Questions tagged [density-operator]

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

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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 miss 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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Expanding a density matrix in terms of operators

In Lukasz's paper: https://arxiv.org/pdf/0909.2654.pdf He writes "consider a density matrix ρ, written as a polynomial of the 2N Majoranas cj in such a way that each cj occurs to the power 0 or 1 in ...
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Direct Derivation of Kraus Operator from Interaction Hamiltonian

For the dynamics of open quantum systems, the Kraus operators $K_\kappa$ can be derived from the unitary orbit $U(t)\rho U(t)^\dagger$ for $\rho=\rho_S\otimes\rho_E$ of the composite system given by ...
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Necessary and sufficient conditions for a pure state

I've seen some claims that idempotency ($\rho^2=\rho$) is necessary and sufficient to guarantee the existence of some state $\psi$ such that $\rho=|\psi\rangle\langle\psi|$, as well as claims on the ...
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Non-diagonal elements in density matrix: interference effect between the statistical mixture

In Cohen-Tannoudji, at page 303, vol 1, it is given that Quote: \begin{array}{l}{\text { A calculation analogous to the preceding one gives the following expression }} \\ {\text { for the non-...
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Entanglement of Werner States

Let the Werner state $$\rho_W = W\mid\Psi^-\rangle\langle\Psi^-\mid + \frac{1-W}{4}\mathbb{I},\ W\in [0,1],$$ where $|\Psi^-\rangle=(|01\rangle-|10\rangle)/\sqrt{2}$. I have repeatedly heard that ...
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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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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 ...