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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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Why does the quantum expectation in Bell's inequality have to take on the form of a quantum probability measure in Valter Moretti's book?

Valter Moretti, Fundamental Mathematical Structures of Quantum Theory Section 5.3.1 on Bell's inequality on p.204 as shown below states in Equation (5.8) that we have to interpret the expectation $E_\...
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How are quantum states of particles represented in particle processes?

For example, lets say we have an electron-positron annihilation scenario. What will be the density matrix representing the quantum state of the electron and the positron? What will be the density ...
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High fidelity implies low entropy

I was reading the security proof of QKD by Lo-Chau. One of their main idea is to bound the entropy of the eavesdropper if the singlet states are of high fidelity. Here I summarized the first lemma: ...
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Jensen's inequality on (super)operator exponential

Let us define the expectation value $\langle A\rangle_{\rho}$ of a superoperator $A$ over a density matrix $\rho$ as $(\rho, A(\rho))$, where the scalar product between operators reads $(O_1,O_2):= Tr[...
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The partition function is given by taking the trace of a density matrix. How do I convert this sum to an integral with correct factors?

I am learning Quantum Statistical Mechanics from Kardar's Book on Statistical Physics of Particles, in that he does the following for the partition function of a 3D gas: $Z_1 = tr(\rho) = \sum_k exp\...
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Is that right that about trace of the two density matrices multiply to each other? [closed]

We have: $$\mathrm{Tr}\,(\rho \rho^{\prime})=1$$ then, is it right to say $\rho=\rho^{\prime}$?
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Does the Born quantum master equation preserve positivity?

The Born quantum master equation (QME), $$ \frac{d \rho}{d t} = -g^2\sum_{\alpha,\beta} \int_{0}^{t} d\tau \mathcal{B}_{\alpha\beta}(\tau)[A_\alpha(t),A_\beta(t-\tau)\rho(t-\tau)] +h.c. $$ is a ...
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Is there any restriction for locally mapping a given 2-qubit density matrix into a desired 2-qubit density matrix with lower entanglement?

Suppose we're given a 2-qubit density matrix($\rho_{4\times4}$). we can apply two local maps on each of these qubits seperatly. So the output is density matrix($\rho^{\prime}_{4\times4}$). I'm ...
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Seperable Quantum States

Some similar questions have been ask before, but I still don't really get the definition of seperable states in quantum mechanics. Consider a bell state of a two qubit system. \begin{align} \left|\Psi\...
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Proof that $\mathrm{Tr} _ {\mathcal{Y}}[(\hat{X} \otimes \mathbf{1} _ {\mathcal{Y}}) \hat{\rho}] = \hat{X} \mathrm{Tr} _ {\mathcal{Y}} (\hat{\rho})$ [closed]

I've been trying to prove a partial trace identity that I need to prove something related to the Stinespring representation of a Completely Positive map, but I haven't been successful. (For some ...
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States of entangled particles after no/partial/full measurement

I'm getting contradictory information from the internet concerning entangled particles, measurement, and state knowledge that I'm hoping can be cleared up with a simple setup. Say we have a source of ...
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Applying operators to quantum states

I am reading a primer on quantum proofs (without having a background in QM) ("Quantum Proofs" by Thomas Vidick and John Watrous, pages 11 and 15) and have a question. For an operator $A\in L(...
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Why does the finite trace of density matrix imply a discrete Schmidt decomposition? [closed]

In the paper defining an average Schmidt number for a particular entangled system, Law and Eberly say: Because density matrices always have finite trace, the Schmidt decomposition is always discrete, ...
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Prove that the conditional von Neumann entropy satisfies $S(A|C)=-S(A|B)$ if computed on a pure $\rho_{ABC}$

Can someone show me how to prove this relation? If $\rho_{ABC}$ is a pure state, then $S(A|C)=-S(A|B)$ I already know that $S(A|B)=S(\rho_{AB})-S(\rho_{B})$ and that I should somehow use that $S(\...
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Probability for a quantum many particle system of temperature $T$ to be found at temperature $T'$

Suppose we have a system of $N$ particles (let's postpone the question re statistics) in thermal equilibrium, described via a density operator $$ \rho_\beta = Z^{-1} e^{-\beta H} $$ $$ Z = \text{tr} \,...
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Density Matrix vs Wavefunction Formalism

What is the main motivation and advantages of the density matrix formalism compared to the wavefunction formalism? From what I understand, the density matrix is more commonly used when you want to ...
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Can the characteristic function $\chi_\rho(\beta)={\rm tr}[\rho D(\beta)]$ be an indicator function?

Given the characteristic function defined as: $$\chi(\beta)=\text{tr}[\rho D(\beta)],$$ with $D(\alpha)=e^{\alpha a^\dagger-\bar\alpha a}$ the displacement operator. Is it possible that for some $\rho$...
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Simple way to see if $n$-qubit Hamiltonian has no interaction terms?

Consider a system of $n$-qubits. The Hilbert space is $\mathcal{H} \cong \mathbb{C}^2 \otimes ... \otimes \mathbb{C}^2$. Fix each tensor factor $\mathbb{C}^2$ to have the $\sigma_z$ vector space basis ...
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What is Decoherence? [duplicate]

What precisely is decoherence? Assume familiarity with the density matrix formalism of quantum mechanics. I read this related question, but I am looking for a more precise answer than the one given. I ...
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Is the partial trace of a Schmidt decomposition of a pure state a spectral decomposition?

I have a doubt about the spectral decomposition of the partial trace of the Schmidt decomposition. $\mathcal{H}_a$ and $\mathcal{H}_b$ are two Hilbert spaces with $\text{dim}(\mathcal{H}_a) = N$ and $...
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Relation between a diagonal element of a density matrix and the off-diagonal elements in the same row/column [duplicate]

In a density matrix $\rho=(\rho_{ij})$, the diagonal elements $\rho_{jj}$ contains the information about the probability $|\rho_{jj}|^2$ that the $j$-th eigenstate is occupied by the system. Meanwhile,...
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Definition of the single-particle density matrix

The single-particle density matrix is defined as $$\rho_{ij } = \langle \psi | a_j^\dagger a_i |\psi \rangle . $$ I am curious about the order of the indices. Why is it not $$ \rho_{ij } = \langle \...
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How to prove that all extensions of a pure state are product states? [closed]

I tried using the $\rho$ operator for this, and I got nothing. I approached it assuming I have two subsystems $A$ and $B$, both inside their own Hilbert space $H_A$ and $H_B$. And even defined $\rho = ...
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Conceptual problem with post-measurement density matrix

I'm asked to derive what the density matrix would be after a measure of $|\phi \rangle$ ($\rho \rightarrow \rho'$). At first I thought that, since $\rho$ has an statistical meaning on an ensemble of ...
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Estimate (time dependent) Hamiltonian from given time evolution of a density matrix

Basically the question is, whether you can give some estimation for the Hamiltonian of a system, given the time evolution of a density matrix $\rho$ under the assumption that it obeys the von-Neumann ...
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Production spin density matrix $R$ in particle physics and quantum state tomography

I want to perform quantum state tomography. For that, I need to construct a production spin density matrix $R$ . But I am unable to write it. I can write the matrix element for the process eg. $\tau^+ ...
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DFT total energy from band energy

I'm using Kohn-Sham DFT as a part of my research. The material is metallic crystal. In the following, you can assume that $\rho$ refers to the density matrix and $H$ refers to a hamiltonian matrix ...
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Does the density matrix still holds for infinite subsytems?

Consider a maximally mixed state composed of $n$ equal qubits. Its density matrix is then:$$\hat{\rho}^{(n)}=\hat{\rho}^{\otimes n}.$$ The eignevalues of such a matrix, $\lambda$ will all be equal (...
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Square of the density operator for a mixed state [duplicate]

Consider a density operator $\hat{\rho}$ for a mixed state defined by $$\hat{\rho} = \sum_k p_k |\psi_k\rangle \langle\psi_k|$$ Here $p_k$ is the probability of finding the $k$th system of the ...
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Schmidt decomposition of density operators

Consider a bipartite quantum system described by the density operator, $\hat{\rho}$, an operator acting on the Hilbert space $\mathcal{H}=\mathcal{H}_{A}\otimes\mathcal{H}_{B}$. This matrix will be a ...
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What are examples of qubit channel with non-singular Choi state?

Are there any obvious (ideally physically reasonable) examples of qubit channels that give rise to non-singular Choi states? I've been exploring the Choi state of a variety of qubit channels but find ...
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In Aaronson's "The Learnability of Quantum States", what do $E_j$ and $E$ represent?

In an interesting paper by Scott Aaronson, he demonstrate the learnability of quantum states in a framework of pac-learning in machine learning. Here what he wrote: But I am wondering what is the ...
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Proving that a positive partial transpose (PPT) state is non-distillable

I am attempting to prove the following statement: A positive partial transpose (PPT) state cannot be distilled. I would like to do so using the following facts: Let $\rho$ represent a bipartite ...
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How to calculate $Tr_{E} \underbrace{[\rho^{bath} \otimes \rho^{bath}......\otimes \rho^{bath}]}_{N \ times}$ if I know $Tr_{E} [\rho^{bath}]$?

One can directly jump to the question and skip the first two paragraphs (motivation for the question) Consider a system in a thermal gas of $N$ particles. If I want to study the reduced dynamics of ...
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Schmidt decomposition of density matrix

For a bipartite system: $\mathcal{H}=\mathcal{H}_{a}\otimes\mathcal{H}_{b}$ described by a density operator $\hat{\rho}_{ab}$, I can promote it to a vector in the Liouville space, $|\hat{\rho}_{ab}\...
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How do I convert this separable state into a product state?

I know two particles in a Bell state cannot be written as a product state as they are entangled. But what if I had a classically correlated state$$\rho = \frac{1}{2}(|11\rangle\langle 11| + |00\rangle\...
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Constructing wavefunction for a mixed state

This question is somehow the reverse of another question. If a quantum system $S$ is in a pure state, then we can find a wavefunction that describes $S$. This wavefunction is unique up to a phase ...
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How do you get from the polarisation operator to particle current?

I'm going through Mahan's "Many Particle Physics", and I'm a bit confused about his reasoning. He introduces the polarisation operator as $$\textbf{P}=\int\textbf{r}\rho(\textbf{r})d^3r$$ ...
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Existence of Glauber-Sudardhan $P$-representation of arbitrary given density operator for light field

In all textbooks on quantum optics I can reach (Scully, Leonhardt, Walls, etc), the Glauber-Sudardhan $P$-representation $P(\alpha)$ is introduced in the following two ways: Fourier transform of $\...
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How to take derivative of density operator?

I was just trying to confirm to myself that the following density operator $$\rho(t) = e^{-iHt/\hbar} \rho(0) e^{iHt/\hbar}$$ fulfills the Liouville-von Neumann equation: $$\frac{d}{dt}\rho(t) = - \...
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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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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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