Questions tagged [density-operator]

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

Filter by
Sorted by
Tagged with
0
votes
1answer
116 views

Can one test an octonionic interpretation for a quantum-information conjecture, apparently valid in the real, complex and quaternionic settings?

For the values $\alpha = \frac{1}{2},1, 2$, corresponding to real, complex and quaternionic scenarios, the formulas (https://arxiv.org/abs/1301.6617, eqs. (1)-(3)) \begin{equation} \label{Hou1} P_1(\...
0
votes
0answers
28 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
57 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?
-1
votes
2answers
160 views

Compact expression of Maxwell's equations: missed minus sign

With much courtesy I ask a simple explanation to be able to obtain a minus sign missing from the compact form of Maxwell's equations: $$\boxed{\square \overleftrightarrow F=\mu_0 \boldsymbol{\mathcal{...
5
votes
2answers
97 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 ...
12
votes
1answer
817 views

How can I write a Gaussian state as a squeezed, displaced thermal state

I would like to write a Gaussian state with density matrix $\rho$ (single mode) as a squeezed, displaced thermal state: \begin{gather} \rho = \hat{S}(\zeta) \hat{D}(\alpha) \rho_{\bar{n}} \hat{D}^\...
2
votes
1answer
162 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 ...
2
votes
1answer
49 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
50 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
66 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
43 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
56 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 ...
1
vote
1answer
58 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 ...
0
votes
0answers
15 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-...
3
votes
0answers
97 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{...
1
vote
2answers
228 views

Time-dependence of density operator in quantum statistical mechanics

I'm struggling to understand a couple of textbook explanations relating to the density operator in quantum statistical mechanics. Firstly, in Huang's book "Statistical Mechanics" it says that "The ...
4
votes
1answer
252 views

Damped quantum harmonic oscillator - evolution of coherent state

I am trying to solve the following Master equation (also similar to damped quantum harmonic oscillator): $$\frac{d\hat{\rho}}{dt} = \frac{\Gamma}{2}\left(2\hat{a}\hat{\rho}\hat{a}^{\dagger} - \hat{a}^...
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
18 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 ...
4
votes
1answer
510 views

Solving the Lindblad quantum master equation in matrix form

I have just started learning density matrix and quantum master equations, and I am given a problem set that asks to find the solution to the Lindblad equation with $H$, $L_+$, $L_-$, $L_z$, and $\rho(...
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
43 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 ...
1
vote
1answer
401 views

Werner versus Isotropic States: Physical Significance

Can someone please explain the difference between Werner and Isotropic states based on physical significance rather than simply ($U\otimes U$) vs. ($U\otimes U^*$) invariant? (Where the $U$'s are ...
8
votes
2answers
305 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 ...
3
votes
1answer
66 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 ...
5
votes
1answer
436 views

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 ...
1
vote
2answers
48 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
48 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 ...
1
vote
1answer
343 views

Lindblad equation solution

I have been trying to solve a Lindblad Equation and then thought about whether there is a closed form Lindblad Equation solution for most types. Googling hasn't lead me to anything useful. So, is ...
0
votes
1answer
60 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{...
4
votes
1answer
1k views

Separability of density operators on tensor product spaces

Consider a composite system $\mathcal{H}=\mathcal{H}_{A}\otimes\mathcal{H}_{B}$ where $\mathcal{H}_{A}$ and $\mathcal{H}_{B}$ are Hilbert spaces of constituent components (say two qubits). Let $\rho_{...
1
vote
0answers
23 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
37 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 \...
3
votes
1answer
214 views

Quantum master equation and off diagonal terms

I have a couple of related questions What is exactly the difference between the quantum master equation and the regular master equation? My understanding is that the normal master equation is used to ...
2
votes
1answer
149 views

Density Matrix approach in Density Functional Theory - interpretation

In a paper describing a Kohn-Sham Density Functional Theory implementation, the authors describe the use of the density matrix for e.g. the calculation of the electronic density and for efficiency ...
5
votes
0answers
831 views

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:= \...
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}...
3
votes
1answer
54 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
26 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
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 ...
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/...
1
vote
3answers
1k views

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(\...
0
votes
0answers
43 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) \...
7
votes
1answer
460 views

Using open system dynamics to define a quantum state

Background The density matrix of a closed quantum system with Hilbert space $\mathscr H$ evolves according to the von Neumann equation \begin{align*} i\hbar\dot\rho=[H,\rho]. \end{align*} Given a ...
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
51 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
1answer
124 views

Probabilities with the Density Matrix

The density matrix of the system is given by: $$ [\rho_{S}(t)]_{mn} = [\rho_{S}(0)]_{mn} e^{-i\omega_{0}(m - n)t} e^{-i \delta(t)(m^2 - n^2) - \gamma(t)(m - n)^2}, ...
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 ...
3
votes
2answers
2k views

Reasoning Check: Trace of squared mixed-state density matrix

It's often written in the QI literature that, for a density operator $\rho$, if $\text{Tr}\left[\rho^{2}\right] < 1$, then $\rho$ describes a mixed state. However, I haven't seen any proofs of this ...