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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How does one generate a random $N \times N$ density matrix with quaternionic entries — with respect to Hilbert-Schmidt measure?

In the article https://arxiv.org/abs/0909.5094 the formula (eq. (1)) \begin{equation} \rho_{HS} =\frac{A A^{\dagger}}{\mbox{Tr} A A^{\dagger}} \end{equation} is given, for generating a random $N \...
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Investigate noise in measures for Quantum-Non-Markovianity

I'm currently working on investigating noise and how it effects measures for non-markovianity in quantum systems...the problem is that i'm not really used to 'doing science' on my own (just recently ...
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One-body reduced density matrix

Assume there is a N-particle state denoted as $|\Psi_N\rangle$, the density operator from its definition reads $\gamma_N = |\Psi_N\rangle\langle\Psi_N|$ and the density matrix elements take the form ...
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understanding resolving power in QM

Consider the definition of resolving power of two states in quantum mechanics as being the absolute value of the difference between the probabilities of two states following the measurement averaged ...
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308 views

Recognizing Entanglement from the Density Matrix in 2-qubit case

We know that the density matrix for a 2-qubit system can be written in the Pauli representation as : $$\rho = \frac{1}{4}\sum_{ij}t_{ij}\sigma_i\otimes\sigma_j$$ where $\sigma_i$ are the Pauli ...
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Normalization of density operator in combined system in QM

If we consider a combined system of quantum systems $A$ and $B$ with probability density $\rho^{C} = \rho^{A} \otimes \rho^{B}$. If we measure system $A$ to be in state $|a_0⟩$. Then the state of the ...
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On Stinespring's Theorem

Wherein the proof of Stinespring's theorem does complete positivity enter? Suppose my map is not completely positive but the Kraus operators follow the trace preservation condition as $\sum_{n = 1}^{...
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454 views

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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261 views

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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283 views

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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1answer
348 views

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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What's the physical meaning of the eigenvalues of the spin-flipped density matrix?

In the computation of the entanglement of formation(EoF) of a 2 qubits mixed state, $\rho$, according to Wooters, we need to compute the concurrence of the state by computing the eigen values $\{\...
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Maximizing entropy in QM (Sakurai's Modern Quantum Mechanics)

There is section in Sakurai's "Modern Quantum Mechanics 2nd edition" page 188 that is quite confusing as to what he is doing. In the section on "Quantum Statistical Mechanics" he defines a quantity $\...
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329 views

Fermi golden rule and density matrix

Fermi Golden Rule expresses up to the first order the rate of departure from a state $|\psi_i>$ under the influence of a perturbation $V$ $$ W=\frac{2\pi}{\hbar} \int dk_f \mathcal{D}(k_f) \, \left|...
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Why do populations only change in second order of the driving field?

In the field of quantum optics when solving master equations it is well know that the populations1 are constants to linear order in the driving field. I.e. weak driving fields will only affect the ...
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Why density matrices in QFT are calculated only by going to Euclidean metric?

I have been reading a few papers on entanglement entropy. I noticed that whenever people calculate either the density matrix or the reduced density matrix of a specific region, it is usually done by ...
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Quantum entanglement for indistinguishable particles

When I encounter the definition of the mathematical definition of quantum entanglement. System composed by many parts $A$, $B$,.., $N$ can be described by a density matrix operator $\hat{\rho}$ acting ...
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479 views

When is the density matrix real & symmetric?

Book: Statistical Mechanics (3rd ed.) by R K Pathria Page 118, Chapter 5, Sec 5.1, Eq. 13 The author says that the density matrix is real & symmetric if the system is in equilibrium. Can somebody ...
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181 views

Off Diagonal Zero Entries in Pure State Density Matrix

Let $$\rvert\psi\rangle = a_u\rvert u\rangle + a_d\rvert d\rangle = \langle u\rvert\psi\rangle | u\rangle + \langle d\rvert \psi\rangle |d\rangle,$$ where $$\rvert u\rangle = \begin{pmatrix} 1 \\ 0 ...
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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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515 views

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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399 views

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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143 views

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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1answer
111 views

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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379 views

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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1answer
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Off-diagonal elements of density matrix, measurement of coherence?

I have a ensemble of systems and each system is made of a single one-dimensional quantum harmonic oscillator. Suppose all systems in the ensemble are in the following quantum state $$ |\Psi\rangle = \...
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1answer
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Derivative of the trace of $e^{-\beta \mathbf{A}}$

I'm trying to compute the derivative with respect to an inverse temperature parameters $\beta$ of a density matrix that has the following form: $$\rho(\beta,\mathbf{A}) = \frac{e^{-\beta \mathbf{A}}}{...
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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(...
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539 views

Time-dependent Hamiltonian and the Liouville-von Neumann equation

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{\...
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Operators in Heisenberg picture [duplicate]

An operator $\hat{Q}(t)$ can be written as $\hat{Q}(t)= e^{iHt} \hat{Q(0)} e^{-iHt}$ in Heisenberg picture. Let us choose $\hat{Q(0)} = |{m(0)}\rangle \langle{m(0)}|$. Then $e^{iHt} \hat{Q(0)} e^{-...
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3answers
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How are linear combinations of qubit states represented in the Bloch sphere?

According to the Wikipedia article on the Bloch sphere, a pure state of a qubit can always be represented as $$| \psi \rangle = \cos \left( \frac{\theta}{2} \right)| 0 \rangle + e^{i \phi} \sin\left(\...
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How to understand Bloch sphere representation?

I'm really new to quantum computation. Now, I'm going through a tutorial article Quantum Computation: a Tutorial (NB: PDF). I was confused by certain points over there. So, on page 5, when the ...
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432 views

Dirac delta function property in a scattering proof

I'm studying the proof for the decoherence of the off diagonal elements of a density matrix through scattering with the environment and I'm stuck at a certain point: My problem is A1.14 relation. (A1....
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1answer
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How one arrives at the Lüder's Rule?

When working with density matrices the collapse axiom of quantum mechanics says that a measurement of $S$, with result $s_i$, transforms the original density matrix $\sigma_{0}$ into the conditional ...
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About the proof of the subadditivity of the von Neumann entropy

I'm trying to understand the proof of the so called subadditivity of the von Neumann entropy, $$S(\rho^{AB}\,)\leq{S}(\rho^A)+S(\rho^B)$$ where $S(\rho)=-\mathrm{tr}\{\rho\log\rho\}$. In the proof I ...
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Reduced density matrix does not reproduce expectation values

It seems I have found a counter example of the theorem that states that the reduced density matrix reproduces the expectation values in that corresponding subsystem: We set $\bar{h}=c=1$. The system ...
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103 views

Maximum entropy of a pure state passed through a quantum channel

What is $\underset{\rho\ \text{pure}}{\text{max}}S(\mathcal{N}^{\otimes N}(\rho))$? where $S$ is the Von Neumann entropy defined as $S=-\text{tr}(\rho\ln\rho)$, $\mathcal{N}$ is a quantum channel (i....
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What does it mean for a density operator to be non-negative?

I am taking a quantum mechanics course, and my professor gave us these general properties of a density operator: Any Hermitian operator $\rho $ is a density operator iff: $$1)\space Tr(\rho) = 1$$...