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5 votes
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Maximum number of "almost orthogonal" vectors one can embed in Hilbert space

There is no general exact formula for $N(\epsilon, d)$ and some special cases (for example SIC-POVM) is an area of active research. However there is a Welch bound that gives $\epsilon^2 \ge \frac{n-...
Danylo Y's user avatar
  • 7,332
5 votes

Are SIC-POVMs optimal for quantum state reconstruction?

First of all, here's a short disclaimer: I'm not an in-depth expert in this field, I'm just currently getting in contact with tomography more and more often :) So take the following with a grain of ...
Markus Heinrich's user avatar
4 votes
Accepted

Are MUBs complex projective 3-designs?

There is a bound for the size $n$ of a complex projective $t$-design (see Eq. 2.5 of Roy and Scott paper): $$ n \ge \binom{d-1+⌊t/2⌋}{⌊t/2⌋}\binom{d-1+⌈t/2⌉}{⌈t/2⌉}. $$ So, for a 3-design, $n\ge \...
Danylo Y's user avatar
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4 votes
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Why does full state reconstruction require at least $N+1$ MUBs?

Denote the projections onto basis elements by $P_j^{(k)}=|u_j^{(k)}\rangle\langle u_j^{(k)}|$, where superscript indexes different bases. Tomography of a density matrix $\rho$ gives us probabilities $\...
Danylo Y's user avatar
  • 7,332
3 votes

Maximum number of "almost orthogonal" vectors one can embed in Hilbert space

This is quite old now, but for the benefit of people coming across this post there is a good discussion of this problem on Terence Tao's blog: https://terrytao.wordpress.com/2013/07/18/a-cheap-version-...
Ian McCulloch's user avatar
3 votes

Maximum number of "almost orthogonal" vectors one can embed in Hilbert space

This seems like it should be a known mathematical property of Hilbert spaces, but I can't immediately lay my hand on any such result. In lieu of that, this is very far from an answer to your question, ...
DaftWullie's user avatar
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2 votes
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Algorithm for Mutually Unbiased Basis Sets Available?

Even though I did not find an implementation, I had a look at mathematical constructions of those MUBs in prime power dimensions - what I've found was this paper and I implemented the described ...
Juri V's user avatar
  • 105
2 votes

Algorithm for Mutually Unbiased Basis Sets Available?

You can use heuristic algorithms or optimization techniques to search for good approximate solutions. These algorithms aim to find suboptimal solutions that are close to the optimal partitioning of ...
Gandalf73's user avatar
  • 133
1 vote
Accepted

Stabilizer Matrices for Mutually Unbiased Bases - what goes wrong here?

It seems that qiskit's unitary_simulator gives another result than doing it manually. For example, the unitary matrix given by the code above for the family $\{ XZ, ZY, YX \}$, the circuit is ...
Juri V's user avatar
  • 105
1 vote

Are MUBs complex projective 3-designs?

Another straightforward approach is to consider the alternative definition of (complex projective) $t$-designs in terms of frame potentials. Namely, as also mentioned e.g. in this other post, a set $X\...
glS's user avatar
  • 25.3k

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