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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)

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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 $\...
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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 ...
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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}||\...
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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 ...
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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 ...
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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 $\|\...
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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 ...
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What is the 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_{\...
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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 ...
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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 $||\mathcal{E}(\rho)-\mathcal{E}(\sigma) ||\leq k ||\rho-\sigma||$...
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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|...
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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 ...
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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 $\epsilon \rightarrow 0$, ...
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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}||\...
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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 $|U-V|=\sqrt{(U-V)^\dagger (U-V)}$...
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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 ...
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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 ...
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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 $\rho$ ...
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Does the trace distance have a geometric 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 $\sigma$...
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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 ...
Confused grad student's user avatar
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1 answer
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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 ...
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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 ...
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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 ...
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