Questions tagged [linear-algebra]

For questions about vector spaces of all dimensions and linear transformations between them, including systems of linear equations, bases, dimensions, subspaces, matrices, determinants, traces, eigenvalues and eigenvectors, diagonalization, Jordan forms, etc.

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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 ...
3
votes
3answers
660 views

How to show that Bell states are orthonormal

I was reading some material on QC online and I found some material that explains how to show that Bell states are orthonormal but without details. I understand that we need to check $\langle state1|...
4
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1answer
257 views

Difference between change of basis in bra-ket notation and matrix notation

In matrix notation, say I have the vector $\begin{bmatrix} 1 \\ 0 \end{bmatrix}$. It is currently represented in the computational basis $\{\begin{bmatrix} 1 \\ 0\end{bmatrix}, \begin{bmatrix} 0 \\ 1\...
4
votes
1answer
114 views

Creating orthogonal quantum states from a set of given (possibly linearly independent) quantum states

I want to understand how to orthogonalize a system of qubits. Suppose I have $n$ sets of quantum states like $$\{ |1_i\rangle|2_i\rangle|3_i\rangle \cdots|k_i\rangle \mid i=1 \dots n \}$$ where $i=1, \...
3
votes
1answer
136 views

Diamond norm distance bound on Stinespring dilations of channels

The diamond distance between two channels $\Phi_0$ and $\Phi_1$ is defined in this answer. $$ \| \Phi_0 - \Phi_1 \|_{\diamond} = \sup_{\rho} \: \| (\Phi_0 \otimes \operatorname{Id}_k)(\rho) - (\Phi_1 ...
5
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2answers
145 views

How are the eigenvalues of $\rho=\frac12(|a\rangle\!\langle a| +|b\rangle\!\langle b|)$ derived?

Let's say I have a density matrix of the following form: $$ \rho := \frac{1}{2} (|a \rangle \langle a| + |b \rangle \langle b|), $$ where $|a\rangle$ and $|b\rangle$ are quantum states. I saw that ...
2
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1answer
87 views

Why can any density operator be written this way? (quantum tomography)

From page 24 of the thesis "Random Quantum States and Operators", where $(A,B)$ is the Hilbert-Schmidt inner product: \begin{aligned} \rho &=\left(\frac{1}{\sqrt{2}} I, \rho\right) \frac{...
2
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1answer
69 views

How does measuring a value of one operator affect the probability of measuring a value for another operator?

Suppose I have two non-commuting operators, $U_1$ and $U_2$ with eigenvalues $\lambda_{1,1}, \lambda_{1,2}$ and $\lambda_{2,1}, \lambda_{2,2}$, respectively. In order to determine how measuring one ...
1
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2answers
96 views

Find a unitary to prepare state $|0\rangle$ to a specific vector

I am working with Variational Quantum Linear Solver (VQLS) algorithm, where it needs to prepare a control_b circuit. Assume b is 1d with $ 2^n $ elements in it. $$ {\bf Ax = b} \tag{1}$$ I need to ...