# Tag Info

Accepted

### What unitary commutes with all local Pauli operators?

TL;DR: The only $U$ that commutes with all $\sigma_{X,i}$ and all $\sigma_{Z,i}$ is a scalar multiple of identity. This follows from the Schur's lemma, but can also be shown using elementary linear ...
• 22.4k
Accepted

### what is square root of a density matrix power two?

If $\rho$ is a density matrix, then $\sqrt{\rho^2} = \rho$. To see why this is, let's start with the definition of the square root of a matrix. Ordinarily, if $A$ is a square matrix, there may be ...
• 5,907
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Accepted

• 6,162
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### Are quantum channels bounded linear maps?

A linear map defined on density operators is uniquely extendible on the set of all linear operators (in finite-dimensional case). Simply because any linear operator is a linear combination of density ...
• 7,304
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### Decomposition of a $4 \times 4$ unitary matrix

That paper appears to do their rotations in a very strange order. The method you're interested in is how to use Givens rotations to perform a QR decomposition (see, e.g. https://en.wikipedia.org/wiki/...
• 58.2k
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### $\mathbb{C}^2 \otimes \mathbb{C}^2$ vs $\mathbb{C}^4$

You don't have to directly express the basis of one space in another to know they are the "same". We say two Hilbert spaces $A$ and $B$ are isomorphic if there exists a linear isomorphism ...
• 5,773
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### A conceptual Query regarding measurement during a Quantum Algorithm

Quantum gates (which I assume you mean by "quantum operations") are reversible: applying a gate to different "input" states results in different "output" states. This ...
• 9,050

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### How to calculate the Haar measure for the middle SU(2), in an SU(3) factorization?

Let us consider the $SU(3)$ decomposition as such: Assuming the matrix we create is Haar-random, it means that when it is applied to $|0\rangle$ it should yield a Haar-random state. So let's see how ...
• 6,162
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### closeness between two unitaries on the bloch sphere

A unitary $U$ corresponds to a rotation on the Bloch sphere. Technically, there is a smooth map from $2 \times 2$ unitaries with determinant 1 (which form the group SU(2)) to $3\times 3$ rotation ...
• 7,304

### closeness between two unitaries on the bloch sphere

There are several ways to quantify the closeness of unitary operators. The best approach will be heavily dependent on context. The approach you laid out is close to an operator norm, which would look ...
• 3,392
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### unitary that transforms one Hilbert space to another Hilbert space

No. Suppose $A$ and $B'$ are qutrits, $A'$ and $B$ are qubits and $U$ is the SWAP unitary that sends $|ab\rangle$ to $|ba\rangle$ for every $a\in\{0,1,2\}$ and $b\in\{0,1\}$. Clearly, SWAP transforms \$...
• 22.4k
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### What is the relationship between gate fidelity and norm?

The gate fidelity can be expressed with the Frobenius norm as follows: \begin{align*} \min_{\varphi\in\mathbb R}\frac1{2^n}\|U-e^{i\varphi}V\|_2^2&=\min_{\varphi\in\mathbb R} \frac1{2^n}{\rm tr}((...

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