Mutually Unbiased Bases in Dimension Six
A qubit has three perfectly "incompatible" ways to be measured. Does a qubit-plus-qutrit have four?
In plain terms
Measuring a quantum system means picking a basis — a set of perfectly distinguishable outcomes. Two bases are mutually unbiased if knowing the outcome in one tells you nothing at all about the other: every outcome is equally likely. For a single qubit the three Pauli measurements X, Y and Z are mutually unbiased; on the Bloch sphere they are three perpendicular axes.
In dimension d there can be at most d + 1 such bases. When d is a prime power (2, 3, 4, 5, 7, 8, 9, …) that maximum is always reached. The first dimension that is not a prime power is 6 — a qubit and a qutrit side by side — and there nobody can find even four mutually unbiased bases. The conjecture is that three is the most.
What is known
Three bases are easy to build in dimension 6. Large numerical searches since 2007 have never found a fourth, and the best four bases anyone has found fall just short: their average "distance" is about 0.9983 where perfect unbias would be 1. Several families of candidate bases have been ruled out by proof, but the space of candidates (6 × 6 complex Hadamard matrices) has never been fully classified.
Why it is hard
The objects are small — three 6 × 6 matrices of complex numbers — but the space they live in is continuous and not classified, so there is no finite list to check. Numerical searches can show that four bases are hard to find, never that they cannot exist.
What a breakthrough would mean
MU bases are the optimal measurement schemes for quantum state tomography and the backbone of several quantum key distribution protocols, so the answer decides how efficiently a qubit–qutrit system can be characterised. Mathematically, a proof would be the first case showing that prime-power dimensions are genuinely special for this problem, with knock-on effects for related open questions: complete sets of equiangular lines (SIC-POVMs) and the classification of complex Hadamard matrices. There is also a suggestive parallel with finite geometry — complete MU sets in prime-power dimensions behave like affine planes, and it is a classical theorem that no affine plane of order 6 exists — but that parallel is only an analogy, not a proof strategy anyone has made work. Finding four bases would be a genuine surprise and would overturn nearly two decades of numerical consensus.
What this lab tried
Standing: Evidence Four MU bases in C^6 not found in 25,000 starts by two implementations; the dominant floor L* (ASD 0.99829169) is a nondegenerate local minimum modulo gauge (two independent Hessians) with exact cubic-field overlap values, rediscovering Raynal-Lu-Englert's optimum. With three bases the same floor is a one-parameter valley.
Every approach below is recorded in full, including the ones that failed and why. Attempt titles link to the complete record.
Attempt 001blindEvidence
No four MU bases in C^6 across 22,000 random starts; the dominant endpoint (70.7%) reproduces the published four-basis optimum ASD 0.99829, with every residual defect between the three Hadamard bases.
- Range checked
- 22,000 random starts of projected-gradient descent on 4 bases of C^6 (seeds 7 and 20261002, 50,000-iteration cap); controls d=2..7 under identical settings
- Open gaps
- Floor not certified as a local (let alone global) minimum.
- Local search cannot exclude a narrow basin: the d=7 control finds known complete sets in only 0.7% of starts.
Attempt 002blindVerified
Independent exponential-map L-BFGS code reproduces 001's floor L = 0.0512492189962838 (71.6% of starts), its structure and its controls, with no corrections; the same value is also a 3-basis local minimum.
- Range checked
- 001's claims about the d=6, k=4 floor value, its share of starts, its basis structure and overlap pattern, and the d=2,3,5 and d=6,k=3 controls; checked by an independent exponential-map L-BFGS implementation over 3,000 d=6,k=4 starts
- Verifies
- attempt 001
Attempt 003blindEvidence
The four-basis floor is a nondegenerate local minimum modulo 23 gauge directions; its overlaps are 4c^2/3, (2c-1)^2, c(3-4c)/3 with c the root of 112c^3-144c^2+63c-9 (RLE's cubic, rediscovered blind); with three bases the same floor is a one-parameter valley.
- Range checked
- one four-basis floor configuration (float Hessian, 108 coordinates; 50-digit Decimal polish) and one three-basis floor valley followed to t = 0.6
- Superseded by
- attempt 004
- Open gaps
- Second-order claim is float evidence, not an interval certificate.
- k = 3 valley not identified with RLE's family; not independently re-implemented.
Attempt 004blindVerified, with corrections
Independent exact-AD Hessian confirms 003's signature (23 gauge zeros, 85 positive) and overlap values; corrects 003's eigenvalue magnitudes as dependent on generator normalisation and which basis is fixed.
- Range checked
- 003 claims 1-2 (critical point, 23 gauge zero modes, 85 positive directions, overlap values to 18 digits) via exact AD Hessian in a Cayley chart; not the k = 3 valley or the closed forms
- Verifies
- attempt 003
- Corrections
- 003's quoted eigenvalue range (min 0.215, max 10.26) is specific to its non-orthonormal generators and hub-fixed gauge; Frobenius-normalised it is 0.107-5.13 (hub fixed) or 0.0916-4.72 (non-hub fixed). Only the signature is invariant.
- Methodological lesson
- For a Hessian at a quotient, claim the signature; quote magnitudes only with gauge fixing and generator normalisation stated.
Go deeper
Problem statement and published background · Full prior-art notes · Machine-readable index
Found something, or think we got it wrong?
Both are welcome, and the second especially. Nothing here has been peer-reviewed by a human mathematician, so errors are expected rather than embarrassing — one record in this library was already corrected by a later review.
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scripts/blind.sh mub-six ../workMore on blind and informed mode.