I have to distinguish between 4 linearly independent vectors belonging to $\mathbb{C^{16}}$ space by creating a set of Positive Operator Valued Measurements (POVM) that will act on these vectors. I have found that the given set of vectors are such that there exists no projectors which can categorize them into mutually orthogonal sub-spaces under $\mathbb{C^{16}}$, so that I cannot simplify the problem as a problem in two vectors within these subspaces. The set of POVM ${\{E_i\}}$ has to follow completeness $\sum{_{i=1}^n E_i=\mathbb{I}}$ for some $n$ and optimally distinguish between these vectors with maximum probability possible through optimization. I tried generalizing the procedure proposed in http://iopscience.iop.org/article/10.1088/0305-4470/31/34/013/pdf for 4 vectors in $\mathbb{R^4}$ but it is not working out because of the fact that the cross product they used in 3 dimensions cannot be generalized in 4 or higher dimensions directly (Hurwitz Theorem, composition algebra). On page 5 of the paper they have proposed to use outer product for its generalization, but that is not satisfying the matrix multiplication. If someone may help doing the same procedure in $\mathbb{R^4}$ for four vectors, would be enough help I seek for its use in $\mathbb{C^{16}}$ space.
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$\begingroup$ Please make precise what you mean by distinguish; it would also be helpful if you gave an example. $\endgroup$– David HandelmanCommented Dec 23, 2017 at 19:31
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$\begingroup$ By distinguishing I mean applying the operators to these 4 vectors gives me one of them uniquely and makes the other vanish or at least gives a linear combination of the 4 with very high probability of one and vanishingly of the other ones (this is optimization distinguishability) when they are not orthogonal. $\endgroup$– Siddhant SinghCommented Dec 23, 2017 at 19:53
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1 Answer
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Suggestion:
Let the independent 4 vectors be $(v_1,...,v_4)$.
For each $1\leq i\leq 4$ let $\phi_i$ be in the span of the four, but orthogonal to $v_i$ (achieved e.g. by Gram-Schmidt). Then $E_i=|\phi_i\rangle\langle\phi_i|$ distinguishes the $i$'th vector, and you may complement by a projection $E_0$ onto the orthogonal complement of the 4 vectors if you want a full decomposition.
Is this sort of in the right direction for you?
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$\begingroup$ Yes, mathematically that will work, but how do I guarantee the maximization the probability of distinction through that? $\endgroup$ Commented Dec 24, 2017 at 9:57
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$\begingroup$ Let $\{\rho_i,1\leq i\leq m\}$ be a set of linearly independent density operators $\rho_i$ with prior probabilities $p_i$. Then the optimal measurement is a von Neumann measurement with measurement operators $\{\Pi_i=\mathcal{P}_{\mathcal{S_i}},1\leq i\leq m\}$ where $\mathcal{P}_{\mathcal{S_i}}$ is an orthogonal projection onto an $r_i$-dimensional subspace $\mathcal{S_i}$ of $\mathcal{H}$ (Hilbert space of $\rho_i$'s) with $r_i= rank(\rho_i)$ and $\mathcal{P}_{\mathcal{S_i}}\mathcal{P}_{\mathcal{S_j}}=\delta^i_j\mathcal{P}_{\mathcal{S_i}}$ (orthonormality of operators). $\endgroup$ Commented Dec 24, 2017 at 9:59
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$\begingroup$ This is a theorem I found which says something similar, but how do I construct the $\mathcal{P}_{\mathcal{S_i}}$ for each $\rho_i$? $\endgroup$ Commented Dec 24, 2017 at 10:00
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$\begingroup$ @SiddhāntSingh By linearly independent do you mean as operators in $H$? Your density operators, do they take the form of a finite sum: $\sum_j \lambda_j |\phi_j\rangle\langle \phi_j|$, $\lambda_j>0$? And the ranges of different $\rho_i$'s may have non-empty intersections, or perhaps this would be inconsistent with your orthonormality of operators (usually not asked for in POVM as far as I know) $\endgroup$ Commented Dec 24, 2017 at 13:25
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$\begingroup$ Yes, they take that form with different eigenvectors but some degenerate eigenvalues $\lambda_j$'s for each $\rho_i$ eigen expansion. But the 4 matrices I have, do not have all orthogonal subspaces (there are intersecting subspaces), hence I need an optimization to operate over them to achieve distinction with maximum probability. How to construct the $\phi_i$ that you initially pointed out? $\endgroup$ Commented Dec 24, 2017 at 17:03