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Questions tagged [density-operator]

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

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Lindblad and Input-Output Formalism in Quantum Optics

I'm confused about how to apply the Lindblad formalism and the input-output formalism in practice, and how one goes between the two. Suppose I have a cavity (C) coupled to a reservoir (R), with the ...
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Definition of the fully depolarizing quantum channel

The fully depolarizing quantum channel in a $d$ dimensional Hilbert space is defined by $$ \mathcal N^D (\rho) = \text{Tr}[\rho]\frac I d $$ I've seen that definition in several places but I don't ...
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“Interaction-Free” measurement involving statistical mixtures

What happens in the standard "interaction free measurement" when the detector connected to a bomb is replaced with attenuation (where light is lost through a semi-transparent medium)? Consider the ...
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Need help understanding weird definition of pure states [duplicate]

So in many sources I have read that A pure state contains only one element, since the only entry on the density matrix will be 1. But what about superpositions?...
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Thermalization and structure formation, starting with a pure state coupled to heat bath

I am looking for tractable quantum mechanical systems which show thermalisation and/or structure formation. One idea is a small quantum system S, e.g. a quantum Heisenberg model in a pure state (the ...
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reduced density matrix of state

given a multi particle state I have to calculate the reduced density matrix where I trace out the third particle for this I first calculate the corresponding 2D density matrix with the bra vector of ...
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What are ' non-trivial transitions' in this context?

So I am reading this paper where the goal is to prove that thermal states are the only completely passive states. To be more exact first the paper consider a single system with let's say a density ...
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Constructing two-qubit density matrix given expectation values of all products of Pauli operators

I think my question breaks down into two parts. Let's say you have a two qubit system and you can perform projective measurements. Each round of measurements will consist of results looking like ...
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A confusion about why can't a statistical mixture be modelled as a superposition of pure states?

I have read Cohen's book, and various posts in this site; however, I'm still not convinced why we can't model a statistical mixture as a superpositions of pure states ? For example, consider the ...
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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 ...
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What does a “pure” state mean in QM? [duplicate]

Question: In Quantum Mechanics, people use the word "pure state" for some states; however, what do they mean exactly ? Thoughts: I mean, a state is a vector in our vector (Hilbert) space, so in that ...
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Kraus operators of a POVM

I would like to know how one finds the Kraus operators of a channel corresponding to a POVM. Consider a POVM of the form $M_i$ such that $\sum_i M_i = \mathbb{I}$. I can represent this by a quantum ...
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Time propagation of density

I need some quick help understanding this equation. $\frac{\partial}{\partial t} \rho(r,t) = \frac{i}{\hbar} \langle [\sum_{i=1}^N \frac{p_i^2}{2m_i}, \hat{\rho_r}] \rangle$ =$ \langle [\sum_{i=1}^...
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Eigenvalues of the thermal state density operator

We define the thermal density operator as $$\tau(\beta) = \frac{e^{-\beta H}}{\mathrm{Tr}(e^{-\beta H})}$$ where $H$ is the systems Hamiltonian. Today I was told that the eigenvalues of the ...
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Why quantum map must be hermitian?

Quantum maps transform a density matrix into another one, Assume we are in the Hilbert space :$ H_A $ the quantum map on the density matrix $\rho_A$ living in $H_A$ is : $\mathcal{L}_A$ Why $\mathcal{...
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Conceptual meaning of Thermal States

Thermal states are generally defined as $$\tau(\beta)= \frac{e^{-\beta H}}{\mathrm{Tr}(e^{-\beta H})}$$ What are some physical statements one can make about them? A system in thermal equilibrium is ...
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Calculating the evolution at any moment $t$ of a density matrix

I was reading the paper https://arxiv.org/abs/1303.4686, where we are given $N$ systems, all with the same Hamiltonian $$H=\sum_i \varepsilon_i \mid i\rangle\langle i\mid ~,$$ such that the joint ...
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Difference between pure and thermal states

As far as I know by inserting a harmonic potential $V(x) = \frac{1}{2}m \omega x^2$ into the time-independent schrödinger equation I can obtain the wave-functions eigenstates and eigenvalues (energies)...
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Are Thermal states Harmonic oscillators?

Excuse me if I use somewhat wrong terminology. But I've always been confused about this. So firstly when we talk about a 2-state system, like a qubit, it has dimension d=2, no? But what if we ...
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Post-measurement density matrix derivation

This is something standard, by I'm trying to redo this with spectral theory. Suppose we start with the usual postulates of quantum mechanics: States are unit rays on a separable Hilbert space. In ...
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Conservation of quantum information in mixed states quantum secret sharing (QSS) schemes?

Consider a $((3,5))$ pure state quantum secret sharing (QSS) scheme. For instance this paper: arXiv:quant-ph/9901025. If I divide any 5 shares to two sets then allways one of those two set are ...
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How to measure off diagonal elements of mixed state density matrix?

Let’s assume I want to do quantum tomography using polarizers, half Waveplates and detectors , it’s obvious for me how we can measure diagonal elements of 2 qubit system density matrix using polarizer ...
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Action of rotation operator on spin 1/2 system

In Sakurai book on QM in chapter 3, he states the following relation $$e^{\frac{iS_z\phi}{\hbar}}[(\rvert+\rangle\langle-\rvert)+(\rvert-\rangle\langle+\rvert)]e^{\frac{-iS_z\phi}{\hbar}}$$ $$=e^{\...
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Expansion of an arbitrary density matrix in terms of coherent states?

It is well-known that any pure state can be expanded in terms of coherent states namely $$\left|\psi\right>=\frac{1}{\pi}\int d^2\alpha\left<\alpha|\psi\right>\left|\alpha\right>$$ due to ...
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Alternate definitions of Thermal states

The definition of thermal states I'm used to is: $$\tau_{\beta} = \frac{1}{Z}\,e^{-\beta H}$$ where $Z$ is the partition function defined as $Z= \mathrm{Tr}(e^{-\beta H})$, $\beta$ the inverse ...
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What exactly are control functions (used for parametrization)?

Let us consider a system in state $\rho$ with an internal hamiltonian $H_0$ on which we apply a cyclic, unitary evolution $H_t = H_0 + V(t)$ Where $V(t)$ is a time dependent external potential for ...
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Does a physical interpretation of density matrix cross-terms exist?

Say we have some state $$|\psi\rangle=\frac{1}{\sqrt{2}}(|0\rangle+i|1\rangle)$$ it is in a quantum superposition of $|0\rangle$ and $|1\rangle$. Its density matrix is $$\rho=\begin{pmatrix}\frac 1 2 &...
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Rank of a density matix

I was just trying to understand the meaning of rank of a density matrix. I came across the following post, which says that the rank of density matrix is the number of non-zero eigenvalues. And for a ...
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Completely positive maps - dimension of the ancilla space

If a map between positive operators $\Phi: X \rightarrow Y$ is also completely positive, it is true that $\Phi\otimes I_A$ is also a positive map for any choice of ancilla operator space $A$. That ...
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What does the energy-resolved spin density averages show?

What does the energy-resolved spin density averages show? Spin density refers to the density of states for spin up and down, my question is what does energy-resolved refer and how can be the energy-...
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2answers
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Why is the partial differentiation of density operator with respect to time zero, for an ensemble in thermal equilibrium?

Sakurai initially says that density operator evolves with time because state kets evolve with time. But for an ensemble in thermal equilibrium, its partial differentiation is zero. As far as I know, ...
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Explanation of why this derivation of Schmidt decomposition works

I'm following Preskill's notes and he derives the Schmidt decomposition in the following way: Let a bipartite state be $\psi_{AB} = \sum_{i,j}\lambda_{ij}\vert i\rangle\vert j\rangle = \sum_{i} \vert ...
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Conditioning a quantum state: why does it have this particular form?

Can someone please explain the phenomenon of conditioning a quantum state, and why it has this particular form? An observable $A$ of a quantum mechanical system, described by the density operator $\...
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How to obtain $Y$ rotation with only $X$ and $Z$ rotation gates on the Bloch sphere?

Let's say you have a system with which you can perform arbitrary rotations around the $X$ and $Z$ axis. How would you then be able to use these rotations to obtain an arbitrary rotation around the $Y$ ...
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Density operator in second quantization form [closed]

In first quantization the particle density operator is $$n(x)=\sum_{\alpha}\delta^{3}(\vec{x}-\vec{x}_{\alpha})$$ In second quantization I have: $$ n(\vec{x})=\sum_{\alpha,i,j}\langle i|_{\alpha}\...
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Programmatically generate valid density matrices of arbitrary dimension

I would like to generate a random $N\times N$ density matrix for a program. My current technique works for qubits but I suspect there are much more elegant ways. For a single qubit state, I write $\...
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1answer
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Can a single-qubit state be nontrivially extended to a non-pure state?

Consider a generic single-qubit state $$\rho=\lambda_1\lvert \lambda_1\rangle\!\langle \lambda_1\rvert+\lambda_2\lvert \lambda_2\rangle\!\langle \lambda_2\rvert\in\mathcal H_S.$$ I am interested in ...
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Entangled state density matrix: Defintion and example

I was looking at this question but still don't fully understand the distinction between classically correlated mixed states and entangled mixed states. I understand that a pure state is considered ...
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1answer
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Proof of strong convexity of trace distance

I'm trying to follow the Nielsen and Chuang proof (equation 9.49 of Chapter 9, page 408). I reproduce it here for completeness. With trace distance defined as $D(\rho, \sigma) = \frac{1}{2}tr(|\rho - ...
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Can we compute the entropy of a subsystem of a quantum state?

Suppose there is some entangled system with 5 subsystems labeled 1 to 5. Can we write the density operator for the subspace of subsystems 1 to 4 and compute entropy and the other relations with it ...
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Does the anti-diagonal of a density matrix have any special interpretation?

Suppose a density matrix $$ \rho= \begin{bmatrix} x_{11} & x_{12} & x_{13} & \cdots & x_{1n} \\ x_{21} & x_{22} & x_{23} & \cdots & x_{2n} \\ \...
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Complete positivity: why is the condition sufficient for quantum maps?

I know that when we define quantum maps, we need the map to be completly positive, to ensure that if our system $A$ is entangled with some extra system $B$, the evolution on $H_A \otimes H_B$ will ...
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1answer
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Is partial trace the inverse operation of Kronecker product?

Computer science student here, who is interested in quantum information theory. Suppose I have these pure states: \begin{bmatrix}1&0\\0&0\end{bmatrix} and \begin{bmatrix}0&0\\0&1\end{...
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1answer
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Spontaneous symmetry breaking in fluids

BACKGROUND: One can think of solids as spontaneously breaking translational symmetries in the sense that each atom in a lattice has to pick a particular position. Yet, as with everything in our ...
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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 ...
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Reduced density matrix of two spins

I am reading this (https://arxiv.org/abs/1209.0062) article about constructing order parameters from reduced density matrix. The author is discussing long-range order by taking antiferromagnetic spin ...
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General properties on reduced density matrix with assumption on the global density matrix

Let's consider $\mathcal{H_1} \otimes \mathcal{H_2}$ the space of the problem. I call $\rho$ a density matrix of the full space and : $\rho_1=Tr_2(\rho)$ the reduced density matrix in $\mathcal{H_1}$...
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How to calculate correlation functions for fermionic operators?

In the paper by Peschel (2003) https://arxiv.org/pdf/cond-mat/0212631.pdf How does one derive the following relation: $$ \langle c_{n}^\dagger c_{m}^\dagger c_{k}c_{l}\rangle = \langle c_{n}^\...
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Why we ignore off-diagonal elements in partition function?

In quantum statistical mechanics, the density operator is $$ \rho = \exp(-\beta H_0)/Z $$ where $$Z = \text{Tr} (\exp(-\beta H_0)) \, .$$ Why do we take the trace over only diagonal elements and ...
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Given density matrices of the subsystems is it in general possible to recover density matrix of the whole system?

Let us consider bipartite system in an entangled mixed state. Since its density matrix can always be diagonalized we can write it in following ways: $$\rho_{AB} = \sum_{j} p_{j} | \psi_j\rangle \...