Questions tagged [density-operator]
The density operator describes a quantum system in an (in general mixed) state.
485
questions
0
votes
1answer
36 views
Positive Semi-Definiteness of a Density Matrix - can the eigenvalues be larger than 1?
I know that one of the requirements for a density matrix is that it is positive-semidefinite. This means that the eigenvalues are non-negative (and sum to 1, so we can assign them the meaning of ...
2
votes
1answer
80 views
Do pure states actually exist?
Is it possible to ever actually measure something so precisely that it actually collapses to a pure state, or do we really just get arbitrarily close? If a wavefunction never actually collapses to a ...
0
votes
2answers
59 views
Can a density matrix have more than two dimensions?
It crossed my mind when reviewing density matrices that if you were looking at a composite system consisting of three subsystems, (indexed by three quantum numbers: $<i,j,k|\rho|i,j,k>$) then ...
0
votes
0answers
55 views
Von neumann entropy for reduced density matrix
I have reduced density matrix of the form (corresponding to 2 coupled harmonic oscillators):
$$\rho_r (x_1, x'_1)= \frac{\text{sech}(\eta ) \sqrt{\frac{m\omega \cosh
(\eta )}{\hbar }} \exp \left(-...
0
votes
1answer
35 views
Could the definition of the experiment result occur before measurement?
Is there any fundamental demonstration that result of an experiment cannot have been determined before the measurement, yet according to the probabilistic rules of quantum mechanics?
I understand ...
0
votes
0answers
29 views
Confusion with mixed state [duplicate]
I have read that mixed state is a collection of pure states ...while a pure sate is a collection ie suoerposition of eigen states is that right?..so it can be thought of as a superposition of ...
-1
votes
2answers
63 views
Why are entanglement and purity non-linear functions of $\rho$?
Any linear function of the density matrix can be related to a proper observable, but is it not the case of entanglement and purity?
0
votes
2answers
208 views
Compact expression of Maxwell's equations: is there a miss minus sign?
The compact form of Maxwell's equations:
$$\boxed{\square\, \boldsymbol{\mathsf{F}}=\mu_0 \boldsymbol{\mathcal{J}}} \tag{1}$$
where the current density quadrivector is given by the relation $\...
5
votes
2answers
100 views
Interpretation of density matrix
In Landau’s Statistical Physics (part 1) , section 5, he writes:" In particular, it would be quite incorrect to suppose that the description by means of the density matrix signifies that the subsystem ...
2
votes
1answer
52 views
What does a coordinate representation of density matrix mean?
A coordinate representation of density matrix $\rho$ is defined as
$$ \rho (x, x') \equiv \left<x\right| \rho \left|x'\right> .$$
When $x = x'$, this expresses a probability where a particle ...
2
votes
2answers
53 views
Do the pure states in the decomposition of a density operator need to be orthonormal to each other?
So, I was studying quantum computation using the book Nielsen and Chuang and it stated a theorem known as "Spectral Decomposition theorem"
$$A=\sum _{i}\lambda _{i} | i \rangle \langle i|$$
I infer ...
2
votes
1answer
82 views
Is a partial trace cyclic?
I want to know if a partial trace keeps the cyclic property of the trace.
The partial trace is defined as
$$ tr_B: \mathcal{B}_1(\mathcal{H}_A\otimes \mathcal{H}_B) \longrightarrow \mathcal{B}_1(\...
2
votes
0answers
28 views
Expanding a density matrix in terms of operators
In Lukasz's paper:
https://arxiv.org/pdf/0909.2654.pdf
He writes "consider a density matrix ρ, written as a polynomial of the 2N Majoranas cj in such a way that each cj occurs to the power 0 or 1 in ...
0
votes
0answers
45 views
Direct Derivation of Kraus Operator from Interaction Hamiltonian
For the dynamics of open quantum systems, the Kraus operators $K_\kappa$ can be derived from the unitary orbit $U(t)\rho U(t)^\dagger$ for $\rho=\rho_S\otimes\rho_E$ of the composite system given by ...
2
votes
1answer
58 views
Necessary and sufficient conditions for a pure state
I've seen some claims that idempotency ($\rho^2=\rho$) is necessary and sufficient to guarantee the existence of some state $\psi$ such that $\rho=|\psi\rangle\langle\psi|$, as well as claims on the ...
0
votes
0answers
17 views
Non-diagonal elements in density matrix: interference effect between the statistical mixture
In Cohen-Tannoudji, at page 303, vol 1, it is given that
Quote:
\begin{array}{l}{\text { A calculation analogous to the preceding one gives the following expression }} \\ {\text { for the non-...
0
votes
1answer
60 views
Entanglement of Werner States
Let the Werner state $$\rho_W = W\mid\Psi^-\rangle\langle\Psi^-\mid + \frac{1-W}{4}\mathbb{I},\ W\in [0,1],$$
where $|\Psi^-\rangle=(|01\rangle-|10\rangle)/\sqrt{2}$.
I have repeatedly heard that ...
2
votes
0answers
34 views
Diagonalisation of quasi-thermal state
I have the following density operator $$\frac{1}{t \pi N} \int_{\mathbb{C}} \mathrm{d}^2\gamma \exp
\left[
-\frac{|\gamma+r\alpha|^2}{t^2 N}
\right]
|{\gamma}\rangle\langle{\gamma}|,$$
where $0\leq t,...
3
votes
0answers
19 views
Signal coherence/correlation vs quantum coherence
In general, I understand a signal $s(t) \in \mathbb{C}$ is called "coherent" when it has a large autocorrelation function. A pair of different signals $s(t)$, $r(t)$ can also be "coherent" if their ...
1
vote
0answers
18 views
Have you ever seen: $\sqrt{\rho_{ee}\rho_{gg}}$-|$\rho_{eg}$|?
The next term appears in my research and it is quite meaningful: $\sqrt{\rho_{ee}\rho_{gg}}$-|$\rho_{eg}$|
Where $\rho_{gg}$ and $\rho_{ee}$ are the populations in the excited and ground states, and $...
3
votes
0answers
44 views
Relation between maximally mixed state and thermal state
Hawking calculated the density matrix of the outgoing radiation to be a thermal state. I have heard people say this is a maximally mixed state. Is this because given a fixed average energy in the ...
3
votes
0answers
100 views
Open Quantum Systems: Born-Approximation and the preservation of Trace, Hermicity and Positivity
This is related to a previous question of mine.
We consider a density matrix $\sigma(t)$ operating on a Hilbert space $\mathscr{H}_{s}\otimes \mathscr{H}_b$ with Hamiltonian $H = H_s \otimes \mathbb{...
8
votes
2answers
317 views
Why can't every quantum state be expressed as a density matrix/operator?
It was my previous impression that all quantum states in a Hilbert space can be represented using density matrices† and that's already the most general formulation of a quantum state. Then I came ...
1
vote
2answers
49 views
About Density Matrix of a Particle
The quantum state of a spin- 1/2 particle can be written, in the momentum representation, as a two-component spinor,
$$\textit{Ψ}(\textbf{p})=\left(\begin{matrix}a_{1}(\textbf{p})\\a_{2}(\textbf{p})\...
0
votes
1answer
49 views
Probability of finding a particle in a superposition
In QM, is it possible to ask what the probability of finding a particle in a superposition will be?
Once a particle is in a superposition, it is possible to find out the probability that it will be ...
0
votes
1answer
63 views
Integrating of von Neumann equation for density matrix
Suppose we are given the Hamiltonian
$$H=f \frac{\text{Tr}\sigma_x \rho}{\text{Tr}\rho}\sigma_x,$$
where $\rho$ is the density matrix, and $\sigma_x$ is the Pauli matrix
$$
\sigma_x=
\begin{...
3
votes
1answer
72 views
Is tracing out a subsystem always akin to discarding all information about it?
Suppose we have some quantum system with sub-systems A and B. It could be, for example, two qubits or groups of qubits. Is it fair to say that tracing out the sub-system A is always akin discarding ...
1
vote
0answers
27 views
Tracing $\rho (t)$ with respect to the Bath when system and bath are coupled in an open quantum system
Consider a system S that is coupled to a bath B. Let {$|s_i\rangle 's$} and {$|b_j\rangle 's$} be the eigen states of the system and bath hamiltonians respectively (i.e)
\begin{align}
\hat{H}_{S}|s_i\...
0
votes
1answer
41 views
How does the measure of purity of a mixed state evolve with time in quantum mechanics?
We know that the Tr() is invariant with respect to unitary transformation. So does the density matrix $\rho(t)$ does not evolve with time?
$\begin{align} \ \rho(t) =&|\psi(t)\rangle \langle \...
1
vote
0answers
39 views
How many tensor product terms are necessary to express a separable state? [duplicate]
Wikipedia (https://en.wikipedia.org/wiki/Separable_state) defines a separable state, as a state $\rho$ which can be written as:
$ \rho = \sum_{k=1}^l p_k \rho_1^k \otimes \rho_2^k $
where $\sum_{k=1}...
0
votes
0answers
27 views
How to calculate a one-body reduced density matrix
I calculated eigenvalues and eigenvectors of a many-body problem for a SBEC (spinorial Bose-Einstein Condensate) in the SMA approximation . Then I can calculate the density matrix of this problem. ...
2
votes
1answer
38 views
Use of Uhlmann representation in proving the strong subadditivity of the von Neumann entropy
I am trying to prove strong subadditivity of the von Neumann-entropy, using joint convexity of the quantum relative entropy.
The procedure is given in https://en.wikipedia.org/wiki/...
3
votes
1answer
55 views
Constraints on higher-dimensional Bloch vectors
I'm interested in the constraints on the $(4^n-1)$-dimensional generalized Bloch vector (the Bloch vector for $n$ qubits). To the best of my knowledge, these are not analytically characterized for ...
0
votes
0answers
52 views
Density matrix expression by path integral
I came across an expression which I don't understand for the density matrix $\rho$ given by the path integral method (Fradkin, p.760) -
$$
\left< \phi(x) \left| \rho\right| \phi\left(x'\right) \...
1
vote
0answers
45 views
Are the ideals in two GNS constructions linked to the equivalence (or not) of the CCR representations?
Starting from the abstract C* algebra $A$ of canonical commutative relations, a state $\rho$ over this algebra enables to construct a Hilbert space $A/I$ where $I$ is the ideal of the elements $a$ ...
0
votes
1answer
52 views
Manipulation of composite density matrices (operators)
Suppose we have two systems with density matrices $\rho_1$ and $\rho_2$. Initially they are non-interacting, and so their composite density matrix looks like:
$$\rho_t = \rho_1 \otimes \rho_2$$
I ...
0
votes
0answers
37 views
Fundamental thermodynamic relation for discrete quantum partition function (density matrix)
In the case of a discrete classical partition function defined as:
$$
Z=\sum_{q \in Q}e^{-\beta (E(q)+pV(q))}
$$
It is straightforward to show that it implies the following fundamental thermodynamic ...
0
votes
1answer
34 views
The equivalent of vector addition for density operators?
In quantum mechanics, pure states may be represented by (subspaces spanned by) vectors in a Hilbert space, which may be added. This is physically meaningful, and in wave mechanics leads to visible ...
0
votes
1answer
92 views
How do I know whether the description of an electron state is complete?
Let's consider an electron as part of a larger system as an atom consisting not only of a nucleus but also of several other electrons. I guess, one can characterize the atom quantum-mechanically in a ...
1
vote
0answers
96 views
Measurement problem: Origin of probabilities in Many-Worlds Interpretation
As far as I can tell there appears to be an active group of academics (including the likes of Sean Carrol) who believe in the Many-Worlds Interpretation of quantum mechanics, but feel that the origin ...
0
votes
0answers
43 views
Name of real-valued representation of density matrix?
This is a specialization of my question https://math.stackexchange.com/q/3157300/ on math.SE.
There are many ways to write the density matrix $\hat \rho$ as vector $\vec \rho$. In the Liouville space,...
0
votes
0answers
151 views
Solving density matrix in a two-level atom
I'm working through some parts of Stephen C. Rand's Lectures on Light: Nonlinear and Quantum Optics, specifically sections in which he works with density matrices. In several places he makes ...
2
votes
1answer
60 views
Does taking partial trace commute with local operators?
Suppose we have a big system with two subsystems $H=A\otimes B$. For a unitary $U$ in the Hilbert space $A$ and a state $\rho$ in the Hilbert space $H$. Is the following statement true?
$$ \text{Tr}_B ...
0
votes
0answers
17 views
Local rotations to make a density matrix visually appealing?
I'm curious if there are general rules for finding the local rotations on a density matrix which make it most visually appealing? I have approached this problem in the past from the perspective of ...
0
votes
2answers
99 views
How can one get the density operator from the characteristic function?
To solve analytically the master equation of two qubits interacting with a cavity mode through their environment we use the charactristic function,
$$\chi (\beta)=\operatorname{tr}[\rho D(\beta)],$$ ...
1
vote
1answer
73 views
Derivation of von Neumann Equation for Density Matrices
Consider an ensemble of systems where each system is in one of a set of states $|\alpha_i\rangle$, with proportions $w_i$, such that the density operator is
$$ \hat{\rho} = \sum_i w_i |\alpha_i\...
1
vote
0answers
56 views
Quantum Coherence in a Two-level System in the Density Matrix Formalism
Dealing with semiclassical light-matter interaction, in particular the interaction between an electromagnetic field and a two level system using the density matrix formalism, I learned that the system ...
1
vote
1answer
115 views
The maximally mixed state is at the center of all quantum states
The set of quantum states $\rho$ in $d$ dimensions is the set of positive semidefinite operators living in a Hilbert space of dimension $d$. Let us denote this set by $\text{Pos}(X)$ and note that ...
1
vote
0answers
53 views
Can we write the Quantum Fidelity between two density operators in terms of Quasi-Probability Distributions: $P$, $Q$ and $W$?
Quantum Fidelity between two density operators, $\hat{\rho}$ and $\hat{\sigma}$, is given by $F(\hat{\rho},\hat{\sigma})=\left(Tr\sqrt{\sqrt{\hat{\rho}}\hat{\sigma}\sqrt{\hat{\rho}}}\right)^2$, where $...
3
votes
1answer
194 views
Lindblad and Input-Output Formalism in Quantum Optics
I'm confused about how to apply the Lindblad formalism and the input-output formalism in practice, and how one goes between the two.
Suppose I have a cavity (C) coupled to a reservoir (R), with the ...
