Questions tagged [density-operator]

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

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Expectation value in a mixed state of two distinguishable spin-$1/2$ particles

Consider two distinguishable spin-$1/2$ particles in a state with total spin $S=1$, described by the density matrix $$\rho=\frac{1}{3}\left[\left|11\right\rangle\left\langle11\right|+\left|10\...
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Aharonov-Bohm using density matrix?

On the one hand, we know that the overall phase of the wave function (of the whole system) is not a measurable quantity, but more an artifact of mathematical description — the physical states are rays ...
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Measurement on density operator in eigenbasis of operator

Suppose I have some density operator, expressed in the position eigenbasis: $$\rho =\int p(x)|x\rangle\langle x | dx,$$ where $p(x)$ is some probability density. Then a projective position ...
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Can we always find a Hilbert space corresponding to a region of spacetime?

In defining entanglement entropy in field theories we take a region A in spacetime. Now if $\mathcal{H}$ is the Hilbert space of the field theory we assume that we can decompose this Hilbert space ...
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Probabilities with density matrices

Given two mixed states $\rho$ and a $\sigma$, does it make sense to say that the probability of $\rho$ being in the state $\sigma$ is given by $Tr(\rho \sigma)$? It seems to me that the answer must ...
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Distributive property of tensor product

I have a homework problem in nuclear magnetic resonance. After a bunch of calculations, I have arrived at the expression: $$\langle M_1(t)\rangle = {\rm tr}\left [\rho(0)\sigma_+^{(1)}\exp\left(i\frac{...
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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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How to derive reduced density matrix of $a|0_A\rangle|0_B\rangle + b|1_A\rangle|1_B\rangle$

Suppose the quantum system has state $$a|0_A\rangle|0_B\rangle + b|1_A\rangle|1_B\rangle.$$ I want to prove that system $A$ has reduced density matrix of $a^2|0_A\rangle\langle 0_A| + b^2 |1_A\rangle\...
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Constructing a CPTP-map on one density matrix using another

My question is: If one is given two density matrices $A$ and $B$, is there a way to use the first to construct a CPTP-map (quantum channel) acting on the on the other? I thought that Stinespring ...
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692 views

Von Neumann entropy: How do we get $S(\rho)=-\sum_i p_i \ln(p_i)$?

I would like to understand why the Von Neumann entropy can be written like this : $$ S(\rho)=-\sum_i p_i \ln(p_i)$$ as written here: https://en.wikipedia.org/wiki/Von_Neumann_entropy Indeed, if I ...
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Finding the density matrix

I'm trying to calculate the density matrix of the state $$|\psi \rangle=4|00\rangle + 3i|11\rangle -i|01\rangle + 2|10\rangle$$ And I've approached the problem by multiplying out $|\psi\rangle \...
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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 \begin{equation} |\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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Are there any problems that can only be tackled with density matrices and not with pure state evolution?

Say I have a state $|\Psi_0\rangle$. I measure observable $\hat{A}$, the wavefunction collapses to one its eigenstates. I can write $|\Psi_0\rangle = \sum_j \alpha_j|\psi^A_j\rangle$, where $|\psi^...
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$P$ representation of the thermal density operator

I'm trying to derive the P representation for the thermal state $$ \rho = \sum_{n=0}^\infty \frac{\mathrm{e}^{-\beta \omega n}}{Z} |n\rangle \langle n | $$ where $\beta$ is the inverse temperature, $...
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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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Why should we diagonalize the density matrix? Why is the diagonal representation more special?

Suppose I have $$\rho=\sum p_k \vert \phi_k \rangle \langle \phi_k\vert = \sum \lambda_k \vert \lambda_k \rangle \langle \lambda_k\vert$$ where I call the second decomposition "orthonormal", meaning $\...
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Can we make the spectrum decomposition of the Hamiltonian and the density matrix with the same basis?

The following screenshot comes from this book: Nonequilibrium many-body theory of quantum systems: A modern introduction. In the discussion, the authors have performed the spectrum decomposition for ...
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What information does the Von Neumann entropy give for mixed states?

Von Neumann entropy is defined as $$ S=-\mathrm{Tr}\left(\rho \ln\rho\right) $$ It can be used to measure the entanglement between two sub-systems, provided that the total system is in pure state. ...
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What's the physical meaning of the kernel of density matrix?

The kernel of this linear map is the set of solutions to the equation A x = 0, where 0 is understood as the zero vector. But what's the physical meaning of the kernel of density matrix?
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Is the partial trace of a mixed state always mixed? If not, are there natural examples where the partial trace of a mixed state is a pure state?

I know that the partial trace of a pure entangled state must be mixed and that of a product pure state must be pure; but I couldn't find an answer to my above question.
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Time Dependent Density Matrix

Given a wave function $\left|\psi\right>$ at time $t=0$ and an Hamiltonian, how does one find the time-dependent density matrix? So far I calculated the density matrix of the wave function, which ...
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How to verify non-negativity of a density matrix?

A density matrix, $\rho$ must be Hermitian, normalized ($Tr[\rho]=1$) and non negative. Non negativity means that it should have non negative eigenvalues. Given a density matrix, first two conditions ...
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From the Heisenberg-Langevin equation to the Lindblad equation

In a open quantum system, one can easily derive the Heisenberg-Langevin equation of motion which describes the time evolution of creation/annihilation operators (in say, a cavity) $$\dot{a}(t) = i[H,...
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1answer
180 views

When constructing Hamiltonian matrix for many spins, what is the significance of the order of factors in the outer product?

I'm trying to learning the Density Matrix Renomalization Group (DMRG) method from the book "Strongly Correlated Systems: Numerical Methods". For a two-spin system they build a Hamiltonian from the ...
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Why do unitary transformations on ensembles of states result in the same density matrix?

This is from Nielsen and Chuang. If we have an ensemble of pure states that obey the following relationship for all $i, j$ $\vert \psi_i \rangle = \sum_{j} u_{ij}\vert \phi_j \rangle$ where $u_{ij}$...
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Entanglement Entropy of two Spin systems

I'm given a quantum system with a density matrix $\rho$ which is broken down into two part systems A and B, which can be entangled. The system has two Spin-$\frac{1}{2}$-degrees of freedom (either up ...
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Density matrices, off-diagonal terms, coherences and correlations

I'm trying to better understand the off-diagonal terms of the density matrix - an often brought up question on this site I realise. Specifically my confusion at the moment concerns the interpretation ...
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Density matrix of a mixture of singlets and triplets

I was wondering how to write the density matrix of two spin-$\frac{1}{2}$ in a mixture of singlets and triplets. By definition for a mixture, $$ \hat{\rho}=\sum_{j}w_j\, |\psi_j \rangle\langle \psi_j| ...
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Reduced Density Matrix Fermions

Suppose we've got a fermionic state $$ \rho=\bigotimes_{i=1}^N|v_i\rangle\langle v_i| $$ where $|v_i\rangle=\left(\sqrt{d_i}c^\dagger(\psi_i)+\sqrt{1-d_i}c^\dagger(\eta_i)\right)|0\rangle_i$. The ...
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Is density matrix a tensor?

Is density matrix a tensor? Would it change if we represent it in another basis as a tensor would? Is there any difference in this regard between pure and mixed quantum states? (I definitely can ...
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Time evolved density matrix

Working with an uncoupled harmonic oscillator Hamiltonian: $H = H_A + H_B$, where $H_A = \hbar \omega (a_+ a_{-} + 1/2 )$ and $H_B = B \sigma_z$. I'm trying to calculate the density matrix $\rho (t)$...
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Does a statistical system go into a pure state as the temperature $T\to 0$ (or $\beta\to\infty$)?

The density matrix $\hat{\rho}$ for a canonical ensemble is given by $$\hat{\rho}=\frac{\sum\limits_{n}e^{-\beta E_n}|n\rangle\langle n|}{Z}\tag{1}$$ $$=\frac{e^{-\beta E_0}}{Z}\Big[\sum\limits_{n=0}|...
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Solvable model with harmonic oscillator

I need help with a mathematical physics question. I have given the following system: A spin is coupled to a single harmonic oscillator mode with Hamiltonian $$H=(\epsilon/2) \sigma_{z} + \omega\, a^{...
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How is tracing out a physical operation?

Suppose $\rho_{AB}$ denotes the density matrix of a bipartite system. Reduced density matrix of A ($\rho_A$) is obtained by tracing out B $$\rho_A\equiv\sum_{i}\langle i_B |\rho_{AB}|i_B\rangle$$ ...
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Will entanglement change if we diagonalize a mixed state?

I know that unitary operations do not alter the eigenvalues of a density matrix, i.e. its purity(mixedness) is conserved. However entanglement is only invariant under local unitary operations of the ...
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How do I plot the decoherence of an open system from its density matrix?

If I have a two qubit state interacting with an environment that will decohere it, how do I model the decoherence from the density matrix? For example, if I start with some state $\Psi(0)=|0>_1|1&...
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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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291 views

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 $|\psi_S\rangle=\alpha|\uparrow\rangle+\beta |\downarrow\rangle$ ...
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Quantum Brownian Motion - Calculation of moments [closed]

The master equation of quantum brownian motion is derived as \begin{equation}\frac{d}{dt} \hat{\rho_s}(t) = -i[\hat{H_S} + \frac{1}{2}M\tilde{\Omega}^2 \hat{X}^2 , \hat{\rho_s}(t)] -i\gamma[\hat{X}, \{...
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Relation between Kraus operators and the Choi matrix

Let $\Phi$ be a CPTP map on density operators for a fixed $n-$dimensional state space and fix a basis $\{ | j\rangle \}$. I'm trying to understand the relationship between the Choi matrix $$M_\Phi:= \...
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Ineffecient continuous measurements in quantum measurement theory

In a quantum measurement theory text "Quantum Measurement Theory and Applications" by K.Jacobs, he defines and explains the idea of a continuous measurement, which is a measurement $$y = x_{true} + \...