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arXiv:1306.1214

Determination of all pure quantum states from a minimal number of observables

Damien Mondragon, Vladislav Voroninski

Published Jun 5, 2013• 20 citations
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Abstract

We show that for any positive integer $n$, the maps $x \in \mathbb{C}^n \mapsto \{\left|\langle x, z_i \rangle \right|^2\}_{i=1}^{4n} \in \mathbb{R}^{4n}$, where $z_i$ are the columns of four $n\times n$ unitary matrices, are generically injective modulo multiplication by a global phase factor, yielding a family of embeddings of $\mathbb{C}P^{n-1}$ into $\mathbb{R}^{4n-4}$. In particular, this implies that distribution measurements about a pure state with four generic full-rank observables are informationally complete, which is sharp for $n \geq 6$. To complement this information-theoretic study, we establish in a companion paper that the PhaseLift algorithm yields efficient phase retrieval from quadratic measurements with $O(1)$ unitary matrices, with high probability, where the unitaries are iid according to Haar measure.

DOI
10.48550/arxiv.1306.1214

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Phase Retrieval in C4\mathbb C^4C4 Requires Exactly Eleven Measurements
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I construct a positive-operator-valued measure (POVM) which has 2d2d2d rank-1 elements and which is informationally complete for generic pure states in ddd dimensions, thus confirming a conjecture made by Flammia, Silberfarb, and Caves (e-print quant-ph∕0404137). I show that if a rank-1 POVM is required to be informationally complete for all pure states in ddd dimensions, it must have at least $3d\ensuremath{-}2$ elements. I also show that, in a POVM which is informationally complete for all pure states in ddd dimensions, for any vector there must be at least $2d\ensuremath{-}1$ POVM elements which do not annihilate that vector.