Quantum state tomography via compressed sensing.

@article{Gross2010QuantumST,
  title={Quantum state tomography via compressed sensing.},
  author={David Gross and Yi-Kai Liu and Steven T. Flammia and Stephen Becker and Jens Eisert},
  journal={Physical review letters},
  year={2010},
  volume={105 15},
  pages={
          150401
        }
}
We establish methods for quantum state tomography based on compressed sensing. These methods are specialized for quantum states that are fairly pure, and they offer a significant performance improvement on large quantum systems. In particular, they are able to reconstruct an unknown density matrix of dimension d and rank r using O(rdlog²d) measurement settings, compared to standard methods that require d² settings. Our methods have several features that make them amenable to experimental… 

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References

SHOWING 1-10 OF 37 REFERENCES

Demonstration of an all-optical quantum controlled-NOT gate

TLDR
An unambiguous experimental demonstration and comprehensive characterization of quantum CNOT operation in an optical system that produces all four entangled Bell states as a function of only the input qubits' logical values, for a single operating condition of the gate.

Strong converse for identification via quantum channels

We present a simple proof of the strong converse for identification via discrete memoryless quantum channels, based on a novel covering lemma. The new method is a generalization to quantum

Scalable multiparticle entanglement of trapped ions

TLDR
The scalable and deterministic generation of four-, five-, six-, seven- and eight-particle entangled states of the W type with trapped ions are reported, which obtain the maximum possible information on these states by performing full characterization via state tomography, using individual control and detection of the ions.

Reconstructing a pure state of a spin s through three Stern-Gerlach measurements

Consider a spin s prepared in a pure state. It is shown that, generically, the moduli of the spin components along three directions in space determine the state unambiguously. These probabilities are

Exact Matrix Completion via Convex Optimization

TLDR
It is proved that one can perfectly recover most low-rank matrices from what appears to be an incomplete set of entries, and that objects other than signals and images can be perfectly reconstructed from very limited information.

Spin tomography

We propose a tomographic reconstruction scheme for spin states. The experimental set-up, which is a modification of the Stern–Gerlach scheme, can be easily performed with currently available

Evenly distributed unitaries: On the structure of unitary designs

We clarify the mathematical structure underlying unitary t-designs. These are sets of unitary matrices, evenly distributed in the sense that the average of any tth order polynomial over the design

A Singular Value Thresholding Algorithm for Matrix Completion

TLDR
This paper develops a simple first-order and easy-to-implement algorithm that is extremely efficient at addressing problems in which the optimal solution has low rank, and develops a framework in which one can understand these algorithms in terms of well-known Lagrange multiplier algorithms.

A rank minimization heuristic with application to minimum order system approximation

TLDR
It is shown that the heuristic to replace the (nonconvex) rank objective with the sum of the singular values of the matrix, which is the dual of the spectral norm, can be reduced to a semidefinite program, hence efficiently solved.

On the rank minimization problem over a positive semidefinite linear matrix inequality

TLDR
This paper considers the problem of minimizing the rank of a positive semidefinite matrix, subject to the constraint that an affine transformation of it is also positive semidfinite, and employs ideas from the ordered linear complementarity theory and the notion of the least element in a vector lattice.