Quantum state tomography via compressed sensing

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The techniques easily generalize to spin- j j particles [ 25 ]

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We use the usual matrix norms ‖ A ‖ tr = ∑ i σ i , ‖ A ‖ 2 2 = tr ⁡ A † ​ A = ∑ i σ i 2 , ‖ A ‖ = max i ⁡ σ i \|A\|_{\tr}=\sum_{i}\sigma_{i},\|A\|_{2}^{2}=\tr A^{\dagger}A=\sum_{i}\sigma_{i}^{2},\|A\|=\max_{i}\sigma_{i} , with σ i \sigma_{i} the singular values of A A . The last definition extends to super-operators: if 𝒜 \mathcal{A} is a super-operator, then ‖ 𝒜 ‖ \|\mathcal{A}\| is its largest singular value, or, equivalently ‖ 𝒜 ‖ = sup σ , ‖ σ ‖ 2 = 1 ‖ 𝒜 ​ σ ‖ 2 \|\mathcal{A}\|=\sup_{\sigma,\|\sigma\|_{2}=1}\|\mathcal{A}\sigma\|_{2} (a.k.a. “ 2 → 2 2\to 2 ”-norm)

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If the term 1 / ( 2 ​ d 2 ) 1/(2d^{2}) were zero, Y Y would be a strict subgradient

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Going beyond [ 12 ] , we bound deviations in 1 1 -norm, as opposed to 2 2 -norm. The former norm gives stronger results and carries an operational meaning in terms of statistical distinguishability

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