12
votes
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 ...
- 5,158
10
votes
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 ...
- 1,251
9
votes
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 ...
- 6,711
9
votes
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 ...
- 5,158
8
votes
Accepted
Closeness of purifications of states
No dimension-independent bound is possible.
Consider states $\rho_A$ and $\sigma_A$ that are close in $p$-norm (for $p>1$) but have relatively low fidelity. Specifically, assume
$$
\|\rho_A - \...
- 5,158
8
votes
What is intuition for the trace distance between quantum states?
Short answer. The trace distance between two states more or less determines how distinguishable they are by any operational means. A trace distance of 0 means that they are indistinguishable (because ...
- 11.9k
8
votes
Prove that $\|p^{\otimes n} - q^{\otimes n}\| \leq n \|p-q\|$ for density operators $p,q$
Marsl is correct, and his "hint" is really more a sketch of a solution than a hint. Rather than viewing the question or its solution as just formal algebra, you can also approach his same solution ...
- 1,251
7
votes
Accepted
Prove that $\|p^{\otimes n} - q^{\otimes n}\| \leq n \|p-q\|$ for density operators $p,q$
Hint: To make your induction work, write
$$\eqalign{p^{\otimes n} - q^{\otimes n} & = & \left(p^{\otimes(n-1)}\otimes p \right)-\left(q^{\otimes (n-1)} \otimes q\right)\\
& = & \left(...
- 899
7
votes
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 ...
- 20.8k
6
votes
Accepted
Is the diamond norm subadditive under composition?
For arbitrary linear super-operators $U_j$ and $V_j\def\D{\mathrm{Diamond}} \def\Dn#1{\lVert #1 \rVert_\diamond}\def\le{\leqslant} $, we have
$$\def\D{\mathrm{Diamond}} \def\Dn#1{\lVert #1 \rVert_\...
- 11.9k
6
votes
Accepted
Trace distance of two classical-quantum states
Yes, since the trace norm is the sum of the absolute value of the singular values, and the singular values can be found for each of the $a$ blocks independently.
- 5,682
6
votes
How can we upper bound the norm of a partial trace?
The $1$-norm decreases under partial trace and so we have an upper bound of $1$ when the states are normalized,
$$
\|\mathrm{Tr}_B[|\psi_1\rangle \langle \psi_2|]\|_1 \leq \||\psi_1\rangle \langle \...
- 5,355
6
votes
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.
- 55.7k
6
votes
Accepted
How to find the distance between a given $\rho$ and the nearest pure state(s)?
Recall that for any Hermitian operator $A$ and any unit vector $|\psi\rangle$ the real number $\langle \psi|A|\psi\rangle$, known as the Rayleigh quotient, is bounded by the largest eigenvalue $\...
- 20.8k
6
votes
If $\rho,\sigma$ are classical-quantum states, can the fidelity $F(\rho,\sigma)$ be expressed in terms of $F(\rho_i,\sigma_i)$?
Observe that, for any collection of matrices $A_i$, we have
$$\sqrt{\sum_i |i\rangle\!\langle i|\otimes A_i}
= \sum_i |i\rangle\!\langle i|\otimes \sqrt{A_i}, \\
{\rm Tr}\left(\sum_i |i\rangle\!\...
glS♦
- 23.4k
5
votes
Accepted
Can the fidelity $F(\rho,\sigma)$ be computed knowing only $\rho - \sigma$?
The answer is no, as the following counter-example reveals. Let $\varepsilon\in(0,1)$ and define
$$
\rho_0 =
\begin{pmatrix}
\frac{1+\varepsilon}{2} & 0 & 0\\
0 & 0 & 0\\
0 & 0 &...
- 5,158
5
votes
Accepted
Relation between trace distance and inner product between pure states
A derivation of this is given in Mark Wilde's book https://arxiv.org/abs/1106.1445 equation 9.173, pages 274-275.
- 5,355
5
votes
Accepted
Prove that the trace norm is dual to the spectral norm
There are different ways to prove what you want to prove, including the solution tsgeorgios has suggested, but for the sake of gaining greater intuition I would suggest starting with the recognition ...
- 5,158
5
votes
Is the trace distance between multipartite states invariant under permutations?
I'd like to add a small addition to the answer of @DaftWullie about why you would expect this operationally to be true -- without knowing permutations correspond to unitary matrices.
It boils down to ...
- 5,355
5
votes
Accepted
Prove that a channel is close to acting on only one system
I suppose you're asking the following: for any $\epsilon\ge0$ and $\Phi$ that satisfy conditions, is there $\delta_\epsilon \rightarrow 0$ when $\epsilon \rightarrow 0$, such that there exists a ...
- 6,711
4
votes
Accepted
Prove that for one-qubit unitaries $\text{Tr}|U-V|=2\max_\psi\|(U-V)|\psi\rangle\|$
Let's start with expanding the calculation of $E$:
$$
E(U,V)=\max_{|\psi\rangle}\sqrt{\langle\psi|(U-V)^\dagger(U-V)|\psi\rangle}.
$$
Clearly, we want $|\psi\rangle$ to be the eigenvector with maximum ...
- 55.7k
4
votes
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?
Here's a concrete example for a single qubit.
We can always change the basis to have $|\psi\rangle=|0\rangle$.
Let us further suppose that $\langle0|\rho|0\rangle=0$, so that
$$\rho=\begin{pmatrix}0&...
glS♦
- 23.4k
4
votes
Accepted
What can be said about the closeness of two states if the difference of their fidelity measured with respect to a fixed state is close to 0?
In general, it would seem no. The quantity
$$
\mathrm{Tr}[(\rho - \sigma)|\psi\rangle\langle\psi|]
$$
is only concerned with the distance between $\rho$ and $\sigma$ on the subspace $\mathrm{span}(|\...
- 5,355
4
votes
Accepted
Does the trace distance between marginals bound the distance between the overall states?
No. Just take two Bell states. They have identical reduced density matrices yet are orthogonal, that is, as distant from each other as it gets.
- 5,682
4
votes
Relation between trace distance and inner product between pure states
$\newcommand{\bra}[1]{\langle #1\rvert}\newcommand{\braket}[2]{\langle #1\rvert #2\rangle}\newcommand{\ket}[1]{\lvert #1\rangle}\newcommand{\on}[1]{\operatorname{#1}}\newcommand{\ketbra}[2]{\lvert #1\...
glS♦
- 23.4k
4
votes
Closeness of unitary dilations of CPTP maps
Suppose that $\Phi_i$ $i=1,2$ are CPTP maps with $\|\Phi_1-\Phi_2\|_\diamond\leq \epsilon$. Let $V_i=U_i^\dagger(A\otimes I_K)U_i$ and $\sigma=\rho\otimes |0\rangle\langle0|$, then we have that $V_i$ ...
- 1,893
4
votes
maximization of trace between two operators with respect to different norm constraints
If I understand your question correctly, you're trying to prove something that is false. Consider the operator
$$
Y = \begin{pmatrix}
2 & 0 & 0\\
0 & -1 & 0\\
0 & 0 &...
- 5,158
4
votes
Bounding inner product of states with distance
If I understand your problem correctly, you can just expand $|| |{\psi}\rangle- |{\phi}\rangle||_2
$ to get the result, i.e. $|||\psi \rangle -|\phi \rangle ||_2=\sqrt{\left( \langle \psi |-\langle \...
- 2,922
4
votes
Accepted
How to show $T(\rho,\sigma)≥\sum_i|r_i − s_i|$ with $r_i,s_i$ eigenvalues of $\rho,\sigma$?
By the min-max theorem, we have$^1$
$$
\begin{align}
t_k&=\max_{\quad U\\\dim U=k}\min_{|x\rangle\in U\\\langle x|x\rangle=1}\langle x|V|x\rangle\tag1\\
&=\max_{\quad U\\\dim U=k}\min_{|x\...
- 20.8k
4
votes
Accepted
Does proximity of two bipartite states in a norm force high overlap between the elements of the Schmidt bases?
No.
Here is an example without small Schmidt coefficients.
To this end, consider
$$
\lvert\phi\rangle = a\lvert0\rangle\lvert0\rangle + b \lvert1\rangle\lvert1\rangle\ ,
$$
and
$$
\lvert\psi\rangle = ...
- 5,682
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