# Questions tagged [textbook-and-exercises]

Applies to questions of primarily educational value - styled in the format similar to that found in textbook exercises. This tag should be applied to questions that are (1) stated in the form of an exercise and (2) at the level of basic quantum information textbooks.

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### How are eigenvectors and eigenvalues expressed in the Bloch sphere?

I'm relatively new to the subject of quantum computing, and I recently came across the idea of eigenvalues and eigenvectors. I believe I understand the relationship between the two, where eigenvalues ...
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### Proof of quantum data processing inequality in N&C on pg 566

On page 566, it states that using $S(\rho^{'})-S(\rho,\varepsilon) \ge S(\rho)$ and combining this with $S(\rho) \ge S(\rho^{'})-S(\rho,\varepsilon))$, we get $S(\rho^{'})=S(\rho)-S(\rho,\varepsilon)$....
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### How does the graphical notation used to denote doubly-controlled gates work?

$\qquad$ $\qquad$ What is the difference between solid and hollow? How to express the corresponding matrix of these figures? In addition, if they are not adjacent, what should be done in the middle of ...
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### Find the Kraus operators of a combined amplitude and phase damping channel

I am going through the paper Surface code with decoherence: An analysis of three superconducting architectures and I have a doubt about how the authors get what they refer to as the combined channel ...
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### What is the representative matrix for a measurement in the Bell-state basis?

I have a few questions about measurement in Bell-state basis. In particular, if $Z = \begin{bmatrix} 1 & 0\\ 0 & -1 \end{bmatrix}$ is for a measurement on the computational basis, then what ...
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### Why is the function $f_s(x)=\sum_i x_i s_i \pmod 2$ balanced?

A parity function $f_s:\{0,1\}^{n}\rightarrow\{0,1\}$, for some $s\in \{0,1\}^n$, is a function of the form $f_s(x) = x \cdot s$, where the inner product is taken modulo 2. Show that $f_s$ is a ...
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### Can $U(\rho_1\otimes\rho_2)U^\dagger$ be entangled if either of $\rho_1$ or $\rho_2$ is a maximally mixed state?

Given two qubit states $\rho_1$ and $\rho_2$. By applying some unitary $U$ we get $\rho = U(\rho_1 \otimes \rho_2)U^\dagger$. Can $\rho$ be entangled if either of $\rho_1$ or $\rho_2$ is a maximally ...
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### Unable to use qiskit_textbook module

When I tried to use the Qiskit textbook module, it threw the following error: Command was: ...
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### What happens when you send a Bell state through depolarizing channel?

For noise parameter $Q$ and a density matrix $\rho$, we know that the depolarization channel $\mathcal{E}$ would act like: $$\mathcal{E}(\rho) = (1 - Q)\rho +Q\frac{I}{2},$$ where $I$ is the ...
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### How to decompose a unitary single qubit gate by universal quantum gate set?

How to decompose a unitary single qubit gate? I have read some paper or books, which told me a unitary single qubit gate could be decomposed by universal quantum gates set. For example {phase gate, ...
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### Bipartite states whose coefficients are entries of a unitary matrix

I've been trying to solve this question It seems that in order to show it has unit length, we must show that $$\frac{1}{d} \sum_{m, n=0}^{d=1} \lvert U_{m, n}\rvert ^2 = 1$$ I've tried searching ...
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