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Product Weyl-Heisenberg covariant MUBs and Maximizers of Magick

Bogdan S. Damski, Rafał Bistroń, Diego Ponterio, Jakub Czartowski, Karol Życzkowski

Abstract

In this work we investigate discrete structures in product Hilbert spaces. For monopartite systems of size $d$ one relies on the Weyl-Heisenberg group $WH(d)$, while in the case of composite Hilbert spaces we identify designs covariant with respect to the product group, $[WH(p)]^{\otimes n}$. In analogy with magic -a quantity attaining its maximum for states fiducial with respect to $WH(d)$ -we introduce a similar notion of magick, defined with respect to the product group. The maximum of this quantity over all equimodular vectors yields fiducial states that generate $d$ $\textit{a priori}$ isoentangled mutually unbiased bases (MUBs), which, when supplemented by the identity, form their complete set. Such fiducial states are explicitly constructed in all prime-power dimensions $p^n$ with $p\ge 3$. The result for $p\ge 5$ extends the construction of Klappenecker and Rötteler, whereas for $p=3$ it is mathematically distinct and is based on Galois rings. The global maximum of magick for $d=2^3$ yields fiducial states corresponding to the symmetric informationally complete (SIC) generalized measurement of Hoggar. Our approach feeds into a unifying perspective in which highly symmetric quantum designs emerge from fiducial states with extremal properties via structured group-orbit constructions.

Product Weyl-Heisenberg covariant MUBs and Maximizers of Magick

Abstract

In this work we investigate discrete structures in product Hilbert spaces. For monopartite systems of size one relies on the Weyl-Heisenberg group , while in the case of composite Hilbert spaces we identify designs covariant with respect to the product group, . In analogy with magic -a quantity attaining its maximum for states fiducial with respect to -we introduce a similar notion of magick, defined with respect to the product group. The maximum of this quantity over all equimodular vectors yields fiducial states that generate isoentangled mutually unbiased bases (MUBs), which, when supplemented by the identity, form their complete set. Such fiducial states are explicitly constructed in all prime-power dimensions with . The result for extends the construction of Klappenecker and Rötteler, whereas for it is mathematically distinct and is based on Galois rings. The global maximum of magick for yields fiducial states corresponding to the symmetric informationally complete (SIC) generalized measurement of Hoggar. Our approach feeds into a unifying perspective in which highly symmetric quantum designs emerge from fiducial states with extremal properties via structured group-orbit constructions.
Paper Structure (18 sections, 12 theorems, 68 equations, 1 figure, 1 table)

This paper contains 18 sections, 12 theorems, 68 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Setting the scene
  3. Extrema of local magick
  4. Fiducial vectors for multipartite isoentangled MUBs
  5. $p\geq 5$ case
  6. $p = 3$ case
  7. $p=2$ case
  8. Concluding Remarks
  9. General features and maxima of magick
  10. Properties of magick
  11. Relation between distances and magick
  12. SIC
  13. MUBs
  14. Total fiducial states have more magick than the product of all local ones
  15. Analytical formulas for $d$ a priori isoentangled MUBs in dimension $d = p^n$
  16. ...and 3 more sections

Key Result

Lemma 1

Let $\rho \in \Omega_d$ be any quantum state on the product of local Hilbert spaces $\mathcal{H}_{d_i}$, with respect to which we calculate magick $M(\rho)$. Then the following statements are true:

Figures (1)

  • Figure 1: WH-covariant MUB and SIC in dimension $d = 2$. The MUB fiducial vector $|f_{\text{MUB}}\rangle$ is generated from the equatorial state $|+\rangle$ with a suitable diagonal $T$ gate (rotating the state about the $Z$ axis), and the remaining equimodular states of mutually unbiased basis are generated by $W_{01} \propto Z,\,W_{10}\propto X,\, W_{11}\propto Y$ (vertices of green square). In order to obtain a full set of MUBs (green spheres) one needs additionally the computational basis, which itself is WH-covariant since $\ket{0} = Z\ket{0}$ and $\ket{1} = X\ket{0} = XZ\ket{0}$. The SIC fiducial vector $\ket{f_{\text{SIC}}}$ and its images under the WH group (orange spheres) lie "above" the centres of octahedron faces. The stabilizer pure states are represented by vertices of an octahedron inscribed in the Bloch ball.

Theorems & Definitions (35)

  • Definition 1: MUB
  • Definition 2: SIC
  • Definition 3: Weyl-Heisenberg group
  • Definition 4: Stabilizer state
  • Definition 5: Clifford group
  • Definition 6: Magic
  • Definition 7: Magick
  • Lemma 1
  • Definition 8
  • Definition 9
  • ...and 25 more