New answers tagged linear-algebra
2
votes
Accepted
Finding the "dual" basis of an overcomplete basis for Quantum State Tomography
A general procedure to find the dual of any given frame $\{v_k\}_k$ is to compute the frame operator $S$, and then the (canonical) dual frame elements as $\tilde v_k\equiv S^{-1} v_k$. The dual frame ...
glS♦
- 23.4k
1
vote
Accepted
Problem with the mathematical definition of the eigenvalue algorithm on a specific exercise
Calculate the wavefunction step by step by applying each gate in sequence
$$
\begin{align}
|000\rangle&\xrightarrow{H\otimes H}\frac12\sum_{a,b=0}^1|a\rangle|b\rangle|0\rangle\tag1\\
&\...
- 20.8k
1
vote
Accepted
Why $\sqrt{\rho} = P \sqrt{\rho}$ in the proof of quantum error correction conditions in Nielsen & Chuang?
Recall from the sentence immediately preceding $(10.20)$ that $\rho$ is a state in the code subspace $C$ and from the statement of theorem $10.1$ that $P$ is a projector onto $C$. We will show that $\...
- 20.8k
1
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Why $\sqrt{\rho} = P \sqrt{\rho}$ in the proof of quantum error correction conditions in Nielsen & Chuang?
$\rho$ is a state in the code. $P$ is the projector onto the codespace. That means (by definition) that for any state $|\psi\rangle$ in the code, $P|\psi\rangle=|\psi\rangle$.
$\rho$ is one such state ...
- 55.7k
2
votes
Accepted
Problem with eigenvalue evaluation algorithm application on matrix $U$
TL;DR: Qubit order in the top register is reversed.
QFT qubit order in Quirk
Quirk's QFT gate treats the top qubit as the least significant and the bottom qubit as the most significant. Thus, if you ...
- 20.8k
6
votes
Accepted
Finding the eigenvalues of a qutrit state
Maybe you forgot about the coefficient $\frac{1}{\sqrt{2}}$. The correct reduced density matrix is $$\frac12 (\left|1\right>\left<1\right| + \left|2\right>\left<2\right|)$$
- 208
4
votes
Accepted
Is possible to write a separable state as a finite or countable infinite sum of product states?
Yes, you can always write an integral of separable states as a (finite) convex combination of product states, owing to the fact that the set of separable states is compact.
A quick way to see this is ...
- 5,158
2
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What is known about the size of the spectral gap of unital quantum channels?
TL;DR: Spectral gap depends on the specific channel. Moreover, for any $g\in[0,1]$ there is a channel with spectral gap $g$.
Non-peripheral eigenoperators are traceless
We can make simple observations ...
- 20.8k
1
vote
Accepted
How to get the Kraus operator $M_0=\sqrt{1-p}\, I$ for the depolarizing channel, from its isometric representation?
The unitary representation specifies how the isometry $U$ acts on pure states. The channel $\Phi$ is related to $U$ by $\Phi(\rho)=\operatorname{tr}_E[U\rho U^\dagger]$. Notice that on pure states ...
glS♦
- 23.4k
0
votes
How is this step performed in Deutsch's algorithm?
The key point here is that the query oracle $U_f$ is a linear transformation.
Linearity means, if $|x\rangle = \alpha|0\rangle+\beta|1\rangle$, then
$$U_f|x\rangle|y\rangle=U_f(\alpha|0\rangle+\beta|1\...
- 9,097
1
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Is there a general method for calculating expectation values for time-dependent wavefunctions?
I won't give the final answer (close to it), but instead try and point you in the general direction.
Initial State: The given initial state is $ | \psi(t=0) \rangle = | 0 \rangle $.
Pauli X Matrix: ...
- 755
1
vote
Accepted
How to know what eigenvalue corresponds to measurements of individual qubits in a multiqubit system?
If you consider the state
$$\Omega_- = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle).\tag{1}$$
you will find that there is NO violation. Let me show this. Note that the approach I will use for solving ...
- 126
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