Questions tagged [density-operator]

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

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How is quantum superposition different from mixed state?

According to Wikipedia, if a system has $50\%$ chance to be in state $\left|\psi_1\right>$ and $50\%$ to be in state $\left|\psi_2\right>$, then this is a mixed state. Now, consider the state $...
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Density matrix formalism

The density matrix $\hat{\rho}$ is often introduced in textbooks as a mathematical convenience that allows us to describe quantum systems in which there is some level of missing information. $\hat{\...
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Classical and quantum probabilities in density matrices

In textbooks, it is sometimes written that a mixed state can be represented as mixture of $N$ (I assume here $N<+\infty$) quantum pure states $|\psi_i\rangle$ with classical probabilities $p_i$: $$...
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Is the density operator a mathematical convenience or a 'fundamental' aspect of quantum mechanics?

In quantum mechanics, one makes the distinction between mixed states and pure states. A classic example of a mixed state is a beam of photons in which 50% have spin in the positive $z$-direction and ...
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1answer
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Differences between pure/mixed/entangled/separable/superposed states

I am currently trying to establish a clear picture of pure/mixed/entangled/separable/superposed states. In the following I will always assume a basis of $|1\rangle$ and $|0\rangle$ for my quantum ...
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What is the entropy of a pure state?

Well, zero of course. Because $S = -\text{tr}(\rho \ln \rho)$ and $\rho$ for a pure state gives zero entropy. But... all quantum states are really pure states right? A mixed state just describes ...
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What is the actual meaning of the density operator?

I am not able to understand the definition of the density operator. I know that if $V$ is a vector space and if I have $k$ states belonging to this vector space, say $|\psi_{i}\rangle$ for $1\le i\le ...
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1answer
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What is the difference between general measurement and projective measurement?

Nielsen and Chuang mention in Quantum Computation and Information that there are two kinds of measurement : general and projective ( and also POVM but that's not what I'm worried about ). General ...
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Are there more entangled states or non-entangled ones?

I'm trying to understand entanglement in terms of scarcity and abundance. Given an arbitrary vector $v$ representing a pure quantum state of, say, dimension 4, i.e. $v \in \mathcal{H}^{\otimes 4}$, ...
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What's the intuition behind the Choi-Jamiolkowski isomorphism?

What is the intuition behind the Choi-Jamiolkowski isomorphism? It says that with every superoperator $\mathbb{E}$ we can associate a state given by a density matrix $$ J(\mathbb{E}) = (\mathbb{E} \...
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What is a completely positive map *physically*?

I am sure this question is really stupid, but I could not refrain from asking it in this forum. This can be considered as a continuation of this question. ...
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2answers
585 views

What does the sum of two qubits tell about their correlations?

How much can I learn about correlations between two quits by measuring the sum of their values? What is the best way to formalize such a question? Below is my original, longer formulation of the ...
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1answer
421 views

Shape of the state space under different tensor products

I am currently studying generalized probabilistic theories. Let me roughly recall how such a theory looks like (you can skip this and go to "My question" if you are familiar with this). Recall: In a ...
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1answer
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Trace of an operator matrix (Quantum computation and quantum information)

I'm reading the book Quantum computation and quantum information by Mike & Ike and I'm stuck at 2.60/2.61. There, the author says that, given the operator $A|ψ⟩⟨ψ|$, its trace is: $${\rm tr}(A|\...
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Is purification physicaly meaningful?

Consider a quantum system with Hilbert space $\mathscr{H}$ and suppose the quantum state is specified by a density operator $\rho$. Since it is hermitian, it has a spectral decomposition: $$\rho = \...
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3answers
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Hilbert-Schmidt basis for many qubits - reference

Every density matrix of $n$ qubits can be written in the following way $$\hat{\rho}=\frac{1}{2^n}\sum_{i_1,i_2,\ldots,i_n=0}^3 t_{i_1i_2\ldots i_n} \hat{\sigma}_{i_1}\otimes\hat{\sigma}_{i_2}\otimes\...
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1answer
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How can I write a Gaussian state as a squeezed, displaced thermal state

I would like to write a Gaussian state with density matrix $\rho$ (single mode) as a squeezed, displaced thermal state: \begin{gather} \rho = \hat{S}(\zeta) \hat{D}(\alpha) \rho_{\bar{n}} \hat{D}^\...
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What is the physical meaning of the Lindblad operator?

I read the wikipedia article on the Lindblad operator, but I still don't understand what this operator is supposed to describe. I therefore considered setting up an example in order to get the idea. ...
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1answer
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Majorana-like representation for mixed symmetric states?

Is there a generalization of the Majorana representation of pure symmetric $n$-qubit states to mixed states (made of pure symmetric $n$-qubit)? By Majorana representation I mean the decomposition of ...
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How does one compute the state of a quantum system following imperfect measurement?

Suppose I have a quantum system $S$ ("system") with Hamiltonian $H_S$ and initial density matrix $\rho_S(0)$. I allow $S$ to interact with another system $P$ ("probe"), which has Hamiltonian $H_P$ and ...
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What is quantum entanglement? [closed]

What is quantum entanglement? Please be pedagogical. Edit: I have updated my background under my profile.
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938 views

Reduced density matrices for free fermions are thermal

Many recent papers study entanglement in eigenstates of fermionic free hamiltonians (normally on a lattice) using the basic assumption that the reduced density matrices are thermal (e.g. Peschel 2003)....
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Unitary evolution even after tracing out degrees of freedom

In quantum mechanics it is usually the case that when degrees of freedom in a system are traced out (i.e. ignored), the evolution of the remaining system is no longer unitary and this is formally ...
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Schrödinger's cat and the difficulty of macroscopic superposition state

The Schrödinger's cat was regarded as peculiar since we seldom encounter a superposition state in macroscopic scale: $$ \mid \mathrm{dead \,\,cat} \rangle + \mid \mathrm{alive \,\, cat}\rangle $$ We ...
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2answers
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Precise meaning of composition of ket and bra, e.g. $|\psi\rangle\langle\psi|$

I'm currently studying density matrices, and have been frequently coming across the construction $$|\psi\rangle\langle\psi| \,.$$ What is the formal meaning of this composition? I understand $\...
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Intuition on positive-operator valued measures (POVM)

I'm having a little trouble understanding what positive-operator valued measure (POVM) are- in particular why/how they are non-negative. For instance, if they just represent measurements, what about ...
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Is there a clear and intuitive meaning to the eigenvectors and eigenvalues of a density matrix?

Is there a clear and intuitive meaning to the eigenvectors and eigenvalues of a density matrix? Does a density matrix always have a a basis of eigenvectors?  
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1answer
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Quantum entanglement for indistinguishable particles

When I encounter the definition of the mathematical definition of quantum entanglement. System composed by many parts $A$, $B$,.., $N$ can be described by a density matrix operator $\hat{\rho}$ acting ...
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1answer
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Examples of Matrix Product States

Matrix product states (MPS) are a way of representing a (many-body) wavefunction. The method has been described, for example, in The density-matrix renormalization group in the age of matrix ...
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1answer
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Liouville-von Neumann equation can be directly derived from Heisenberg picture?

The Liouville-von Neumann equation for the density matrix is: $$ i\hbar\frac{\partial\rho}{\partial t}=[H,\rho],$$ while in the Heisenberg picture: $$ \frac{d}{dt}A(t)=\frac{i}{\hbar}[H,A(t)] +\...
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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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3answers
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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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Green functions and density matrix

tl;dr The single particle density matrix is directly related to NEGF as shown here, I wish to find a way to relate NEGF also to density matrices which describe probability distribution of many body ...
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Entanglement of Mixed Quantum State

As per Wikipedia: Quantum entanglement is a physical phenomenon that occurs when pairs or groups of particles are generated or interact in ways such that the quantum state of each particle cannot ...
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367 views

How to connect these two formulations regarding the need for a density matrix in quantum mechanics?

I found these two formulations: The density matrix is: 1) "needed if we consider a system that is part of a larger closed system." 2) "needed for a system to be in a statistical ensemble of ...
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1answer
468 views

What's the physical meaning of the kernel of density matrix?

The kernel of this linear map is the set of solutions to the equation A x = 0, where 0 is understood as the zero vector. But what's the physical meaning of the kernel of density matrix?
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If an isolated quantum system consists of only one particle, is it possible for it to be in a mixed state?

Mixed states are defined as the statistical ensemble of pure states. Classically, I understand the word, "statistical" referring to a system with a large number of microscopic particles. So if I go ...
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What is a Gibbs state?

What is a Gibbs state and what does it differ from a pure state? Say I have a two-level atom and it is described by a Gibbs state $\rho_G = \dfrac{e^{- \frac{H}{kT}}}{Z}$. I know $Z$ is a partition ...
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States versus ensembles in quantum mechanics

In quantum mechanics, we talk about (1) vectors, (2) states, and (3) ensembles (e.g., a beam in a particle accelerator). Suppose we want to translate this into mathematical definitions. If I'd never ...
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1answer
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Hilbert space for Density Operators (instead of Banach spaces)

Is it possible to construct a well defined inner-product (and therefore orthonormality) within the set of self-adjoint trace-class linear operators? In the affirmative case, dynamics could be analyzed ...
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Completely positive maps and symmetric states

Let $\mathcal{N}$ be a completetely positive trace preserving map (aka a quantum channel) acting on a finite dimensional system $\mathrm{A}$, and let $\pi$ denote the maximally mixed state on $\mathrm{...
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Using open system dynamics to define a quantum state

Background The density matrix of a closed quantum system with Hilbert space $\mathscr H$ evolves according to the von Neumann equation \begin{align*} i\hbar\dot\rho=[H,\rho]. \end{align*} Given a ...
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Interpretation of a density matrix as an observable

In quantum mechanics, any density matrix (or density operator) is Hermitian. Observables are also represented by Hermitian operators. So it follows that a density matrix can also be interpreted as ...
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Can we always find a Hilbert space corresponding to a region of spacetime?

In defining entanglement entropy in field theories we take a region A in spacetime. Now if $\mathcal{H}$ is the Hilbert space of the field theory we assume that we can decompose this Hilbert space ...
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Entropy of subsystem maximized?

Let $A$ and $B$ be 2 subsystems of a quantum mechanical system, so a state of the whole system is a vector in $A \otimes B$. As far as I understand, a density operator $ \rho $ in general can't be ...
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1answer
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When does Quantum Mechanics Measurement lead to Mixed State

Consider the context of the Stern-Gerlach experiment. As is stated on numerous sources (e.g. Feynman Lectures, MIT Lecture), a silver atom in the state of $\vert+z\rangle$ that is put through a Stern-...
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Lindblad equation for heisenberg operators?

Very related to this question: Is it possible to go from the Master Equation formalism to Heisenberg-Langevin equations I don't yet have enough reputation to comment so I'm asking the new question ...
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1answer
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Density matrix: error with diagonalization claim and fixing it

On page 174 of Townsend's "A Modern Approach to Quantum Mechanics", 2nd edition, it says the following: "For a mixed state, one for which $p_k$ is the probability that a particle is in the state $|\...
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1answer
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Coherences in the density matrix

It is said that the off-diagonal elements of density matrix are "coherence". When a system interacts with its environment the off-diagonal elements decay and the final density matrix is the diagonal ...
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1answer
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Trying to understand mixed states

I took a basic quantum chemistry course (McQuarrie's "Quantum Chemistry"), but it never dealt with mixed states -- only pure states (or if it did, we never got to it in class). So I'm trying to ...