Sample-Optimal Tomography of Quantum States

By Jeongwan Haah, Aram Wettroth Harrow, Zhengfeng Ji, Xiaodi Wu, Nengkun Yu. IEEE Transactions on Information Theory, 2015

470 citations recorded by the source corpus.

It is a fundamental problem to decide how many copies of an unknown mixed quantum state are necessary and sufficient to determine the state. Previously, it was known only that estimating states to error <inline-formula> <tex-math notation="LaTeX">$\epsilon $ </tex-math></inline-formula> in trace distance required <inline-formula> <tex-math notation="LaTeX">$O(dr^{2}/\epsilon ^{2})$ </tex-math></inline-formula> copies for a <inline-formula> <tex-math notation="LaTeX">$d$ </tex-math></inline-formula>-dimensional density matrix of rank <inline-formula> <tex-math notation="LaTeX">$r$ </tex-math></inline-formula>. Here, we give a theoretical measurement scheme (POVM) that requires <inline-formula> <tex-math notation="LaTeX">$O (dr/ \delta) \ln ~(d/\delta) $ </tex-math></inline-formula> copies to estimate <inline-formula> <tex-math notation="LaTeX">$\rho $ </tex-math></inline-formula> to error <inline-formula> <tex-math notation="LaTeX">$\delta $ </tex-math></inline-formula> in infidelity, and a matching lower bound up to logarithmic factors. This implies <inline-formula> <tex-math notation="LaTeX">$O((dr / \epsilon ^{2}) \ln ~(d/\epsilon))$ </tex-math></inline-formula> copies suffice to achieve error <inline-formula> <tex-math notation="LaTeX">$\epsilon $ </tex-math></inline-formula> in trace distance. We also prove that for independent (product) measurements, <inline-formula> <tex-math notation="LaTeX">$\Omega (dr^{2}/\delta ^{2}) / \ln (1/\delta)$ </tex-math></inline-formula> copies are necessary in order to achieve error <inline-formula> <tex-math notation="LaTeX">$\delta $ </tex-math></inline-formula> in infidelity. For fixed <inline-formula> <tex-math notation="LaTeX">$d$ </tex-math></inline-formula>, our measurement can be implemented on a quantum computer in time polynomial in <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>.

Subfields: Artificial Intelligence, Atomic and Molecular Physics, and Optics

View Paper: https://www.semanticscholar.org/paper/de247019a9a3cb9362105068f90f30738066f8f7

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Citations in the corpus

Predicting many properties of a quantum system from very few measurements (2020, background): “Constructive protocols [37, 57] saturate this bound, but require entangled circuits and measurements that act on all state copies simultaneously.”
Shadow tomography of quantum states (2017, methodology): “tomography is easy to achieve using k= Oe D2/ε2 copies of the state ρ, by just ignoring the Ei’s and doing full tomography on ρ, using the recent protocols of O’Donnell and Wright [25] or Haah et al. [17]. At a diﬀerent extreme of parameters, shadow tomography is also easy using k= Oe M/ε2 copies of”
Efficient quantum tomography (2015, unknown): “We also thank Aram Harrow for pointing us to [Key06].”