Questions tagged [density-operator]

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

Filter by
Sorted by
Tagged with
23
votes
1answer
12k views

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 ...
10
votes
3answers
3k views

What is quantum entanglement? [closed]

What is quantum entanglement? Please be pedagogical. Edit: I have updated my background under my profile.
31
votes
4answers
3k views

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 $p_i$: $$...
70
votes
8answers
25k views

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 $...
20
votes
1answer
8k views

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 ...
27
votes
6answers
2k views

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 ...
6
votes
1answer
1k views

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 ...
4
votes
3answers
778 views

Seemingly a paradox on the eigenstate thermalization hypothesis (ETH)

In the research field of Many-body Localization (MBL), people are always talking about the eigenstate thermalization hypothesis (ETH). ETH asserts that for a isolated quantum system, all many-body ...
21
votes
4answers
3k views

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 ...
9
votes
6answers
1k views

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 ...
7
votes
3answers
366 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 ...
5
votes
3answers
2k views

Why does the density matrix $\rho$ obey a wrong-signed Heisenberg equation of motion?

The density matrix is defined as $$ \rho_\psi ~:=~ \frac{\lvert\psi(t)\rangle \langle \psi(t)\vert}{ \langle \psi(t) |\psi(t)\rangle }$$ in the Schrödinger picture. $\rho_\psi$ is obviously a time ...
11
votes
0answers
273 views

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 ...
4
votes
2answers
367 views

Proving the unitary relation of ensemble decompositions

In my class it was told that ensemble decompositions of a density operator $\rho$ are not unique, but that the ones that exist are related by a unitary operator. I'm trying to prove this, but I get ...
5
votes
4answers
483 views

Set of states $\{|\phi_n\rangle\}$ in the density operator $\rho=\sum\limits_n p_n|\phi_n\rangle\langle\phi_n|$

The set of quantum states $\{|\phi_n\rangle\}$ in the definition of the density operator $$\rho=\sum\limits_n p_n|\phi_n\rangle\langle\phi_n|$$ need not be orthonormal, and need not form a basis. But ...
2
votes
1answer
724 views
-2
votes
1answer
142 views

Why is the partial trace of this subsystem equal to this? [closed]

I am doing my bachelors dissertation based on an article by David Deutsch. He defines the action of a quantum gate as: $$ U = \sum_{x, y \in \mathcal{Z}_{2}} |x \dot{+}y\rangle|y\rangle\langle x|\...
16
votes
2answers
4k views

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} \...
36
votes
5answers
5k views

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{\...
22
votes
5answers
3k views

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 ...
13
votes
3answers
3k views

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. ...
3
votes
1answer
923 views

How to write a generic density matrix for multi qubit system

I was reading the paper device independent outlook on quantum mechanics. The author defines a generic two qubit density matrix as $$ \rho=\frac{1}{4}\left( I \otimes I + \vec{r_{\rho}} \cdot \vec{\...
17
votes
4answers
2k views

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. $v \in \mathcal{H}^{\otimes 4}$, ...
5
votes
4answers
3k views

How does a state vector be projected onto an eigenspace after measurement

In http://en.wikipedia.org/wiki/Measurement_in_quantum_mechanics#Degenerate_spectra, it is said that If there are multiple eigenstates with the same eigenvalue (called degeneracies),..., The ...
4
votes
2answers
1k views

Is it possible to go from the Master Equation formalism to Heisenberg-Langevin equations

If I have derived a master equation (e.g. in the Lindblad form) and solved for the density matrix, $\rho(t)$ I can get the mean value of an operator, A as: $ <A> = \mathrm{Tr}A\rho $. But ...
2
votes
1answer
600 views

Uhlmann's Theorem: proof of $\text{tr}(A^{\dagger} B) = \langle m | A \otimes B |m\rangle $ [closed]

In p228, Chapter 9 of Mark Wilde's text , in the course of proving Uhlmann's theorem for quantum fidelity, it claims $$\sum_{i,j} \langle i|^R \langle i|^A (U^R \otimes (\sqrt{\rho}\sqrt{\sigma})^A) |...
11
votes
3answers
6k views

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. ...
3
votes
0answers
208 views

Is there a physical significance to non-normal states of the algebra of observables?

Quantum theory may be formalized in several different ways. Generally, the physical discussion of different states of a quantum system distinguishes pure and mixed states, and then subsumes both in a ...
5
votes
2answers
16k views

What is the Reduced Density Matrix?

The difference between pure and mixed states is the difference in their density matrix structure. For density matrix $\rho$ of mixed state the trace of $\rho^{2}$ should be less than 1. For pure ...
5
votes
3answers
2k views

What is the physical interpretation of the density matrix in a double continuous basis $|\alpha\rangle$, $|\beta\rangle$?

(a) Any textbook gives the interpretation of the density matrix in a single continuous basis $|\alpha\rangle$: The diagonal elements $\rho(\alpha, \alpha) = \langle \alpha |\hat{\rho}| \alpha \...
5
votes
1answer
321 views

Algebraic Formalism of Quantum Mechanics

In the algebraic formalism, the physical system is described by its observables, viewed as self-adjoint elements in a certain *algebra and a state is a linear functional, if I understand right. Can ...
3
votes
1answer
2k views

Fermi-Dirac distribution derivation?

I am trying to derive the Fermi-Dirac statistics using density matrix formalism. I know that $$<A>= Tr \rho A.$$ So I started from $$<n(\epsilon_i)>= Tr \rho n(\epsilon_i)=\frac {1}{Z} ...
9
votes
2answers
2k views

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?  
4
votes
1answer
2k views

Off-diagonal terms in density operator of pure state

The density operator for a pure state $| \psi \rangle = \sum c_i | \psi_i \rangle$ (where $\{|\psi_i\rangle\}$ is a basis) is: $$\rho = \sum_{i,j} c_j^* c_i | \psi_i \rangle \langle \psi_j |$$ On ...
4
votes
9answers
1k views

Does the wave function/density state actually exist?

I have been reading with interest the debates here on whether the wave function/density state actually collapses or not, or whether it is subjective Bayesian or objective with actual complex numbered ...
4
votes
3answers
593 views

Takhatajan's mathematical formulation of quantum mechanics

So I began skimming L. Takhatajan's Quantum Mechanics For Mathematicians, and saw the mathematical formulation of QM that he uses (page 51). (The PDF file is available here.) I've only taken a basic ...
0
votes
1answer
941 views

Relationship between the Lindblad Equation and Redfield Equation

Both the Lindblad and Redfield Equation both model the open quantum system dynamics given a Hamiltonian and some operators. What is the relationship between the two equations? How can they transformed ...
7
votes
3answers
539 views

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 ...
4
votes
1answer
2k views

What information does the Von Neumann entropy give for mixed states?

Von Neumann entropy is defined as $$ S=-\mathrm{Tr}\left(\rho \ln\rho\right) $$ It can be used to measure the entanglement between two sub-systems, provided that the total system is in pure state. ...
2
votes
5answers
231 views

Definition of Entanglement

The definition of quantum entanglement, found on the internet and the literature is: On a bipartite system $\mathcal{H}_A \otimes \mathcal{H}_B$, let $\rho$ be a mixed state. It is said to be ...
2
votes
2answers
2k views

Density Operator, Expectation Value, Coherent States

How would I go about evaluating expectation values like $\langle X \rangle$ and $\langle P \rangle$? Work I've done: I've done the integration over $\phi$ and rewrote $\rho$ as: $\rho = e^{-|\alpha|...
1
vote
4answers
2k views

State vector vs density operator

We formulate quantum mechanics using language of state vectors. One alternative formulation is possible using density operator or density matrix. Why we are doing this alternative approach? Is the ...
1
vote
1answer
167 views

Quantum map and preservation of trace

I am currently learning about quantum maps, ie maps that transform a density matrix into another one. Assume we are in the Hilbert space : $H_A \otimes H_B$. I call the quantum map on the density ...
4
votes
2answers
317 views

Does a statistical system go into a pure state as the temperature $T\to 0$ (or $\beta\to\infty$)?

The density matrix $\hat{\rho}$ for a canonical ensemble is given by $$\hat{\rho}=\frac{\sum\limits_{n}e^{-\beta E_n}|n\rangle\langle n|}{Z}\tag{1}$$ $$=\frac{e^{-\beta E_0}}{Z}\Big[\sum\limits_{n=0}|...
4
votes
2answers
204 views

Is entanglement *not* intrinsic to state, but dependent on division into subsystems? (Susskind QM)

I'm working through Susskind's "Quantum Mechanics" book (TTM series), which I quite like. Background In Lecture 7 (Chapter 7), he studies a 2-spin system. A single spin has eigenvectors: $$|u\...
3
votes
1answer
160 views

von Neumann Entropy of a joint state

Definition 1 The von Neumann entropy of a density matrix is given by $$S(\rho) := - \mathrm{Tr}[\rho \ln \rho] = H[\lambda (\rho)] $$ where $H[\lambda (\rho)] $ is the Shannon entropy of the set of ...
3
votes
2answers
188 views

Why do populations only change in second order of the driving field?

In the field of quantum optics when solving master equations it is well know that the populations1 are constants to linear order in the driving field. I.e. weak driving fields will only affect the ...
2
votes
1answer
165 views

Classical and quantum correlations in bipartite system

I would like to know how to answer following questions: Is there classical/quantum correlations in given bipartite pure/mixed state? I have gathered several definitions. Some of them (it seems) ...
1
vote
1answer
474 views

Deriving the optical Bloch equations from the von Neumann equations

Is it possible to derive the optical Bloch equations for a 2-level-system driven by an oscillating EM-Field from the von Neumann equation for the density operator? I'm assuming a system consisting ...
1
vote
1answer
130 views

What sort of operations can be applied on a Hilbert spaces?

I was reading the paper No Universal Flipper for Quantum States. In this paper they have tried to prove by contradiction that a universal flipping machine cannot exist. By flipping I mean if I have a ...