Stack Exchange SOORAJ SOMAN

Engaging with platforms like Mathematics and Quantum Computing Stack Exchange has been a vital part of my independent learning journey. By actively solving problems, participating in discussions, and sharing detailed explanations, I’ve not only deepened my understanding of complex mathematical and quantum computing concepts but also contributed to the broader academic community. These collaborative exchanges have helped me develop a structured approach to tackling challenging topics, and many of the insights and clarity reflected in the notes within the Learning Journey section of this website are a direct result of such discussions. Below, I’ve highlighted a selection of posts that represent some of my most meaningful contributions and explorations on these platforms.

Mathematics Stack Exchange

$f(y) - f(x) \leq f(|y - x|)$ if $|y - x| \leq \frac{1}{2}$ given $f(x) = -x \log_2 x$: Analyzed the behavior of the inequality and proved a bound with detailed steps and reasoning.
Understanding Filippov’s Inductive Proof for Jordan Canonical Form: Provided a detailed explanation and visual aid for the proof after extended discussions.
Understanding the Expression $\text{tr}(\rho(X \otimes I)) = \sum_{a,b,a',b'} \rho_{ab,a'b'} X_{a,a'} \delta_{b,b'}$: Initiated and contributed to a detailed discussion on the trace properties of tensor products.
The median minimizes the sum of absolute deviations (the $\ell_1$ norm): Provided a detailed explanation of a standard proof with an original example.
Real roots of the equation $\log_{(5x+4)}(2x+3)^3-\log_{(2x+3)}(10x^2+$$23x+12)=1$: Investigated the real roots of a logarithmic equation, providing cases and constraints, and collaborated on refining the solution for accuracy and completeness.
Determinant using factor theorem: Explored the factor theorem's application to determinants, collaboratively analyzing symmetric polynomial factors and their implications.

Quantum Computing Stack Exchange

Derivation of Efficiency of Phase Estimation Algorithm: Explored and engaged deeply with the derivation of efficiency bounds for the phase estimation algorithm.
Why does the $\chi$ matrix have $d^4-d^2$ independent parameters?: Initiated and actively contributed to an in-depth discussion related to the $\chi$ matrix in quantum process tomography.
Derive the Concavity of Quantum Conditional Entropy from Strong Subadditivity: Initiated a detailed inquiry and contributed to a collaborative understanding of the derivation using strong subadditivity.
Show that $E_k=(I\otimes\langle e_k|)U(I\otimes|e_0\rangle)$ implies $U=\begin{bmatrix}[E_1]&\cdots\\ [E_2]&\cdots\\\vdots&\ddots\end{bmatrix}$: Initiated a structured query and worked collaboratively to clarify the connection between Kraus operators and the block matrix structure of unitary transformations.
Show the linearity of $(\langle a_m|\otimes I_B\otimes I_C\otimes \langle d_q|) U(I_{A}\otimes I_B\otimes $$|0_{C}\rangle\otimes |0_{D}\rangle)$: Analyzed the linearity of operators in the Kraus representation, focusing on their role in mapping composite quantum states under unitary evolution