# Questions tagged [mathematics]

Use this tag for questions about mathematics relevant to quantum computing and/or quantum information theory. DO NOT use this tag for general mathematics questions.

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### Is there a general parametric transformation matrix form in bloch-space corresponding to the unitary operations on qutrits?

I've been looking into the structure of the Bloch sphere for qudits, and I am wondering if there is a transformation matrix (or rotation matrix) formula corresponding to high-dimensional quantum ...
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### Specific relation between the classical Fourier transform for finite abelian groups and the QFT for finite abelian groups

$\newcommand{\C}{\mathbb{C}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\ket}{|#1\rangle} \newcommand{\bra}{\langle#1|}$ I am a math undergrad (with admittedly minimal background in quantum ...
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### Can the Bloch sphere representation be applied to many-qubit states with an iterative approach?

By ignoring the global phase, we can represent a single qubit state as \begin{equation} |\psi\rangle = \cos(\theta)|0\rangle + e^{i\phi}\sin(\theta)|1\rangle \end{equation} which very much looks like ...
75 views

### Equality condition on Holder's inequality for matrix for infinity norm

The equality condition for Holder's inequality, $\text{Tr}A^*B \leq ||A||_p||B||_q$ is $|A|^p = \lambda |B|^q$ for scaler $\lambda > 0$. What happens when $p$ or $q$ is $\infty$? I found out that ...
779 views

### If two unitary operators commute, do their roots also commute?

This is probably a pretty basic linear algebra question, but suppose we have two unitary operators $A$ and $B$, acting on the same $n$ qubits of $|\psi\rangle$, with $[A,B]=0$ - that is, $A$ and $B$ ...
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### Does the 4x4 matrix $|00\rangle\!\langle00|+|11\rangle\!\langle11|$ have a decomposition?

Can the diagonal matrix $$\begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 0& 0 \\0&0&0&0 \\ 0&0&0&1 \end{pmatrix}$$ be written as a tensor product $A\otimes B$...
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### What does $A - \langle A \rangle$ mean?

I've seen the uncertainty of $A$ written as $$(\Delta A)^2 = \langle (A - \langle A \rangle)^2 \rangle.$$ But what does this even mean since $A$ is an operator and $\langle A \rangle$ is a ...
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### Why do we need Hilbert spaces when talking about qubits and quantum computation?

I was just curious to know why do we need Hilbert Spaces when talking about the qubits and quantum computation in general. I mean why can't we just work with inner product spaces, rather than going ...
439 views

### Are complex amplitudes really needed?

Qubit amplitudes are defined as complex numbers. But in all tutorials I have recently read, only real numbers are used and everything works. So, if I completely forget the official 'complex' ...
1 vote
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### Two-qubit Bell measurement matrix where the two qubits are not contiguouis

In the answer here, it is explained that where the measurement operates on only a subset of the qubits of the system (for example qubits 2 and 3 out of five), the matrix can be constructed using the ...
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### Prove that $|(\langle \psi|_{A} \otimes \langle \phi|_{B})|\theta\rangle_{AB}|^{2}<1$ for entangled $|\theta\rangle_{AB}$

I am trying to show that $|\langle \psi|_{A} \otimes \langle \phi|_{B}|\theta\rangle_{AB}|^{2}<1$ given $|\theta\rangle$ is an entangled state, and as such has schmidt rank >1. Decomposing it, ...
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### How to find the eigenstates of a general $2\times 2$ Hermitian matrix?

Given a measurement operator in the general Hemitian form $$M = \begin{pmatrix} z_1 & x+iy \\ x-iy & z_2\end{pmatrix},$$ where $x,y,z_1,z_2 \in \mathbb{R}$, show that the eigenvalues are  ...
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### Why does $H^2=X^2 =I$ not imply $H=X$?

if $HH = I$ and $XX =I$, then is $H=X$? $HH = I = XX$ or, $HH = XX$ then, taking under root, is $H = X$? This is absurd but how to disprove it?
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### Does conjugation by a Clifford send each non-identity Pauli to every other non-identity Pauli with equal frequency?

I see here in Olivia DeMatteo's notes, she states: When we consider the action of the entire Clifford group on a single non-identity Pauli, it maps that Pauli to each of the $d^2 − 1$ other possible ...
### Why are orthogonal spins $(1,0)$ and $(0,1)$ represented as collinear vectors in the Bloch sphere?
Recently, I am studying some topics related to product formula, and I am curious about how to implement such formula on real quantum devices. The $(2k)$-th order product formula can be witten as \...