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Why is the trace distance between two density matrices not always $0$?

If $|A|_{tr}=Tr(\sqrt{A^\dagger A})$ then surely $$ |\rho_1-\rho_2|_{tr}=Tr(\sqrt{(\rho_1-\rho_2)^\dagger (\rho_1-\rho_2)}) $$ $$ =Tr(\sqrt{(\rho_1^\dagger -\rho_2^\dagger)(\rho_1-\rho_2)}) $$ $$ =Tr(\...
mrepic1123's user avatar
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How to view operator norms on open-system representation of quantum channels

I know how the operator norms $\| X\|_{1}$,$\| X\|_{2}$, and $\| X\|_{\infty}$ are defined for any operator $X\in B(\mathcal{H})$. My question is about how to view$\| T(X)\|_{1}$,$\| T(X)\|_{2}$, and ...
Pink Elephants's user avatar
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Improving Quantum State Distinguishability through Embedding

Consider the following map, $\mathcal{E}:\mathcal{L}(H_A) \rightarrow \mathcal{L}(H_{AB})$, $$ \mathcal{E}(\rho_A|U_{AB}, U_{AC}) = {\rm Tr_C} \left[ U_{AC}U_{AB}(\rho_A\otimes |0_B\rangle\langle 0_B|\...
Sowmitra Das's user avatar
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Prove $\|{\cal E}(\rho-\sigma)\|_1\leq\|({\cal E}\otimes{\rm id})(U\{(\rho - \sigma)\otimes |0⟩⟨ 0|\}U^{\dagger})\|_1$ with $U$ a CNOT

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 - ...
Sowmitra Das's user avatar
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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 ...
Anindita Sarkar's user avatar
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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=...
Sooraj S's user avatar
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Upper bound on trace distance of subsystems based on full system

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 ...
QuestionEverything's user avatar