# 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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### Finding the "dual" basis of an overcomplete basis for Quantum State Tomography

This question is related to this stack exchange post: What does the POVM corresponding to single-qubit state tomography look like? From what I understand, when we are interested in reconstructing a ...
1 vote
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### Problem with eigenvalue evaluation algorithm application on matrix $U$

Once I get to the end of the algorithm, I can't understand how to calculate the eigenvalue using formulas. Bear in mind that it is an exercise to be carried out with pen and paper. the matrix of $U$ ...
196 views

### Finding the eigenvalues of a qutrit state

I am interested in the state: $\frac{1}{\sqrt{2}} (\left|11\right> + \left|22\right>)$ If I find the density matrix of this, I find the $9 \times 9$ matrix $\rho$. If I want to find the reduced ...
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### What is known about the size of the spectral gap of unital quantum channels?

I am interested in the spectrum of unital quantum channels $\Phi$ (which act on finite dimensional spaces). The literature is extremely vast on such objects so I hope some experts can point me along ...
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### Is possible to write a separable state as a finite or countable infinite sum of product states?

Let us consider the tensor product of two finite Hilbert spaces $\mathcal{H}_1\otimes \mathcal{H}_1$. According to Watrous book, the set of separable states is the convex hull of the set of product ...
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### Relationship between the eigenvalues of a Laplacian matrix and the eigenvalues of the Hamiltonian of a graph for Max-Cut

Is there a relationship between the eigenvalues of a Laplacian matrix of a graph and the eigenvalues of the Hamiltonian for Max-Cut? It is shown here, that Max-Cut can be written as a maximization ...
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### Is there a general method for calculating expectation values for time-dependent wavefunctions?

Is there a general method for calculating expectation value? I have a workshop question, and I'm sure what a good process to follow is. It is given that $$|\psi(t = 0)\rangle = |0\rangle\,,\tag{1}$$ ...
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### What do the values in a unitary matrix represent/what do they mean? How do you figure out what gate a unitary matrix represents?

I am trying to learn Qiskit on my own. I am struggling with unitary matrices. I understand what a unitary matrix is, and why a matrix is unitary. But, I don't understand what the values inside of the ...
1 vote
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### Prove that the eigenvectors of a Hermitian operator form a basis

While I was reading the book Quantum Mechanics The Theoretical Minimum, the author said that if a vector space is $N$ dimensional, an orthonormal basis of $N$ vectors can be constructed from ...
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### What is the correct order to multiply vectors?

I saw the equation: $$|A\rangle =\sum_i\alpha_i|i\rangle$$ Which could be also written like that:$$|A\rangle=\sum_i|i\rangle\langle i|A\rangle$$ I know that $\langle i|A\rangle=\alpha_i$, but can I ...
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### Is the inner product operation commutative or associative?

I am currently reading Quantum Mechanics The Theoretical Minimum by Leonard Susskind. In the second lecture he says that for a given state of a spin $|A\rangle = a|u\rangle + b|d\rangle$: The ...
1 vote
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### Obtaining the unitary matrix of a gate in analytic form in qiskit

Is there any way to obtain the unitary matrix of single way in analytic form? In numerical form, I could use a workaround like this (i.e. for an X Gate): ...
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### How to predict state vector of a quantum circuit in IBM qiskit

How to Predict the state vector output of the following quantum circuit? ...
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### Why is the following exponential ignored (or equals 1) in the probability amplitude?

I'm reading Ronald de Wolf's lecture notes and when explaining Shor's algorithm on page 40 after applying a QFT to $$\frac{1}{\sqrt{m}} \sum_{j=0}^{m-1} |jr+s\rangle$$ the following expression is ...
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### How to achieve the controlled rotation in the HHL algorithm

I'm trying to implement the HHL algorithm generally for any 2 x 2 hermitian matrix, but I'm having trouble with the implementation of the controlled rotation of the ancilla qubit. I've read very many ...
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### Schur's lemma for quantum states

I am trying to understand Lemma 2 in this paper. Consider a state $\tau_{H^n}=\int \sigma^{\otimes n}_{H} \mu(\sigma)$ where $\mu(\sigma)$ is the measure on the space of density operators on a single ...
1 vote
Let $d$ be a natural number. Given $A_1,\dots,A_r\in M_d(\mathbb{C})$, define a linear operator $\Phi(A_1,\dots,A_r):M_d(\mathbb{C})\rightarrow M_d(\mathbb{C})$ by letting \$\Phi(A_1,\dots,A_r)(X)=...