# The role of topology in quantum tomography

@article{Kech2015TheRO, title={The role of topology in quantum tomography}, author={Michael Kech and P{\'e}ter Vrana and Michael M. Wolf}, journal={Journal of Physics A: Mathematical and Theoretical}, year={2015}, volume={48} }

We investigate quantum tomography in scenarios where prior information restricts the state space to a smooth manifold of lower dimensionality. By considering stability we provide a general framework that relates the topology of the manifold to the minimal number of binary measurement settings that is necessary to discriminate any two states on the manifold. We apply these findings to cases where the subset of states under consideration is given by states with bounded rank, fixed spectrum, given…

## 12 Citations

### Constrained Quantum Tomography of Semi-Algebraic Sets with Applications to Low-Rank Matrix Recovery

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Upper bounds on the minimal number of measurement settings or outcomes that are required for discriminating all states within the given set of von Neumann measurements and sets of local observables are provided.

### Quantum Tomography of Semi-Algebraic Sets with Constrained Measurements

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Upper bounds on the minimal number of measurement settings or outcomes that are required for discriminating all states within the given set are provided and results are obtained for low-rank matrix recovery of hermitian matrices and the phase retrieval problem.

### Constrained Quantum Tomography

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Frameworks to find both lower and upper bounds on the minimal number of measurement settings or outcomes needed to discriminate any two quantum states of such a subset are provided and applied to several concrete scenarios.

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We consider quantum state tomography with measurement procedures of the following type: First, we subject the quantum state we aim to identify to a known time evolution for a desired period of time.…

### Stable pure state quantum tomography from five orthonormal bases

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For any finite-dimensional Hilbert space, we construct explicitly five orthonormal bases such that the corresponding measurements allow for efficient tomography of an arbitrary pure quantum state.…

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This work forms the task as a membership problem related to a partitioning of the quantum state space and connects it to the geometry of the state space, and proves various sufficient criteria that force informational completeness.

### On the classification of two-qubit group orbits and the use of coarse-grained 'shape' as a superselection property

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Recently a complete set of entropic conditions has been derived for the interconversion structure of states under quantum operations that respect a specified symmetry action, however the core…

### How many orthonormal bases are needed to distinguish all pure quantum states?

- Mathematics, Physics
- 2015

We collect some recent results that together provide an almost complete answer to the question stated in the title. For the dimension d = 2 the answer is three. For the dimensions d = 3 and d ≥ 5 the…

### From quantum tomography to phase retrieval and back

- Mathematics2015 International Conference on Sampling Theory and Applications (SampTA)
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This letter is devoted to the cross-fertilization between the fields of phase retrieval and quantum tomography. In the first part we discuss topological aspects of quantum tomography which turn out…

### Stable low-rank matrix recovery via null space properties

- Mathematics, Computer ScienceArXiv
- 2015

It is shown that nuclear norm minimization uniformly and stably reconstructs Hermitian rank-$r$ matrices with high probability and discusses applications in quantum physics and the phase retrieval problem.

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