# 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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### 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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### Is the trace distance upper bounded by the Euclidean distance?

Suppose we have two pure state $|\psi\rangle$ and $|\phi\rangle$. I was wondering whether the statement: $\||\psi\rangle\!\langle\psi|- |\phi\rangle\!\langle\phi|\|_{\rm tr}$ is at most the Euclidean ...
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### How to show $T(\rho,\sigma)≥\sum_i|r_i − s_i|$ with $r_i,s_i$ eigenvalues of $\rho,\sigma$?

The proof of the Fannes' inequality replies on the formula $T(ρ, σ)≥\sum_i|r_i − s_i|$, where $r_i,s_i$ are the eigenvalues of $\rho,\sigma$, in the descending order. In the proof given in Box 11.2, ...
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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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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 $\|\... • 3,075 2 votes 1 answer 44 views ### If states are close together does there always exist a channel close to the identity mapping one to the other? Question: Given states$\rho,\omega\in\mathbb C^{n\times n}$and$\varepsilon>0$such that$\rho$and$\omega$are$\varepsilon$-close in trace norm does there exist a channel$\Phi$with$\Phi(\...
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Suppose $\rho'_{AB} \approx_\varepsilon \rho_{AB}$ in trace distance. Is there an explicit construction of some state $\tilde{\rho}_{AB}$ using $\rho'_{AB}, \rho'_A, \rho'_B, \rho_A$ and $\rho_B$ (but ...
I want to know that there is a relation between the distance of two vectors and the corresponding elements of the Schmidt bases. We assume that two bipartite vectors $|\phi\rangle^{AB}$ and \$|\psi\...