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. � 9 � Possible solution: = 84 dinners. 3 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: CP d − 1 – rays in C d or, equivalently, P ( C d ) – rank-1 orthogonal projections. General quantum measurement – POVM Special case: A normalized rank-1 POVM Π = { Π j } k j = 1 is a set of k subnormalized rank-one projections Π j = ( d / k ) | ψ j �� ψ j | satisfying the identity decomposition: k d � | ψ j �� ψ j | = I . k j = 1 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 outcomes? 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 ( ρ Π j ) = ( d / k ) � ψ j | ρ | ψ j � . The Shannon entropy of measurement Π is defined as the Shannon entropy of the probability distribution of the measurement outcomes: k � H ( ρ, Π) := η ( tr ( ρ Π j )) , j = 1 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 := {| e j �� e j |} d j = 1 , where {| e j �} d j = 1 is an orthonormal basis in C d , then: ρ ∈P ( C d ) H ( ρ, Π) = ln d and the only minimizers are states | e j �� e j | , min ρ ∈P ( C d ) H ( ρ, Π) = 0 and the maximizers form a ( d − 1 ) -torus: max √ j = 1 e i θ j | e j � , where θ j ∈ R . d ) � d ( 1 / 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 measurements 3 , including the ’tetrahedral’ SIC-POVM 1 , 2 , and one infinite series 2 , 3 , all SIC-POVMs in dimension three 4 , the POVM consisting of complete set of MUBs in dimension three 5 the Hoggar SIC-POVM in dimension eight 6 The cases in which the maximizers of the entropy has been computed analytically so far: all highly symmetric POVMs in dimension two 7 , all SIC-POVMs in any dimension 7 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 d 2 subnormalized rank-one projections Π j = | φ j �� φ j | / d with equal pairwise Hilbert-Schmidt inner products: tr (Π i Π j ) = |� φ i | φ j �| 2 1 = d 2 ( d + 1 ) for i � = j d 2 . MUB, complete MUB Two orthonormal bases in C d , B 1 = {| e 1 � , . . . , | e d �} and B 2 = {| f 1 � , . . . , | f d �} , are said to be mutually unbiased if |� e i | f j �| 2 = 1 / d for i , j = 1 , . . . , d . The set {B 1 , . . . , B m } of orthonormal bases in C d 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 2 d ( d + 1 ) , . . . , d ( d + 1 ) , 0 , . . . , 0 with d ( d − 1 ) zeros. But it can be not achievable! 2 . Harremoës and F. Topsøe, IEEE Trans. Inform. Theory 47 , 2944 (2001) P case of complete 3-MUB by M. Dall’Arno Anna Szymusiak IM UJ Entropy, special geometric configurations and combinatorial designs
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