Questions tagged [density-operator]

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

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Can a non-unitary quantum operation be reversed?

Say a non-unitary completely positive trace preserving (CPTP) map $\Phi$ takes a state $\rho$ to $\sigma = \Phi(\rho)$. Does there exist some other CPTP map $\Lambda$ where $\Lambda \Phi \neq \mathbf{...
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If subsystem $s$ is in the pure state $ρ$ must have the form $P_s ⊗ ρ_R$

Consider a system composed of two parts s (subsystem) and R (reservoir), and let ρ be the density matrix for some state of the combined system. Show that if subsystem s is in the pure state, ρ must ...
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Prove that for any state $\rho$ there is a basis $|e_k\rangle$ corresponding to uniformly distributed probabilities

I am new to quantum mechanics, and I'm trying to learn about Majorization in quantum states. These are the notes I am following http://michaelnielsen.org/blog/talks/2002/maj/book.ps by Michael Nielsen....
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When is a quantum state pure and when mixed?

Every definition of the two is always very abstract to me. Like, A pure state is located on the surface of the bloch sphere while the mixed state is somewhere within. First of all, what is an ...
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How to express the anti-commutator in the form of a density operator?

By the way, I don't know why the following Latex syntax does not compile properly in Stackexchange. So, I type them in Overleaf, where I use \usepackage{physics} and take a photo. Let $\{ \ket{1} \ket{...
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Why is $\langle \hat{A} \rangle = tr(\hat{\rho} \hat{A})$? [duplicate]

Given that $$\langle \hat{A} \rangle = \langle \psi|\hat{A}|\psi \rangle$$ Why does $\langle \hat{A} \rangle = \mathrm{tr}(\hat{\rho} \hat{A})$, where $\hat{\rho}$ is the density operator, $\hat{\rho} ...
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Transformation operator in Sequence of linear QND measurements

I am following the book Braginsky and Khalili. Consider a measurement scheme where we connect a object to be measured to another quantum system which is then measured by classical devices.(Example: ...
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How to represent the evolution operator of Jaynes-Cummings model interaction hamiltonian in two level basis

I am reading "Quantum optics" by Gerry and Knight. By the JCM(Jaynes-Cummings model), the interaction hamiltonian of the field and the atom can be written $ \hat{H}_I = \hbar \lambda(\hat{a}\...
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Double slits experiment: how does the which-way entropy vary with the distance between slits and screen?

When the screen is just behind the slits we have perfect which way information. The entropy is log 2. It decreases if I place the screen in the Fresnel region (not to far). It decreases much more in ...
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Matrix Exponential [closed]

i'm working on a paper about quantum entropy and i'm stuck on an analytical step about computing a matrix exponential. I have the following Hamiltonian $$H=\frac{1}{2}\sum_ip_i^2+\frac{1}{2}\sum_{i,j}...
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How to find the unitary operation of a depolarizing channel? [duplicate]

Suppose we have a depolarizing channel operation $$E(\rho)=\frac{p}{2}\textbf{1}+(1-p)\rho$$ acting on a Spin$\frac{1}{2}$ density matrix of the form $\rho=\frac{1}{2}(\textbf{1}+\textbf{s}\cdot\...
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Uniqueness of Spectral Decomposition

In one of the papers (related to Quantum Computing) I am reading, I came across this statement which says, An elementary result is that sets of orthogonal rank-one eigenprojectors of Hermitian ...
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Pauli matrices Measurements

When performing a measurement On qubits with the Pauli matrices, They all correspond to outcomes +1,-1 because that's their eigenvalues? In my notes it says that the $z$ Pauli matrix can be broken ...
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Motivation and advantage of C$^*$-algebra formulation of quantum statistical mechanics

I am studying C$^*$-algebras and the formulation of quantum statistical mechanics by them, mostly from the book by Bratteli and Robinson Operator Algebras and Quantum Statistical Mechanics. I could ...
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Gradient of fidelity!

If we have the quantum fidelity function between two states as: $F(\rho, \sigma)= (Tr \sqrt {\sqrt {\rho } \sigma \sqrt {\rho }})^2$ how we can calculate the gradient/derivative of the function with ...
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Difference between Mixed and Pure states [duplicate]

Suppose that there is a system of two photons 1 and 2, each of which is in a mixed state $1/2|R\rangle\langle R| + 1/2 |L \rangle\langle L|$, where $|R \rangle$ and $\langle L|$ are two orthonormal ...
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In non-relativistic Quantum Physics are there equations other than the Schrödinger equations that can be used to model wave functions?

The general form for the time-dependent Schrödinger equations is $$i\hbar\frac{\partial\Psi}{\partial{t}}=\hat{H}\Psi$$ with $\hat{H}$ being the Hamiltonian operator, which as I understand it ...
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In the theory of open quantum systems, what is really meant by *information backflow* from the environment to the system?

In the context of an Open Quantum System (OQS) i.e. a quantum system coupled to a quantum environment, what is really meant by information backflow from the environment to the system? I'm newly ...
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How to find the partial transpose of bipartite states from their matrix representation?

Suppose we have a density operator given by $\rho=\mid \Psi \rangle \langle \Psi \mid$ with $\mid \Psi \rangle = \frac{1}{\sqrt{2}}(\mid 1 \rangle\mid \ 0 \rangle-\mid \ 0 \rangle \mid 1 \rangle)=\...
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Is it possible to have a negative density matrix, and how would you calculate entropy?

I am trying to calculate the Von Neumann Entropy of a quantum state. Given a state $ | \psi \rangle$, I am calculating the Von Neumann entropy by doing the following: $$ S = -\mathrm{tr}(\rho \ln(\rho)...
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Shake water and seed oil: color and density

These days I am spraying the leaves of my garden, to prevent insects, mixing strongly water and soya oil. The color has become all white. Soy oil has a relative density of $0.915 \div 0.925$ kg/dm$^3$...
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Calculate probability of a state after depolarization

Let's say I have a particle in the quantum state $|+\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)$, represented as a density operator (1st matrix) that went through a depolarizing chanel (2nd ...
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Baker–Campbell–Hausdorff and Gaussian density operators for fermions

I'm trying to understand a passage from the paper "Gaussian operator bases for correlated fermions", J. F. Corney and P. D. Drummond (https://arxiv.org/abs/quant-ph/0511007), specifically ...
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Projecting Density Matrix into Edge States Subspace

I'm reading and trying to understand the paper by Diehl, S., Rico, E., Baranov, M. et al. "Topology by dissipation in atomic quantum wires". Nature Phys 7, 971–977 (2011). https://doi.org/10....
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Reduced density matrix for entangled state [closed]

Given state $|\psi\rangle = \frac{|00\rangle + |01\rangle + |11\rangle}{\sqrt{3}}$ I was calculating the reduced density matrix $\rho^A$. The given answer was $\rho^A = \frac{1}{3}\begin{pmatrix} 2 &...
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Density matrix of coherent state

The eigenstate of the annihilation operator $a$ is given by the state $a\mid \alpha \rangle = \alpha \mid \alpha \rangle$. In the Fock state basis, we can expand this state as $$\mid \alpha \rangle = ...
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Space of density matrix

If we say that the $|\psi\rangle$ is in the Hilbert space what does the space of density matrix? Is this phrase true? $|\psi\rangle \langle\psi| \in V \otimes V_\mathrm{dual}$ $V$ is vector space.
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Density operator of a system $S$ coupled to a bath $B$

In the second equation of section 8.1 in this MIT OCW lecture notes, I can't understand how they went from $$\rho_{S}(t)=Tr_{B}\{\rho_{SB}(t)\}=\sum_{k}\langle k|U_{SB}(\rho_{S}(0)\otimes|0\rangle\...
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Single-Qubit Density Matrix Evolution

In this paper general form of reduced single-qubit density matrix evolution interacting with bosonic reservoir can be cast in the following form: $$ \rho^s(t)=\begin{pmatrix} \rho^s_{11}(t) & \rho^...
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Is partial trace method valid for highly entangled state? [closed]

Suppose there are four wires of qubits and the state vector is highly entangled. I want to know the density matrix (or state vector) of wire-$1$ only. In this case, is it available to use partial ...
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Photon number basis representation of a displaced single-mode squeezed thermal state

I am looking for a relatively clean expression for matrix elements for states of the form $$\rho_{\alpha,r,\bar{n}} = \hat{D}(\alpha)\hat{S}(r)\rho_{th}(\bar{n})\hat{S}^\dagger(r)\hat{D}^\dagger(\...
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Is fidelity of teleportation a particular case of quantum correlations?

Fidelity is measure of distance between density operators. it is a measure of the "closeness" of two quantum states, the input state $\rho_{in}$ and the teleported state $\rho_{out}$. where ...
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What is the physical intuition for Bloch Sphere? [duplicate]

I am very confused about how to think about the Bloch Sphere. How can we relate the concept of expectation value to the Bloch sphere? If my state lies in let's say $yz$ plane how can we say that ...
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How to represent the square root of a density matrix via the Glauber-Sudarshan representation?

I am trying to calculate the quantum Fisher Information of some quantum states which are represented via their P (Glauber-Sudarshan) representation, $$\rho = \int P_\rho(\alpha) |{\alpha}\rangle \...
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How to tighten the Fuchs-Van de Graaff inequalities for pure and mixed states?

Defining the trace-distance as $D(\rho , \sigma) = \dfrac12 tr|\rho - \sigma|$ and the Fidelity between two quantum states as $F(\rho , \sigma) = tr\sqrt{\sqrt\rho\sigma\sqrt\rho}$ I need to show the ...
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Free evolution of a density matrix in position space

I have a density matrix $\rho$ in momentum representation at time $t=0$: \begin{equation} \langle p' |\rho(0) |p\rangle = \sum_{n=1}^{1000} p_n \Psi_n^*(p',0) \Psi_n(p,0) \end{equation} ...
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Predictability in decoherence theory to find the classical states: at which time must we evaluate?

I have read Decoherence, einselection, and the quantum origins of the classical, end a way to quantify the classicality of states is the following. We have the system $S$ and its environment $E$. The ...
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Are density states more fundamental than wavefunctions? [duplicate]

Some interpretations, like the many-worlds interpretation, treat the wavefunction (modulo an overall phase factor) as objective and fundamental. But consider the following example for a qubit: a ...
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A proof involving derivatives of Dirac delta functions

Let us define $$\tag{1} Q=i\nabla_\mathbf{k}\delta(\mathbf{k-k'})\left[\rho_{nm}(\mathbf{k'})-\rho_{nm}(\mathbf{k})\right], $$ where $\delta(\mathbf{k})$ is a Dirac delta, and $\rho_{nm}(\mathbf{k}...
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Entanglement and Mixed States

The Wikipedia page for "Density Matrix" (https://en.wikipedia.org/wiki/Density_matrix) takes each of a pair of entangled photons as an example of a mixed state: A radioactive decay can emit ...
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What does it mean for a projection operator to represent a state?

I can understand that an idempotent operator can be represented as a projection operator, such as $|x\rangle\langle x|$. But some authors seem to use projection operators, instead of vectors, to ...
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Time evolution of a tripartite quantum state

Question: How do we write the unitary evolution of a tripartite system in Hilbert Space $\mathcal{H}_A \otimes \mathcal{H}_B \otimes \mathcal{H}_C$ when it is subject to two unitary evolution ...
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Doubts about when a continuous wavefunction can be written as a density matrix

Imagine I have a 1D wavefunction $\Psi(x)$. If I sample this wavefunction at $n$ points, I will have a $n*1$ vector $\left|\Psi\right>$. Now, if I take an outer product of this vector with itself, ...
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Product states in quantum statistical mechanics

I'm a physical chemist so bear with me. We are going over quantum statistical mechanics, and to motivate and derive the density matrix, my advisor used the following explanation: We have a system of ...
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Is it possible to differentiate between pure states and mixed ones in the laboratory?

I am wondering if it is possible to differentiate between pure states and mixed ones in the laboratory? For example consider a quantum system define relative to the orthonormal basis {up,down}. We can ...
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Conditional density matrix

Let there be a composite system $\mathcal{H}=\mathcal{H}_{A}\otimes\mathcal{H}_{B}$, where $\mathcal{H}_{A}$ and $\mathcal{H}_{B}$ are Hilbert spaces of two subsystems of $A$ and $B$. Suppose the ...
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How to distinguish a quantum superposition from a mixed state (ensemble)? Does a single measurement provide any information? [closed]

I will measure an object that is either in a mixed state of $\vert A\rangle$ and $ \vert B \rangle$ or a superposition $\vert A\rangle + \vert B\rangle$, and I am trying to find out which. I have set ...
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How to solve differential equation involving commutator and anti-commutator?

In one of my exercise, I got following differential equation for density matrix $\rho$, $$ \frac{d\rho}{dt}=-i[H_1,\rho]+\{H_2,\rho\} $$ where $H_1$ and $H_2$ are the Hermitian Hamiltonian, and $[.,.]$...
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How many measurements does it take to determine a quantum state?

Suppose I have a wave-function over a Hilbert-space of (complex) dimension $N$. It has $2 N-2$ real degrees of freedom, after normalization and removing the phase. It seems to me that I can measure ...
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What is the anti-normally-ordered representation $\hat \rho_A$ of a state $\hat\rho$?

While reading the Wikipedia page on the P function, I came across the following consideration (paraphrasing from there): Given a state $\rho$, if we write it in anti-normal order as $\rho_A=\sum_{...

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