4 votes

Bounds on local expectation values for two states close in trace distance

We first have: $$|\mathrm{tr}(A(\rho-\sigma))|\leqslant\mathrm{tr}(|A(\rho-\sigma)|)=\|A(\rho-\sigma)\|_1$$ We can then use Hölder's inequality: $$\|A(\rho-\sigma)\|_1\leqslant\|\rho-\sigma\|_1\|A\|_{\...
Tristan Nemoz's user avatar
  • 6,162
2 votes

Are quantum channels bounded linear maps?

To complement Danylo's great answer, let me go into a bit more details about the infinite-dimensional case and point out that the correct extension of a channel $\mathcal{N}:D(\mathcal{H}_A) \...
Frederik vom Ende's user avatar
1 vote

If $\text{tr}_B \rho \in A$, then $\rho \in A \otimes B$?

It's true in an appropriate formulation. First of all, don't confuse pure states $|\psi\rangle$ from $H$ and density matrices $\rho$ on $H$ (which are $|\psi\rangle\langle\psi|$ for pure states). Let $...
Danylo Y's user avatar
  • 7,279
1 vote

Show that quantum channels act as affine transformations in the Bloch sphere

There is already mistake in the very first line of your computation: When you write $\mathcal{E}(\rho)=[\ldots]=\frac12(I+\sum_l E_l(\vec{r}.\vec{\sigma})E_l^\dagger)$ you forgot to apply the channel ...
Frederik vom Ende's user avatar
1 vote

How's quantum noise and fault-tolerance related to symplectic geometry and geometric quantization?

Given $p$ an odd prime, the symplectic vector space $(\mathbb{F}_p^{2n}, \omega)$ is a projective representation of the odd prime dimensional $n$-qudit Pauli group. The $n$-qudit Pauli group is a ...
Cole Comfort's user avatar

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