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 ...
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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 ...
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8 votes
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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 - \...
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  • 4,568
8 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 ...
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  • 5,533
8 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 ...
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  • 4,568
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(...
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  • 819
7 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 ...
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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 ...
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  • 14.3k
6 votes
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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.
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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 \...
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  • 4,192
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.
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  • 47.5k
6 votes
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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 $\...
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  • 14.3k
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\!\...
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  • 19k
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 ...
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  • 4,568
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 &...
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  • 4,568
5 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_\...
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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 ...
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  • 4,192
4 votes
Accepted

Is the quantum state fidelity defined as $F(\rho, \sigma)=\text{tr}\sqrt{\rho^{1/2}\sigma\rho^{1/2}}$ or its square?

Both definitions are used and authors usually make it clear which one they mean. Wikipedia also points this out under the Alternative Defintion section.
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  • 2,119
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&...
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  • 19k
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}(|\...
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  • 4,192
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.
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4 votes
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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 ...
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  • 47.5k
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 &...
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  • 4,568
3 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.
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  • 4,192
3 votes

Prove that the trace norm is dual to the spectral norm

We still have $ \big| \langle B, A \rangle \big| = \big|\text{Tr}(AB^{\dagger}) \big| \leq \text{Tr}|A| $ for any operator $B$ with operator norm $ ||B|| \leq 1 $. First observe that $ ||B|| \leq 1 $ ...
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  • 1,326
3 votes
Accepted

Trace distance bound after partial trace

Yes, the trace distance can only decrease under partial trace. One can see this via the variational characterization of the trace norm $$ \|\rho\|_1 = \max_{-I \leq M \leq I} \mathrm{Tr}[M\rho] $$ ...
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  • 4,192
3 votes
Accepted

Quantum marginal problem - constructing a global state from reduced states

If perturbations are sufficiently small and $\rho_{AB}$ has sufficiently broad support then a desired global state $\rho_{AB}'$ exists. Define $$ \rho_{AB}' = \rho_{AB} + (\rho_A' - \rho_A) \otimes \...
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  • 14.3k
3 votes
Accepted

How do I prove that $\newcommand{\tr}{\operatorname{Tr}}\tr(A \sqrt{B} A \sqrt{B}) = \tr\Big[\Big(\sqrt{\sqrt{B}} A \sqrt{\sqrt{B}}\Big)^2\Big]$?

This follows from the cyclicity of trace. \begin{align} \text{tr}\left(\left(\sqrt{\sqrt{B}}A\sqrt{\sqrt{B}}\right)^2\right) &= \text{tr}\left(\sqrt{\sqrt{B}}A\sqrt{\sqrt{B}}\sqrt{\sqrt{B}}A\sqrt{...
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  • 518
3 votes
Accepted

Relationship between trace distance and total variation distance

A bound on the total variation distance Rammus already provided a short answer, but I'd like to elaborate a bit on why this is the case. This is basically the proof of theorem $9.1$ on page $405$ of ...
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  • 4,748
3 votes
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Properties of composition of isometry and a perturbed adjoint

After @Rammus answer explaining that the given inequality does not hold in general, i'll try to prove a weaker statement. Define $ \Delta = V - \tilde{V} $. The assumption is equivalent to $$ \text{Tr}...
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  • 1,326

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