Questions tagged [density-operator]

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

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One-body entropy?

In theoretical studies of Bose-Einstein condensates, it's common to look at the one-body density matrix: $$ n_{ab}=\langle\hat{c}_a^\dagger\hat{c}_b\rangle $$ where the $\hat{c}_j$ are annihilation ...
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Multiplying inner products in QM

Getting introduced to mixed states in QM, I have trouble with a proof. In particular there was a line where multiple quantum states $\Psi_i, \Psi_j$ and an arbitrary base $n$ in Hilbert space was ...
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Density operator in momenta representation using Fourier transform

I'm determining the density operator in momentum space and by working out the Fourier transform from the coordinate representation I get to: $$\hat n_q=\sum_{kk'ss'}\left<k,s|e^{-iq\hat r}|k's'\...
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Non-diagonal matrix element in density matrix

Is it possible to determine non-diagonal element of density matrix? It will be cool, if somebody present few examples. What is meaning of such element?
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What is density matrix in QFT?

In quantum mechanics exist fundamental object Density matrix. (See for introduction last chapter in Principles of Quantum Mechanics by David Skinner). Density matrix nesesary to describe systems even ...
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What is algebraic structure, operation or relation explains relationship between the 2 diagonal terms of the 2 density matrices?

Quantum decoherence therefore prescinds from the observer and from the measurement process in a certain way preceding it and simulating the collapse of the wave function. In particular, "...
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How to get the evolution equation of reduced density operator? [closed]

Here is a simple system such that we can get the exact evolute equation for reduced density operator, but I'm not sure how to derive it from the result proved in the first part of this question. ...
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Why is the standard definition of fidelity unnecessarily complicated?

The (Uhlmann-Josza) fidelity of quantum states $\rho$ and $\sigma$ is defined to be $$F(\rho, \sigma) := \mathrm{tr} \left[\sqrt{\sqrt{\rho} \sigma \sqrt{\rho}} \right]^2.$$ Every article I've ever ...
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Clarification on the notation of a paper about hybrid

Here is a screenshot from this paper by J. P. Foster and F. Weinhold. This paper focuses on a model of hybridization. It therefore considers movement of electrons in three dimensions. The author does ...
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Fourier Transform of Density Operator boosting momentum

I am currently reading Girvin and Yang's book on Condensed Matter Physics. I am reading up on the dynamical structure factor and in Section 4.3 where they talk about how $\rho_{+q}$, which is defined ...
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If $\text{Tr}_B \rho^{AB}$ is almost pure, then $\rho^{AB}$ is almost a product state?

Let $\rho^{AB}$ be a bipartite state, and let $\rho^A$ denote the partial trace. Suppose $$ \lVert \rho^A - |\sigma\rangle\langle\sigma|^A \rVert_1 \leq \varepsilon $$ for some pure state $|\sigma\...
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Derive Density Matrix of Canonical Ensemble by Tracing out the Reservoir

I would like to ask about a possible way to derive the (quantum mechanical) density matrix of the canonical ensemble: $$ \rho = \frac{1}{Z} e^{-\beta H} $$ I want to proceed as follows: It is known ...
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Given two states which are close in trace distance, are there purifications of these states which are equally close?

Given a pair of state $\rho_A$ and $\sigma_A$ such that they are close in trace distance i.e. $$|\rho_A - \sigma_A| \leq \epsilon,$$ can one find purifications of these states which are also $\...
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Statistical mechanics, partition function, and probability

I would like to confirm if the distribution $\rho$ of natural parameter $\varphi$ written as $\rho$ =$\frac{1}{Z}$$H(\beta)$$e^{(\varphi T(\beta))}$, where $Z=\int H(\beta)$$e^{(\varphi T(\beta))}d\...
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Defining the modular Hamiltonian

In Casini's proof of the Bekenstein bound a crucial step is the use of the modular Hamiltonian $K$ defined $$\rho^0_V = \frac{e^{-K}}{\text{tr} e^{-K}}$$ where $\rho_V^0 = \text{tr}_{-V} |0\rangle\...
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Are states with same von Neumann entropy the same up to unitary equivalence?

Let $H$ be a finite-dimensional Hilbert space with identity operator $I$. It is well-known that the von Neumann entropy $S$ of a density operator $\rho$ on $H$ is maximized if and only if $\rho = I$. ...
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Restriction master equation to single-particle subspace

In Anastopoulos and Hu - A Master Equation for Gravitational Decoherence: Probing the Textures of Spacetime (https://arxiv.org/abs/1305.5231), they restrict the system to a single particle subspace (p....
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Calculating expectations values from density matrix in position basis

given I know the density matrix elements in position basis as a function of time. $ \langle x | \rho(t) |x' \rangle$ How do I calculate the expection values like $\langle x^2(t) \rangle $ or $ \...
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Can any bipartite state be written as $\rho_{AB}=\sum_{ij} p_{ij}\rho_A^i\otimes \rho_B^j$?

I'm having a silly doubt. If $\rho_{AB}$ is separable then we can write it as $$ \rho_{AB}=\sum_i p_i\rho_A^i\otimes \rho_B^i$$ but can we write a general (maybe entangled) state as $$ \rho_{AB}=\sum_{...
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Understanding the density matrix for pure states vs. mixed states

So, there's a question in my textbook that explains the following scenario: A physicist runs two experiments A and B to prepare quantum systems in a variety of initial states. In experiment A he ...
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How to determine the purity and nature of a composite system?

So for the combined state of a pair of two-level atoms, A and B, with a density matrix $$\rho =\frac{1}{2}\lvert g_A, g_B \rangle\langle g_A, g_B \rvert + \frac{1}{2}\lvert g_A, e_B \rangle\langle ...
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non-pure states

I understand that states are associated with a state operator with certain mathematical properties. There is a subset of states, called pure states, that are represented by a state operator of the ...
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Proof that $\langle \dot{O} \rangle = \text{Tr}(\dot{\rho}O)$

I'm reading a paper ([PRB],[arXiv]; first paragraph of section IV) that uses the identity $$\langle \dot{O} \rangle = \text{Tr}(\dot{\rho}O),$$ where $\rho$ is the density matrix, and $O$ is an ...
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The $2$'nd law when a subsystem's entropy decreases due to a measurement?

Background So let's say there's system $1$ and system $2$ where system $1+2$ is an isolated system. Let the density matrix of system $2$ be: $$\rho_2 = \begin{pmatrix} 3/4 & 1/4 \\ 1/4 & ...
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Why do the degrees of freedom of a density operator not match up with the degrees of freedom of a state vector? [duplicate]

The density operator $\rho$ of a mixed 2-qubit system has $4^2-1=15$ degrees of freedom. We can require Tr[$\rho^2$] $ =1$ so that the system is in a pure state. Now we have 14 degrees of freedom. If ...
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Deriving the expression for one body density

My textbook (Richard M. Martin - Electronic structure) has the following equation for the one body density of a system of $N$ electrons: $$ \langle \Psi | \Psi \rangle n(r) = \langle \Psi | \hat n(r) |...
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What is the interpretation of a non-diagonal density matrix?

Suppose an ensemble of a spin $\frac{1}{2}$ particle in the $S_z$ basis is described by a diagonal density operator $$\rho=w_{11}|+\rangle\langle+|+w_{22}|-\rangle\langle-|$$ where $w_{ii}$ are real ...
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Quantum Mechanics “inside out”

Let us assume that we know only some basic QM notions which are part of the Heisenberg picture of quantum mechanics and Dirac quantization Physical observables are represented by Hermitian operators $...
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Formalism for an open system with non-adiabatic (non periodic) time dependence

Most non-equilibrium statistical processes of open time dependent systems are approached by Markovian dynamics of a system where time dependence of the system is assumed to be adiabatic (if Floquet ...
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Representation of spin-1 density matrices

Pauli matrices, together with the identity matrix can generate any $2\times 2$ matrix. By adding the condition that the matrices must be hermitian and with trace 1, we can represent density matrices ...
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Two qubit system and the notion of mixed states

In this paper, pages 6-9, the author discusses mixed states for two entangled qubits. After solving for Alice's density matrix, the author keeps referring to Alice's qubit state as a mixed state. But, ...
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Symmetry of the scattering super-operator

Suppose we have an initial ensemble described by a density matrix $\rho$ and any given member of the ensemble scatters from one of some set of scattering matrices $\{S_g \equiv O_g S O_g^\dagger : g \...
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Three dimensional visualization of a qutrit

My question is in reference to the paper "Three dimensional visualization of a qutrit"(https://arxiv.org/abs/1601.07361). The author's start with a symmetric two qubit density matrix written in the ...
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Density operator diagonal terms during decoherence

On page 7 of this paper by Sebens and Carroll they discuss the diagonal terms of the reduced density matrix during decoherence. I am very confused. I thought the diagonal terms are the only ones that ...
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Density matrix and wigner function from first and second moments

Let's say I know the first and second moments of position and momentum for all times. $\langle x\rangle $, $\langle p\rangle $,$\langle x^2\rangle $, $\langle p^2\rangle $, $\langle xp\rangle $, $\...
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How do I prove that reduced density matrix is Hermitian?

It is known that density matrix $\rho$ is hermitian. How do I prove that for a bipartite system $AB$, reduced density matrix of $A$, $\rho_A = Tr_B\{\rho_{AB}\}$, is hermitian, given that $\rho_{AB}$ ...
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Does reduced density matrix in a bipartite system evolve unitarily such that $\rho(t) = U(t)\rho_0U^{*}(t)$?

It is known that for density matrix, the von Neumann equation holds. $$\dot{\rho} = -i[H,\rho]$$ and thus $$\rho(t) = U(t)\rho_0 U^{*}(t)$$ where $U$ refers to a unitary matrix. But what happens for ...
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Trace of the quantum map $ A^n_m (\rho) = \sum_{ij} | i…i \rangle^n \langle i…i|^m \rho | j…j \rangle^m \langle j…j|^n$

We define some quantum map $ A^n_m (\rho) $ and let it act on density matrix $\rho$: $$ A^n_m (\rho) = \sum_{ij} | i...i \rangle^n \langle i...i|^m \rho | j...j \rangle^m \langle j...j|^n.$$ ...
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Positive Semi-Definiteness of a Density Matrix - can the eigenvalues be larger than 1?

I know that one of the requirements for a density matrix is that it is positive-semidefinite. This means that the eigenvalues are non-negative (and sum to 1, so we can assign them the meaning of ...
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Do pure states actually exist?

Is it possible to ever actually measure something so precisely that it actually collapses to a pure state, or do we really just get arbitrarily close? If a wavefunction never actually collapses to a ...
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Can a density matrix have more than two dimensions?

It crossed my mind when reviewing density matrices that if you were looking at a composite system consisting of three subsystems, (indexed by three quantum numbers: $<i,j,k|\rho|i,j,k>$) then ...
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Von neumann entropy for reduced density matrix

I have reduced density matrix of the form (corresponding to 2 coupled harmonic oscillators): $$\rho_r (x_1, x'_1)= \frac{\text{sech}(\eta ) \sqrt{\frac{m\omega \cosh (\eta )}{\hbar }} \exp \left(-...
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Could the definition of the experiment result occur before measurement?

Is there any fundamental demonstration that result of an experiment cannot have been determined before the measurement, yet according to the probabilistic rules of quantum mechanics? I understand ...
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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 ...
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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?
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Compact expression of Maxwell's equations: is there a missing minus sign?

The compact form of Maxwell's equations: $$\boxed{\square\, \boldsymbol{\mathsf{F}}=\mu_0 \boldsymbol{\mathcal{J}}} \tag{1}$$ where the current density quadrivector is given by the relation $\...
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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 ...
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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 ...
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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 ...
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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(\...

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