[Paper Review] MUBs and SIC-POVMs of a spin-1 system from the Majorana approach

P. K. Aravind|arXiv (Cornell University)|Jul 9, 2017

Quantum Information and Cryptography5 references3 citations

TL;DR

This paper uses the Majorana (stellar) representation to geometrically derive mutually unbiased bases (MUBs) and symmetric informationally complete positive operator-valued measures (SIC-POVMs) for a spin-1 system. By expressing the overlap of two spin-1 states in terms of their Majorana vectors, the study constructs MUBs and SICs as symmetric configurations of vectors in 3D space, offering a novel geometric perspective on these quantum structures, though the method does not extend to higher spin systems.

ABSTRACT

In the Majorana or stellar representation of quantum states, an arbitrary (pure) state of a spin-1 system is represented by a pair of points on the unit sphere or, equivalently, by a pair of unit vectors. This paper presents an expression for the squared modulus of the inner product of two spin-1 states in terms of their Majorana vectors and uses it to give a geometrical construction of the MUBs and SIC-POVMs of a spin-1 system. The results are not new and duplicate those obtained earlier by other methods, but the Majorana approach nevertheless illuminates them from an unusual point of view. In particular, it reveals the MUBs and SICs as symmetrical collections of vectors in ordinary three-dimensional space, rather than as rays in a projective Hilbert space. While it does not appear feasible to extend this treatment to higher spin systems, the spin-1 case exhibits sufficient subtlety and complexity to be worth spelling out for its pedagogical and historical interest.

Motivation & Objective

• To provide a geometric derivation of mutually unbiased bases (MUBs) and symmetric informationally complete positive operator-valued measures (SIC-POVMs) for a spin-1 system using the Majorana representation.
• To express the overlap between two spin-1 states in terms of their Majorana vectors, enabling geometric analysis of orthogonality, unbiasedness, and equiangularity.
• To demonstrate that MUBs and SICs can be understood as symmetric configurations of vectors in ordinary 3D space, rather than abstract rays in Hilbert space.
• To explore the limitations of the Majorana approach in generalizing to higher spin systems while highlighting its pedagogical and historical value for spin-1.

Proposed method

• Utilizes the Majorana representation, where a pure spin-1 state is represented by a pair of unit vectors (or points) on the unit sphere.
• Derives a closed-form expression for the squared overlap |⟨ψ₁|ψ₂⟩|² in terms of the angles and relative orientations of the two Majorana vectors.
• Applies geometric constraints—orthogonality (overlap = 0), unbiasedness (overlap = 1/3), and equiangularity (overlap = 1/4)—to identify MUB and SIC configurations.
• Constructs MUBs by identifying four bases whose Majorana vectors satisfy mutual unbiasedness, forming a symmetric configuration related to a regular octahedron.
• Attempts to derive SICs by solving the equiangularity condition, leading to a quartic equation whose roots define double-cone structures for candidate SIC states.
• Validates results by mapping Majorana vector configurations to rays in ℂP² using the standard Majorana-to-ray transformation, confirming unitary equivalence to known SICs.

Experimental results

Research questions

• RQ1 Can the Majorana representation be used to geometrically construct mutually unbiased bases (MUBs) for a spin-1 system?
• RQ2 How can the overlap between two spin-1 states be expressed purely in terms of their Majorana vectors to enable geometric analysis?
• RQ3 Can the equiangularity condition for SIC-POVMs be solved geometrically using the Majorana formalism, and what configurations emerge?
• RQ4 What are the symmetries and geometric structures underlying known SICs and MUBs in the spin-1 case when viewed through the Majorana lens?
• RQ5 To what extent can the Majorana approach be generalized to higher spin systems, and what are its inherent limitations?

Key findings

• The squared overlap between two spin-1 states is expressed as a function of their Majorana vectors' angles and relative azimuthal angles, enabling geometric analysis.
• The MUBs of a spin-1 system are shown to correspond to four bases whose Majorana vectors form a symmetric configuration equivalent to the vertices of a regular octahedron.
• The construction yields a unique set of four mutually unbiased bases up to unitary equivalence, consistent with known results.
• For SIC-POVMs, the equiangularity condition leads to a quartic equation whose solutions define double-cone structures for the Majorana vectors of SIC states.
• Only two solutions to the quartic equation yield valid SICs, both of which are known from earlier work and correspond to symmetric configurations in 3D space.
• The Majorana approach reveals that MUBs and SICs are not just abstract quantum structures but correspond to highly symmetric point configurations in ordinary 3D space.