# Questions tagged [density-operator]

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

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### Identity Matrix in other bases

Suppose I have the density matrix: $$\rho = p|\psi^-\rangle\langle\psi^-| + (1-p)\times \frac14 \mathbb{I}_4 \,,$$where $p$ is some probability $<1$, $\mathbb{I}_4$ is the $4\times 4$ identity ...
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### Preparing a state $|\chi\rangle=\frac{1}{\sqrt{2}}|\frac{1}{2},+\frac{1}{2}\rangle+\frac{1}{\sqrt{2}}|\frac{1}{2},-\frac{1}{2}\rangle$

Consider the spin state of a particle of spin $s=\frac{1}{2}$. As far as the spin degrees of freedom are concerned, the operators $\textbf{S}^2$ and $S_z$ form the complete set of commuting ...
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### Density operator matrices of pure and completely mixed ensembles

I want to confirm that the density matrix corresponding to a pure ensemble depends on the basis you choose in the following sense and then extend the idea to completely mixed states: We have the ...
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### Difference between green's function and density matrix

What is the basic difference between green's function or propagator of given system and density matrix (in the position basis) of the same system ? Can some one explain the difference between with ...
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### POVM for sequential interactions

Consider a state $\rho_{A}\otimes\rho_{B}$ where both $\rho_{A}$ and $\rho_{B}$ are qubit system, undergoes unitary evolution as follows \begin{equation} \rho_{AB}'= (\cos\theta\mathbb{I}_{A}\otimes\...
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### Numerically Calculate expectation of $xp+px$?

I'm curious if there is a quick way to numerically calculate $\langle xp + px \rangle$ if we had the density function of our system. For example, if for x we can take the density function in the x-...
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### How does the addition of two wavefunctions develop in time?

Two time dependent wavefunctions: $\Psi _1(t)= \psi_1*exp(\frac{-i * E_1}{\hbar}*t)$ $\Psi _2(t)= \psi_2*exp(\frac{-i * E_2}{\hbar}*t)$ Both a solution to the timeindependent (note "in") ...
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### Algebraic Formalism of Quantum Mechanics

In the algebraic formalism, the physical system is described by its observables, viewed as self-adjoint elements in a certain *algebra and a state is a linear functional, if I understand right. Can ...
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### Criterion for 3-qubit separability

Is there a necessary and sufficient criterion to verify if a 3-qubit density matrix is separable? I understand that for 2-qubit systems one can use the Peres-Horodecki criterion which is a ...
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### How to decompose a density matrix of a mixed ensemble into a sum of pure ensembles [closed]

I'm trying to solve a problem where I am given a few matrices and asked to determine if they could be density matrices or not and if they are if they represent pure or mixed ensembles. In the case of ...
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### Multiplying two different density matrices, what physical situation will such need arises?

This question arises from a reading in quantum chemistry: In this link, the natural bonding orbitals (NBO) $\Theta_k$ (basically localised versions of molecular orbitals) are the eigenfunctions of a ...
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### Positive partial transposition for systems of more than 2 qubits?

Say we start with a 4-qubit pure state that does not satisfy the Peres-Horodecki criterion, i.e., its partial transposition has negative eigenvalues. If I trace out one of its qubits, I will (most ...
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### Can one ensure the separability of 3-qubit density matrices by requiring a positive partial transpose plus a range criterion?

From what I understand, the positive partial transpose (PPT) criterion is not a sufficient separability condition for 3-qubit states. I have found that the range criterion is also a necessary ...
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### Analogy expectation of an observable / random variable

I'm trying to figure out the analogies between the expectation of a random variable $X$ and the expectation of an observable of a quantum mechanical system $A$ (using this wikipedia article). The ...
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### Measurement on density operator

Question: A system in a mixed state $\rho$ is measured with the measurement described by a projection operator $P$. What is the probability of the outcome? What is the density operator ...
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### Spectral decomposition of a time dependent operator

Let $M$ be an operator with spectral decomposition $$M(0) = \sum_{i = 1}^n \lambda_i \left|{m_i(0)}\right\rangle\left\langle{m_i(0)}\right|.$$ To find $M(t)$, I know that $M(t) = U^{\dagger} M(0) U$. ...
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### Help With Deriving Caldeira-Leggett's Influence Functional

I am attempting to retrace steps performed numerous times before and derive Caldeira-Leggett's influence functional found in their paper "Path Integral Approach To Quantum Brownian Motion". However, ...
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### What cannot be calculated using density matrix?

Let's say I have a compound quantum system (CQS) in an (unknown to me) pure state $\left|\Psi\right>$. If an operator $\mathbf{A}$ acts only on variables of a subsystem (S) of CQS, then I can ...
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### How to tell if a density matrix is separable?

Consider the following 4-qubit entangled state $$\left|\psi\right>=\left|0000\right>+\left|1110\right>+\left|1101\right>+\left|1011\right>$$ By tracing out qubits A and B (where I am ...
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### What is the maximum number of linearly independent density matrices in a 2D space?

I think for a generic 2 by 2 matrix we would require 4 linearly independent matrices to span the whole space, 1 per element. Then I would assume that the two constraints for a legitimate density ...
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### How do we arrive that the form of the thermal density matrix?

I'm following the notes by Pieter Kok, the relevant material for this question is around page 29. I learned about the density matrix for just a single spin-1/2 particle with unknown initial conditions,...
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### On explaining QM by modeling the observation mechanism using a density operator

I just have some elementary thoughts on the foundations of QM, based on modeling the observation mechanism by a density operator. A density operator is a weighted sum of orthogonal projections on the ...
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### Is this density operator unphysical?

I have been told that this form of density function: $$\rho(q)=\sum_{\mathbf p} M(\mathbf p ,\mathbf q)h^\dagger_\mathbf ph_{\mathbf p - \mathbf q}$$ where $h^\dagger(h)$ is the holon creation(...
If I have a quantum system described by a time-independent Hamiltonian $\hat{H}$, then the Liouville-von Neumann equation is \begin{align} i\hbar\frac{\partial\hat{\rho}}{\partial t}=[\hat{H},\hat{\...