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

learn more… | top users | synonyms

1
vote
1answer
47 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 ...
4
votes
2answers
243 views

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 ...
0
votes
0answers
6 views

Correct basis for a bosonic bipartite system

Suppose I have two interacting bosonic systems in a double-well potential. They interact, if you want, via a Bose-Hubbard hamiltonian $H_1$ and $H_2$ (where 1 and 2 labels the corresponding ...
1
vote
2answers
134 views

How can I prove following density matrices have same eigenvalues?

I have the following two density operators, the paper I am reading says that these two operators have same eigenvalues $$\rho^i = \frac{1}{3} ( |0\rangle \langle 0 | +|1\rangle \langle 1 |+|2\rangle ...
0
votes
1answer
34 views

Distinguishing density operators with the same diagonal elements

If I have two sources of qubits and one source produces the density matrix: $$\rho_1 = \begin{pmatrix} 1 & 0 \\ 0 & 1\end{pmatrix}$$ and the other source produces: $$\rho_2 = ...
1
vote
1answer
40 views

Solving for the density operator in the quantum Brownian motion master equation

I want to solve for the density operator in the quantum Brownian motion master equation, \begin{align} \begin{aligned} ...
0
votes
2answers
72 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 ...
2
votes
3answers
125 views

Expression of density operator

States in Quantum Mechanics can be thought of as density operators, i.e., positive semi-definite, normalized trace class operators on a Hilbert Space $\mathcal{H}$. In the case ...
1
vote
0answers
25 views

Are the Wigner and Husimi transforms injective?

I am wondering if the Wigner function is injective. By injective I mean, that, for every density matrix $\rho$, there is a different Wigner distribution. The same question applies to the Husimi ...
0
votes
1answer
31 views

Wigner-Yanase skew information [closed]

I am reading Eric Carlen's paper on Trace Inequalities and Quantum Entropy. I am currently reading about the Wigner-Yanase skew information which is defined as: $$I_{WY}(\rho)=-\frac{1}{2} ...
2
votes
2answers
83 views

Finding the matrix representation of a superoperator

I am trying to express superoperator (e.g. the Liouvillian) as matrices and am having a hard time finding a way to do this. For instance, given the Pauli matrix $\sigma_y$, how do I find the matrix ...
2
votes
1answer
62 views

Different kinds of trace for statistical ensembles

In the chapter 7 of the book "A Modern Course in Statiscal Physics" by L. Reichl, we found $Tr[\hat{\rho}]=1$ for microcanonical ensembles and $Tr_N[\hat{\rho}]=1$ for canonical and grandcanonical ...
3
votes
1answer
41 views

Why is the frequency bandwidth of the environment important for Markovianity?

In the derivation of Spontaneous Emission in two level systems in Quantum Optics (be it Wigner Weisskopf or a different approach, such as density operators to find the master equation), one makes ...
2
votes
2answers
132 views

What does density operator being same for two sytems tells us?

Yesterday I asked a question. I got it that if a density operator is given as $$\rho=\sum_{i=1}^{i=k}p_i|\psi_i\rangle \langle\psi_i| \tag{1}$$ then it means that the system is one of the states ...
7
votes
3answers
505 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 ...
3
votes
2answers
97 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 ...
1
vote
1answer
119 views

Importance of zero and non-zero eigenvalues of density matrix

What can we say about the quantum state from the number of zero and non-zero eigenvalues of the corresponding density matrix? Anything related to entanglement or any other properties? Does they vary ...
1
vote
1answer
43 views

Probabilities of pure states and density operators

According to my skript: A pure state is a ray: $\quad$ $\{λψ\}$, where $ψ ∈ \mathcal H$, $||ψ|| =1$ fixed and $λ ∈ \mathbb C$, $|λ| = 1$. Pure states are uniquely given by 1-dimensional orthogonal ...
0
votes
0answers
20 views

Criterion of tangled state by using particular transposition of density matrix

Let's have (for simplicity) two-qubit density matrix $\rho $. If we make the particular transposition only by one qubit, we'll get some other matrix $\tilde \rho $. Then the criterion of the mixed ...
1
vote
0answers
23 views

Sufficient criterion for su(2) invariant spin-1*spin_s bipartite density matrix

SU(2) invariant spin-1 and spin-S bipartite density matrix is given by $\rho ^{1,S}=\frac1{3*(2S+1)}[1+\alpha {S^A_i\times S^B_i}+\beta S^A_{ij}\times S^B_{ij}]$, i j varies from ...
5
votes
1answer
105 views

Equivalence classes in a Hilbert space

I'm reading something about quantum information/quantum computing theory, and I've run into a wall. I know what is meant by an equivalence class and how something can be partitioned into equivalence ...
0
votes
0answers
30 views

Eigen value, matrix, Quantum game

In this paper, on the page 5 http://math.ucsd.edu/~dmeyer/research/publications/qstrat/qstrat.pdf in the second paragraph: his first action puts the penny into a simultaneous eigenvalue 1 eigenstate ...
1
vote
1answer
91 views

What are density matrices and how do they work?

I have looked in Stack Exchange about density matrices but haven't found any answers. What are density matrices and how do they work? What are they used for? (Also, please tell me what is wrong with ...
0
votes
1answer
55 views

What is the density operator for an isothermal–isobaric ensemble (T,p,N)?

In the microcanonical ensemble $(E,V,N)$, the density operator is $$\hat{\rho}=\frac{\delta(\hat{H}-E\,\hat{I})}{Tr(\delta(\hat{H}-E\,\hat{I}))}$$ Where $\hat{H}$ is the Hamiltonian of the system and ...
1
vote
1answer
47 views

Density matrix of a single qubit as a function of its Stokes Parameters

$\newcommand{\bra}[1]{\left\langle#1\right|} \newcommand{\ket}[1]{\left|#1\right\rangle} \newcommand{\prom}[1]{\langle{#1}\rangle} \newcommand{\matrixel}[3]{\bra{#1}{#2}\ket{#3}}$ How can I prove ...
1
vote
1answer
47 views

Strange definition of a two-level system by the Bloch vector

A two-level system can be described by a density operator involving the Bloch vector $$ \vec{r}; \quad r_x = Tr(\rho X); \quad r_y = Tr(\rho Y); \quad r_z = Tr(\rho Z) $$ as $$ \rho = \frac{I + ...
1
vote
3answers
147 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
92 views

Quantization from density of states

The density of states in quantum mechanics is obtained via the rather not so complicated relation below: $$\rho(E)=\delta (E-E_n)$$ Which means that if we know the energy quantization condition for a ...
15
votes
5answers
852 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 ...
3
votes
2answers
117 views

How does one diagonalize a density operator that has exponential elements?

What is the diagonal form of the density operator $\hat\rho$, of which I know that $$\langle x\left|\hat\rho\right|x'\rangle\propto \exp\left[{-\frac{\gamma}{2}(x^2+x'^2)+\beta xx'}\right]$$where ...
1
vote
1answer
74 views

Entangled vectors in hilbert space

We consider a system of two particles of spin $\frac{1}{2}$, each described by the two-dimensional one-particle Hilbert space $\mathcal{H}$. Let $|\pm\rangle\in\mathcal{H}$ denote the eigenvectors of ...
0
votes
1answer
51 views

Example of a state which is positive but its partial transpose is not positive

Could any one give me an example of a state whose density matrix is positive semidefinnite but partial transpose is not positive semidefinnite?
2
votes
3answers
334 views

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 ...
1
vote
1answer
96 views

How do we find the canonical ensemble density matrix for two spins?

A compound system is constructed by two coupling spins, and the Hamiltonian is $$ \hat H = -J\hat\sigma_1·\hat\sigma_2 - \mu_\mathbf{B}\big( \hat\sigma_{1z}+\hat\sigma_{2z} \big)B. $$ So, how ...
5
votes
0answers
112 views

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}} ...
0
votes
1answer
92 views

How can I solve an equation involving partial trace?

I am unable to find the solution to the following equation: Tr$_{2}[U(|\psi\rangle \langle\psi|\otimes \rho)U^{\dagger}]=\rho$ Here $\psi$ is state vector representing a qubit and $\rho$ state of ...
1
vote
1answer
60 views

Prove the solution of von Neumann equation will never stabilize if Hamiltonian and initial density matrix commutes

Given von Neumann equation $$\frac{d}{dt} \rho(t) = -i [H, \rho(t)] = -i e^{-iHt}[H, \rho(0)]e^{iHt}.$$ If we know that $[H, \rho(0)] \neq 0$, how do we prove in details the solution of von Neumann ...
5
votes
1answer
203 views

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 ...
0
votes
0answers
50 views

Non-unqiue basis sets of reduced density matrix in quantum mechanics/decoherence

In Why decoherence solves the measurement problem by Art Hobson: $|\psi \rangle _{SA} = c_1|s_1 \rangle |a_1 \rangle + c_2 |s_2\rangle |a_2 \rangle$ which is a wavefunction that describes non-local ...
5
votes
2answers
226 views

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 ...
3
votes
1answer
59 views

Linearity of the time evolution operator for the reduced density matrix of an entangled state

Suppose to have a system $S$ immersed in an enviroment; the pure states are elements of $H_S \otimes H_E$, where $H_S$ is the hilbert space of the system and $H_E$ is the hilbert space of the ...
18
votes
4answers
952 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. ...
1
vote
1answer
46 views

Jump Method and the Lindblad Equation

I am studying the time evolution of a density matrix using the Lindblad equation. My initial density matrix is $\rho(0)=|\alpha\rangle\langle\alpha|$, where $|\alpha\rangle$ is a coherent state. Then ...
3
votes
2answers
116 views

Solution of dynamics of density matrix

Given the dynamics of the density matrix: $ \frac{d}{d t}\begin{pmatrix} \rho_{00} & \rho_{01} \\ \rho_{10} & \rho_{11} \end{pmatrix} = \begin{pmatrix} \lambda ...
0
votes
1answer
67 views

Meaning of the Reduced Density Operator

I am confused about what it is exactly that a reduced density operator describes. To illustrate, I came across the following seemingly paradoxical argument. Consider a biparte system $AB$, described ...
2
votes
2answers
57 views

Derivation of a quantum dynamical map on open quantum system

Let us consider a initial total quantum system as $\rho(0) = \rho_S(0)\otimes\rho_B$ where $\rho_S(0)$ is initial open system and $\rho_B$ is density matrix for environment. We can use partial trace ...
4
votes
1answer
865 views

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)] ...
1
vote
1answer
248 views

Heisenberg XXX time evolution operator for three qubits

I've a problem to reproduce the result in equation (4) on page three of this paper: http://arxiv.org/abs/0802.2588. So far I've understood that they apply a Heisenberg XXX interaction between ...
0
votes
0answers
96 views

Reduced density matrix of coupled oscillators and entanglement entropy

This paper describes a way to find the entanglement entropy of $N$ entangled harmonic oscillators, after tracing out the first $n$. A few statement made within have royally confused me, and I haven't ...
2
votes
1answer
164 views

Eigenstates of a density matrix of continuous variables

Consider a system of two entangled harmonic oscillators. The normalised ground state is denoted by $\psi_0(x_1,x_2)$. The reduced density matrix of the second oscillator is given by: $$\rho_2 = ...