# Questions tagged [hilbert-space]

This tag is for questions relating to Hilbert Space, a vector space equipped with an inner product, an operation that allows defining lengths and angles. It arises naturally and frequently in mathematics and physics, typically as infinite-dimensional function spaces having the property that it is complete or closed. Applies also to pre-Hilbert spaces, rigged Hilbert spaces, and spaces with negative norm or zero-norm states.

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### How can you calculate a reduced matrix element $\langle 0|\varepsilon|0\rangle$? [closed]

How can you calculate a reduced matrix element $\langle 0|\varepsilon|0\rangle$ ? $\varepsilon$ is a polarization vector. I also want to know the lists on the webpage that says some reduced matrix ...
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### XOR/CNOT on 2 qubits in superposition [closed]

What is $(|0⟩ - |1⟩) ⊕ (|0⟩ + |1⟩) = ?$ or more elaborate what is the result of the CNOT gate when a and b are in superpositions CNOT$|a,b⟩→|a,a⊕b⟩$, having $a = (|0⟩ - |1⟩)$ and $b = (|0⟩ + |1⟩)$
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### Proof that joint-measurability means commutativity

For $i=1,2$, two measurements $m_{i}:\mathcal{X}_{i}\to\mathcal{L}(\mathcal{H})$, from alphabet $\mathcal{X}_{i}$ to set of bounded linear operators on Hilbert space $\mathcal{H}$, are compatible or ...
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### What is the dimension of two quantum systems with bases $\{a_1,a_2\}$ and $\{b_1,b_2,b_3\}$, combined? [closed]

Quantum system A has a basis $\{a_1, a_2\}$. System B has a basis $\{b_1, b_2, b_3\}$. A and B evolve according to their own Hamiltonian and do not interact at all. If I consider A and B as one large ...
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### Commutation between hermitian operator $H$ and the operator $U(m,n) = |\varphi_m\rangle \langle \varphi_n|$

I was trying to do this exercise from Cohen-Tannoudji Quantum Mechanics book: $|\varphi_n\rangle$ are the eigenstates of a Hermitian operator $H$ ( $H$ is, for example, the Hamiltonian of some ...
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### What does cohomology of $Q_B$ mean in BRST quantization in Polchinski?

While proving no-ghost theorem ($4.4$ Polchinski) the term cohomology of $Q_B$ is used quite a lot of time. From what I understand this has to be a set since "cohomology of $Q_B$" is ...
### Matrix representation of the operator $\vec{Z}: \langle l^{'} , m^{'} | \vec{Z} | l, m \rangle$
I know that $[L_z, z] = 0$, which means that the operators $L_z$ and $\vec{Z}$ share a common eigenspace. So, I wonder how I can get the matrix representation of the operator $\vec{Z}$ in the ...