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

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Difference between DMRG (density matrix renomalization group) and MPS (matrix product states)?

I am learning DMRG recently. I noticed there are many papers both in the DMRG approach and MPS (such as variational matrix product state (VMPS) by F.Verstraete and J.I.Cirac) approach. In my eyes, ...
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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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Trace representation of density matrix question [closed]

System $A$ and system $B$ form a composite system. https://en.wikipedia.org/wiki/Partial_trace I wonder why $\rho_{AB}$ cannot be represented as $(\text{tr}_{B}(\rho))\otimes (\text{tr}_{A}(\rho))$....
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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 ...
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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 ...
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Does this quote from my textbook imply that not all states are superpositions?

I read this at a book; The difference between bits and qubits is that a qubit can be in a state other than $|0\rangle$ or $|1\rangle$. It is also possible to form linear combinations of states,...
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Definition of the “support” of the reduced density matrix

Some of the papers in condensed matter physics use the word "support" (space). For example, the following papers use the support especially for the reduced density matrix. http://journals.aps.org/prb/...
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Separability of density operators on tensor product spaces

Consider a composite system $\mathcal{H}=\mathcal{H}_{A}\otimes\mathcal{H}_{B}$ where $\mathcal{H}_{A}$ and $\mathcal{H}_{B}$ are Hilbert spaces of constituent components (say two qubits). Let $\rho_{...
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Purity of a maximally mixed quantum state

The trace of the square of a density matrix(which is also called the Purity of the quantum state) is (lower)bounded by the inverse of the dimension of the Hilbert space and (upper) bounded by 1. I ...
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Density Matrix in Quantum Optics

I am studying laser physics and didn't get the whole idea of the density matrix (in this case for two-level systems). I know this is really basic so bear with me! I understand that the diagonal ...
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68 views

One-electron reduced density matrix: Argument for positive semidefiniteness

I cannot follow an argument for positive-semidefiniteness of the one-electron density matrix given in "Molecular Electronic-Structure Theory" by Helgaker/Jorgensen/Olsen. First some definitions: $F(M)$...
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Lindblad equation solution

I have been trying to solve a Lindblad Equation and then thought about whether there is a closed form Lindblad Equation solution for most types. Googling hasn't lead me to anything useful. So, is ...
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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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How does one describe a state with a density matrix after measuring position?

My question is about position measurement in non relativistic quantum mechanics. I've been taught that when you measure the value of an observable for some state of a system described by $|\psi\rangle$...
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Convex combinations of states yielding a pure state

Wikipedia states that Geometrically, when the state is not expressible as a convex combination of other states, it is a pure state.[9] The family of mixed states is a convex set and a state ...
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How can all quantum measurement statistics be seen just as projective measurements on pure states?

Let $\rho$ be the density matrix for a system and let the POVMs be $\{E_m\}$ such that $\sum_i {E_m} = I$. The probability of getting the outcome $m$ is $\operatorname{Tr}(E_m \rho)$. The source I ...
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Entropy change in Heisenberg picture

If we stick with Heisenberg picture where density matrix $\rho$ is constant, how do we account for entropy increase? I've read the answer to State collapse in the Heisenberg picture but I don't see ...
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Bounds on dimension of a purification?

Let $\rho \in H_A$ be a density operator, $H_A$ is finite dimensioal, it is well known that $\rho$ has a purification in some larger hilbert space. Let $b$ be the minimum dimension for $H_B$ such ...
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Time dependence of the displacement operator

I am following the derivation of the master equation (and application of this) in these lecture notes. Unfortunately I do not follow the step of eliminating the driving terms of the harmonic ...
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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 ...
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Kubo formula for general observables

In the wiki page about Kubo formula, the expectation of some observable under weak time-dependent perturbation is derived. However, from my point of view, some crucial steps are missing. I did the ...
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What is the effect of squeezing on the Husimi phase space representation or Q-function?

The effect of the squeezing operator \begin{equation} S = e^{- r (a^2 + a^{\dagger 2}) / 2} \end{equation} on a Wigner phase space representation or W-function of a system with density matrix $\rho$ \...
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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{\...
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What kind on transformations can be applied on density matrices?

Completely positive trace preserving maps ( CPTP ) transform a valid density matrix to another, then why do we only talk about unitary transformations on density matrices ( $\rho \to U\rho U^{\dagger}...
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Why reduced density operator being same is necessary sufficient for no signalling?

Problem Statement : Two parties $A$ ( Alice ) and $B$ ( Bob ) ( in order ) share an entangled pair $\frac{1}{\sqrt{3}}(|00\rangle+|01\rangle +|11\rangle)$. Bob does a measurement in basis $\{ |0\...
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How to see validity of no signalling principles in case of entangled parties?

From what I understood the density operator $\rho$ is a mathematical tool which tells us about the probabilities of getting a particular output after measurement. I have two parties entangled with ...
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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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Why is $\langle \textbf R \rangle =Tr \rho R$ here $\rho$ is the state density matrix

I came across this, \begin{post} To each state there corresponds a unique state operator.The average value of a dynamical variable \textbf{R}, represented by the operator $R$ ,in the virtual ensemble ...
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Kraus operators + path integrals = Lindblad equation?

The other day our professor was talking about Kraus representation of density operator and the derivation of Lindblad equation. He told that this was related to the Feynman path integrals and that we ...
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What is the reason for 2 qubits no longer being entangled after interaction with a causality violating qubit?

Background : I was reading the following paper on closed timelike curves ( CTC ) : Quantum Mechanics Near Closed Timelike Curves. The Deutsch consistency equation for CTC is $$\rho_{CTC}=Tr_{CR}( U \...
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Density matrix in Quantum Statistical Mechanics

What is the connection between the density matrix in quantum statistical mechanics and the probability of being a particular state in classical statistical mechanics? It would seem that the elements ...
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How can I calculate the partial trace for a combined state of a pair of two-level atoms to get a reduced state? [closed]

Let's say I have a combined state of a pair of two-level atoms, $A$ and $B$, given by the density matrix: $$ \rho = \frac{1}{2}\mid g_A, g_B \rangle \langle g_A, g_B\mid + \frac{1}{2} \mid g_A, e_B \...
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Can reduced density matrices of sub systems of an entangled composite system be different?

In a 4-dimensional hilbert space, only 4 entangled states( normalized ) are possible ( if I am not wrong ), the bell basis. In each of the state in bell basis the reduced density matrix is $\frac{I}{2}...
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Methods to distinguish between pure/mixed states and entangled/separable states

I'm a little confused about how we can distinguish between pure/mixed states and entangled/separable states and I would really appreciate some help! I understand a density operator $\rho$ represents ...
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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 ...
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Amount of entanglement in terms of greatest eigen value for hermitian matrices?

I was reading the paper No Universal Qubit Flipper. In this the paper they show inability to create a universal flipping machine. The method they follow is they take an entangled state between Alice ...
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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 ...
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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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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 ...
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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 = \begin{pmatrix}...
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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} \frac{d\rho_S(t)}{dt}=&-\left(\frac{i}{\hbar}\right)\Big[H_S+\frac{1}{2}M\...
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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 ...
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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 $\mathcal{H}=\mathbb{C}^...
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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 ...
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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} Tr([\sqrt(...
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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 ...
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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 ...
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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 (...
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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 $|\...
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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 ...