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)
27
questions
4
votes
1answer
43 views
Why do probablity distribution with orthogonal suppor have maximal Kolmogorov distance?
Can anyone explain why the $l_1$ distance has the property that probability distributions $P,Q$ with orthogonal support (meaning that the product $p_iq_i$ vanishes for each value of $i$) are at a ...
4
votes
2answers
119 views
Is the trace distance between multipartite states invariant under permutations?
Consider two multipartite states $\rho_{A_1A_2..A_L}$ and $\sigma_{A_1A_2..A_L}$ in $\mathcal{H}_{A_1} \otimes\mathcal{H}_{A_2} \otimes...\mathcal{H}_{A_L} $. For an arbitrary permutation $\pi$ over $\...
3
votes
1answer
53 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| < ...
5
votes
1answer
145 views
Closeness of purifications of states
Uhlmann's theorem states that if two states $\rho_A, \sigma_A$ satisfy $F(\rho_A, \sigma_A)\geq 1 - \varepsilon$, then there for any purification $\Psi_{AR}$ of $\rho_A$, one can find a purification $\...
2
votes
2answers
82 views
Properties of composition of isometry and a perturbed adjoint
Suppose $\vert\Phi\rangle_{AR} = \frac{1}{\sqrt{|D|}}\sum_{i\in D} \vert ii\rangle_{AR}$ is the maximally entangled state. Let $V_{A\rightarrow BE}$ and $\tilde{V}_{A\rightarrow BE}$ be two isometries ...
1
vote
1answer
49 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:
$$
\frac{1}{2}||\...
3
votes
2answers
127 views
Prove that the trace norm is dual to the spectral norm
Suppose $A\in L(X,Y)$. $||\cdot||$ denotes spectral norm and denotes the largest singular value of a matrix, i.e. the largest eigenvalue of $\sqrt{A^*A}$.
$||\cdot||_{tr}$ denotes trace norm. We have ...
1
vote
1answer
42 views
Trace distance bound after partial trace
Let's say I have a pair of states among three parties Alice(A), Bob(B) and Eve(E), $\rho_{ABE}$ and $\rho_{UUE}$ where the first two parties hold uniform values U.}
I know that the trace distance ...
3
votes
1answer
89 views
Quantum marginal problem - constructing a global state from reduced states
Consider a bipartite state $\rho_{AB}$ with reduced states $\rho_A = \text{Tr}_B(\rho_{AB})$ and $\rho_B = \text{Tr}_A(\rho_{AB})$.
Suppose one obtains states $\rho'_{A}$ and $\rho'_{B}$ such that $\|\...
3
votes
1answer
57 views
How do I prove that $\newcommand{\tr}{\operatorname{Tr}}\tr(A \sqrt{B} A \sqrt{B}) = \tr\Big[\Big(\sqrt{\sqrt{B}} A \sqrt{\sqrt{B}}\Big)^2\Big]$?
Let's say I have 2 density operators $A$ and $B$. Now, here is what I am trying to calculate:
$$\newcommand{\tr}{\operatorname{trace}}
\tr(A \sqrt{B} A \sqrt{B}).
$$
I saw that this trace can be ...
1
vote
1answer
93 views
Relationship between trace distance and total variation distance
Consider two quantum states $\rho$ and $\sigma$ and the probability distributions induced by measuring both of them in the standard basis. Let’s call the probability distributions $p_{\rho}$ and $p_{\...
1
vote
2answers
76 views
Relation between trace distance and inner product between pure states
Let $|\phi\rangle,|\psi\rangle$ be two state vectors, and let $d=\frac{1}{2}\mathrm{Tr}(\sqrt{(|\phi\rangle\langle\phi|-|\psi\rangle\langle\psi|)^2})$ be their trace distance. Then it will always hold ...
1
vote
1answer
40 views
Is the restriction of a strictly contractive channel (SCC) to a subspace necessarily still SCC? (impossibility of perfect QEC for SCCs)
This paper shows the impossibility of perfect error correction for strictly contractive quantum channels, i.e., for channels such that $||\mathcal{E}(\rho)-\mathcal{E}(\sigma) ||\leq k ||\rho-\sigma||$...
1
vote
1answer
109 views
Can the fidelity $F(\rho,\sigma)$ be computed knowing only $\rho - \sigma$?
The motivation for this question comes from trace distance. For any two states $\rho, \sigma$, the trace distance $T(\rho, \sigma)$ is given by
$$T(\rho, \sigma) = |\rho - \sigma|_1,$$
where $|\cdot|...
3
votes
1answer
115 views
Saturating the Fuchs-van de Graaf inequality
It is well-known that one side of the Fuchs-van de Graaf inequality is saturated for pure states, i.e. $F(\rho,\sigma)^2 = 1-d(\rho,\sigma)^2$ when $\rho$ and $\sigma$ are pure (here we are using the ...
2
votes
2answers
55 views
What can be said about the closeness of two states if the difference of their fidelity measured with respect to a fixed state is close to 0?
Suppose I have two states $\rho$ and $\sigma$. We are given that,
$$Tr((\rho - \sigma)|\psi\rangle\langle\psi|) \geq \epsilon$$
where $|\psi\rangle$ is a fixed state and $\epsilon \rightarrow 0$,
...
3
votes
1answer
36 views
Does the trace distance between marginals bound the distance between the overall states?
If the quantum states of the subsystems of two systems are close (for example: in terms of trace distance), are the states of the larger systems also close, i.e., if
$$ ||\rho_A - \rho_{A^\prime}||\...
5
votes
2answers
54 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
97 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
104 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:
$$
F(\rho, \sigma) := \...
2
votes
1answer
71 views
Continuity bounds on $D_{\max}(\rho_{AB}\|\rho_A\otimes\rho_B)$
The max-relative entropy between two states is defined as
$$D_{\max }(\rho \| \sigma):=\log \min \{\lambda: \rho \leq \lambda \sigma\},$$
where $\rho\leq \sigma$ should be read as $\sigma - \rho$ is ...
2
votes
1answer
62 views
It two unitaries are delta apart in trace norm, then what is the trace norm of outputs states when the same input state is applied to two unitaries?
Suppose we are given two unitary matrices $U$ and $V$, with the following guarantee,
$$||U - V||_1 \geqslant \delta$$
for some $\delta \geqslant 0$.
We apply an input density state $\rho$ ...
7
votes
1answer
250 views
Does the trace distance have a geometrical interpretation?
Consider the trace distance between two quantum states $\rho,\sigma$, defined via
$$D(\rho,\sigma)=\frac12\operatorname{Tr}|\rho-\sigma|,$$
where $|A|\equiv\sqrt{A^\dagger A}$.
When $\rho$ and $\...
6
votes
2answers
140 views
Prove that $\|p^{\otimes n} - q^{\otimes n}\| \leq n \|p-q\|$ for density operators $p,q$
I've been trying to figure this out for a while and I'm totally lost.
My goal is to show that for two density operators $p$, $q$, that $$||p^{\otimes n} - q^{\otimes n}|| \leq n ||p-q||$$
So far ...
6
votes
1answer
166 views
Is the diamond norm subadditive under composition?
The diamond norm distance between two operations is the maximum trace distance between their outputs for any input (including inputs entangled with qubits not being operated on).
Is it the case that ...
9
votes
2answers
922 views
What is intuition for the trace distance between quantum states?
Given two mixed states $\rho$ and $\sigma$, the trace distance between the states is defined by $\sum_{i=1}^n |\lambda_i|$, where $\lambda_i$'s are eigenvalues of $\rho - \sigma$.
I know the ...
3
votes
1answer
131 views
Trace distance of two classical-quantum states
I have these two classical-quantum states:
$$\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 ...
