Questions tagged [density-operator]
The density operator describes a quantum system in an (in general mixed) state.
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Density of states for non-interacting Bosons
I am tasked with calculating the density of states in terms of the angular frequency given the dispersion relation. But I couldn't help but think: why can't we calculate the density of states by ...
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I'm stuck with the conversion of the 3 qubit $W$-state in terms of Pauli's spin matrices so that i can H-P transform them! [duplicate]
Given a state $|\psi\rangle=\frac{1}{\sqrt{3}}(|001\rangle+|010\rangle+|100\rangle)$. What is the density matrix in term of spin operators? Kindly give me detailed solution of this?
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Understanding density operator of the bath state for quantisation volume larger than de-broglie wavelength
I have been reading the paper "Collisional decoherence reexamined" by K. Hornberger and J.E. Sipe.
In the sub-section II-B titled "Convex decompositions of the bath density operator&...
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Perfect correlations in bipartite (impure) density operators
Consider a bipartite system, defined on a Hilbert space $\mathcal{H}= \mathcal{H}_A \otimes \mathcal{H}_B$. Consider a basis $\{|A_i\rangle \otimes |B_j\rangle\}$. What is the general form of a ...
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Wigner transform of $O_1 O_2$ in terms of Wigner transforms of $O_1$ and $O_2$?
The Wigner-Weyl transform of a quantum operator $O$ is defined as
$$
W[O](q,p) = 2 \int_{-\infty}^{\infty} dy\ e^{- 2 i p y} \langle q + y | O | q - y \rangle \ dy
$$
and then given a density matrix $\...
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Reduced density matrix of 4 qubits
I am trying to find the reduced density matrix of a four qubit quantum system. The Hamiltonian is
$$H = -B(\sigma_z \otimes \mathbb{I}_8 + \mathbb{I}_2 \otimes \sigma_z \otimes \mathbb{I}_4 + \mathbb{...
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Time-ordered matrix exponential quasi-static limit
Define a matrix differential equation $$\dot{X}=A(t)X(t),$$ where $X=[x1,x2,...]^T$ is a 1D vector and $A(t)$ is a complex-valued time-dependent matrix.
This system can be solved by $$X(t)= \mathcal{T}...
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Reduced density matrices and relation to entanglement
I've read that if a state is a product state, the reduced density matrices are pure and if the state is entangled, the reduced density matrices are both mixed.
What would it mean if you had a system ...
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Why are eigenvectors of the density matrix orthogonal pure states?
I am a first year PhD student studying condensed matter theory. I'm working with entanglement entropy and am having some trouble understanding the diagonal form of the density matrix. I get why we ...
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Do completely positive maps have a leading eigenoperator with nonvanishing trace?
A completely positive map $\mathbb{W}$ is a map from $\mathbb{C}^{n\times n} \to \mathbb{C}^{n\times n}$ that can be written in terms of $n\times n$ matrices $K$ as
$$ \mathbb{W}(\rho) = \sum_{i=1}^N ...
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Spin/Helicity-Density Matrices in Particle Physics
I am searching for literature explaining the Spin/Helicity-Density Matrix formalism in order to compute "spin-dependent" matrix elements and cross-sections in particle physics.
I would ...
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How to prove the orthogonality of spherical tensor operators?
$\def\mybra#1{\langle{#1}|}$ $\def\myket#1{|{#1}\rangle}$
$\def\Tr{\mbox{Tr}}$ $\def\mybraket#1{\langle{#1}\rangle}$ $\def\B#1#2{B^{#1}_{#2}}$
$\def\C#1#2{C^{#1}_{#2}}$
$\def\r#1#2{\hat\rho^{#1}_{#2}}$...
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About the definition of mixed states
My question is really straightforward. Can we define a mixed state $\varphi$ to be a linear combination of pure states of the form:
$$\varphi = \sum_{n=1}^{\infty}p_{n}\psi_{n}$$
where each $\psi_{n}$ ...
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In what sense is the set of density matrices compact in infinite dimensions?
Consider a complex, infinite-dimensional and separable Hilbert space $H$ and let $\mathcal I(H)$ denote the space of trace-class operators. The set of density operators $$\mathcal S(H):= \{\rho\in \...
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Entanglement entropy and reduced density matrix
I'm learning about the definition of von Neumann entanglement entropy
$$S(\rho_1)=-\text{Tr}[\rho_1\ln\rho_1]$$
where $\rho_1$ is the reduced density matrix $\rho_1=\text{Tr}_2(\rho)$.
I was confused ...
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"Entropy" of a set of correlators in a quantum system
Please forgive the ill-posedness of this question; I am hoping someone can help me formulate what I am asking more clearly.
Consider the ground state of a one-dimensional quantum spin chain on $N$ ...
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Stuck while deriving the Lindblad Master Equation
I was following Quantum Markov Processes from the book The Theory of Open Quantum Systems by Breuer and Petruccione. In the section The Markovian Quantum Master Equation they proceeds to 'construct ...
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How can valley coherence be defined if the crystal is initially in a mixed state?
In the field of valleytronics, they refer to valley coherence as: "the phase relationship between a particle in a superposition of two different valleys" [S. Vitale et al., Small 1801483 (...
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Similarity transformation of operators
Consider a $2\times 2$ Hamiltonian as $H=p_x\sigma_x$.
Which is not in its diagonal form, which implies it is not written in its eigenbasis. Now consider I have an observable, which has the matrix ...
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How do we prove that POVMs are the most general measurements?
It is often claimed that POVMs represent the most general measurement statistics possible. But what is the justification for this claim? Textbooks and university courses generally try to build up to ...
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What is the interpretation of the eigenvalues of $e^{-\beta (H-\mu N)}$?
In quantum statistical mechanics, the equilibrium state is characterized by a density matrix $\rho$. Let me focus on the grand canonical ensemble, although the question also holds for the canonical ...
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One-body density matrix has occupation numbers greater than 1
TL;DR - I computed the one-body density matrix (OBDM) stochastically via a method in a paper listed below, and it generates non-physical occupation numbers that 1) either has some negative values or 2)...
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Factorization of density matrices
I'm currently reading through the following document about quantum noise and open quantum systems: https://courses.cs.washington.edu/courses/cse599d/06wi/lecturenotes13.pdf. On page 6 of the document, ...
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Why $\sum_n|E_n \rangle \operatorname{exp}(-\beta\mathcal{H})\langle E_n| = \operatorname{exp}(-\beta \mathcal{H})$? (Quantum statistics)
I am reading James P. Sethna, Statistical Mechanics : Entropy, Order Parameters, and Complexity, p.137 and stuck at some equality. I think that this question is like homework-question ( In fact this ...
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Unsure of $|A \times A|$ vector notation in quantum informatics [closed]
I am new to quantum informatics and am doing a course on it. The guy who runs the course keeps writing down this notation - $|0 \times 0|$ or $|1 \times 1|$ - although it is completely foreign to me....
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A general theorem about state operators as a convex set?
I am trying to convince myself of a general theorem which fully "defines" the set of state operators. It is easy to prove that any convex combination of valid state operators is also a valid ...
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If a state operator factors, then it factors into partial state operators in particular -- why?
Suppose that we have a state operator for a bipartite quantum system $\rho = A \otimes B$. As far as I know, one must then have in particular $\rho^{(1)} = A$; that is, if a state operator factors ...
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Is the set of density matrices on a $d$-dimensional Hilbert space compact?
The set of the density matrices is a set with elements $\rho$ satisfying the following two conditions: $0\preceq \rho $ and $\mathrm{tr}\left( \rho \right) =1$. I wonder whether the set is compact?
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Is it possible to relate the expectation values of an operator w.r.t. two different density matrices if the matrices are related up to a displacement?
I'm trying to derive a general relationship between the expectation values of two different operators with respect to some unspecified state $E_{j}(\lambda)=\mathrm{Tr}[\rho \hat{O}_{j}(\lambda)]$ (...
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Hastings' "two" definitions of trivial mixed states
In https://arxiv.org/abs/1106.6026 (Definition 3), Hastings defines a density matrix $\rho$ in Hilbert space $\mathcal{H}$ to be trivial if one can tensor in additional degrees of freedom $\mathcal{K}$...
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How to write the density matrix for a state in the basis $\{ |H\rangle,|V\rangle \}$? [closed]
Suppose we have a set of photons in a mixed state with probabilities $P_1=0.2$ and $P_2=0.8$, respectively, of being in the pure states;
$$|\phi_1\rangle=|H\rangle; \; |\phi_2\rangle=\frac{3}{5}|H\...
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What is the space of density matrices?
In quantum mechanics, a pure state of any quantum system is determined by a state $|\psi\rangle$ living in a Hilbert space $\mathcal{H}$. Any Hilbert space has a basis of states and any state can be ...
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Why does $Z \otimes Z$ have only two eigenvalues?
The observable $Z = \begin{bmatrix}1 & 0\\\ 0 & -1\end{bmatrix}$ on a single qubit system has two eigenvalues, 1 and -1, which means when measured, the system can give one of two possible ...
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Stern-Gerlach and density operators
The setup is the following:
We have a particle beam of spin-up (:= $|+\rangle$) particles coming in a Stern-Gerlach apparatus which measures spin in x-direction.
After passing through, the beam splits ...
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Exact form of two-time correlators in out of equilibrium evolution from factorized distributions
Suppose that the have an initial product state at some time $t=0$ written in a computational basis $|0\rangle,|1\rangle$, for instance the state $|1011010\rangle$. The associated density matrix $\...
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Prove that the state fidelity for pure states has the form $\langle\psi|\sigma|\psi\rangle$
For two density operators $\rho$ and $\sigma$, the fidelity is given by
$$
F(\rho,\sigma)=\left(\mathrm{Tr} \sqrt{\rho^{\frac{1}{2}}\sigma\rho^{\frac{1}{2}}}\right)^2
$$For a pure state $\rho=|\psi\...
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Von Neumann Algebra decomposition
I am trying to understand the Von Neumann decomposition, according to which every Von Neumann Algebra can be uniquely decomposed as integral (or direct sum) of factors. More specifically, I am trying ...
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How to derive or understand the quantum detailed balance condition for Markov open system?
In the "On The detailed balance conditions for non-Hamiltonian systems", I learned that for a Markov open quantum system to satisfying the master equation with the Liouvillian superoperators,...
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Help clarifying confusion about many-particle states on Quantum Mechanics
Suppose I have many particles distributed in two states $a$ and $b$, if a fraction of particles in the $a$ is denoted by $n_a$, and the fraction of particles in the $b$ state denoted by $n_b$, some of ...
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How does the Stosszahlansatz work in quantum mechanics?
Is there a formulation emergence of the Stosszahlansatz in quantum mechanics with density matrices?
Stosszahlansatz is the assumption that the velocities of colliding
particles are uncorrelated, and ...
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Does the sum of all entries ("grand sum") of a density matrix have any meaning or a particular value?
Title says it all really. As is commonly known, the trace of the density matrix must be one and the trace over the density matrix times an operator is the expectation value of that operator. If you ...
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Bell State Density Operators in Continuous Systems
My question is about how density operators work in continuous systems, specifically maximally entangled states. I've outlined the math below but the TLDR is:
Does it make sense to talk about the ...
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Mixed quantum states represented by density matrices - proof [duplicate]
I have often heard that two physically distinguishable mixed quantum states produce different density matrices. However, how would I prove it? I know that they have to differ on the main diagonal, ...
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Mass density of photons in a refractive medium
The effective mass density of photons in a vacuum $\rho^{vac}_M$ is related to the photon energy density $\rho^{vac}_E$ by
$$\rho^{vac}_M=\frac{\rho^{vac}_E}{c^2}.$$
Is it true that the mass density ...
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Is it enough to recovery density matrix for error correction in quantum computing?
When we discuss error correction, we always talk about the recovery of the density matrix of a single qubit. But somehow I feel that this is not enough. Consider a circuit that contains more than 1 ...
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Properties and physical meaning of disjointness of reducible representations
I have the following doubt. Let's assume we have two mixed states $\rho_1 = \Sigma_i a_i \omega_i^{1}$ and $\rho_2 = \Sigma_i b_i \omega_i^{2}$ on the same algebra, where the states $\omega$ are all ...
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Multiple preparations of the same density operator
In complement EIII on density operators in Quantum Mechanics by Cohen-Tannoudji, it is stated that
An ensemble of pure states $|\psi_k\rangle$ with probabilities $p_k$ leads to a single density ...
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How do we determine when a modular Hamiltonian is local?
I'm trying to deepen my understanding of Von Neumann entropy (out of interest in the quantum Bekenstein bound) by learning more about the reduced density matrix (aka local density matrix), which is ...
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How can a dephasing channel be formally represented by a unitary operation?
The dephasing channel is often represented as
$$\mathcal{E}\left(\rho\right)=p_0\rho+(1-p_0)\sigma_z\rho\sigma_z$$ that acts on a single qubit.
Another way of writing it down is by using Kraus-...
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How to find the density operator of two joint systems given the density operator of the individual systems?
If I have two systems A and B and I want to find the density operator of the joint systems $\rho_{AB}$, is it just the tensor product of the density matrix of the individual systems, $\rho_{AB} = \...
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