All Questions
Tagged with textbook-and-exercises density-matrix
63
questions
4
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2
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82
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Why do minimal ensemble decompositions for $\rho$ contain $|\psi⟩\in{\rm supp}(\rho)$ with probability $1/\langle\psi|\rho^{-1}|\psi⟩?$
I came across the following exercise (2.73) in Nielsen & Chuang and am trying to understand it intuitively.
Here is my reasoning of what is going on:
The purpose of this exercise:
Let’s say we are ...
- 163
1
vote
2
answers
56
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What is meant with "different ensembles can give rise to the same density matrix?"
I am reading the Nielsen & Chuang section on density matrices and I don't understand the example given to demonstrate a concept. Here is what I am reading:
First, they said these two different ...
- 163
2
votes
1
answer
73
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What's the Schmidt decomposition of $|\psi\rangle = 1/ \sqrt{3}( |0\rangle| 0\rangle + |0\rangle |1\rangle + |1\rangle |1\rangle)$?
$|\psi\rangle = 1/ \sqrt{3}( |0\rangle| 0\rangle + |0\rangle |1\rangle + |1\rangle |1\rangle) $
I absolutely cannot figure out the Schmidt decomposition of this state. I have looked at a ton of ...
- 21
0
votes
2
answers
77
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Why is a density matrix an orthogonal projector?
Suppose I have a density matrix like $\rho = \frac{1}{2}[I + \hat{n}\vec{\sigma}]$.
The claim is that $\rho$ is an orthogonal projector for the state $|+\rangle$ in an arbitrary direction $\hat{n}$.
...
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-1
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2
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45
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Define a traceless part of $\rho$ [closed]
I saw in a paper: $|\bar{\rho}\rangle\rangle=|\rho\rangle\rangle-|\hat{I}\rangle\rangle / 2^{n / 2}$ for the $4^n$-dimensional vector representing the traceless part of $\rho$. https://arxiv.org/abs/...
- 639
3
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1
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133
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calculate the reduced density matrix of a 2 qubit state and compare the eigenvalues
So I have the exercise to apply a Cz gate to the following 2 Qubit state
$|a\rangle \otimes |b\rangle = (a_0 |0\rangle + a_1 |1\rangle) \otimes (b_0 |0\rangle + b_1 |1\rangle)\\\\$
Afterwards, I ...
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1
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1
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65
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How do I show that a reduced density matrix of $1$ is $\rho_{12}^{1} = \text{Tr}_{2}[\rho_{12}] = \sum_{i}\langle i_{2} | \rho | i_{2} \rangle$?
Let the system be a 2 - qubit system and let $\rho_{12}$ be a density matrix of some state for this 2 - qubit system.
How do I show that a reduced density matrix of $1$ is $\rho_{12}^{1} = Tr_{2}[\...
- 518
4
votes
1
answer
113
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How do I find the reduced density matrix of a system where two people share one qubit and have one qubit of their own?
I have the following problem and have attempted to find a solution to it, but to no avail.
Alice and Bob have one qubit each, say $|\psi\rangle$ with Alice and $|\phi\rangle$ with Bob. They also share ...
- 165
1
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1
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102
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Can a density operator be written equivalently as $\rho=\sum_i p_i|\psi_i〉\!\langle\psi_i|$ and $\rho=\sum_i\lambda_i|\psi_i\rangle\!\langle\psi_i|$?
My doubt arises from page 99, 101 of the book Quantum Computation and Quantum Information by Michael A.Nielson and Issac L.Chung.
Let {${p_{i}, | \psi_{i} \rangle }$} be an ensemble of pure states.
...
- 518
1
vote
1
answer
60
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What is the probability of a state $|0\rangle$ being in another state $\alpha|0\rangle+\beta|1\rangle$?
I am trying to calculate the probability of a state (density matrix) being in a specific other state.
Lets say I have a 2-dimensional state with the states given by the orthonormal basis states $|0\...
- 95
2
votes
2
answers
2k
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What is the density matrix of a pure state?
By definition of the density matrix for example the density matrix of $|0\rangle$ state (pure state) is:
$$\rho=|0\rangle \langle 0| =
\begin{pmatrix}
1 & 0 \\
...
- 267
2
votes
1
answer
262
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What physical quantity does a density operator represent as an observable?
The density operator is a representation of a state of a quantum system $\rho=|\psi\rangle\langle\psi|$, so it's just an alternative characterisation of a state (or more generally a statistical ...
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How to prove that ${\rm tr}(A|\psi\rangle\langle\psi|)=\langle\psi| A|\psi\rangle$?
How can one prove that $tr(A\mid\psi\rangle\langle\psi\mid)=\langle\psi\mid A\mid\psi\rangle$? In Nielsen/Chuang they mention this is due to Gram-Schmidt decomposition but I can’t understand how.
- 101
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1
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500
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How to prove that the trace of a density matrix is $1$?
Equation 2 gives the following proof:
$$
\text{Tr}[\rho] = \sum_x \langle x\vert \rho\vert x\rangle = \sum_x \langle x\vert
\sum_i p_i\vert \psi_i\rangle \langle \psi_i\vert\vert x\rangle = \sum_i ...
- 822
2
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1
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124
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How to compute the maximum possible coherence of a two-particle Bell state?
I am reading through some notes and am stuck on a bit of math that shows the max possible coherence. Our wave function is $|\psi\rangle =\frac{|01\rangle+|10\rangle}{2}$ and doing $|\psi\rangle \...
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