# Questions tagged [density-operator]

The density operator describes a quantum system in an (in general mixed) state.

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### Tensor product between a density matrix and a ket vector

What's the tensor product of the $2\times2$ matrix $\rho = \begin{bmatrix} 2/3 & 0.3 \\ 0.3 & 1/3 \\ \end{bmatrix}$ and $|\Psi\rangle =$ cos$(\theta)|0\rangle +$ sin$(\theta)|1\rangle$? ...
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### How do we determine what is the temperature (or beta or energy) of a quantum system?

In statistical physics, we learn about the "inverse temperature of the system" as $\beta = \frac{1}{k_B T}$. Now in most cases we'd leave $\beta$ as a free parameter, and then calculate the (say) the ...
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### What is the difference between complex and real coherneces?

If we have one density matrix $$\rho=\begin{bmatrix}1/2&&1/2\\1/2&&1/2\end{bmatrix}$$ for the state $\lvert\psi\rangle=\frac{1}{\sqrt2}(\lvert g\rangle+\lvert e\rangle)$ and a ...
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### Quantum Gates on Density Matrices

Let's say we have some general mixed state $$\Big(\begin{matrix} \alpha & \beta \\ \beta^\dagger & \delta\end{matrix}\Big)$$ Purely a mechanical question, how might I apply, say, a NOT ...
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### Time evolution operator for system -environment interaction

I am reading a paper https://journals.aps.org/prb/pdf/10.1103/PhysRevB.96.224302. In this paper the initial state of the system and environment is given as |\Psi(0)\rangle=|\phi_{s}(...
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### How to express random spin up / spin down particle or beam in spin z basis?

If I express it like this: $$\psi = \frac{1}{\sqrt 2} \lvert +z \rangle + \frac{1}{\sqrt 2} \lvert -z \rangle$$ that will give a $50\%$ spin up / $50\%$ spin down measurement along $z$ which is ...
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### Discarding a quantum system in joint state

If you have a joint quantum state given by the density operator $$\rho^{(XYZ)} = \sum_{k}p_k\rho_{k}^{(XY)} \otimes |k^{(Z)} \rangle \langle k^{(Z)}|$$ then am I correct in stating that if we want to ...
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### von Neumann Entropy of a joint state

Definition 1 The von Neumann entropy of a density matrix is given by $$S(\rho) := - \mathrm{Tr}[\rho \ln \rho] = H[\lambda (\rho)]$$ where $H[\lambda (\rho)]$ is the Shannon entropy of the set of ...
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### How to prove that $\mathrm{tr}(\rho^2)=1$ if and only if the state is pure?

How can I prove that $\mathrm{tr}(\rho^2)$ = 1 if and only if the state is pure? My idea: I know how to show that $\mathrm{tr}(\rho^2) \leq 1$ and from there I am trying to show by contradiction ...
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### What is the state of a single electron in an entanglement?

If we consider a bipartie system that is entangled, $$|\psi^{AB}\rangle=\frac{|{\uparrow_z\downarrow_z}\rangle-|{\downarrow_z\uparrow_z}\rangle}{\sqrt{2}}$$ And we want to know the probability ...
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### Is there a clear and intuitive meaning to the eigenvectors and eigenvalues of a density matrix?

Is there a clear and intuitive meaning to the eigenvectors and eigenvalues of a density matrix? Does a density matrix always have a a basis of eigenvectors?
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### Magnitude of off-diagonal terms in density matrix

I want to prove that if I have a density matrix of the form: $$\begin{pmatrix} p_{++}& p_{+-}\\ p_{-+}&p_{--} \end{pmatrix}$$ then $|p_{+-}|^2 = |p_{-+}|^2 \le p_{++}p_{--}$. (This was ...
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### Green functions and density matrix

tl;dr The single particle density matrix is directly related to NEGF as shown here, I wish to find a way to relate NEGF also to density matrices which describe probability distribution of many body ...
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### What is the importance of a diagonal density matrix after measurement/decoherence?

Let $|\psi_{SA}\rangle$ be the state of a system and an apparatus, for example an electron spin and a Stern-Gerlach apparatus. If $|\psi_S\rangle=\alpha|\uparrow\rangle+\beta |\downarrow\rangle$ ...