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Computing Numerical and Exact SIC-Povms Computing Numerical and Exact SIC-POVMs Chaos and Quantum Information Seminar Jagiellonian University Krak´ow Computing Numerical and Exact SIC-POVMs Markus Grassl International Centre for Theory of Quantum Technologies University Gdansk [email protected] sicpovm.markus-grassl.de 29 March 2021 in collaboration with Marcus Appleby, Ingemar Bengtsson, Michael Harrison, Gary McConnell, Andrew Scott additional support by the Max Planck Institute for the Science of Light, Erlangen, and MPG 29.03.2021 – 1– Markus Grassl Computing Numerical and Exact SIC-POVMs Positive Operator Valued Measures (POVMs) General measurement device (from a quantum information processing point of view) single-shot experiment ✲ “1” ✲ “2” one “click” ✲ ✲ ρin “3” per input ✲ “4” ✲ “5” 29.03.2021 – 2– Markus Grassl Computing Numerical and Exact SIC-POVMs Positive Operator Valued Measures (POVMs) General measurement device (from a quantum information processing point of view) single-shot experiment probabilities ✲ “1” p1 Tr(ρinΠ1) ✲ “2” p2 Tr(ρinΠ2) one “click” ρ ✲ ✲ in “3” p3 = Tr(ρinΠ3) 0 = Πi 0 per input ≥ ⇒ ≥ ✲ “4” p4 Tr(ρinΠ4) ✲ “5” p5 Tr(ρinΠ5) 29.03.2021 – 2– Markus Grassl Computing Numerical and Exact SIC-POVMs Positive Operator Valued Measures (POVMs) General measurement device (from a quantum information processing point of view) single-shot experiment probabilities ✲ “1” p1 = Tr(ρinΠ1) ✲ “2” p2 = Tr(ρinΠ2) one “click” ρ ✲ ✲ in “3” p3 = Tr(ρinΠ3) 0 = Πi 0 per input ≥ ⇒ ≥ ✲ “4” p4 = Tr(ρinΠ4) ✲ “5” p5 = Tr(ρinΠ5) 29.03.2021 – 2– Markus Grassl Computing Numerical and Exact SIC-POVMs Positive Operator Valued Measures (POVMs) General measurement device (from a quantum information processing point of view) single-shot experiment probabilities ✲ “1” p1 = Tr(ρinΠ1) 0 ≥ ✲ “2” p2 = Tr(ρinΠ2) 0 ≥ one “click” ρ ✲ ✲ in “3” p3 = Tr(ρinΠ3) 0 = Πi 0 per input ≥ ⇒ ≥ ✲ “4” p4 = Tr(ρinΠ4) 0 ≥ ✲ “5” p5 = Tr(ρinΠ5) 0 ≥ 29.03.2021 – 2– Markus Grassl Computing Numerical and Exact SIC-POVMs Positive Operator Valued Measures (POVMs) General measurement device (from a quantum information processing point of view) single-shot experiment probabilities ✲ “1” p1 = Tr(ρinΠ1) 0 ≥ ✲ “2” p2 = Tr(ρinΠ2) 0 ≥ one “click” ρ ✲ ✲ in “3” p3 = Tr(ρinΠ3) 0 = Πi 0 per input ≥ ⇒ ≥ ✲ “4” p4 = Tr(ρinΠ4) 0 ≥ ✲ “5” p5 = Tr(ρinΠ5) 0 ≥ 29.03.2021 – 2– Markus Grassl Computing Numerical and Exact SIC-POVMs Positive Operator Valued Measures (POVMs) General measurement device (from a quantum information processing point of view) single-shot experiment probabilities ✲ “1” p1 = Tr(ρinΠ1) 0 ≥ ✲ “2” p2 = Tr(ρinΠ2) 0 ≥ one “click” ρ ✲ ✲ in “3” p3 = Tr(ρinΠ3) 0 = Πi 0 per input ≥ ⇒ ≥ ✲ “4” p4 = Tr(ρinΠ4) 0 ≥ ✲ “5” p5 = Tr(ρinΠ5) 0 ≥ 1= pi = Tr(ρin Πi) P P 29.03.2021 – 2– Markus Grassl Computing Numerical and Exact SIC-POVMs Positive Operator Valued Measures (POVMs) General measurement device (from a quantum information processing point of view) single-shot experiment probabilities ✲ “1” p1 = Tr(ρinΠ1) 0 ≥ ✲ “2” p2 = Tr(ρinΠ2) 0 ≥ one “click” ρ ✲ ✲ in “3” p3 = Tr(ρinΠ3) 0 = Πi 0 per input ≥ ⇒ ≥ ✲ “4” p4 = Tr(ρinΠ4) 0 ≥ ✲ “5” p5 = Tr(ρinΠ5) 0 ≥ 1= pi = Tr(ρ Πi) = Πi = 1 in ⇒ P P P 29.03.2021 – 2– Markus Grassl Computing Numerical and Exact SIC-POVMs Positive Operator Valued Measures (POVMs) m • set of m positive semidefinite operators Πi with Πi = 1 iP=1 • implementation with the help of an m-dimensional auxiliary system and a projective (von Neumann) measurement on the auxiliary system † † ρ 0 0 U (ρ 0 0 ) U Aiρ A i i in ⊗| ih | −→ in ⊗| ih | −→ in i ⊗| ih | † with Ai Ai = Πi (not unique), depending on U Neumark extension • note: the number m of outcomes is independent of the dimension d of the Hilbert space of ρin 29.03.2021 – 3– Markus Grassl Computing Numerical and Exact SIC-POVMs (Minimal) Informationally-Complete POVMs 2 • POVM with m = d linearly independent operators Πi = unique representation of any state as ⇒ d2 ρ = cjΠj Xj=1 • measuring the POVM yields probabilities d2 pi = Tr(ρ Πi)= cj Tr(ΠiΠj ) Xj=1 = reconstructing ρ by inverting the matrix with elements Tr(ΠiΠj) ⇒ 29.03.2021 – 4– Markus Grassl Computing Numerical and Exact SIC-POVMs Symmetric Informationally-Complete POVMs • informationally complete POVM with rank-one elements Πi = α ψi ψi | ih | • symmetry: overlaps tr(ΠiΠj )= const = β for i = j 6 • 1 1 using Πi = 1 it follows that α = d and β = d2(d+1) P 2 1+ δijd = ψi ψj = ⇒ |h | i| 1+ d • reconstruction d2 ρ = d(d + 1) Tr(ρ Πj) 1 Πj − Xj=1 =cj | {z } = coefficient cj is an affine function of the probability pj = tr(ρ Πj) ⇒ 29.03.2021 – 5– Markus Grassl Computing Numerical and Exact SIC-POVMs A Simple to State Problem 2 d Are there d vectors v , v ,..., v 2 C in the complex vector space of 1 2 d ∈ dimension d such that: 2 (i) vj vj = 1 for j = 1,...,d h | i 2 1 2 (ii) vj vk = for 1 j<k d |h | i| d + 1 ≤ ≤ The vectors vj form an equiangular tight frame/finite unit norm tight frame. 29.03.2021 – 6– Markus Grassl Computing Numerical and Exact SIC-POVMs A Simple to State Problem 2 d Are there d vectors v , v ,..., v 2 C in the complex vector space of 1 2 d ∈ dimension d such that: 2 (i) vj vj = 1 for j = 1,...,d h | i 2 1 2 (ii) vj vk = for 1 j<k d |h | i| d + 1 ≤ ≤ The vectors vj form an equiangular tight frame/finite unit norm tight frame. All solutions form a real algebraic variety, using 2d real variables per vector T vj =(aj + ibj ,aj + ibj ,...,aj,d + ibj,d) (i = √ 1) 1 1 2 2 − 2 3 2 d 2d variables, d equations (i) of degree 2 and 2 equations (ii) of degree 4. 29.03.2021 – 6– Markus Grassl Computing Numerical and Exact SIC-POVMs Weyl-Heisenberg Group • generators: Hd := X,Z h i d−1 d−1 where X := j + 1 j and Z := ωj j j | ih | d| ih | jP=0 jP=0 (ωd := exp(2πi/d)) • relations: ′ ′ c a b c′ a′ b′ a b−b a c′ a′ b′ c a b ωdX Z ωd X Z = ωd ωd X Z ωdX Z • basis: a b Hd ζ(Hd)= X Z : a,b 0,...,d 1 = Zd Zd ∈ { − } ∼ × trace-orthogonal basis of all d d matrices × 29.03.2021 – 7– Markus Grassl Computing Numerical and Exact SIC-POVMs Constructing SIC-POVMs Ansatz: SIC-POVM that is the orbit under the Weyl-Heisenberg group Hd, i.e., v(a,b) := XaZb v(0,0) | i | i ′ ′ ′ ′ 1 for (a,b)=(a ,b ), v(a,b) v(a ,b ) 2 = |h | i| 1/(d + 1) for (a,b) =(a′,b′) 6 d−1 (0,0) v = (x j + ix j ) j , | i 2 2 +1 | i Xj=0 (x0,...,x2d−1 are real variables, x1 = 0) = we have to find only one fiducial vector v(0,0) instead of d2 vectors ⇒ | i = polynomial equations with 2d 1 variables, but already quite complicated ⇒ − for d = 6 29.03.2021 – 8– Markus Grassl Computing Numerical and Exact SIC-POVMs Jacobi Group (or Clifford Group) • automorphism group of the Weyl-Heisenberg group Hd, i.e. † T Jd : T HdT = Hd ∀ ∈ • the action of Jd on Hd modulo phases corresponds to the symplectic group SL(2, Zd), i.e. ′ ′ ′ a a † a b c a b ˜ ˜ T X Z T = ωdX Z where = T , T SL(2, Zd) b′ b ∈ = homomorphism Jd SL(2, Zd) ⇒ → • additionally: complex conjugation (anti-unitary) 1 0 XaZb XaZ−b corresponding to 7→ 0 1 − 29.03.2021 – 9– Markus Grassl Computing Numerical and Exact SIC-POVMs Zauner’s Conjecture [G. Zauner, Dissertation, Universit¨at Wien, 1999] Conjecture: For every dimension d 2 there exists a SIC-POVM whose elements are the ≥ orbit of a rank-one operator E0 under the Weyl-Heisenberg group Hd. What is more, E0 commutes with an element S of the Jacobi group Jd. The action of S on Hd modulo the center has order three. support for this conjecture (to date): • numerical solutions for all dimensions d 193, plus a few more ≤ • exact algebraic solutions for some dimensions (see below) one of the prize problems in [Pawe l Horodecki,Lukasz Rudnicki, Karol Zyczkowski,˙ Five open problems in quantum information, arXiv:2002.03233] 29.03.2021 – 10– Markus Grassl Computing Numerical and Exact SIC-POVMs Numerical Search for SIC-POVMs • for any state ψ Cd | i∈ d d 2 f( ψ )= ψ j + ℓ ℓ ψ ψ k + ℓ j + k + ℓ ψ | i h | ih | ih | ih | i j,kX=1 Xℓ=1 d d 2 2 = ψj+ℓ ψℓ ψk+ℓ ψj+k+ℓ ≥ d + 1 j,kX=1 Xℓ=1 with equality iff ψ is a fiducial vector for a Weyl-Heisenberg SIC-POVM | i • gradient descent to minimize f( ψ ), subject to unit norm | i 29.03.2021 – 11– Markus Grassl Computing Numerical and Exact SIC-POVMs Numerical Search for SIC-POVMs • for any state ψ Cd | i∈ d d 2 2 f( ψ )= ψj+ℓ ψℓ ψk+ℓ ψj+k+ℓ | i ≥ d + 1 j,kX=1 Xℓ=1 with equality iff ψ is a fiducial vector for a Weyl-Heisenberg SIC-POVM | i • gradient descent to minimize f( ψ ), subject to unit norm | i P ~x • use F (~x)= f for an arbitrary vector ~x Cd, P ~x ∈ where P is the projectionk k onto a subspace (prescribed symmetry) • chain rule yields a relatively simple formula for the gradient of F (~x) in terms of the gradient of f • complexity O(d3) for both the function and the gradient when storing O(d2) intermediate values 29.03.2021 – 11– Markus Grassl Computing Numerical and Exact SIC-POVMs Numerical Search for SIC-POVMs • efficient implementation of F (~x) and its gradient in C++ by Andrew Scott • parallel computation of the function/gradient using OpenMP/CUDA • minimization using limited-memory Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm • search runs into local minima, we need many random initial points • running many instances on HPC clusters by MPG and GWDG • – for d = 189: approx.
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