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I would like to know if an analytic solution is known for the Neumark dilation of the qubit tetrahedron SIC-POVM defined by $$ M_0= \frac{1}{4\sqrt{3}} \Big( \sqrt{3}I + X +Y +Z \Big), \qquad M_1= \frac{1}{4\sqrt{3}} \Big( \sqrt{3}I + X -Y -Z \Big),$$ $$M_2 = \frac{1}{4\sqrt{3}} \Big( \sqrt{3}I - X -Y +Z \Big), \qquad M_3=\frac{1}{4\sqrt{3}} \Big( \sqrt{3}I - X +Y -Z \Big), $$ where $X$, $Y$ and $Z$ are the Pauli matrices. What I'm looking for is a 4-outcome rank-one projective measurement $\Pi$ on a qubit state $\rho$ composed with some pure auxiliary qubit state $|\phi\rangle$ such that the measurement probabilities are preserved for any $\rho$, $$ \textrm{tr}\bigg( \rho M_i \bigg) = \textrm{tr} \bigg( \big(\rho \otimes |{\phi}\rangle \langle{\phi}|\big) \Pi_i \bigg), $$ i.e. I don't need that the actual measured states are the same between the tetrahedron and the dilated measurement. According to Theorem 2 of https://arxiv.org/abs/1609.06139v2, such a measurement $\Pi$ and state $|\phi \rangle$ exist in this case. Are these known for the tetrahedron, or is there a method (aside from trial and error) that I can use to find them? Thanks in advance.

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  • $\begingroup$ There are several methods to derive the dilation, for example the one described here and the Asher Peres book it cites. $\endgroup$ Apr 3, 2023 at 14:13
  • $\begingroup$ writing the dilation for a given POVM is easy: you want the isometry $V$ with action $V|\psi\rangle=\sum_a \sqrt{M_a}|\psi\rangle\otimes|a\rangle$, with $a$ labelling the possible measurement outcomes and $M_a$ the POVM elements. More concretely, you can imagine $V$ as the (non-square) matrix obtained stacking $\sqrt{M_a}$ one on top of the other. The projective measurement is then just measuring in the computational basis. Of course, dilations aren't unique. I'm not sure if this gives you much insight though $\endgroup$
    – glS
    Apr 3, 2023 at 15:48
  • $\begingroup$ though note that the method discussed in the paper seems quite different than a standard dilation. So which one are you asking about? $\endgroup$
    – glS
    Apr 3, 2023 at 23:22
  • $\begingroup$ Thank you both for your comments. These are useful references for dilations but yes, the method described above with the isometry involves an auxiliary system of dimension $dn$, with $d$ the dimension of the primary system and $n$ the number of outcomes, and the approach in Peres involves an auxiliary system of dimension $n(d-n+1)$, whereas in my case, with a $d^2$-outcome rank-one measurement ($d=2$), I want to find a rank-one projective measurement over the system plus $d$-dimensional (qubit) auxiliary system. $\endgroup$ Apr 4, 2023 at 7:20

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