Spin tomography

We consider the problem of characterizing genuine multiparticle entanglement for permutationally invariant states using the approach of PPT mixtures. We show that the evaluation of this necessary biseparability criterion scales polynomially with the number of particles. In practice, it can be evaluated easily up to ten qubits and improves existing criteria significantly. Finally, we show that our approach solves the problem of characterizing genuine multiparticle entanglement for permutationally invariant three-qubit states.

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“…Analogously, we can also symmetrize Q (1...k)|k+1...N for permutations of the last N −k particles to obtain Eq. (21). This concludes the proof.…”

Section: Ppt Mixtures Of Pi States a Characterization Of Pi Pptsupporting

confidence: 66%

Genuine multiparticle entanglement of permutationally invariant states

2013

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“…Such states arise in very common physical settings, e.g. a pure state subject to a local noise process [20].A standard implementation of tomography [5,6] would use d 2 or more measurement settings, where d = 2 n for an nqubit system. But a simple parameter counting argument suggests that O(rd) settings could possibly suffice -a significant improvement.…”

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confidence: 99%

Quantum State Tomography via Compressed Sensing

et al. 2010

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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(rd log 2 d) measurement settings, compared to standard methods that require d 2 settings. Our methods have several features that make them amenable to experimental implementation: they require only simple Pauli measurements, use fast convex optimization, are stable against noise, and can be applied to states that are only approximately low-rank. The acquired data can be used to certify that the state is indeed close to pure, so no a priori assumptions are needed. We present both theoretical bounds and numerical simulations.The tasks of reconstructing the quantum states and processes produced by physical systems -known respectively as quantum state and process tomography [1] -are of increasing importance in physics and especially in quantum information science. Tomography has been used to characterize the quantum state of trapped ions [2] and an optical entangling gate [3] among many other implementations. But a fundamental difficulty in performing tomography on many-body systems is the exponential growth in the state space dimension. For example, to get a maximum-likelihood estimate of a quantum state of 8 ions, Ref.[2] required hundreds of thousands of measurements and weeks of post-processing.Still, one might hope to overcome this obstacle, because the vast majority of quantum states are not of physical interest. Rather, one is often interested in states with special properties: pure states, states with particular symmetries, ground states of local Hamiltonians, etc., and tomography might be more efficient in such special cases [4].In particular, consider pure or nearly pure quantum states, i.e., states with low entropy. More precisely, consider a quantum state that is essentially supported on an r-dimensional space, meaning the density matrix is close (in a given norm) to a matrix of rank r, where r is small. Such states arise in very common physical settings, e.g. a pure state subject to a local noise process [20].A standard implementation of tomography [5,6] would use d 2 or more measurement settings, where d = 2 n for an nqubit system. But a simple parameter counting argument suggests that O(rd) settings could possibly suffice -a significant improvement. However, it is not clear how to achieve this performance in practice, i.e., how to choose these measurements, or how to efficiently reconstruct the density matrix. For instance, the problem of finding a minimum-rank matrix subject to linear constraints is NP-hard in general [7].In addition to a reduction in experimental complexity, one might hope that a post-processing algorithm which takes as input only O(rd) ≪ d 2 numbers could be tuned to run considerably faster than standard methods. Since the output of the procedure is a low...

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“…First, the map → (M) ULIN projects − 1 d 1 onto the span of the first M hyperplanes in the vector space discussed above; applying the map a second time has no effect. This projection property implies 15) for any two density operators and . Second, we note that, since Eq.…”

Section: Linear Inversion For Incomplete Mub Tomographymentioning

confidence: 86%

Least-bias state estimation with incomplete unbiased measurements

et al. 2015

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Measuring incomplete sets of mutually unbiased bases constitutes a sensible approach to the tomography of high-dimensional quantum systems. The unbiased nature of these bases optimizes the uncertainty hypervolume. However, imposing unbiasedness on the probabilities for the unmeasured bases does not generally yield the estimator with the largest von Neumann entropy, a popular figure of merit in this context. Furthermore, this imposition typically leads to mock density matrices that are not even positive definite. This provides a strong argument against perfunctory applications of linear estimation strategies. We propose to use instead the physical state estimators that maximize the Shannon entropy of the unmeasured outcomes, which quantifies our lack of knowledge fittingly and gives physically meaningful statistical predictions.

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Spin tomography

Cited by 87 publications

References 18 publications

Genuine multiparticle entanglement of permutationally invariant states

Genuine multiparticle entanglement of permutationally invariant states

Quantum State Tomography via Compressed Sensing

Least-bias state estimation with incomplete unbiased measurements

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