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9 votes

Represent Hadamard gate in terms of rotations and reflections in Bloch sphere

but how about γ? Gamma doesn't show up on the Bloch sphere. It's a global phase. It's unobservable without conditioning the operation on a second qubit, in which case it turns into phase kickback ...
Craig Gidney's user avatar
  • 37.9k
4 votes

What's the Schmidt decomposition of $|\psi\rangle = 1/ \sqrt{3}( |0\rangle| 0\rangle + |0\rangle |1\rangle + |1\rangle |1\rangle)$?

One way of computing the decomposition is through density matrices, but then you will have to diagonalize those density matrices. This requires the eigenvalue decomposition of each density matrix. ...
MonteNero's user avatar
  • 2,666
4 votes

Represent Hadamard gate in terms of rotations and reflections in Bloch sphere

The Hadamard gate is a $\pi$ rotation about the diagonal axis in the XZ-plane. It is not a $\pi/2$ rotation about the $y$ axis. This can be easily seen from the fact that the Hadamard squares to the ...
Norbert Schuch's user avatar
2 votes

Finding the effect of conjugate transpose on a state $|b\rangle$

For a unitary $U$, the conjugate transpose $U^\dagger$ is the inverse of $U$, i.e. the linear operator such that$^1$ $U^\dagger U=UU^\dagger=I$. Guess and check The inverse is unique$^2$, so a general ...
Adam Zalcman's user avatar
  • 22.9k
2 votes
Accepted

In the QECC condition $\langle\psi|E_a^\dagger E_b|\phi\rangle=C_{ab}\langle\psi|\phi\rangle$, what is $C_{ab}$?

The concept of $C_{ab} $ in the context of the Quantum Error-Correcting Code (QECC) conditions as described in Theorem 2.7 can indeed be confusing due to the mathematical notation and the terminology ...
Bram's user avatar
  • 654
2 votes

How does measuring a density matrix give Kraus operators?

TLDR If you measure a qubit in the computational basis without getting the result of the measurement the corresponding channel can be described by the Kraus operators $K_0 = |0\rangle \langle 0|$ and $...
qubitzer's user avatar
  • 317
2 votes

What is meant with "different ensembles can give rise to the same density matrix?"

An ensemble in this context is a set of states with attached probabilities. In your example the ensembles would be written as $\{(\frac12,|a\rangle\langle a|),(\frac12,|b\rangle\langle b|)\}$ and $\{(\...
glS's user avatar
  • 25.4k
1 vote

What is meant with "different ensembles can give rise to the same density matrix?"

The two ensembles are $|0\rangle, |1\rangle$ and $|a\rangle, |b\rangle$. The point of this example is to show you that the same density matrix $\rho = \begin{pmatrix} 3/4 & 0 \\ 0 & 1/4 \end{...
David Dentelski's user avatar
1 vote

In the QECC condition $\langle\psi|E_a^\dagger E_b|\phi\rangle=C_{ab}\langle\psi|\phi\rangle$, what is $C_{ab}$?

I imagine it's supposed to be a matrix $C$ with elements $C_{ab}$, i.e. $a$ indexes the row, and $b$ the column of $C$.
DaftWullie's user avatar
  • 58.8k
1 vote

How large does the isometry in Naimark's theorem need to be for a 3-outcome POVM?

Let $\boldsymbol\mu\equiv\{\mu_j\}_{j=1}^m\subset \operatorname{Pos}(\mathbb{C}^d)$ be a POVM whose elements act on a $d$-dimensional space and have unit rank, that is, $\mu_j = w_j \mathbb{P}_{\psi_j}...
glS's user avatar
  • 25.4k
1 vote

How to compute the post-measurement state when measuring only the first of a three-qubit system?

tl;dr: The approach is correct but OP's calculation features multiple errors. Just to be precise about what I'll be doing: In the POVM formulation a quantum measurement is described by a collection $\{...
Frederik vom Ende's user avatar

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