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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's user avatar
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1 vote
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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\\ &\...
Adam Zalcman's user avatar
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1 vote
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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 $\...
Adam Zalcman's user avatar
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1 vote

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 ...
DaftWullie's user avatar
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2 votes
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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 ...
Adam Zalcman's user avatar
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6 votes
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Finding the eigenvalues of a qutrit state

Maybe you forgot about the coefficient $\frac{1}{\sqrt{2}}$. The correct reduced density matrix is $$\frac12 (\left|1\right>\left<1\right| + \left|2\right>\left<2\right|)$$
Vladimir Lysikov's user avatar
4 votes
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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 ...
John Watrous's user avatar
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2 votes

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 ...
Adam Zalcman's user avatar
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1 vote
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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's user avatar
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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\...
Egretta.Thula's user avatar
1 vote

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: ...
banercat's user avatar
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1 vote
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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 ...
Alex's user avatar
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