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Hey yeah, that’s true. I placed the mats where I was most likely to land if I fell instead of directly under the rock

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Hey! Thanks for your question! Sorry for the late reply, I've been pretty busy transitioning to graduate school life. I'm not sure I'm following your notation, what are you using p1 and n1 to denote? Yes it is true that 3<= M(d) <= 7! And yes I've explicitly constructed 3 matrices with distinct column vectors (up to column permutation). The bases formed by the column vectors are pairwise mutually unbiased, along with the standard basis. Note here that in order to begin 'comparing bases' we have to fix a UNIQUE standard basis, i.e. we have fixed a single standard basis (bases are kinda like combinations in our case because we don't care about the 'ordering' of the basis elements). This leaves us with 4 mutually unbiased bases (up to permutation of basis elements and rotations of C^6)! If you end up seeing this reply, I hope it was somewhat helpful! Thanks for leaving a comment, I love talking about math with others. If you have any other questions, feel free to leave more and I'll try to get to them when I can : )

@@dcqin Thank you for replying, here I was using p1 = 2, from the product 2 * 3, where p2 = 3, and n1 is the power on p1. So, according to your explanation, can we just conclude that for d=6, there are only 4 MUBs or, are there other methods from which we get more MUBs. I understand that, here, we compare only a pair of basis, where 1 is the fixed standard basis.

@@anishsrinath9647 Ah gotcha, okay! Unfortunately, we cannot conclude that there are only 4 MUBs :/ MUBs are similar to an "orthonormal basis of bases." I say this because a maximal set of MUBs is a maximal object in the same way that a basis is a maximal object (what we mean by maximal is that it is the 'biggest' possible object I can find with that property). So let's consider an analog to the constructed MUBs in terms of orthonormal basis vectors. Assume we are in R^6. I can give you an linear independent set of vectors (0,1,0,0,0,0), (0,0,1,0,0,0), (0,0,0,1,0,0). These vectors are pairwise orthogonal so they form an orthonormal set. Along with our "fixed" (standard) basis vector (1,0,0,0,0,0), this forms an orthonormal set of 4 vectors. However, just because we were able to construct an orthonormal set of 4 vectors does not mean there DOES NOT exist a BIGGER set :D Note that (0,0,0,0,1,0) and (0,0,0,0,0,1) are also pairwise orthonormal. Here's some more food for thought: In general, when seeing new math, it is really helpful (at least for me) to categorize a theorem by its content, i.e. what kind of theorem it is: is it a correspondence theorem? --- ex: Galois correspondence, ideal-variety correspondence, covering maps-fundamental group correspondence is it an existence theorem? -- ex: four-coloring theorem, minkowski's theorem on existence of non-zero lattice point in a convex set, pigeonhole principle is it an inequality theorem? -- Cauchy Schwarz, Jensen's, triangle inequality, Noether's bound (this one is also an existence theorem) is it a characterization theorem? -- Characteristic property of quotient topology, characteristic property of product topology is it a uniqueness theorem? unique homotopy lifting...uhm I can't think of any others off the top of my head.... Of there are many more categories that you can think of, and they're definitely not strict boxes--definitely overlap like Noether's bound is both an inequality and an existence theorem in my head--but nonetheless useful heuristics! The point I'm trying to make is that identifying what type of theorem it is, i.e. what is the content, can be massively helpful even if the objects of study are fairly abstract (like maximal sets of mutually unbiased bases!).

For the Hadamard approach, there wasn't an obvious construction to 'find' more MUBs... In composite dimensions, there might be a way we can construct a maximal set of MUBs--but its also possible that it doesn't exist; It is a conjecture on existence after all! To my knowledge, it's still an open problem, and it has been around for quite a while now so there is a ton of literature on the subject and what people have tried (much of it is beyond the scope of my math knowledge). If you hop on arxiv.org and search up "mutually unbiased bases" you can find what researchers have been working on. I know there are some combinatorial and geometric approaches that I didn't mention. For example, there is a combinatorial relationship between finite projective planes and MUBs for prime dimensions, and I have a video on that on my channel that you can check out! Otherwise, good luck to you on your studies! : )

@@dcqin thank you so much, cleared my doubt