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12 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 ...
Greg Kuperberg's user avatar
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 ...
John Watrous's user avatar
  • 6,107
9 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 - \...
John Watrous's user avatar
  • 6,107
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 ...
Danylo Y's user avatar
  • 7,342
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 ...
John Watrous's user avatar
  • 6,107
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 ...
Niel de Beaudrap's user avatar
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 ...
Greg Kuperberg's user avatar
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(...
Marsl's user avatar
  • 929
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 ...
Adam Zalcman's user avatar
  • 22.9k
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_\...
Niel de Beaudrap's user avatar
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.
Norbert Schuch's user avatar
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 \...
Rammus's user avatar
  • 5,863
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.
DaftWullie's user avatar
  • 58.8k
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 $\...
Adam Zalcman's user avatar
  • 22.9k
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's user avatar
  • 25.4k
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 &...
John Watrous's user avatar
  • 6,107
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.
Rammus's user avatar
  • 5,863
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 ...
John Watrous's user avatar
  • 6,107
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 ...
Rammus's user avatar
  • 5,863
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 ...
Danylo Y's user avatar
  • 7,342
5 votes

Is the trace distance upper bounded by the Euclidean distance?

You always have the opposite inequality: $\|A\|_1\ge \|A\|_2$. You see it easily from the fact that $\|A\|_1$ is the sum of the singular values of $A$, while $\|A\|_2^2$ is the sum of the squares of ...
glS's user avatar
  • 25.4k
5 votes
Accepted

Bounds on local expectation values for two states close in trace distance

We first have: $$|\mathrm{tr}(A(\rho-\sigma))|\leqslant\mathrm{tr}(|A(\rho-\sigma)|)=\|A(\rho-\sigma)\|_1$$ We can then use Hölder's inequality: $$\|A(\rho-\sigma)\|_1\leqslant\|\rho-\sigma\|_1\|A\|_{\...
Tristan Nemoz's user avatar
  • 6,462
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 ...
DaftWullie's user avatar
  • 58.8k
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's user avatar
  • 25.4k
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}(|\...
Rammus's user avatar
  • 5,863
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.
Norbert Schuch's user avatar
4 votes
Accepted

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?

For the way round that you've got your inequalities, I don't think there's much that can be said. To see why, let's consider the first expression $$ \|U-V\|_1=\text{Tr}(\sqrt{2I-VU^\dagger-UV^\dagger})...
DaftWullie's user avatar
  • 58.8k
4 votes

Relation between trace distance and inner product between pure states

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glS's user avatar
  • 25.4k
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 $V_i$ ...
Condo's user avatar
  • 2,048
4 votes
Accepted

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 &...
John Watrous's user avatar
  • 6,107

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