Timeline for How to optimally distinguish between linearly independent vectors in higher dimensional complex/real space?
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| Dec 30, 2017 at 18:09 | comment | added | Siddhant Singh | Thank you for your response H.H. Rugh! However, I am forbidden to use any orthogonalization method such as Gram-Schmidt, because here the vectors are physically "Quantum States" which cannot be disturbed and turned into Orthogonal vectors. Perhaps, I found one solution to the problem by proposing a set of POVM here, but the upper bound on the probability of distinctions isn't proved. For these vectors, only a POVM can be applied which corresponds to a measurement device for a quantum state (these 4 vectors in the Hilbert Space). So, still a general solution to my problem is unknown. | |
| Dec 24, 2017 at 23:30 | comment | added | H. H. Rugh | For $\phi_1$, I would take Gram-Schmidt orthonormalise $v_2,v_3,v_4,v_1$ (in that order) to obtain an orthonormal base $e_2,e_3,e_4,\phi_1$, then $\phi_1$ has a non-zero projection on $v_1$ and is orthogonal to each of $v_2,v_3,v_4$. Then repeat and orthonormalize $v_1,v_3,v_4,v_2$ (in that order) to get $e_1,e_3,e_4,\phi_2$, etc... Would the $\phi_i$'s have the desired properties? | |
| Dec 24, 2017 at 17:03 | comment | added | Siddhant Singh | 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? | |
| Dec 24, 2017 at 13:25 | comment | added | H. H. Rugh | @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) | |
| Dec 24, 2017 at 10:00 | comment | added | Siddhant Singh | This is a theorem I found which says something similar, but how do I construct the $\mathcal{P}_{\mathcal{S_i}}$ for each $\rho_i$? | |
| Dec 24, 2017 at 9:59 | comment | added | Siddhant Singh | 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). | |
| Dec 24, 2017 at 9:57 | comment | added | Siddhant Singh | Yes, mathematically that will work, but how do I guarantee the maximization the probability of distinction through that? | |
| Dec 23, 2017 at 23:00 | history | answered | H. H. Rugh | CC BY-SA 3.0 |