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Questions tagged [density-operator]

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

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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,...
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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 ...
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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 $...
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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 ...
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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{...
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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 ...
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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})\...
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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 ...
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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{...
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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 ...
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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\...
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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 \...
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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}...
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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. ...
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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/...
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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 ...
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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) \...
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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 $a$ ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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,...
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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 ...
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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 ...
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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 ...
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2answers
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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, $$\chi (\beta)=\operatorname{tr}[\rho D(\beta)],$$ ...
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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\...
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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 ...
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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 ...
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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 $...
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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 ...
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Definition of the fully depolarizing quantum channel

The fully depolarizing quantum channel in a $d$ dimensional Hilbert space is defined by $$ \mathcal N^D (\rho) = \text{Tr}[\rho]\frac I d $$ I've seen that definition in several places but I don't ...
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“Interaction-Free” measurement involving statistical mixtures

What happens in the standard "interaction free measurement" when the detector connected to a bomb is replaced with attenuation (where light is lost through a semi-transparent medium)? Consider the ...
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Need help understanding weird definition of pure states [duplicate]

So in many sources I have read that A pure state contains only one element, since the only entry on the density matrix will be 1. But what about superpositions?...
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Thermalization and structure formation, starting with a pure state coupled to heat bath

I am looking for tractable quantum mechanical systems which show thermalisation and/or structure formation. One idea is a small quantum system S, e.g. a quantum Heisenberg model in a pure state (the ...
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1answer
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reduced density matrix of state

given a multi particle state I have to calculate the reduced density matrix where I trace out the third particle for this I first calculate the corresponding 2D density matrix with the bra vector of ...
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1answer
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Constructing two-qubit density matrix given expectation values of all products of Pauli operators

I think my question breaks down into two parts. Let's say you have a two qubit system and you can perform projective measurements. Each round of measurements will consist of results looking like ...
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A confusion about why can't a statistical mixture be modelled as a superposition of pure states?

I have read Cohen's book, and various posts in this site; however, I'm still not convinced why we can't model a statistical mixture as a superpositions of pure states ? For example, consider the ...
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Definition of Entanglement

The definition of quantum entanglement, found on the internet and the literature is: On a bipartite system $\mathcal{H}_A \otimes \mathcal{H}_B$, let $\rho$ be a mixed state. It is said to be ...
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What does a “pure” state mean in QM? [duplicate]

Question: In Quantum Mechanics, people use the word "pure state" for some states; however, what do they mean exactly ? Thoughts: I mean, a state is a vector in our vector (Hilbert) space, so in that ...
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1answer
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Kraus operators of a POVM

I would like to know how one finds the Kraus operators of a channel corresponding to a POVM. Consider a POVM of the form $M_i$ such that $\sum_i M_i = \mathbb{I}$. I can represent this by a quantum ...
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Time propagation of density

I need some quick help understanding this equation. $\frac{\partial}{\partial t} \rho(r,t) = \frac{i}{\hbar} \langle [\sum_{i=1}^N \frac{p_i^2}{2m_i}, \hat{\rho_r}] \rangle$ =$ \langle [\sum_{i=1}^...
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Eigenvalues of the thermal state density operator

We define the thermal density operator as $$\tau(\beta) = \frac{e^{-\beta H}}{\mathrm{Tr}(e^{-\beta H})}$$ where $H$ is the systems Hamiltonian. Today I was told that the eigenvalues of the ...
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Why quantum map must be hermitian?

Quantum maps transform a density matrix into another one, Assume we are in the Hilbert space :$ H_A $ the quantum map on the density matrix $\rho_A$ living in $H_A$ is : $\mathcal{L}_A$ Why $\mathcal{...
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Conceptual meaning of Thermal States

Thermal states are generally defined as $$\tau(\beta)= \frac{e^{-\beta H}}{\mathrm{Tr}(e^{-\beta H})}$$ What are some physical statements one can make about them? A system in thermal equilibrium is ...
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Calculating the evolution at any moment $t$ of a density matrix

I was reading the paper https://arxiv.org/abs/1303.4686, where we are given $N$ systems, all with the same Hamiltonian $$H=\sum_i \varepsilon_i \mid i\rangle\langle i\mid ~,$$ such that the joint ...
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Difference between pure and thermal states

As far as I know by inserting a harmonic potential $V(x) = \frac{1}{2}m \omega x^2$ into the time-independent schrödinger equation I can obtain the wave-functions eigenstates and eigenvalues (energies)...
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Are Thermal states Harmonic oscillators?

Excuse me if I use somewhat wrong terminology. But I've always been confused about this. So firstly when we talk about a 2-state system, like a qubit, it has dimension d=2, no? But what if we ...