Questions tagged [trace-distance]
In quantum mechanics, and especially quantum information and the study of open quantum systems, the trace distance T is a metric on the space of density matrices and gives a measure of the distinguishability between two states. It is the quantum generalization of the Kolmogorov distance for classical probability distributions. (Wikipedia)
21
questions
2
votes
2answers
86 views
Trace norm is dual norm to spectral norm
Suppose $A\in L(X,Y)$. $||\cdot||$ denotes spectral norm and denotes the largest singular value of a matrix, i.e. the largest eigenvalue of $\sqrt{A^*A}$.
$||\cdot||_{tr}$ denotes trace norm. We have ...
1
vote
1answer
23 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
vote
0answers
36 views
Quantum marginal problem - constructing a global state from reduced states
Consider a bipartite state $\rho_{AB}$ with reduced states $\rho_A = \text{Tr}_B(\rho_{AB})$ and $\rho_B = \text{Tr}_A(\rho_{AB})$.
Suppose one obtains states $\rho'_{A}$ and $\rho'_{B}$ such that $\|\...
2
votes
1answer
51 views
How do I prove that $\newcommand{\tr}{\operatorname{Tr}}\tr(A \sqrt{B} A \sqrt{B}) = \tr\Big[\Big(\sqrt{\sqrt{B}} A \sqrt{\sqrt{B}}\Big)^2\Big]$?
Let's say I have 2 density operators $A$ and $B$. Now, here is what I am trying to calculate:
$$\newcommand{\tr}{\operatorname{trace}}
\tr(A \sqrt{B} A \sqrt{B}).
$$
I saw that this trace can be ...
1
vote
1answer
70 views
Relationship between trace distance and total variation distance
Consider two quantum states $\rho$ and $\sigma$ and the probability distributions induced by measuring both of them in the standard basis. Let’s call the probability distributions $p_{\rho}$ and $p_{\...
1
vote
2answers
38 views
Relation between trace distance and inner product between pure states
Let $|\phi\rangle,|\psi\rangle$ be two state vectors, and let $d=\frac{1}{2}\mathrm{Tr}(\sqrt{(|\phi\rangle\langle\phi|-|\psi\rangle\langle\psi|)^2})$ be their trace distance. Then it will always hold ...
1
vote
1answer
38 views
Is the restriction of a strictly contractive channel (SCC) to a subspace necessarily still SCC? (impossibility of perfect QEC for SCCs)
This paper shows the impossibility of perfect error correction for strictly contractive quantum channels, i.e., for channels such that $||\mathcal{E}(\rho)-\mathcal{E}(\sigma) ||\leq k ||\rho-\sigma||$...
1
vote
1answer
103 views
Can the fidelity $F(\rho,\sigma)$ be computed knowing only $\rho - \sigma$?
The motivation for this question comes from trace distance. For any two states $\rho, \sigma$, the trace distance $T(\rho, \sigma)$ is given by
$$T(\rho, \sigma) = |\rho - \sigma|_1,$$
where $|\cdot|...
2
votes
1answer
65 views
Saturating the Fuchs-van de Graaf inequality
It is well-known that one side of the Fuchs-van de Graaf inequality is saturated for pure states, i.e. $F(\rho,\sigma)^2 = 1-d(\rho,\sigma)^2$ when $\rho$ and $\sigma$ are pure (here we are using the ...
2
votes
2answers
52 views
What can be said about the closeness of two states if the difference of their fidelity measured with respect to a fixed state is close to 0?
Suppose I have two states $\rho$ and $\sigma$. We are given that,
$$Tr((\rho - \sigma)|\psi\rangle\langle\psi|) \geq \epsilon$$
where $|\psi\rangle$ is a fixed state and $\epsilon \rightarrow 0$,
...
3
votes
1answer
30 views
Does the trace distance between marginals bound the distance between the overall states?
If the quantum states of the subsystems of two systems are close (for example: in terms of trace distance), are the states of the larger systems also close, i.e., if
$$ ||\rho_A - \rho_{A^\prime}||\...
5
votes
2answers
52 views
Prove that for one-qubit unitaries $\text{Tr}|U-V|=2\max_\psi\|(U-V)|\psi\rangle\|$
Given two 1-qubit rotations $U=R_n (\theta)$ and $V=R_m(\phi)$ with $n$ and $m$ vectors defining a rotation and $\theta, \phi$ angles, define $D(U,V)=Tr(|U-V|)$ where $|U-V|=\sqrt{(U-V)^\dagger (U-V)}$...
0
votes
2answers
92 views
Prove that $\operatorname{Tr}_B(O_A M)=O_A\operatorname{Tr}_B(M)$
Can someone help me with the following question?
Let $M$ be a general operator on the composite system $\mathcal{H}_A\otimes \mathcal{H}_B$ and let $O_A$ be an operator on $\mathcal{H}_A$. Using ...
4
votes
1answer
85 views
Is the quantum state fidelity defined as $F(\rho, \sigma)=\text{tr}\sqrt{\rho^{1/2}\sigma\rho^{1/2}}$ or its square?
I have seen two different definition of Fidelity in different sources. For example, Nielsen & Chuang QCQI, 10th edition, page 409 defines Fidelity like the following:
$$
F(\rho, \sigma) := \...
2
votes
1answer
65 views
Continuity bounds on $D_{\max}(\rho_{AB}\|\rho_A\otimes\rho_B)$
The max-relative entropy between two states is defined as
$$D_{\max }(\rho \| \sigma):=\log \min \{\lambda: \rho \leq \lambda \sigma\},$$
where $\rho\leq \sigma$ should be read as $\sigma - \rho$ is ...
2
votes
1answer
59 views
It two unitaries are delta apart in trace norm, then what is the trace norm of outputs states when the same input state is applied to two unitaries?
Suppose we are given two unitary matrices $U$ and $V$, with the following guarantee,
$$||U - V||_1 \geqslant \delta$$
for some $\delta \geqslant 0$.
We apply an input density state $\rho$ ...
7
votes
1answer
220 views
Does the trace distance have a geometrical interpretation?
Consider the trace distance between two quantum states $\rho,\sigma$, defined via
$$D(\rho,\sigma)=\frac12\operatorname{Tr}|\rho-\sigma|,$$
where $|A|\equiv\sqrt{A^\dagger A}$.
When $\rho$ and $\...
6
votes
2answers
137 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 ...
4
votes
1answer
136 views
Is the diamond norm subadditive under composition?
The diamond norm distance between two operations is the maximum trace distance between their outputs for any input (including inputs entangled with qubits not being operated on).
Is it the case that ...
8
votes
2answers
734 views
What is intuition for the trace distance between quantum states?
Given two mixed states $\rho$ and $\sigma$, the trace distance between the states is defined by $\sum_{i=1}^n |\lambda_i|$, where $\lambda_i$'s are eigenvalues of $\rho - \sigma$.
I know the ...
3
votes
1answer
121 views
Trace distance of two classical-quantum states
I have these two classical-quantum states:
$$\rho = \sum_{a} \lvert a\rangle \langle a\lvert \otimes q^a \\
\mu = \sum_{a} \lvert a\rangle \langle a\lvert \otimes r^a $$
Where $a$ are the classical ...
