# Questions tagged [trace-norm]

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Let $\rho, \sigma$ be two states of a qubit, and let $U$ be the $CX_{12}$-gate (control on 1st qubit, target on 2nd qubit). Prove that, for an arbitrary CPTP Map $\mathcal{E}$, $$\|\mathcal{E}(\rho - ... 2 votes 1 answer 73 views ### How to show that the trace distance equals the maximal total variation distance? Let \rho and \sigma be two density operators such that probability of obtaining a is tr(\rho E_a) if the state before measurement was \rho and tr(\sigma E_a) if the state before ... 2 votes 1 answer 96 views ### The matrix norm \|A\|=\max_{\langle u|u\rangle=1}|\langle u|A|u\rangle| in the proof of Lieb's theorem In Exercise A6.4, Appendix 6: Proof of Lieb’s theorem, Page 645, Quantum Computation and Quantum Information by Nielsen and Chuang, A matrix norm of A is defined as$$\|A\|=\max_{\langle u|u\rangle=...
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If we have the following upper bound on the sum of trace distances: $$\frac{1}{N} \sum_{a, b}||p_1(a | b) \rho_{ab} - p_2(a | b) \sigma_{ab}|| \le \epsilon,$$ where $p_1$ and $p_2$ are two ...