (d ))|τ ∈ S(C )} k BUT for any 2-design POVM E = {Ej }j=1 and r(j|i) = tr(Ei Fj )(d/k) we also have k X qτ (j) = (αpτ (i) − β)r(j|i) i=1 with α = d + 1 and β = d/k

Anna Szymusiak Generalized measurements, quantum designs and ‘quantum simplices’ Properties of qplex: contained in outer ball, contains inner ball the same as quantum state space Can geometrical properties and abstract deﬁnitions be transferred into general 2-design POVM case?

Anna Szymusiak Generalized measurements, quantum designs and ‘quantum simplices’ How general qplex can be located in the large simplex?

Can this better understanding of the structure of ∆Π lead to progress in the minimal entropy problem?

Anna Szymusiak Generalized measurements, quantum designs and ‘quantum simplices’ M. Appleby, C.A. Fuchs, B.C. Stacey & H. Zhu, Eur. Phys. J. D 71: 197 (2017) P. Harremoës & F. Topsøe, IEEE Trans. Inform. Theory 47, 2944–2960 (2001) S.G. Hoggar, European J. Combin. 3, 233–254 (1982)

A.J. Scott, J. Phys. A: Math. Gen. 39, 13507–13530 (2006)

Anna Szymusiak Generalized measurements, quantum designs and ‘quantum simplices’