Questions tagged [density-operator]
The density operator describes a quantum system in an (in general mixed) state.
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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) -
$$
\left< \phi(x) \left| \rho\right| \phi\left(x'\right) \...
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I'm confused to understand von Neumann entropy and purification [closed]
In my textbook, for all density matrices, von Neumann entropy has the following properties:
(1)$\forall \rho \in S(\mathbb{H}), H(\rho )\ge 0$ and $H(\rho )=0 \iff \rho$:pure state
(2)$\forall \rho _1,...
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Are the ideals in two GNS constructions linked to the equivalence (or not) of the CCR representations?
Starting from the abstract C* algebra $A$ of canonical commutative relations, a state $\rho$ over this algebra enables to construct a Hilbert space $A/I$ where $I$ is the ideal of the elements $a$ ...
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1answer
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Manipulation of composite density matrices (operators)
Suppose we have two systems with density matrices $\rho_1$ and $\rho_2$. Initially they are non-interacting, and so their composite density matrix looks like:
$$\rho_t = \rho_1 \otimes \rho_2$$
I ...
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32 views
Fundamental thermodynamic relation for discrete quantum partition function (density matrix)
In the case of a discrete classical partition function defined as:
$$
Z=\sum_{q \in Q}e^{-\beta (E(q)+pV(q))}
$$
It is straightforward to show that it implies the following fundamental thermodynamic ...
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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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1answer
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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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1answer
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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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1answer
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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
$$ \hat{\rho} = \sum_i w_i |\alpha_i\...
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Quantum Coherence in a Two-level System in the Density Matrix Formalism
Dealing with semiclassical light-matter interaction, in particular the interaction between an electromagnetic field and a two level system using the density matrix formalism, I learned that the system ...
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1answer
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The maximally mixed state is at the center of all quantum states
The set of quantum states $\rho$ in $d$ dimensions is the set of positive semidefinite operators living in a Hilbert space of dimension $d$. Let us denote this set by $\text{Pos}(X)$ and note that ...
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Can we write the Quantum Fidelity between two density operators in terms of Quasi-Probability Distributions: $P$, $Q$ and $W$?
Quantum Fidelity between two density operators, $\hat{\rho}$ and $\hat{\sigma}$, is given by $F(\hat{\rho},\hat{\sigma})=\left(Tr\sqrt{\sqrt{\hat{\rho}}\hat{\sigma}\sqrt{\hat{\rho}}}\right)^2$, where $...
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Lindblad and Input-Output Formalism in Quantum Optics
I'm confused about how to apply the Lindblad formalism and the input-output formalism in practice, and how one goes between the two.
Suppose I have a cavity (C) coupled to a reservoir (R), with the ...
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2answers
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Definition of the fully depolarizing quantum channel
The fully depolarizing quantum channel in a $d$ dimensional Hilbert space is defined by
$$
\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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1answer
43 views
“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
43 views
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
32 views
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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2answers
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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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5answers
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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
50 views
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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1answer
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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})}$$
where $H$ is the systems Hamiltonian.
Today I was told that the eigenvalues of the ...
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1answer
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Why quantum map must be hermitian?
Quantum maps transform a density matrix into another one, Assume we are in the Hilbert space :$ H_A $
the quantum map on the density matrix $\rho_A$ living in $H_A$ is : $\mathcal{L}_A$
Why $\mathcal{...
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Conceptual meaning of Thermal States
Thermal states are generally defined as
$$\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 ~,$$
such that the joint ...
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Difference between pure and thermal states
As far as I know by inserting a harmonic potential $V(x) = \frac{1}{2}m \omega x^2$ into the time-independent schrödinger equation I can obtain the wave-functions eigenstates and eigenvalues (energies)...
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Are Thermal states Harmonic oscillators?
Excuse me if I use somewhat wrong terminology. But I've always been confused about this.
So firstly when we talk about a 2-state system, like a qubit, it has dimension d=2, no?
But what if we ...
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1answer
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Post-measurement density matrix derivation
This is something standard, by I'm trying to redo this with spectral theory. Suppose we start with the usual postulates of quantum mechanics:
States are unit rays on a separable Hilbert space. In ...
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1answer
33 views
Conservation of quantum information in mixed states quantum secret sharing (QSS) schemes?
Consider a $((3,5))$ pure state quantum secret sharing (QSS) scheme. For instance this paper: arXiv:quant-ph/9901025. If I divide any 5 shares to two sets then allways one of those two set are ...
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How to measure off diagonal elements of mixed state density matrix?
Let’s assume I want to do quantum tomography using polarizers, half Waveplates and detectors , it’s obvious for me how we can measure diagonal elements of 2 qubit system density matrix using polarizer ...
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1answer
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Action of rotation operator on spin 1/2 system
In Sakurai book on QM in chapter 3, he states the following relation $$e^{\frac{iS_z\phi}{\hbar}}[(\rvert+\rangle\langle-\rvert)+(\rvert-\rangle\langle+\rvert)]e^{\frac{-iS_z\phi}{\hbar}}$$
$$=e^{\...
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Expansion of an arbitrary density matrix in terms of coherent states?
It is well-known that any pure state can be expanded in terms of coherent states namely
$$\left|\psi\right>=\frac{1}{\pi}\int d^2\alpha\left<\alpha|\psi\right>\left|\alpha\right>$$ due to ...
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2answers
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Alternate definitions of Thermal states
The definition of thermal states I'm used to is:
$$\tau_{\beta} = \frac{1}{Z}\,e^{-\beta H}$$
where $Z$ is the partition function defined as $Z= \mathrm{Tr}(e^{-\beta H})$, $\beta$ the inverse ...
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1answer
34 views
What exactly are control functions (used for parametrization)?
Let us consider a system in state $\rho$ with an internal hamiltonian $H_0$ on which we apply a cyclic, unitary evolution
$H_t = H_0 + V(t)$
Where $V(t)$ is a time dependent external potential for ...
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Does a physical interpretation of density matrix cross-terms exist?
Say we have some state $$|\psi\rangle=\frac{1}{\sqrt{2}}(|0\rangle+i|1\rangle)$$
it is in a quantum superposition of $|0\rangle$ and $|1\rangle$.
Its density matrix is $$\rho=\begin{pmatrix}\frac 1 2 &...
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2answers
83 views
Rank of a density matix
I was just trying to understand the meaning of rank of a density matrix. I came across the following post, which says that the rank of density matrix is the number of non-zero eigenvalues. And for a ...
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1answer
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Completely positive maps - dimension of the ancilla space
If a map between positive operators $\Phi: X \rightarrow Y$ is also completely positive, it is true that $\Phi\otimes I_A$ is also a positive map for any choice of ancilla operator space $A$.
That ...
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What does the energy-resolved spin density averages show?
What does the energy-resolved spin density averages show?
Spin density refers to the density of states for spin up and down, my question is what does energy-resolved refer and how can be the energy-...
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2answers
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Why is the partial differentiation of density operator with respect to time zero, for an ensemble in thermal equilibrium?
Sakurai initially says that density operator evolves with time because state kets evolve with time. But for an ensemble in thermal equilibrium, its partial differentiation is zero.
As far as I know, ...
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2answers
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Explanation of why this derivation of Schmidt decomposition works
I'm following Preskill's notes and he derives the Schmidt decomposition in the following way:
Let a bipartite state be $\psi_{AB} = \sum_{i,j}\lambda_{ij}\vert i\rangle\vert j\rangle = \sum_{i} \vert ...
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1answer
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Conditioning a quantum state: why does it have this particular form?
Can someone please explain the phenomenon of conditioning a quantum state, and why it has this particular form?
An observable $A$ of a quantum mechanical system, described by the density operator $\...
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How to obtain $Y$ rotation with only $X$ and $Z$ rotation gates on the Bloch sphere?
Let's say you have a system with which you can perform arbitrary rotations around the $X$ and $Z$ axis. How would you then be able to use these rotations to obtain an arbitrary rotation around the $Y$ ...
