Questions tagged [density-operator]
The density operator describes a quantum system in an (in general mixed) state.
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0answers
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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 ...
2
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2answers
28 views
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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1answer
38 views
“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 ...
0
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0answers
22 views
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?...
0
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0answers
18 views
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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1answer
21 views
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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0answers
35 views
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 ...
1
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1answer
28 views
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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2answers
63 views
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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5answers
198 views
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 ...
2
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0answers
48 views
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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1answer
30 views
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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0answers
13 views
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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votes
1answer
28 views
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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1answer
41 views
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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0answers
25 views
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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0answers
44 views
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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0answers
42 views
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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0answers
40 views
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 ...
2
votes
1answer
51 views
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 ...
1
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1answer
33 views
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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0answers
27 views
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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1answer
31 views
Action of rotation operator on spin 1/2 system
In Sakurai book on QM in chapter 3, he states the following relation $$e^{\frac{iS_z\phi}{\hbar}}[(\rvert+\rangle\langle-\rvert)+(\rvert-\rangle\langle+\rvert)]e^{\frac{-iS_z\phi}{\hbar}}$$
$$=e^{\...
2
votes
0answers
43 views
Expansion of an arbitrary density matrix in terms of coherent states?
It is well-known that any pure state can be expanded in terms of coherent states namely
$$\left|\psi\right>=\frac{1}{\pi}\int d^2\alpha\left<\alpha|\psi\right>\left|\alpha\right>$$ due to ...
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votes
2answers
49 views
Alternate definitions of Thermal states
The definition of thermal states I'm used to is:
$$\tau_{\beta} = \frac{1}{Z}\,e^{-\beta H}$$
where $Z$ is the partition function defined as $Z= \mathrm{Tr}(e^{-\beta H})$, $\beta$ the inverse ...
0
votes
1answer
33 views
What exactly are control functions (used for parametrization)?
Let us consider a system in state $\rho$ with an internal hamiltonian $H_0$ on which we apply a cyclic, unitary evolution
$H_t = H_0 + V(t)$
Where $V(t)$ is a time dependent external potential for ...
8
votes
3answers
118 views
Does a physical interpretation of density matrix cross-terms exist?
Say we have some state $$|\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 &...
0
votes
2answers
73 views
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 ...
1
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1answer
43 views
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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0answers
13 views
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
54 views
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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2answers
85 views
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 ...
0
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1answer
46 views
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 $\...
3
votes
1answer
76 views
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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1answer
103 views
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:
$$ n(\vec{x})=\sum_{\alpha,i,j}\langle i|_{\alpha}\...
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2answers
32 views
Programmatically generate valid density matrices of arbitrary dimension
I would like to generate a random $N\times N$ density matrix for a program. My current technique works for qubits but I suspect there are much more elegant ways.
For a single qubit state, I write $\...
2
votes
1answer
56 views
Can a single-qubit state be nontrivially extended to a non-pure state?
Consider a generic single-qubit state
$$\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.$$
I am interested in ...
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0answers
76 views
Entangled state density matrix: Defintion and example
I was looking at this question but still don't fully understand the distinction between classically correlated mixed states and entangled mixed states.
I understand that a pure state is considered ...
2
votes
1answer
64 views
Proof of strong convexity of trace distance
I'm trying to follow the Nielsen and Chuang proof (equation 9.49 of Chapter 9, page 408). I reproduce it here for completeness.
With trace distance defined as $D(\rho, \sigma) = \frac{1}{2}tr(|\rho - ...
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1answer
69 views
Can we compute the entropy of a subsystem of a quantum state?
Suppose there is some entangled system with 5 subsystems labeled 1 to 5. Can we write the density operator for the subspace of subsystems 1 to 4 and compute entropy and the other relations with it ...
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1answer
33 views
Does the anti-diagonal of a density matrix have any special interpretation?
Suppose a density matrix
$$
\rho=
\begin{bmatrix}
x_{11} & x_{12} & x_{13} & \cdots & x_{1n} \\
x_{21} & x_{22} & x_{23} & \cdots & x_{2n} \\
\...
3
votes
3answers
100 views
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 ...
2
votes
1answer
89 views
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{...
3
votes
1answer
80 views
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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1answer
104 views
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 ...
0
votes
0answers
36 views
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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0answers
36 views
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 $\mathcal{H_1}$...
2
votes
0answers
46 views
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:
$$
\langle c_{n}^\dagger c_{m}^\dagger c_{k}c_{l}\rangle = \langle c_{n}^\...
1
vote
1answer
79 views
Why we ignore off-diagonal elements in partition function?
In quantum statistical mechanics, the density operator is
$$ \rho = \exp(-\beta H_0)/Z $$
where
$$Z = \text{Tr} (\exp(-\beta H_0)) \, .$$
Why do we take the trace over only diagonal elements and ...
3
votes
2answers
248 views
Given density matrices of the subsystems is it in general possible to recover density matrix of the whole system?
Let us consider bipartite system in an entangled mixed state. Since its density matrix can always be diagonalized we can write it in following ways:
$$\rho_{AB} = \sum_{j} p_{j} | \psi_j\rangle \...
