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

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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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States after measurement

I am learing about quantum mechanics, and I am quite unclear about the the role of measurement and the density operator. Let $\mathcal H$ be a Hilbert space, $\psi(t) \in \mathcal H$, $O$ be a ...
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SU(2) Coherent State and Probabilities

The SU(2) Corherent State for a for more than 1 two level system is given by: $$ | \eta, J \rangle = (1 + |\eta|^2)^{-J} \sum_{m=-J}^{J} \sqrt{\binom{2J}{J + m}} \times \eta^{J + m} |J, m \rangle, ...
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Entanglement of bi- and tripartite pure and mixed states

since I'm not sure on how to find out whether a system is entangled or not I thought about examples that could clarify the whole thing. first example: system is in the state $\rho=1/2 (| 000 \rangle \...
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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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expectaction value of coordinates on bloch sphere under special time evolution

We consider an example in which the time evolution of a system is given by the following density matrix $\rho(t)=\frac{1}{a_0+c_0}\begin{pmatrix} \eta^2(t)a_0 & \eta(t) b_0\\ \eta(t)b_0^{*} &...
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Density operator - Coherence relativ to a different basis

Consider the density matrix of the pure state $\left|\psi\right>$. Relativ to the basis, whose eigenvector is $\left|\psi\right>$, there are no coherences. Do a basis transformation and show ...
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Examples of Matrix Product State(s)

Matrix product states(MPS) is a way of representing a (many-body) wavefunction. The method has been described in, https://arxiv.org/abs/1008.3477 However, would it be possible to see a concrete ...
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On what conditions is a matrix a density matrix (of a pure state)?

I've been reading Weinberg's Lectures on Quantum Mechanics and the topic of the density operator (or density matrix, whatever you may call it) baffles me. Basically we have, $\hat{\rho}=\sum_ip_i\...
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Quantum master equation and off diagonal terms

I have a couple of related questions What is exactly the difference between the quantum master equation and the regular master equation? My understanding is that the normal master equation is used ...
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how to get the matrices in partial trace

Good day, I want to ask the matrix that obtained from this link How to take partial trace?. answer from @Nontriviality below In principle what you do is to multiply the square matrix by rectangular ...
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Two qubits system in polar co-ordinates

I know that I can write a single qubit state in terms of polar co-ordinates $(r,\theta,\phi)$ on a Bloch sphere. \begin{equation} \rho = \begin{pmatrix} \frac{1+r \cos\theta}{2} &\frac{r \exp(-i\...
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2answers
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In quantum double slit experiment what is the state vector or density matrix of the electron after the electron passes through the two slits?

Also I would like to confirm my thinking on quantum double slit experiment. Before it passes through the two slits (slit 1 and slit 2), is the electron state vector $\frac{1}{\sqrt{2}}\left(\left|...
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1answer
103 views

How can we define the distance for a pair quantum states in phase space?

In condensed matter physics, we know that two degenerate ferromagnetic ground states $|\uparrow\uparrow ...\uparrow \rangle$ and $|\downarrow\downarrow...\downarrow\rangle$ are far from each other in ...
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1answer
40 views

Probabilities with the Density Matrix

The density matrix of the system is given by: $$ [\rho_{S}(t)]_{mn} = [\rho_{S}(0)]_{mn} e^{-i\omega_{0}(m - n)t} e^{-i \delta(t)(m^2 - n^2) - \gamma(t)(m - n)^2}, ...
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43 views

How to calculate the partial trace [duplicate]

Can anyone help me in explaining how this example below get the reduced density matrix from the density matrix in bipartite system. $$\rho =\frac{1}{4}\begin{pmatrix} 1 & 1 & cos(\frac{\alpha}...
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1answer
58 views

Reduced Density operator in matrix form

I already read book of Quantum Computation and Quantum Information by Nielsen and Chuang according to reduced density operator and I already understand how to do the reduced density using Dirac ...
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Link Between the Density Operator and the Partition Function and Boltzmann Distribution in Quantum Statistical Mechanics

I have a very limited knowledge of statistical mechanics, but I seem to running into some related concepts for my background readings for the research project this summer. For example, see the ...
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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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1answer
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Dirac notation - trace of product of (bipartite) density matrices

I'm getting confused by the Dirac notation. Suppose I have the following two objects. $$\rho = \sum_k p_k (\rho_A \otimes \rho_B) = \sum_k p_k |k \rangle \langle k | \otimes |k\rangle \langle k | ,$$...
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Questions involving seperable states

I'm reading Mark Wilde's book Quantum Information Theory, and I'm stuck on two parts. I'm unable to prove: The state $\sum_{z} p_{Z}(z) \; \rho_z \; \otimes \sigma_z $, where $\rho_z$ and $\sigma_z$ ...
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Equivalence condition of lindblad operators

So from Nielsen and Chuang th. 8.2 we know the equivalence condition of Kraus operators (quantum operation) What is the equivalence condition of lindblad operators of master equation?
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1answer
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Undergraduate quantum book treating density operators, mixed states, and entanglement [duplicate]

I'm working on a project on quantum measurement theory - in particular, relating to the quantum Zeno effect - over the summer. Right now, I'm in the process of doing background readings that'd enable ...
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1answer
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What is an incoherent state?

I am reading through a recent paper which speaks frequently of "incoherent states" without ever defining what such a state is. I gather from the context of the paper that it has something to do with ...
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2answers
114 views

Does Unitary operator take a pure state to a pure state or can it take a pure state to a mixed state? [closed]

Does Unitary operator take a pure state to a pure state or can it take a pure state to a mixed state? I think so but why? I assume the Unitary operator acts on a pure state only.
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Can every density operator be written as an outer product of two vectors?

I have a feeling this is a very basic question. I apologize if it is. Using Dirac's notation, can every (mixed) density operator $\rho_A$ of system $A$ be written as the ket-bra (outer) product $|a_1 ...
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Density Matrix representation of excited atoms

I'd like to get an answer to this question from someone who knows his density matrix theory. I want to compare two different systems and ask how their density matrix representation looks. First look ...
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1answer
55 views

Two definitions of the density matrix?

There seems to be two different definitions of definitions of density matrices in Physics. In Quantum Information we define a the density matrix associated with a wave function $ | \psi \rangle$ as $...
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1answer
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Diagonalisation: Schmidt vs eigenvalue - when to use which?

In physics we encounter diagonalisation of matrices or operators in a variety of areas. But there are different kinds, the main two being Schmidt decomposition and eigenvalue diagonalisation. The two ...
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1answer
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A seemingly paradox for Eigenstate Thermalization Hypothesis (ETH)

ETH states that for a system, all of its eigenstates thermalize. To be more specific, consider an energy eigenstate of the full system $H|n\rangle=E_n|n\rangle$. If the full system is in this ...
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What is the qualitative difference between quantum superpostion and mixed states? [duplicate]

As I understand it, if one has a complete knowledge of the state of a quantum system (insofar as one knows the statistical distributions of all the observables associated with the state) then one can ...
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Quantum computing entanglement dimensions question

While trying to understand the basics of how quantum computers work, I recently read this statement. "...consider that single-qubit states can be represented by a point inside a sphere in 3-...
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2answers
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Physical meaning of $Tr(\rho ^2)$

If $\rho$ is the density matrix of a system then $Tr(\rho ^2) \leq 1$. If the equality holds the system is in a pure state and it is in a mixed state otherwise. But, what is the physical meaning of $...
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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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Probability distribution of a pretty-good measurement

Let $\rho_{XE}$ be a classical-quantum state. That is, $$ \rho_{XE} = \sum_{x}\Pr[X=x] \cdot |x\rangle \langle x | \otimes \rho_{x} $$ where every $\rho_{x}$ is a density matrix with $\mathrm{Tr}(\...
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Berry phase with density matrix approach

Berry phase, coming from Schrodinger equation, has well known form in terms of closed integral $$\gamma = \int_C A(\xi) d\xi $$ with Berry connection $$A(\xi) = i < \psi(\xi) | \partial_{\xi} | \...
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the density matrix in QFT on a cylinder

My question regards the density matrix in quantum field theory on a cylinder. The partition function is given by $Z=\text{Tr} e^{-\beta H}$. The elements of this thermal density matrix become \begin{...
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1answer
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Can mixed states be treated in the second quantization formalism? [closed]

In the first quantization formalism, mixed states can be handled using density matrices. When treating many-body quantum systems however, the second quantization formalism often comes handier, ...
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1answer
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Normal Ordering and Smearing

I read on Wikipedia two different descriptions of the "Husimi-Q representation." One is that it is the Wigner function convolved with a Gaussian, which in particular results in a positive definite ...
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1answer
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How to represent a Liouville projection superoperator in Hilbert space?

Is there a general way to represent a Liouville projection operator in Hilbert space, or can they take on any form so long as they satisfy the required properties of a projector? e.g. The thermal ...
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1answer
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Understanding the optimal approximate qubit cloning method

I'm trying to understand the operation used for optimal cloning of pure qubits states from the paper Optimal Cloning of Pure States by R. F. Werner. The paper describes the optimal cloning method $\...
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1answer
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2x2 Matrices that are not valid quantum states

Given a 2-dimensional Hilbert space, quantum states can be expressed as $2\times 2$ density matrices. In terms of the Pauli matrices, or Bloch representation, they can be written as \begin{equation} \...
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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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How to include temperature effect in optical bloch equations (optical pumping)?

My problem is about the optical pumping of Alkali atoms by circularly polarized pump light. Consider a circular polarized light ($\Delta m=+1$) $$\vec{E}(z,t)= \vec{E}^{(+)}_0 e^{-i\nu t}+c.c. $$ ...
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Density matrix from Wigner distribution

Density matrix or Wigner function can be defined from the other with Fourier (or inverse) transformation. equivalently the value of W(q,p) can be seen as the mean value of the displaced parity ...
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What are fragmented condensates?

It is defined that if more than one eigenvalue of the one-body density matrix are macroscopically occupied the condensate is said to be fragmented. $$ n^{(1)},n^{(2)},...=\mathcal{O}(\mathcal{N}) $$ ...
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reduced density matrix of multiparticle system

Let the state of a composite system is $|\psi_{i}\rangle$ then the density matrix is defined as, $\rho=\sum P_{i}|\psi_{i}\rangle\langle\psi_{i}|$ Consider a composite system of two particles and its ...
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Complex vector on Block sphere [closed]

I've the following problem. Given this vector on a 3D complex space: $$\\\\ {\phi_{1}} = \begin{matrix}% 1/2(-1, & i\sqrt{2}, &1)^{T} \end{matrix}\\ $$ Is it possible to draw it on the ...
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Reduced density matrix

During a course on quantum mechanics we've been talking about density matrices. Now I came across the following exercise. Consider a two spin $\frac{1}{2}$ systems, labeled 1 and 2. Calculate: ...