Product Weyl-Heisenberg covariant MUBs and Maximizers of Magick

1. Problem Statement

The paper addresses the construction of Mutually Unbiased Bases (MUBs) and Symmetric Informationally Complete (SIC) measurements in composite (multipartite) quantum systems.

• Context: While full sets of $d+1$ MUBs are known to exist for prime power dimensions ($d=p^n$) using global Weyl-Heisenberg (WH) group actions, their existence in composite dimensions (e.g., $d=6$) remains an open problem.
• Gap: Previous work established that fiducial states for MUBs and SICs in single-partite systems maximize a quantity called "magic" (a measure of non-stabilizerness) under the global WH group. However, this framework had not been extended to multipartite systems where the symmetry is defined by the product of local WH groups ($[WH(p)]^{\otimes n}$).
• Goal: The authors aim to:
  1. Define a "product magic" (or magick) for composite systems.
  2. Determine if fiducial states generating product-WH-covariant MUBs and SICs maximize this magick.
  3. Explicitly construct such fiducial states for prime-power dimensions $d=p^n$ (specifically $p \ge 3$).
  4. Investigate the entanglement properties of the resulting bases, specifically seeking isoentangled structures (where all states in the set share the same entanglement).

2. Methodology

The authors employ a combination of algebraic number theory, group representation theory, and quantum information theory.

• Definition of Magick: They define magick $M(\rho)$ for a composite system as the sum of absolute values of the traces of the state $\rho$ against the product WH group elements:
  $$M(\rho) = \sum_{k,l} |\text{Tr}[W_{kl} \rho]|$$
  where $W_{kl}$ are tensor products of local Weyl-Heisenberg operators.
• Extremization: They prove that maximizing magick over all pure states yields SIC fiducial vectors, while maximizing it over equimodular vectors (states with equal magnitude components) yields MUB fiducial vectors.
• Algebraic Construction:
  • For $p \ge 5$: They utilize Galois fields ($\mathbb{F}_{p^n}$) and introduce a new parameter $a \in \mathbb{F}_{p^n}$ to generalize the construction of Klappenecker and Rötteler. The fiducial vector is constructed using a trace function over the field: $|f\rangle = \sum \omega^{\text{tr}(a j^3)} |j\rangle$.
  • For $p = 3$: Standard constructions based on Galois fields fail for characteristic 3 due to the vanishing of quadratic terms in cubic expansions (related to the non-existence of Alltop sequences). To circumvent this, the authors utilize Galois rings ($GR(9, n)$) and their residue fields. They construct fiducial vectors using the trace over the Galois ring with the 9th root of unity ($\omega_9$).
  • For $p = 2$ (Qubits): They provide explicit examples for $d=2$ and $d=4$ but conjecture that no such product-WH-covariant fiducial MUB vectors exist for $d=2^n$ with $n > 2$.
• Entanglement Analysis: They analyze the reduced density matrices of the constructed fiducial states to demonstrate that the resulting MUBs are a priori isoentangled (generated by local operations on a single state, ensuring equal entanglement by construction).

3. Key Contributions

A. Theoretical Framework: Magick and Extremization

• Lemma 1-7: The authors rigorously prove that the magick measure is convex, invariant under local Clifford operations, and bounded.
• Main Theorem: They establish that:
  • The global maximum of magick is achieved by SIC fiducial states.
  • The maximum of magick over equimodular states is achieved by MUB fiducial states.
  • This extends the "magic maximization" principle from global WH groups to product WH groups.

B. Explicit Constructions

• Prime Power $p \ge 5$: They present a family of fiducial vectors parameterized by $a \in \mathbb{F}_{p^n}$:
  $$|f_{MUB, p^n}\rangle = \sum_{j=0}^{p^n-1} \omega_p^{\text{tr}_{\mathbb{F}}(a j^3)} |j\rangle$$
  This generalizes previous solutions and generates a family of inequivalent constructions depending on the properties of the map $a \mapsto a^3$.
• Prime Power $p = 3$: They introduce a novel construction based on Galois Rings $GR(9, n)$:
  $$|f_{MUB, 3^n}\rangle = \sum_{j=0}^{3^n-1} \omega_9^{\text{tr}_{GR}(a j^3)} |j\rangle$$
  This successfully bypasses the known non-existence of Alltop sequences for characteristic 3, providing the first explicit product-WH-covariant MUB construction for multi-qutrit systems.

C. Entanglement and Isoentangled Structures

• The constructed MUBs are a priori isoentangled. Unlike "a posteriori" isoentangled sets (where entanglement is found to be equal after construction), these are generated by local operations on a single fiducial state, guaranteeing the property.
• They show that the average purity of the reduced states in these sets matches the theoretical average for 2-designs.

D. Sporadic Solutions and Conjectures

• $d=9$ (Two Qutrits): They found a sporadic set of MUBs generated by three fiducial states under a subgroup of the product WH group, rather than a single fiducial state. This suggests a broader class of "fiducial structures" beyond single-vector generation.
• Conjecture: For $d=2^n$ with $n > 2$, no product-WH-covariant MUB fiducial vectors exist (analogous to the non-existence of product-WH-covariant SICs for $n>1, n \neq 3$).

4. Results

• Maximization: Confirmed that MUB and SIC fiducial states are global maximizers of magick (and product magick) in their respective domains.
• New Families: Successfully constructed infinite families of product-WH-covariant MUBs for all $d=p^n$ where $p \ge 3$.
• Galois Ring Breakthrough: The $p=3$ construction is a significant mathematical novelty, resolving a barrier that prevented standard Galois field methods from working in characteristic 3.
• Butson Class: The constructed Hadamard matrices belong to specific Butson classes ($BH(d, p)$ for $p \ge 5$ and $BH(d, 9)$ for $p=3$), linking quantum designs to combinatorial matrix theory.
• Hoggar Lines: The paper notes that for $d=8$ ($2^3$), the global maximum of magick corresponds to the Hoggar lines (a known SIC), reinforcing the connection between extremal magick and symmetric structures.

5. Significance

• Unifying Perspective: The work provides a unified framework where highly symmetric quantum designs (MUBs, SICs) emerge from fiducial states with extremal properties (maximizing magick) via structured group orbits.
• Quantum Information Applications:
  • State Tomography & Cryptography: MUBs are essential for quantum state tomography and QKD. Providing explicit constructions for composite systems (especially $p=3$) expands the toolkit for high-dimensional quantum protocols.
  • Entanglement Detection: The isoentangled nature of these bases offers new resources for detecting and characterizing entanglement in multipartite systems.
  • Magic State Distillation: By linking MUBs to the maximization of "magick," the paper strengthens the theoretical link between quantum computational advantage (magic states) and geometric structures in Hilbert space.
• Mathematical Impact: The use of Galois rings to solve problems in quantum design theory opens new avenues for applying algebraic number theory to quantum information, particularly in overcoming limitations of finite fields in specific characteristics.

In summary, this paper successfully extends the theory of Weyl-Heisenberg covariant structures to multipartite systems, introduces a new measure of non-stabilizerness ("magick") tailored for product groups, and provides explicit, novel constructions for MUBs in prime-power dimensions, overcoming previous algebraic obstructions.