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Entropy, special geometric configurations in the quantum state space and combinatorial designs

Anna Szymusiak in collaboration with Wojciech Słomczy´ nski

Institute of Mathematics, Jagiellonian University, Kraków, Poland

49 Symposium on Mathematical Physics Toru´ n, June 17-18, 2017

Anna Szymusiak IM UJ Entropy, special geometric configurations and combinatorial designs

• DR. HESSE INVITES FOR A DINNER

The story idea: "d=3 SIC POVMs and Elliptic Curves" by Lane Hughston, lecture available online on PIRSA

Anna Szymusiak IM UJ Entropy, special geometric configurations and combinatorial designs

• Dr. Hesse wants to invite his 9 friends for a dinner.

Unfortunately, he has just 4 chairs so he can invite just 3 guests at once. All his friends like each other so Dr. Hesse would like every two of them to meet in his house exactly the same number of times. To be fair, he would like to host each friend the same number of times. Possible solution: 9

3

• = 84 dinners.

That’s quite a lot! Is the minimal solution achievable?

Anna Szymusiak IM UJ Entropy, special geometric configurations and combinatorial designs

The minimal solution:

• Dr. Hesse wants to invite his 9 friends for a dinner.

Unfortunately, he has just 4 chairs so he can invite just 3 guests at once. All his friends like each other so Dr. Hesse would like every two of them to meet in his house exactly the same number of times once. To be fair, he would like to host each friend the same number of 4 times.

Anna Szymusiak IM UJ Entropy, special geometric configurations and combinatorial designs

Anna Szymusiak IM UJ Entropy, special geometric configurations and combinatorial designs

Anna Szymusiak IM UJ Entropy, special geometric configurations and combinatorial designs

Anna Szymusiak IM UJ Entropy, special geometric configurations and combinatorial designs

Anna Szymusiak IM UJ Entropy, special geometric configurations and combinatorial designs

Anna Szymusiak IM UJ Entropy, special geometric configurations and combinatorial designs

Anna Szymusiak IM UJ Entropy, special geometric configurations and combinatorial designs

Anna Szymusiak IM UJ Entropy, special geometric configurations and combinatorial designs

Anna Szymusiak IM UJ Entropy, special geometric configurations and combinatorial designs

9 guests 12 dinners each guest comes 4 times 3 guests at each dinner every two guests meet once v = 9 points arranged in b = 12 blocks any point is contained in r = 4 blocks each block consists of k = 3 elements any pair of distinct points is contained in λ = 1 block (v, b, r, k, λ)-design

Anna Szymusiak IM UJ Entropy, special geometric configurations and combinatorial designs

QUANTUM STATES? ENTROPY?

Anna Szymusiak IM UJ Entropy, special geometric configurations and combinatorial designs

Pure quantum states: CPd−1 – rays in Cd

• r, equivalently,

P(Cd) – rank-1 orthogonal projections. General quantum measurement – POVM Special case: A normalized rank-1 POVM Π = {Πj}k

j=1 is a set of k subnormalized

rank-one projections Πj = (d/k)|ψjψj| satisfying the identity decomposition: d k

k

• j=1

|ψjψj| = I. One can identify such POVMs with configurations of quantum states.

Anna Szymusiak IM UJ Entropy, special geometric configurations and combinatorial designs

Anna Szymusiak IM UJ Entropy, special geometric configurations and combinatorial designs

How to quantify the indeterminacy of the measurement

• utcomes?

Anna Szymusiak IM UJ Entropy, special geometric configurations and combinatorial designs

MEASURE OF RANDOMNESS – THE SHANNON ENTROPY

The state before measurement: ρ. The probability of obtaining j-th outcome is given by tr(ρΠj) = (d/k)ψj|ρ|ψj. The Shannon entropy of measurement Π is defined as the Shannon entropy of the probability distribution of the measurement outcomes: H(ρ, Π) :=

k

• j=1

η(tr(ρΠj)), for an initial state ρ, where η(x) := −x ln x (x > 0), η(0) = 0. H(·, Π) attains its minima in pure states. H(ρ, Π) is maximal for maximally mixed state ρ∗ := 1

d I.

Our aim: to find minimizers and maximizers of the entropy among pure states.

Anna Szymusiak IM UJ Entropy, special geometric configurations and combinatorial designs

Related problems, studied already: informational power of POVM – under some additional assumptions (e.g. group covariance) equivalent to the problem of minimizing entropy

Oreshkov, O, Calsamiglia, J., Muñoz-Tapia, R., Bagan, E., New J. Phys. 13, 073032 (2011) Dall’Arno, M., D’Ariano, G.M., Sacchi, M.F ., Phys. Rev. A 83, 062304 (2011)

entropic (un)certainty relations

Wehner, S., Winter, A., New J. Phys. 12, 025009 (2010)

the Wehrl entropy minimization

Lieb, E.H., Solovej, J.P ., Acta Math. 212, 379–398 (2014) Anna Szymusiak IM UJ Entropy, special geometric configurations and combinatorial designs

What are we looking for?

Anna Szymusiak IM UJ Entropy, special geometric configurations and combinatorial designs

PVM case If Π is a projective valued measure (PVM),i.e. Π := {|ejej|}d

j=1, where

{|ej}d

j=1 is an orthonormal basis in Cd, then:

min

ρ∈P(Cd ) H(ρ, Π) = ln d and the only minimizers are states |ejej|,

max

ρ∈P(Cd ) H(ρ, Π) = 0 and the maximizers form a (d − 1)-torus:

(1/ √ d) d

j=1 eiθj |ej, where θj ∈ R.

Not an easy task in general case!

Anna Szymusiak IM UJ Entropy, special geometric configurations and combinatorial designs

The cases in which the minimizers of the entropy has been computed analytically so far: all highly symmetric POVMs in dimension two: seven sporadic measurements3, including the ’tetrahedral’ SIC-POVM1,2, and one infinite series2,3, all SIC-POVMs in dimension three4, the POVM consisting of complete set of MUBs in dimension three5 the Hoggar SIC-POVM in dimension eight6 The cases in which the maximizers of the entropy has been computed analytically so far: all highly symmetric POVMs in dimension two7, all SIC-POVMs in any dimension7

• 1M. Dall’Arno, G.M. D’Ariano, and M.F

. Sacchi, Phys. Rev. A 83, 062304 (2011)

• 2O. Oreshkov, J. Calsamiglia, R. Muñoz Tapia, and E. Bagan, New J. Phys. 13, 073032 (2011)
• 3W. Słomczy´

nski and AS, Quantum Inf. Process. 15, 565-606 (2016) 4AS, J. Phys. A 47, 445301 (2014)

• 5M. Dall’Arno, Phys. Rev. A 90, 052311 (2014)

6AS and W. Słomczy´ nski, Phys. Rev. A 94, 012122 (2016) 7AS, arXiv:1701.01139 [quant-ph] Anna Szymusiak IM UJ Entropy, special geometric configurations and combinatorial designs

SIC-POVM A symmetric informationally complete POVM (SIC-POVM) consists of d2 subnormalized rank-one projections Πj = |φjφj|/d with equal pairwise Hilbert-Schmidt inner products: tr(ΠiΠj) = |φi|φj|2 d2 = 1 d2(d + 1) for i = j . MUB, complete MUB Two orthonormal bases in Cd, B1 = {|e1, . . . , |ed} and B2 = {|f1, . . . , |fd}, are said to be mutually unbiased if |ei|fj|2 = 1/d for i, j = 1, . . . , d. The set {B1, . . . , Bm} of orthonormal bases in Cd is called a set of mutually unbiased bases (MUB) if every two bases from this set are mutually unbiased. A complete MUB consists of (d + 1) bases.

Anna Szymusiak IM UJ Entropy, special geometric configurations and combinatorial designs

POVM MIN MAX 2-SIC twin 2-SIC 2-SIC Hesse 3-SIC complete 3-MUB Hesse 3-SIC generic 3-SIC 3-ONB generic 3-SIC Hoggar 8-SIC twin Hoggar 8-SIC Hoggar 8-SIC d-SIC ? d-SIC complete 2-MUB complete 2-MUB two twin 2-SICs complete 3-MUB Hesse 3-SIC ? complete d-MUB (d>3) ? complete d-MUB ???

Table: Extremal configurations for some SIC-POVMs

Note that: d = 3 – infinite family of nonequivalent SIC-POVMs the Hoggar SIC-POVM (d = 8) is the only known SIC-POVM that is not group-covariant with respect to the finite Weyl-Heisenberg group there are exactly 3 supersymmetric SIC-POVMs (any two rays can be transformed into any two rays via symmetry group’s action): d = 2, d = 3 (known as the Hesse configuration), d = 8 (the Hoggar lines); Zhu (2015)

Anna Szymusiak IM UJ Entropy, special geometric configurations and combinatorial designs

POVM MIN MAX 2-SIC twin 2-SIC 2-SIC Hesse 3-SIC complete 3-MUB Hesse 3-SIC generic 3-SIC 3-ONB generic 3-SIC Hoggar 8-SIC twin Hoggar 8-SIC Hoggar 8-SIC d-SIC ? d-SIC complete 2-MUB complete 2-MUB two twin 2-SICs complete 3-MUB Hesse 3-SIC ? complete d-MUB (d>3) ? complete d-MUB ???

Table: Extremal configurations for some SIC-POVMs

Proof methods Hermite interpolation method alternative way: results by Harremoës and Topsøe: The optimal probability distribution for SIC-POVM should be of the form:

• 2

d(d + 1), . . . , 2 d(d + 1), 0, . . . , 0

• with d(d−1)

2

• zeros. But it can be not achievable!

P . Harremoës and F. Topsøe, IEEE Trans. Inform. Theory 47, 2944 (2001) case of complete 3-MUB by M. Dall’Arno Anna Szymusiak IM UJ Entropy, special geometric configurations and combinatorial designs

POVM MIN MAX 2-SIC twin 2-SIC 2-SIC Hesse 3-SIC complete 3-MUB Hesse 3-SIC generic 3-SIC 3-ONB generic 3-SIC Hoggar 8-SIC twin Hoggar 8-SIC Hoggar 8-SIC d-SIC ? d-SIC complete 2-MUB complete 2-MUB two twin 2-SICs complete 3-MUB Hesse 3-SIC ? complete d-MUB (d>3) ? complete d-MUB ???

Table: Extremal configurations for some SIC-POVMs

Anna Szymusiak IM UJ Entropy, special geometric configurations and combinatorial designs

Hesse 3-SIC: (0, 1, −1) (0, 1, −ω) (0, 1, −ω2) (1, 0, −1) (1, 0, −ω) (1, 0, −ω2) (1, −1, 0) (1, −ω, 0) (1, −ω2, 0) This is known as Hesse configuration: 9 points in CP2 lying on 12 projective lines in such a way that 3 points lie on each line and 4 lines intersect in each point (Hesse’s discovery: from cubic curves in complex projective space) Every triple of collinear points defines a unique state orthogonal to

• them. This states are the minimal configuration for Hesse SIC.

Moreover they form a complete set of mutually unbiased bases.

Anna Szymusiak IM UJ Entropy, special geometric configurations and combinatorial designs

Coming back to Dr. Hesse’s dinner...

Anna Szymusiak IM UJ Entropy, special geometric configurations and combinatorial designs

9 guests 12 dinners each guest comes 4 times 3 guests at each dinner every two guests meet once guests = states 3 states "come to the same dinner" iff one of them is a superposition of two others

Anna Szymusiak IM UJ Entropy, special geometric configurations and combinatorial designs

But the minimal configuration for complete 3-MUB is the Hesse SIC...

Anna Szymusiak IM UJ Entropy, special geometric configurations and combinatorial designs

12 dinners (points, states) 9 guests (blocks) each guest comes 4 times (blocks consist of 4 points) 3 guests at each dinner (every point in 3 blocks) no one comes for the same dinner (same menu) twice (points of the same type do not appear in one block) if there are two dinners with different menu there is 1 guest invited for both (points of different type appear in one block) partially balanced incomplete block design with 2 associate classes

Anna Szymusiak IM UJ Entropy, special geometric configurations and combinatorial designs

When do we obtain such combinatorial designs? NOT ALWAYS! Example: generic SIC-POVM in dimension 3 Sufficient (necessary?) condition: Set of states S constituting the POVM forms a 2-point homogeneous space. Then relations Rc := {(ψ, φ) ∈ S2 : |ψ|φ|2 = c} define a commutative association scheme. For fixed p > 0 and v ∈ CPd−1 sets Bg := {ψ ∈ S : |gv|ψ|2 = p}, g ∈ Sym(S), define a partially balanced incomplete block design. Examples: "tetrahedral" SIC-POVM Hoggar SIC-POVM complete 2-MUB (octaheral POVM)

Anna Szymusiak IM UJ Entropy, special geometric configurations and combinatorial designs

CONCLUSION POVM MIN MAX 2-SIC twin 2-SIC 2-SIC Hesse 3-SIC complete 3-MUB Hesse 3-SIC generic 3-SIC 3-ONB generic 3-SIC Hoggar 8-SIC twin Hoggar 8-SIC Hoggar 8-SIC d-SIC ? d-SIC complete 2-MUB complete 2-MUB two twin 2-SICs complete 3-MUB Hesse 3-SIC ? complete d-MUB (d>3) ? complete d-MUB ???

Anna Szymusiak IM UJ Entropy, special geometric configurations and combinatorial designs

THANK YOU

Anna Szymusiak IM UJ Entropy, special geometric configurations and combinatorial designs