arXiv subjects

Jiangwei Shang

Publications and source records attributed to Jiangwei Shang.

43 records · Page 3Linked to original sources

arXiv2015

Monte Carlo sampling from the quantum state space. II

High-quality random samples of quantum states are needed for a variety of tasks in quantum information and quantum computation. Searching the high-dimensional quantum state space for a global maximum of an objective function with many local maxima or evaluating an integral over a region in the quantum state space are but two exemplary applications of many. These tasks can only be performed reliably and efficiently with Monte Carlo methods, which involve good samplings of the parameter space in accordance with the relevant target distribution. We show how the Markov-chain Monte Carlo method known as Hamiltonian Monte Carlo, or hybrid Monte Carlo, can be adapted to this context. It is applicable when an efficient parameterization of the state space is available. The resulting random walk is entirely inside the physical parameter space, and the Hamiltonian dynamics enable us to take big steps, thereby avoiding strong correlations between successive sample points while enjoying a high acceptance rate. We use examples of single and double qubit measurements for illustration.

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arXiv2014

Quantum state tomography: Mean squared error matters, bias does not

Because of the constraint that the estimators be bona fide physical states, any quantum state tomography scheme - including the widely used maximum likelihood estimation - yields estimators that may have a bias, although they are consistent estimators. Schwemmer et al. (arXiv:1310.8465 [quant-ph]) illustrate this by observing a systematic underestimation of the fidelity and an overestimation of entanglement in estimators obtained from simulated data. Further, these authors argue that the simple method of linear inversion overcomes this (perceived) problem of bias, and there is the suggestion to abandon time-tested estimation procedures in favor of linear inversion. Here, we discuss the pros and cons of using biased and unbiased estimators for quantum state tomography. We conclude that the little occasional benefit from the unbiased linear-inversion estimation does not justify the high price of using unphysical estimators, which are typically the case in that scheme.

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arXiv2013

Optimal error regions for quantum state estimation

Rather than point estimators, states of a quantum system that represent one's best guess for the given data, we consider optimal regions of estimators. As the natural counterpart of the popular maximum-likelihood point estimator, we introduce the maximum-likelihood region---the region of largest likelihood among all regions of the same size. Here, the size of a region is its prior probability. Another concept is the smallest credible region---the smallest region with pre-chosen posterior probability. For both optimization problems, the optimal region has constant likelihood on its boundary. We discuss criteria for assigning prior probabilities to regions, and illustrate the concepts and methods with several examples.

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arXiv2012

Experimental proposal for symmetric minimal two-qubit state tomography

We propose an experiment that realizes a symmetric informationally complete (SIC) probability-operator measurement (POM) in the four-dimensional Hilbert space of a qubit pair. The qubit pair is carried by a single photon as a polarization qubit and a path qubit. The implementation of the SIC POM is accomplished with the means of linear optics. The experimental scheme exploits a new approach to SIC POMs that uses a two-step process: a measurement with full-rank outcomes, followed by a projective measurement on a basis that is chosen in accordance with the result of the first measurement. The basis of the first measurement and the four bases of the second measurements are pairwise unbiased --- a hint at a possibly profound link between SIC POMs and mutually unbiased bases.

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arXiv2012

Symmetric minimal quantum tomography by successive measurements

We consider the implementation of a symmetric informationally complete probability-operator measurement (SIC POM) in the Hilbert space of a d-level system by a two-step measurement process: a diagonal-operator measurement with high-rank outcomes, followed by a rank-1 measurement in a basis chosen in accordance with the result of the first measurement. We find that any Heisenberg-Weyl group-covariant SIC POM can be realized by such a sequence where the second measurement is simply a measurement in the Fourier basis, independent of the result of the first measurement. Furthermore, at least for the particular cases studied, of dimension 2, 3, 4, and 8, this scheme reveals an unexpected operational relation between mutually unbiased bases and SIC POMs; the former are used to construct the latter. As a laboratory application of the two-step measurement process, we propose feasible optical experiments that would realize SIC POMs in various dimensions.

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arXiv2011

SeCQC: An open-source program code for the numerical Search for the classical Capacity of Quantum Channels

SeCQC is an open-source program code which implements a Numerical Search for the classical Capacity of Quantum Channels (SeCQC) by using an iterative method. Given a quantum channel, SeCQC finds the statistical operators and POVM outcomes that maximize the accessible information, and thus determines the classical capacity of the quantum channel. The optimization procedure is realized by using a steepest-ascent method that fol- lows the gradient in the POVM space, and also uses conjugate gradients for speed-up.

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arXiv2011

SOMIM: An open-source program code for the numerical Search for Optimal Measurements by an Iterative Method

SOMIM is an open-source code that implements a Search for Optimal Measurements by using an Iterative Method. For a given set of statistical operators, SOMIM finds the POVMs that maximizes the accessed information, and thus determines the accessible information and one or all of the POVMs that retrieve it. The maximization procedure is a steepest-ascent method that follows the gradient in the POVM space, and uses conjugate gradients for speed-up.

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