Questions tagged [density-operator]

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

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Pure state vs mixed state in this example

Consider, I have a quantum state $|\Psi\rangle$, such that : $$|\Psi\rangle=c_1|\psi_1\rangle+c_2|\psi_2\rangle$$ This is defined as a pure state, since I have complete information about the system. ...
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Master equation with a coherent bath

When we consider an oscillator $a$ acting with a bath of oscillators $b_i$ with the interaction Hamiltonian reads $$H_{int}=\sum_{i}g_ia b_i^{\dagger}+g_i^*a^{\dagger}b_i,$$ with the free Hamiltonian: ...
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Finding the time dependency of the matrix elements of the density operator [closed]

So I have been asked to do the above task but I am not sure how to do it correctly. A hamiltonian is given as: $$ -((0 A), (A, 0)) $$ $A$ being a matrix element. I decided to do: $$ p_{11} = \tfrac12 \...
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What is an intuitive or simple proof of Gleason's theorem and how it relates to the Born rule?

What is an intuitive or simple proof of Gleason's theorem and how it relates to the Born rule? I tried to read the articles, but the proof seemed big and the kind that are unintuitive (im not ...
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Problem understanding expectation value of operators defined with density operator in quantum mechanics

I have a problem in understanding why we can write the expectation value of an operator $\hat{O}$ as the trace of $\hat{\rho}\hat{O}$ where $\hat{\rho}$ is the density matrix defined for pure state. ...
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Why an energy constraint gives the canonical ensemble (NVT) when we maximize von Neumann entropy, when energy is not conserved in NVT?

In Sakurai and Napolitano's Modern Quantum Mechanics 2nd Edition, I'm learning Section 3.4, page 188 where they derive the canonical ensemble by maximizing von Neumann entropy with energy constraint. ...
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How to write an arbitrary qubit density operator? [duplicate]

Physics noob here: I am reading the Wikipedia on Density Matrices (https://en.wikipedia.org/wiki/Density_matrix), and in the section labeled "Pure and mixed states", it states "An ...
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Position representation of density operator in collapse models

I have been studying this paper for the past few days but really get stuck on one specific derivation that seems easy but I can't comprehend it. It's about quantum collapse models, so quite niche. We ...
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Pure and mixed density operators of a Schmidt decomposition

Suppose we have Hilbert space factorisable in to K subsystems $$ \mathcal{H} = \mathcal{H}_1\otimes...\otimes\mathcal{H}_K $$ in which we can express a pure state as the Schmidt decomposition $$ |\...
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Variational principle of statistical mechanics

I was just reading an article called "Extractable work from ensembles of quantum batteries". in the third section of the article, it refers to 'variational principle of statistical ...
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Why is the state of a quantum system called "Density $\textbf{Operator}$"?

In quantum mechanics, a $d$-dimensional pure state is represented by a vector belonging to a $d$-dimensional Hilbert space $\mathcal{H}^d$. A mixed state is represented by a density matrix $\rho \in \...
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Free particle in momentum representation

In Greiner, density operator for a free particle has been calculated in momentum basis. They consider a large box of vilume $V=L^3$ and periodic boundary condition. $$\phi_\vec{k}(\vec r)=\frac{1}{\...
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Expression of density operator for Microcanonical ensemble

Consider a quantum microcanonical ensemble,with a fixed energy $E$. In Greiner, the expression for its density operator is given as. $\displaystyle\hat\rho=\frac{\delta(\hat H-E.1)}{Tr(\delta(\hat H-E....
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How do I prove that a reduced density matrix has properties of a density matrix?

The properties of a density matrix are defined as follows: $(1) \ \ \mathrm{Tr}\rho = 1 $ $(2) \ \ \rho^\dagger = \rho $ $(3) \ \ \rho \ge 0 $ $(4) \ \ \mathrm{Tr}\rho^2 \le 1 $ A reduced density ...
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Degrees of freedom of a $d$ level system and the dimension of its Bloch manifold

A qubit is described by two independent degrees of freedom that parametrize the Bloch sphere. Question: For a level $d$ level system, i.e., a qu$d$it, what are the corresponding degrees of freedom? ...
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Partial trace of local operators applied to maximally entangled states

I was looking at a problem where two invertible local operators were applied to a maximally entangled state, and didn't quite understand how some of it works out. We have local operators $A \otimes \...
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Definition of tensor product in Nielsen & Chuang "Quantum Computation and Quantum Information"

Nielsen and Chuang (Quantum Computation and Quantum Information, 2010) define the tensor product of linear operators acting on vector spaces on page 73. Therefore the tensor product used in the ...
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Off-diagonal elements of density matrix in three-level system

given is a three-level system, let's say an atom with three possible states 1 being the lowest and 3 the highest ($E_1<E_2<E_3$), where $$ \Psi(t)= C_1(t)|1\rangle+C_2(t)|2\rangle+C_3(t)|3\...
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Physical Interpretation of eigenvalues/eigenvectors of the density matrices of a given order

I was reading this paper by Per-Olov Löwdin and it discusses how density matrices can be used to represent/interpret the wavefunction. And, I had a question regarding how the eigenvalues and ...
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On the language use of quantum mechanics: "state $\rho$" or "density matrix $\rho$ of the mixed state"?

For pure states one usually uses the bra-ket Notation and then uses language e.g. "the state $|\psi>$..." Is it also common to say similarly for mixed states, which are usually written as ...
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Order in taking density matrix trace

I remember that for infinite dimensional Hilbert space, the trace is not cyclic, for example, for harmonic oscillators, then we have $$Tr(aa^{\dagger})\neq Tr(a^{\dagger}a)$$ Then, when we calculate ...
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Is there a relation between the density matrix and the density of the position probability? Are they the same concept?

We have a system of two bosons particles and we are interested in calculating the one-particle density and two-particle-density when both are in different states. So, to do that, I consider the ...
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Construct the Density Matrix of a Gaussian State from its First and Second Moments / Wigner Function

Borrowing some description for the setup from a question I posted earlier here; Suppose we have $N$ bosonic modes (or quantum harmonic ocsillators) with the usual commutation relations. Now define the ...
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Necessary and sufficient conditions for operator on $\mathbb C^2$ to be a density matrix

Consider a one-qubit system with Hilbert space $\mathscr H\simeq \mathbb C^2$. Define the hermitian operator $$\rho := \alpha\, \sigma_0 + \sum\limits_{i=1}^3 \beta_i\, \sigma_i \quad , \tag{1}$$ ...
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Find coefficient for pure and mixed states

Consider a generic $2\times 2$ Hermitian matrix written as $$\rho =\alpha\sigma_0+\beta\hat{\vec n}\cdot\vec\sigma\quad ,$$ where $\hat{\vec n}$ is a unit vector and the coefficients are real numbers. ...
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How to prove that every mixed one-qubit state admits a Bloch-sphere representation? [duplicate]

A mixed state $\rho$ can be written as $$\rho=\frac{1}{2}\left(I+r_x\sigma_x+r_y\sigma_y+r_z\sigma_z\right)\qquad\left(\vec{r}:=\left(r_x,r_y,r_z\right)^T\in\mathbb{R}^3; ||\vec{r}||\leq 1\right)$$ ...
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Given the Symplectic Matrix acting on phase space, find the Gaussian Unitary acting on the Hilbert space

In Gaussian Quantum Mechanics, a unitary preserving the Gaussian nature of the state is a called a Gaussian Unitary. In the phase space picture, a Gaussian state is fully characterized by its first ...
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Can quantum Zeno dynamics be written as a Lindblad master equation?

The interaction of the system with the environment is considered weak and Markovian in the Lindblad master equation. Whereas the measurements involved in the Zeno effect seem quite drastic, I think. ...
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Why is the density matrix of a system has this block form?

In Ficek's paper (http://zon8.physd.amu.edu.pl/~tanas/spis_pub/pdf/04-joptb-S90.pdf), the density matrix of a two two-level atom system has a block form like this. Why does it make sense to assume ...
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Analog of the Pauli vector for $SU(4)$

In quantum mechanics and representation theory, it is well known that the Pauli matrices transform as a vector due to the special relationship between $SU(2)$ and $SO(3)$. For example, suppose we have ...
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Trouble proving properties of the density matrix

So let's say you've got a Hilbert space that's $n$-dimensional with some vector $\phi$, such that $\langle \phi | \phi \rangle = 1$; let $|\phi \rangle \langle \phi | = \rho$. Also say we've got some ...
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Are there cases where given an ensemble, we cannot blindly replace the ensemble by another that has the same density matrix?

It is known that if two ensembles result in the same density matrix, then they give rise to the same observable statistics and the two ensembles are related by a transformation described in this post. ...
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Finding the Kraus Operators of a Quantum Channel from its Choi Matrix

TLDR : What is the form of the projector I need to use to attain a 2X1 vector from $P_i v_k$ with which I can build my Kraus Operator? I am calculating the Kraus Operators for a Quantum Channel ...
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How to test if a bipartite density matrix violates Bell's inequality?

For a given density matrix $$\rho = \sum_{ijkl=0}^1 r_{ijkl} |i,j\rangle \langle k,l|$$ describing a bipartite two-qubit system, how can I prove for what values $r_{ij}$ the density matrix violates ...
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Difference in density matrix where prob. 1/2 in both $|0\rangle$ and $|1\rangle$ [duplicate]

In my course, we had an example: give the density matrix of a system arising from a proces that generates a [0> with probability 1/2 and a [1> with prob. 1/2. The answer is $1/2[0\rangle\langle0]...
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Compressing the Hilbert Space in Traditional DMRG

The traditional, non Matrix Product State, formulation of the Density Matrix Re-normalization Group (DMRG) algorithm can be coded in python. Such a code can be found in the following link: https://...
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Quantum Mechanical Pictures and Operator Expansion

In Quantum Mechanics, there are 3 pictures: Schrodinger, Heisenberg and Dirac (Interaction) picture. In Schrodinger picture, the states evolve in time and operators are constant in time, and vice ...
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Measurement on mixed states

I have a conflict between my lecture notes on quantum mechanics, where it is stated that the probability of measuring an eigenvalue $a_i$ on a mixed state with desnsity matrix $\rho$ is $$ \...
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Perturbative calculation of the number of particles

We know that in QFT particle production is modeled by adding an external source, $j(x)$, where $x$ is the space-time point, the interaction term in the Hamiltonian density is of the form $j(x)\phi(x)$,...
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Current distribution in hydrogen atom

In the Bohr model of hydrogen, the electron orbits the proton at selected radii such that the angular momentum is quantized in integer multiples of $\hbar$. We can then associate a current with such a ...
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Preserving normalisation for non-unitary dynamics in the Heisenberg picture

As the title says. Say I'm trying to calculate some 2 point correlator of some operator $\hat{\sigma}$: $$ \langle \hat{\sigma}(t)\hat{\sigma}(t+\tau) \rangle = \text{tr}[e^{L^{\dagger}\tau}(\hat{\...
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Does a subsystem being mixed imply the state is entangled?

If a pure state, $\rho_{AB}$, has subsystems described by mixed density matrices, the overall state is entangled (as far as I understand). Can you conclude the same with an initially mixed bipartite ...
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Concurrence between 2 qubits of tripartite system

If we consider a tripartite system (say of 3 qubits) described by density matrix $\rho_{ABC}$, does the concurrence $C(\rho_{BC})$ still accurately measure the entanglement between subsystems B and C? ...
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Interpretation of the Bose-Einstein Condensate Wavefunction squared and Density Matrix

I'm currently trying to get some background information about the theoretical treatment of Bose-Einstein Condensate, and I'm reading through this paper on the topic by J. Rogel Salazar. I have also ...
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Definition of density operator/ Matrix

In Sakurai's book, the density operator is defined as $$\rho\equiv \sum_i w_i|i\rangle \langle i|$$ where $w_i$ is statistical weight. Now, I'm reading a book by Parisi, In which it says in Sec 5.1, ...
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The Density Matrix of a Pure State

It is my understanding that any pure quantum state $|\psi\rangle$ can be represented by the density matrix $|\psi\rangle\langle\psi|$. It is also my understanding that $|\psi\rangle\langle\psi|$ ...
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Partial Trace of Density Operator

To find the reduced density matrix, $\rho_A$, of a composite quantum system with two subsystems A and B, I've seen that the procedure is to take the partial trace of the full density matrix, $\rho_{AB}...
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In what way is conditional quantum probability restrictive, and why?

This is close to a duplicate of https://mathoverflow.net/q/412327/ but with a different emphasis. Unlike the mathoverflow equivalent, here I want to ask for your informed intuition as physicists. To ...
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What's the physical meaning of a reduced density matrix in EPR?

Consider an EPR situation in which there are two particles, a and b, of which state is given by $\Psi = \frac{1}{\sqrt2}(|1\rangle|0\rangle + |0\rangle|1\rangle)$, where $|0\rangle$ and $|1\rangle$ ...
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Density matrix element in Jaynes-Cummings model

In the Jaynes-Cummings model, when using the density matrix to describe mixed states for the atom-field system, after some calculations I got to this matrix element: $$ \rho_{ee}^A = \sum_{n=0}^{\...
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