# Tag Info

Accepted

### Prove that the trace distance is upper-bounded by the Hilbert-Schmidt distance

Let $P - Q = \rho - \sigma$ be a Jordan-Hahn decomposition, meaning that $P$ and $Q$ are the unique positive semidefinite operators that satisfy that equation and have orthogonal images. We'll start ...
Accepted

### Does the trace distance have a geometric interpretation?

There is a geometric interpretation that you certainly can take seriously, but the geometry that you get is not as clean as you might have hoped. Trace distance between operator states is an example ...

### Given an orthogonal projection $\Pi$, is $\|\Pi(\sigma-\rho)\Pi\|_1\le\|\sigma-\rho\|_1$ true?

Yes, you can use Cauchy interlacing theorem to prove that. Let $M = \sigma - \rho$ and dimension of the space is $n$. In an appropriate basis $\Pi = I_k \oplus 0_{n-k}$. Assume $k=n-1$ (in general ...

### Given an orthogonal projection $\Pi$, is $\|\Pi(\sigma-\rho)\Pi\|_1\le\|\sigma-\rho\|_1$ true?

Here is a slightly more general alternative to Danylo's answer. The trace norm satisfies the inequality $$\| X Y Z \|_1 \leq \|X\| \|Y\|_1 \|Z\|$$ for all choices of operators $X$, $Y$, and $Z$ that ...
Accepted

Accepted

### How to prove that $\frac{| x_0 \rangle + | x_1 \rangle}{\sqrt{2}}$ hides one of $x_0$ or $x_1$?

We can bound the amount of information that can be retrieved from $|\psi\rangle$ using Holevo's bound. Alice and Bob Let us first reformulate the situation in the terms usually employed in the context ...
Accepted

Accepted

### Is the trace distance between multipartite states invariant under permutations?

A permutation of the qubits is a unitary operation. The trace distance is invariant under unitaries (https://en.wikipedia.org/wiki/Trace_distance#Properties). Thus, statement 1 is true.