All Questions
Tagged with density-matrix trace-distance
14 questions
4
votes
1
answer
204
views
Is the closest diagonal state to a given state always the dephased original state?
This question is about the following optimization problem:
Given some density matrix $\rho\in\mathbb C^{n\times n}$ find the diagonal state which is closest to it in trace norm. More precisely, find
$...
- 3,494
2
votes
1
answer
60
views
Why is the trace distance between two density matrices not always $0$?
If $|A|_{tr}=Tr(\sqrt{A^\dagger A})$ then surely
$$
|\rho_1-\rho_2|_{tr}=Tr(\sqrt{(\rho_1-\rho_2)^\dagger (\rho_1-\rho_2)})
$$
$$
=Tr(\sqrt{(\rho_1^\dagger -\rho_2^\dagger)(\rho_1-\rho_2)})
$$
$$
=Tr(\...
- 123
2
votes
1
answer
55
views
If states are close together does there always exist a channel close to the identity mapping one to the other?
Question: Given states $\rho,\omega\in\mathbb C^{n\times n}$ and $\varepsilon>0$ such that $\rho$ and $\omega$ are $\varepsilon$-close in trace norm does there exist a channel $\Phi$ with $\Phi(\...
- 3,494
1
vote
1
answer
78
views
Is it true that $|r_i-s_i| \le 1/2$ for all $i$, where $r_i$ and $s_i$ are the eigenvalues of density matrices $\rho$ and $\sigma$?
In Nielsen and Chuang's Box 11.2: Continuity of the entropy, in the process of proving the Fannes' inequality, it says:
A moment’s thought shows that $\left|r_i − s_i\right| \le 1/2$ for all i,
The ...
- 237
2
votes
1
answer
89
views
Upper bounding the trace distance between a noisy and noiseless quantum state
Consider a quantum state
$$ \rho = \begin{pmatrix}
\rho_{00} & \rho_{01} \\
\rho_{10} & \rho_{11} \\
\end{pmatrix}. $$
Now, consider the effect of the amplitude damping noise $\mathcal{N}$ of ...
- 1,515
2
votes
1
answer
168
views
How to prove the strong convexity of the trace distance?
On page $408$ of Nielsen & Chuang in the step going from equation $(9.48)$ to $(9.49)$, I don't see how:
$$\sum\limits_i (p_i - q_i)tr(P \sigma_i) \leq D(p_i, q_i)$$
I proceed as follows:
$$\sum\...
- 197
8
votes
2
answers
239
views
Given an orthogonal projection $\Pi$, is $\|\Pi(\sigma-\rho)\Pi\|_1\le\|\sigma-\rho\|_1$ true?
Suppose I have an arbitrary orthogonal projector $\Pi$ and two density operators $\rho, \sigma$. Is it true that:
$$ ||\Pi (\sigma - \rho) \Pi||_1 \le || \sigma - \rho ||_1 $$
where $||\cdot||_1$ ...
- 459
3
votes
1
answer
192
views
Helstrom Measurement when two quantum states are close
I've been reading a paper about Entangled-quantum GAN (see this PDF) and wondering why descriptions below Eq.(3) in the paper are in fact true.
To summarize the description, suppose we have two ...
- 85
3
votes
2
answers
137
views
Closeness of $\rho$ such that $\text{Tr}(|\psi\rangle\langle\psi|\rho)\le1/2^n+{\cal O}(2^{-2n} )$ for all $|\psi\rangle$ to the maximally mixed state
Consider an $n$ qubit density matrix $\rho$ such that
$$\text{Tr}(|\psi\rangle\langle \psi| ~\rho) \leq \frac{1}{2^{n}} + \mathcal{O}\left(\frac{1}{2^{2n}} \right), $$
for every $n$ qubit pure state $|...
- 1,515
5
votes
3
answers
741
views
How to find the distance between a given $\rho$ and the nearest pure state(s)?
I have a $d$-dimensional state $\rho$. Is there any way to find the (possibly not unique) trace distance to the nearest pure state:
$$
\min_{|\psi\rangle} \,\,\lVert \rho - |\psi\rangle\langle \psi| \...
- 7,646
3
votes
1
answer
318
views
How can we upper bound the norm of a partial trace?
Suppose we have the normalised states $|\phi_{1}\rangle,|\phi_{2}\rangle \in A \otimes B$ where $A$ and $B$ are $d$-dimensional complex vector spaces.
Suppose $|\langle\phi_{2}|\phi_{1}\rangle| < ...
- 31
1
vote
1
answer
131
views
Trace distance of two classical-quantum state with hashing
Let's say I have a classical-quantum(cq) state $\rho_{XE}$, where the classical part $(X)$ is orthogonal. It's trace distance from another uniform density operator is defined to be:
$$
\frac{1}{2}||\...
- 1,785
1
vote
1
answer
727
views
Trace distance bound after partial trace
Let's say I have a pair of states among three parties Alice(A), Bob(B) and Eve(E), $\rho_{ABE}$ and $\rho_{UUE}$ where the first two parties hold uniform values U.}
I know that the trace distance ...
- 1,785
6
votes
2
answers
203
views
Prove that $\|p^{\otimes n} - q^{\otimes n}\| \leq n \|p-q\|$ for density operators $p,q$
I've been trying to figure this out for a while and I'm totally lost.
My goal is to show that for two density operators $p$, $q$, that $$||p^{\otimes n} - q^{\otimes n}|| \leq n ||p-q||$$
So far ...
